Nabil Absi

Génie Industriel Cr

Avril Cmp

Stéphane

Balbo Flavien

Jean-François Bassereau

Batton-Hubert Mireille

Douce Berger

Sandrine

Bernache-Assollant Didier

Jean Bigot

MR(DR2 Pierre

Génie

Appliquees O Mathematiques

Maître-Assistant Roustant

O Informatique

Professeur IMAGE, VISION, SIGNAL J C Boissier

Professeur Genie Pinoli

A Industriel

Professeur Microelectronique S Dolgui

Professeur Dauzere Peres

SMS

in French)

Ce chapitre présente un court extrait en français de toute la thèse 1 . Ces travaux ont été menés dans l'industrie des semi-conducteurs. La thèse est en deux parties. La première partie concerne la gestion des qualifications menées dans le but d'augmenter l'utilisation de la capacité de production dans des îlots de production. La seconde partie étudie la planification de production de deux lignes de productions couplées.

1 According to French universities regulation, in case a thesis is written in English, a French summary must be provided.

CHAPTER 1. RÉSUMÉ (SUMMARY IN FRENCH)

exclusive to Soitec. Chapter 9 sets the industrial and scientific context of this part. Chapter 10 presents the production planning model with all industrial constraints. Chapter 11 presents and analyzes several closed-loop production problems in which raw materials may be reused several times after remanufacturing. Both parts of the dissertation are related. We do not study this link in this thesis. However, some suggestions are presented at the end of the thesis in Chapter 12.

To my family for their unconditional love and support for lifelong development ...

ACKNOWLEDGEMENTS

At the end of this project, I would like to thank those who supported me throughout the course of this thesis and made it an enjoyable and extremely rewarding experience.

In the first place, I would like to thank the ANRT (Association Nationale de la Recherche et de la Technologie) and the Soitec company who financed this thesis.

I would like to express my appreciation to my academic thesis advisor, Mr Stéphane Dauzère-Pérès for his in-depth guidelines, analytical insights and conveying continuously a spirit of adventure. His presence at all stages of the thesis guaranteed its success.

I had the pleasure to have Mr Bernard Cassini as the industrial supervisor at Soitec to whom I am indebted. His professional experience, willingness to explore new ways and supportive attitude provided the foundation of a successful industrial implementation.

My sincere gratitude goes to Mr Nabil Absi who co-advised the second part of this thesis that deals with production planning. His initiative, insightful comments and deep knowledge in lot-sizing made this challenge exciting and valuable.

Warm words of thanks go to the committee members: Mr Bernard Penz, Mr Farouk Yalaoui, Mr Jan Fransoo and Mr Marc Sevaux for their constructive comments and suggestions. I appreciate their extensive and extremely detailed examination.

I owe a great deal of thanks to my University, École des Mines de Saint-Étienne, and its personnel for their invaluable support; especially to the members of my department SFL (Sciences de la Fabrication et Logistique) at Centre Microélectronique de Provence.

Many thanks go to the Soitec company, where this thesis was conducted. I especially benefited from the valuable support of my colleagues in Industrial Engineering, Supply Chain Management, ex-PDM and Production Control departments.

Mehdi ROWSHANNAHAD Grenoble, May 2015

God bless that man whose warp is wisdom and whose weft is justice.

Shahnameh of Ferdowsî

List of Figures

Introduction

L'électronique basée sur des matériaux semi-conducteurs a révolutionnée notre quotidien. L'industrie des semi-conducteurs vit dans un constant souci de miniaturisation, de performance et d'efficacité, de baisse de consommation d'énergie et finalement de rapidité de production et d'industrialisation tout en réduisant les coûts. L'évolution constante de marché, la diversité accrue des gammes de produits, la baisse de cycle de vie des produits et le coût élevé de fabrication exigent des lignes de production flexibles et agiles. Ces travaux de thèse porte sur l'optimisation de la capacité et la planification de production. Ils ont été menés au sein de l'entreprise Soitec, le leader des produits semi-conducteurs de hautes-performances, nommés SOI (Silicium sur Isolant). Bien que ces concepts soient déployés chez Soitec, ils peuvent être adaptés à d'autres lignes de production, dans l'industrie des semi-conducteurs ou autres. Dans la Section 1.2, le cadre des travaux est posé en définissant le contexte industriel. Ensuite, différents aspects concernant la gestion des qualifications (Partie I) sont explorés sous la Section 1.3. La Partie II traite de la planification d'un système de production à boucle fermée. Ce sujet est résumé sous la Section 1.4. On tire des conclusions relatives aux deux parties tout en indiquant des perspectives dans la Section 1.5.

Contexte Industriel

Des composants à base de matériaux semi-conducteurs sont utilisés en abondance dans les appareils électroniques qui forment notre vie de tous les jours. Ces composants sont en grande partie à base de silicium. Le dioxyde de silicium se trouve en abondance sous la forme de sable. Après un procédé spécifique, des tranches ou des plaques de silicium ("wafer" en anglais) sont fabriquées à partir de silicium d'une extrême pureté. Des composants électroniques, tels que des diodes, transistors, circuits intégrés et des puces sont ensuite gravés sur ces plaques de silicium. Vu la taille miniature des composants, la moindre impureté dans une étape de production les endommagerait, d'où la fabrication dans des salles extrêmement propres dites salle blanche ("clean room" en anglais). On appelle aussi des unités de fabrication 1.2 Contexte Industriel de semi-conducteurs, des fabs.

Les appareils ont d'autant plus de problèmes de batterie. En même temps, les exigences en terme de performances augmentent. Par conséquent, des matériaux semi-conducteurs de hautes performances ont été développés. Des tranches de SOI (Silicium-sur-Isolant et "Silicon-on-Insulator" en anglais) répondent à ces besoins. Une tranche de SOI est composée d'une mince couche de silicium active. Une couche de dioxyde de silicium et un support à base de silicium (voir Figure 1. Les étapes de fabrication des plaques de SOI sont [START_REF] Kononchuk | Silicon-On-Insulator (SOI) Technology: Manufacture and Applications[END_REF]:

1). Des

• La plaque de silicium A (aussi appelée "Top") est oxydée thermiquement.

L'idée est de créer une couche d'oxyde à la bonne épaisseur et avec la bonne uniformité;

• À l'étape d'implantation, le rôle est de faire pénétrer des ions d'hydrogène H+ en profondeur dans le silicium oxydé. La profondeur est déterminée en fonction de l'épaisseur de Si actif souhaitée. Des ions implantés créent une zone de fracture;

• La plaque A est nettoyée avant de passer à l'étape de collage. Les plaques sont traitées dans des bains chimiques afin d'ôter les contaminations métalliques et particulaires. Le nettoyage avant le collage rend les plaques hydrophiles pour faciliter le collage. Une fois le Top nettoyé, il est collé avec la plaque B (appelée le Base). Le collage est dit "hydrophile par adhésion moléculaire". Il est basé sur l'attraction de couches d'eau présentes à la surface des plaques;

• Ensuite, les plaques sont repassées dans des fours à l'étape recuit. Lors de cette étape, les ions implantés se recombinent et se dilatent de façon à générer la fracture nécessaire à l'obtention de la plaque SOI. Au niveau de l'interface de collage, les liaisons hydrogène sont remplacées par des liaisons atomiques plus solides. Le décollement se fait après de manière mécanique;

La Gestion des Qualifications

• Les trois dernières étapes de finition sont le polissage, le RTA et la stabilisation et l'amincissement. Le polissage a pour objectif de diminuer la rugosité de la plaque. C'est un procédé mécano-chimique qui combine l'action du frottement sur un pad avec l'action chimique du slurry. Le RTA ("Rapid Thermal Annealing" en anglais) est un recuit rapide qui permet de lisser la surface de la plaque après décollement. La stabilisation est une étape d'oxydation superficielle qui a pour but d'éliminer une couche de silicium abîmée lors du décollement ainsi que de stabiliser l'interface de collage créée lors du recuit. L'amincissement est également une oxydation superficielle qui permet d'amener le SOI à l'épaisseur finale demandée par le client.

La plaque de silicium qui reste après le décollement peut être retravaillée afin d'être réutilisée comme un nouveau Top. Une plaque Top déjà utilisée est appelée un négatif. Sachant que seulement une couche très mince de plaque Top reste sur la plaque SOI, la plaque négatif générée peut être retravaillée afin de revenir dans la ligne de fabrication de SOI pour en faire d'autres. Le processus de refabrication d'une plaque négatif est appelé le processus de refresh ou tout court le refresh.

Cette spécificité représente un des grands avantages économiques de la Technologie Smart-Cut™.

La Gestion des Qualifications 1.3.1 La Gestion des Qualifications dans la Fabrication Semiconducteur

Dans l'industrie des semi-conducteurs, de multiples opérations sont exécutées dans différentes étapes de production pour fabriquer un produit. Chaque étape est réalisée dans un poste de travail ("workcenter" ou "workstation" en anglais). Chaque poste de travail est un ensemble composé d'opérateurs, d'appareils de manutention et d'un parc d'équipements ("toolset" en anglais). Ce dernier est constitué de machines parallèles et souvent non-identiques qui exécutent des opérations similaires.

CHAPTER 1. RÉSUMÉ (SUMMARY IN FRENCH)

Par exemple, un ensemble de fours font le parc d'équipements Traitement Thermiques (tout court: TTH). Les machines dans un parc d'équipements peuvent avoir différentes caractéristiques en ce qui concerne, le débit de production ("throughput" en anglais) ou le temps de processus ("process time" en anglais), la taille de lot, la configurabilité, le logiciel, etc. Une recette est associée à chaque opération. Elle définit les instructions nécessaires pour obtenir le processus souhaité. Par exemple, une recette de TTH définit les différentes phases de la montée ou descente en température, les paliers de températures ainsi que la pression de gaz à chaque phase aussi bien que la vitesse de refroidissement.

Afin de pouvoir exécuter une opération sur une machine, la recette correspondant à l'opération doit déjà être qualifiée sur celle-ci. À cause des restrictions matérielles et informatiques, il n'est pas toujours possible de faire tourner toutes les opérations sur toutes les machines. Autrement dit, toutes les recettes ne sont pas qualifiables sur toutes les machines d'un parc d'équipements. L'exécution d'une nouvelle qualification peut être rapide, par exemple l'équivalent d'une séquence de production ("production run" en anglais). Toutefois, elle peut s'avérer longue, par exemple aussi longue qu'un cycle de production. Quelle que soit la durée, une qualification coûte en temps et en énergie. En règle générale, une configuration de qualification recette-machine pourrait prendre un des trois statuts suivants. Quand une recette n'est pas autorisée sur une machine, elle est non-qualifiable. Si la qualification est autorisée, elle est qualifiable. Et finalement, une recette qualifiable peut aussi être (déjà) qualifiée.

L'idéal serait que toutes les recettes soient qualifiées sur tout le parc d'équipements. Ainsi on pourrait allouer le volume de production (la charge ou "workload" en anglais) de chaque recette sur n'importe quelle machine sans aucune restriction. Pourtant, comme cela a déjà été dit, toutes les recettes ne sont pas qualifiables sur toutes les machines. En outre, une qualification reste coûteuse. Outre cela, l'état des qualifications est très dynamique, par exemple de nouvelles recettes sont souvent créées avec l'apparition de nouveaux produits; les recettes évoluent dans le temps créant de nouvelles recettes; la perte des qualifications (souvent dite "disqualification") surviendra suite aux maintenances, pannes machines ou absence de tournage d'une recette pendant un temps excessif sur une machine, etc. Toutes ces contraintes 1.3 La Gestion des Qualifications démontrent l'importance de la gestion des qualifications [START_REF] Johnzén | Importance of Qualification Management for Wafer Fabs[END_REF].

Un des défis du management est d'ajuster les contraintes de capacité avec le plan de production afin de répondre à la demande. La capacité devrait être optimisée en équilibrant au mieux la charge suivant le temps productif des équipements et la main-d'oeuvre disponible. La configuration des qualifications a un impact direct sur l'utilisation de la capacité d'un parc d'équipements. Il est impossible d'allouer une charge de production associée à une recette à un équipement non-qualifié. Donc, une configuration des qualifications adéquate permet l'optimisation de l'utilisation de la capacité disponible et garantit le bon fonctionnement de la fab. Une telle configuration ajoute de la flexibilité au système manufacturier en donnant plus de choix pour l'allocation des charges.

Par conséquent, une stratégie de gestion des qualifications est nécessaire pour garantir la satisfaction de la demande client et l'optimisation de l'utilisation de la capacité des équipements.

La Gestion des Qualifications sous Contrainte de Capacité

Les machines ne sont pas toujours disponibles pour la production. Le temps total disponible des machines est découpé en différentes catégories (voir Figure 1.3). Seul le créneau nommé "productive time" est vraiment utilisé pour la production. Cette donnée doit être considérée lors de l'équilibrage de charge et donc dans la gestion des qualifications. Des mesures de flexibilité ont été définies dans [START_REF] Johnzén | Flexibility Measures for Qualification Management in Wafer Fabs[END_REF] pour la gestion des qualifications. Ces indicateurs qui varient entre 0 et 1 (0% et 100%) mesurent le lissage de l'équilibrage optimal des charges. On les appelle mesures de flexibilité à capacité infinie ("uncapacitated flexibility measures" en anglais) car elles ne tiennent pas compte de la capacité des machines. L'idée dans la gestion des qualifications est de mesurer le gain de flexibilité potentiel associé à la qualification d'un couple recette-machine qualifiable. Pour calculer ce gain, la flexibilité de la configuration actuelle des qualifications dans un parc d'équipements est mesurée. Ensuite à chaque fois, un couple recette-machine qualifiable est virtuellement qualifié, et la flexibilité associée à cette nouvelle configuration est calculée. Ces calculs constituent la matrice Par la suite, les mesures de flexibilité à capacité finie (F Capa ) sont définies.

Paramètres

R

Le nombre total des recettes à faire, M Le nombre total des machines dans un parc d'équipements, W IP r Le volume de production total associé à chaque recette r, T P r,m

Le débit de production de la recette r sur la machine m (nombre de plaques traité par heure), Capa m La capacité de la machine m (en heures),

Q r,m     
1 si recette r est qualifiée sur machine m, 0 si recette r n'est pas qualifiée sur machine m. γ L'exposant de l'équilibrage de charge (γ ≥ 1).

La Gestion des Qualifications

Variables W IP r,m Le volume de production de la recette r alloué à la machine m, C r,m

Le temps de production de la recette r alloué à la machine m, W IP m Le volume de production total alloué à la machine m (W

IP m = R r=1 W IP r,m ), C m
Le temps de production total alloué à la machine m

(C m = R r=1 W IPr,m
T Pr,m ). L'équilibrage de charge peut se faire en terme de volume de production ou temps de production. Afin de calculer la flexibilité associée à une configuration de qualification d'un parc d'équipements, un modèle d'optimisation (1.1) doit être résolu. La mesure de flexibilité souhaitée constitue la fonction objectif. La seule contrainte assure que tout le volume de production de chaque recette est fabriqué et cela seulement sur les machines qualifiées. En maximisant la fonction objectif, l'équilibrage de charge optimal et la valeur de flexibilité sont obtenus.

F W IP Capa (1.
2) mesure la flexibilité en terme de volume de production ("WIP" en anglais pour en-cours).

F W IP Capa =   M m=1 (W IP m /Capa m ) M   γ M m=1 (W IP m /Capa m ) γ M ∈ (0, 1] (1.2)

CHAPTER 1. RÉSUMÉ (SUMMARY IN FRENCH)

Même avec un équilibrage imparfait, la mesure de flexibilité F W IP Capa pourrait atteindre son maximum. Cette information peut induire les décideurs en erreur lors de la prise de décisions pour faire de nouvelles qualifications. Pour contrer cette conséquence non-désirable l'exposant γ est prévu. En l'augmentant, la valeur de flexibilité baisse sans aucun impact sur l'équilibrage de charge. Donc, à l'aide de γ, il faut ajuster la valeur de flexibilité de manière qu'elle représente au mieux l'équilibrage réel de la charge dans la fab.

En considérant la capacité finie des équipements, la mesure de flexibilité ne peut seule conduire le décideur vers le meilleur choix de qualification. En effet, elle nous informe seulement sur la qualité de l'équilibrage. À titre d'exemple, elle atteint son maximum si la charge est parfaitement équilibrée même si chaque machine est disons deux fois plus (ou moins) chargée que sa capacité. Donc, des indicateurs complémentaires, nommés ratio de déviation de la capacité DR Capa , sont introduites pour assurer une meilleure prise de décision. Le Ratio de déviation de la capacité pour le volume de production DR W IP Capa est défini dans (1.3).

DR W IP Capa = M m=1 |W IP m -Capa m | M m=1
Capa m

(1.3)

Gestion des qualifications à capacité finie en terme de temps de production

La mesure de flexibilité à capacité finie en terme de temps de production F T ime Capa est définie dans (1.4).

F T ime Capa = Ideal Ratio Capa M m=1 R r=1 W IP r,m T P r,m • Capa m γ ∈ (0, 1] .
(1.4)

Le calcul de F T ime Capa se fait en deux phases. Dans un premier temps, tous les couples recette-machine qualifiables sont virtuellement qualifiés pour calculer la constante Ideal Ratio Capa (1.5). Il représente le meilleur équilibrage qu'on pourrait atteindre

La Gestion des Qualifications

en qualifiant tous les couples recette-machine qualifiables. Pour cela il suffit de considérer (1.5) comme la fonction objectif du Modèle (1.1) sachant que cette fois, il faut la minimiser.

Ideal Ratio

F T ime

Capa est une fonction convexe sous γ ≥ 1. L'augmentation de γ non seulement baisse la valeur nominative de flexibilité mais elle a aussi un impact sur l'équilibrage de charge. En l'augmentant, on diminue l'écart entre la capacité de chaque machine et le temps de production total qu'on y alloue. Ceci au détriment de l'allègement de charge des machines rapides au profit des machines lentes.

Sur le même principe que pour le volume de production, le ratio de déviation de la capacité pour le temps de production est défini dans (1.6).

DR T ime Capa = M m=1 R r=1 W IP r,m T P r,m -Capa m M m=1
Capa m (1.6) Des exemples et extensions concernant la gestion des qualifications à capacité finie sont décrits dans le Chapitre 4. Ce chapitre comporte aussi des méthodes de résolution dédiées à l'équilibrage de charge en terme de volume de production et temps de production.

En résumé, la capacité des machines doit être prise en compte lors de l'équilibrage de charge ainsi que la gestion des qualifications.

La Gestion des Qualifications sous Contrainte de Taille de Batch

Le regroupement des produits lors de la production ou batching est une pratique commune en industrie. En industrie des semi-conducteurs, les plaques de silicium parcourent l'ensemble des étapes de production par lots de 25. Suivant l'opération ou l'équipement, les lots sont regroupés (ou parfois même dégroupés) pour faire des

CHAPTER 1. RÉSUMÉ (SUMMARY IN FRENCH)

batches. Les lots dans un batch reçoivent la même opération. Le batching réduit la variabilité de fabrication tout autant que le coût de production. Donc, il est préférable de faire tourner les processus longs ou avec un coût de lancement élevé en batch.

La taille de batch peut varier dans le même parc d'équipements d'une recette à l'autre et pour une même recette d'une machine à l'autre. La variation de taille de batch d'un parc d'équipements TTH peut aller de 4 à 10 lots. Cette contrainte constitue une source importante de la variabilité et a un impact direct sur l'allocation dans l'équilibrage de charges et par conséquent sur la gestion des qualifications.

Le modèle d'optimisation tenant compte des tailles de batch est présenté au Chapitre 5. L'analyse de complexité montre que le problème est NP-difficile. Plusieurs heuristiques sont développés pour l'équilibrage de charges en terme de volume ainsi que de temps de production. Grâce aux expérimentations sur les instances industrielles de Soitec, les performances des heuristiques sont analysées notamment en comparaison avec le logiciel commercial Cplex.

En conclusion, la contrainte de la taille de batch doit être considérée lors de l'équilibrage de charge et de la gestion des qualifications pour assurer le meilleur choix de qualification.

La Gestion des Qualifications et la Variabilité des Charges

La variabilité dans un système manufacturier comprend les variations dans les flux de production. Elle peut avoir des sources stochastiques ou déterministes. Les sources stochastiques restent non contrôlables, par exemple: la demande, la panne des équipements, etc. En revanche, les sources déterministes sont maîtrisables, par exemple: les contraintes liées aux processabilités des produits sur les équipements, le batching, les contraintes liées au lancement, les flux ré-entrants, etc. La variabilité est l'ennemie de la production. Elle a un impact négatif sur la planification de production, l'ordonnancement et toute gestion liée à la production. Cette influence négative est plus grave en ce qui concerne les goulots d'étranglement. Il faut donc maîtriser la variabilité, surtout dans les étapes critiques, afin d'éviter sa propagation dans toute la ligne de production. La variabilité conduit aussi à une 1.3 La Gestion des Qualifications perte de capacité. Vu le coût élevé de production en industrie des semi-conducteurs, une utilisation optimale de la capacité disponible est souhaitée.

La variabilité diminue avec l'augmentation de la flexibilité. La flexibilité dans un parc d'équipements est la conséquence des possibilités liées à l'allocation des produits aux machines. Plus de flexibilité permettrait une meilleure allocation des charges et par conséquent l'optimisation de la capacité.

L'impact de la gestion des qualifications sur la variabilité est étudié au Chapitre 6. Des mesures sont introduites pour évaluer la variabilité des charges dans un parc d'équipements. Elles sont utilisées pour proposer de nouvelles qualifications afin de réduire la variabilité et augmenter l'utilisation de la capacité du parc d'équipements.

La mesure de la variabilité (1.7) est introduite pour évaluer la variabilité des charges dans un parc d'équipements. Pour chaque nouvelle qualification possible, la réduction en terme de variabilité est calculée. La meilleure qualification est choisie suivant la réduction de variabilité.

V ar T ime Capa = M m=1 (C m -Capa m ) γ (1.7)
Le rôle de γ est le même que dans la mesure de flexibilité pour les temps de production (1.4). La mesure de variabilité (1.7) est similaire au ratio de déviation de capacité (1.6) non-normalisée et avec l'exposant de l'équilibrage de charge. Elle tente de diminuer l'écart de la charge allouée à chaque machine et sa capacité disponible. D'autres variantes des mesures de variabilité sont définies au Chapitre 6. min V ar

W IP r,m = W IP r ∀r (1.8b) W IP r,m ≥ 0 ∀r, m
Les mesures de variabilité doivent être minimisées (1.8a) tout en assurant que tout le volume de production de chaque recette est fabriqué et cela seulement sur les machines qualifiées (1.8b). On explique ensuite comment adapter les méthodes de résolution au Chapitre 4 pour résoudre le modèle (1.8).

CHAPTER 1. RÉSUMÉ (SUMMARY IN FRENCH)

Des expérimentations sont menées sur le parc d'équipements TTH de Soitec pour étudier l'impact d'une nouvelle qualification sur la variabilité des charges. En premier lieu, on considère une instance de 22 machines et 37 recettes pour une seule période. Le diagramme 1.4 montre l'équilibrage de charge avec la configuration de qualification actuelle. Les lignes horizontales définissent la capacité des machines. Chaque barre verticale montre la charge associée à chaque machine sachant que chaque couleur représente le volume d'une recette. Le dépassement des barres des lignes horizontales indique une surcharge et à l'inverse, une sous-charge. En calcu-Figure 1.4: L'Équilibrage de Charge pour la Configuration Actuelle des Qualifications lant la réduction de variabilité associée à chaque nouvelle qualification (ce qui signifie ajouter de la flexibilité), la qualification qui réduit le plus possible la variabilité est choisie. Le diagramme de l'équilibrage de charge (Figure 1.5) illustre comment la variabilité des charges est réduite après une nouvelle qualification, autrement dit après l'augmentation de la flexibilité. En continuant à faire de nouvelles qualifications qui diminuent le plus la variabilité, l'équilibrage de charge s'améliore. Au lieu de montrer les diagrammes de l'équilibrage de charge pour chaque nouvelle qualification, quelques indicateurs de performance sont utilisés dans le Tableau 1.1. On constate qu'en faisant de plus en plus de nouvelles qualifications diminuant la vari-Figure 1.5: L'Équilibrage de Charge pour la Configuration après une Qualification Meilleure abilité, on diminue la surcharge et cela en utilisant la capacité non-utilisée et en répartissant mieux la charge totale. En même temps, on remarque que l'impact de nouvelles qualifications baisse graduellement. Donc, à un certain moment un compromis doit être fait entre la diminution de la variabilité souhaitée et l'augmentation de la flexibilité. La Figure 1.6 présente les résultats de Tableau 6.1. Pour dix instances industrielles, le Tableau 1.2 montre l'impact d'une nouvelle qualification sur la réduction de la variabilité. En général, la meilleure qualification réduisant le plus la variabilité, réduit la surcharge et la capacité non-utilisée des machines considérablement, tout en augmentant légèrement la charge totale sur l'ensemble de toolset. À titre d'exemple, dans la première instance, la surcharge est entièrement éliminée (-100%), l'utilisation de la capacité non-utilisée est considérablement augmentée (20, 20%) tandis que la charge totale est légèrement augmentée de 3, 40%. Les expérimentations montrent aussi que les variations ne suivent pas un profil linéaire. Suivant le mix des produits, les caractéristiques et la configuration du parc d'équipements, le même montant de réduction de la variabilité aura plus ou moins d'impact sur les indicateurs de performances. Les Instances 1 et 10 ou 4 et 9 en sont l'exemple. En ce qui concerne les Instances 1 et 10, la variabilité est ré- En résumé, plus de flexibilité absorbe la variabilité. Donc, une plus grande flexibilité est requise là où la variation des charges est la plus élevée mais pas forcément là où la charge est la plus élevée. En d'autres mots, si la variabilité est basse mais qu'en parallèle il subsiste des surcharges, de nouvelles qualifications n'apporteraient pas de flexibilité supplémentaire et ne résoudraient pas le problème. Dans ce cas, d'autres scénarios comme l'achat de nouveaux équipements doivent être considérés.

CHAPTER 1. RÉSUMÉ (SUMMARY IN FRENCH)

Industrialisation de la Gestion des Qualifications

Les concepts expliqués dans cette partie de thèse ont été industrialisés au sein de l'entreprise Soitec. Un nouveau processus de prise de décision a été introduit et mis en place. Ce processus implique le recueil des données qui consiste en états des qualifications, débits de production, tailles de batch, volumes de production et capacités des machines (voir Figure 1.7). La matrice des états de qualification comporte trois codes pour désigner l'état des qualifications des couples recette-machine: 0 pour non-qualifiable, 1 pour qualifiable et 2 pour (déjà) qualifié.

Les données d'entrée sont chargées dans l'application de gestion des qualifica- La Figure 1.9 illustre les différentes interfaces pour la récupération des données. Les volumes par recette sont extraits en croisant les volumes de production par produit et les routings associés à chacun des produits. La capacité des machines est déterminée en considérant l'historique de l'uptime des machines, la cible de l'uptime et les maintenances en vue. Les états des qualifications, les débits de production et les tailles de batch sont fournis par le département "process".

Une application nommée "Input File Creator" rassemble toutes ces données en une feuille Excel dans un format prédéfini. Ensuite, l'utilisateur charge cette feuille dans l'application de gestion des qualifications (appelée "Nexus" chez Soitec). Les

La Gestion des Qualifications

Input Data résultats de l'optimisation ont différentes utilisations. Les propositions des qualifications servent à choisir les meilleures qualifications pour assurer une utilisation optimale de la capacité des machines. En considérant le futur mix de production, l'équilibrage optimal des charges sert pour la planification de capacité. L'équilibrage optimal des charges en terme de volume de production sert comme guideline pour l'ordonnancement.

Conclusions et Perspectives

Dans la première partie de la thèse, nous avons étudié l'impact de la gestion des qualifications sur l'optimisation de l'utilisation de la capacité des équipements. Cette gestion apporte aussi de la flexibilité dans le système manufacturier. L'impact de la capacité limitée des équipements sur l'équilibrage de charge et par conséquent sur la gestion des qualifications a été étudié dans le Chapitre 4. Le Batching est une contrainte importante dans certains parcs d'équipement en industrie des semiconducteurs. Lors de l'allocation des charges, cette contrainte doit être respectée. Plusieurs algorithmes d'équilibrage de charge sous contrainte de batch ont été introduits dans le Chapitre 5. De nouvelles qualifications apportent potentiellement plus de flexibilité au parc d'équipements. Plus de flexibilité réduit la variabilité des charges. Ce sujet est discuté dans le Chapitre 6. Les concepts de cette partie de Dans le Chapitre 9, on défini le cadre scientifique et industriel de la seconde partie de la thèse. Un des éléments de la gestion de la chaîne logistique concerne la planification de production. Elle fait référence aux décisions prises au niveau tactique dans l'objectif de déterminer les périodes et les quantités de production. La planification de production (autrement dit dimensionnement des lots) s'exprime par un modèle d'optimisation souvent avec l'objectif de minimisation des coûts (ou maximisation des profits) sujet aux contraintes. Les contraintes concernent la satisfaction de la demande à un taux de service souhaité, l'équilibrage des flux de production et les stocks, la capacité de production, la nomenclature, etc.

CHAPTER 1. RÉSUMÉ (SUMMARY IN FRENCH)

Le problème de dimensionnement des lots discuté dans cette seconde partie modélise le processus décrit dans la Section 1.2. Ce processus donne naissance à un système manufacturier avec deux lignes de production avec flux ré-entrants d'où l'expression planification de production en boucle fermée. Le dimensionnement des lots est fait sur la base d'un horizon discret avec de longues périodes et la nomenclature bi-niveau, un niveau concerne les matières premières et l'autre le produit fini.

Dimensionnement des Lots Multi-Produits Bi-Niveau à Capacité Finie avec Multiples Refabrications des Produits Dérivés Réutilisables

Dans le Chapitre 10, nous avons modélisé le processus de fabrication de plaques SOI en utilisant la Technologie Smart-Cut™ ainsi que le processus de "refresh" associé. Dans la suite, le problème est décrit rapidement sans rentrer dans le détail de la modélisation.

La Technologie Smart-Cut™ est décrite en Section 1.2. Les matières premières achetées sont les plaques de Fresh et de Base. En utilisant ces deux plaques, une plaque SOI est fabriquée. La plaque Fresh est aussi nommée le Top. La fabrication

CHAPTER 1. RÉSUMÉ (SUMMARY IN FRENCH)

de cette plaque SOI génère une plaque "négatif" (notée Négatif 0 ou Neg0 ) qui est considérée comme un produit dérivé. Le Neg0 peut être recyclé (communément appelé "refreshé") pour revenir dans le cycle de fabrication de SOI. La plaque Neg0 recyclée est appelée Refresh 1 ou R1. Le R1 peut être utilisé avec une autre plaque de type Base pour la production d'une autre plaques SOI. Ce processus de réutilisation des plaques de Top peut continuer jusqu'à une limite maximum qui dépend des produits considérés. Ce processus est illustré pour 7 niveaux de refresh dans la de production et de commande. Les coûts de stockage concernent les matières premières achetées (les plaques de Fresh et de Base), les produits dérivés (les plaques négatifs), les plaques de Refresh et les produits finis (les plaques SOI). Les coûts de lancement concernent l'achat des matières premières (les plaques de Fresh et de Base), la production des produits finis (les plaques de SOI) et le processus de refresh. Les contraintes industrielles sont aussi considérées, par exemple: le rendement de production, l'aspect refresh hors-site, la capacité, etc.

Le modèle mathématique introduit est NP-difficile. Des jeux de données ont été construits à partir de données industrielles. Le logiciel CPLEX a été utilisé pour résoudre ce problème. Il se montre très performant et résout les plus grandes instances en moins de 10 minutes. En augmentant l'horizon de planification, l'importance du processus de refresh devient plus claire. En passant de 6 à 46 périodes, le coût

CHAPTER 1. RÉSUMÉ (SUMMARY IN FRENCH)

d'achat des plaques de Fresh diminue considérablement. Ce phénomène est dû au fait que le coût d'achat des plaques de Fresh est nettement inférieur à celui de refresh. Pour 6 périodes, le coût d'achat des plaques de Fresh et celui de processus de refresh occupent respectivement 18, 63% et 5, 08% du coût total du plan de production. En passant à 46 périodes, le pourcentage du coût d'achat des plaques de Fresh diminue fortement, passant à 2, 72%. En contrepartie, le coût de refresh augmente légèrement, à savoir: 8, 49%. Les expérimentations démontrent l'intérêt de la planification simultanée des deux processus de production de SOI et de refresh. Un autre point observé est que la production de plaques SOI ne sert pas seulement à satisfaire la demande mais aussi à générer des plaques de type négatif afin d'assurer la disponibilité des plaques de refresh pour des productions de SOI dans les périodes futures. Par ailleurs, la capacité de la ligne de refresh n'est pas seulement contrainte par elle-même mais aussi par celle de la ligne de SOI. Car c'est bien cette dernière qui assure la génération des plaques de type négatif utilisées ensuite dans la ligne refresh.

Dimensionnement de Lots Mono-Produit Bi-Niveaux à Capacité Infinie avec Multiples Refabrications des Produits Dérivés Réutilisables

En se basant sur le problème de planification de production du système manufacturier de la Technologie Smart-Cut™, un modèle de dimensionnement de lots mono-produit et à capacité infinie est défini dans le Chapitre 11. Ce modèle permet une étude académique plus approfondie sur des caractéristiques et des propriétés du problème. La maîtrise d'un tel modèle ouvre la voie pour proposer des heuristiques qui utilisent la décomposition et la relaxation Lagrangienne pour le cas multi-produits et à capacité finie. La Figure 11.1 illustre le système manufacturier composé des lignes de fabrication SOI et refresh. Dans ce système, seule la gestion des plaques de Top et SOI est considérée. Le niveau de refresh est défini par l et il peut atteindre l max . Le modèle mathématique comporte trois contraintes de lancement, à savoir celle de l'achat des plaques de Fresh, de processus de fabrication de produits finis (les plaques SOI) et Afin de proposer une méthode de résolution, on a seulement considéré la contrainte de lancement d'achat de matières premières. Ensuite, on a défini la structure des coûts basée sur les réalités industrielles. Enfin, on a introduit des hypothèses de "Full Push" pour simplifier le modèle. Ces hypothèses assurent que les matières premières seront systématiquement transformées en produits finis. Ainsi, les produits dérivés sont recyclés immédiatement et ne seront disponibles qu'au début de la période suivante. Ces hypothèses entraînent la suppression de plusieurs stocks sauf celui des produits finis (voir Figure 1.13). On propose un algorithme de ré- l'algorithme reste exponentiel.

Planification de Production en Boucle Fermée

Conclusions et Perspectives Générales

Les deux parties de cette thèse sont complémentaires. Dans la première, nous avons considéré une contrainte particulièrement forte dans l'industrie des semiconducteurs comme un levier de l'optimisation de la capacité. Dans la seconde partie, la planification de production de deux lignes de production en boucle fermée a été étudiée. La capacité est une des entrées d'un modèle de planification de production. Donc, comme perspective, ces deux idées peuvent être combinées dans un seul modèle. Dans un tel modèle, les nouvelles qualifications assurent la capacité nécessaire pour satisfaire la demande client lors de la planification de production. Un modèle a été proposé dans la Section 12.2.

General Introduction

This PhD project grew out of the collaboration between industry and academia. Soitec, a pioneer in building revolutionary semiconductor materials, is the world leader in producing high performance Silicon-On-Insulator (SOI) wafers. Soitec invests in research and development to make breakthroughs in advanced semiconductor materials. The company has specific production lines to satisfy the fabrication needs of the developed products. That is why, Soitec constantly seeks to improve its industrial practices to cope with market changes and satisfy demand. The industrial engineering department of Soitec decided to start the first PhD project of Soitec in industry by beginning a collaboration with Manufacturing Sciences & Logistics (SFL) department of École Nationale Supérieure des Mines de Saint-Étienne. The SFL department in CMP (Center of Microelectronics in Provence) has extensive experience in collaborating with industry, in particular with semiconductor manufacturing companies, such as STMicroelectronics. The French national research and technology agency (ANRT) has set up a program entitled CIFRE (standing for "Conventions Industrielles de Formation par la REcherche") to promote high quality research and development projects in the framework of a PhD project. This PhD project has received financial aid of ANRT. The main subject treated is capacity and production planning. Each subject is developed in a separate part.

The first part deals with capacity planning and optimization in the context of semiconductor manufacturing. We take a binding restriction, called qualification, present in semiconductor manufacturing as a lever for increasing and optimizing capacity utilization. The unstable business environment, and complex and extremely dynamic production environment require to adapt a flexible approach for qualification management. Interest in flexibility has grown rapidly in the last years both among practitioners and academicians. A large number of definitions has been given for the concept of flexibility. In Part I, we clarify what we mean by flexibility while showing its importance in capacity planning. In the second part of the thesis (Part II), we are interested in production planning of two related production lines in Soitec. We tackle the industrial problem by proposing an innovative mathematical model. By taking the core challenge of the industrial case, we define and analyze an original academic problem.

In each chapter of

The PhD project was launched with the subject of qualification management and capacity utilization. As the objectives of the industry were met, we started the second part of the thesis which is dedicated to a specific closed-loop manufacturing system.

Thesis Outline and Reading Plan

The manuscript is designed to be flexible. It is divided into an introductory chapter and two parts which are organized in a way to make each of them selfcontained and independent of the other (see Figure 1.14). The introductory Chapter 2 describes the general industrial context of the whole study. Its purpose is to give an initial understanding of this topic to the readers who may not be familiar with semiconductor manufacturing (in particular SOI manufacturing using the Smart-Cut™ Technology).

Part I (Chapters 3 to 8) discusses capacity optimization and planning through flexible qualification management. In Chapter 3, the qualification management and its impact on capacity planning are discussed. Chapter 4 studies qualification management while considering the limited capacity of the machines. Batching is an important characteristic for some machines in the industry in general and in semiconductor manufacturing in particular. The impact of batching on qualification management is discussed in Chapter 5. Chapter 6 discusses the relationship of qualification management and production variability. The introduced concepts are industrialized. The industrialization is explained in Chapter 7. Extended conclusions and perspectives of Part I are presented in Chapter 8.

In Part II, the focus is on the production planning for SOI fabrication and refresh process lines in Soitec. The described production process in Section 2.3 is

Industrial Context

This chapter introduces the industrial context of both parts of the thesis. Almost all aspects of the first part concerning capacity planning through flexible qualification management can be applied to most semiconductor fabrication facilities. The focus of the second part is on production planning of a Silicon-On-Insulator (SOI) wafer fabrication supply chain using the specific Smart-Cut™ Technology. The technology described briefly in this chapter is exclusive to the company Soitec.

Introduction

The history of semiconductor manufacturing dates back to the invention of fieldeffect transistor (FET) by the physicist Julius Edgar Lilienfeld in 1925 in Canada. At that time, it did not find practical use because of the lack of high-quality semiconductor material. However, the electronics industry was born with the invention of vacuum tubes in the early 19s and devices such as radios. Transistors rapidly replaced vacuum tubes as the latter consume a lot of power while producing heat. They are fragile, relatively big and also wear over time. Transistors are made up of semiconductor material. Semiconductor material is a substance which is conductor under some conditions else an insulator. The electrical conductivity of a semiconductor changes with the variation of voltage or current of the control electrode. Several elements and compounds show semiconductor characteristics. However, most of the semiconductor materials are built on silicon. The small size of the transistors has revolutionized the world. The electronics revolution has been following the Moore's law which states that the number of components in an integrated circuit doubles every 18 to 24 months. The goal of the semiconductor industry has been to keep up with this pace of evolution with the creed "smaller, faster and cheaper". To follow this slogan, constant research is conducted on semiconductor components material, design and production process. While the manufacturing processes become more complex, the cycle time and production cost must decrease, which call for a better capacity utilization of equipment and labor. Besides, the life cycle of the products shortens while their diversity increases. Therefore, the production process must be efficient, cost competitive, flexible and agile. In the remainder of this chapter, we paint a global picture of semiconductor manufacturing with an emphasis on SOI manufacturing in Section 2.2. Section 2.3 focuses on the main theme of this dissertation, i.e. production and capacity planning.

Semiconductor Manufacturing

Semiconductor components are made of semiconductor elements or compounds; the most used semiconductor substance is silicon. Silicon (Si) can be found by mass

Semiconductor Manufacturing

in dusts, sands, planetoids and planets under the form of silicon dioxide (silica) or silicates (Figure 2.1a). However, very clean and good form sand is used for making silicon. The sand (SiO 2 ) is heated just above its melting point to obtain silicon (Si). After several chemical processes, blocs of polycrystalline silicon are obtained (2.1b). Monocrystalline Silicon (single-crystal silicon) ingots (Figure 2.1c), produced with high purity, are used in the fabrications of semiconductor components. They are grown mostly using the Czochralski process. Silicon wafers are obtained using mechanical-chemical procedure from the silicon ingot. Firstly, the silicon ingot is ground to the desired diameter. Then a notch or a flat edge is given to the ingot serving as the future wafer orienting guide. The ingot is sliced into silicon wafers (Figure 2.1d). Several mechanical and chemical steps guarantee the flatness, thickness and smoothness of the wafers. The silicon wafers can then be used for making semiconductor components such as chips and ICs (integrated circuits). However, some devices require higher performance and specific characteristics which cannot be obtained by traditional silicon wafers. That is where Silicon-on-Insulator wafers are used. In the next section, we discuss more about SOI wafers and specially one of the most interesting techniques available to make them, the Smart-Cut™ Technology.

Semiconductor components made of silicon are used in most electronic devices. The semiconductor components are etched on silicon or SOI wafers. As even a tiny bit of dust can damage an integrated circuit or a chip, the whole semiconductor manufacturing process is done in so-called clean rooms. A clean room is upto 10, 000 times cleaner than an operating theater. The air inside the clean room is continu-Chapter 2. Industrial Context ously filtered and cleaned. Operators, technicians and anyone who enters the clean room must wear special clothing to keep the area free from particles. The clean room garments include face mask, hood, coverall, shoe cover, glasses and gloves.

The whole process of semiconductor manufacturing can be divided into two sequential sub-processes: front-end and back-end (see Figure 2.2). The front-end concerns all fabrication steps for the creation of circuits on a blank wafer. The back-end begins when the chips and all circuits are created on the wafer. It concerns testing, dicing, wiring, assembly and packaging.

The front-end or wafer fabrication produces silicon chips on a blank wafer (typically silicon). A chip is a set of electronic circuits manufactured in layers. A microchip may contain over a billion transistors. Photomasks are used to optically transfer patterns to wafers. The photolithography process transfers the geometric pattern of the photomask on the wafer. After a series of chemical treatments, the desired patterns are engraved on the wafer and the undesired exposed material is etched away. The photolithography is a very precise process which contains several re-entrant steps. At the end of the front-end process, the proper functioning of the chip is electronically tested.

The first step of the back-end operation is die preparation in which the wafer is sawed into individual chips or dices. Then each dice is attached to the metallic support structure of the package (e.g. the leadframe) by means of an alloy or an adhesive. Then the leads on the leadframe are wire bonded to the input/output electrical terminals of the package. Packaging or encapsulation is the final step of the semiconductor manufacturing process in which the chip is encapsulated in a plastic or ceramic mold. The package protects the chip from corrosion and physical damage. Before shipping the integrated circuits to the customers, they are tested electronically.

Silicon-On-Insulator (SOI) Wafer Fabrication

Semiconductor components have revolutionized our daily life. Silicon wafers are extensively used in semiconductor manufacturing to produce microelectronic components such as integrated circuits and chips. However, some devices require higher SOI is widely used in the microelectronics industry such as radiofrequency devices, microelectro-mechanical systems (MEMS), photonics and biotechnological chips.

Silicon-On-Insulator (SOI) Wafer Fabrication

Chapter 2. Industrial Context

An SOI wafer is composed of three layers (see Figure 2.4):

• The active silicon layer called Top;

• The buried Silicon Oxide layer called BOX;

• The monocrystalline silicon which serves as mechanical support called Base. The thickness of the active silicon layer (Top) varies depending upon the SOI application between 0.01µm and 1.5µm. The thickness of the dielectric insulator (BOX) varies between 0.05µm and 3µm. The thickness of the silicon substrate (Base) varies between 375µm and 775µm based on the wafer diameter.

The main advantage of SOI wafers compared to silicon bulk wafers lies in the buried oxide (BOX). In a MOSFET, only the superficial layer of the silicon (0.006 to 0.1µm) is used for the transport of carriers. The rest of the substrate which represents about 99.99% of its thickness, causes current leakage and undesirable parasite effects. SOI not only eliminates the parasite effects but also decreases the current consumption [START_REF] El | Novel Pseudo-MOSFET Methods for the Characterization of Advanced SOI Substrates[END_REF].

Circuits based on SOI technology are more compact, faster, more power-efficient and more heat-resistant compared to traditional silicon wafers. SOI technology is challenging more and more the traditional bulk silicon wafers. For instance, more than 60% of mobile devices and 80% of game consoles produced in 2012 use SOI chips [START_REF] Kononchuk | Silicon-On-Insulator (SOI) Technology: Manufacture and Applications[END_REF].

The main advantages of the SOI technology are [START_REF] Kononchuk | Silicon-On-Insulator (SOI) Technology: Manufacture and Applications[END_REF]:

Silicon-On-Insulator (SOI) Wafer Fabrication

• High performance, both in speed and power in comparison to traditional silicon bulk;

• Smaller chip area due to better scaling; and

• CMOS process simplification.

A group of important companies in the microelectronics industry have formed the SOI Industry Consortium to promote the benefits of the SOI technology in the market and enhance its usage.

SOI wafers may be produced using different technologies such as SIMOX™ (Separation by IMplantation of OXygen), wafer bonding or Seed methods. The production process studied in this thesis considers a type of wafer bonding technology called the Smart-Cut™ Technology.

Smart-Cut™ Technology

Smart-Cut™ Technology has been developed since 1993 to obtain SOI materials [START_REF] Bruel | Silicon on Insulator Material Technology[END_REF]. The Smart-Cut™ Technology is based on direct bonding of two wafers. One of the wafers is implanted by light gas ions such as hydrogen. At the mean depth of the ion penetration, a weakened zone is formed. Once the implanted wafer is bonded with the second wafer, the splitting occurs at the implanted zone. Therefore, a thin silicon layer is transferred from the implanted wafer to the second wafer. The second wafer acts only as a mechanical support. The semiconductor components are built on the transferred thin silicon layer.

The Smart-Cut™ Technology steps used to produce SOI Wafers are as follows [START_REF] Kononchuk | Silicon-On-Insulator (SOI) Technology: Manufacture and Applications[END_REF] (see Figure 2.5):

• The silicon Wafer A is oxidized thermally. The Buried Oxide (BOX) is now formed;

• The oxidized Wafer A is implanted by ions of Hydrogen creating microcavities. The micro-cavities constitute the weakened zone. The weakening is defined by the energy and the implantation dose. The implantation energy determines the hydrogen ion penetration depth through the BOX;

Chapter 2. Industrial Context

• The implanted Wafer A is cleaned and bonded to Wafer B at room temperature. The cleaning eliminates the particles from the surface of the wafers. Moreover, it covers the wafers with radicals of OH which facilitate the bonding. The very plane surface of the wafers allow molecular adhesion of two wafers;

• The bonded Wafers A and B are split, leaving a thin layer of Wafer A on Wafer B. The splitting takes place in an oven. By increasing the temperature, the pressure of the Hydrogen (H 2 ) in the micro cavities cause horizontal fracture at the weakened zone;

• Finally, using thermal annealing, the bonding surface is consolidated.

The roughness left on the SOI Wafer is removed using mechanical-chemical process.

The silicon bulk which remains after the splitting can be reworked again to be used as another Wafer A. The process of reworking the used Wafer A is called the refresh process. The second Part of this dissertation (Chapters 9 to 11) is dedicated to production planning of the SOI Production and Refresh Process. One of the most economical advantages of the Smart-Cut™ Technology is the possibility of reusing Wafer A. The recycling (or "refresh" process) contains several steps. First the oxide layer of the used Top Wafer is removed. Then the edge is polished. Using Double Side Polishing (DSP), the wafer is polished roughly. After cleaning, the Chemical-Mechanical Planarization (CMP) process smooths the wafer which is ready for the final cleaning. Once cleaned, the wafer is graded. If the wafer fits the specifications, it is used to make SOI wafers. If it fails the inspection, depending on the status, it returns either to the refresh line for a second try or it is discarded or used as a test wafer. A detailed modeling of the refresh process line can be found in Appendix B.

As Wafer A is recycled several times, its quality must be excellent. While the quality of Wafer B is not critical. It serves only as a mechanical support (from which the name handle wafer is derived). Because of the low cost of Wafer B and that Wafer A can be recycled multiple times, the Smart-Cut™ Technology is cost competitive. It is almost similar to a mono-wafer technology. Among other Smart-Cut™ Technology advantages, we can mention the high quality and the thickness homogeneity of the transferred active layer. Furthermore, the Smart-Cut™ Technology can be adapted to transfer single-crystal substrates on different supports [START_REF] Kononchuk | Silicon-On-Insulator (SOI) Technology: Manufacture and Applications[END_REF]. SOI wafers are produced in clean rooms like other semiconductor components.

Silicon

Chapter 2. Industrial Context

Production and Capacity Planning in Semiconductor Manufacturing

The goal of any production system is to transform raw materials into finished products by adding more value to the initial raw material at each production step. This is also the case in semiconductor manufacturing with the difference that its production and business environment are very complex. The production complexity comes from the sophisticated nature of the products and associated manufacturing systems. The relatively high product and production cost and the competitive business environment call for an efficient and effective production planning system.

The objective of a production system is to satisfy the customer demand over a time horizon in the most efficient and effective way while considering multiple constraints and optimizing different production indicators. Some production measures which are to be optimized are cycle time, (work-in-process) WIP levels and machine capacity utilization.

The second part of this study deals with the production planning of the SOI and refresh production lines. Special constraints associated with the SOI fabrication using the Smart-Cut™ Technology are analyzed and considered in the problem modeling. In particular, the refresh process, which is one of the exclusive features of SOI manufacturing using the Smart-Cut™ process is detailed.

The capacity planning is often categorized into long-, medium-and short-term horizons. The time dimension differs from industry to industry [START_REF] Vollmann | Manufacturing Planning and Control for Supply Chain Management[END_REF].

The focus in the first part (Chapters 3 to 7) of this thesis is on medium-to short-term horizon capacity planning which involve more operational decisions on a monthly to weekly basis.

The short-term capacity planning determines the expected machine utilization of (a single machine or a toolset) for some determined production plan or loading [START_REF] Mönch | Production Planning and Control for Semiconductor Wafer Fabrication Facilities: Modeling, Analysis, and Systems[END_REF]. This machine utilization is used to see if the capacity meets the production plan. If not, either the capacity must be increased or the production plan must be revised. The interested reader can refer to [START_REF] Mönch | Production Planning and Control for Semiconductor Wafer Fabrication Facilities: Modeling, Analysis, and Systems[END_REF] for more literature on subjects related to production planning in semiconductor manufacturing.

Conclusion

Conclusion

In this chapter, we have set the stage for the remainder of the dissertation. The importance and essential role of semiconductor manufacturing has been pointed out. This study is conducted in Soitec, a company which produces high performance wafers used for high demanding applications. Rapid expansion of the semiconductor industry and introduction of new technologies have created high-mix low-volume production lines. In order to cope with rapid changes, to lower the cycle time and production costs, efficient and flexible capacity and production planning is required.

Part I Qualification Management: Optimizing Flexibility and Capacity Utilization

Chapter 3

Qualification Management in Semiconductor Manufacturing

Semiconductor manufacturing is one of the most complex industries in the world. The increased diversity of products and short product life cycles call for re-configurable machines. Qualification is a kind of setup which gives a piece of equipment new capability. Nevertheless, qualifications consume time and energy. All of these constraints in the dynamic fab environment suggest an efficient and effective qualification management. This chapter opens up the discussion about flexible qualification management and capacity planning in semiconductor manufacturing.

Introduction

The semiconductor industry is one of the most complex and modern industries in the world. Expensive manufacturing equipment and the ever-growing competition call for an optimal capacity utilization of the fabrication facilities (called "fabs"). In order to cope with the fast-changing business environment, the production system must be flexible enough to produce a wide range of products. As the life cycle time of products shortens more and more, production systems must be agile to rapidly manufacture new products or adapt to product mix changes.

In semiconductor manufacturing, wafers undergo operations at workstations called "toolset". Each toolset is a collection of nonidentical multi-purpose parallel machines that are reconfigurable. In order to perform an operation, a recipe must be executed on the product. A recipe is the machine instructions to obtain the desired process. In order to perform an operation on a product, its corresponding recipe must be qualified on the machine. However, due to multiple hardware and software restrictions, maintenance or retrofit costs, it is not possible to qualify all recipes on every machine.

Qualification is one of the characteristics of the semiconductor manufacturing. It is a kind of setup with the difference is that a qualification is performed once and not before each production run. And the qualified machine remains qualified for that recipe until a disqualification occurs. Based on the toolset, several reasons may cause recipe disqualification on a machine. Machine breakdown or maintenance may require the re-qualification of all previously qualified recipes. Not running a recipe for a long time on a machine may also lead to automatic disqualification.

Qualifications have a direct impact on production capacity. If a recipe is not qualified on a machine, it is not possible to allocate the production volume of this recipe to the machine. The impact of the recipe-to-machine qualification configuration in a toolset is discussed in 3.3.

Besides, qualifying a recipe on a machine can be very time-and energy-consuming. Test products must be used for test runs. During test runs, the machines are under scheduled downtime status, therefore in a non-productive status. Metrology and defect inspection resources must also be extensively used. Hence, it is not economi-

Qualification in Semiconductor Manufacturing

cally wise to perform a great number of qualifications. If a poor recipe-to-machine configuration results in loss of capacity, too many qualifications (and maintaining them) are very costly and also cause scheduled downtimes. These two contradictory constraints call for an efficient qualification management policy. Qualifications are also one of the sources of variability. Variability often lead to the increase of the buffer stocks in the fab. Variability and hence stock and cycle times are reduced by a better qualification management. This leads to a smoother product flow in the production line and cycle time reduction.

Qualification in Semiconductor Manufacturing

In the course of production, each wafer undergoes operations at each workstation (named as "toolset"). A toolset is a collection of parallel and mostly nonidentical machines (or "tools") performing similar operations. For instance, a collection of furnaces, with eventually different specifications (throughput, batch size, hardware and software) constitutes the "Thermal Treatment Toolset" or a collection of "furnaces" constitutes the "Thermal Treatment Toolset". Each operation (which corresponds to a production step) is associated with a recipe which corresponds to the machine instructions to obtain the desired process. A thermal treatment recipe may specify the ramp up and ramp down temperature values and duration, cool-down rate and gas pressure at each phase. Or each "implantation recipe" defines the implantation energy and duration besides other technical specifications.

In order to perform an operation on a machine, its recipe must be qualified beforehand on the machine. Due to machine hardware and software restrictions, all operations cannot be performed on all machines. In other words, all recipes cannot be qualified on every machine of the toolset. Therefore, in order to qualify a recipe on a machine, several setups (both software and hardware) and tests may be needed. A qualification can be relatively quick to perform (e.g. equivalent to a production run), but also very time-consuming (e.g. equivalent to a production cycle time) and hence costly.

Chapter 3. Qualification Management in Semiconductor Manufacturing

The ideal configuration would be that all recipes are qualified on all machines, so that we can freely allocate the production volume of any recipe to any machine. However, due to process restrictions, some of the recipe-to-machine qualifications are not authorized. Qualifying "qualifiable recipes" on machines is costly and time consuming, causing scheduled downtimes. Therefore, in practice only a few number of qualifications can be performed.

In order to reduce qualification costs and to avoid performing unnecessary qualifications, an efficient qualification strategy must be set up. In [START_REF] Johnzén | Importance of Qualification Management for Wafer Fabs[END_REF], the importance of qualification management in wafer fabs is pointed out. As already stated, a poor qualification configuration may result in an unbalanced toolset making production planning and also scheduling difficult. Although, if this were the case (i.e. all recipes were authorized to be qualified on all machines), it would be too costly to qualify all recipes on all machines. The recipes are sorted into three categories: Unauthorized (or unqualifiable or non-qualifiable), authorized (or qualifiable) and qualified. This qualification classification links the recipes and the machines. In this thesis, without loss of generality, the recipes are considered to be either "(already) qualified" or "non-qualifiable" on machines. In Chapter 7, we explain the industrialization of all cases. The recipes however may also be sorted into other categories: Normal recipes which are qualified at least on one machine and Not previously Qualified Recipes (termed hereafter NQR) which are mostly new recipes corresponding to new products which have not been previously qualified on any machine. The case of not previously qualified recipes may occur when a new product is launched or when a recipe has been disqualified from all machines. For some toolsets, such as "thermal treatment", disqualifications may occur automatically due to not processing a recipe during a certain time window on the machine. However, disqualifications may be caused after changing the hardware, corrective maintenance operation, (rarely) uninstalling some software, etc.

Qualification Management and Capacity Planning

Qualification Management and Capacity Planning

One of the main management challenges is to match capacity constraints to forecast plans either by increasing capacity (or capacity utilization) or by adapting the plans to the capacity constraints by delaying the delivery date or canceling the orders. Other possible courses of action to match capacity and forecast plans are subcontracting, purchasing manufactured parts instead of raw materials and choosing possible alternate routing ( [START_REF] Vollmann | Manufacturing Planning and Control for Supply Chain Management[END_REF] and [START_REF] Silver | Inventory Management and Production Planning and Scheduling[END_REF]). In a production line or a workcenter, the capacity is the output rate, i.e. the the processing work amount per time. The capacity must be optimized through workload balancing according to available workforce and equipment productive time and capabilities [START_REF] Kingsman | Modeling Input -Output Workload Control for Dynamic Capacity Planning in Production Planning Systems[END_REF].

The recipe-to-machine qualification configuration has a direct impact on the capacity utilization of the toolset. If the recipe of a product is not qualified on a machine, it is not possible to allocate quantities of the product to that machine. Therefore, a poor qualification configuration causes loss of capacity. The effect is intensified for a bottleneck toolset. Qualification constraints are like machine eligibility restrictions (also called machine dedications). However, as the machines can be qualified or disqualified, they can be considered as re-configurable. Due to product mix, dynamic fab environment and toolset limited capacity, this may lead to backlog or unsatisfied demand. Therefore, an adequate recipe-to-machine qualification configuration is necessary for the smooth running of the fab [START_REF] Johnzén | Importance of Qualification Management for Wafer Fabs[END_REF].

Generally speaking, capacity planning (or capacity allocation) is seen as critical in semiconductor manufacturing and is still investigated in the literature (see for instance [START_REF] Kim | Heuristics for Capacity Planning Problems with Congestion[END_REF], [START_REF] Chen | Coordinated Capacity Planning in Two-Stage Thin-Film-Transistor Liquid-Crystal-Display (Tft-Lcd) Production Networks[END_REF] and [START_REF] Zhang | Capacity Planning with Reconfigurable Kits in Semiconductor Test Manufacturing[END_REF]). Equipment qualification is considered in capacity planning in semiconductor manufacturing. For instance, in [START_REF] Rodriguez-Verjan | Optimized Allocation of Defect Inspection Capacity with a Dynamic Sampling Strategy[END_REF], the authors calculate the required inspection capacity in a dynamic sampling strategy. However, as the inspection machines are only qualified for some inspection process operations (inspection recipes), the machine qualification is considered in the sampling strategy to optimize the capacity allocation.

Flexibility in Semiconductor Manufacturing

Today's economy is marked by globalization and as a result increased competition. The technological breakthrough has shortened the product life cycle requiring the companies very short manufacturing response time. Therefore, flexible production systems are required to quickly respond to market fluctuation, increase the production agility and answer the diverse market demand. Since 1980s flexibility in manufacturing decision making has become more into focus. Dedicated machinery and non integrated general purpose machine tools are two traditional forms of manufacturing systems [START_REF] Gerwin | Do's and Don'ts of Computerized Manufacturing[END_REF]. The first system allows mass production of very limited products (or mostly a single product) with almost no flexibility. The second system permits very small batch production of a variety of products. The flexibility of this type of production system is very high but it is usually not suitable for large production volumes. One other disadvantage of such a system is the high production cost per unit and required highly qualified operators.

Nowadays, most production systems are computer-driven, making them flexible and more capable of producing large volumes of multiple products at a low unit cost. Flexible Manufacturing Systems (FMS), Computer Aided Design (CAD) or Computer Aided Manufacturing (CAM) systems are supporting the development of this third type of manufacturing systems [START_REF] Gupta | Flexibility of Manufacturing Systems Concepts and Measurements[END_REF].

According to [START_REF] Buzacott | Flexibility and Productivity in Manufacturing Systems[END_REF], flexibility is defined as the ability of a manufacturing system to cope with changing circumstances. It is one of the main objectives and critical measures of any manufacturing system [START_REF] Gupta | Flexibility of Manufacturing Systems Concepts and Measurements[END_REF].

Manufacturing flexibility (MF) can be categorized from different points of view, such as the time horizon, software or hardware flexibility, etc. Flexibility is very important in semiconductor manufacturing due to high investment, labor, facilities and equipment costs.

The investigations conducted in this part of the thesis, is an effort, in continuation of [START_REF] Johnzén | Modeling and Optimizing Flexible Capacity Allocation in Semiconductor Manufacturing[END_REF], to quantitatively measure manufacturing flexibility in a workcenter (toolset) in semiconductor industry. Several types of manufacturing flexibilities are defined in [START_REF] Browne | Types of Flexibilities and Classification of Flexible Manufacturing Systems[END_REF]: Machine Flexibility, Process Flexibility, Product Flexibility, Routing Flexibility, Volume Flexibility, Expansion Flexibility, Process Sequence Flexibility

Literature Review

and Production Flexibility.

The discussed flexibility measures in [START_REF] Johnzén | Modeling and Optimizing Flexible Capacity Allocation in Semiconductor Manufacturing[END_REF] do not fit exactly one single category. The ultimate objective of each flexibility measure is defined in its place.

A comprehensive literature synthesis on flexibility is available in [START_REF] Gupta | Flexibility of Manufacturing Systems Concepts and Measurements[END_REF]. Flexible manufacturing system features have also been taken into account while production planning and scheduling [START_REF] Ghosh | Routing Flexibility and Production Scheduling in a Flexible Manufacturing System[END_REF].

In this thesis, flexibility is to gain very much with very little (or even no) effort. The criterion that we consider, is the levelness of the workload in a toolset. By reaching full flexibility, we mean that the toolset is well-balanced, i.e. all machines are equally loaded, according to their capacity of course.

Literature Review

In the following chapters, flexible qualification management is considered from different aspects. We try to gather the related literature in this section.

Flexibility

For many years, manufacturing flexibility has drawn interest from researchers and practitioners as a key factor of competitivity in dynamic and uncertain environments. Operations flexibility is defined in [START_REF] Ruiz-Torres | Makespan and Workstation Utilization Minimization in a Flowshop with Operations Flexibility[END_REF] as the assignment of production tasks to workcenters, assuming that the number of tasks are larger than the number of workcenters and that the workcenters are able to perform all tasks. Benefits of the defined operations flexibility for a flowshop environment with the objective of minimizing the completion time of all jobs and also maximizing workcenter utilization is studied further. The solution approach is based on a two-phase heuristic: Job to workcenter allocation and job sequencing.

Early studies consider manufacturing flexibility between plants and products, with various capacity limitations and demands [START_REF] Boyer | Manufacturing Flexibility at the Plant Level[END_REF]. However, in this study, we focus on manufacturing flexibility and capacity utilization at workcenter level, i.e. to which extent the flexibility of recipe-to-machine assignments affect toolset capacity utilization.

Chapter 3. Qualification Management in Semiconductor Manufacturing

Qualification Management

Recipe-to-machine qualification management has come into attention in recent years due to its importance in the semiconductor industry. Here, we discuss the main studies on the subject. Recipe-to-machine qualification configuration is studied as a configuration problem for a parallel multi-purpose machines workshop in [3]. The study takes two features into account: Demand uncertainty and qualification cost. To hedge against demand uncertainty, the recipe-to-machine qualification configuration must be robust while at the same time, the qualification cost must be minimized. A bi-objective optimization model is proposed which maximizes the robustness level with a certain fixed qualification cost, i.e. a fixed number of new qualifications. Then the model is solved with a branch-and-bound method.

Finding the qualification configuration at minimum cost in order to balance the workload on the toolset while meeting demand, termed as load-balanced production plan is presented in [START_REF] Aubry | Minimizing Setup Costs for Parallel Multi-Purpose Machines Under Load-Balancing Constraint[END_REF]. They present a mixed integer linear program and they show that the problem is NP-hard in the strong sense. Modeled as a transportation problem, lower and upper bounds on the setup costs for uniform parallel machines with identical setup times are estimated. The recipe-to-machine qualification configuration of a parallel multi-purpose machine is studied in [START_REF] Aubry | Maximizing the Configuration Robustness for Parallel Multi-Purpose Machines Under Setup Cost Constraints[END_REF]. Models and resolution approaches are proposed to find the best compromise between the number of qualifications (cost) and workload balance robustness. It is argued that as the demand variability is high in forecasts, the deviation from the forecast plan (robustness) must be maximized while maintaining a load-balanced production plan without exceeding a given total setup. New indicators, called Flexibility Measures are proposed in [START_REF] Johnzén | Flexibility Measures for Qualification Management in Wafer Fabs[END_REF] and [START_REF] Johnzén | Modeling and Optimizing Flexible Capacity Allocation in Semiconductor Manufacturing[END_REF] to estimate the flexibility of recipe-to-machine qualification configuration of the whole toolset depending upon two different objectives. Several extensions of the measures and optimization approaches for multiple qualifications are also proposed and validated in [START_REF] Johnzén | Modeling and Optimizing Flexible Capacity Allocation in Semiconductor Manufacturing[END_REF]. We continue these studies. Therefore, the measures are recalled in Section 4.2.1 in detail.

An optimization model with binary variables is proposed for a photolithography toolset in [START_REF] Ignizio | Cycle Time Reduction via Machine-to-Operation Qualification[END_REF] with the objective of balancing the workload on all the machines. To do this, for each new qualification, the qualification binary variables are set to 1. In

Literature Review

the model, it is possible to define a minimum and/or a maximum number of allowed qualifications.

Discrete-event simulation has been used in [START_REF] Kabak | Impact of Recipe Restrictions on Photolithography Toolsets in an ASIC Fabrication Environment[END_REF] to investigate the impact of recipe-to-machine qualification configuration and production start volume on the workload of each machine in a toolset. Different simulations are performed with different production start volumes and recipe-to-machine qualification configurations. Using the results of these simulations, the workload associated with recipes with only one qualification is compared to the workload of recipes qualified on several machine. For instance, they show that the overall workload is higher when each recipe has only one qualified machine. It is also stated that the immature products are more likely to have only one or few machines qualified compared to high-volume mature products.

Batching

Batching is an important industrial constraint which has significantly be studied in the scientific literature, especially in the context of semiconductor manufacturing. In Chapter 5, in order to evaluate the recipe-to-machine qualification configuration, we solve a problem which is close to a scheduling problem on nonidentical unrelated parallel batch machines where splitting recipes is allowed. However, our objectives are quite different from the ones considered in the literature. Some relevant papers on batch machine scheduling are cited below.

In the semiconductor industry context, [START_REF] Gupta | Optimization of Due-Date Objectives in Scheduling Semiconductor Batch Manufacturing[END_REF] propose an algorithm to minimize three due-date objectives (average tardiness, maximum tardiness, and number of tardy jobs in batch processes) simultaneously in scheduling a single batch processing machine. [START_REF] Damodaran | Minimizing Makespan on a Batch-Processing Machine with Non-Identical Job Sizes Using Genetic Algorithms[END_REF] propose an efficient genetic algorithm to minimize the makespan for a single batch-processing machine with nonidentical job sizes. [START_REF] Yugma | A Batching and Scheduling Algorithm for the Diffusion Area in Semiconductor Manufacturing[END_REF] solve the flexible job-shop scheduling problem on nonindentical batching machines with a metaheuristic based on a disjunctive graph representation and for different objectives functions. [START_REF] Dauzère-Pérès | Scheduling Jobs on a Single Batch Processing Machine with Incompatible Job Families and Weighted Number of Tardy Jobs Objective[END_REF] study the scheduling problem of minimizing the weighted and unweighted number of tardy jobs on a single batch processing machine with incompatible job families. The scheduling problem on unrelated parallel batch ma-

Chapter 3. Qualification Management in Semiconductor Manufacturing

chines with capacity restrictions is treated in [START_REF] Li | Scheduling Unrelated Parallel Batch Processing Machines with Non-Identical Job Sizes[END_REF]. Several heuristics for minimizing the makespan are proposed. [START_REF] Bilyk | Scheduling Jobs with Ready Times and Precedence Constraints on Parallel Batch Machines Using Metaheuristics[END_REF] propose and compare metaheuristics to solve the scheduling problem of jobs with ready times on identical parallel machines and when the total weighted tardiness is minimized. Again in the semiconductor industry, [START_REF] Jia | A Meta-Heuristic To Minimize Makespan for Parallel Batch Machines with Arbitrary Job Sizes[END_REF] consider the scheduling problem of a set of n jobs with different job sizes on a set of m identical and parallel batch machines with the objective of minimizing the makespan. A metaheuristic in two phases is proposed: In the first phase, the jobs are grouped into batches and, in the second phase, the batches are scheduled on the machines.

Chapter 5 deals with the impact of batch size constraint on qualification management. In all cited qualification management studies, batch sizes are ignored.

Variability

Stochastic modeling has been widely used to measure the production variability of production lines [START_REF] He | Production Variability of Production Lines[END_REF]. Many studies have been done on the measurement of the variability of the production flows in a fluid modeling network [START_REF] Ciprut | On the Variance of the Production Output of Transfer Lines[END_REF]. In our study, we only consider one workcenter in one period and evaluate the variability using an optimization model.

The benefits of process flexibility in capacity utilization and sales increase in supply chains is extensively studied in [START_REF] Jordan | Principles on the Benefits of Manufacturing Process Flexibility[END_REF]. [START_REF] Graves | Process Flexibility in Supply Chains[END_REF] define a flexibility measure for supply chain systems. At plant level, manufacturing flexibility between plants and products, with various capacity limitations and demands is studied in [START_REF] Boyer | Manufacturing Flexibility at the Plant Level[END_REF]. At workcenter level, [START_REF] Johnzén | Flexibility Measures for Qualification Management in Wafer Fabs[END_REF] propose flexibility measures to evaluate the manufacturing flexibility of a given workcenter configuration according to the production volume or production time. The industrial implementation and consideration of special cases for new and alternate recipes and their impact on toolset capacity is further studied in [START_REF] Rowshannahad | Qualification Management and its Impact on Capacity Optimization[END_REF].

The relationship between manufacturing flexibility and production variability is studied in [START_REF] Muriel | Impact of Partial Manufacturing Flexibility on Production Variability[END_REF]. The study is conducted on a multi-plant multi-product maketo-order manufacturing supply chain. Based on an optimization-based simulation model, it is shown that, by increasing the manufacturing flexibility, the production

Conclusions

variability is reduced. Production throughput variability is calculated for a single workstation with deterministic process times and random downtimes in [START_REF] Kim | Estimating the Distribution and Variance of Time to Produce a Fixed Lot Size Given Deterministic Processing Times and Random Downtimes[END_REF]. Probability density function and variance of time to produce are developed for a fixed lot size. Finally, it is shown how the proposed probability density function can be used in discrete event simulation to generate a cycle time distribution of a lot size of one.

In Chapter 6, we study the impact of qualification management on production variability.

Conclusions

In this chapter, we defined the recipe-to-machine qualification as a characteristic of semiconductor manufacturing. Qualifications provide the possibility of workload allocation of a recipe to a machine. An appropriate recipe-to-machine qualification configuration allows an efficient capacity utilization of the toolset. Flexibility is seen as an important manufacturing element in semiconductor industry. With this background, we attack the core of the subject. In the remainder of this part of the thesis, we discuss flexible qualification management and capacity planning in semiconductor industry from different angles. In the next chapter, we will see how limited machine capacity affects qualification management.

Introduction and Motivation

A machine is not always productive. The Total Time of a machine is usually categorized under these superstates [START_REF]SEMI-E10-0304E. Specification for Definition and Measurement of Equipment Reliability, Availability, and Maintainability (RAM)[END_REF] (Figure 4.1): Non-Scheduled Time, Unscheduled Downtime, Scheduled Downtime, Engineering Time, Standby Time and Productive Time. The only state which is really used for production is the Productive Time. Each machine, depending upon its condition, usage and importance in the production system, has a target productive time. This productive time is referred hereafter to as the maximum capacity (or shortly, the capacity) of the machine. The capacity of machines in the same toolset may be different, and is dynamic over time due to for instance maintenance plans. At each time interval and planning phase, the available capacity of each machine must be considered. Ignoring this important factor while planning may lead to infeasible or inefficient plans. In this chapter, we take into consideration the capacity of each machine in a toolset for qualification management (QM). We extend the WIP and Time Flexibility measures introduced in [START_REF] Johnzén | Flexibility Measures for Qualification Management in Wafer Fabs[END_REF] and, after analyzing these extended measures, we show that additional measures are required. According to place is the "Productive Time". This status is a percentage of the total time which we refer to as the maximum available time or the maximum capacity. Maximum

Capacitated Flexibility Measures

capacity may also be limited when the machine is shared with another production line.

Capacitated Flexibility Measures

Flexibility Measures based on two different criteria (recipe-to-machine configuration robustness and toolset workload balance) are defined in [START_REF] Johnzén | Flexibility Measures for Qualification Management in Wafer Fabs[END_REF] for QM. TheToolset Flexibility Measure evaluates the robustness of a recipe-to-machine qualification configuration. Taking into account capacity in this flexibility measure is not critical and will not be discussed in this thesis. The two other flexibility measures are WIP (Work-In-Process) Flexibility and Time Flexibility. They aim at balancing the workload on the machines in a toolset. The WIP Flexibility Measure evaluates the recipe-to-machine qualification configuration with regard to the workload balance in terms of production volumes (or WIP). The Time Flexibility Measure evaluates the qualification configuration with regard to the workload balance in terms of production times. Flexibility Measures vary between 0 and 1. Higher flexibility values indicate a more effective qualification configuration. In order to evaluate the impact of each new qualification, the flexibility value of the current qualification configuration is calculated and stored. Then each qualifiable recipe-to-machine couple is virtually qualified, and the resulting qualification configuration is recalculated and stored. By subtracting the flexibility values for each new configuration from the current flexibility value, the flexibility gain of each new qualification is computed.

In Section 4.2.1, we recall the WIP and Time Flexibility Measures proposed Chapter 4. Flexible Qualification Management under Capacity Constraint in [START_REF] Johnzén | Flexibility Measures for Qualification Management in Wafer Fabs[END_REF]. These measures assume that all machines have (unlimited) equal capacity. Hereafter, we refer to these Flexibility Measures as Uncapacitated Flexibility Measures. By modifying these measures, we define in Section 4.2.2 new flexibility measures which consider the capacity of each machine. These new flexibility measures are referred to as Capacitated Flexibility Measures. While taking into account capacity constraints, we show that complementary measures, called Capacity Deviation Ratio, are required to appropriately evaluate the qualification configuration of a toolset. These measures are introduced in Section 4.2.3. In Section 4.4, we discuss how Capacitated Flexibility and Capacity Deviation Ratio measures may be used to interpret the workload balancing diagram used for capacity planning.

Below, the parameters and variables used throughout the chapter are defined.

Parameters

R

Total number of recipes to be processed, M Total number of machines in the toolset, W IP r Total production volume of recipe r, T P r,m

Throughput rate of recipe r on machine m (number of wafers per hour), Capa m Capacity of each machine m (in hours),

Q r,m     
1 if recipe r is qualified on machine m, 0 if recipe r is not qualified on machine m. γ Workload balancing exponent (γ ≥ 1).

Variables W IP r,m

The production volume of recipe r assigned to machine m, C r,m

The production time of recipe r on machine m, W IP m

The total production volume assigned to machine m

(W IP m = R r=1 W IP r,m ), C m
The total production time assigned to machine m

(C m = R r=1 W IPr,m
T Pr,m ).

Uncapacitated Flexibility Measures

Two flexibility measures are defined in [START_REF] Johnzén | Flexibility Measures for Qualification Management in Wafer Fabs[END_REF] to evaluate how well the workload can be balanced on a toolset with its qualification configuration. These measures do not take into account the capacities of machines in a toolset. Following, the Uncapacitated WIP and Time Flexibility Measures are recalled and briefly discussed.

Uncapacitated WIP Flexibility Measure

By balancing the workload in terms of WIP in the toolset, the Uncapacitated WIP FM F W IP U ncapa evaluates the recipe-to-machine qualification configuration of a toolset [START_REF] Johnzén | Flexibility Measures for Qualification Management in Wafer Fabs[END_REF]:

F W IP U ncapa =   M m=1 W IP m M   γ M m=1 (W IP m ) γ M ∈ (0, 1] . (4.1)
For any γ, F W IP U ncapa attains its maximum value, i.e. 1, when the numerator and denominator are equal. The term ( M m=1 W IP m ) in the numerator represents the overall workload on all machines, and is equal to the total production volume of all recipes ( R r=1 W IP r ). This is because all of the production volume of each recipe must be produced. Hence, the numerator in (4.1) represents an equal distribution of the total production volume over all machines ( R r=1 W IP r /m). This is the case of a perfectly-balanced toolset. The denominator represents the same value from a machine perspective. It corresponds to an equal distribution of the total workload over all machines. If the numerator and denominator are equal, it means that the average production volume distribution and the average workload distribution are the same. Therefore, by maximizing F W IP U ncapa under a given recipe-to-machine qualification configuration, the optimal workload balance of the toolset is obtained.

Chapter 4. Flexible Qualification Management under Capacity Constraint

Uncapacitated Time Flexibility Measure

The Uncapacitated Time FM F T ime U ncapa evaluates the recipe-to-machine qualification configuration with respect to the workload balance on the toolset in terms of production times. The definition from [START_REF] Johnzén | Flexibility Measures for Qualification Management in Wafer Fabs[END_REF] is as follows:

F T ime U ncapa = Ideal Ratio U ncapa M m=1 R r=1 W IP r,m T P r,m γ ∈ (0, 1] . (4.2)
As with F W IP U ncapa , F T ime U ncapa varies between 0 and 1. The numerator called IdealRatio U ncapa is the minimum value of the total production times M m=1 ( R r=1 W IPr,m T Pr,m ) γ when all of the qualifiable recipe-to-machine couples are "virtually" qualified.

Contrary to F W IP

U ncapa , γ not only decreases the flexibility value but also affects the workload balancing in F T ime U ncapa . A reduced example based on industrial data is used throughout the article to demonstrate (a) the behavior of Capacitated and Uncapacitated FMs, namely capacitated versus uncapacitated workload balancing, (b) the impact of γ while workload balancing using Time FMs and (c) the impact of a qualification on the toolset workload balance. The dataset is composed of 8 recipes and a toolset of 5 machines. Figure 4.3a shows the optimal workload balance diagram for the current toolset qualification configuration using F T ime U ncapa , and with γ = 2. After calculating the flexibility gains for all qualifiable recipe-to-machine couples, the best qualification gain is evaluated at 15.4%. This brings F T ime U ncapa to 97.7%. The optimal workload balance for the toolset qualification configuration after performing the best qualification is shown in Figure 4.3b. Now, the total workload is better distributed and more evenly balanced in the toolset. Note that, as the capacity of all machines are implicitly considered to be equal and unlimited, the total process time of all machines is centered around the average (about 100 hours). Let us evaluate again the Uncapacitated Time FM using a larger γ (γ = 10). Figure 4.4 shows the optimal workload balance on the toolset before and after the best qualification. It is worth to note that the flexibility value for the current qualification configuration has decreased from 82.3% (for γ = 2) to 11.3% (for γ = 10). The most important change is in the optimal workload balance. with lower throughput (M 1 and M 2). Therefore, increasing γ leads to an increase of the total process time and at the same time to a decrease of the maximum process time. These remarks call for a careful choice of γ via observation of the toolset workload in the fab and the priority of managers.

Capacitated Flexibility Measures

In this section, by modifying the Uncapacitated FMs, we introduce new FMs which take into account the limited and unequal capacity of each machine of a toolset. As Uncapacitated FMs, Capacitated FMs vary between 0% and 100%.

Capacitated WIP Flexibility Measure

The Capacitated WIP FM may be used for toolsets with homogeneous machines or machines with similar process times for all recipes. The capacity of each machine is restricted. The capacity restriction may be due to several reasons. For example, the Preventive Maintenance (PM) counter may be limited to a certain level and it is not desired to trigger the PM before the next planning period. In this case, the maximum capacity of the machine(s) in question must be limited to the desired level. In the same category, we can mention the case of machines using consumable materials such as Chemical-Mechanical Polishing (CMP) machines. The pads used for polishing are worn out over time. These pads must be changed after polishing a given number of wafers. In order not to interrupt the production in a planning period, the maximum capacity of the machines may be restricted to a fixed number of wafers. F W IP Capa evaluates the flexibility of a recipe-to-machine qualification configuration of a toolset from the standpoint of WIP workload balance under capacity constraint.

F W IP Capa =   M m=1 (W IP m /Capa m ) M   γ M m=1 (W IP m /Capa m ) γ M ∈ (0, 1] (4.3)

Capacitated Flexibility Measures

Since W IP m = R r=1 W IP r,m , (4.3) can be reformulated as in (4.4):

F W IP Capa = M •   M m=1 R r=1 (W IP r,m /Capa m ) M   γ M m=1 R r=1 W IP r,m /Capa m γ ∈ (0, 1] . (4.4) When γ ≥ 1, F W IP Capa is a convex function.
For any value of γ = 1, F W IP Capa attains its maximum value of 1 when the numerator and the denominator are equal. This means that the workload of each machine against its maximum capacity is equal. As with F W IP U ncapa , γ has no impact on the toolset workload balancing. The increase of γ just decreases the value of F W IP U ncapa . γ is an interesting parameter to adjust the WIP FMs (both Uncapacitated and Capacitated) according to the real workload distribution on the shop floor. With a low γ, for instance 1, the WIP FM may exceed 90% even with a poor workload balance. This high flexibility value may be misleading. Therefore γ must be adjusted using historical data and careful observation of the actual workload balance on the shop floor. In the numerical experiments presented in this paper, γ is set to 6, which according to our experience, leads to relevant WIP Flexibility values.

Capacitated Time Flexibility Measure

Many toolsets are composed of heterogeneous machines with different throughputs for each recipe. Therefore, it does not make sense to evaluate the qualification configuration of such toolsets based on their workload balance using WIP FMs. Therefore, production times must be considered. Below, we introduce the Capacitated Time FM (F T ime Capa ):

F T ime Capa = Ideal Ratio Capa M m=1 R r=1 W IP r,m T P r,m • Capa m γ ∈ (0, 1] . (4.5)

Chapter 4. Flexible Qualification Management under Capacity Constraint

The constant Ideal Ratio Capa is the minimum value of relative production times when all of the qualifiable recipe-to-machine couples in the toolset are qualified. This is calculated by "virtually" setting all Q r,m to 1:

Ideal Ratio Capa = min M m=1 R r=1 W IP r,m T P r,m • Capa m γ with Q r,m = 1 ∀r, m. (4.6) F T ime
Capa is a convex function when γ ≥ 1. Ideal Ratio Capa being a constant, by maximizing F T ime Capa , we minimize the denominator. The denominator represents the sum of relative production times which can at best be equal to Ideal Ratio Capa when the best qualification configuration is obtained.

The way capacity is modeled in (4.5) is straightforward. The term (T P r,m • Capa m ) states that the machine throughput is relevant to its capacity. For instance, a machine with two times more capacity is the same as a machine with two times larger throughputs.

Let us reconsider the same example under capacity constraint. Figure 4.5 shows the capacitated workload balance diagram when γ = 2 for the current qualification configuration. The line depicts the maximum capacity of each machine. In this example, the Capacitated FM value is lower than the Uncapacitated FM value: F T ime U ncapa = 82.3% and F T ime Capa = 61.1% for the current configuration, and F T ime U ncapa = 95.5% and F T ime Capa = 97.7% for the toolset configuration after the best qualification. With capacity constraint, the best qualification proposition may be different than in the uncapacitated case. This is the case in our example: "Recipe 8" on "Machine 5" for capacitated QM instead of "Recipe 4" on "Machine 4" for uncapacitated QM. The impact of γ on capacitated QM using Time Flexibility is illustrated in Figures 4.5 and 4.6. By increasing γ, the total process time is increased due to a workload shift from machines with high throughput rates to machines with lower throughput rates (as in F T ime U ncapa ). However in the capacitated case, the increase of γ decreases the difference between the process time on each machine and the capacity of the machine; meaning that the capacity is better respected. By increasing the Load Balancing Exponent (γ), the value of the total production time (in F T ime U ncapa ) and the total relative production time (in F T ime Capa ) increase exponentially, which leads to a better leveling between each individual production time (in F T ime U ncapa ) or each it is not only interesting but also decisive to measure these objectives in QM under capacity constraint.

Capacity Deviation Measurement

Let us define new indicators called Capacity Deviation Ratios, which are required to complement the Capacitated FMs. When balancing workload under capacity constraint, some machines may be over-or underloaded. Overloaded machines slow down the production flow and increase the variability while underloaded machines lead to loss of capacity and productivity. These elements must be carefully considered in QM.

WIP Capacity Deviation Ratio (DR W IP

Capa )

As the name suggests, the WIP Capacity Deviation Ratio measures the workload deviation of each machine from its maximum capacity in terms of production volume. It computes, at the same time, machine under-and over-utilization.

Consider the Capacitated WIP FM defined in (4.3). When the ( W IPm

Capam ) ratio is the same for all machines in the toolset, F W IP Capa reaches its maximum value, regardless of the possible capacity over-or under-utilization of each machine. This shows that only considering F W IP Capa may be misleading for QM under capacity constraint. Below, we define the WIP Capacity Deviation Ratio DR W IP Capa which evaluates the absolute mean deviation from the maximum capacity of each machine. Note that DR W IP Capa ∈ R + and that lower DR W IP Capa values indicate a better machine capacity utilization.

DR W IP Capa = M m=1 |W IP m -Capa m | M m=1 Capa m (4.7)

Time Capacity Deviation Ratio (DR T ime Capa )

As for F W IP Capa , F T ime Capa may reach 100% although the toolset is more (or less) loaded than its maximum capacity. Hence, F T ime Capa must be combined with the Time Capacity

Capacitated Flexibility Measures

Deviation Ratio DR T ime Capa .

DR T ime Capa = M m=1 R r=1 W IP r,m T P r,m -Capa m M m=1 Capa m (4.8)
This case is illustrated in Figure 4.7 where we have artificially doubled the production volume of all recipes in our example. While the flexibility values remain the same (F T ime Capa = 0.9% for the current toolset qualification configuration and F T ime Capa = 73.4% for the toolset qualification configuration after the best qualification), the deviation ratio has shot up (DR T ime Capa = 15.7% versus DR T ime Capa = 107% for the current configuration and DR T ime Capa = 3.1% versus DR T ime Capa = 98.8% for the configuration after the best qualification). The best case is with 100% flexibility and 0% deviation ratio. This means that the workload is perfectly balanced while using the overall capacity of the toolset without over-or under-utilizing any machine. Using 27 industrial instances, the impact of γ on the average toolset capacity Because F T ime Capa depends on γ, it may not be a good measure to evaluate the impact of γ. However, DR T ime Capa is independent of γ. Figure 4.8 depicts how increasing γ leads to increasing the toolset capacity utilization by shifting the workload from high throughput machines to machines with lower throughput rates. This is visualized with a decrease of DR T Ratio Indicators are essential in QM. While evaluating the qualification flexibility gain under capacity constraints, the selected qualification should increase the most the desired FM and at the same time reduce as much as possible the corresponding capacity deviation ratio indicator. This requires a bi-objective optimization of these measures which goes beyond the scope of this study.

Weighted Capacitated Flexibility Measures

Capacitated flexibility measures are proposed because machines do not have the same capacity. In the capacitated flexibility measures by increasing the workload balancing exponent (γ), the machine capacity utilization is increased, i.e. machine overload and underload are penalized at the same level although their impact is different. There is a loss of money related to non fully utilized machines (labor cost, overhead cost, ...) while, if a machine is overloaded, production is delayed and cycle times are consequently increased. Moreover, an overloaded machine tends to increase the production variability. Hence, it is relevant to weigh differently overloaded and underloaded machines.

The flexibility measures allow to identify qualifications which tend to level the workload. It may happen that the (first) new qualification(s) level more the workload of the underloaded machine while removing the overload is more important. That is why we propose weighted capacitated flexibility measures in which over-or underloaded machines are penalized linearly (using weights o m and u m ) and nonlinearly using different workload balancing exponents (γ o and γ u ).

The additional notations used in this section are listed below.

Parameters γ o

Overload Balancing Exponent,

γ u Underload Balancing Exponent, o m Overload weight of machine m, u m
Underload weight of machine m.

The weighted capacitated Time Flexibility measure is defined:

F T ime Capa = = IdealRatio Capa = M m=1;Cm>Capam o m ( Cm Capam ) γo + M m=1;Cm≤Capam u m ( Cm Capam ) γu . ( 4.9)
Since C m = R r=1 (W IP r,m /T P r,m ), F T ime Capa = can be written:

F T ime Capa = = IdealRatio Capa = M m=1|Cm>Capam o m ( R r=1 W IPr,m T Pr,m Capam ) γo + M m=1|Cm≤Capam u m ( R r=1 W IPr,m T Pr,m

Capam

) γu (4.10) where

IdealRatio Capa = = min M m=1|Cm>Capam o m ( Cm Capam ) γo + M m=1|Cm≤Capam u m ( Cm Capam ) γu with Q r,m = 1 ∀r, m.
(4.11) Overload and underload weights (o m and u m ) can be chosen empirically for example such that o m + u m = 1. However, they can also be determined based on the cost or profit of being over-or underloaded on each machine m, e.g. by considering the cycle time decrease or increase; or the opportunity cost for not producing a product (loss of revenue).

In the same way, we define also the Weighted Deviation Measures:

DR T ime Capa = = IdealDeviationCapa = M m=1|Cm>Capam o m |C m -Capa m | γo + M m=1|Cm≤Capam u m |C m -Capa m | γu (4.12) IdealDeviationCapa = = min M m=1|Cm>Capam o m |C m -Capa m | γo + M m=1|Cm≤Capam u m |C m -Capa m | γu with Q r,m = 1 ∀r, m. (4.13) C m > Capa m ⇒ o m = 1, u m = 0 (4.14a) C m ≤ Capa m ⇒ o m = 0, u m = 1 (4.14b) DR T ime Capa = = IdealDeviationCapa = M m=1 (o m + u m )|C m -Capa m | γo+γu (4.15)

Solution Approaches: Capacitated Workload Balancing

IdealDeviationCapa = = min M m=1 (o m + u m )|C m -Capa m | γo+γu with Q r,m = 1 ∀r, m.
(4.16)

Solution Approaches: Capacitated Workload Balancing

The introduced Capacitated Flexibility Measures are based on the optimal workload balance of the toolset. However, there is a noticeable difference between workload balancing in terms of production volumes (or WIP) and that of production times. Concerning the workload balance in terms of WIP, the sum of production volume ( M m=1 W IP m ) is always the same regardless of the way the load is distributed on the toolset. Whereas, when balancing the workload in terms of production times, the total production time ( M m=1 W IPr,m T Pr,m ) depends on how the production volumes are distributed on the toolset. This makes the problem resolution considerably more difficult.

Capacitated Workload Balancing in terms of Production Volumes (F W IP

Capa )

The production volume workload balancing with capacity constraints is done in two phases. In the first phase, an initial (and most often non-optimal) solution is constructed. The second phase consists of improving the initial solution obtained in the first phase until finding an optimal solution. Let us first present the algorithm used for the construction of the initial workload balance and then, the algorithm used for improving this initial workload balance.

Constructing Initial WIP Workload Balance

In order to construct a realvalued initial solution, the production volume of each recipe is divided among all of the qualified machines for that recipe. Following is a formalization of the algorithm:

Step 0. Set r = 1.

Chapter 4. Flexible Qualification Management under Capacity Constraint

Step 1. For each machine in the toolset, allocate the production volume of recipe r to its qualified machines by equally distributing the production volume of recipe r (W IP r ) among all of the qualified machines for r

(W IP r,m = W IP r / M m=1 Q r,m ).
Step 2. If r < R, set r = r + 1 and go to Step 1.

WIP Workload Balancing Algorithm with Capacity Constraints

The initial WIP workload balance (initial solution) must be improved in order to find the optimal (or precise-enough near-optimal) workload balance. The main idea of this algorithm has been presented in [START_REF] Johnzén | Flexibility Measures for Qualification Management in Wafer Fabs[END_REF] for uncapacitated production volume workload balancing. In this paper, we formalize and explain in detail the adapted algorithm for capacitated production volume workload balancing.

The algorithm begins with an initial solution x 0 , generated by the initial WIP distribution algorithm.

Before describing the algorithm, let us define the set of loading machines for recipe r (L M r ) to be the set of qualified machine(s) for recipe r, with the smallest W IP m /Capa m ratio among all of the qualified machines for r, i.e.:

m * ∈ L M r if W IP m * Capa m * = min ∀m|Qr,m=1
W IPm Capam (4.17)

Step 0. Start with initial solution x 0 . Set r = 1 and k = 1.

Step 1. For all of the qualified machines for recipe r, remove the WIP of recipe r initially allocated to machine(s) m:

If W IP r,m > 0 then W IP m = W IP m -W IP r,m ∀m (4.18)
Sort the machines according to their W IP m /Capa m ratio in an ascending order.

Step 2. Allocate (equally) WIP quantities to loading machine(s) until the W IP m * /Capa m * (or R r=1 W IP r,m * /Capa m * ) ratio on one (or more) loading machine(s)

Solution Approaches: Capacitated Workload Balancing

m * in L M r is equal to the W IP m /Capa m ratio of a machine m qualified for recipe r but not in L M r :

W IP m * Capa m * = W IP m Capa m (4.19)
Then, m is added to the set of loading machines, i.e. L M r ≡ L M r ∪ {m }.

If r < R, set r = r + 1 and go to Step 1.

Step 3.

If F W IP Capa k -F W IP Capa k-1 > 0, then set k = k + 1, r = 1

and go to

Step 1.

Capacitated Workload Balancing in terms of Production Times (F T ime

Capa )

The resolution method for the Time Flexibility Measure is more complex than the WIP Flexibility Measure. F T ime Capa can be solved using the same approach proposed in [START_REF] Johnzén | Modeling and Optimizing Flexible Capacity Allocation in Semiconductor Manufacturing[END_REF] for F T ime U ncapa . In order to find the optimal workload balance for the Capacitated Time Flexibility Measure (F T ime Capa ), the optimization problem (4.20) must be solved when F T ime Capa is maximized. The only constraint of the model ensures that the production volume of each recipe r is entirely allocated to qualified machines (Q r,m = 1). max

F T ime Capa = Ideal Ratio Capa M m=1 ( R r=1 W IP r,m T P r,m • Capa m ) γ subject to (4.20) M m=1|Qr,m=1 W IP r,m = W IP r ∀r W IP r,m ≥ 0 ∀r, m
Since Ideal Ratio capa is constant, by maximizing F T ime Capa , the sum of the relative Chapter 4. Flexible Qualification Management under Capacity Constraint production times (4.21) must be minimized.

min f = M m=1 ( R r=1 W IP r,m T P r,m • Capa m ) γ (4.21)
Since (4.21) is a continuous function, the problem is reduced to an unconstrained minimization problem. All algorithms that minimize unconstrained continuous functions start from an initial solution (x 0 ) and improve the solution at each iteration until either no more improvement is made or the solution is good enough [START_REF] Nocedal | Numerical Optimization[END_REF]. The algorithm described hereafter is a line search method called active-set method. In line search methods, at each iteration k (at solution (x k )), the algorithm finds a search direction called p k . Then the algorithm searches a new solution (x k+1 ) by moving along the direction (p k ) at a length (α k ) with a lower function value [START_REF] Nocedal | Numerical Optimization[END_REF]. In the active-set method, we assume that a feasible solution to the problem exists.

The idea behind the active-set method is to partition the set of constraints into active (binding) and inactive (non-binding) constraints, where the inactive constraints are essentially ignored. At each iteration (solution point k, x k ), the active-set method determines a set of constraints called the working set (W k ) to be active. The surface defined by the working set is called the working surface. At each iteration, by moving on the working surface, the solution is improved until the optimal solution is found. Moving on the working surface consists of finding a nonzero search direction (p k ) and a nonnegative step length (α k ) over which the objective value is decreased. There exist several methods and approaches for finding a search direction and a feasible step length. While moving on the working surface, the inactive constraints must not be violated to avoid getting out of the feasible region and the solution to become infeasible. While moving on the working surface and defining the step length, one or several constraints may limit the step length. This or these constraints are called blocking constraint(s). If one (or several) blocking constraint is encountered, it is added to the working set making a surface of lower dimension than before [START_REF] Luenberger | Linear and Nonlinear Programming[END_REF]. At each iteration, the Lagrangian multipliers (λ) of the working set are evaluated. If all are nonnegative, the optimal solution is obtained else we drop one or more of the constraints having a negative Lagrangian multiplier. At each iteration of the algorithm, either the solution x k or the working set W k is modified, where either x k = x k+1 or W k = W k+1 . Therefore for a non-degenerate case, the algorithm terminates after a finite number of iterations [START_REF] Wong | Active-Set Methods for Quadratic Programming[END_REF].

An investigation of the active-set method is out of the scope of this chapter, therefore we refer the interested reader to more specialized textbooks on the subject ( [START_REF] Nocedal | Numerical Optimization[END_REF], [START_REF] Gill | Practical Optimization[END_REF] and [START_REF] Griva | Linear and Nonlinear Optimization[END_REF]).

The workload balancing in terms of production times is done in two phases (like workload balancing in terms of production volume). In the first phase, an initial (and most often nonoptimal) solution is constructed. In the second phase, the initial solution is improved until an optimal (or precise-enough near-optimal) solution is reached.

Note that the structure of the algorithm constructing the initial solution is the same as the one used for WIP initial workload balancing with the difference that production times are considered.

Step 0. Set r = 1.

Step 1. For each machine in the toolset, allocate the production volume of recipe r to its qualified machines by equally distributing the production volume of recipe r (W IP r ) among all of the qualified machines for r

(W IP r,m = W IP r / M m=1 Q r,m
). Calculate and allocate the corresponding production time for each load allocation (W IP r,m /T P r,m ).

Step 2. If r < R, set r = r + 1 and go to Step 1.

The initial solution is then improved in an iterative procedure. In each iteration, only one recipe (r * ) is considered. Therefore (4.21) is decomposed into the relative production time of recipe r * on machine m ( W IP r * ,m T P r * ,m •Capam ) plus the relative production time of all other recipes on machine m ( fr * ,m ) (4.22) ( [START_REF] Johnzén | Modeling and Optimizing Flexible Capacity Allocation in Semiconductor Manufacturing[END_REF] and [START_REF] Griva | Linear and Nonlinear Optimization[END_REF]).

f = W IP r * ,m T P r * ,m • Capa m + fr * ,m γ (4.22)
Step 0. Set r * = 1 and k = 1.

Chapter 4. Flexible Qualification Management under Capacity Constraint

Step 1. Remove the initially allocated load W IP r * ,m of recipe r * from all machines m in the toolset (4.23).

If W IP r * ,m > 0 then W IP m = W IP m -W IP r * ,m ∀m (4.23)
Step 2. Decompose (4.21) for recipe r * to obtain (4.22). Re-distribute W IP r * using the active-set method. If r * < R, set r * = r * + 1 and go to Step 1.

Step 3. The active-set method is the core of the algorithm discussed above for the redistribution of recipe r * . Before starting the description of a simple active-set method, we introduce the notations and some necessary elements of the algorithm.

If f k -f k-1 > 0 (4.

Notations W k

Working set of active constraints at solution k, Ā Constraint matrix for active constraints, Ār Right inverse for Ā, Z Null-space matrix for Ā, ∇f Gradient vector of f , ∇ 2 f Diagonal elements of the Hessian matrix of f .

In order to calculate the Lagrangian multipliers and step length, we need the gradient vector (4.24) and diagonal elements of the Hessian matrix (4.25) of (4.22).

∇f = γ T P r * ,m • Capa m W IP r * ,m T P r * ,m • Capa m + fr * ,m γ-1 (4.24) ∇ 2 f = γ (γ -1) (T P r * ,m • Capa m ) 2 W IP r * ,m T P r * ,m • Capa m + fr * ,m γ-2 (4.25)
An algorithm of a simple active-set method is provided below ( [START_REF] Griva | Linear and Nonlinear Optimization[END_REF], [START_REF] Johnzén | Modeling and Optimizing Flexible Capacity Allocation in Semiconductor Manufacturing[END_REF], [START_REF] Nocedal | Numerical Optimization[END_REF], [START_REF] Fletcher | Practical Methods of Optimization[END_REF]).

Step 0. Start with an initial feasible solution x 0 and working set of active constraints at x 0 (W 0 ). Set k = 1.

Solution Approaches: Capacitated Workload Balancing

Step 1. The Optimality Test

If ZT ∇f (x k ) = 0
Step 1a. If there are no active constraints (W IP r * ,m = 0), STOP -Local Stationary Point.

Step 1b. Else, compute Lagrangian Multipliers:

λ = ĀT r ∇f (x k ) . (4.26)
Step 1c. If λ ≥ 0, STOP -Local Stationary Point. Else delete the constraint corresponding to the most negative Lagrangian multiplier from the working set W k . Update W k , Z, Ā and Ār .

Step 2. The Search Direction

Compute a descent feasible search direction p k with respect to the active constraints in W k . The Reduced Newton Search Direction (5.7) [START_REF] Hoyer | Variants of the Reduced Newton Method for Nonlinear Equality Constrained Optimization Problems[END_REF] is an efficient search direction.

p k = -Z ZT ∇ 2 f (x k ) Z -1 ZT ∇f (x k ) (4.27)
Step

The Step Length

Compute a step length α k such that f (x k + α k p k ) < f (x k ) subject to retaining feasibility with respect to all constraints. For that α k ≤ ᾱk , where ᾱ is the maximum feasible step length along p k :

α k = min i / ∈W k ,p k <0 W IP k -p k (4.28)
Step 3a. If α k < ᾱk , α k is an unconstrained step, i.e. no blocking constraint is encountered and

x k+1 = x k + α k p k remains feasible.
Step 3b. If α k ≥ ᾱk , then the step along p k is blocked by one (or more) (inactive) constraint(s) not in W k but which are active in x k+1 . In this case, W k is modified to include the blocking constraint to the

working set (W k+1 ← W k ∪ i).
Update Z, Ā and Ār . Set k = k + 1 and go to Step 1.

If more than one blocking constraint exists (i.e. more than one constraint boundary is reached), the solution is degenerate and only one of the blocking constraints is added to the working set.

Capacitated Workload Balancing in terms of Production Volumes (F W IP Capa ) -Alternative Approach

An alternative approach to the algorithm proposed in Section 4.3.1 is to substitute T P r,m • Capa m with the capacity of each machine (Capa m ) in the Capacitated Time Flexibility Measure (F T ime Capa ). Note that F T ime

• reduces to F W IP
• when the throughput of all recipes are equal on all machines.

Managerial Insights

The proposed methodology is being applied in industry. Based on the industrial experiments, some of the managerial insights are pointed out.

The Flexibility Measures described in this chapter, were implemented with the programming language VBA in Microsoft Excel ® . The application solves the optimization problem for the current qualification configuration and saves the current flexibility level according to the selected flexibility measure. Then it virtually qualifies each of the qualifiable recipes and saves it in a matrix. Finally the flexibility gain is calculated by subtracting each element of the matrix from the current flexibility level while highlighting the best qualification.

The application has been used for more than one year for the fab's main bottleneck toolset as a decision support system for qualification management and capacity planning.

The results show that, on average, by performing the best qualification with γ = 6, the Capacitated Time Flexibility Measure (F T ime Capa ) increased by 23.7%, the Time

Managerial Insights

Capacity Deviation Ratio (DR T ime Capa ) decreased by 9.6% and the toolset maximum workload decreased also by 4%. The indicators show the considerable improvements in Toolset Flexibility and capacity utilization with only one qualification.

The capacitated flexibility measures proposed in this study are based on toolset workload balance. Several outputs of the model can be used and interpreted by different users and from different aspects. Thanks to the flexibility gain table, the process department can perform a limited number of qualifications bringing more flexibility. At the same time, useless and costly qualifications with poor flexibility gains are avoided. Used as a decision support system, among several qualification alternatives, the best qualification can be chosen by making a trade-off between the flexibility gain and the required qualification effort. In a more long-term vision, necessary qualifications could be anticipated by taking into account future production volumes. This reduces the obligation of performing the so-called rush qualifications. By identifying in advance the set of new qualifications to be performed, the incurred downtime is minimized and the impact on the production line can be better managed.

Using the Capacitated Time Flexibility Measure, the optimal workload balance in terms of process time is calculated. It is possible to observe the workload balance improvement after performing a new qualification. Therefore, the workload balance diagram is used by capacity planners, which serves also as a guideline for capacity improvement actions.

Thanks to Capacitated Flexibility Measures, we are able to study the impact of preventive maintenance or machine breakdown on the toolset qualification configuration and workload balance. The proposed model can also be used in the case of production ramp-up and new machine acquisition. By creating several scenarios consisting of dummy machine(s) and running the application, the impact of each scenario is calculated on flexibility level and toolset workload. By plotting the graph of the flexibility gain and/or deviation ratio against the Return On Investment (ROI) of each scenario, the best solution can be chosen.

In the model, the volumes are allocated without considering scheduling constraints, in other words as if preemption and splitting of recipes on machines were allowed. Therefore the effect of scheduling has not been taken into account. How-Chapter 4. Flexible Qualification Management under Capacity Constraint ever the detailed WIP allocation on the machines can be used as a rough guideline for load allocation. It is also possible to use this output as the input of a scheduler and dispatcher. In addition, note that a new qualification with high flexibility gain helps to better balance the workload on the machines and gives more alternatives for processing a recipe. This gives more flexibility for dispatching lots on the machines and levels the workforce of the toolset.

The toolset qualification configuration changes over time. This is due to new qualifications, disqualifications, machine breakdowns, machine acquisitions) but also the change of the production volume and mix. The flexibility level of a recipe-tomachine qualification configuration, when monitored continuously, can serve as a warning. Preventive measures (such as (re-)qualification and at a strategic level, new machine acquisition) can be triggered when the Flexibility Measures falls below a certain level.

Conclusions and Perspectives

In this chapter, we investigated an approach for qualification management in semiconductor manufacturing. Indicators called Capacitated Flexibility Measures were proposed to evaluate the workload balance of a toolset under equipment capacity restriction for a given qualification configuration. It was shown that Flexibility Measures alone are not enough to select new qualifications. So, complementary indicators called Deviation Ratios were introduced. In the development of solution approaches, two main problems have been considered. In the first one, the toolset consists of homogeneous machines where all process times are equal. The second problem considers heterogeneous machines with different recipe-to-machine process times. For both optimization models (WIP and Time), dedicated solution approaches were presented. Using real fab data, we illustrated the relevance of dedicated Capacitated Flexibility Measures and Capacity Deviation Ratios for qualification management and capacity planning.

Due to lack of space, some other practical aspects of qualification management are not discussed in this chapter. As stated in the introduction, some recipes are nonqualifiable. This makes qualification management a bit more complicated. Besides,

Conclusions and Perspectives

due to new product launch or disqualifications, some recipes may need to be qualified first on the machines. A study of these recipe-types called not previously qualified recipes for Uncapacitated Flexibility Measures can be found in [START_REF] Rowshannahad | Qualification Management and its Impact on Capacity Optimization[END_REF]. Another aspect of qualification management is recipe priority. Some recipes may be more important than others because of deadlines, customer needs, etc.

As a perspective for future investigations concerning this chapter, it is interesting to study bi-objective optimization of the both flexibility and deviation ratio measures.

In order to reduce setup and production costs, and obtain a uniform result on the products, batch processing is used in some toolsets in semiconductor manufacturing. Batch size therefore impacts the toolset workload balance and must be considered for qualification management. The next chapter is centered around this industrial issue.

Chapter 5

Flexible Qualification Management under Batch Size Constraint

Qualification and batch size constraints are two important characteristics of many toolsets in semiconductor manufacturing. In this chapter, we consider the impact of considering batch sizes on two flexibility measures used for qualification management. We propose several solution approaches for balancing the workload on machines in terms of production volume or production time. Using numerical experiments conducted on real fab data, the solution approaches are compared and the impact of batch sizes on qualification management is discussed.

Introduction and Motivation

Until now, we have demonstrated that qualifications have a direct impact on production capacity. In this chapter, we study the impact of batching on qualification management. Wafers of the same type travel together in units called lots. Each lot consists of normally 25 wafers. Depending upon the machine, lots are grouped to form batches. The lots of a batch undergo the same process. Batching is used for the reduction of manufacturing process variability which can result in defective parts. It is also used to reduce the manufacturing costs. Hence, it is mostly appropriate for processes with high setup or changeover costs, for instance long-run processes such as those in furnaces. Besides, batching assures a uniform processing condition for all lots. Small batch size increases manufacturing cost (setup, labor, tool replacement, etc.) while a batch size being too large, may lead to increased tolerance and product variability. Therefore an optimal batch size must be determined according to machine specifications and manufacturing process variability [START_REF] Shin | A Stochastic Model for the Optimal Batch Size in Multi-Step Operations with Process and Product Variability[END_REF].

Batch processing is widely used in semiconductor manufacturing as well as other industries such as the metallurgical industry. Some typical batch processing steps in semiconductor manufacturing are thermal treatment, cleaning, implantation, etc. Batching may be either serial (called s-batching) or parallel (called p-batching). In semiconductor manufacturing, p-batching is more frequent than s-batching [START_REF] Mönch | A Survey of Problems, Solution Techniques, and Future Challenges in Scheduling Semiconductor Manufacturing Operations[END_REF]. In this chapter, we consider p-batching.

In some cases, lots are ungrouped when a machine processes less than 25 wafers at each production run. In the toolset for which the numerical experiments in Section 5.4 are conducted, the batch size varies from 4 to 10 lots at each production run (equivalent to 100 to 250 wafers per run)! The wide range of the batch size motivates the study of its impact on qualification management and hence capacity planning. As the machine specifications are different, the batch size of the same recipe may vary depending upon the machine. Since the throughput of each machine differs from recipe to recipe, an efficient qualification management considering these constraints optimizes the capacity utilization of the toolset while reducing the production variability and the cycle time.

Both recipe-to-machine qualification configuration and batches are two impor-

Optimization Model and Complexity Analysis

tant deterministic sources of variability ( [START_REF] Rowshannahad | Qualification Management to Reduce Workload Variability in Semiconductor Manufacturing[END_REF] and see Chapter 6). To avoid variability propagation through the production line, mastering variability is more crucial for bottleneck toolsets.

In this chapter, we investigate the impact of the batch size on the recipe-tomachine allocation used in qualification management and toolset capacity. In Section 5.2, the optimization model for batch size constraints is presented and the complexity is analyzed. In Section 5.3, several solution approaches for the flexibility measures considering batch size constraint are proposed. In Section 5.4, numerical experiments on real fab data for a "Thermal Treatment" toolset are presented and discussed. Finally, we draw conclusions and mention some research perspectives in Section 5.5.

Optimization Model and Complexity Analysis

In the following sections, we discuss the related mathematical model and complexity analysis.

Optimization Model

In this chapter, we propose solution approaches for workload balancing for both the WIP and time Flexibility measures. In addition, recipe-to-machine WIP allocations must be an integer multiple of the batch size of that recipe-to-machine, except for at most one recipe-to-machine allocation for each recipe. The latter is because it is not always possible to find batch sizes for recipe r so that their sum is exactly equal to W IP r . Let us denote by Y r,m a binary variable which is equal to 1 if the quantity of recipe r assigned to machine m is not an integer multiple of the batch size and 0 otherwise. Let us also denote by W IP C r,m the WIP allocation of recipe r to machine m which can be continuous, i.e. not an integer multiple of the batch size.

The total W IP allocated to each machine can constitute full batch(es) (if any) and/or a non full-batch allocation. Therefore, in the flexibility measures (F

(Run r,m BS r,m + W IP C r,m ) = W IP r ∀r (5.1b) Run r,m BS r,m + W IP C r,m ≤ Q r,m W IP r ∀r, m (5.1c) W IP C r,m ≤ (BS r,m -1)Q r,m ∀r, m (5.1d) W IP C r,m ≤ Q r,m W IP r Y r,m ∀r, m (5.1e) M m=1 Y r,m ≤ 1 ∀r (5.1f) Y r,m ∈ {0, 1} ∀r, m Run r,m ∈ N + ∀r, m W IP C r,m ≥ 0 ∀r, m
The objective function F (5.1a) is either F W IP or F T ime . Constraint (5.1b) ensures that all of the production volume of each recipe is allocated to the toolset. Constraint (5.1c) guarantees that recipe r is only allocated to machines m that are qualified for r, i.e. such that Q r,m = 1. Constraint (5.1d) states that the non full-batch allocation must be smaller than or equal to the batch size (else a full batch can be made). Constraint (5.1e) and Constraint (5.1f) ensure that at most only one variable W IP C r,m can be strictly positive for each recipe r, i.e. at most one non full-batch run is allowed for each recipe r.

Note that γ is even more important in F T ime when considering batch sizes. By varying γ, the WIP allocation of recipes to machines may change. Since the batch size of each recipe on each machine may be different, if the workload balancing is done with batch size constraints, significant changes in γ may significantly impact the recipe-to-machine qualification configuration. This issue is discussed in Section 5.4.

Complexity Analysis

It is possible to show that the problem with F W IP or F T ime as a criterion is NP-hard in the strong sense. This is because, even if all recipes are qualified on all machines, i.e. Q r,m = 1 ∀r, m, then the problem can be simplified to the scheduling problem on parallel machines P ||C max , which is known to be NP-hard in the strong sense [START_REF] Garey | Computers and Intractability: A Guide to the Theory of NP-Completeness[END_REF]. The transformation is done by noting that

F W IP = 1 or F T ime = 1 means that R r=1 W IP r,m is the same for each machine m, i.e. R r=1 W IP r,m = R r=1 W IPr,m M
. The decision problem "Is there a solution to the qualification problem such that F W IP = 1 or F T ime = 1" can be transformed to solving a related scheduling problem

P ||C max and checking that C max = R r=1 W IPr,m M .

Workload Balancing with Batch Size Restrictions

The algorithms for real-valued workload balancing for the WIP and Time Flexibility measures are discussed in [START_REF] Johnzén | Flexibility Measures for Qualification Management in Wafer Fabs[END_REF] and in [START_REF] Johnzén | Modeling and Optimizing Flexible Capacity Allocation in Semiconductor Manufacturing[END_REF], respectively. Here, we discuss two types of algorithms for workload balancing for the WIP and Time Flexibility measures with batch size constraints.

The first type algorithms construct a full-batch solution directly. While, the second type algorithms, called Batch-Feasibility Algorithms takes the real-valued optimal solution and makes it feasible according to batch constraints.

In the next section, we compare the results of proposed algorithms with the standard commercial solver, IBM ILOG CPLEX 12.5.1.

Workload Balancing in terms of Production Volumes

In order to find the workload balance with batch size constraint on the toolset in terms of the production volume (WIP), we consider two main approaches. The first approach is a to solve the problem while considering batch sizes. The second approach consists of a heuristic which takes the optimal real-valued solution as Constraint an initial solution and then, according to the batch size of each recipe-to-machine couple, generates a solution considering batch size.

For balancing the production volume on the toolset, we maximize (F W IP ) while distributing the production quantity of each recipe among the qualified machines for that recipe. The problem is formalized in (5.1).

The full-batch resolution is composed of two algorithms. The first algorithm constructs a real-valued or full-batch initial solution and the second algorithm progressively improves the initial solution to obtain either the optimal or a precise-enough near-optimal full-batch solution.

The algorithm used to construct the real-valued initial solution equally distributes the WIP of each recipe to their corresponding qualified machines. This yields a feasible and (almost always) non-optimal solution. The initial solution can be constructed according to the batch sizes or not.

The second (and the main) algorithm takes this initial solution as an input and improves it to find the optimal solution. The results in [START_REF] Rowshannahad | Qualification Management with Batch Size Constraint[END_REF] show that the second algorithm gives the same final full-batch solution regardless of the initial solution, i.e. whether constructed according to batch size or not.

Full-Batch Algorithm

Here, we modify the algorithm described in [START_REF] Johnzén | Flexibility Measures for Qualification Management in Wafer Fabs[END_REF] to consider batch sizes. Several strategies are possible which are described along the section. We may consider batch sizes from the beginning, i.e. while constructing the initial feasible solution or only when improving the initial feasible solution, constructed considering or ignoring batch sizes.

Constructing Full-Batch Initial Solution For calculating a real-valued initial feasible solution, we equally divide the WIP of each recipe among all of the qualified machines for that recipe. To consider batch sizes, in each loop, we start allocating the WIP by making as many batches as possible, i.e. W IPr,m BSr,m . The new WIP allocation is composed of one or more batches which represent one or more production runs of recipe r on machine m.

Workload Balancing with Batch Size Restrictions

Step 0. Set recipe index r to 1 (r = 1).

Step 1. For each machine in the toolset, determine the production volume allocation of recipe r to machines by equally distributing the production volume of recipe r (W IP r ) among all of the qualified machines for r.

If for each of the qualified machines for recipe r, i.e. where Q r,m = 1, the calculated W IP r,m is more than a full batch (W IP r,m ≥ BS r,m ) and forms an integer multiple of the batch size (BS r,m ), then attribute the WIP, else attribute only the integer part of the calculated W IP r,m , i.e. W IPr,m BSr,m . However, if the calculated W IP r,m is less than a full batch (W IP r,m ≤ BS r,m ), then allocate W IP r,m to the less loaded machine among the machines qualified for recipe r.

Step 2. Set r = r + 1, if r ≤ R, then go to step 1.

Full-Batch Optimization Algorithm

The algorithm begins with an initial solution S 0 , generated by the initial WIP distribution algorithm. The initial solution is improved iteratively until the optimal or precise-enough near-optimal solution is obtained.

One case that often happens is that the overall production volume of each recipe does not correspond to an integer number of full batches, which is also difficult to consider from the beginning (when production planning is done). This is because the batch size of each recipe depends on the machine and we do not know exactly in advance how much production volume of which recipe is allocated to which machine considering that the machine statuses and qualification configuration change over time. In this case, even by attributing W IP r,m considering the batch size of recipe r on machine m (BS r,m ), the rest of the production volume for each recipe r is not equal to a production run, i.e. one full batch equivalent to BS r,m . The question is to which of the qualified machines of recipe r, the remaining production volume must be assigned. We will allocate this volume to the less loaded machine among all qualified machines for recipe r. However it is better, if possible, to determine the production volumes from the beginning to form full batches. This is because processing only one wafer or a full batch of for instance 250 wafers will take the same Chapter 5. Flexible Qualification Management under Batch Size Constraint process time. Besides, if full batches cannot be made, in some toolsets, including Thermal Treatment toolset, filler wafers must be used to fill the vacant places in the carrying boats. By making full batches, we eliminate the cost of filler wafers and reduce the non productivity in the fab. Another case that must be considered is when the production quantity of a recipe is smaller than the smallest batch size of any machine qualified for this recipe. In this case, the production quantity of this recipe is produced only on the machine with the smallest total workload (W IP m ).

In summary, while allocating the workload, three cases may happen: (a) One (or several) full batch(es) and one real-valued WIP allocation; (b) One real-valued WIP allocation and no full batch and finally (c) One (or several) full batch(es).

Let us define by LM r the set of loading machines for recipe r as the set of qualified machine(s) for r, with the smallest WIP quantity among all of the qualified machines for r, i.e.

m * ∈ LM r ⇒ W IP m * = min ∀m|Qr,m=1
{W IP m } .

Step 0. Start with initial solution S 0 . The recipe index r and the iteration k are set to 1 (r = 1 and k = 1).

Step 1. Remove the WIP of recipe r initially allocated to machine m (W IP r,m ) from machine m, (W IP m = W IP m -W IP r,m ). Then, the total WIP quantities on the machines are sorted in a decreasing order.

Step 2. Calculate the WIP allocation quantities (W IP r,m * ) if we had equally distributed the WIP of r on the loading machine(s) m * in LM r until the WIP quantity on one (or more) loading machine(s) m * in LM r , (i.e. W IP r,m * ) is equal to the WIP quantity of a machine m qualified for recipe r and not in LM r , i.e. W IP m * = W IP m with m * ∈ LM r and m / ∈ LM r .

Step 3. WIP Re-Allocation

Step 3a. Step 3b. If W IP r,m * < BS r,m * , then there is no full batch. The realvalued WIP allocation is equal to W IP r,m * .

If W IP r,m * > W IP r,m * BS r,m * BS r,
Step 3c.

If W IP r,m * = W IP r,m * BS r,m * BS r,m * , then there is (are) one (or sev- eral) full batch(es) which is (are) equal to W IP r,m * ( W IP r,m * BS r,m * BS r,m * ).
Then, m is added to the set of loading machines, i.e. LM r ≡ LM r ∪ {m }. If r < R, then go to Step 1.

Step 4.

If F W IP k -F W IP k-1 > 0, then set k = k + 1, r = 1 and go to Step 1.
As the proposed algorithms are based on progressive improvement of an initial solution, it is not possible to implement the condition

F W IP k -F W IP k-1 > 0 of
Step 3 in a strict manner. Therefore, the flexibility value of the previous solution S k-1 is compared with the flexibility value of the new solution S k by defining an acceptable gap called . is a very small value, for example 1 × 10 -20 . The algorithm iterates until precise-enough near-optimal solution is reached.

Batch-Feasibility Algorithm

When taking into account batch sizes, the problem becomes NP-hard in the strong sense for both flexibility measures as discussed in Section 5.2.2. This is why we propose a heuristic which takes as input the optimal real-valued workload balance, and makes it feasible by considering the batch size of each recipe on each machine. The resolution approach is different from the previous algorithms in the sense that we first choose the machine and then the recipe, whereas in the previous methods, a recipe is first chosen from which WIP values are assigned to its qualified machines. The advantages of our approach are that we already know the optimal solution of the real-valued problem and we have more flexibility in the problem Chapter 5. Flexible Qualification Management under Batch Size Constraint resolution. Furthermore, the real-valued solution serves as an upper bound of the full-batch problem solution.

Step 0. The optimal solution generated by the real-valued WIP distribution algorithm is the initial solution (S 0 ).

Step 1. In the whole toolset, find the most loaded machine (m ). From the set of all qualified recipes for this machine to which WIP values have been allocated (Q r,m = 1 and W IP r,m > 0), remove the load from the WIP allocation which is nearest (or in a second version: "farthest") to form a full batch. The removed value is equal to

{ W IP r,m
BS r,m }. Attribute this value to the less loaded machine which has already a non-integer WIP allocation for the same recipe (i.e. { W IP r,m BS r,m } > 0). Note that this last condition (i.e. non-integer WIP allocation) has been added to avoid cycling.

Step 2. Once all the WIP allocations are redistributed (if necessary), the stop criterion is checked.

If F W IP k -F W IP k-1 > 0, then k = k + 1, r = 1 and go to step 1.
Two versions of the algorithm have been implemented. In the first version, at the beginning of Step 1, the preliminary workload on the toolset is ordered decreasingly and then all WIP attributions are done using this order. In the second version, the ordering is redone each time a WIP allocation is performed. The numerical results in this chapter are based on the second version which yields better results on average.

Workload Balancing in terms of Production Times

As with the production volumes, two approaches for workload balancing in terms of process times are presented. In the first approach, the batch constraint is considered from the beginning. The second approach is based on the same heuristic used for WIP workload balancing. Batch-Feasibility Algorithm takes the optimal realvalued load balancing solution of the Time Flexibility and makes it feasible with respect to the batch size constraint (see Model 5.1).

Workload Balancing with Batch Size Restrictions

The active-set method has been used in [START_REF] Johnzén | Modeling and Optimizing Flexible Capacity Allocation in Semiconductor Manufacturing[END_REF] to find the optimal real-valued workload balance for the Time Flexibility measure (F T ime ). The method is adapted below to consider the batch size constraint by limiting the step length.

Full-Batch Resolution: Adapted Active Set Method

In order to find the optimal workload balance for the Time Flexibility measure (F T ime ), the optimization problem (5.1) must be solved when F T ime is maximized. Since C ideal is constant, by maximizing F T ime , the sum of the production times (5.2) must be minimized.

min f = M m=1 ( R r=1 W IP r,m T P r,m ) γ (5.2)
The essential part of the resolution algorithm is based on a line search method called active-set method described in Section 4.3.2.

In the active-set method, the solution is improved at each iteration k (corresponding to solution (x k )). The solution improvement consists of finding a nonzero search direction (p k ) and a nonnegative step length α k on the working surface to reach a new solution new solution (x k+1 ). A solution in our problem corresponds to a WIP reallocation (W IP k+1 r * ,m ). The working surface is defined by the working set (W k ) which corresponds to a subset of constraints which are binding (active). Here, we will restrict the step length (α k ) in a way not to violate the batch size constraint (BS r,m * ). While moving on the working surface and defining the step length, one or several blocking constraint(s) may limit the step length. If one (or several) blocking constraint is encountered, it is added to the working set making a surface of lower dimension than before [START_REF] Luenberger | Linear and Nonlinear Programming[END_REF]. In our problem, the variables (W IP r,m ) are upper-bounded by their batch size (BS r,m ). Therefore, the variable becomes active, each time a bound constraint is encountered, i.e. W IP k r * ,m = Run r,m BS r * ,m where Run r,m ∈ N + ∀r, m [START_REF] Gill | Practical Optimization[END_REF]. At each iteration, the Lagrangian multipliers (λ) of the working set are evaluated. If all are nonnegative, the optimal solution is obtained else we drop one or more of the constraints having a negative Lagrangian multiplier. Step length definition makes the algorithm not to violate the batch size constraint. As the method is already presented in Section 4.3.2, we only present the adapted active-set method, the decomposition, the gradient and the Hessian functions.

Chapter 5. Flexible Qualification Management under Batch Size Constraint

Instead of Equations (4.22), (4.24) and (4.25), Equations (5.3), (5.4) and (5.5) must be used, respectively.

f = W IP r * ,m T P r * ,m + fr * ,m γ (5.3) ∇f = γ T P r * ,m W IP r * ,m T P r * ,m + fr * ,m γ-1
(5.4)

∇ 2 f = γ (γ -1) (T P r * ,m ) 2 W IP r * ,m T P r * ,m + fr * ,m γ-2
(5.5)

Step 0. Start with an initial feasible solution x 0 and working set of active constraints at x 0 (W 0 ). Set k = 1.

Step 1. The Optimality Test

If ZT ∇f W IP k r * ,m = 0
Step 1a. If there are no active constraints (W

IP k r * ,m = 0), Local Sta- tionary Point -STOP.
Step 1b. Else, compute Lagrangian Multipliers:

λ = ĀT r ∇f W IP k r * ,m .
(5.6)

Step 1c. If λ ≥ 0, Local Stationary Point -STOP. Else delete the constraint corresponding to the most negative Lagrangian multiplier from the working set W k . Update W k , Z, Ā and Ār .

Step 2. The Search Direction Compute a descent feasible search direction p k with respect to the active constraints in W k . The Reduced Newton Search Direction (5.7) [START_REF] Hoyer | Variants of the Reduced Newton Method for Nonlinear Equality Constrained Optimization Problems[END_REF] is an efficient search direction.

p k = -Z ZT ∇ 2 f W IP k r * ,m Z -1 ZT ∇f W IP k r * ,m
(5.7)

Step 3. The Step Length Compute a step length

α k such that f (W IP k r * ,m + α k p k ) < f (W IP k r * ,m
) subject to retaining feasibility with respect to all constraints, i.e. α k ≤ ᾱk , where ᾱk is the maximum feasible step length

Workload Balancing with Batch Size Restrictions

along p k . The objective of this Step is to redistribute W IP k r * ,m according to its corresponding batch size, i.e. BS r * ,m . Three cases may happen:

Step a. If W IP k r * ,m > W IP k r * ,m
BS r,m * BS r,m * , then one (or several) full-batch allocation(s) and just one real-valued allocation is done. The step length α k is defined in 5.8.

α k = min i / ∈W k ,p k <0 W IP k r * ,m BS r,m * BS r,m * -W IP k r * ,m p k . (5.8)
Since the variable lies on its upper bound, the bound constraint is active. Therefore, W k is modified to include the bound constraint to the working set (W k+1 ← W k ∪ i). Update Z, Ā and Ār .

Step b. If W IP k r * ,m < BS r,m * , then no full-batch allocation can be made and only a real-valued allocation is done. The step length α k is defined in 5.9.

α k = min i / ∈W k ,p k <0 -W IP k r * ,m p k . ( 5.9)
[Step 3b-I.] If α k < ᾱk , α k is an unconstrained step, i.e. no blocking constraint is encountered and x k+1 remains feasible (x k+1 = x k + α k p k ).

[Step 3b-II.] If α k ≥ ᾱk , then the step along p k is blocked by one (or more) (inactive) constraint(s) not in W k but which are active in x k+1 . In this case, W k is modified to include the blocking constraint to the working set (W k+1 ← W k ∪ i). Update Z, Ā and Ār .

Step c.

If W IP k r * ,m = W IP k r * ,m
BS r,m * BS r,m * , then one (or several) full-batch allocation(s) are done. The step length α k is zero. This can be verified by considering 5.8. Since the variable lies on its upper bound, the bound constraint is active. Therefore, W k is modified to include the bound constraint to the working set (W k+1 ← W k ∪ i). Update Z, Ā and Ār .

Set k = k + 1 and go to Step 1.

Chapter 5. Flexible Qualification Management under Batch Size Constraint

Batch-Feasibility Algorithm

When taking into account batch sizes, the problem becomes NP-hard in the strong sense for both flexibility measures as discussed in Section 5.2.2. That is why we propose a heuristic which takes as input the optimal real-valued workload balance, and makes it feasible by considering the batch size of each recipe on each machine. The resolution approach is different from the previous algorithms in the sense that we first choose the machine and then the recipe, whereas in the previous methods, a recipe is first chosen from which WIP values are assigned to its qualified machines. The advantages of this approach are that as the real-valued optimal solution of the problem is known, more flexibility exists in the problem resolution. Furthermore, the real-valued solution serves as an upper bound of the full-batch problem solution.

Step 0. Start with the optimal real-valued solution (S 0 ). Set the solution index to one (k = 1).

Step 1. Order the machines in the toolset according to their workload in terms of production times in a decreasing order. Choose the most loaded machine (m ) (potential contributor machine). From the set of all qualified recipes for this machine to which WIP values have been allocated (Q r,m = 1 and W IP r,m > 0), remove the load from the WIP allocation which is nearest to form a full batch (if any). The removed value is equal to

{ W IP r,m BS r,m }.
Step 2. Attribute this value to the less loaded machine which has already a not full batch WIP allocation for the same recipe (i.e.

{ W IP r,m
BS r,m } > 0). Note that this last condition (i.e. non-integer WIP allocation) is added to avoid cycling.

Re-order the machines in the toolset according to their workload in a decreasing order.

Repeat Step 1 and

Step 2 until all WIP allocations are redistributed (if necessary).

Numerical Experiments

Step 3.

If F W IP k -F W IP
k-1 > 0, then update the solution index (k = k + 1) and go to Step 1.

Numerical Experiments

The quality of the solution provided by our approaches and the impact of the batch size constraint are studied in this section. As the concept is industrialized, we use 25 industrial instances of Soitec for the experimentation.

Comparing with Exact Solutions

The commercial solver IBM ILOG CPLEX 12.5.1 has been used to evaluate the proposed algorithms. However, IBM ILOG CPLEX can only solve quadratic programs and not other types of non-linear programming models. Therefore, γ is set to 2 in this section. For WIP flexibility, the optimal solution is obtained by minimizing the denominator (

M m=1 ( R r=1
W IP r,m ) γ ). Then, F W IP is calculated. The same approach is used for F T ime . The best flexibility values obtained by our methods are bold-faced.

WIP Flexibility Measure

Table 5.1 presents the WIP flexibility values for each instance using our resolution approaches and IBM ILOG CPLEX for γ = 2. Note that, because of the small value of γ, the WIP flexibility measures are relatively big and very close for each instance. For all instances (except one), the full-batch algorithm (Section 5.3.1.1) outperforms the batch-feasibility heuristic. Moreover, the WIP flexibility values of the full-batch algorithm are very close to the optimal solutions. 2) also yields good results but not as good as the adapted active-set method.

Time Flexibility Measure

Numerical Experiments

Impact of Batch Size Constraint on Qualification Decisions

In the industrial decision support system, our methods are implemented in VBA using Microsoft Excel. For the experiments of this section, γ is set to 6. The choice of γ is based on historical experiments which corresponds the best to the actual toolset workload in the fab. The WIP and time flexibility measures for the Chapter 5. Flexible Qualification Management under Batch Size Constraint current recipe-to-machine qualification configuration and the qualification configuration after the best qualification are presented in Tables 5.3 and 5.5. As the optimal real-valued solutions for both WIP and time flexibility measures are available, they are used as upper bounds to evaluate the quality of our algorithms with batch size constraints. Tables 5.4 and 5.6 present the qualification choice correspondence using the full-batch algorithms compared to the real-valued solutions for the WIP and time flexibility measures.

WIP Flexibility Measure

Table 5.3 shows the WIP flexibility value for the current recipe-to-machine configuration and the WIP flexibility for the new configuration after the best new qualification. The second and third columns show the optimal real-valued WIP flexibility values for all instances. The fourth and fifth columns show the WIP flexibility values taking batch sizes into account using the full-batch algorithm. The sixth and seventh columns show the flexibility measures taking batch sizes into account using the batch-feasibility heuristic. When looking at Table 5.3 and although, in most cases, the WIP flexibility is very close with or without batch size constraints, the difference is quite large in Instance 12 (from 79.24% down to 64.54% for the current configuration and from 93.88% down to 75.30% for the configuration after the best qualification). This result could be due to the fact that our solution approaches considering batch sizes do not necessarily find the optimal solution. Table 5.3 also shows that the full-batch algorithm strongly dominates the batch-feasibility heuristic.

An important question is whether the best qualification proposed when solving the problem with real-valued variables is also relevant if the problem is solved with batch size constraints. Table 5.4 compares the WIP flexibility measure for the best qualification proposed by our batch solution approaches with the best qualification proposition obtained using the real-valued resolution but solved with our batch solution approaches. This shows the flexibility evaluation deviation if we had applied the qualification decision without considering batch sizes.The columns "Best" contain the WIP flexibility measure for the recipe-to-machine configuration after the best (new) qualification. The columns "Best (RV)" show the WIP flexibility measure for the recipe-to-machine configuration after the best (new) qualification proposed

Numerical Experiments

Real value

Full when using the (optimal) real-valued solution approach. The columns "Match" indicate whether the next (new) qualification of the corresponding approach is the same as the (new) qualification proposition using the (optimal) real-valued resolution approach. It is interesting to note that there are multiple instances for which the proposed qualifications differ and with substantial differences in terms of WIP flexibility, such as Instances 5 (

Conclusions and Perspectives

Time Flexibility Measure

Looking at Table 5.5, with γ = 6, the adapted active-set method still yields considerably better results than the batch-feasibility heuristic. In most cases where the batch-feasibility heuristic gives better results, the difference is negligible. For example, in the first instance, the current flexibility using the batch-feasibility heuristic is 40.43% while the result obtained by the adapted active-set method is 40.40%. To get better results, it would be interesting to calculate the time flexibility considering the batch sizes using both approaches and choose the best solution. However, this would be very time consuming.

The batch-feasibility heuristic is less efficient with higher values of γ. In the realvalued resolution, by increasing gamma, we increase the number of recipe splitting while allocating workload. As the batch-feasibility heuristic "blindly" re-allocates the production volumes to make full batches (without considering the workload of the machines based on their process times), the efficiency of the algorithm decreases. Therefore, for high values of γ, it is preferable to solve the problem in one shot using the adapted active-set method.

The results of Table 5.5 show that considering batch sizes may lead to large difference in the time flexibility value of the toolset. This is the case for many instances (Instances 7 to 12 and 14 to 25). Table 5.6 shows the qualification matches using the real-valued resolution and the full-batch resolution. As it can be seen, in most cases, the qualification propositions do not match. But what would have happened if we had chosen the real-valued qualification proposition? According to Table 5.6, it could have led to a large misevaluation of the time flexibility. Some instances (7 to 9, 19, 22, 24 and 25) illustrate this phenomenon. Our numerical experiments on industrial data shows that, by ignoring batch sizes, wrong qualification decisions may be taken, leading to potential capacity loss.

Conclusions and Perspectives

Batching is an important characteristic of various workshops in the semiconductor manufacturing industry. In this chapter, we have proposed different workload Chapter 5. Flexible Qualification Management under Batch Size Constraint balancing algorithms (in terms of production quantities and production times) considering batch sizes. The workload allocations are used in qualification management. Using real fab data, we have shown that ignoring batch size constraints may lead to inappropriate qualification decisions which in turn leads to a non-optimal toolset capacity utilization.

Various perspectives are possible. Having near-optimal recipe-to-machine WIP A general perspective for qualification management is to propose flexibility measures which consider the yield for each recipe on each machine and propose relevant qualifications taking this parameter into account. Another aspect which may be considered directly or indirectly in flexibility measures is recipe priorities, i.e. more priority could be given to the recipes related to more important products or "hot lots". Auxiliary resource management (for example masks in the lithography toolset) is also a subject related to qualification management which can enrich future studies.

Conclusions and Perspectives

Instance

Chapter 6

Qualification Management and Workload Variability

Variability is an inherent component of all production systems. To prevent variability propagation through the whole production line, variability must be constantly monitored, especially for bottleneck toolsets. In this chapter, we propose measures to evaluate workload variability for a toolset configuration. Using industrial data, we show how making the toolset configuration more flexible by qualifying products on machines decreases variability. By quantifying the toolset workload variability, our variability measures makes it possible to estimate the variability reduction associated to each new qualification. The industrial results show significant workload variability reduction and capacity improvement. 16.1 Introduction and Motivation 6.2 Variability Measures 6.4 Resolution Approaches 6.5 Numerical Experiments 6.6 Conclusions and Perspectives

Introduction and Motivation

The dynamic business environment in semiconductor manufacturing industry calls for an increasing product-mix flexibility. Besides, production systems are restricted by machine capabilities. The ultimate goal of every manufacturing system is to satisfy customer demand while making the best possible use of the production facilities.

Variability is the uncertainties and variations in the production flow. Variability has stochastic and deterministic sources. The stochastic variability sources are uncontrollable, and the most well-known are demand, machine breakdown, rework, operator delay, etc. On the other hand, deterministic variability sources are controllable. They include machine eligibility, batches, setups, re-entrant flows, etc. The controllable deterministic variability source considered in this chapter is product-tomachine eligibility, called qualification.

Production variability influences production planning and scheduling, capacity planning, inventory management, equipment and labor cost, etc. [START_REF] Kim | Estimating the Distribution and Variance of Time to Produce a Fixed Lot Size Given Deterministic Processing Times and Random Downtimes[END_REF]. Even little variability in bottleneck workcenters can cause high variability in the whole production line. Therefore, production variability reduction at bottleneck workcenters is crucial to prevent variability propagation to the whole production line. Production variability leads to loss of capacity. Therefore, due to expensive equipment cost, mastering variability is critical in semiconductor manufacturing. Production variability decreases as manufacturing systems become more flexible. The flexibility of a manufacturing system is determined based on to which extent a product can be allocated to a machine. Hence, capacity allocation determines the flexibility of a production system [START_REF] Muriel | Impact of Partial Manufacturing Flexibility on Production Variability[END_REF].

This chapter is a continuation of our investigations about qualification management and production flexibility. In the previous chapters, variability is not explicitly considered. This is why, in this chapter, we aim at reducing production variability by defining variability measures. The workload variations in a single workcenter in one period are considered. We want to show how variability is reduced when the flexibility of the toolset is increased by performing new qualifications. The variability measures proposed can be used to measure the impact of batch or single

Toolset Workload Variability and Manufacturing Flexibility

wafer processing. However additional constraints must be added to the optimization model. Chapter 5 specifically address the problem of batch processing. Through experiments with industrial data, we illustrate that production variability and manufacturing flexibility are two sides of the same coin. This chapter is organized as follows. Section 6.2 details the framework of the study. The measures that we propose to evaluate the workload variability are presented in Section 6.3. Some extensions and resolution approaches to the problem are presented in 6.4. In Section 6.5, numerical experiments on industrial data are discussed. Finally, Section 6.6 provides conclusions and proposes future investigation paths.

Toolset Workload Variability and Manufacturing Flexibility

Recipe-to-machine qualification restrictions are comparable to product-to-machine eligibility constraints. The qualification concept is also alike the configuration concept. Reconfigurable manufacturing systems have more flexibility. By qualifying a qualifiable recipe, the toolset manufacturing flexibility increases. Figure 6.1 shows three configurations of a toolset. Figure 6.1a shows dedicated machines to recipes. This dedicated strategy is sometimes referred to as no flexibility. In Figure 6.1b, each recipe is qualified on at least two machines. The setting is called partial or sparse flexibility. Figure 6.1c depicts a totally flexible toolset where all recipes are qualified on all machines. For us, we reach full-flexibility when all machines are equally loaded based on its capacity. So, it is not absolutely necessary to qualify all recipes on all machines to attain full flexibility. Fortunately this is not the case, because qualifying all recipes on all machines is firstly not always possible and then not economically feasible. This is the miracle of flexibility that we gain a lot by no or little effort. Note that "theoretically", we may reach flexibility even in case where one recipe is qualified only on one machine, i.e. the dedicated strategy. However this case is scarce in industry. In the next section, we define variability measures to evaluate to which extent the increase of the manufacturing flexibility contributes to Chapter 6. Qualification Management and Workload Variability Increasing manufacturing flexibility implies at least one new qualification, enabling the qualified machine to process another recipe by creating a new link between the qualified recipe-machine couple. As already specified in the introduction, the notion of manufacturing flexibility in this study refers to [START_REF] Johnzén | Flexibility Measures for Qualification Management in Wafer Fabs[END_REF], where flexibility measures are defined to evaluate the flexibility gain associated with performing qualifications. One of the main objectives of their measures is to estimate the impact of qualifications on workload balancing. We have further discussed these measures in previous chapter and in the remainder of this part. They are used to evaluate new qualifications in the daily fab operations. However, by only considering these flexibility measures, the impact of a qualification on the toolset workload variability is not clear to decision makers. This study aims at throwing light on the other side of qualification management via variability measurement. Moreover, compared to [START_REF] Johnzén | Flexibility Measures for Qualification Management in Wafer Fabs[END_REF], we also consider machine capacity restrictions.

Variability Measures

In order to obtain a unique variability value, the optimal workload balance of the toolset according to its configuration must be calculated. By minimizing the proposed measures, which are inspired from statistical moments, the optimal workload

Variability Measures

balance of a toolset must be calculated.

The variability measures introduced in the next sections (V ar T ime U ncapa in Section 6.3.1, V ar T ime Capa in Section 6.3.2, and V ar T ime Capa = in Section 6.3.3) are used as the objective function (V ar •

• ) of the mathematical model below. The only set of constraints (6.1b) of the model guarantees that the production volume of recipe r is only allocated to machines that are qualified for r, i.e. machines m such that Q r,m = 1. By minimizing the selected variability measure subject to the set of constraints, the optimal toolset workload balance is obtained.

min V ar • • (6.1a) Subject to M m=1|Qr,m=1 W IP r,m = W IP r ∀r (6.1b) W IP r,m ≥ 0 ∀r, m
This model can be solved by adapting the Active Set method described in [START_REF] Johnzén | Modeling and Optimizing Flexible Capacity Allocation in Semiconductor Manufacturing[END_REF] and Chapter 4. In Section 6.4, the resolution is briefly addressed.

Uncapacitated Time Variability Measure (V ar T ime U ncapa )

For the current toolset qualification configuration, by minimizing the sum of the total process time of each machine (6.2), the workload variability is calculated. Then, for each qualifiable recipe-to-machine couple, we re-calculate the variability (V ar T ime U ncapa ) by virtually qualifying the qualifiable couple. In order to evaluate the variability reduction associated with each new qualification, the variability of each new configuration is subtracted from the variability of the initial configuration.

V ar T ime U ncapa = M m=1 (C m ) γ (6.2)
By increasing the workload balancing exponent (γ), the load of high speed machines is shifted to slower machines where qualification is allowed. Increasing γ creates a smoother workload distribution on the toolset.

Capacitated Time Variability Measure (V ar T ime Capa )

Machine failures, operator unavailability, scheduled and unscheduled maintenance are sources of variability which affect the uptime of machines. The uptimes of each machine in the same toolset can be different and are considered to be deterministic in this paper. V ar T ime Capa (6.3) evaluates the workload variability of a toolset while considering the capacity of machines.

V ar T ime Capa = M m=1 (C m -Capa m ) γ (6.3)
In order to calculate the variability reduction of each new qualification, as explained in Section 6.3.1, the variability of the current qualification configuration and the configuration after each new qualification must be calculated.

By increasing the workload balancing exponent (γ), the model tries to fit better the toolset workload to the available capacity by shifting workload from overloaded machines to less loaded machines.

Weighted Capacitated Time Variability Measure (V ar T ime Capa = )

As in Section 4.2.4 for flexibility measures, we propose a Weighted Capacitated Time Variability Measure (V ar T ime Capa = ) in which overloaded machines and underloaded machines are linearly and non-linearly penalized. The linear penalizers of a machine m are o m and u m and nonlinear penalizers correspond to different workload balancing exponents γ o and γ u .

V ar T ime Capa = = M m=1|Cm>Capam o m (C m -Capa m ) γo + M m=1|Cm≤Capam u m (C m -Capa m ) γu
The qualification that reduces the most the production variability depends on the values chosen for the penalizers (o m , u m , γ o and γ u ).

Resolution Approaches

Resolution Approaches

Model 6.1 can be solved by adapting the resolution approach based on [START_REF] Johnzén | Modeling and Optimizing Flexible Capacity Allocation in Semiconductor Manufacturing[END_REF] and [START_REF] Griva | Linear and Nonlinear Optimization[END_REF] and described in Section 4.3.2. Only a summary of the functions and the necessary changes in the resolution approach are mentioned below.

Capacitated Time Variability Measure (V ar T ime

Capa )

V ar T ime Capa = M m=1 (C m -Capa m ) γ (6.5)
Equations (4.22), (4.24) and (4.25) are respectively replaced by (6.6), (6.7) and (6.8).

f = (C m -Capa m ) γ + fm (6.6a) f = W IP r * ,m T P r * ,m -Capa m γ + fm (6.6b) ∇f = γ T P r * ,m (C m -Capa m ) γ-1 (6.7) ∇ 2 f = γ (γ -1) (T P r * ,m ) 2 (C m -Capa m ) γ-2 (6.8)

Weighted Capacitated Time Variability Measure (V ar

T ime Capa = ) V ar T ime Capa = = M m=1|Cm>Capam o m (C m -Capa m ) γo + M m=1|Cm≤Capam u m (C m -Capa m ) γu
f =    o m (C m -Capa m ) γo + fm if C m > Capa m u m (C m -Capa m ) γu + fm if C m ≤ Capa m
Chapter 6. Qualification Management and Workload Variability

∇f =    om•γo T P r * ,m (C m -Capa m ) γo-1 if C m > Capa m um•γu T P r * ,m (C m -Capa m ) γu-1 if C m ≤ Capa m ∇ 2 f =    om•γo(γo-1) (T P r * ,m ) 2 (C m -Capa m ) γo-2 if C m > Capa m um•γu(γu-1) (T P r * ,m ) 2 (C m -Capa m ) γu-2 if C m ≤ Capa m

Numerical Experiments

In this section, we study the concept of workload variability by conducting experiments on industrial data of a Thermal Treatment Toolset in an "SOI" (Silicon-On-Insulator) production line. The Thermal Treatment Toolset, which consists of non-homogeneous furnaces, is a bottleneck toolset. Each standard SOI product must at least visit three times this toolset. For the industrial experiments, the value of γ is set to 4. Careful observations of the shop floor workload allocation shows that any value between 4 to 6 suits the model for practical purposes. As the capacity of the machines are different, we use the Capacitated Time Variability Measure (V ar T ime Capa ) (6.3) as the objective function of the mathematical model.

First, we consider a data set for one period and study the impact of performing a single qualification on the percentage of the workload variability reduction. Other independent performance indicators used to interpret the workload balance are: The overall toolset workload variation percentage (6.10), overload (6.11) and unused capacity (underload) (6.12) variation percentages. The variation comparison for each performance indicator is simply calculated as shown in (6.13) for overload (OL) variation comparison. Using the static workload balance diagram, we show the impact of one new qualification on the workload variability. Finally, we discuss the impact of new qualifications on the reduction of the production variability for some industrial instances taken from the daily fab operations. i.e. creating new links between recipe set and machine set. It can be observed that performing more and more new qualifications reduces less and less the production variability. A trade-off must be made between the cost of performing new qualifications and the benefit of reducing workload variability. Figure 6.4 depicts the results of Table 6.1. It is worth to note that workload variations are not linear as the number of new qualifications increases linearly. Some qualifications decrease variability more than others. However, too many new qualifications do not decrease variability very much. Table 6.2 presents how one new recipe-to-machine qualification affects production variability for ten industrial instances. In general, one new qualification reduces variability, overload and unused capacity drastically while only slightly increasing the total workload. Note that, in the first instance, the overload is completely eliminated (-100%), capacity utilization is highly increased (20.20%) while the total workload only increases by 3.40%. The experiments illustrate again equal to the variability, overload and unused capacity decrease. This implies that, by performing one new qualification, only a small increase of workload leads to a high decrease of variability. The reason is that the process times of recipes are different from machine to machine and, by creating a new link between a recipe-machine couple via qualification, a better capacity allocation becomes possible.

W orkload Sum =

Numerical Experiments

Conclusions and Perspectives

In this chapter, we have studied the relationship between toolset qualification configuration (toolset manufacturing flexibility) and workload variability. A product can only be processed in a toolset when its associated recipe is qualified on at least one machine. Increasing the number of qualifications adds flexibility to the toolset and allows better capacity allocation.

Variability measures are presented to evaluate the workload variability of a toolset qualification configuration. Based on industrial data, we showed that, by

Conclusions and Perspectives

performing the best qualification according to the proposed variability measures, toolset overload, unused capacity and also variability are reduced.

It was shown that that without incurring the high qualification cost to achieve a full flexible system (Figure 6.1c), we can possibly reach the desired flexibility at a very lower cost by having a partial flexibility configuration (Figure 6.1b).

In conclusion, more manufacturing flexibility is required where the workload variability is the largest and not simply where the workload is the largest. In other words, more manufacturing flexibility absorbs workload variability. If the workload variability is low, meaning that (almost) all machines of a toolset are loaded equally according to their capacity, more manufacturing flexibility does not reduce variability. In this case, acquiring new machines might be necessary.

Several perspectives are possible for this study. An important source of variability is batching. The same variability measures can be used to evaluate how batches affect workload variability. One concept closely related to variability is robustness. It is interesting to study the relative quality of the solution for specific recipe-tomachine qualification configuration or data variability. The proposed approach leads to local variability reduction. Using stochastic modeling, it would be interesting to integrate the present toolset variability measurement to decrease the global production variability. In this case, the impact of manufacturing flexibility on buffer stock requirements between workcenters can be studied. In the same context, it is interesting to answer the following questions: how does the variability propagates in the production line? how does the variability pattern changes after crossing other workcenters? Can variability create a Bullwhip Effect? If yes, how to quantify the effect? Machine failure is often a major element of toolset variability. Although they reflect machine breakdowns, the machine capacities used in our measures are deterministic. It could be interesting to explicitly consider machine breakdown probabilities.

Finally, it could also be relevant to formalize the trade-off between the costs associated with performing and maintaining new qualifications and the gains related to the improved flexibility and variability quantified with our measures. This could lead to an interesting bi-criteria optimization problem. It remains to be seen if this will bring enough added value to the decision makers that are currently using our decision support system, since they will have to provide more information.

Introduction

In previous chapters, we discussed qualification management and its impact on capacity planning from different aspects. When industrializing our approaches, we had to deal with some additional industrial constraints which are detailed in this chapter.

Not previously qualified recipes are discussed in Section 7.2. The industrial implementation for two different toolsets and the decision making process are presented and discussed in Section 7.3. Some results obtained with the application are shown in Section 7.4. We close the chapter with some conclusions and perspectives.

Not Previously Qualified Recipes

When dealing with real data, it is necessary to consider the case in which at least one recipe is initially not (or no longer) qualified on any machine. A typical case, is a recipe associated with the introduction of a new product. Not previously qualified recipes are denoted NQRs. NQRs may also result from (automatic) disqualifications when a recipe is not processed over a certain time window on the machine. Below, we modify uncapacitated flexibility measures to consider NQRs. We define one additional parameter P as the penalty coefficient for not previously qualified recipes (NQRs).

WIP Flexibility Measure

The following WIP flexibility measure first proposes qualifications for the not previously qualified recipes (NQRs) and, once all the NQRs are qualified, helps to suggest further qualifications for other recipes.

Using the penalty coefficient (P ) in F W IP

N QR as defined below, we artificially put a large load on the machines until all of the NQRs are qualified. When all NQRs are 7.2 Not Previously Qualified Recipes qualified, the measure proposes qualifications for other recipes.

F W IP N QR = M × M m=1 (W IP m /M ) M m=1 (W IP m + P R r=1|N QR W IP r ) ∈ (0, 1] (7.1)
P must be large enough to make the maximum W IP of the set of non previously qualified recipes larger than the maximum value of all W IP s. Therefore, in each iteration, as NQRs are qualified, the value of P is updated.

M axW IP = max ∀r {W IP r } (7.2a) M axW IP N QR r = max ∀r|N QR {W IP r } (7.2b) P = M axW IP M axW IP N QR r (7.2c)

Time Flexibility Measure

The same idea as with the F W IP N QR is used to derive F T ime N QR . The penalty coefficient (P ) artificially puts a large load (in terms of process time) on the machines. The term P ×C N QR disappears when all NQRs are qualified and F T ime N QR reduces to F T ime . P must be large enough to make the maximum process time of the set of NQRs larger than the maximum process time of any normal recipe. In other words, the penalty coefficient (P ) makes the maximum of the sum of the process times on each tool ( R r=1 W IPr T Pr,m ) of NQRs (C N QR ) larger than the maximum of all C m s. Therefore, in each iteration, as NQRs are qualified, the value of P is updated.

F T ime N QR = C ideal M m=1 (C m + P × C N QR ) γ ∈ (0, 1] (7.3)

Chapter 7. Industrialization of Qualification Management

Since C m = R r=1 (W IP r,m /T P r,m ), (7.3) can be reformulated as in (7.4).

F T ime N QR = C ideal M m=1   R r=1 W IP r,m T P r,m + P R r=1|N QR W IP r,m T P r,m   γ (7.4)
Contrary to the Toolset Flexibility, in order to calculate both the WIP and time flexibility measures, the optimal workload balance must be calculated. Note that capacitated flexibility measures as well as variability measures can be modified in the same way to consider NQRs. The usage of the modified flexibility measures are very important when there are NQRs in the recipe set. Using normal flexibility measures, the flexibility gain obtained for NQRs is negative! Therefore, these recipes do not appear among qualification alternatives although they should be the first for qualification.

Implementation and Industrialization

In this section, we describe how the concept has been deployed in industry.

Decision Making Process

In this thesis, we have succeeded to put in place a new workflow process in Soitec for qualification management.

Performing a qualification can be a time-consuming and costly task (based on the toolset). It may disturb production and several departments might have to be mobilized. As a result, it is preferable to know in advance the required new qualifications and of course the best ones. The required data (see Section 7.3.3) are collected for the next three weeks on a weekly basis. The simulation for the next three weeks is used to avoid rush qualifications (mostly due to not previously qualified recipes) and to provide a rough vision of the toolset capacity and flexibility. When using the application on a one-week data, the scheduled downtimes are identified and the production plan is more stable. The results of the analysis give guidelines for workload allocation and scheduling at the operational level.

Implementation and Industrialization

Figure 7.1 shows where the data is collected and which departments use the results. The process department uses the flexibility gain values to determine the best qualification(s) to perform. This ensures a more robust and balanced toolset leading to capacity optimization. The capacity planning service must provide (if possible) the required capacity to realize the production plan determined in collaboration with the supply chain department. Once the desired qualification configuration is determined, the optimal workload balance is calculated using the application. This optimal workload distribution serves as guidelines to allocate the recipe volumes to machines by the production control department.

Industrialization

The flexibility measures described in [START_REF] Johnzén | Flexibility Measures for Qualification Management in Wafer Fabs[END_REF] were implemented with the programming language VBA™ (Visual Basic for Applications) under Microsoft Excel™ originally in [START_REF] Johnzén | Modeling and Optimizing Flexible Capacity Allocation in Semiconductor Manufacturing[END_REF]. The application has been completed to cover all of the subjects discussed in this thesis. The application solves the optimization model for the current qualification configuration and saves the current flexibility level according to the selected flexibility measure. Then it virtually qualifies each of the qualifiable recipes and saves it in a matrix. Finally the flexibility gain is calculated by subtracting each element of the matrix from the current flexibility level while highlighting the best qualification. The flexibility gain is shown in both a tabular and listed manner. In the latter, the possible qualifications are listed in a decreasing order of their flexibility measure. In a separate worksheet, the optimal workload balance according to process

Input Data

One of the important aspects of industrialization is input data preparation. In general, four data categories are used: Toolset qualification status, production volume (or WIP) per recipe, throughput and batch size for each qualified and qualifiable recipe for each machine and the maximum capacity per machine. Toolset qualification status defines the qualification status of each recipe-to-machine couple. It can be non qualifiable (or unqualifiable), qualifiable or (already) qualified. So, the application calculates the flexibility gain if each qualifiable recipe was qualified.

Input Data Extraction and Treatment

Figure 7.2 shows a snapshot of the Microsoft Excel™-based application, developed using the programming language VBA™. The data are extracted from an image of the MES (Manufacturing Execution System) database. Using SQL (Structured Query Language), the production route of each part is identified. The steps and the recipes are extracted. Then the production volume per product reference is crossreferenced with the recipes to give the production volume (or WIP) per recipe.

Results

Input Data

Results

The application is being used for two bottleneck toolsets as a decision support system for qualification management and capacity planning.

Thermal Treatment Toolset

The thermal treatment toolset consists of 22 machines belonging to four different machines types designated respectively from types A to D. Each machine type has similar hardware and software characteristics. Some of the recipes have a corresponding alternate recipe which can be used to better balance the workload. As the alternate recipes are performed on different machine types, they may be faster or slower than the original recipe. With low values of γ, the optimization balancing algorithm tends to allocate more load to faster machines. When increasing γ, the algorithm tend to allocate more load to slower machines to better balance the workload among all machines.

As an example, the results of the simulation of one week are used. The System flexibility measure is chosen as the flexibility measure. As workload balancing is more important for the company, we chose 70% of Time Flexibility and 30% of Toolset Flexibility. After extracting and preparing data, the application is run in order to the same machine category have similar characteristics, their flexibility gain is also more or less the same. Another specificity of this recipe as that it is actually composed of two recipes, which are both equivalent but processable on different machine types. "Recipe 3" is processable on machines types A, C and D while its equivalent recipe ("Alternate recipe 3") is executable on machine Type B. By referring to the load balance (Figure 7.7), the machines of type A are the most loaded. Several reasons could explain this load allocation. Due to their hardware or software configuration, they are already qualified for a significant number of recipes. Therefore more load could be allocated to them. Another reason could be their throughput. However, as already mentioned, the behavior of the algorithm can be controlled using the balancing exponent (γ).

The best qualification is specified in green (i.e. to qualify "Recipe 3-Alternate Recipe 3" on "Machine 16"). It is important to note that the application is used as a Decision Support System (DSS). This implies that the decision maker can choose his/her best qualification among different alternatives, according to his/her qualitative criteria. The role of the decision maker becomes more important when the differences between flexibility gains are not significant. By qualifying the new recipe, the same machines are still over-loaded despite the unused capacity on the majority of the machines (Figure 7.9). The reason is that by qualifying a new recipe, we add more load on the toolset and due to current qualification settings and restrictions, we are unable to allocate production volumes to less loaded machines.

The flexibility gain table after the first qualifications is shown in Figure 7.10. The best qualification increases the System flexibility (F Sys ) by 12.90%. By decomposing F Sys , F T ime increases by 17.43% (i.e. from 79.4% to 96.8%) and F T S increases by 2.4% (i.e. from 68.1% to 70.5%). The workload balance is depicted in Figure 7.11. It is worth noting that only one new qualification can increase the machine utilization and eliminate overload. Finally, one of the other outputs of the application is the optimal WIP distribution diagram corresponding to each optimal workload balance. In Figure 7.12, the optimal WIP distribution diagram corresponding to the optimal workload balance of Figure 7.11 is shown. Each color represents a recipe, the x-axis shows the machines of the toolset and the y-axis represents the production volume (for confidentiality reasons, the figures are removed).

Implantation Toolset

The application has also been used for the implantation toolset. The number of recipes and the constraints in this toolset are far less compared to the thermal treatment toolset. Besides, the implantation toolset consists of only ten machines of two different types. The implantation recipes are sorted under categories (named families). By creating new recipes or introducing new products, the new recipe is considered to be qualified on all machines already qualified for the recipe family to which it belongs. Therefore, the qualification proposals are useful when we intent to qualify a new recipe family on a machine type. The workload balance diagram is used to visualize the workload balance on different machines.

Conclusion and Perspectives

In this chapter, we showed how the research conducted in this part of the thesis is applied in industry. In particular, we discussed an important industrial constraint, namely not previously qualified recipes. Using modified flexibility measures, the impact of new products can be estimated on the variability and the configuration flexibility of the toolset. By omitting a machine from the toolset list and calculating the configuration flexibility, we can estimate to which extent the machine is important for the production of the current product-mix. This machine criticality can be calculated before planning preventive maintenance to study the impact of stopping a machine. Each toolset may have specific restrictions. Some of the most current in semiconductor manufacturing are auxiliary resource management, yield issues of running some recipes on some machine, consumable usage reduction, etc.

Chapter 8

Conclusions and Perspectives of

Qualification Management

Conclusions

In the first part of the thesis, we studied various facets of flexible qualification management in semiconductor industry and its relationship with capacity optimization and planning. Here we give a summary of the main achievements together with some conclusions.

Qualification Management

In Chapter 3, we discussed the importance of qualification management in semiconductor manufacturing industry. It enables a machine to process a new recipe. Qualifications in a toolset are like machine eligibility restrictions. Qualifications allow workload allocation of recipes to machines. Thus, qualifications are important in production management. Capacity planning tries to match the production plan with the available labor and equipment. Qualification management is one of the possibilities of increasing equipment capacity utilization and capacity planning. Our aim is to find the most appropriate recipe-to-machine qualification configuration. The qualification configuration changes over time, with recipe changes, disqualifications, machine breakdown, etc. Besides, a qualification configuration may be suitable for a certain product-mix and volume while with product-mix changes, the configuration may turn out to be poor. Performing each new qualification adds flexibility to the toolset qualification configuration. However, some new qualifications may not improve capacity utilization due to existing qualifications, low workload on the Chapter 8. Conclusions and Perspectives of Qualification Management newly qualified machines, low production volume of the newly qualified recipes, etc. Therefore qualification configuration flexibility must be increased intelligently.

Capacitated Qualification Management

The flexibility measures introduced in [START_REF] Johnzén | Flexibility Measures for Qualification Management in Wafer Fabs[END_REF] do not consider the unequal limited capacity of the equipment while qualification management. QM considering the toolset maximum capacity was discussed in Chapter 4. Limited equipment available capacity affects the optimal workload balance and hence QM. We have modified the uncapacitated flexibility measures to consider unequal and limited equipment capacity. However, in order to make sound judgments, we need other indicators which we have called Capacity Deviation Ratios. These two complementary measures are used for capacitated QM. Flexibility and deviation ratio measures increase the flexibility to fight against unbalanced workload, regardless if equipment overor under-utilization. The equipment capacity under-utilization or over-utilization have different impacts on the production line. While underloaded machines cause loss of productivity, overloaded machine increase the production line variability and endanger demand satisfaction. Therefore, we have introduced unequally penalized flexibility and capacity deviation measures to emphasize either equipment under-or over-utilization. Numerical experiments demonstrate that the equipment capacity is to be considered in QM to find the best qualification configuration.

Qualification Management with Batch Size Constraint

Batching is one of the characteristics of many processes. Similar products are grouped to form a batch. In batch machines, a batch is processed at a production run instead of a single product. Batching helps to reduce production and setup costs while yielding products of the same quality. Batching is preferable where long and expensive runs are needed. In Chapter 5, we studied the impact of batch sizes on qualification decisions and hence capacity utilization. In order to calculate the configuration flexibility of a toolset, we have solved a problem similar to the scheduling of unrelated nonidentical parallel machines with eligibility constraints. The problem is known to be NP-hard in strong sense. Several heuristics are proposed

Conclusions

to resolve the problems. Numerical experiments show that batch sizes must be considered in QM to optimize the toolset capacity utilization.

Qualification Management and Production Variability

Variability is known to be the enemy of production. It should be constantly monitored and reduced to get a smooth production flow. Equipment qualification, and capacity and batching are among sources of variability. In Chapter 6, we developed variability measures to show to which extent an efficient equipment qualification management strategy can reduce the variability. It was shown that too many qualifications tend to reduce less and less the workload variability. Qualification management helps to increase toolset capacity utilization where the variability is the highest and not necessarily where the equipment workload is the highest. Variability control helps to decrease inventory and in-line WIP while increasing production performance.

Industrialization

The concept is used in Soitec as a decision support system for qualification management and capacity planning. Some snapshots of the developed applications and the decision making process are presented in Chapter 7.

Flexibility and Cost

In Chapter 6, we showed that we may achieve full flexibility at very little cost by performing a minimum number of qualifications. This means that we can achieve "full flexibility", by not qualifying all recipes on all machines as illustrated in Figure 6.1c. In summary, by using the developed models in Soitec, flexibility has increased the capacity utilization without any capital investment in machine purchase. Meaning that small flexibility increase can bring about a huge amount of improvements in KPIs such as workload, overload and unused capacity (see Table 6.1).

Chapter 8. Conclusions and Perspectives of Qualification Management

Perspectives

At the end of each chapter, we have listed various perspectives. Here, we present some indications for future investigations.

Capacitated Time Flexibility Measure (F T ime Capa ) Considering the Required Qualification Duration

Setup times and costs are not always negligible. The required qualification time may consist of a considerable time portion relative to the period under study. Consider for instance a new qualification which takes about 35 hours in a study scope of one week i.e. 168 hours per week. For a toolset of 10 tools and assuming an equal uptime of 80% for each machine, this qualification represents about 2.6% of the total available time. Now consider another qualification which requires only about 10 hours, i.e. about 0.7% of the total time available in this toolset. The qualification time should be taken into account if the differences are significant in order to calculate the correct configuration flexibility, variability and capacity. Besides the qualification time, several other costs such as product inspection and incurred downtimes may also be considered.

Let us define the following parameters and variables:

Parameter cq r,m qualification duration of recipe r on machine m.

Variable OQ r,m     
1 if recipe r is to be qualified on machine m, 0 otherwise.

In the following, the Capacitated Time Flexibility measure is slightly modified to take this feature into account:

F T ime Capa = Ideal Ratio Capa M m=1 ( R r=1 ( W IPr,m T Pr,m + cq r,m OQ r,m ) Capa m ) γ ∈ (0, 1] . (8.1)

Perspectives

The constant Ideal Ratio Capa is the minimum value of the total production time C m plus the required qualification duration ( R r=1 cq r,m OQ r,m ) divided by Capa m when all qualifiable machines are "virtually" qualified :

Ideal Ratio Capa = min M m=1 ( R r=1 ( W IPr,m T Pr,m + cq r,m OQ r,m ) Capa m ) γ with Q r,m = 1 ∀r, m. (8.2)

Disqualification

The opposite action of qualification is called disqualification. When a recipe (previously qualified) is disqualified from a machine, it is no longer possible to allocate production volume of that recipe to a machine. A disqualification may occur automatically after hardware change, maintenance, etc. Maintaining qualifications may be costly. Therefore, in some cases, we may need to disqualify recipes such as production ramp-down. By disqualifying a recipe, we can evaluate the importance or criticality of each qualification. Qualification criticality is calculated the same way as qualification flexibility. With the difference that we now virtually disqualify recipes instead of qualifying them. First, the flexibility value of the current configuration is calculated. Then each (already) qualified recipe is virtually disqualified. After each virtual disqualification, the flexibility of the toolset configuration is saved in a matrix. The initial configuration flexibility is subtracted from each flexibility element of the matrix. The values will be nonpositive contrary to flexibility values which are nonnegative.

Maintaining Qualifications

In order not to perform a re-qualification, it would be interesting to minimize the number of machine disqualifications. The cost of re-qualification and maintaining a qualification must be considered over time. The problem becomes dynamic and more sophisticated. This subject is partially treated in [START_REF] Obeid | Ordonnancement et Contrôle Avancé des Procédés en Fabrication de Semiconducteurs[END_REF].

Further Industrial Deployment

Other industrial restrictions affect qualification management. Among these, we can name auxiliary resources, (perishable) consumables, material handling system restrictions, sequencing of the tasks, etc.

As an example of an interesting industrial and academic challenge, we describe some of the cleaning process constraints in SOI manufacturing. One of the recurrent process steps in semiconductor manufacturing is cleaning. The toolset is also very used in SOI fabrication process. The toolset is composed of several machines. Each machine has 10 tanks which are filled with liquid chemical materials, such as HF (Hydrofluoric acid). The robot can at most handle one lot (of 25 wafers) at a time. A stage corresponds to a production step in the SOI fabrication process. Depending on the product, a stage consists of different recipes. A recipe defines the different chemicals which must be used and the time the wafers are exposed to the chemical(s). During the execution of the recipes, wafers in the cassettes are dipped in the tanks for some minutes. Not all tanks must be visited for all recipes. The execution of a recipe in a stage may cause contamination in the consumable chemicals. Contamination levels are identified for each stage. Higher contamination levels add additional constraints to the problem. Chemicals in the tanks are consumables. Tank chemical must be changed after a certain amount of time a chemical has been used on wafers. Sometimes, it can be expressed based on the number of wafers (or lots) cleaned in the chemical(s). The batch constraint does not exist in a strong sense. However, wafers move from one tank to another in lots of 25 wafers.

Part II

Closed-Loop Production Planning: Lot-Sizing with Multiple Remanufacturing of Reusable By-Products

Chapter 9

Motivations and State-of-the-Art

In this introductory chapter, we provide the general framework of the second part of the thesis which addresses an original production planning problem. We begin the study by a brief description of the activities in a supply chain with a focus on production planning. The Smart-Cut Technology used for SOI manufacturing is a patented and exclusive production process. This unique manufacturing process motivates our study.

Introduction

The open market and globalization require a global strategy of the whole enterprise instead of local separated decisions. The context is more challenging in the semiconductor industry. One of the main elements of the supply chain of every manufacturing company is production management. A sound production management system assures the continuity of the enterprise and a stable and healthy turnover. Due to the hierarchical structure of a supply chain, production management is declined in several concepts. At the tactical level, production management concerns in particular capacity and production planning while at the operational level, it mainly concerns scheduling and dispatching. The concept of capacity planning was discussed in the Part I of this thesis. We also approached scheduling problems, especially when considering batches in Chapter 5. This part of the thesis is dedicated to production planning. The topic is also called lot sizing in the academic literature. The focus is on production planning modeling dedicated to SOI manufacturing based on Smart-Cut™ Technology. We discuss a lot sizing model considering SOI fabrication and refresh process lines.

First, the industrial context of the study is set by pointing out the importance of an efficient production planning in Section 9.2. Then, a picture of the concept of supply chain management is panted in Section 9.3 with a particular focus on production planning in Section 9.4. It is shown that the addressed problem has not been treated in the scientific literature by mentioning some related papers in Section 9.5. We conclude this introductory chapter in Section 9.6.

Motivations

The global SOI production process is explained in Section 2.3. In this section, this process is recalled while underlining the motivation of the study. The exclusivity of the production process is based on a patented thin layer deposition method called Smart-Cut™ Technology. In this method, two wafers are bonded together while a thin layer of oxide separates them. The bonded wafers are then split, leaving a thin silicon layer on the oxide. The remaining wafer can be recycled several

Motivations

times for the production of more SOI wafers. The recycling (called refreshing in the industrial jargon) needs another remanufacturing line. Due to confidentiality reasons, we cannot detail the refresh process steps. Just we note that less steps are required than in the SOI fabrication. However, constant monitoring of the yield and cycle times of the refresg line are necessary for smooth running of the SOI production line.

In the following, the extreme importance of the refresh process is pointed out.

The purchased Fresh Wafer is quite expensive. This is why the goal is to recycle it more and more. The high price of the Fresh Wafer is due to its quality. The quality of Wafer A (Top Wafer) must be excellent for two reasons: First, it constitutes the active layer of the SOI Wafer and second, it must keep its characteristics after several recycling processes. On the other hand, the quality of Wafer B (support, handle or Base Wafer) is not critical. On the other hand, the recycling (refresh) process is far less expensive than the purchase of new Top Wafers (Fresh Wafers). So, the refresh process helps reducing the costs.

The refresh process (especially the intern refresh process) gives the possibility of negotiating the purchase of the bulk wafers from the suppliers and better absorb the bulk market fluctuations. The purchase negotiation concerns not only the purchase price but also the purchase commitment term. The refresh process creates more independence from suppliers.

The internal refresh process is also important because the bulk suppliers prefer to use their capacity to produce Fresh Wafers (prime wafers) rather than for the refresh process. The Smart-Cut™ Technology allows Soitec to be the only company which uses Refresh Wafers as well as Fresh Wafers.

Besides, the worldwide wafer market is out of control but the internal refresh process remains under control of Soitec. Namely, Soitec can use its expertise to create more alternatives in the BOM and manage better its material requirements. By better mastering the refresh and SOI fabrication processes, more refresh levels can be qualified for the fabrication of SOI Wafers.

Using the internal refresh process, we increase the internal capability. As the refresh can also be subcontracted, the refresh process line can be used as a buffer zone for labor planning. Meaning that if the activity of the SOI production line

Chapter 9. Motivations and State-of-the-Art

increases, it is possible to decrease the activity of the refresh line for a short while by subcontracting the refresh process (or even buying more Fresh Wafers).

When purchasing Fresh Wafers, the purchase lead time may not be completely under control due to negotiations, ordering, processing and shipment procedures. When using the refresh process, not only the cycle time is decreased but we also better plan the supply chain. Securing the Top Wafer supply leads to a more reliable supply chain and a decrease of the supply chain management risk.

Supply Chain Management

A supply chain is a collection of at least two legally distinct organizations with the aim of satisfying the final customer demand by producing products or providing services [START_REF] Stadtler | Supply Chain Management and Advanced Planning[END_REF]. The present organizations in a supply chain are linked together through material, information and financial flows. Usually a supply chain consists of suppliers, manufacturers, distributors, retailers and customers (Figure 9.1). The suppliers provide raw material and components for the manufacturer. Note that, the suppliers may be broken down to several levels or "tiers" which collaborate between them and make another supply chain. However, most manufacturers procure raw materials and components from "Tier 1 Suppliers". In the same way, the other building blocks of a supply chain may contain other components.

Logistics is an integrating function which assures the forward and reverse product or service and information flow from the suppliers to the final customers. It is mostly divided into inbound and outbound logistics. While inbound logistics deals with the inbound movements (materials, components, information, etc.) to the business, outbound logistics refers to all flows between the business and the customers [START_REF] Coyle | Supply Chain Management: A Logistics Perspective[END_REF]. Figure 9.1 shows an example of the information, material (or service) and financial flows in a supply chain. The information flow starts from the customers by the demand for a product (or service), the delivery date, the quantity, etc. Customers procure their needs from retailers who keep a small stock of products. Distributors (national, regional, etc.) supply retailers. In some cases, notably via internet sales, distributors directly supply the end customer. Distributors transfer the requirements information to the manufacturers. The requirements, translated in terms of

Supply Chain Management

raw materials and component, are transferred to the suppliers. The material flow begin from suppliers and continue the value chain to reach the end customers. At each level, the financial flow exists from the end customer to the suppliers. However, these flows may be reverse for some other types of supply chain. For example, in a supply chain where product return and repair is also possible, the material flow is reversed from the customer towards the manufacturer. Or in case of reimbursement, the financial flow is also reversed towards the customer. Supply Chain Management (SCM) is a set of approaches used to efficiently integrate organizational units across a supply chain while coordinating materials, information and financial flows in order to minimize system wide costs while meeting service level requirements [START_REF] Simchi-Levi | Designing and Managing the Supply Chain[END_REF]. The aim of supply chain management is to improve global competitiveness by reducing costs (improving efficiency) and increasing customer service (improving effectiveness) [START_REF] Coyle | Supply Chain Management: A Logistics Perspective[END_REF]. While Manufacturing Planning and Control (MPC) tends to optimize the performance while coordinating information and materials flows inside the company (intrafirm), SCM focuses on coordinating and optimizing the interactions across companies (interfirm) [START_REF] Vollmann | Manufacturing Planning and Control for Supply Chain Management[END_REF].

The efficiency and effectiveness of a supply chain are measured using Key Performance Indicators (KPIs). Using SCM, we expect to improve KPIs. A list of KPIs and the expected improving trend can be found in [START_REF] Knolmayer | Systems, Supply Chain Management Based on SAP: Architecture and Planning Processes[END_REF], [START_REF] Frazelle | Supply Chain strategy: The Logistics of Supply Chain Management[END_REF] and [START_REF] Stadtler | Supply Chain Management and Advanced Planning[END_REF].

Decision Levels in Supply Chain Management

Decisions in Supply Chain Management have different importance and are updated periodically. Therefore, they are usually classified into three levels according to their importance and time horizon [START_REF] Anthony | Planning and Control: a Framework for Analysis[END_REF] and [START_REF] Silver | Inventory Management and Production Planning and Scheduling[END_REF].

Strategic Level

The strategic decisions, made at the highest management level, define long-term policies of the supply chain, namely for several years. They influence all decisions of lower levels. Typical examples of strategic decisions concern the design of the supply chain, new products, number and location of production facilities and warehouses and research and development projects.

Tactical Level Tactical decisions concern mid-term planning. The planning horizon is usually between one month and two years. It mostly determines rough and aggregates resource utilization such as budgeting, demand, production, distribution and sales planning.

Operational Level Operational decisions have short term effects, ranging from hours to weeks. They specify the most detailed and precise instructions for execution of tasks, such as scheduling, sales plan, inventory planning, distribution scheduling.

The decisions at different levels are linked and have mutual impact. Therefore, in order to obtain feasible decisions, the coherence between the decisions of different levels should be ensured. The production planning decisions discussed in Part II fall in the tactical level while the qualification decisions discussed in Part I have a more operational nature.

Closed-Loop Supply Chain

The problem tackled in this Part of the thesis has similarities to what is called a Closed-Loop Supply Chain (CLSC ). A CLSC considers the backward product flow (reverse supply chain) as well as the traditional forward product flow. The backward product flow focuses on the return of the product to the company from the customers or the market. A Closed-Loop Supply Chain aims at recovering the entire product

Supply Chain Management

or at least some of its modules or parts. Key additional elements in CLSCs are ( [START_REF] Guide | The Reverse Supply Chain[END_REF]):

Product acquisition is the task of retrieving the right quantity of used products at the right time and with a certain quality level directly from the customers or from the market.

Reverse logistics is the planning framework for the reverse flow of the collected products (and their related information) for product recovery (such as remanufacturing, recycling, etc.), see [START_REF] Dowlatshahi | A Strategic Framework for the Design and Implementation of Remanufacturing Operations in Reverse Logistics[END_REF] for an interesting discussion about reverse logistics.

Product disposition consists of sorting, testing and grading the returned products. Based on the result of this step, the returned product may be considered for some sort of recovery, resell or landfill.

Product recovery is the process of adding value to the selected used products. The recovery may be at the product, module or component level. Based on the recovery level and the required effort, the product recovery may be called remanufacturing, repair, part or module recovery, reconditioning, refurbishing or material recycling [START_REF] Guide | Production Planning and Control for Remanufacturing : A State-Of-The-Art Survey[END_REF].

Remarketing is the process of reintroducing the recovered products into the market. It concerns selling and redistribution.

CLSCs are important due to enormous potential savings, environmental regulations and strategic business issues related to procurement of raw materials.

Production System

A production or manufacturing system is a collection of technologies (patent, know-how, etc.), equipment and labor with the aim of satisfying the customer need by the end product. The production (or manufacturing) of goods is the process of adding value to raw materials or components through transformation or assembly.

The end products are composed of raw materials, components or subassemblies. A Bill of Materials or BOM is a hierarchical list of the needed quantities to manufacture the end product. A Single-Level BOM displays the end product with only one level of children (the so-called immediate raw materials, subassemblies or parts). However, if there are more parent-children relationship between the sub-assemblies, it concerns a Multi-Level BOM. Using the BOM, based on the demand for the end products (independent demand), we make a projection of the requirements, in terms of components (dependent demand). In a production system, the raw materials or components are processed or assembled in workcenters. A production (or manufacturing) line consists of a series of workcenters. A workcenter consists of people and equipment at shop floor level and performs one or several production operations. Each operation adds value to semi-finished products. The output of a workcenter is picked up by the subsequent workcenter. The sequence of production operations and the associated workcenters, production process (or recipe), equipment, required labor, setup and process time are defined in the production routing. The manufacturing time, setup time and waiting times constitutes the lead times of each operation. All the described elements of a production system are necessary for production and capacity planning.

Closed-Loop Manufacturing Systems

In order to classify our problem, we define a Closed-Loop Manufacturing System as a system in which by-products, coproducts and/or components can be returned to the manufacturing system after a process. The return process may concern a recovery activity such as remanufacturing, refurbishment, recycling, etc.

Production Environment

A production (or manufacturing) environment must be chosen based on the product or service which is offered by the firm and the company's strategy in how to respond to the market. The production environment type is important in supply chain planning, especially in demand and requirements planning. The production environment (or strategy) can be categorized as follows (Figure 9.2) ([100], [START_REF] Vollmann | Manufacturing Planning and Control for Supply Chain Management[END_REF] and [START_REF] Higgins | Manufacturing Planning and Control: Beyond MRP II[END_REF]). The Make-to-Stock (MTS) production environment offers standard products or services based on anticipation of demand. The demand of the customer is satisfied directly from products in stock. Therefore the level of service to the customer depends directly on the stock level. The crucial question is the replenishment policy of the stock. Higher customer service level requires high on-hand inventory or a good knowledge of future demand. Hence, a trade-off between the inventory level and customer service level must be determined. Accuracy in demand forecasting is essential for the efficiency of a Make-to-Stock production strategy. The number of products is limited with fairly long and predictable life cycles [START_REF] Higgins | Manufacturing Planning and Control: Beyond MRP II[END_REF].

Supply Chain Management

In the Make-to-Order (MTO) production environment, the production starts once a firm order of the customer is placed. The strategy is mostly suitable for customized products with relatively high inventory cost produced in relatively low volume. Using this policy, the excessive inventory holding expenses (common in Make-to-Stock strategy) decrease at the cost of increased waiting time for the customer to receive the product. Lead time for the final product can be reduced by procuring to forecast long lead time, highly used items with low degree of obsolescence [START_REF] Higgins | Manufacturing Planning and Control: Beyond MRP II[END_REF].

The Engineer-to-Order (ETO) production environment is similar to MTO with the difference that partial or the whole product design is according to the customer specifications. Therefore, a high level of customer participation in both design and manufacturing phases is required. This production policy is suitable for low volume complex or one-off (one-of-a-kind) products. The engineered-to-order products are expensive and time-consuming to manufacture. The raw materials or basic components are often difficult to source or to design. On the other hand,

Chapter 9. Motivations and State-of-the-Art

very low inventory is carried. The production time and the required manufacturing capacity are difficult to estimate. High manufacturing flexibility is also required.

In the Assemble-to-Order (ATO) production environment, the product is manufactured according to the customer specifications from standard modules on-hand. ATO is a hybrid strategy between Make-to-Stock and Make-to-Order strategies. The modules (or components) are kept in inventory (MTS) while the final assembly occurs after the customer order placement (MTO). Therefore, ATO requires a modular structure of the final product. The lead time between the order receipt and product delivery is lower in comparison to the MTO strategy. Besides, the products are more customizable compared to the MTS strategy and the on-hand module inventory is lower.

The production environment depends upon the customer requirements, final product design, product life cycle, demand behavior and market behavior, production facilities, production cost, inventory management and production cycle time. Different production policies may be applied for different products in the same company. A combination of these policies is also possible. A vital concept in planning is the customer order decoupling point which defines which material is forecast-driven or customer order-driven. Figure 9.2 shows that as we move more towards ETO policy, the production is sales-driven and on the inverse direction, manufacturingdriven.

The production environment of Soitec is located in the range of Engineer-to-Order or Make-to-Order production environments.

Remanufacturing

Until now, our discussion was centered on manufacturing. However, an emerging and promising field in both research and industry considers product recovery. Product recovery aims at restoring products while eliminating waste to a large degree. Some of the product recovery options are depicted in Figure 9.3. However, the options may go beyond this framework. Several recovery options may be applied for different components of the same product. More and more, product recovery is considered in the design phase of the products. The following concepts are defined by [START_REF] Guide | Building Contingency Planning for Closed-Loop Supply Chains with Product Recovery[END_REF]: Remanufacture-to-Stock (RMTS), Reassemble-to-Order (RATO), and Remanufacture-to-Order (RMTO) based on the classification proposed in [START_REF] Wheelwright | Link Manufacturing Process and Product Life Cycles[END_REF].

Supply Chain Management

In our research, we consider a specific type of by-product recovery. The industrial jargon of Soitec for this type of recovery is refreshing. However, to avoid confusion and due to similarities of this type of recovery to remanufacturing, the latter term is used.

Types of Production Systems

Production is the main element of the value chain in a supply chain. In order to determine an adapted production management policy, the production systems are classified into two main categories: Continuous and Intermittent Production Systems (Figure 9.4).

Continuous Production System

In a continuous production system, the inputs, outputs, processes and sequences are standard. It follows a Make-to-Stock policy based on anticipation of demand. The factory size is normally large with highly specialized automated machinery and material handling systems. The material requirement is very predictable. The production is without interruption and at a high rate. The production process generates usually by-products. The bottlenecks are stationary and known [START_REF] Silver | Inventory Management and Production Planning and Scheduling[END_REF]. Therefore, production planning and control is easy. It can be categorized into Mass Production Flows and Process Production Flows.

Supply Chain Management

cialized machinery production or prototype building. In Assembly Production Flows, interchangeable parts and modules are assembled together sequentially to make the final product. The products are customizable to some extent as the parts or modules are standardized. Lot (or Batch) Production Flows allows the production of similar products in production campaigns, in other words in lots (or batches). A lot (or batch) is a group of identical products which follow the same routing and production operations. Organizing big production campaigns (lots) generates a lot of stock while small size production campaigns must be more frequent and require more setups. The production policy is mostly Make-to-Stock or Make-to-Order depending on the firmness of the orders or demand predictions. A production system can be a combination of these production systems. Lot Production Flows and Assembly Production Flows can be combined together as a single production system.

Production planning of a Lot Production System is referred to as a lot-sizing problem. Generally speaking, a lot-sizing problem determines the size of the production campaigns of each product over a planning horizon (defined in days, weeks or months) in order to satisfy the customer demand. The objective is mostly expressed as the minimization of production setup and inventory costs while meeting demand and capacity restrictions. Of course, additional constraints may be considered in a lot-sizing optimization problem.

The SOI fabrication and Refresh process, considered in this thesis, are classified under intermittent production system with lot production flows. We discuss this lot-sizing problem in more details in Section 9.4.

Manufacturing Configuration

A manufacturing system is composed of human and machine resources. Until leaving the production line, the product visits different machines according to operations routing. One or several machines may be grouped in a workcenter. A workcenter (Aka workstation or toolset) adds value to semi-final products by performing operations. The sequence of visiting the machines inside a workcenter or between the workcenters defines the manufacturing routing. The routing between the workcenters (or the machines) can be seen as the workshop configuration. The

Chapter 9. Motivations and State-of-the-Art

scheduling and capacity planning depends upon the workshop and workcenter configurations. In the following classifications, we distinguish resource characteristics and routing characteristics. None of these classifications are necessarily reserved to a workcenter nor to a workshop. The operations routing (sequence) in a workshop (or a workcenter with parallel machines) may be classified to Flow Shop, Job Shop or Open Shop. In the open shop configuration, no special routing and sequence are defined for the operations. This type of workshop is rarely found in industry. In flow shop configuration, all products visit the machines (or workcenters) in the same order. The flow shop configuration may exist in both continuous and intermittent manufacturing environment. Examples are cement industry (continuous manufacturing environment) and automobile industry (intermittent manufacturing environment). In the job shop configuration, each product (or product family) has its own routing and sequencing on the machines or workcenters. The job shop configuration can be found in semiconductor manufacturing or pharmaceutical industry.

Resource Characteristics

The workstations in SOI fabrication line consist of unrelated parallel machines with a job shop configuration. The refresh process line has a flow shop configuration with several workcenters with identical machines. The machines in the workcenters are multi-purpose and mostly configurable. Multi-purpose or flexible machines (in comparison to single-purpose machines) are able to perform several jobs. A multipurpose machine may be configurable or non-configurable. As the name suggests, configurable multi-purpose machines can be configured to perform jobs that they could not perform before configuration. However, the configuration degree and

Supply Chain Management

possibilities are not the same.

Production Planning

In supply chain planning, production planning concerns the tactical decision answering the questions when to produce which product and how much (or how many). The production planning problem is expressed as an optimization problem with the objective of minimizing the cost (or maximizing the profit) subject to several constraints. With a cost minimization objective, the challenge is to find the best trade-off between opposing costs such as inventory cost -which depends upon production quantity-and setup costs. However, depending on the problem, several other cost items may be considered, e.g. material procurement costs, production cost, maintenance cost, labor cost, backlog or lost sales costs, overtime costs, etc. The main constraint of a production planning problem is customer demand satisfaction. Additional constraints concern safety stock, capacity constraints (in terms of human or production resources), market share, etc.

In the following, we discuss some concepts of supply chain planning, with more focus on production planning.

Enterprise Resource Planning (ERP)

An enterprise can be divided into major functional areas: Finance, manufacturing and logistics, sales and marketing and human resources. Enterprise Resource Planning or ERP systems provide a cross-functional integration bed for the business activities of an organization. In other words, ERPs are comprehensive information systems which support integrated planning and execution among all functions of an organization. Manufacturing, inventory management and logistics are among the major modules in ERP systems. Production planning and scheduling is the core of manufacturing and logistics function. One of the basic tools used in production planning in manufacturing systems with lot production flows is Material Requirements Planning or simply MRP.

Chapter 9. Motivations and State-of-the-Art

Material Requirements Planning (MRP) Intermittent production systems with lot production flows have been evolved with basic material planning tools such as Material Requirements Planning (or MRP). MRP has the objective of determining the right quantities of the right part at the right time [START_REF] Vollmann | Manufacturing Planning and Control for Supply Chain Management[END_REF]. The major drawback of the MRP approach is the lack of capacity checks from other departments, making the plans hardly feasible operationally. To rectify the production plans of MRP, Manufacturing Resource Planning (or MRP-II ) has been created. Closed-loop MRP is an enhancement of MRP systems in which capacity checks are carried out in an iterative process.

Manufacturing Resource Planning (MRP-II)

MRP II is a planning method for medium-term manufacturing resource planning and control. MRP II explicitly considers the capacities of resources while interacting more with other departments by converting some outputs of production planning and control. MRP II can be seen as a closed-loop MRP while considering financial flows in the enterprise. The decision and data consistency are more precise in MRP II. MRP II has given birth to ERP systems.

Advanced Planning System (APS)

Advanced Planning Systems have been created to optimize and synchronize the whole set of activities in a supply chain in a collaborative and global way. Based on the service level and financial objectives, the activities are optimized from demand planning until final delivery at all decision levels and in line with all interfaces. APS solutions consist of several software modules, in which each module covers a planning scope in the supply chain. The supply chain planning matrix depicted in Figure 9.5 shows the possible modules in an APS. However, the cutting and the scope of each module may be different from one software provider to another. APS take advantage of the integrated and consistent ERP bed across the enterprise. Therefore APS supplement the ERP system used as a transaction and execution system but do not substitute it [START_REF] Stadtler | Supply Chain Management and Advanced Planning[END_REF]. In this thesis, we consider a supply chain in which there is also a reverse material flow from the manufacturer to the supplier or between two distinct production lines of the same manufacturer. The concept is similar to closed-loop supply chain in which there is a possible return material flow from the customer to the manufacturer for remanufacturing. Nevertheless, it is not the final product which is returned to the remanufacturing process but the used raw material (called by-product).

Production Planning

Production Planning

In this part, we are interested in production planning. The term mostly used in industry and academia is lot-sizing.

Optimization Problems

Models are used to represent a given real-world problem. [START_REF] Bradley | Applied Mathematical Programming[END_REF] defines four classes of models: Operational exercise, gaming, simulation and analytical model. In this thesis, we consider analytical models in which the model is entirely represented in mathematical terms bringing a high degree of simplification. Analytical models give birth to optimization problems in which a function (called objective function) is typically minimized or maximized over a set of points, called feasible region. Production

Chapter 9. Motivations and State-of-the-Art

planning problems involve a sequence of decisions over multiple-periods. Industrial requirements generate a great number of variables and constraints complicating more the resolution approach.

Lot-Sizing Problems

We mentioned in Section 9.3.3.3 that the lot (or batch) production flow is a strategy used in intermittent production systems. This strategy consists of organizing production (or purchasing) campaigns (called "lots") per period (normally on a daily or weekly basis) over a planning horizon. By organizing production by lot, we may anticipate the demand satisfaction of several periods in order to incur less (fixed) production setup costs. However, by anticipating production, we pay more inventory costs. That is why a lot-sizing problem usually consists of finding the best trade-off between setup and inventory costs. However, the trade-off may lie among setup, production or ordering costs and inventory costs.

A detailed examination of lot-sizing problems is out of the scope of this dissertation. Moreover, in many scientific documents and textbooks, detailed state-of-theart discussions about lot-sizing problems are presented (see [START_REF] Pochet | Production Planning by Mixed Integer Programming[END_REF]). So, we just look over some main ideas to determine where we stand.

A special category of lot-sizing problems deals with manufacturing processes with a potential return of material flow. In order to compare our case with cases studied in literature, we open the discussion about Closed-Loop Lot-Sizing later in Section 10.6.3.

Classification of Lot-Sizing Problems

The history of lot-sizing dates back to 1913 ( [START_REF] Erlenkotter | Ford Whitman Harris and the Economic Order Quantity Model[END_REF]), by the Economic Order Quantity (EOQ) (or Economic Lot Size Model) introduced in [START_REF] Harris | Operations and Cost. Factory management series[END_REF]. EOQ and all its extensions consider a constant and deterministic demand rate over an infinite period. However, in most practical cases, the average demand rate varies over time and the planning horizon is limited. While the EOQ model recommends the same replenishment quantities for all periods, this would not be optimal with time-varying demand. The problems addressing time-varying demand are termed as lot-sizing problems.

Production Planning

The first basic lot-sizing model was analyzed in [START_REF] Wagner | Dynamic Version of the Economic Lot Size Model[END_REF]. The proposed dynamic programming algorithm is named after the authors, Wagner and Whitin (hereafter WW). With very low demand rate variations, it would make sense to use the EOQ model. If not, exact solution methods can be used to obtain the optimal solution. However, lot-sizing problems are mostly difficult to solve, therefore heuristics are developed to get approximate solutions. Lot-sizing problems can be classified based on different characteristics, such as number of levels in BOM, length of the periods, demand behavior, production capacity and so on.

For a classification of lot-sizing problems for single-level and multi-level BOM (Figure 9.6), see [START_REF] Wolosewicz | Approche Intégrée en Planification et Ordonnancement de la Production[END_REF]. In single-level lot-sizing problems, the demand of each product is independent of other products. While in multi-level lot-sizing, the problem becomes more complicated because of the parent-children relationship between products at different levels. In lot-sizing, the time scale may be discrete or continuous. The planning horizon is generally considered to be either finite or infinite. Mostly,

Single Level

Multi

Chapter 9. Motivations and State-of-the-Art

problems with discrete time periods have a finite planning horizon while continuous time problems have an infinite planning horizon. The EOQ model is namely a continuous time model with infinite planning horizon. From the resource point of view, a problem may be capacitated or uncapacitated. In real-world problems, resources have limited capacity. However, as capacitated lot-sizing problems are considerably more difficult to solve, for simplicity, the resource capacity restrictions may be ignored. A literature review of the models and the algorithms for uncapacitated and capacitated single level lot sizing problems can be found in [START_REF] Karimi | The Capacitated Lot Sizing Problem: A Review of Models and Algorithms[END_REF]. Capacitated lot sizing problems with deterministic demand considering at each time one of the following extensions: Back-order, setup carryover, sequencing and parallel machines are discussed in [START_REF] Quadt | Capacitated Lot-Sizing with Extensions: A Review[END_REF]. For a classification and review of solution algorithms for capacitated lot-sizing problems, see [START_REF] Buschkühl | Dynamic Capacitated Lot-Sizing Problems: a Classification and Review of Solution Approaches[END_REF].

Based on the number of items, lot-sizing problems might be single-item or multiitem. In single-item models, the items (products) do not compete for resource capacity utilization. In multi-item models, because of the interdependency between products, they compete to use (scarce) capacity. Both EOQ and WW are single-item models. For a literature review of single-item lot-sizing problems, see [START_REF] Brahimi | Single Item Lot Sizing Problems[END_REF].

The discrete time lot-sizing problems are classified into extra big time bucket, big time bucket and small time bucket depending on the period length. The planning periods length of extra big time bucket are of the order of months, for big time buckets are about a few days or weeks and for small time bucket problems are about a few hours. Longer time buckets can be considered as aggregations of smaller time buckets. In this case, we talk about hierarchical lot-sizing (see [START_REF] Bitran | Hierarchical Production Planning[END_REF]).

Interested readers can find a concise discussion and state-of-the-art about supply chain management with a focus on production planning in [START_REF] Gomez Urrutia | Optimisation Intégrée des Décisions en Planification et Ordonnancement dans une Chaine Logistique[END_REF] (in French) and [START_REF] Brahimi | Production Planning : New Lot-Sizing Models and Algorithms[END_REF] (in English).

In this part of the thesis, we treat big time bucket discrete lot-sizing problems. The industrial problem considered in Chapter 10 is multi-item and capacitated. In Chapter 11, we consider a single-item uncapacitated problem. The problems are bi-level; with one level of raw materials (level 1) and one level of end product (level 0).

Literature Review

Literature Review

The problem studied in this part is related to different domains of lot-sizing. Some related studies are recalled to distinguish our research from previous investigations.

Our study concerns a multi-item supply chain. The classical capacitated multiitem lot-sizing problem with non-stationary costs, demands and setup times is considered in [START_REF] Trigeiro | Capacitated Lot Sizing with Setup Times[END_REF]. The problem is decomposed into a set of uncapacitated single product lot-sizing problems using a Lagrangian relaxation. The single-item problems are solved using a dynamic programming algorithm. A smoothing heuristic is used to make the dual solution feasible.

If there is a parent-component relationship in the item structure, the problem is classified as a multi-level lot-sizing problem. In single-level problems, only independent demands (from external customers) are considered while, in multi-level problems, the production of each final item generates a dependent demand for its components. The problem studied in this thesis is a bi-level lot-sizing problem. The problem of minimizing the setup and inventory costs in a capacitated multi-level lot-sizing problem is discussed in [START_REF] Billington | Mathematical Programming Approaches to Capacity-Constrained MRP Systems: Review, Formulation and Problem Reduction[END_REF]. An interesting literature review on multi-level capacitated lot-sizing problems together with a solution approach for the dynamic multi-level capacitated lot-sizing problem are discussed in [START_REF] Helber | A Fix-and-Optimize Approach for the Multi-Level Capacitated Lot Sizing Problem[END_REF].

In the literature, the difference between by-product (by-production) and coproduct (co-production) is not clear. Both co-products and by-products are generated during the production process or at each production run. Co-products have their own independent demand and cannot be used interchangeably. Whereas byproducts are undesired "sub-products" generated during production. They must normally be reworked before being able to reuse them.

Terms such as "by-product", "co-product", "remanufacturing" and "recycling", used to designate different concepts in the literature, are close to our research. In our research, the raw material once used for production is considered as "byproduct". This by-product cannot fulfill any demand and must be reworked before coming back to the manufacturing cycle. The process of restoring the generated by-products makes them reusable again as raw materials. Therefore, this process Chapter 9. Motivations and State-of-the-Art can be considered as a "remanufacturing process". In the literature, remanufacturing concerns the returned products from the customers whereas in our case the customer is not involved.

A multi-item uncapacitated lot-sizing problem in which co-products are produced at each production run is treated in [START_REF] Agrali | A Dynamic Uncapacitated Lot-Sizing Problem with Co-Production[END_REF]. In this paper, it is considered that the coproducts have their own demand and cannot fulfill the demand of the main product. Several MIP formulations are presented for the problem. Using a variant of the zero-inventory property, a dynamic program is used to solve the problem in the single-item case.

Other studies consider the co-production of a range of products with different performances in a single production run. The co-products are then sorted according to their key performance to satisfy demands of each co-product [START_REF] Tomlin | Pricing and Operational Recourse in Coproduction Systems[END_REF].

Remanufacturing in reverse logistics is considered in [START_REF] Retel Helmrich | Economic Lot-Sizing with Remanufacturing: Complexity and Efficient Formulations[END_REF]. By remanufacturing in reverse logistics, it is meant that there is not only a one-way flow of the products to the customers but materials and products may be returned to the manufacturer for product recovery. This is what is called a "closed-loop supply chain" (see Section 9.3.2). In the proposed model, known quantities of used products are returned from customers in each period. Once reworked, the returned products are used to satisfy the customer demand as new products. Therefore, in each period, it is determined whether to produce new products or to remanufacture returned products to satisfy the demand. The supply chain is modeled using Mixed-Integer Programming (MIP). The studied problem is shown to be NP-hard. Several alternative reformulations of the original formulation are presented. The solution qualities of the LP relaxations are compared.

In a more recent study, a closed-loop supply chain with setup costs, product returns and remanufacturing is considered in [START_REF] Zhang | A Lagrangian Relaxation Based Approach for the Capacitated Lot Sizing Problem in Closed-Loop Supply Chain[END_REF]. The study is inspired from the paper manufacturing industry in which both virgin and deinked pulps are used to make papers. A MIP model and a Lagrangian relaxation based solution approach are further proposed. A manufacturing -remanufacturing closed-loop supply chain in a dynamic continuous time stochastic context is studied in [START_REF] Kenné | Production Planning of a Hybrid Manufacturing-Remanufacturing System under Uncertainty within a Closed-Loop Supply Chain[END_REF].

Spengler et al. [START_REF] Spengler | Environmental Integrated Production and Recycling Management[END_REF] propose production planning models for recycling generated by-products during production, and dismantling and recycling products at the end

Conclusion

of their lifetime. Several other studies use the term remanufacturing to denote the restoring or recycling of products [START_REF] Ferrer | Material Planning for a Remanufacturing Facility[END_REF].

None of the cited articles treat all aspects of our research. The problem modeling is original and to the best of our knowledge, it is not addressed in the literature. In the following chapters, the problem is unfolded in more detail and the notations are introduced.

Conclusion

In this chapter, an overview of supply chain management and production systems was presented to set the stage for the forthcoming chapters.

We are interested in production planning of two related production lines of Soitec in which the by-products generated in one production line is reworked in the second to come back to the first production line. The rework is considered as a kind of remanufacturing. That is why the production system is similar to closed-loop manufacturing systems. The production environment follow an Engineer-to-Order or Make-to-Order policy. The whole production system is intermittent with a batch production flow. The workcenters (toolsets) of the main production line consist of multi-purpose unrelated configurable parallel machines, with toolsets in a job shop configuration. The remanufacturing process line has in turn workcenters with mostly identical machines with a flow shop configuration. The production planning is done on a finite discrete time horizon with big time buckets. The mathematical models are large scale and multi-period with constrained solution space, linear functions, deterministic parameters, continuous and binary variables.

In Chapter 10, we discuss the original industrial problem with all industrial constraints. The industrial problem gives birth to an original academic problem covered in Chapter 11.

Chapter 10

Multi-Item Bi-Level Capacitated Lot-Sizing with Multiple Remanufacturing of Reusable By-Products

In this chapter, we investigate a multi-item production planning problem in which reusable by-products are generated during final product fabrication. After further processing, the generated by-products can be reused as raw materials. However, by-products can be "recycled" only a given number of times. The production and recycling processes are performed in internal and external sites with limited capacity. Each product may be produced using specific raw material (newly purchased or recycled) references. The proposed model represents a part of the supply chain of "SOI" (Silicon-On-Insulator) fabrication units. Using numerical examples based on industrial data, the model is validated and some of its characteristics are discussed. Finally, some perspectives of this work are proposed.

Introduction

Silicon wafers are extensively used in semiconductor manufacturing to produce microelectronic components such as chips and integrated circuits. However, some devices require higher performance which cannot be delivered by traditional silicononly wafers. Components built on Silicon-On-Insulator (SOI) wafers offer much more performance while consuming less energy compared to components on silicononly wafers. SOI wafers may be produced using different technologies: SIMOX™ (Separation by IMplantation of OXygen), wafer bonding or Seed methods. The production system studied in this chapter concerns a type of wafer bonding technology called the Smart-Cut™ Technology described in Section 2.3.

The chapter is organized as follows. The main aspects of the Smart-Cut™ Technology and the studied supply chain are concisely described in Section 10.2. The mathematical model corresponding to the supply chain under study is introduced in Section 10.3. The complexity analysis of the mathematical model is to be found in 10.4. Some industrial extensions and additional constraints are presented in 10.5. Using numerical experiments on test instances generated based on industrial data, the model is validated and some managerial insights are discussed in Section 10.6. Finally, conclusions are drawn and some perspectives of the study are presented in Section 10.7.

Problem Definition

In this study, we consider the supply chain of a Silicon-on-Insulator (SOI) Wafer production unit using the Smart-Cut™ Technology. In SOI Wafers, a thin layer of silicon is laid on a silicon Wafer which serves only as a physical support (or handle). These two silicon layers are separated by an insulator: The oxide. Figure 10.1 illustrates the main steps of the Smart-Cut™ Technology. Once Wafer A is oxidized and implanted, it is ready to be bonded with Wafer B. After the Wafers are bonded, they are split to form the SOI Wafer. Wafer A is the "donor" Wafer in the sense that a thin silicon layer of this substrate is deposited on Wafer B. In the industrial jargon, Wafer A is called "Top" while Wafer B is called "Base". As

Problem Definition

only a thin layer of the Top Wafer is deposited on the Base Wafer, it is possible to reuse the Top Wafer several times to produce other SOI Wafers. This is one of the main advantages of the Smart-Cut™ Technology which makes the process cost competitive.

A "used Top Wafer", called "Negative Wafer", must be reworked before returning to the SOI fabrication process. This remanufacturing process is called the refresh process or shortly refresh. In industrial terminology, a new Top Wafer used for the SOI fabrication is called a "Fresh Wafer". A Fresh Wafer is purchased from silicon Wafer suppliers. After the first utilization, the generated Negative of the Fresh Wafer is called "Negative 0" or shortly "Neg 0". Refreshing "Negative 0" gives a newly usable Top Wafer called "Refresh Wafer 1". A Wafer may be refreshed only a maximum number of times (called the "maximum refresh level"). It is economically interesting to refresh a Top Wafer as many times as possible. However there is an end to the refresh process because of quality and yield constraints. Mostly, mature products have the highest refresh level because of a better understanding of the characteristics of the product and a higher expertise of the refresh process. Using this logic of numbering, a Fresh Wafer can also be called "Refresh Wafer 0". The Top Wafer can be refreshed only a limited number of times. Performing too many refresh processes leaves a deteriorating impact on the final SOI [START_REF] Robert | Impact des Recyclages Successifs du Substrat Donneur sur la Qualité Finale du SOI[END_REF]. Multiple thermal treatment processes cause defectivity in the Wafers and makes the silicon transfer more and more difficult. The defectivity in the refresh process must be the least in order to keep high quality for future SOI Wafers. Hence, continuous defect inspection monitoring of the refresh process line is important. Special yield and quality constraints or specific customer specifications may cause the SOI-Refresh planning to be more complicated. Some products may only use a Fresh Wafer as a Top Wafer and not its Refresh Wafers. Or some customers may require to use only up to a certain refresh level for their products. A rare situation which may also occur is that some products can only be produced from the Refresh Wafers of a Fresh Wafer. In this case, in order to obtain the required Refresh Wafers, Fresh Wafers are first used to produce another product. Then the generated Negative Wafers are refreshed and used further.

The refresh process, of limited capacity, can be done internally or externally.

Mathematical Model

The objective of our model is to decide when and how much to produce final products (SOI), when and how much to purchase raw materials (Base Wafer and Fresh Wafer), when and how much to refresh used Top Wafers in order to satisfy demand. The demand satisfaction is done over a discrete time horizon while minimizing the total cost (production, purchase, refresh and inventory costs). The plan must satisfy inventory, bill of materials as well as capacity constraints. The parameters, the decisions variables and the mathematical model are presented below.

The production planning model corresponding to the studied supply chain is The goal is to determine the optimal raw material procurement, refreshing and production policies as well as the associated inventory levels. Yield of the refresh process for Top Wafer f at site m,

Parameters

Mathematical Model

a i,b     
1 if Base Wafer b can be used in product i, 0 otherwise.

a i,f     
1 if Top f (Fresh or Refresh) Wafer can be used in product i, 1 if production occurs in period t, 0 otherwise.

0 otherwise. b f ,f            1 if Top f (Refresh)
V B t      1 if Base Wafer procurement occurs in period t, 0 otherwise. V F t      1 if Fresh Wafer procurement occurs in period t, 0 otherwise. W m,t     
1 if refresh process is performed in site m in period t, 0 otherwise.

Mathematical Model

Mathematical model

min ∀f |n f =0 ∀t cp f,t F f,t + ∀b ∀t cp b,t B b,t + ∀m ∀f |n f >0 ∀f ∀t cr f,m,t R f,f ,m,t + ∀i ∀t cg i,t G i,t + ∀t sp B t V B t + ∀t sp F t V F t + ∀m ∀t sr m,t W m,t + ∀t sg t Y t + ∀b ∀t h b,t S b,t + ∀f ∀t h + f,t S + f,t + ∀f |n f =n max f ∀t h - f,t S - f,t + ∀i ∀t h i,t S i,t (10.1)
subject to

S i,t-1 + G i,t = d i,t + S i,t ∀i, t (10.2) ∀b a i,b X B i,b,t = G i,t ∀i, t (10.3) ∀f a i,f X F i,f,t = G i,t ∀i, t (10.4) Bb,t + S b,t-1 + B b,t - ∀i X B i,b,t = S b,t ∀b, t (10.5) Ff,t + S + f,t-1 + F f,t - ∀i X F i,f,t = S + f,t ∀f |n f = 0, t (10.6) Rf,t + S + f,t-1 + ∀m ∀f |b f,f =1 α f,m R f,f ,m,t-l - ∀i X F i,f,t = S + f,t ∀f |n f = 0, t (10.7) S - f,t-1 + ∀i X F i,f,t - ∀m ∀f b f,f R f,f ,m,t = S - f,t ∀f |n f = n max f ,
∀f |n f =0 ∀f β f ,m R f,f ,m,t ≤ rc m,t ∀m, t (10.10) ∀i G i,t ≤ M • Y t ∀t (10.11) ∀b B b,t ≤ M • V B t ∀t (10.12) ∀f |n f =0 F f,t ≤ M • V F t ∀t (10.13) ∀f ∀f |n f >0 R f ,f,m,t ≤ M • W m,t ∀m, t (10.14) R f,f ,m,t = 0 ∀m, t, f, f |b f,f = 0 (10.15) R f,f ,m,T = 0 ∀m, f, f (10.16) Y t , V B t , V F t ∈ {0, 1} ∀t (10.17)
W m,t ∈ {0, 1} ∀m, t (10.18)
G i,t ≥ 0 ∀i, t (10.19)
S b,t , B b,t ≥ 0 ∀b, t (10.20)
S + f,t , S - f,t , S i,t , F f,t ≥ 0 ∀f, t (10.21) R f,f ,m,t ≥ 0 ∀f, f , m, t (10.22) X B i,b,t ≥ 0 ∀i, b, t (10.23) X F i,f,t ≥ 0 ∀i, f, t (10.24)
The objective function (10.1) minimizes the total cost which is the sum of the purchase cost of new Top Wafers (Fresh Wafers) as well as Base Wafers, refresh cost at all sites, final products (SOI Waferss) production cost, raw material (Bulk, i.e. Fresh and Base Wafers) procurement cost, refresh setup cost, production setup cost, and inventory costs of Base Wafers, Top Wafers (either Fresh or Refresh Wafers), generated Negative Wafers and final products (SOI Wafers).

Constraint (10.2) models the flow conservation of finished goods. Constraints (10.3) and (10.4) respectively determine the amount of Base Wafers and Top Wafers (either Fresh or Refresh Wafers) which are used to satisfy the production plan of

Chapter 10. Multi-Item Bi-Level Capacitated Lot-Sizing with Multiple

Remanufacturing of Reusable By-Products

Complexity Analysis

An instance of our problem is NP-hard without considering its bi-level nature and all refresh constraints, only because resource capacities are time-dependent. In fact, [START_REF] Florian | Deterministic Production Planning: Algorithms and Complexity[END_REF] and [START_REF] Bitran | Computational Complexity of the Capacitated Lot Size Problem[END_REF] demonstrate that the classical capacitated single-item problem is NP-hard in the weak sense.

Moreover, in Section 11.3.1, we show that the problem in the mono-level and single-item case (i.e. only one raw material with its refresh levels) without capacity constraints is still NP-hard.

Industrial Extensions

Here, we address some industrial specificities and restrictions in the studied SOI supply chain. Some of them are already treated in the model. Otherwise, the necessary constraints are presented.

• Market share rules. The procurement department negotiates based on long-term agreement the bulk consumption demand from a supplier. These agreements define the "market share" rules which aim at increasing material procurement reliability, decreasing lead-time variability and guaranteeing a better acquisition price. The market share rules define that the bulk purchased from a supplier must at least be x percent of the total bulk consumption. While planning, it is desired that the market share is considered and, in case of violation, that the gap is minimized.

The • character represents F and B.

Industrial Extensions

Parameters Q Total number of suppliers,

a q,b     
1 if Base Wafer b can be purchased from supplier q, 0 otherwise.

a q,f     
1 if Fresh Wafer f can be purchased from supplier q, 0 otherwise.

U B • q,t
Market share upper bound for supplier q in period t

( Q q=1 U B • q,t ≥ 1 ∀t), LB • q,t
Market share lower bound for supplier q in period t

( Q q=1 LB • q,t ≤ 1 ∀t). Variables B q,b,t
Ordered quantity of Base Wafer b to supplier q in period t, F q,f,t

Ordered quantity of Fresh Wafer f (f |n f = 0) to supplier q in period t.

If market share is considered for Top and Base Wafers separately.

Q q=1 a q,b B q,b,t = B b,t ∀b, t (10.25a) B b=1 LB B q,t B b,t ≤ B b=1 a q,b B q,b,t ≤ B b=1 U B B q,t B b,t ∀t, q (10.25b) Q q=1 a q,f F q,f,t = F f,t ∀f, t (10.26a) F f =1 LB F q,t F f,t ≤ F f =1 a q,f F q,f,t ≤ F f =1 U B F q,t F f,t ∀t, q (10.26b)

Chapter 10. Multi-Item Bi-Level Capacitated Lot-Sizing with Multiple Remanufacturing of Reusable By-Products

If market share is considered at the same time for Top and Base Wafers.

Q q=1 a q,b F q,b,t = B b,t ∀b, t (10.27a) Q q=1 a q,f F q,f,t = F f,t ∀f, t (10.27b) LB q,t ( F f =1 F f,t + B b=1 B b,t ) ≤ F f =1 a q,f F q,f,t + B b=1 a q,b B q,b,t ∀t, q (10.27c) F f =1 a q,f F q,f,t + B b=1 a q,b B q,b,t ≤ U B q,t ( F f =1 F f,t + B b=1
B b,t ) ∀t, q (10.27d)

• Purchase-refresh constraint Some bulk suppliers also propose the refresh process (external refresh). The suppliers do not agree that their competitors manipulate their wafers. Therefore, Fresh Wafers bought from one specific supplier may only be refreshed internally or by the same supplier; and not by one of its competitors.

We add the index q to the refresh variable to consider the origin of the Top Wafer. We also need a matrix which defines the relationship between suppliers and production or refresh sites.

Parameters

a q,m           
1 if supplier q and production or refresh site m are compatible, 0 otherwise.

Variables R f,f ,q,m,t
Refreshed quantity f (Refresh Wafer) obtained from Negative f purchased originally from supplier q at site m in period t.

In constraints (10.7) and (10.8) as well as the objective function (10.1), the terms R f,f ,m,t must be replaced with a q,m R f,f ,q,m,t while summing over q.

Industrial Extensions

• Raw materials, remanufactured by-products and final product connectivity It may happen that the Refresh Wafers (at all levels) of one Fresh Wafer are used to produce an SOI product but the Fresh itself is not used. This is due to yield issues. A Fresh Wafer has a different geometry from all its Refresh Wafers. The refresh process acts as a normalizing process. This means that once a Negative Wafer 1 is refreshed, it has a different geometry than its previous Fresh Wafer and all the subsequent Refresh Wafers have the same geometry. Therefore, for some Top Wafers, the Fresh Wafer is used to produce one SOI reference and all its Refresh Wafers are used to produce other SOI reference(s). In this way, the yield is increased. In other words, once the Negative of a Fresh is refreshed, it becomes "connectable" to all references for which it was prohibited. In another case, there exist products which use only Fresh Wafers and their refreshes may not at all be used for any other product. All these constraints can be modeled using the parameters a i,b , a i,f and b f ,f .

For some products, a Base Wafer b is associated with a Top Wafer f for the production of the SOI Wafer i.

a b,f     
1 if Base Wafer b can be used with Top Wafer f , 0 otherwise.

a b,f,i           
1 if Base Wafer b can be used with Top Wafer f to produce SOI product i, 0 otherwise.

• Maximum refresh level Due to SOI customer requirements or product characteristics, not all refresh levels of a Fresh Wafer are allowed to be used. Therefore the maximum refresh level is determined from two standpoints. First, the Fresh Wafer f may be refreshed n max f times. Second, an SOI Wafer i may be produced from the Refresh Wafers of the Fresh Wafer

f until n max f →i level (n max f →i ≤ n max f ). If n max f →i < n max f
, the Refresh Wafers of f not used for SOI product i may be used to produce another SOI product i . This constraint can be modeled using

Chapter 10. Multi-Item Bi-Level Capacitated Lot-Sizing with Multiple

Remanufacturing of Reusable By-Products the parameter a i,f .

• Yield monitoring -Minimum Fresh Wafer usage In the SOI production line, there must be a minimum level of Fresh Wafers. In other words, the SOI production line must not be run with only Refresh Wafers. This is related to the SOI production line yield monitoring. Once a Fresh Wafer is refreshed (even once), its geometry and characteristics change. Having this in mind, when only Refresh Wafers are used in the SOI production line, if the line performance drops, it is difficult to judge whether the problem comes from the SOI production line or from Refresh Wafers. Hence, for securing the yield monitoring of the SOI production line, a minimum level of Fresh Wafers are to be used with Refresh Wafers. This constraint can be satisfied using a minimum production level as in Constraint (10.28).

∀i

X F i,f,t ≥ F min ∀f |n f = 0, t (10.28)
• Monitor wafers. The Negative Wafer of the last level may also be refreshed and used as monitor wafer. The monitor wafers may be used internally or sold to external companies. However, as monitor wafers have less value than Refresh Wafers, the refresh process line capacity must first be allocated to get Refresh Wafers and then -if capacity remains unused-to get monitor wafers.

Numerical Experiments 10.6.1 Data Sets

Based on industrial data, small, medium and large test instances are constructed to run experiments on the model and study its behavior. The parameters for generating data sets are listed in Table 10.1. In order to avoid infeasibility, coefficients 10.6 Numerical Experiments called capacity tightness factors (CTF), q p and q r are used for adjusting production and refresh capacities respectively. Both of the capacity tightness factors are fixed to 1.0 (tight), 1.2 (normal), 1.6 (large), and 2.0 (very large). Initial inventories and scheduled wafers to be received (in transit) are set to zero. The refresh process can be performed in different sites. Therefore, the refresh cost and refresh setup cost (cr f,m,t and sr m,t ) are defined based on the refresh site m. The refresh site can be internal, external, close or remote.

The instances are generated by fixing one of the parameters and by considering all the combinations of the other parameters. In total, 2304 instances are generated. The reduced Mixed Integer linear Program (MIP) of the smallest instance has 2383 constraints and 9822 variables from which 280 are binaries, whereas the reduced MIP of the largest instance has 16452 constraints and 149356 variables from which 526 are binaries.

Experimental Results

The test instances are solved using IBM ILOG CPLEX™ 12.5.1. All computational experiments have been run on an AMD Phenom II X2 B57 3.20 GHz with 3.24GB of RAM. The relative MIP gap tolerance is set to 0.5%. A summary of the results can be found in Tables 10.2 and 10.3, where the average results of all instances are provided for a given value of each of the parameters CT F , T , I, F , B and M . Table 10.2 shows the average resolution time. Large CPU Times are observed and rather independently of the variations of most parameters. A significant increase of the CPU time is observed when the length of the planning horizon increases. The average CPU Time is multiplied by more than 17 when the number of periods increases from 6 to 48. The CPU Time also significantly increases when the number of Top Wafer references increases. Table 10.3 shows the percentage of each cost component in the total optimal cost. The percentage of Base Wafer purchase and procurement cost is significantly larger than the percentage of the Fresh Wafer purchase and procurement cost. The reason is that less Fresh Wafer procurement is needed as Refresh Wafers are produced using the refresh process. However, since

Chapter 10. Multi-Item Bi-Level Capacitated Lot-Sizing with Multiple

Remanufacturing of Reusable By-Products the refresh process is relatively cheaper than Fresh Wafer procurement, relatively small refresh process and setup costs are incurred. The largest production and setup costs concern the SOI fabrication. Inventory costs are negligible for each cost component even if inventory costs are slightly larger for SOI production. Except for the length of the planning horizon (T ), the variation of all other parameters do not significantly alter the percentages of the cost components. By increasing the planning horizon, the Fresh Wafer purchase (and also procurement and inventory) cost drastically reduces while the refresh process and setup costs increase. Note that the maximum refresh level n max f is 5. This means that a newly bought Fresh Wafer can only be refreshed five times. As the lead time is equal to one period

n max f 5 l 1 a i,b a i,b ∈ [0, 1]|P (a i,b = 1) = 0.90 a i,f a i,f ∈ [0, 1]|P (a i,f = 1) = 0.90 b f ,f b f ,f ∈ [0, 1]|P (b f ,f = 1) = 0.70 α f,m 0.98 h b,t 1 h + f,t 2 h - f,t 2 h i,t 4 rc m,t q r ( ∀i d i,t /(α f,m n max f )) β f,m 1 pc t q p ( ∀i d i,t ) η i 1 cg i,

Chapter 10. Multi-Item Bi-Level Capacitated Lot-Sizing with Multiple

Remanufacturing of Reusable By-Products (without any SOI and refresh capacity restrictions), it takes 7 periods to fully use a purchased Fresh Wafer (one period for Negative generation and one for the refresh process). Therefore, as the planning horizon increases, the refresh process becomes more important. Therefore, when the planning horizon increases, the Fresh Wafer procurement cost decreases and the refresh process cost increases. However, as the Fresh Wafer purchase cost is relatively larger than the refresh process cost, the decrease of the Fresh Wafer procurement cost components is larger than the increase of the refresh process cost components. This illustrates the economical importance of the refresh process and that an efficient production planning contributes to a substantial cost decrease. Note that, as the percentage of the Fresh Wafer procurement cost decreases, the Base Wafer procurement and SOI production cost percentages increase in the total planning cost.

Closed-Loop Production Planning

Now that we have a clear idea of the problem, it is possible to make a more detailed comparison of the presented research with the literature. We skimmed through 10.6 Numerical Experiments the ideas of Closed-Loop Supply Chain (9.3.2) and product recovery (9.3.3.2). In production systems with remanufacturing or some kind of reverse material flow, the production planning system must be consequently adapted. [START_REF] Guide | Production Planning and Control for Remanufacturing : A State-Of-The-Art Survey[END_REF] and [START_REF] Guide | Production Planning and Control for Remanufacturing: Industry Practice and Research Needs[END_REF] enumerate the complicating characteristics of remanufacturing. We merely mention them by telling to which extend the problem we tackle is different.

The complicating characteristics of remanufacturing based on [START_REF] Guide | Production Planning and Control for Remanufacturing: Industry Practice and Research Needs[END_REF] are:

• The uncertain timing and quantity of returns. This is not true in our case, since by-products are generated after the splitting step. So the timing can be estimated, and the quantity depends on the yield of the production steps until splitting.

• The need to balance returns of items from consumers with demand for remanufactured items. As the demand can be satisfied using the Fresh Wafers as well as Refresh Wafers, this problem does not really exist.

• The need to disassemble the returns. There is no disassembly in our case.

• The uncertainty in materials recovered from returned products. According to the refresh line yield, we can exactly estimate the successfully refreshed Negative Wafers (by products).

• The requirement for a reverse logistics network. In case of refresh of by-products generated in the same site, there is no need for a logistics network.

• The complications of material matching restrictions. The Top Wafer-Finished product matching restrictions are modeled using parameters (a i,b , a i,f and b f ,f ) in Section 10.3.

• The problems of stochastic routings for materials for remanufacturing operations and highly variable processing times. In our case, all routings are deterministic.

Chapter 10. Multi-Item Bi-Level Capacitated Lot-Sizing with Multiple Remanufacturing of Reusable By-Products

For a more recent survey, analysis and classification of the literature treating production planning and remanufacturing, see [START_REF] Junior | Production Planning and Control for remanufacturing literature Review and Analysis[END_REF].

Conclusions and Perspectives

In this chapter, the supply chain of a SOI fabrication unit using the Smart-Cut™ Technology was modeled. Using this technology, one of the two purchased raw materials can be used several times after reprocessing. The reprocessing which is considered as a kind of "remanufacturing" can be done internally or externally. A mixed-integer linear program (MILP) is proposed to model the production planning problem. The model is validated and the optimal solution behavior is studied using generated data sets based on industrial data.

The refresh capacity is constrained by both the refresh line capacity and the SOI production line capacity. In fact, the SOI production determines the rate of the generation of Negative Wafers. Therefore, the refresh process throughput depends on the available refresh capacity and the SOI production (generation of Negative Wafers) rate.

Due to the purchase and refresh cost structure, optimal solutions in our numerical experiments usually include purchase and refresh campaigns, which cause an irregular and fluctuating cost profile along the planning horizon. This may not be desirable both financially and for workforce management, for which a more stable cost expense over the whole planning horizon is preferable. A possible approach is to use a non-linear cost objective function in order to make the sum of all costs closer to their average. However, it makes the model non-linear and its resolution much more difficult.

Introduction

Sustainable development has come into attention due to environmental and economical issues. Closed-loop supply chain and product recovery are emerging fields in the domain of sustainable development. Reverse flow of materials and remanufacturing add complexity to production systems. Planning strategies must be adapted to globally optimize production flows.

We introduce a problem inspired from an industrial case in semiconductor manufacturing. Silicon-On-Insulator (SOI) wafers are used instead of silicon-only wafers in semiconductor manufacturing where high performance and efficiency is needed. SOI wafers can be produced using different technologies. Among them, Smart-Cut™ Technology is a very economic and efficient way of fabricating SOI Wafers. Using this technology, a thin crystalline layer of a so-called donor wafer (Top Wafer) is laid on another silicon wafer (called the handle, support or Base Wafer) using bonding and splitting processes [START_REF] Bruel | Silicon on Insulator Material Technology[END_REF]. As only a thin layer of the Top Wafer is transferred to the final product (SOI Wafer), the Top Wafer can be reused to produce other SOI Wafers. Before reusing the used Top Wafer (the by-product) in the SOI production line, it must be reprocessed (or remanufactured). The production and remanufacturing processes form a so-called "closed-loop manufacturing system".

In this chapter, we discuss the production planning of closed-loop manufacturing systems, an economic lot-sizing problem with multiple remanufacturing of reusable by-products. The discussed problem can be enhanced to model more complicated real world problems.

In the next section, the problem is formally set. The mathematical model is presented and analyzed in Section 11.3. In Sections 11.4 and 11.5 simplified versions of the problem are presented and thoroughly discussed. The chapter is closed with conclusions and multiple perspectives.

Problem Definition

The academic problem we tackle is an uncapacitated single-item bi-level lot-sizing problem in which by-products are generated during production. The generated by-whole cycle is shown in Figure 11.2a. The core of the problem which makes the production planning complicated is the internal cycle (Figure 11.2b). We consider only one product and one raw material which is reusable (refreshable) l max times. A newly purchased Top Wafer is called a Fresh Wafer. Once used, it is called Negative Wafer 1x or Neg 1. The remanufacturing process to make the Negative Wafer usable in the SOI manufacturing is called the refresh process or shortly refresh. The obtained wafer from Neg 1 is called Refresh Wafer 2x. The parameter l (refresh level) indicates how many times the raw material has been used and how many more times before reaching its maximum refresh level l max . Concerning Top Wafers, l = 0 indicates that the raw material is a Fresh Wafer (has never been used), and l > 0 that the raw material is a Refresh Wafer. In Negative Wafers, l = 0 designate the by-product generated after SOI production using a Fresh Wafer. Negative Wafers of level l max are no longer refreshable.

It is assumed that the replenishment, production and refresh alternatives are restricted to the beginning of each period2 . The lead time of the refresh process is known with certainty and is considered to be one period. We suppose that the Negative Wafer generated during a production period can be refreshed at the same period but will only be available at the beginning of the next period. The generation (return) of the by-products is deterministic. By fabricating one final product, one by-product is generated. The demand rate may vary from period to period, but it is

Uncapacitated Lot-Sizing with Multiple Remanufacturing (ULS-MR)

known. The unit variable purchase, production and refresh costs do not depend on the purchase, production or refresh quantity; i.e. no discounts or economy of scale exist. No capacity restriction is considered for production or for refresh process. Fresh Wafer purchase cost is implicitly considered to be larger than the refresh process cost. Otherwise, the refresh process loses its economic interest and the problem reduces to a classical economic lot-sizing problem. All Fresh Wafer ordering, final product demand, final product fabrication occur at the beginning of the period. While inventory costs are charged based on the end of the period inventory. No shortages are permitted. The entire replenishment, production and refresh happen at once. With these assumptions, we define a basic model which can be used for more complicated studies.

Uncapacitated Lot-Sizing with Multiple Remanufacturing (ULS-MR)

In this section we first present an Uncapacitated Single-Item Bi-Level Lot-Sizing Problem with Multiple Remanufacturing of Reusable By-Products model (hereafter, ULS-MR). We show that the ULS-MR problem is NP-hard. Let us simplify the model by omitting two binary setup variables (corresponding to final product fabrication and by-product remanufacturing) while leaving only one (corresponding to purchase) to obtain a problem with a single setup. The properties of the optimal solution are discussed after. In order to make the model still more tractable and find the optimal solution through a dynamic programming algorithm, we make some assumptions, called "Full Push Policy Assumptions" (ULS-MR 1 FP ). ULS-MR The objective function (11.1) minimizes the raw material (Fresh Wafer) purchase cost, the refresh cost at all levels, the production cost using the raw materials (Top Wafers) at all levels (Fresh and Refresh Wafers), the raw material procurement cost, the refresh setup cost, the final product (SOI) production setup cost, the Negative

s l + t-1 + p t = x l t + s l + t ∀t, l = 0 (11.3) s l + t-1 + z l-1 t-1 = x l t + s l + t ∀t, l > 0 (11.4) s l - t-1 + x l t = z l t + s l - t ∀t, l|l = l max (11.5) p t ≤ M 1 • v t ∀t (11.6

Chapter 11. Single-Item Bi-Level Uncapacitated Lot-Sizing with Multiple Remanufacturing of Reusable By-Products

Wafer inventory cost at each level, the Fresh and Refresh Wafer inventory cost and the final product inventory cost. Constraint (11.2) models the final product (SOI) inventory balance. Constraint (11.3) models the purchased raw material (Fresh Wafer) flow conservation. Each time production takes place in period t, (x l t > 0 in Constraint (11.2)), a Negative Wafer is generated. The generated Negative Wafer enters the Negative Wafer flow conservation Constraint (11.5). If the Negative Wafer has not yet reached its maximum refresh level, it can possibly be refreshed. Constraint (11.4) models the Refresh Wafer flow conservation. The correctness of the flow conservation constraints can be verified by summing Constraints (11.2), (11.3) and (11.4) to obtain T t=1 l max -1 l=0

z l t + T t=1 p t = T t=1 d t ∀t. The z l-1
t-1 and x l t variables of the Refresh Wafer flow conservation constraint (11.4) appear for the first time with t ≥ 2. As an example, the Negative Wafers x 0 1 are refreshed in the same period z 0 1 (Constraint (11.5)). The refreshed Wafers become available in the next period z 0 2 to be used in SOI production x 1 2 (Constraint (11.4)). Note that the flow conservation constraints (11.4) and (11.5) do not allow to record the Refresh Wafer inventories which are consumed right after leaving the refresh process. Therefore, the term ( ∀t

l max l=1 ch l + t z l-1 t-1
) is added to the objective function to take into account the immediately used Refresh Wafers.

Constraints (11.6), (11.7) and (11.8) are respectively raw material procurement, final wafer production and refresh process setup constraints. M 1 , M 2 and M 3 are large positive numbers, which can be substituted as follows to give a tighter formulation: M 1 = M 2 = T t =t d t and M 3 = T t =t+1 d t . Constraints (11.9) and (11.10) are added to prevent the boundary (edge) effects. Constraint (11.9) ensures that the Negative Wafer of the last level is not refreshed. It can also be replaced with z l max t = 0 ∀t. Constraint (11.10) ensures that no refresh occurs in the last period. It can also be replaced with l max -1 l=0 z l t=T = 0. As we only produce to satisfy the demand, stationary production costs have no impact on the production plan. Hence, the term ∀t ∀l cp t x l t in the objective function (11.1) can be substituted by the constant ∀t cp t d t .

In this chapter, a resolution method is proposed. Hence, an example of the decision variables is presented. Given l max = 2, diagrams 11.14 and 11.15 show how the variables (in particular x l t , z l t , s l + t and s l - t ) evolve with each final product

Uncapacitated Lot-Sizing with Multiple Remanufacturing (ULS-MR)

fabrication and Negative Wafer refresh (remanufacturing) process. The notations below the arrows refer to the corresponding constraints. Fresh (R0)

s 0 + t --→ SOI 0 st -→ Neg0 s 0 - t --→ R1 s 1 + t --→ SOI 1 st -→ Neg1 s 1 - t --→ R2 s 2 + t --→ SOI 2 st -→ Neg2 (11.15)
The material flow representation of the model is illustrated in Figure 11.3. Note that it is not a classical network flow in which the flow conservation constraint is satisfied in the nodes representing the final product (SOI Wafers) fabrication.

Complexity Analysis of ULS-MR

In the following, it is demonstrated that the ULS-MR problem is NP-hard even for only one level of refresh (l max = 1) and without considering the production setup of the final product (in a mono-level configuration). To prove the NP-hardness of ULS-MR, we perform a polynomial reduction from the Partition Problem.

Partition Problem: Given the set S containing N positive integers a 1 , a 2 , • • • , a N . Can the set S be partitioned into two separated subsets S 1 and S 2 (S = S 1 ∪ S 2 and S 1 ∩ S 2 = ∅) such that the sum of the elements of each subset are equal to A (i.e. Proof. Let us consider an instance I U LS-M R of ULS-MR with T = 2N + 3 periods (see Figure 11.4). Let l max = 1 and ∀t: First, assume that a solution to the instance I U LS-M R has a cost of at most 2A + N + 2. In this case, the inventory costs are sufficiently large to avoid keeping stocks of SOI and Top (Fresh and Refresh) Wafers. Let us begin with the first period (t = 1). As the demand must be satisfied, for the first period, A final products must be fabricated (x 0 1 = A). Having no Top Wafers in initial inventory, A Fresh Wafers are ordered (p 1 = A). The final product fabrication generates A Negative Wafers (s 0 - 1 = A) which does not cost anything (ch l - 1 = 0). The cost associated with the 200

cf t = 1, cr t = 0, cp t = 0, csf t = csr t = 1, csp t = 0, ch l - t = 0, ch l + 2k-1 = 3 k = 1, • • • , N + 1, , ch l + 2k = 0 k = 1, • • • , N + 1, ch t = 3. The final product demand associated with each period t is d t . d 1 = A, d 2k = 0 k = 1, • • • , N + 1, d 2k+1 = a k k = 1, • • • , N
S 0- 1 S 0- 2 S 0- 3 S 1- 1 S 1- 2 S 1- 3 0 v 1 v 2 v 3 l max = 2 Z 0 1 Z 1 1 Z 0 2 Z
t 3 X 0 1 X 1 1 X 2 1 X 0 2 X 1 2 X 2 2 X 0 3 X 1 3 X 2 3

Uncapacitated Lot-Sizing with Multiple Remanufacturing (ULS-MR)

first period is equal the Fresh Wafer purchase and procurement setup costs (A + 1). The demand of the first period is now satisfied.

As d 2k = 0 k = 1, • • • , N + 1 and d 2k+1 = a k ∀k, the sum of the demands of the periods t = 2, • • • , 2N + 2 is equal to 2A. Now, the question is whether to satisfy the demand by (re-)procuring Fresh Wafers or refreshing the A Negative Wafers (generated in period 1) or both. The answer is that, as the refresh process cost (cr t ) is zero and the Fresh Wafer procurement setup and refresh process setup costs are equal (csf t = csr t = 1) (and as we are minimizing the costs), the refresh process is more interesting than Fresh Wafer re-procurement. But, by refreshing the A Negative Wafers generated in the first period, only A Top Wafers are produced. So the requirements for other A Top Wafers until period 2N + 2 must still be satisfied. Note that the final period (2N + 3) has also a demand of A units. We show that the remaining demand of 2A until the end of the planning horizon must be satisfied with purchase and refresh of A Fresh Wafers. Let us first calculate the cost after the first period. Note that large inventory costs avoid to keep stocks (making campaigns). So the A generated Negative Wafers resulting from the first period are refreshed in even periods to satisfy the demand of uneven periods. The demand of each period in the interval 2, • • • , 2N + 2 is satisfied by either refreshing A Negative Wafers or purchasing A Fresh Wafers. Finally, the demand of the last period is satisfied by refreshing the A Negative Wafers generated during the periods before 2N + 2. The refresh occurs in period 2N + 2 and satisfies the Top Wafer need of the final period.

As neither SOI Wafers nor Top Wafers are kept in stock and that the demand of every period is positive, either purchase or refresh occur in each period. As we minimize the cost and tend to satisfy only the demand (and not more), even one period with both purchase and refresh makes the total cost going beyond 2A+N +2. Hence, purchase and refresh periods form the sets S 1 and S 2 . Now, it is explained why any other production plan will be either infeasible or more costly. Ordering less Fresh Wafers would lead to unsatisfied demand. However, one may suggest to order more Fresh Wafer and refresh less. We show that the cost of such a plan exceeds 2A + N + 2. Ordering more Fresh Wafers in the first period is penalized with high Top Wafer holding cost of 3 units per each extra item. This penalty will never be equal to the setup cost of the refresh process, which is only

Chapter 11. Single-Item Bi-Level Uncapacitated Lot-Sizing with Multiple

Remanufacturing of Reusable By-Products 1 unit. Now imagine the case where the demand of the final period (2N + 3) is not purely satisfied from refreshing. In this case, suppose that we purchase Fresh Wafers in the last period (p 2N +3 = ) to complete the Top Wafer need. So, we do not need to purchase exactly A Fresh Wafers in periods 2, • • • , 2N + 2; and we order A -Fresh Wafers. We must satisfy the remaining demand of A of periods 2, • • • , 2N + 1 using the ordered A -Fresh Wafers. This is only possible if we refresh of the generated Negative Wafers. This leaves us with A -2 Negative Wafers which we refresh in period 2N + 2. As we must provide A Top Wafers for the satisfaction of the final period demand, we must order the missing 2 . So is equal to 2 .

The cost associated with this production plan is calculated as follows: A + 1 for purchasing A Fresh Wafers in the first period; N +(A -) for periods 2, • • • , 2N +1; 1 cost unit for the setup cost of refreshing A -2 in period 2N + 2 and 2 + 1 for purchase unit cost and ordering setup of 2 Fresh Wafers in final period 2N + 3. The total cost is 2A + N + 2 + + 1, which is larger than 2A + N + 2. Hence, it is proved that if I U LS-M R is positive, then I is positive.

Conversely, it is shown that if I is positive, then I U LS-M R is positive. In other words, if an instance I is positive, then it exists S 1 and S 2 such that i∈S 1 a i = i∈S 2 a i = A. Having I, we can build a valid solution at the cost of 2A + N + 2. The solution is to procure A Fresh Wafers to satisfy the demand of the first period, to refresh A generated Negative Wafers and to purchase A Fresh Wafers to satisfy the Top Wafer need of periods 2, • • • , 2N + 1, and to refresh A generated Negative Wafers in period 2N +2 to satisfy the final period demand. The refresh and purchase make sets S 1 and S 2 , separately.

Considering the fact that the optimal solution to ULS-MR and to the Partition Problem is independent of the ordering of a i ∀i, the ULS-MR problem is still NPhard even with decreasing or increasing demand trends over the planning horizon. The Fresh Wafer Life Cycle is defined as the interval between the Fresh Wafer order until its full utilization (i.e. the last refresh level l max ), obviously if T ≥ l max + 2. One period is needed for the Negative Wafer generation and refreshing. The refreshed wafer is available at the beginning of the next period. Proof. First consider the right hand side of the inequality ( T t=1 p t ≤ T t=1 d t ) which states that the total raw material purchase cannot exceed the total demand over the whole planning horizon. By contradiction, assume there is an optimal policy in which the total purchase exceeds the total demand of all periods, i.e. T t=1 p t > T t=1 d t . Let us demonstrate the existence of another policy with lower total cost (or at least the same cost). As no (final product) production and refresh process setup costs are considered and since the refresh cost is lower than the Fresh Wafer purchase cost, if we instead order a portion of the total demand and use the by-products (Negative Wafers) to satisfy the Top Wafer need, we save the difference between the

ULS-MR with only Procurement Setup (ULS-MR 1 )

refresh process and the Fresh Wafer purchase costs multiplied by the amount saved in purchase and sent to refresh. Now consider the left hand side of the inequality ( T t=1 dt min{T,l max +1} ≤ T t=1 p t ). It is technically impossible to refresh more than min{T, l max + 1} times. This means that, in the best case, we may use the Fresh Wafer and all its Refresh Wafers (if the planning horizon length allows) to satisfy the demand and no more. Note 1. Here, the aim is not to discuss the optimal solution properties of the original model. However, Proposition 11.4.1 also holds for the original problem. In the same way, the bounds for the refresh process are defined as

T t=1 dt min{T,l max +1} ≤ T t=1 l max -1 l=0 z l t ≤ T t=1 l max l max +1 d t .
The following proposition states that if Fresh Wafers are ordered in period t, then final product fabrication (using the purchase) takes place in the same period t.

Proposition 11.4.2. In an optimal solution of ULS-MR

1 , if p t > 0, then x 0 t > 0.
Proof. The proof is done by contradiction. Let us consider an optimal solution in which the production takes place in period t after a Fresh Wafer purchase campaign in period t. In this case, the Fresh Wafer quantity (relating to the purchase campaign in period t) can be moved to period t (where the production takes place), saving the Fresh Wafer inventory cost between t and t since the Fresh Wafer purchase and holding costs are stationary. This leads to a total cost at least equal to the total cost of the given optimal solution contradicting the initial assumption.

Proposition 11.4.3. In an optimal solution of ULS-MR

1 , if x l+1 t > 0, then z l t-1 = x l+1 t ∀l|l = l max .
The proposition states that in case of SOI production from Refresh Wafers in a given period t, the required Refresh Wafer will be satisfied through the refresh process of Negative Wafers of the previous level in the previous period t-1 available in t.

Proof. As no refresh process setup cost is considered, with stationary unitary refresh cost and as the Negative Wafer inventory cost is lower than any Refresh Wafer

Chapter 11. Single-Item Bi-Level Uncapacitated Lot-Sizing with Multiple

Remanufacturing of Reusable By-Products inventory cost, it is profitable not to refresh a Negative Wafer just to keep its Refresh Wafer in stock. Hence, if we refresh Negative Wafers, it is only to satisfy the SOI production need.

Note 2. In general, if x l t > 0 ∀l, then (z l-1 t-1 > 0) and/or (p t > 0 or s 0 + t-1 > 0). The conclusion can be detailed as follows:

• if x l t > 0, l = 0, then either p t > 0 or s 0 + t-1 > 0;

• if x l t > 0, l > 0, then z l-1 t-1 > 0 and z l-1 t-1 = x l t .

Proposition 11.4.4.

There exists an optimal solution to ULS-MR 1 in which s l + t = 0 ∀t, l > 0.

Proof. It follows from Proposition 11.4.3 and Constraint (11.4).

The Refresh Wafer inventory must be defined as s l + t + z l-1 t where s l + t is only to maintain the inventory balance. s l + t alone does not allow to record the immediately consumed Refresh Wafers. Considering Proposition 11.4.3, for the single setup model, only z l-1 t allows Refresh Wafer inventory cost charging.

Proposition 11.4.5.

There exists an optimal solution to ULS-MR 1 in which s 0 + t-1 p t = 0 ∀t.

Proof. This is the newly purchased raw material Zero Inventory Ordering property. The proof is done by contradiction. Given an optimal solution for which this property does not hold. It is possible to construct a feasible solution with a lower or equal total cost by moving s 0 +

• from the last purchase period (before t) to the current purchase period t. This contradicts the assumptions. Note that this property holds even with stationary purchase costs. Proposition 11.4.5 can also be proved intuitively. Remember that Top Wafers of higher levels are first used to make SOI Wafers, i.e. firstly Fresh Wafer, if not available R1, if not available R2 and so on. If Fresh Wafer stock is on hand in period t, either no production has occurred in the precedent periods (before t) or production has occurred in the precedent periods (before t). First case where no production

ULS-MR with only Procurement Setup (ULS-MR 1 )

has occurred in the precedent periods: It means that no refreshable Negative Wafers are on-hand, so that they can be refreshed and used in the SOI production. In this case, the Fresh Wafer purchase p t can be grouped with the precedent Fresh Wafer purchase. Second, if production has occurred in the precedent periods: In this case, refreshable Negative Wafers must be on-hand, as Fresh Wafers are still available on stock. Therefore, in order to satisfy the demand in terms of SOI, on-hand Negative Wafers must be refreshed before any order is placed, since the refresh cost is lower than Fresh Wafer purchase cost. Some conclusions can be drawn from Proposition 11.4.5. Note 3. Constraint (11.3) of the Model is expanded using Proposition 11.4.3. ---→ 11.4.4

s l + t-1 + p t = x l t + s l + t ∀t, l = 0 (11.16a) s 0 + t-1 -s 0 + t = x 0 t -
0 t = 0 → s 0 + t-1 -s 0 + t = 0 →    s 0 + t-1 = s 0 + t =
l max -1 l=0 z l t-1 > 0 (11 .21)
(11.20b) comes from the fact that Refresh Wafers will not be used for the production of SOI Wafers until Fresh Wafers are on hand. Therefore, if no SOI production using Fresh Wafers are made (x 0 t = 0) while Fresh Wafer inventory is positive s 0 + t > 0, there must have been no demand for SOI wafers.

Chapter 11. Single-Item Bi-Level Uncapacitated Lot-Sizing with Multiple

Remanufacturing of Reusable By-Products Proposition 11.4.6. There exists an optimal solution to ULS-MR 1 in which

l max -1 l=0 s l - t-1 p t = 0 ∀t.
Proof. As no refresh process setup cost is considered, it is possible to refresh in period t -1 and order Fresh Wafers in t to complete the raw material need in order to satisfy d t . But it is not possible to keep refreshable Negative Wafers as soon as the refresh process cost is less expensive than Fresh Wafer procurement. Therefore, it is economically interesting to refresh all refreshable Negative Wafers on hand to satisfy the SOI production need instead of purchasing any Fresh Wafers and keeping refreshable Negative Wafers in stock.

Note than Fresh Wafer procurement (say in period t, i.e. p t ) may occur within a Fresh Wafer Life Cycle, and when z l t-1 > 0. Nevertheless, if refreshable Negative Wafer inventory exists, it would be preferable to refresh these Negative Wafers instead of acquiring Fresh Wafers which cost considerably more than refreshing the Negative Wafers. To conclude, it is possible to order Fresh Wafers and to refresh Negative Wafers at the same time. But it is not possible to order Fresh Wafers while keeping refreshable Negative Wafers in inventory. Note 4. As no SOI production setup and refresh process setup costs are considered and also because the refresh process lead time is assumed to be equal to 1 period, the Fresh Wafer purchase is to complete the Top Wafer need of the final product demand (see (11.22)).

l max -1 l=0 z l t-1 + p t -d t ≥ 0 ∀t (11.22)

Mathematical Model of ULS-MR 1

The optimal solution properties allow us to simplify ULS-MR by removing the refresh variables z l t as follows. Based on Propositions 11.4.3 and 11.4.4, we get:

z l t = x l+1 t+1 = s l - t-1 -s l - t + x l t ∀t, l|l = l max (11.23)

ULS-MR with only Procurement Setup (ULS-MR 1 )

The mathematical model for ULS-MR 1 is written:

min ∀t (cf + cp)p t + ∀t l max l=1 cpx l t + ∀t l max l=1 cr(s l - t-1 + x l t -s l - t ) + ∀t csf v t + ∀t l max -1 l=0 ch l -s l - t + ∀t ch 0 + s 0 + t + ∀t l max l=1 ch l + (s l - t-1 + x l t -s l - t ) + ∀t chs t (11.24)
subject to Regarding Fresh Wafers, at each procurement or production period, we must take two more decisions. Decision 1: How many more Fresh Wafers are to be bought to keep in stock for future use? Decision 2: How many of the stocked Fresh Wafer inventory are to be used for production in a production period? Constraint (11.26)

s t-1 + l max l=0 x l t = d t + s t ∀t (11.25) s 0 + t-1 + p t = x 0 t + s 0 + t ∀t (11.26) s l - t-1 + x l t = x l+1 t+1 + s l - t ∀t, l|l = l max (11.27) p t ≤ M 1 • v t ∀t (
v 1 v 2 v 4 l max = 2 Z 0 1 Z 0 2 Z 1 2 Z 0 3 Z 1 3 X 0 1 X 0 2 X 1 2 X 0 3 X 1 3 X
S 0- 1 S 0- 2 S 0- 3 S 0- 4 S 1- 2 S 1- 3 S 1- 4 R Z 0 4 Z 1 4 Negative Wafers l = 1 l = 0 t 2 t 3 t 4 X 0 1 X 0 2 X 1 2 X 0 3 X 1 3 X 1

ULS-MR 1 under "Full Push Policy" (ULS-MR

1 FP )
is at the origin of these decisions.

Concerning Negative Wafers, at each production or refresh period, we must take two decisions. Decision 3: How many final products must be fabricated to assure that Negative Wafer generation is enough for future raw material requirements? Decision 4: How many of the generated Negative Wafers in the inventory are to be sent to refresh? Constraint (11.27) is at the origin of these decisions. Note that Decision 3 is one of the most complicating characteristics of this problem, i.e. final product fabrication is not only to satisfy the final product demand but also to create the by-products needed to satisfy the final product demand later.

In order to simplify these decisions and the model, we introduce "push" policies in the next section.

ULS-MR 1 under "Full Push Policy" (ULS-MR 1 FP )

To make the problem more tractable, we consider three additional assumptions which eliminate decisions regarding three SKUs: Fresh, Negative and Refresh Wafers. We "push" the manufacturing system to produce and remanufacture in order to avoid keeping any inventory of these three SKUs.

"Full Push Policy" Assumptions

1. All Fresh Wafers (purchased raw material) are systematically transformed to SOI Wafers (final products).

2. The generated Negative Wafers are directly refreshed after generation.

3. The Refresh Wafers are also systematically transformed to final products.

It follows that neither Top Wafer inventory nor Negative Wafer inventory is kept (see Figure 11.8). Hence, imposing these assumptions eliminates decisions on whether to keep or to use inventories of these SKUs. By ordering p t units of Fresh Wafer, p t units of final products systematically appear the following periods until period t + l max + 1. ∀l and s t-1 are on-hand, we can pre-treat the final products demands after t. Under the Full Push assumptions, we know with certainty when the final products are fabricated. This makes the data pre-treatment of the final demand possible. After data pre-treatment, the stock vector is null and we have new demands (d • ). Proposition 11.5.5. If the demand is non-decreasing over the planning horizon, there exists an optimal solution to ULS-MR 1 FP such that p t s t-1 = 0 ∀t.

ULS-MR

Proof. The proof is done by contradiction. Assume two periods t and t , where t < t and t is a procurement period. With non-decreasing demand, let us consider an optimal solution for which this property does not hold. This means that we have satisfied demand and the final product inventory is still on-hand at the end of period ). It is possible to construct a feasible solution with a lower than or equal total cost by ordering less Fresh Wafers of amount s t l max +1 in the last purchase period (t) while still satisfying demand. This contradicts the assumptions. This property is a kind of Zero Inventory Ordering Property. It allows regeneration intervals to be defined. In general, we end up with a minimum final product inventory level.

Note that this property does not necessarily hold for decreasing demand trend. Proposition 11.5.6. If the demand is non-decreasing over the planning horizon, there exists an optimal solution to ULS-MR 1 FP such that s T = 0.

Proof. The proof is done by contradiction. With non-decreasing demand, let us consider an optimal solution for which this property does not hold. This means that the demand is satisfied and a final product inventory is still on-hand at the end of the planning horizon, i.e. s T > 0. It is possible to construct a feasible solution with a lower or equal total cost by ordering less Fresh Wafers of amount s T l max +1 in Chapter 11. Single-Item Bi-Level Uncapacitated Lot-Sizing with Multiple Remanufacturing of Reusable By-Products the previous purchase period while still satisfying demand. This contradicts the assumptions.

In other words, Proposition 11.5.6 states raw materials are exhausted by making full Fresh Wafer life cycles, leading to zero final product inventory at the end of the planning horizon. Note that Propositions 11.5.5 and 11.5.6 also hold for ULS-MR 1 . (11.38). Let us consider a simple numerical example to see to which extent we may still reduce the constraints. The purchased Fresh Wafers, in say period 1, are used to make final products (based on Constraint (11.40): p 1 = x 0 1 ). The final products that are fabricated are used to satisfy the demand in period 1 (d 1 ) in Constraint (11.38). The fabrication of final products generates Negative Wafers (Neg0) which are sent to refresh directly (based on Constraint (11.42): x 0 1 = z 0 1 ). The refreshed Negative Wafers (Refresh Wafers at level 1, or simply R1) are used at the beginning of the next period to fabricate final products (based on Constraint (11.41): z 0 1 = x 1 2 ). Final Wafers produced from Refresh Wafer at level 1 (x 1 2 ) is used in Constraint (11.39) to satisfy the demand of the second period (d 2 ). This cycle continues. Constraint (11.43) shows the Top Wafer need for final product fabrication.

Mathematical Model for ULS-MR

l max l=1 z l-1 t + p t = l max l=0 x l t ∀t (11.43)
The flow conservation Constraints (11.38) to (11.42) reduce to Constraints (11.44) and (11.45). As the by-product generation and refresh process happen step by step in successive periods after a procurement, the upper limit of the sum is defined as (min{t, l max + 1}) -1. This phenomenon can be observed in the beginning of the planning horizon in Figure 11.9. Constraint (11.42) is needed to charge the Refresh Wafer inventory cost at the end of each period. In order to get rid of x l t , let us re-write the constraint as (11.46

v 1 v 2 v 4 l max = 2 Z 0 1 Z 0 2 Z 1 2 Z 0 3 Z 1 3 X 0 1 X 0 2 X 1 2 X 0 3 X 1 3 X 2 3 d 4 t 4 S 4 P P P X 1 4 X 2 4 X 0 4 v 3 R R R R R Refresh Process F Fresh Wafer Purchase F F F F P
Final Product Fabrication (By-product Generation) A dynamic programming (DP) algorithm is presented to solve the model. Let C(t) be the total minimum cost of solving the problem over the first t periods. It corresponds to the best Fresh Wafer procurement strategy that satisfies the final product (SOI Wafer) demand requirements in periods 1, • • • , t.

The Shortest Path Representation

The DP states can be presented by a directed graph with T t=1 2 (t-1) nodes. Each node represents a status. Each arc (i, j) where i ≤ j represents the regeneration interval [i -1, j]. In such a regeneration interval, with the raw material (Fresh Wafer) procurement setup in period i, the final product (SOI Wafer) demand of the interval [i -1, j] is covered. A cost C(i -1, j) is associated with each arc (i, j). It corresponds to the purchase unitary and setup costs and the resulting inventory costs of remanufactured by-products and final products.

The number of nodes increases exponentially. Even three indices are not enough for unique labeling of the nodes. To show this, let use define the indices of each node as (i, j, k) with i < j and j ≤ k, where i is the previous production period, j is the current period and j is the final period covered by this replenishment campaign.

Chapter 11. Single-Item Bi-Level Uncapacitated Lot-Sizing with Multiple

Remanufacturing of Reusable By-Products Figure 11.10 shows an instance of four periods. Even in this simple example, we see that at the period 4, problems with duplicates in labeling arise. Therefore, for 0,0,0 ). Each position may take either no value or a value of 1 or 0. If a purchase setup occurs in period t, its position takes value 1. If the demand of a period is covered but no setup occurs, its position is set to 0. Otherwise the position is vacant. For instance, C(1, 0, 1, -, • • • ) is the cost associated with a node in the third period. It means that, with a procurement setup in the first period, we cover the demand of the first and second periods. A setup in the third period also occurs to cover the demand of the third period. An example of this type of labelling for a tree representing the states of the DP for four periods is depicted in 11.11. In order to calculate the recurrence equations of the DP, we need to define the state at each node. A state is defined by a vector π = (p i,j , s i,j , s l+ i,j ). We define each element of the status vectors as follows.

0,1,1 0,1,2 0,1,3 0,1, 4 1,3,3 2,3,3 1,4,4 3,4,4 3,4,4 1,2,2 1,2,3 1,2,4 1,3,4 2,3,4 2
The procurement quantity p i,j (with i ≤ j) placed in period i to cover the demand 11.5 ULS-MR 1 under "Full Push Policy" (ULS-MR 1 FP )

l max = 2 T = 4 (-,-,-,-) (1,-,-,-) (1,0,0,-) (1,0,-,-) (1,0,0,0) (1,1,-,-) (1,1,0,0) (1,1,0,1) (1,1,1,0) (1,1,1,1) (1,1,1,-) (1,1,0,-) (1,0,1,-) (1,0,0 ,1)
(1,0,1,0)

(1,0,1,1)

p i,j = max{0; max 1≤k≤j-i+1 { k-1 k =0 d i+k -s i-1 - k-1 k =0 l max +1-k l=1 s l + i-1 min{k, l max + 1} }} (11.53)
The final product inventory at the end of period j resulting from ordering p i,j is calculated using (11.54) below.

s i,j = (min{j -i + 1, l max + 1})p i,j + s i-1 + j-i k =1 l max -k +1 l=1 s l + i-1 - j u=i d u (11.54)
The Refresh Wafer inventory at level l (with l = 0) at the end of period j resulting from ordering p i,j is calculated using (11.55) below.

s (min{j-i+1,l max }) + i,j = p i,j (11.55)

Recurrence equation

The costs are calculated based on vector π at each state. We begin with (11.56a) and define each position one by one (for instance, the first position is filled as and substituting in the objective function (11.48).

The optimal solution is obtained when all positions of C(•) are set to 1 or 0. The minimum value of C(•) provides the optimal solution value and purchasing periods.

Dominance Rules

The number of nodes increases exponentially in the tree representing the DP. Some filtering rules may help to discard some nodes and branches.

• With non decreasing demand over the planning horizon, Proposition 11.5.5 and 11.5.6 help to define the regeneration intervals more easily.

• If for all periods (t ) between two periods of t and t , we have (and generate) enough raw materials to satisfy the demand, then we do not need new raw material procurement. Formally, we set periods (t + 1, • • • , t ) as non-purchase periods if (11.57) holds:

t-t +1 k=1 l max -k l=1 s l + t-1 + s t-1 + (min{t -t + 1, l max })p t ≥ t u=t d u t ≤ t ≤ t (11 .57)
• The raw materials (Top Wafer) stock of the arcs departing from the same node exhausts after l max + 1 periods. However, the final product stock of the nodes may not always be null. If the stock levels are equal, the node with the minimum cost is kept and the other nodes are discarded. If the final product stock levels are different, the node with the least final product stock value is kept and the others are discarded.

• In each period, among the nodes ending with the same inventory vector, the least expensive is kept and the others are discarded.

Conclusions and Perspectives

Using Proposition 11.5.6 a backward DP can be proposed. An analysis should then be conducted on several variants of DP to determine their efficiency.

As argued before, by relaxing "push" assumptions new decisions arise. In order to solve ULS-MR 1 , the proposed DP can be adapted. The maximum number of the states of the DP will be the same. However at each node, the mentioned decisions regarding Fresh Wafer and Negative Wafers must be taken.

Conclusions and Perspectives

In this chapter, we introduced in detail a novel closed-loop production planning problem derived from a real world industrial case. The model can be extended to cover other real applications, for instance in recycling or circular economy. The lot-sizing model contains three setup constraints associated with purchase, remanufacturing and production. The complexity analysis shows that the uncapacitated single-item model is NP-hard. Note that one characteristic which makes the problem difficult is that the SOI Wafer (final product) fabrication is not only driven by the demand satisfaction but also by Negative Wafer generation. The generated Negative Wafers are then used to satisfy SOI Wafers.

In a first attempt to solve the problem, we simplified it by eliminating two setup constraints of production and remanufacturing and keeping only purchase setup constraints. Properties of this problem were explored and a model with less variables and constraints was proposed. In order to develop a dynamic programming to solve the problem, we considered new assumptions, called "Full Push Policy". It implies that once a replenishment (or refresh) occurs, we push the production system to fabricate final products. In the same way, we immediately remanufacture the generated by-products. We proposed a dynamic programming resolution algorithm. Note that the "Full Push Policy" assumptions help to simplify the problem by eliminating several decisions.

Chapter 11. Single-Item Bi-Level Uncapacitated Lot-Sizing with Multiple

Remanufacturing of Reusable By-Products algorithm, based on extensive experimentation, it could be analyzed whether the number of states explodes in practice. The idea is to check the number of explored states in comparison to the worst case.

As the original model is NP-hard, heuristics can be proposed for its resolution. It is worth to observe that some of the properties of an optimal solution discussed for the model with single purchase setup are also valid for the original problem. An adaptation of the Silver-Meal heuristic, or least cost heuristics or other classic heuristics with worst case performance guarantee could be derived. The traditional formulation proposed in this chapter provide weak lower bounds. In order to obtain better lower bounds, alternative formulations of the problem can be proposed. The tightness of formulations can be evaluated by comparing the LP relaxations and MIP computational times.

Chapter 12

General Conclusions and Perspectives

With this final chapter, we come to the end of our journey, even if an endless way is still to explore. Conclusions and perspectives relevant to each subject are presented at the end of the chapters. Sections 3.6, 4.5, 5.5, 6.6, 7.5 together with Section 8.1 of the conclusive Chapter 8 are dedicated to Part I. Therefore, in this brief chapter, we do not aim to repeat all conclusions and perspectives, but to present a more global and comprehensive picture of the various contributions of the thesis while pointing out how the two parts can be connected.

Conclusions

Early in Chapter 2, the framework of the thesis was outlined. It was conducted in the company Soitec, a world leader for high performance semiconductor material. The flagship product of the company, called SOI (Silicon-On-Insulator) wafer, is used to make efficient microelectronic components.

The microelectronics business environment is unstable and highly competitive. Moreover, the fabrication costs are high and the production is dynamic. Therefore, flexibility and agility are identified as the ants to play any game in semiconductor manufacturing systems.

In semiconductor manufacturing, at each operation step, a recipe is defined for each product. A recipe must be qualified on a machine in order to be able to allocate the production volume of the recipe to the machine. This restriction has a direct impact on capacity utilization of toolsets. Chapter 3 discusses the impact Chapter 12. General Conclusions and Perspectives of qualification management (QM) on capacity optimization and flexibility increase which is the subject of the first part of this thesis. By the concept "flexibility", we try to evaluate the workload balance on a toolset; the better the workload balance, the higher the flexibility. An adequate qualification configuration gives flexibility for recipe workload allocations to machines. Hence, we aim at increasing flexibility in workload allocation. Increasing flexibility leads to capacity optimization.

Industrial constraints affect the QM. First, the influence of the limited equipment capacity on capacity allocation and QM was studied in 4. Capacitated flexibility measures were presented to evaluate the flexibility of a workload balance on a toolset. Additional measures, called "deviation ratio" are necessary to help the decision making process for performing new qualifications.

The presented work is interesting from the point of view of workload balancing and capacity allocation. Chapter 6 also considers workload balancing with capacity restrictions. Our criteria is to minimize the deviation of the total workload allocated to a machine from the maximum available time of each machine while balancing the workload on the whole toolset. In Section 6.4.2, weighted variability measures were discussed. It is possible modify the criteria to minimize the number of overloaded (or underloaded) machines by considering a big weight for corresponding cases.

Batching is a frequent production characteristic. The influence of batching on qualification management was studied in Chapter 5. The sequencing of allocations were not considered while workload balancing. But when making full batches, the production volume of each recipe (of product) is as if it is "packed" in a package, which is the batch size. Therefore, the batches allocated to each machines can be sequenced freely on the same machine. Hence, workload allocation under batch size constraint can be used for scheduling. In scheduling literature, qualification restrictions are equivalent to machine eligibility restrictions. Qualifications management is similar to configurability in flexible manufacturing systems (FMS).

Finally, the industrialization of the concept at Soitec was discussed in Chapter 7. This decision making process is new in the company. Therefore, data exchange interfaces were created to facilitate the continuous usage of the concept.

In conclusion, by a better QM, we try to decreases the WIP on the shopfloor (leading to more floor space), decrease the inventory level, increase machine capacity Chapter 12. General Conclusions and Perspectives proach can be used to measure the incurred variability due to other variability factors such as batching. Section 7.5 proposed other industrial constraints while QM, some examples are recipe priority, production yield, auxiliary resource management, consumables, material handling system, etc.

Losing a qualification may decrease the toolset flexibility. Using the discussed approach, it is possible to evaluate the flexibility loss related to a disqualification. We named this recipe-to-machine criticality. In the same way it is possible to calculate machine criticality. This valuable measure can help for preventive maintenance (PM) planning. By calculating the machine criticality in advance, it is possible to schedule PM in a period where machine criticality is at the least. Another interesting perspective is to study QM over time instead of considering one production period. The qualification setup cost was not taken into account. In dynamic QM, the setup costs and the disqualification issue should also be considered. The perspectives relating to Part II of the thesis were examined in Sections 10.7 and 11.6. Now perspectives relating both parts of the thesis are presented. In the first part, the optimal toolset capacity utilization is calculated to determine new qualification(s). Therefore, new qualifications are levers for increasing capacity. On the other hand, we know that production planning is constrained by production capacity. Therefore, the two concepts are complementary and can be combined.

Qualification Management and Production Planning

So far, it was shown that the recipe-to-machine qualification configuration directly affects the production capacity. Here, we open the discussion of how qualification management and production planning may be combined by adding new constraints to a capacitated lot-sizing model. It is not our goal to re-write a complete capacitated lot-sizing model. Merely necessary constraints are mentioned. In Part I, we have always used throughput T P and not process time P T . Both are equivalent, i.e. T P = 1/P T . If a recipe is not qualified (or qualifiable) on a machine, its throughput tends to infinity or, in other words, its process time is zero.

Dynamic versions of the QM model and QM models for multiple qualifications were proposed in [START_REF] Johnzén | Modeling and Optimizing Flexible Capacity Allocation in Semiconductor Manufacturing[END_REF]. Without rewriting the whole model, we open the discussion 1 if recipe r is to be qualified on machine m, 0 otherwise. Two cases may be considered. First, we do not intend to perform new qualifications. In this case the qualification must be considered in the setup constraint and workload allocation. However, if additional new qualifications are allowed, the The objective function (12.1a) tries to minimize the production, holding, new qualification and production setup costs. Constraint (12.1b) guarantees the flow conservation balance. Production capacity is respected using Constraint (12.1c). Constraint (12.1d) represents the production setup.

Note that performing new qualifications is costly. Therefore, the maximum number of qualifications can be directly determined in each period. The qualification cost in the objective function also limits the number of qualifications. The term (Q r,m + OQ t r,m )Y t r,m in (12.1d) contains the product of two binary variables which can be replaced by an additional binary variable, say Z t r,m . The term must be substituted by Q r,m Y t r,m + Z t r,m and the set of Constraints (12.2) are to be added to the model. Constraint (12.1e) defines the allowed number of qualifications (k). Note that k must be smaller than or equal to the number of qualifiable recipe-to-12.2 Perspectives machine couples. Constraint (12.1f) ensures that if a recipe-to-machine couple is not qualifiable (P Q r,m = 0), it is not considered to be qualified. Based on the qualification difficulty, monetary cost, time consumption, importance, etc., the qualification decision may belong to two different decisions levels, tactical or more operational. Lot-sizing decisions are made at a tactical level. So if the qualification decision is more operational, it must be carefully though of how to relate these two types of decisions. Studies considering the integration of decisions at different levels attract the attention of researchers and practitioners, for instance integration of lot-sizing and distribution, or lot-sizing and scheduling (see [START_REF] Dauzère-Pérès | An Integrated Approach in Production Planning and Scheduling[END_REF] and [START_REF] Gomez Urrutia | Optimisation Intégrée des Décisions en Planification et Ordonnancement dans une Chaine Logistique[END_REF]). Therefore, it may be possible to inspire from similar researches to relate both decision levels.

Z t

Flexible Production Planning -Limit Changes

Demands, customer expectations, production factors, lead times and other parameters are subject to constant changes. Therefore, the production plan must be revised constantly. However, abrupt changes in the production plan are not desired. It makes the scheduling, material sourcing and in general the functioning difficult. Moreover, the users of the production plans lose their confidence in the results. Hence, it is preferable to have robust production plans throughout the planning horizon. A similar approach as what was presented in Part I and in particular in Section 6.4.2 could be used i.e. to penalize the differences between a new production plan and the current one. By adjusting linear and non-linear weights, the discrepancies of the current production plan from the previous can be reduced. In this case, the objective function needs to be adapted consequently.

Flexible Production Planning -Cost Profile

Due to purchase and refresh cost structure, it is possible that the model recommends campaigns of Fresh Wafer purchase and refresh process. These campaigns cause irregular cost profiles along the whole planning horizon. In order to better manage the financial elements of the planning, it is preferable to create an almost constant cost expense over the whole horizon instead of periodic costs. Here also, it is possible to extend the approach proposed in Chapter 6 to overcome the undesired abrupt cost fluctuations.

Flexible Production Planning -End-of-horizon Effects

One of the undesired effects which one may encounter while planning production is the End-of-horizon Effects. In a lot-sizing problem, the model tends to end up with zero inventory levels. As we all know that the world does not come to its end with the end of planning horizon, it causes difficulties for planner. Several approaches are proposed to overcome this undesired end-of-horizon effect. One of them is to consider sufficiently long (to be also defined!) planning horizons, i.e. more than needed. Another approach consists in considering some kind of safety stock or minimum ending inventory position.

In the case where backlogging is possible, the model may tend to backlog instead of satisfying demand as we approach the end of horizon. One possible way is to penalize more the backlogging cost as we approach the end of horizon. Of course, this penalization can be controlled by some sort of linear and/or nonlinear weights (see for instance Section 6.3.3).

Flexibility in Supply Chain Management

The concept of flexibility increasingly attracts the attention of practitioners and researchers, in particular in supply chain management. Some reasons motivating why a supply chain must be flexible due to new conditions are discussed in [START_REF] Simchi-Levi | Operations Rules: Delivering Customer Value through Flexible Operations[END_REF]. Globalization, increasing customer expectations as well as lead times, labor cost increase in developing countries, energy cost fluctuations, increase in logistic costs,

Perspectives

increased risk in supply and many other factors cause that decisions in supply chain management may need to be revised in relatively short times. That is why supply chains must be more flexible and agile enough to quickly respond to changes. The new competition environment enumerated by [START_REF] Simchi-Levi | Operations Rules: Delivering Customer Value through Flexible Operations[END_REF] is dynamic, complex and uncertain. Note that these elements exist in a fab in semiconductor manufacturing. The answer of is to increase flexibility to overcome the described supply chain environment. The flexibility is defined by [START_REF] Simchi-Levi | Operations Rules: Delivering Customer Value through Flexible Operations[END_REF] as the ability to respond quickly and in a cost effective way to changes which may come in many forms such as demand volume and mix, prices, costs, etc. The objective is defined as to increase the service level (decrease the amount of unsatisfied demand) by improving the capacity utilization while reducing cost, of course as quickly as possible. Note that we treated in Part I exactly the capacity utilization increase, while reducing (or even eliminating) overloaded machines (which decrease the service level by leaving some demand unsatisfied). All this by reducing costs incurred by under-utilization of machines, operator performance decrease, increase of WIP and buffer stock at the shop floor, unsatisfied demand, induced variability in the production line, etc. with the least possible time by performing the best (or a good) qualification. [START_REF] Simchi-Levi | Operations Rules: Delivering Customer Value through Flexible Operations[END_REF] mentions three ways to achieve flexibility: flexibility through product design, process design and system design. Two mentioned points under system design are to increase service level by coping with high forecast error and better resource utilization. Note that these points are treated in Part I. Interestingly, similar results are obtained in case studies in SCM as in Section 6.5, i.e. too much manufacturing flexibility does not necessarily decrease the cost (in this section the results are illustrated by other KPIs such as variability, overload decrease, etc.). As in SCM, the objective is to match supply with demand, in Part I, it was tried to match production capacity with demand, which come actually to the same concepts.

A.3 Mathematical Model using Semi-Continuous Variables

step length. The answer is that, if with the WIP on-hand, we make (if possible) a maximum batch. In case some remainder is still on-hand, if a minimum batch can be made, we do it. Else we distribute the remainder to the least loaded machine.

A.3 Mathematical Model using Semi-Continuous Variables

The model proposed in Section 5.2.1 can be adapted to consider minimum batch sizes. However, we present here another modeling approach using Semi-Continuous Variables. First, the single machine case is considered and then the case with multiple machines is developed.

A.3.1 Case of a Single Machine

The production volume of each production run must lie between Minimum Batch Size (minBS) and Maximum Batch Size (maxBS).

Parameters

A.3 Mathematical Model using Semi-Continuous Variables

imum amount. In transportation is the same, no shipment is done until a minimum shipment amount is reached.

For our problem, the semi-continuous variable Z r,k is defined as follows:

Z r,k ∈ {0} ∪ [minBS r , maxBS r ] ⇔ minBS r Y r,k ≤ Z r,k ≤ M axBS r Y r,k . (A.6)
The semi-continuous variable is transformed under the form of a system of mixed inequalities:

                   Y r,k X r,k = X r,k if Y r,k = 1 Y r,k X r,k = 0 if Y r,k = 0 a = M in r (X r,k ) b = M ax r (X r,k ) ⇔                    Y r,k X r,k ≤ bX r,k Y r,k X r

Chapter B. SOI-Refresh Lines Modeling

Fresh 1x depends only on its precedent state, i.e. Negative 1x and not two states before, i.e. SOI production.

A special type of Markov chain are absorbing chains. An absorbing chain is a Markovian chain in which some of the states are absorbing and the rest are transient states. In such a chain, we begin in a transient state and eventually we are sure to leave the transient state and finishes finally in an absorbing state.

A Fresh Wafer enters for the first time the SOI production line. After the splitting step, a Negative Wafer is generated. The Negative wafer is then sent to the refresh line in order to be used again in the SOI production. The generated Negative Wafer is refreshed successfully with a defined probability which is equal to the refresh line yield. And the Negative Wafer will be scraped with a probability equal to the complement of the success probability. If the Negative Wafer is successfully refreshed, if reenters the SOI production line and has the chance to be refreshed again until its maximum allowed (qualified) level. Note that rework exists only for refresh process but not SOI production. The refresh process has several states: Fresh, first level Refresh, second level Refresh until last level Refresh, Monitor and Scrap. Refresh last level, Monitor and Scrap are absorbing states while all other states are transient states. For example, first level Refresh is a transient state while there is a path from first level Refresh to Scrap state but there is no path returning from Scrap to first level Refresh.

The transition matrix is written in the following order, first the listed transient states and finally the absorbing states.

P = Q R 0 I
The transition matrix P consists of fours sub-matrices. Q represents the transition between the transient states. R represents the transition from the transient to absorbing states. 0 is a matrix consisting of zero elements meaning that the transition from an absorbing state to a transient state is impossible. I is an identity matrix meaning that it is never possible to leave an absorbing state. Some facts about absorbing chains can be answered using this kind of matrix decomposition [START_REF] Kemeny | Finite Markov Chains[END_REF]. The matrix (I n -Q) -1 denoted by F is called the Markov chain's fundamental matrix. I n has the same size as Q. The ijth element of Matrix F gives the expected number of visits to state j before absorption occurs, given that the current state is state i.

F can be expanded as:

F = I + Q + Q 2 + ... = (I -Q) -1 (B.1)
F is a square matrix where rows and columns correspond to the non-absorbing states.

Equation B.1 implies:

F = Q 0 + Q 1 + Q 2 + ... (B.2)
Therefore Q t (i, j) gives the expected proportion of period t spent in the jth state or in other words, the probability that the process occupies the jth non-absorbing state in period t, given that the process began in the ith non-absorbing state.

The product F R gives the probability that a particular initial non-absorbing state will end up in a particular absorbing state, in case several absorbing states exist.

The expected number of steps (t i ) before the chain is absorbed, given that the starting state is state i, is called Time to Absorption. It is calculated as follows:

t = F c (B.3)
t is a column vector whose ith element is t i and c is a column vector with all elements equal to 1. t is simply the sum of all elements of each row of the fundamental matrix F . The corresponding fundamental matrix (F ) is as follows: wafers are to be refreshed again. Therefore, there is a chance to stay in the refresh state and repeat the process before going to the next step. However, when the wafers repeat the refresh process, the associated yield factor is With λ = 0.9

F = F R1 R2     F
P = F R1 R 1x R2 R 2x R3 Monitor Scrap                             F λ 0 0 0 0 0 1 -λ R1 0 λ0.
P = F N1 N 1x R1 N2 N 2x R3 N3 Monitor Scrap                                      
F = F N1 N'1 R1 N2 N'2 R2                         F 1

Abstract:

The thesis is composed of two parts. In the first part, we take a binding restriction, called "qualification", present in semiconductor manufacturing as a lever for increasing flexibility and optimizing capacity utilization. A qualification determines the processing authorization of a product on a machine. It acts like an eligibility constraint that allows production volume allocation of a product to a machine. In order to define the best qualification, the production volume should be allocated to parallel non-identical machines which are partially reconfigurable. Capacitated flexibility measures are introduced to define the best qualification which increases machine capacity utilization at most. Batching is another industrial constraint encountered in semiconductor industry. It influences workload balancing and hence qualification management. Several workload balancing algorithms are proposed to find the optimal workload balance of a workcenter. Variability measures are also proposed to evaluate the workload variability of a workcenter. The concept is industrialized and is continuously used at Soitec.

The second part of the thesis deals with closed-loop production planning. Soitec uses Smart-Cut™ Technology to fabricate SOI wafers. Using this technology, one of the two raw materials used to fabricate SOI wafers can be reused several times to make other SOI wafers. However, before coming back to the SOI fabrication line, the used raw material (designated as by-product) must be reworked (remanufactured) in another production line. An original closed-loop production planning model adapted to the supply chain specificities of Soitec is proposed, and is validated using industrial data. Based on this industrial model, a single-item uncapacitated closed-loop lotsizing model is defined, analyzed, and a dynamic programming algorithm is proposed for a simplified version of the problem. Mots Clés : Industrie des semi-conducteurs, gestion des qualifications, flexibilité, variabilité, équilibrage de charge, batching, planification de la capacité, dimensionnement de lots bi-niveau à capacité finie, produits dérivés, refabrication, planification de production en boucle fermée, optimisation.

École Nationale

Résumé: La thèse est composée de deux parties. La première partie traite de la gestion des qualifications dans l'industrie des semi-conducteurs. La contrainte de qualification définit l'éligibilité d'une machine à processer un produit. La gestion des qualifications nécessite de résoudre un problème d'allocation et d'équilibrage des charges sur des machines parallèles non-identiques et partiellement reconfigurables. Nous avons défini et introduit des indicateurs pour la gestion des qualifications en tenant compte de la capacité des équipements ainsi que la contrainte de regroupements de lots (batching). Plusieurs algorithmes d'équilibrage de charge sont proposés et validés pour le calcul de la charge optimale sur un parc d'équipements. Ce concept est industrialisé au sein de l'entreprise Soitec et fait partie du processus de prise de décision.

La deuxième partie de la thèse porte sur la planification de production en boucle fermée. Le processus de fabrication des plaques SOI à Soitec s'appuie sur la Technologie Smart-Cut™. En utilisant cette technologie, une des deux matières premières peut être réutilisée à plusieurs reprises pour la fabrication des produits finis. Le couplage de deux lignes de production crée un système manufacturier en boucle fermée. Nous avons proposé un modèle de dimensionnement de lots original pour la planification de production de ce système manufacturier, que nous avons validé avec des données industrielles. En se basant sur le problème industriel, un problème monoproduit et sans contrainte de capacité est défini, analysé et résolu pour une version simplifiée du problème.

Figure 1 . 1 :

 11 Figure 1.1: SOI ou Silicium-sur-Isolant

Figure 1 . 2 :

 12 Figure 1.2: La Technologie Smart-Cut™ Utilisée pour Produire les Plaques SOI

CHAPTER 1 .Figure 1 . 3 :

 113 Figure 1.3: États d'un Équipement ([86])

Wm 1 . 3 . 2 . 1

 1321 IP r,m = W IP r ∀r W IP r,m ≥ 0 ∀r, Gestion des qualifications à capacité finie en terme de volume de production

W

  IP r,m T P r,m • Capa m γ with Q r,m = 1 ∀r, m. (1.5)

Figure 1 . 7 :

 17 Figure 1.7: Le Format des Données d'Entrée

Figure 1 . 8 :

 18 Figure 1.8: Les Flux de Données et d'Information

Figure 1 . 10 .Figure 1 . 10 :Figure 1 . 11 :

 110110111 Figure 1.10: Fabrication de SOI avec un Exemple de Processus de Refresh jusqu'à 7 Niveaux (l max = 7)

Figure 1 . 12 :

 112 Figure 1.12: Le Système Manufacturier Simplifié de la Fabrication de SOI et du Processus de Refresh en Utilisant la Technologie Smart-Cut™

  Part I, we discover different facets of qualification management and capacity planning by considering new restrictions. The successful industrial GENERAL INTRODUCTION implementation of this concept in Soitec was recognized internationally by receiving the Best Applied Paper Award during the Winter Simulation Conference 2013.

Figure 1 . 14 :

 114 Figure 1.14: Structure of the Thesis and the Precedence Relationship between Chapters

Figure 2 . 1 :

 21 Figure 2.1: From Sand to Silicon Wafers

Figure 2 . 2 :

 22 Figure 2.2: Front-End and Back-End Process ([70])

Figure 2 . 3 :

 23 Figure 2.3: Semiconductor Manufacturing and the Daily Life

Figure 2 . 4 :

 24 Figure 2.4: Silicon-On-Insulator

Figure 2 . 5 :

 25 Figure 2.5: Smart-Cut™ Technology Used to Produce SOI Wafers

4. 1

 1 Introduction and Motivation 4.2 Capacitated Flexibility Measures 4.3 Capacitated Workload Balancing 4.4 Managerial Insights 4.5 Conclusions and Perspectives

Figure 4 . 2 ,

 42 the only time slot in which production takes

Figure 4 . 1 :

 41 Figure 4.1: Equipment States Stack Chart ([86])

Figure 4 . 2 :

 42 Figure 4.2: Total Time Components

Figures 4. 3

 3

Figure 4 . 3 :

 43 Figure 4.3: Toolset Workload Balance Using Uncapacitated Time FM (γ = 2)

Figure 4 . 4 :

 44 Figure 4.4: Toolset Workload Balance Using Uncapacitated Time FM (γ = 10)

Figure 4 . 5 :Figure 4 . 6 :

 4546 Figure 4.5: Toolset Workload Balance Using Capacitated Time FM (γ = 2)

Figure 4 . 7 :

 47 Figure 4.7: Toolset Workload Balance Using Capacitated Time FM (γ = 10) -Twice More Load

Figure 4 . 8 :

 48 Figure 4.8: Workload balancing exponent (γ) variation versus Capacitated Time FM (F T ime Capa ), Time Capacity Deviation Datio (DR T ime Capa ) and toolset capacity utilization percentage

  [START_REF] Damodaran | Minimizing Makespan on a Batch-Processing Machine with Non-Identical Job Sizes Using Genetic Algorithms[END_REF], then set k = k + 1 and r * = 1 and go to Step 1.

1

 1

5. 1

 1 Introduction and Motivation 5.2 Optimization Model and Complexity Analysis 5.3 Workload Balancing with Batch Size Restrictions 5.4 Numerical Experiments 5.5 Conclusions and Perspectives

  m * , then there are one (or several) full-batch(es) and one real-valued WIP allocation. The full-batch 92 5.3 Workload Balancing with Batch Size Restrictions WIP allocation(s) is (are) equal to W IP r,m * = W IP r,m * BS r,m * BS r,m * . The remainder (W IP rr W IP r,m * where W IP r,m * = W IP r,m * BS r,m * BS r,m * ) is added in the last loop to the least-loaded loading machine in the loading machine set.

Figure 6 . 1 :

 61 Figure 6.1: Dedicated Strategy (a), Partial or Sparse Flexibility (b) and Full Flexibility (c) Recipe-to-Machine Configurations (adapted from[START_REF] Graves | Process Flexibility in Supply Chains[END_REF] and[START_REF] Muriel | Impact of Partial Manufacturing Flexibility on Production Variability[END_REF])

(6. 9 )

 9 Equations (4.22), (4.24) and (4.25) are respectively replaced by:

12 )

 12 OL Comparison = (OL N ew Conf ig. -OL Current Conf ig. ) OL Current Conf ig. × 100 (6.13) First, we consider an industrial instance of a toolset consisting of 22 machines and 37 recipes for a single period. Diagram 6.2 depicts the current toolset workload balance. The vertical lines correspond to machine capacities. Each horizontal bar represents the workload of a machine. Bars above the capacity lines show the overloaded machines, and the opposite for underloaded machines. By calculating the

Figure 6 . 2 :

 62 Figure 6.2: Toolset Workload Balance for the Current Recipe-to-Machine Qualification Configuration

Figure 7 . 1 :

 71 Figure 7.1: Overall Decision Making Process

Figure 7 .Chapter 7 .

 77 2 is a snapshot of the control sheet of the application. Using the control panel, it is possible to choose the necessary functions (flexibil-Industrialization of Qualification Management ity measures or variability measures, capacitated or uncapacitated, considering or ignoring not previously qualified recipes, different resolution approaches, etc.)In a separate sheet the required parameters of the model are entered. These include: Load Balancing Exponent (γ), Overload Balancing Exponent (γ o ), Underload Balancing Exponent (γ u ), Overload Weight (o), Underload Weight (u) and System Flexibility Weight (a).

Figure 7 . 2 : 7 . 3

 7273 Figure 7.2: Control Panel of the Application: Qualification Management Optimization Application

Figure 7 .

 7 Figure 7.2 shows a snapshot of the Microsoft Excel™-based application, developed using the programming language VBA™. The data are extracted from an image of the MES (Manufacturing Execution System) database. Using SQL (Structured Query Language), the production route of each part is identified. The steps and the recipes are extracted. Then the production volume per product reference is crossreferenced with the recipes to give the production volume (or WIP) per recipe. Figure 7.3 shows the control sheets of the data extractor. Throughputs, batch sizes and qualification statuses are extracted automatically via another Excel-based application (called Input file Creator, Figure 7.4) to separate worksheets. Finally all data are exported via one Excel workbook containing four worksheets to the Qualification Management Optimization Application (Figure 7.5).Figure 7.6 shows an example of a data extraction.

  Figure 7.2 shows a snapshot of the Microsoft Excel™-based application, developed using the programming language VBA™. The data are extracted from an image of the MES (Manufacturing Execution System) database. Using SQL (Structured Query Language), the production route of each part is identified. The steps and the recipes are extracted. Then the production volume per product reference is crossreferenced with the recipes to give the production volume (or WIP) per recipe. Figure 7.3 shows the control sheets of the data extractor. Throughputs, batch sizes and qualification statuses are extracted automatically via another Excel-based application (called Input file Creator, Figure 7.4) to separate worksheets. Finally all data are exported via one Excel workbook containing four worksheets to the Qualification Management Optimization Application (Figure 7.5).Figure 7.6 shows an example of a data extraction.

Figure 7 . 3 :

 73 Figure 7.3: Data Extractor (WIP per Recipe) Application

Figure 7 . 4 :

 74 Figure 7.4: Input File Creator

Figure 7 . 5 :

 75 Figure 7.5: Data and Information Flow

Figure 7 . 6 : 4 ResultsFigure 7 . 7 :

 76477 Figure 7.6: Input Data Format

Figure 7 . 8 :

 78 Figure 7.8: Flexibility GainTable Containing one Not Previously Qualified Recipe (NQR), i.e. "Recipe 3-Alternate Recipe 3"

Figure 7 . 9 :Figure 7 . 10 :Figure 7 . 12 :

 79710712 Figure 7.9: Workload Balancing after Qualifying "Recipe 3-Alternate Recipe 3" on "Machine 16"

Figure 9 . 1 :

 91 Figure 9.1: Supply Chain (An example)

Figure 9 . 2 :

 92 Figure 9.2: Different Production Environments and Customer Order Decoupling Point ([49])

  A workcenter (or a workshop) may consist of one or several machines with different characteristics. A single machine workcenter (workshop) is the most basic resource configuration. In a parallel machines configuration, several similar machines are gathered together in a workcenter (or workshop). A configuration with parallel machines may be further classified into Identical Machines, Uniform Machines and Unrelated Machines. Identical Machines have the same throughput. Uniform Machines have proportional throughputs. Unrelated Machines have different throughputs.

Figure 9 . 5 :

 95 Figure 9.5: Supply Chain Planning Matrix (Adapted from [98])

Figure 9 . 6 :

 96 Figure 9.6: Classification of Lot-Sizing Problems for single-and multi-level BOM ([102])

Figure 10 . 2 :

 102 Figure 10.2: SOI Fabrication and an Example of Refresh Process up to 7 Levels (l max = 7)

Chapter 10 . 1 (Figure 10 . 3 :

 101103 Figure 10.3: A Simple SOI Production-Refresh Supply Chain

Figure 11 . 2 :

 112 Figure 11.2: SOI Fabrication and Refresh Process Cycles

i∈S 1 aProposition 11 . 3 . 1 .

 11131 i = i∈S 2 a i = A)? The ULS-MR problem is NP-hard.

Figure 11 . 3 :

 113 Figure 11.3: Material Flow Representation

Chapter 11 .

 11 Single-Item Bi-Level Uncapacitated Lot-Sizing with Multiple Remanufacturing of Reusable By-Products SOI Inventory cost ≥ Top Wafer Inventory cost ≥ Negative Wafer Inventory cost Fresh Wafer (R0) Inventory cost ≥ Refresh Wafer 1 (R1) Inventory cost ≥ Refresh Wafer 2 (R2) Inventory cost ≥ ... Negative Wafer (Neg0) Inventory cost ≥ Negative Wafer 1 (Neg1) Inventory cost ≥ Negative Wafer 2 (Neg2) Inventory cost ≥ ...

Figure 11 . 6 : 3 .

 1163 Figure 11.6: Inventory Cost Assumptions

Proposition 11 . 4 . 1 .

 1141 In an optimal solution of ULS-MR 1 , with l max ≥ 1 and T > 1,T t=1 dt min{T,l max +1} ≤ T t=1 p t ≤ T t=1 d t .

Figure 11 . 7 :

 117 Figure 11.7: Material Flow Representation of ULS-MR 1 for l max = 2 and T = 4

Figure 11 . 9 : 4 11. 5 . 3 1 FP

 1194531 Figure 11.9: Material Flow Representation of ULS-MR 1 FP for l max = 2 and T = 4

,4, 4 Figure 11 . 10 :

 41110 Figure 11.10: Tree Representing the States of the DP for an Instance of Four Periods

Figure 11 . 11 :

 1111 Figure 11.11: Tree Representing the States of the DP for an Instance of Four Periods with Correct Labelling

Chapter 11 .

 11 Single-Item Bi-Level Uncapacitated Lot-Sizing with Multiple Remanufacturing of Reusable By-Products (11.56b)) until all positions are determined. C(-, • • • , -) = 0 (11.56a) C(1, • • • , -) = By ordering p 1,1 using (11.53) and (11.54) (11.56b)

12. 2

 2 Perspectivesof how on combine lot-sizing and qualification management. Consider the following parameters and variables. r is qualifiable on machine m,0 if recipe r is not qualifiable on machine m. pt r,mProcess time of recipe r on machine m: recipe r is qualifiable or already qualified on machine m,Capacityof machine m in period t, d t r Demand of recipe r in period t, cp r,m Production cost of recipe r in period t, h r Holding cost of recipe r, sp t r,m Setup cost if recipe r is produced on machine m in period t, sq t r,m Setup cost if recipe r is qualified on machine m. Production volume of recipe r assigned to machine m in period t, Y t r,m Setup binary variable of recipe r assigned to machine m in period t, s t r Inventory of recipe r in the end of period t,

Chapter 12 .

 12 General Conclusions and Perspectivesmathematical programming would be as follows:

  R set of all recipesKproduction run (k ∈ 1, ..., ⊆ r wipr minBSr ), M set of all machines, r is qualified on machine m, r is qualifiable on machine m, 0 if recipe r is not qualifiable on machine m. wip r total production volume of recipe r, minBS r,m minimum batch size of recipe r on machine m, maxBS r,m maximum batch size of recipe r on machine m, v number of desired qualifications.

  ,k ≥ aX r,k a = M in r (X r,k ) b = M ax r (X r,k ) (A.7) Y r,k X r,k ≤ bY r,k ∀r (A.8) Y r,k X r,k ≥ aY r,k ∀r (A.9) Y r,k X r,k ≤ X r,k -a(1 -Y r,k ) ∀r (A.10) Y r,k X r,k ≥ X r,k -b(1 -Y r,k ) ∀r (A.11) k Y r,k X r,k = wip r ∀r (A.12) r Y r,k = 1 ∀k (A.13) X r,k ≥ minBS r Y r,k ∀k (A.14) X r,k ≤ maxBS r Y r,k ∀k (A.15)a and b, defined respectively as the minimum and maximum values of the variable X r,k over r, corresponds to minBS r and maxBS r .According to A.7, Constraints A.14 and A.15 can be written as:0 ≤ Y r,k X r,k ≤ maxBS r . (A.16)

95 *250

 95 λ(.95 * .98) 0 λ(.95 * .015) λ(.95 * .005) + (1 -λ) .98) λ(.95 * .015) λ(.95 * .005) + (1 -λ)Let λ be 90%, then F and F R are calculated as follows:

Figure B. 4 :

 4 Figure B.4: Graphical Representation of Transition Matrix

Figure B. 5 :

 5 Figure B.5: Graphical Representation of Transition Matrix

  1.1 SOI ou Silicium-sur-Isolant . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 La Technologie Smart-Cut™ Utilisée pour Produire les Plaques SOI . 4 1.3 États d'un Équipement ([86]) . . . . . . . . . . . . . . . . . . . . . . 8 1.4 L'Équilibrage de Charge pour la Configuration Actuelle des Qualifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Structure of the Thesis and the Precedence Relationship between Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.1 From Sand to Silicon Wafers . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 Front-End and Back-End Process ([70]) . . . . . . . . . . . . . . . . . 35 2.3 Semiconductor Manufacturing and the Daily Life . . . . . . . . . . . 35

	1.1 Introduction
	1.2 Contexte Industriel
	1.3 Gestion des Qualifications
	1.4 Planification de Production à Boucle Fermée
	1.5 Conclusions et Perspectives Générales
	15

1.5 L'Équilibrage de Charge pour la Configuration après une Qualification Meilleure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.6 Les Variations de la Variabilité des Charges, des Surcharges, et de la Capacité Non-Utilisée versus le Nombre de Nouvelles Qualifications . 16 1.7 Le Format des Données d'Entrée . . . . . . . . . . . . . . . . . . . . 18 1.8 Les Flux de Données et d'Information . . . . . . . . . . . . . . . . . . 19 1.9 La Vue Globale du Processus de Prise de Décision, l'Application de la Gestion des Qualifications et ses Interfaces . . . . . . . . . . . . . . 20 1.10 Fabrication de SOI avec un Exemple de Processus de Refresh jusqu'à 7 Niveaux (l max = 7) . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.11 Une Simple Chaîne Logistique de Fabrication de SOI et de Processus de Refresh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.12 Le Système Manufacturier Simplifié de la Fabrication de SOI et du Processus de Refresh en Utilisant la Technologie Smart-Cut™ . . . . . 25 1.13 Le Schéma du Système Manufacturier SOI-Refresh Simplifié après les Hypothèses "Full Push" . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.14

Table 1 .

 1 1: Nombre de nouvelles qualifications versus la réduction de la variabilité et les variations des indicateurs de performance

	Variation

Table 1 .

 1 2: La réduction de la variabilité en faisant une nouvelle qualification

			Variation
	Numéro	Variabilité Charge Surcharge	Capacité
	d'Instance				Non-Utilisée
	1	-4,13%	3,40% -100,00%	-20,20%
	2	-34,52%	2,16%	-46,76%	-7,36%
	3	-76,11%	4,43%	-89,42%	-22,41%
	4	-43,39%	3,99%	-28,97%	-16,76%
	5	-3,29%	2,20%	-21,47%	-18,06%
	6	-21,90%	2,29%	-20,05%	-15,80%
	7	-62,82%	1,06%	-5,80%	-5,03%
	8	-99,76%	4,57%	-30,10%	-87,41%
	9	-49,65%	1,75%	-18,86%	-18,23%
	10	-4,71%	1,20%	-1,41%	-5,49%

de la flexibilité au parc d'équipements et diminuerait la variabilité.

Planification de Production en Boucle Fermée 1.4 Planification de Production en Boucle Fermée 1.4.1 Motivations et État de l'Art

	1.4
			Toolset Capacity
			Maintenance
	Production Volume Per Recipe		Process time
	Fab DataBase	Weekly Operational	Input File Creator	Batch size Qualification Status
		Plan		Process Department
		QUALIFICATION MANAGEMENT
		DECISION SUPPORT SYSTEM
	Qualification	Optimized	Optimized WIP
	Propositions	Capacity	Distribution
	Process		Capacity Planning	Production Control
				sont pas
	prises en compte dans ces travaux. Entre autres, on peut citer des ressources aux-
	iliaires, des consommables (périssables ou non), la séquence de passage des lots sur
	les machines, le système de manutention, etc. Ces contraintes ont un impact sur
	l'allocation des charges et la gestion des qualifications.

Figure 1.9: La Vue Globale du Processus de Prise de Décision, l'Application de la Gestion des Qualifications et ses Interfaces

thèse ont été mis en place au sein de l'entreprise Soitec. Le processus de prise de décision et d'industrialisation est le sujet du Chapitre 7. Plusieurs perspectives sont imaginables comme suite de ces travaux. Les coûts de qualifications ne sont pas considérés lors de la proposition de nouvelles qualifications. Aujourd'hui, ce sont les utilisateurs qui choisissent les meilleures qualifications selon les gains en terme de flexibilité tout en considérant les coûts et les difficultés associés. Donc, la considération du coût de qualification (en fonction du temps, du coût monétaire ou des difficultés), apporte une dimension de plus à la gestion des qualifications. Chaque parc d'équipements pourrait imposer des contraintes industrielles qui ne

Base Silicon (Si) Ultra-Thin Buried Oxide (BOX) (SiO 2 ) Thin or Ultra-Thin Top Silicon (Si) Handle (Support) Substrate Active Substrate Insulator (Dielectric)

4.2 Capacitated Flexibility Measures

		150						150
	hours)	125					hours)	125
	Time (in	75 100					Time (in	75 100
	Process	50					Process	50
	Total	25					Total	25
		0						0
		M1	M2	M3	M4	M5		M1	M2	M3	M4	M5
			Machines					Machines
	Recipe 1	Recipe 2		Recipe 3		Recipe 1	Recipe 2		Recipe 3
	Recipe 4	Recipe 5		Recipe 6		Recipe 4	Recipe 5		Recipe 6
	Recipe 7	Recipe 8		Maximum Capacity	Recipe 7	Recipe 8		Maximum Capacity
	(a) Current Process Time Distribution	(b)	Process	Time	Distribu-
	(F T ime Capa = 61.1% and DR T ime Capa = 22.9%)	tion	after	Best		Qualification
							(F T ime Capa = 95.5% and DR T ime Capa = 20.4%)

  ime Capa . However, DR T ime Capa and the average toolset capacity utilization variations stabilize for large values of γ. Capacity Deviation

	100%
	80%
							Capacitated Time
	60%						Flexibility after Best
							Qualification
							Capacity Deviation Ratio
	40%						after Best Qualification
							Capacity Utilization
							Percentage after Best
	20%						Qualification
	0%
	2	3	4	5	6	7	8
		Workload Balancing Exponent

W IP and Chapter 5. Flexible Qualification Management under Batch Size Constraint F

  T ime ), W IP r,m is to be substituted with Run r,m BS r,m + W IP C r,m . Run r,m BS r,m states that the WIP allocation of recipe r on machine m is an integer multiple (i.e. number of full-batch runs) of the batch size BS r,m of recipe r on machine m.

	max	F	(5.1a)
		M
	Subject to
		m=1

Table 5 .

 5 2 shows that the adapted active-set method (Section 5.3.2.1) yields very good solutions that are close to the optimal ones. The batch-feasibility heuristic

Chapter 5. Flexible Qualification Management under Batch Size ConstraintTable 5 .

 5 1: WIP flexibility with γ = 2 for the current toolset recipe-to-machine qualification configuration

	Instance Real-value Full batch Batch feasibility CPLEX
	1	95.10%	95.05%	94.92%	95.10%
	2	94.89%	90.55%	90.35%	90.58%
	3	96.99%	96.97%	96.83%	96.99%
	4	95.53%	94.26%	94.06%	95.53%
	5	87.94%	82.47%	82.42%	83.94%
	6	99.41%	94.85%	94.66%	94.89%
	7	76.59%	76.34%	76.32%	76.59%
	8	88.58%	67.23%	67.08%	67.49%
	9	96.26%	68.50%	68.55%	68.75%
		99.81%	99.03%	96.00%	99.81%
		96.60%	96.02%	94.49%	96.60%
		88.72%	88.58%	86.69%	88.72%
		96.31%	95.12%	91.33%	96.31%
		90.80%	86.14%	85.22%	86.48%
		94.80%	94.25%	92.83%	94.80%
		92.60%	87.27%	84.92%	88.19%
		95.66%	90.29%	87.55%	91.10%
		90.92%	85.94%	83.39%	86.59%
		93.84%	84.65%	83.12%	84.88%
		84.42%	76.38%	75.29%	76.74%
		86.31%	81.75%	79.89%	82.39%
		83.67%	82.64%	81.75%	83.67%
		85.43%	85.17%	83.53%	85.43%
		90.08%	89.73%	88.48%	90.08%
		89.57%	89.08%	85.69%	89.57%
	(Section 5.3.2.

Table 5 .

 5 2: Time flexibility with γ = 2 for the current toolset recipe-to-machine qualification configuration

	Instance Real-value Batch feasibility Adapted Active-Set CPLEX
	1	84.79%	84.51%	84.63%	84.67%
	2	83.02%	82.74%	82.85%	83.01%
	3	98.06%	97.92%	96.54%	97.96%
	4	99.56%	98.42%	92.91%	98.92%
	5	89.87%	89.70%	89.62%	89.70%
	6	97.24%	96.97%	94.94%	97.03%
	7	98.58%	97.82%	97.82%	97.95%
	8	82.90%	78.35%	82.45%	82.45%
	9	76.67%	75.64%	76.44%	76.44%
		97.57%	91.41%	96.34%	97.14%
		99.86%	99.39%	99.54%	99.79%
		96.68%	95.50%	96.31%	96.31%
		93.60%	92.67%	90.74%	92.90%
		95.92%	92.36%	91.46%	92.36%
		99.65%	95.63%	98.88%	99.09%
		96.83%	91.83%	96.14%	96.38%
		95.21%	92.57%	94.80%	95.01%
		96.77%	94.26%	96.57%	96.74%
		94.82%	92.89%	93.54%	93.85%
		96.50%	91.76%	94.66%	96.31%
		96.16%	92.96%	92.06%	92.96%
		99.86%	97.27%	97.85%	97.85%
		99.14%	94.18%	98.18%	98.86%
		99.20%	96.55%	98.80%	99.11%
		98.31%	95.22%	98.23%	98.25%

Table 5 . 3

 53

	batch	Batch feasibility

: Comparing solution approaches for WIP flexibility measure (γ = 6).

Flexible Qualification Management under Batch Size Constraint 12

  (75.30% and 69.25%). This illustrates that not considering batch sizes may induce less effective qualification decisions. Note that the full-batch algorithm is quite efficient, considering the (relatively small) gap with the results of the real-valued resolution. As a conclusion, among the workload balancing algorithms and when considering batch sizes for the WIP flexibility measure, the full-batch algorithm is the most efficient.

	Chapter 5. Full batch		Batch feasibility
	Instance	Best	Best (RV) Match	Best	Best (RV) Match
	1	86.53%	86.53%	*	85.46%	85.46%	*
	2	67.22%	67.22%	*	65.86%	65.86%	*
	3	77.61%	77.61%	*	76.37%	74.56%
	4	70.66%	70.66%	*	69.81%	69.81%	*
	5	31.46%	0.70%		5.09%	3.41%
	6	96.96%	96.96%	*	93.69%	93.69%	*
	7	14.23%	14.07%		14.16%	14.09%
	8	20.41%	20.41%	*	19.55%	19.55%	*
	9	91.72%	83.59%		64.39%	46.92%
	10	60.88%	60.88%	*	59.87%	59.72%
	11	32.87%	32.87%	*	31.63%	31.63%	*
	12	75.30%	69.25%		75.11%	66.39%
	13	49.36%	49.36%	*	45.80%	45.80%	*
	14	49.01%	47.31%		47.28%	46.83%
	15	33.05%	33.05%	*	33.61%	33.61%	*
	16	63.63%	63.63%	*	46.71%	46.71%
	17	44.62%	44.62%	*	29.28%	29.28%	*
	18	37.32%	37.32%	*	26.75%	21.62%	*
	19	23.75%	23.75%	*	16.56%	13.80%
	20	35.13%	35.13%	*	32.51%	29.29%
	21	18.76%	18.13%		19.14%	18.42%
	22	21.34%	21.34%	*	21.35%	21.07%
	23	32.23%	32.23%	*	28.69%	26.79%
	24	31.22%	31.22%	*	30.05%	24.75%
	25	67.67%	67.54%		66.40%	64.56%
	*: Proposed qualifications are identical.
			31.46% and 0.70%), 9 (91.72% and 83.59%) and

Table 5 .

 5 4: Comparing best qualifications for WIP flexibility measure.

Table 5 .

 5 5: Comparing solution approaches for the Time flexibility measure.

	Instance	Real value	Batch feasibility	Adapted Active-Set
	number Current	Best	Current	Best	Current	Best
	1	40.91% 72.68% 40.43% 66.89% 40.40% 69.78%
	2	34.29% 49.75% 32.83% 45.60% 33.79% 48.08%
	3	85.53% 93.43% 82.92% 88.27% 80.54% 90.93%
	4	33.89% 38.44% 32.94% 36.84% 33.10% 37.45%
	5	34.42% 63.71% 33.44% 61.90% 33.48% 61.94%
	6	82.58% 94.82% 80.03% 92.72% 79.29%	92.49%
	7	98.00% 99.99% 62.11% 81.03% 66.91% 85.87%
	8	29.34% 41.40% 13.37% 38.72% 25.68% 37.24%
	9	28.02% 49.44% 17.36% 34.27% 17.16%	31.52%
		67.82% 85.11% 20.89% 49.74% 18.36%	49.12%
		99.22% 99.55% 43.64% 49.50% 47.00% 66.12%
		81.14% 96.45% 49.58% 57.28% 67.04% 69.57%
		48.91% 68.81% 45.06% 49.68% 45.01% 53.91%
		70.41% 97.50% 29.66% 47.30% 38.26% 48.66%
		99.64% 99.96% 37.27% 58.74% 55.10% 58.68%
		74.42% 88.54% 27.43% 27.72% 26.28%	27.43%
		61.70% 76.55% 13.68% 32.44% 13.09% 57.54%
		75.33% 92.15% 38.60% 52.36% 44.72% 58.34%
		74.70% 83.46% 26.95% 51.60% 26.95% 64.95%
		69.94% 84.04% 54.98% 61.85% 59.28% 63.86%
		71.49% 85.33% 13.32% 41.37% 33.40% 46.46%
		95.20% 98.17% 45.70% 59.68% 47.06% 71.61%
		97.70% 99.13% 42.96% 78.09% 68.30% 72.30%
		96.03% 98.01% 58.23% 68.45% 58.06% 73.84%
		92.74% 95.00% 26.52% 55.44%

29.82% 61.93%

Table 5 .

 5 6: Comparing best qualifications for Time flexibility measure. allocations considering batch sizes, it may be possible to use the workload balancing allocations as an input for a scheduler or a dispatcher of lots. The batch-feasibility heuristic might be improved by taking the flexibility measure value (objective function value) into account. This is achievable by adding another condition while workload re-allocating. This means that we calculate the flexibility value if we have

		Batch feasibility		Adapted Active-Set
	number	Best	Best (RV) Match	Best	Best (RV) Match
	1	66.89%	66.89%	*	69.78%	69.78%	*
	2	45.60%	45.60%	*	48.08%	48.08%	*
	3	88.27%	88.27%	*	90.93%	90.93%	*
	4	36.84%	36.84%	*	37.45%	37.45%	*
	5	61.90%	61.90%	*	61.94%	61.94%	*
	6	92.72%	92.72%	*	92.49%	92.49%	*
	7	81.03%	78.64%		85.87%	68.68%
	8	38.72%	25.56%		37.24%	29.21%
	9	34.27%	19.42%		31.52%	18.19%
		49.74%	49.74%	*	49.12%	49.12%	*
		49.50%	49.19%		66.12%	60.73%
		57.28%	52.51%		69.57%	68.57%
		49.68%	45.28%		53.91%	47.21%
		47.30%	47.30%	*	48.66%	48.66%	*
		58.74%	58.74%	*	58.68%	57.70%
		27.72%	27.46%		27.43%	26.60%
		32.44%	25.99%		57.54%	57.54%	*
		52.36%	52.36%	*	58.34%	58.34%	*
		51.60%	27.04%		64.95%	37.47%
		61.85%	57.83%		63.86%	60.58%
		41.37%	26.21%		46.46%	46.46%	*
		59.68%	48.69%		71.61%	49.62%
		78.09%	67.44%		72.30%	71.57%
		68.45%	63.22%		73.84%	64.23%
		55.44%	36.00%		61.93%	30.93%

*: Proposed qualifications are identical.

Chapter 5. Flexible Qualification Management under Batch Size Constraint had

  distributed the (removed) WIP for the machines of the same recipe having a non-integer WIP value. And if several alternatives exist, the WIP allocation is chosen which yields the largest flexibility measure.

Recipe 1 Recipe 2 Recipe 3 Alternate Recipe 3 Recipe 4 Alternate Recipe 4 Recipe 5 Recipe 6 Recipe 7 Alternate Recipe 7 Recipe 8 Alternate Recipe 8 Recipe 9 Alternate Recipe 9 Recipe 10 Recipe 11 Recipe 12 Alternate Recipe 12 Recipe 13 Recipe 14 Recipe 15 Recipe 16 Alternate Recipe 16 Recipe 17 Recipe 18 Recipe 19 Alternate Recipe 19 Recipe 20 Alternate Recipe 20 Recipe 21 Recipe 22 Recipe 23 Recipe 24 Alternate Recipe 24 Recipe 25 Recipe 26 Recipe 27 Recipe 28 Recipe 29 Recipe 30 Recipe 31 Recipe 32 Recipe 33 Recipe 34 Alternate Recipe 34 Recipe 35 Alternate Recipe 35 Recipe 36 Recipe 37 Maximum Capacity

Recipe 10 Recipe 11 Recipe 12 Alternate Recipe 12 Recipe 13 Recipe 14 Recipe 15 Recipe 16 Alternate Recipe 16 Recipe 17 Recipe 18 Recipe 19 Alternate Recipe 19 Recipe 20 Alternate Recipe 20 Recipe 21 Recipe 22 Recipe 23 Recipe 24 Alternate Recipe 24 Recipe 25 Recipe 26 Recipe 27 Recipe 28 Recipe 29 Recipe 30 Recipe 31 Recipe 32 Recipe 33 Recipe 34 Alternate Recipe 34 Recipe 35 Alternate Recipe 35 Recipe 36 Recipe 37 Maximum Capacity

	Recipe 1	Recipe 2	Recipe 3	Recipe 4	Recipe 5	Recipe 6	Recipe 7	Recipe 8	Recipe 9
			Alternate Recipe 3	Alternate Recipe 4			Alternate Recipe 7	Alternate Recipe 8	Alternate Recipe 9

Figure 6.3: Toolset Workload Balance for the Configuration After Performing One New Qualification to perform new qualifications, the toolset capacity allocation improves. However, instead of showing the workload diagram, variability and performance indicators are presented in Table 6.1. It shows the workload variability, overload and unused capacity reduction and used capacity increase percentages after new qualifications,

Table 6 .

 6

	1: Number of new qualifications versus variability reduction and perfor-
	mance indicators variations
									Variation
		New		Variability Workload Overload Unused
	Qualification(s)								Capacity
		1			-77.44%		3.33%		-37.33% -32.72%
		2			-86.32%		4.08%		-47.11% -40.02%
		3			-89.94%		4.55%		-57.58% -44.66%
		4			-94.41%		4.94%		-59.48% -48.48%
		5			-95.58%		5.22%		-62.22% -51.20%
		6			-96.08%		5.28%		-61.99% -51.83%
		7			-96.12%		5.23%		-65.82% -51.31%
		8			-96.60%		5.48%		-66.75% -53.78%
		9			-97.55%		5.84%		-72.56% -57.33%
		10			-97.55%		5.84%		-72.56% -57.33%
		10%
		0%
		-10%
		-20%
	Variation	-50% -40% -30% -60%										Variability Workload Overload
		-80% -70%										Unused Capacity
		-90%
		-100%
		0	1	2	3	4	5	6	7	8	9	10
					Number of New Qualifications
	Figure 6.4: Toolset Variability, Workload, Overload and Unused Capacity Variations
	versus Number of New Qualifications
	that the variations do not follow a linear pattern. Depending upon the data set and
	toolset configuration, nearly the same amount of variability reduction can lead to
	more or less impact on the performance measures. This is illustrated in Instances
	1 and 10. While both instances record a variability decrease of about four percent
	4% (-4.13% and -4.71%), the variations of workload (3.40% and 1.20%), overload

Chapter 10. Multi-Item Bi-Level Capacitated Lot-Sizing with Multiple Remanufacturing of Reusable By-Products

	1
	10.1 Introduction
	10.2 Problem Definition
	10.3 Mathematical Model
	10.4 Complexity Analysis
	10.6 Numerical Experiments
	10.5 Industrial Extensions
	10.7 Conclusions and Perspectives

Chapter 10. Multi-Item Bi-Level Capacitated Lot-Sizing with Multiple Remanufacturing of Reusable By-Products

  Wafer can be obtained via the refresh process from f (Negative of either a Fresh or Refresh Wafer), Unitary refresh cost (∀f|n f ∈ {1, • • • , n max f }) of Top Wafer f at site m in period t (The refresh cost of all refresh levels are the same), Used quantity of Top Wafer f in period t to produce product i, S i,t Inventory level of product (SOI) i at the end of period t, S b,t Inventory level of Base Wafer b at the end of period t, S + Inventory level of Top Wafer f at the end of period t, S - Inventory level of Negative of Top Wafer f (used Fresh or Refresh Wafer) at the end of period t,

	0 otherwise. Refresh resource capacity at site m in period t, Process time of refreshing one unit of Top Wafer f at site m, Production resource capacity in period t, Process time of producing one unit of product i, Initial inventory level of Base Wafer b, Initial inventory level of Top Wafer f , Initial inventory level of Negative of Top Wafer f (used Fresh or Refresh Wafer), Initial inventory level of product i, Quantities of Base Wafer b already planned to be received (in tran-sit) at the beginning of period t, Fresh quantities of f already planned to be received (in transit) at the beginning of period t, Refresh quantities of f already planned to be received (in transit) at the beginning of period t, Unitary inventory cost of Base Wafer b at the end of period t, Unitary inventory cost of Top Wafer f at the end of period t, Unitary inventory cost of Negative of Top Wafer f at the end of period t, Unitary inventory cost of product i at the end of period t, Purchase cost of Top Wafer f (∀f |n f = 0) (Fresh Wafer) in period t, Purchase cost of Base Wafer b in period t, Fresh Wafer purchase order cost in period t, Base Wafer purchase order cost in period t, sr m,t Refresh setup cost at site m in period t, rc m,t β f,m pc t η i S b,0 S + f,0 S -f,0 S i,0 Bb,t Ff,t Rf,t h b,t h + f,t h -f,t h i,t cp f,t cp b,t sp F t sp B t sg t Production setup cost in period t. Variables G i,t Produced quantity of product (SOI) i in period t, B b,t Ordered quantity of Base Wafer b in period t, F f,t Ordered quantity of Fresh Wafer f (f |n f = 0) in period t, R f,f ,m,t Refreshed quantity f (Refresh Wafer) obtained from Negative f at site m in period t, X B i,b,t Used quantity of Base Wafer b in period t to produce product i, X F i,f,t f,t f,t Y t    cr f,m,t cg i,t Unitary production cost of product i in period t,  

Chapter 10. Multi-Item Bi-Level Capacitated Lot-Sizing with Multiple Remanufacturing of Reusable By-Products

  t(10.8)

∀i η i G i,t ≤ pc t ∀t (10.9)

Table 10 .

 10 1: Parameters for generating data sets

	Parameter	Value
	I	10, 20, 50, 100
	F	6, 12, 18
	B	4, 5, 6, 7
	T	6, 12, 24, 48
	M	2, 3, 4
	d i,t Uniformly drawn from[1000, 3000]

Table 10 .

 10 3: Cost components for different data sets

	Parameters	Top (Fresh) Wafer Procurement	Results Base Wafer Procurement	SOI Production	Refresh Process
	CTF	Purchase Procurement	Inv.	Purchase Procurement Inv. Production Setup Inv. Refresh Setup Neg. Inv.
	1	9.36%	0.03%	0.01%	20.43%	0.20%	0.00%	61.30%	1.02% 0.34%	7.07%	0.23%	0.00%
	1.2	9.33%	0.03%	0.01%	20.43%	0.21%	0.00%	61.29%	1.04% 0.36%	7.07%	0.23%	0.00%
	1.6	9.06%	0.03%	0.01%	20.50%	0.20%	0.00%	61.49%	1.01% 0.34%	7.13%	0.23%	0.00%
	2	9.17%	0.03%	0.01%	20.46%	0.21%	0.00%	61.39%	1.03% 0.36%	7.11%	0.24%	0.00%
	T
	6	18.63%	0.06%	0.02%	18.65%	0.18%	0.00%	55.95%	0.92% 0.34% 5.08%	0.16%	0.00%
	12	10.17%	0.03%	0.01%	20.28%	0.20%	0.00%	60.83%	1.01% 0.35% 6.89%	0.22%	0.00%
	24	5.32%	0.02%	0.01%	21.20%	0.21%	0.00%	63.60%	1.07% 0.37% 7.93%	0.26%	0.00%
	48	2.72%	0.01%	0.00%	21.71%	0.22%	0.00%	65.14%	1.08% 0.34% 8.49%	0.28%	0.00%
	I
	10	9.05%	0.05%	0.02%	20.19%	0.31%	0.01%	60.57%	1.54% 0.91%	7.01%	0.35%	0.00%
	20	9.33%	0.04%	0.01%	20.37%	0.26%	0.00%	61.11%	1.29% 0.26%	7.05%	0.29%	0.00%
	50	9.23%	0.02%	0.00%	20.67%	0.12%	0.00%	62.01%	0.62% 0.00%	7.18%	0.14%	0.00%
	100	9.38%	0.01%	0.00%	20.74%	0.06%	0.00%	62.23%	0.31% 0.00%	7.19%	0.07%	0.00%
	F
	6	9.32%	0.04%	0.03%	20.22%	0.27%	0.00%	60.66%	1.35% 0.78%	6.98%	0.33%	0.01%
	12	9.18%	0.03%	0.01%	20.45%	0.21%	0.00%	61.36%	1.03% 0.39%	7.10%	0.23%	0.00%
	18	9.26%	0.03%	0.01%	20.48%	0.20%	0.00%	61.45%	0.98% 0.27%	7.10%	0.22%	0.00%
	B
	4	9.20%	0.03%	0.01%	20.47%	0.20%	0.00%	61.40%	1.00% 0.35%	7.10%	0.23%	0.00%
	5	9.14%	0.03%	0.01%	20.47%	0.21%	0.00%	61.42%	1.03% 0.34%	7.11%	0.23%	0.00%
	6	9.30%	0.03%	0.01%	20.44%	0.21%	0.00%	61.32%	1.03% 0.36%	7.08%	0.23%	0.00%
	7	9.27%	0.03%	0.01%	20.45%	0.21%	0.00%	61.34%	1.03% 0.34%	7.08%	0.23%	0.00%
	M
	2	9.38%	0.03%	0.01%	20.42%	0.20%	0.00%	61.27%	1.02% 0.37%	7.06%	0.23%	0.00%
	3	9.11%	0.03%	0.01%	20.48%	0.20%	0.00%	61.45%	1.02% 0.34%	7.12%	0.23%	0.00%
	4	9.20%	0.03%	0.01%	20.46%	0.21%	0.00%	61.39%	1.02% 0.34%	7.10%	0.23%	0.00%

Single-Item Bi-Level Uncapacitated Lot-Sizing with Multiple Remanufacturing of Reusable By-Products

  Inventory cost of Top (Fresh or Refresh) Wafer at level l at the end of period t, ch t Inventory cost of the final product (SOI) at the end of period t.

	Chapter 11. Parameters 11.3
	T	Total number of periods in the planning horizon,
	l max	Maximum refresh level of the Top Wafer,
	d t	Demand of final product (SOI) in period t,
	cf t	Fresh Wafer purchase cost in period t,
	cr t	Refresh process cost in period t,
	cp t	Production cost of the final product (SOI) in period t,
	csf t	Setup cost of Fresh Wafer purchase in period t,
	csr t	Setup cost of the refresh process line in period t,
	csp t	Setup cost of the final product (SOI) production line in period t,
	ch l -t	Inventory cost of Negative Wafer at level l at the end of period
		t,
	ch l +
	Variables
	p t	Quantity of Fresh Wafers purchased in period t,
	x l t	Production quantity of the final product (SOI) using Top Wafer
		at level l in period t,
	z l t	Quantity of Negative Wafers at level l to be refreshed in period
		t,
	s t	Final product (SOI) inventory at the end of period t,
	s l + t	Top (Fresh or Refresh) Wafer inventory at level l at the end of
		period t,
	s l -t	Negative Wafer inventory at level l at the end of period t,
		
	v t	  1 if raw materials procurement occurs in period t,   0 otherwise.
		
	w t	  1 if the refresh process occurs in period t,   0 otherwise. 	1	FP of-
	fers additional properties. Based on these properties, a model is presented in Section 11.5. The resolution approach is given in Section 11.5.3.   1 if final product (SOI) production occurs in period t, y t   0 otherwise.

t

Uncapacitated Lot-Sizing with Multiple Remanufacturing (ULS-MR) Mathematical Model of ULS-MR min

					l max -1
			cf t p t +		cr t z l t +	cp t x l t
			∀t	∀t	l=0		∀t ∀l
			+	csf t v t +	csr t w t +	csp t y t
			∀t		∀t		∀t
	+	l max -1	ch l -t s l -t +		ch l + t s l + t +	l max	ch l + t z l-1 t-1
	∀t	l=0	∀t ∀l		∀t	l=1
							+	ch t s t	(11.1)
							∀t
							subject to
						l max
					s t-1 +	x l t = d t + s t	∀t	(11.2)
						l=0

  )

	l max
	x l t ≤ M 2 • y t	∀t	(11.7)
	l=0
	l max -1
	z l t ≤ M 3 • w t	∀t	(11.8)
	l=0
	z l max t	= 0		(11.9)
	∀t
	w T = 0		(11.10)
	x l t , z l t , s l + t , s l -t ≥ 0	∀t, l	(11.11)
	p t , s t ≥ 0	∀t	(11.12)
	y t , v t , w t ∈ {0, 1}	∀t	(11.13)

Chapter 11. Single-Item Bi-Level Uncapacitated Lot-Sizing with Multiple Remanufacturing of Reusable By-Products

  and d T = A. We show that

		t 1			t 2			t 3
	Negative	l = 1
	Wafer	l = 0	t 1			t 2		t 3
		l = 2	t 1	S 2+	1	t 2	S 2+	2	t 3	S 2+	3
		l = 1	t 1	S 1+	1	t 2	S 1+	2	t 3	S 1+	3
		l = 0	t 1	S 0+	1	t 2	S 0+	2	t 3	S 0+	3
	SOI Wafer
	(Final Products)		t 1	S 1		t 2	S 2		t 3	S 3
			d 1			d 2				d 3

  0, then d t ≥ 0 →(11.21)

			(11.20a)
	s 0 + t-1 , s 0 +
	l max	x l t > 0	P rop.
	l=1

t > 0, then d t = 0. However, d t-1 might be larger than 0. (11.20b)

if d t > 0, then

Chapter 11. Single-Item Bi-Level Uncapacitated Lot-Sizing with Multiple Remanufacturing of Reusable By-Products

  11.28) Note that in the objective function(11.24), we have immediately counted the SOI production cost of all purchased Fresh Wafers ((cf + cp)p t ). The reason is that no Fresh Wafer is bought only for stocking purpose. The first two terms could also be written as ∀t cf p t + ∀t The material flow representation of ULS-MR 1 is illustrated in Figure11.7. By only considering the raw material procurement setup and the cost structure assumptions, the problem is simplified. However, still important decisions must be made in each period regarding the SKUs (Fresh Wafers, Negative Wafers and Refresh Wafers). In fact, the Refresh Wafers do not really create complications since according to Proposition 11.4.4, the Refresh Wafer inventory is null over the whole

							0
				S 1+	1	x l max +1 t	S 1+ = 0 S 2+	2 2			S 1+ S 2+	3 3	(11.29)
	P			∀t P	x l t , s l -t ≥ 0 P	P	P	P	∀t, l	(11.30)
	Final Products	d 1	t 1	S 1	p t , s t , s 0 + t ≥ 0 v t ∈ {0, 1} d 2 t 2 S 2		d 3	t 3	S 3	∀t ∀t	(11.31) (11.32)
				l max l=0 cpx l t .

1 under "Full Push Policy" (ULS-MR 1 FP ) Proposition 11.5.4.

  There exists an optimal solution to ULS-MR 1 FP in whichs l - t = 0 ∀t, l|l = l max .Based on Proposition 11.5.4, Proposition 11.4.6 always holds as no Negative Wafer inventory exists. Without loss of generality, the initial inventory vector is considered to be null. If stocks s l +

	t-1

1 FP

 1 When considering the Full Push Policy Assumptions, flow conservation constraints can be simplified. Flow conservation Constraints (11.2) to (11.5) are repeated below (Constraints(11.33) to(11.37)).

	11.5
	s t-1 + x 0 t = d t + s t	∀t	(11.33)
	l max
	s t-1 +	x l t = d t + s t	∀t	(11.34)
	l=1
	s l + t-1 + p t = x l t + s l + t	∀t, l = 0	(11.35)
	s l + t-1 + z l-1 t-1 = x l t + s l + t	∀t, l > 0	(11.36)
	s l -t-1 + x l t = z l t + s l -t	∀t, l|l = l max	(11.37)
	Top Wafer and Negative Wafer inventories are eliminated. Since no refresh process
	is possible in period 0, by changing Constraint (11.36) to Constraint (11.41), t must
	be at least 2.
	s t-1 + x 0 t = d t + s t	∀t	(11.38)
	l max
	s t-1 +	x l t = d t + s t	∀t	(11.39)
	l=1
		p t = x 0 t	∀t	(11.40)
		z l-1 t-1 = x l t	∀t > 1, l > 0	(11.41)
		x l t = z l t	∀t, l|l = l max	(11.42)

ULS-MR 1 under "Full Push Policy" (ULS-MR 1 FP )

  Constraint(11.44) is obtained by replacing x 0 t with p t (using Constraint (11.40)) in Constraint

Chapter 11. Single-Item Bi-Level Uncapacitated Lot-Sizing with Multiple Remanufacturing of Reusable By-Products As

  ). SOI production and refresh process can be expressed through Fresh Wafer purchase, the objective function can also be simplified.

	11.5
	crp t + l max	cpp t +	csf v t +	l max	ch l + z l-1 t	+	chs t
	∀t	∀t		∀t	∀t	l=1		∀t
								(11.47)
	Constraints (11.44) and (11.45) yield the single flow conservation Constraint (11.49).
	Without loss of generality, we consider all initial inventories equal to zero. The model
	is presented below:
	Mathematical Model for ULS-MR 1
					chs t			(11.48)
					∀t
				subject to
	(min{t,l max +1})-1
	s t-1 + p t = d t + s t s t-1 +	p t-l = d t + s t		∀t ∀t	(11.44) (11.49)
	(min{t,l max +1})-1	l=0
	s t-1 +	p t-l = d t + s t	p t ≤ M 1 • v t		∀t ∀t	(11.45) (11.50)
	l=1	p t-l = z l t		∀t, l|l = l max p t , s t ≥ 0 ∀t	(11.46) (11.51)
				v t ∈ {0, 1}		∀t	(11.52)
	The material flow representation of the model is illustrated in Figure 11.9. It is
	not a classical network flow in which flow conservation constraint is satisfied in the
	nodes representing the final product fabrication.

∀t cf p t + (l max -1) FP min (cf -cr + l max (cr + cp)) ∀t p t + ∀t csf v t + ∀t (min{t,l max +1})-1 l=1 ch l + p t-l +

ULS-MR 1 under "Full Push Policy" (ULS-MR 1 FP )

				0
			S 1+	1	S 1+	2				S 1+	3
	P			P P	S 2+	2	P	P	P	S 2+	3
	Final Products	t 1	S 1	t 2	S 2			t 3	S 3
	d 1			d 2				d 3

  Thus, about 76.37% of the Fresh wafers which enter the SOI-and-Refresh Lines are fully refreshed, i.e. until N 3, about 3.85% can be used as monitor wafers and 1.97% Semiconductor manufacturing, qualification management, flexibility, variability, workload balancing, batching, capacity planning, bi-level capacitated lot-sizing, by-production, remanufacturing, closed-loop production planning, optimization.

	École Nationale Supérieure des Mines de Saint-Étienne
	NNT : 2015 EMSE 0784
	F OHT Overhead Hoist Transfer R1 R1 R1 1st pass 2nd pass Mehdi ROWSHANNAHAD	R2	R2 1st pass	R2 2nd pass	R3	Monitor	Scrap
	F N1 QUALIFICATION MANAGEMENT AND CLOSED-LOOP PRO-4.40843671 PLSP Proportional Lot-Sizing and Scheduling Problem DUCTION PLANNING IN SEMICONDUCTOR MANUFACTUR-3.7871519 N'1x 3.25512 PM Preventive Maintenance ING
					R1		2.8189
	N2 Speciality: Industrial Engineering QM Qualification Management		2.021
	RATO Reassemble-to-Order Keywords:		N'2x R2		1.8 1
	P = RMTO Remanufacture-to-Order
	RMTS Remanufacture-to-Stock
	ROI Return on Investment
	SCC Supply Chain Council
	SCM Supply Chain Management
	SCOR Supply Chain Operations Reference-Model
			0.9 0.045 0.8739 0.78651 0.0393255 0.76370121
	N1 SKU Stock Keeping Unit 0 1	0.05	0.971		0.8739	0.043695	0.8485569
	N'1	0	0	1	0.8		0.72	0.036	0.69912
	R1 SQL Structured Query Language 0 0 0	1		0.9 1-λ	0.045	0.8739
	N2 N'2x TP Throughput	0 0 SOI Production 0 0 Capacity	0 0	0 0 λ Production Yield	1 0	0.05 1	0.971 0.8
	R2 ULS-MR Uncapacitated Single-Item Bi-Level Lot-Sizing with Multiple Remanu-0 0 0 0 0 0 Neg x 1 1|x = l max
	F R = facturing of Reusable By-Products N3 Monitor             F Rx (SOI-x) 0.05 0.95*0.98 1 0.687331089 0.036681593 0.275987319 Scrap N1 0.76370121 0.040757325 0.195541465 0.95* F ULS-MR 1 ULS-MR with only Procurement Setup 0.005 Neg' x N'1x 0.629208 0.16566 0.205132 R1 0.78651 0.019575 0.8 0.95* 0.015 ULS-MR 1 FP ULS-MR 1 under "Full Push Policy" Assumptions 0.193915 Refresh N2 0.8739 0.02175 0.10435 N'2x 0.72 0.15 0.13 0.05 0.15 Capacity WIP Work-In-Progress or Work-In-Process	           
	WW Wagner-Whitin	R2	0.9	M	0	S	0.1
					1				1
	Figure B.6: Graphical Representation of Transition Matrix

Part of this chapter has been published in OMEGA[START_REF] Rowshannahad | Capacitated Qualification Management in Semiconductor Manufacturing[END_REF].

Part of this chapter has been published in[START_REF] Rowshannahad | Qualification Management with Batch Size Constraint[END_REF] and submitted to IJPE. The article presented in Winter Simulation Conference

has won the Best Applied Paper Award.

Part of this chapter has been presented in Winter Simulation Conference

[START_REF] Rowshannahad | Qualification Management to Reduce Workload Variability in Semiconductor Manufacturing[END_REF].

Part of this chapter has been published in[START_REF] Rowshannahad | Qualification Management and its Impact on Capacity Optimization[END_REF].

Part of this chapter has been published in the proceedings of the conference MOSIM

[START_REF] Rowshannahad | A Multi-Item Bi-Level Production Planning Problem with Reusable By-Products[END_REF].

The case where replenishment decisions can be made at any time is discussed in[START_REF] Silver | A Simple Modification of the EOQ for the Case of a Varying Demand Rate[END_REF].

If the initial and ending inventory values and stock lower bounds are fixed, the demand requirements can be reformulated to set the initial and ending inventory values as well as lower bounds to zero[START_REF] Pochet | Production Planning by Mixed Integer Programming[END_REF].

Multiple perspectives can be imagined. First, it is necessary to study the complexity of the reduced models. Then, improving the dynamic programming algorithms may lead to pseudo-polynomial algorithms. Also, the problem may have other properties which could be identified. Regarding the dynamic programming

Chapter 7

Industrialization of Qualification Management

In this chapter, we present and discuss to which extent and how the approaches proposed in the previous chapters have been industrialized. Special industrial restrictions, workflow procedure and industrial applications are presented. 1 7. [START_REF] Agrali | A Dynamic Uncapacitated Lot-Sizing Problem with Co-Production[END_REF]

.4: Types of Production Systems (Adapted from [START_REF] Akrani | Types of Production System -Intermittent and Continuous[END_REF])

In Mass Production Flows, several products are produced or assembled on a very large scale based on demand forecast, e.g. chemicals production, final car assembly, etc. In Process Production Flows, the production system is designed for the production of normally a single product, e.g. glass, paper or steel production.

Intermittent Production System

An Intermittent Production System is designed flexible enough to handle a variety of different products. The production flow is intermittent. The material handling system must be adapted for a variety of raw materials, components, semi-finished products and final products. The products are more customized and the production volume is relatively lower than in a Continuous Production System. The production operations, routing and process times differ from product to product. Usually, few by-products are generated using this type of production system. It is classified into Project Production Flows, Jobbing Production Flows, Lot (Batch) Production Flows and Assembly Production Flows [START_REF] Silver | Inventory Management and Production Planning and Scheduling[END_REF].

In Project Production Flows, one time complex projects are realized by the company at an estimated cost and delay. The product volume is very low and the system is generally very flexible. The production strategy is Engineer-to-Order. Examples are construction projects, ship building, etc. The Jobbing Production Flows is designed to manufacture very few products according to the specifications of the customer. The workstations are flexible and multifunctional. The followed production strategies are Engineer-to-Order or Make-to-Order. Examples are spe-

Chapter 10. Multi-Item Bi-Level Capacitated Lot-Sizing with Multiple

Remanufacturing of Reusable By-Products Internal refresh may also be performed in a different site than the one where the SOI Wafers are produced. Therefore, the shipping, planning, extra packaging, possible deterioration and increased and less certain cycle time must be considered when the refresh process is not done at the same site where SOI is produced. A yield factor is associated with the refresh process because of the manufacturing line scrap. The demand is assumed to be known over a discrete time horizon. It is possible to stock products to satisfy future demand but no backlogging is allowed.

In Figures 10.2 and 10.3, the landscape of the supply chain under study is depicted. Figure 10.2 illustrates different levels of the refresh process until the Top Wafer can no longer be refreshed. At this time, it is used as a test Wafer for qualification or test purposes or simply as a filler Wafer. Figure 10.3 depicts a simple SOI-Refresh supply chain. Top (Fresh) and Base Wafers are purchased from a bulk supplier. The SOI production site includes also a refresh line. An external refresh supplier and a customer are also present.

This study deals with production planning decisions on a discrete time horizon. The model is with big time bucket periods as multiple items can be produced in the same time period [START_REF] Graves | Manufacturing Planning and Control[END_REF] and the products manufactured in a period can be used to

Mathematical Model

product i in period t (G i,t ). Constraints (10.5) and (10.6) respectively model the flow conservation for Base Wafers and Fresh Wafers. Constraint (10.7) refers to Refresh Wafer inventory balance. It indicates that the inventory of the Refresh Wafers f in period t (S + f,t ) is equal to the Refresh Wafers to be received in this period (in transit Refresh Wafers) ( Rf,t ) plus the Refresh Wafer inventory in the previous period (S + f,t-1 ) plus the Refresh Wafers of that period in all sites (refreshed Negative Wafers obtained in period t)

). Constraint (10.8) models the inventory balance of Negative Wafers in period t. It specifies that the inventory of the Negative of the Top Wafer f in period t-1 (S - f,t-1 ) is equal to the Negative inventory of the Top Wafer f at the previous period (S - f,t-2 ) plus the Negatives generated at period t -1 ( ∀i X F i,f,t-1 ) minus the Negative Wafers sent to be refreshed in all sites ( ∀m ∀f b f,f R f,f ,m,t ). Note that the Negative Wafers of f taken in Constraint (10.8) are returned back refreshed in (10.7). However, because of the refresh line scraps, not all of the Negative Wafers are transformed into Refresh Wafers. This is why the yield factor α f,m is used in Constraint (10.7). Constraints (10.9) and (10.10) respectively restrict the production and refresh line capacities in each period t. Constraints (10.11) through (10.14) respectively model the production setup cost, raw material (Base Wafers and Fresh Wafers) procurement cost and refresh setup cost, where M is a large positive number (classical big-M in mixed integer linear programming formulations). Constraint 10.15 is added to prevent refreshing non-refreshable Negative Wafers. Constraint 10.16 avoids refreshing in the last period of the planning horizon. Binary and non-negativity sign restrictions are ensured using Constraints (10.17) through (10.24).

Note that, if Wafers f can be obtained from several Wafers f ( ∀f b f,f > 1), the condition b f,f = 1 is added to avoid removing several times f to obtain Refresh Wafer f . However if ∀f b f,f = 1 ∀f , there exists only one f for each f and, therefore, only one unit of f is removed by removing one unit of f . In this case, the refresh terms in Constraints (10.7), (10.8), (10.10) and the objective function could simply be written as R f,m,t instead of R f ,f,m,t . In this chapter, we study bi-level lot-sizing problems in which reusable by-products are generated during production. The generated by-products can be reused as raw materials after further processing. However, by-products can be "recycled" only a given number of times. The industrial problem concerns the production system of "SOI" (Silicon-On-Insulator) fabrication units. Based on the industrial model discussed in Chapter 10, an uncapacitated single-item version of the problem is derived and analyzed. We also propose a dynamic programming algorithm for the resolution of a restricted version of the problem. 1

Numerical Experiments

Problem Definition

products can be remanufactured to be reused again to satisfy the final product demand. Final products fabricated from both newly purchased and remanufactured raw materials can be used to fulfill the same demand. However, a newly purchased raw material may be remanufactured and reused only a limited number of times.

As stated, we define the problem based on SOI production using the Smart-Cut™ Technology. The interested reader can refer to Section 10.2 for a more detailed discussion of the manufacturing process. Here, we only define the necessary elements for our discussion in this chapter.

In SOI production, a thin layer of a donor wafer (Top Wafer) is transferred to a support wafer (Base Wafer). As only a thin layer of the Top Wafer is deposed on the final product (SOI), the Top Wafer may be reused several times. The purchased raw material -which is not yet used in production-is called "Fresh Wafer". The generated by-product during SOI production is called "Negative Wafer". The Negative Wafers are not immediately reusable as raw material. The process of making by-products reusable is called the "refresh process" or shortly "refresh". The refreshed Negative Wafer which can be used again as raw material in production is called "Refresh Wafer". A purchased raw material (Fresh Wafer) has a limited refresh (or remanufacturing) life. It means that the generated by-product (Negative Wafer) can be reused (refreshed) only a limited number of times. The production and refresh flows are is depicted in Figure 11

ULS-MR with only Procurement Setup (ULS-MR 1 )

In a first step, we simplify the ULS-MR problem by relaxing two setup constraints, i.e. the SOI production and refresh process setup constraints (Constraints (11.7) and (11.8), respectively). The only setup constraint considered is the Fresh Wafer procurement setup cost (Constraint (11.6)). We refer to this new model as ULS-MR 1 .

Inventory cost determination

The inventory cost is composed of physical holding cost and capital cost. The physical holding cost is negligible and considered to be the same for final product, Fresh, Refresh and Negative Wafers. However, the capital cost decreases for Negative Wafers as the refresh level approaches l max . The Negative Wafer inventory value is illustrated in Figure 11.5. Among raw materials, Fresh Wafer has the highest value. As it is used and refreshed further, its value decreases by losing its potential to be refreshed and reused again in production. In reality, a Negative Wafer which is no longer refreshable is used further as test wafer with a scrap value. However, for simplicity the Negative Wafer scrap value is considered to be zero.

Chapter 11. Single-Item Bi-Level Uncapacitated Lot-Sizing with Multiple Remanufacturing of Reusable By-Products

Each amortization value can be determined as Fresh Wafer value-Negative Wafer scrap value

. The same reasoning holds for Top (Fresh and Refresh) Wafers. Therefore, in the Assumptions Before tackling the structure of the optimal solutions, we will make the following assumptions. They are well-grounded and correspond to industrial facts.

• In the reduced model, we have three stock keeping units (SKUs): SOI Wafer (final product), Top Wafer (raw material) and Negative Wafer (byproduct). A Base Wafer and a Top Wafer are used to make an SOI Wafer.

In view of the used raw material and numerous process steps to make an SOI Wafer, it is the most expensive SKU in comparison to Top and Negative Wafers. Between Top and Negative Wafers, the Top Wafer is more valuable as it can directly be used in the final product fabrication. The least value goes to the Negative Wafer, as it needs remanufacturing (refreshing) to return to the SOI production process. By virtue of the values of the SKUs, the inventory cost of the SOI Wafers is larger than the Top Wafer inventory cost. And the Top Wafer inventory cost is larger than the Negative Wafers inventory cost (see Figure 11.5).

ULS-MR with only Procurement Setup (ULS-MR 1 )

• Each Top Wafer has a potential value stored in it. At each refresh process, this value decreases. Thus, a Fresh Wafer (R0) is worthier than a Refresh Wafer 1 (R1) as the Fresh Wafer keeps in itself R1 (and also all other levels of Refresh Wafer, if any). As a result, the Top Wafer inventory cost decreases with the refresh level (see Figure 11.5).

• The same reasoning holds for the Negative Wafers. The Negative Wafers of the higher levels are more valuable than those of the lower levels. Therefore, the fixed capital (and hence inventory) cost associated with higher level Negative Wafers is larger (see Figure 11.5).

• As the value of Negative wafers reduces after each refresh process (byproduct value depreciation), in terms of the inventory cost, higher Top Wafer levels are used for SOI production when a choice can be made. By choosing Top Wafers of higher levels, we also guarantee the Negative Wafer generation. For instance, first we would satisfy the Top Wafer requirements (to produce SOI Wafers) from R1 rather than R2, rather than R3 and so on.

• For simplicity reasons, we consider that the Negative Wafer generation and refresh process can occur in the same period.

• An implicit assumption which is considered is that the refresh process cost is lower than the Fresh Wafer purchase cost. Otherwise the refresh process loses its economical value and the model reduces to a classical lot-sizing problem.

• The costs are assumed to be stationary over time.

A summary of the inventory cost comparison can be found in Figure 11.6.

Structure of the Optimal Solutions for ULS-MR 1

Here, we discuss the optimal solution properties of ULS-MR 1 . To simplify the presentation of this section, we suppose that no initial and final inventories are

Proposition 11.5.3. There exists an optimal solution to ULS-MR 1 FP in which

According to Proposition 11.5.3, Proposition 11.4.5 always holds.

As Negative Wafers are systematically refreshed, no Negative Wafer inventory is kept.

12.2 Perspectives utilization, reduce the cycle time, improve the machine investment amortization, decrease of production time (for instance, by allocating products to machines with shorter process times). The objective is to increase flexibility at a minimum cost to accommodate demand variability, product mix variations and process and design changes.

In the second part of the thesis, the production planning of the SOI fabrication and refresh process lines were discussed. The production planning problem tackled is novel as Soitec disposes of its exclusive the production procedure for SOI fabrication. The two production lines are related and make a so-called closed-loop manufacturing system. While fabricating final products, by-products are generated in the main production line. Once remanufactured, the by-products return to the main production line to make final products. The manufacturing system was modeled with all specific constraints of Soitec.

Based on the industrial problem, a single-item uncapacitated lot-sizing model was defined to study the production planning of a special type of closed-loop manufacturing systems. To the best of our knowledge, this problem has never been treated in previous studies. The problem is NP-hard. Therefore, several reduced variants of the original problem were defined and studied. For one of the reduced models, based on the optimal solution structure, an exact resolution method using dynamic programming was proposed.

Perspectives

Various perspectives were discussed in the end of each chapter. In Section 4.5, the bi-objective optimization of capacitated flexibility and deviation ratio measures was proposed. Bi-objective optimization aims at finding a trade-off compromise between flexibility (workload balance) increase and decrease of the workload allocation deviation from the equipment capacity. In Section 5.5, possible improvements the proposed workload balancing algorithms with batching were discussed. Moreover, Appendix A is dedicated to batching when considering minimum and maximum batch sizes. Section 6.6 considers (local) workload variability in a workcenter and the qualification configuration flexibility in workload allocation. The proposed ap-

Appendices

Appendix A

Batching: Minimum and Maximum Batch Sizes

A.1 Minimum Batch Size

When we talk about batch size, normally the maximum batch size is considered. By maximum batch size, we mean the maximum number of wafers (or products in general) which can be loaded into a machine during a production run. In semiconductor manufacturing, for furnaces, the maximum batch size is determined according to boat slots, yield and quality issues. For implantors, the size and layout of the wheel determine the maximum batch size.

Apart from the maximum batch size, sometimes the minimum batch size is also defined for each production run. Minimum batch size is defined due to several reasons:

The production run becomes economically feasible Several costs are incurred at each production run:

• Setup time,

• Qualification cost, (if qualification required),

• Filler wafers (for filling vacant places),

• Consumables (if used).

The production quality is guaranteed Minimum batch size also guarantees the production yield and quality.

Chapter A. Batching: Minimum and Maximum Batch Sizes

Avoiding frequent setups Minimum batch size sets a minimum profitability production threshold while avoiding frequent setups. Keeping machine qualified Qualification consumes time and energy. Some machines are disqualified automatically if the recipe is not executed after a defined time period. Therefore, it is sometimes logical not to wait to make full (maximum) batches and to make a minimum batch size run, in order to keep the machine qualified.

Avoiding WIP bubbles and reduction of variability Full (maximum) batch size runs make the production more profitable. However waiting for the lots to arrive and constitution of a full batch may slow the production flow, clutter the fab and cause WIP bubbles. WIP bubbles propagate the variability in the fab. The larger the batch size, the more intense the impact. So, it is sometimes interesting to make runs with minimum batch sizes.

Not enough production volume

A simple reason to use minimum batch size is that the whole production volume does not constitute a full batch run but may fit a minimum batch run.

Increasing machine utilization while decreasing operator performance

The minimum batch size guarantees a minimum work level while avoiding the machine being idle and loosing operators performance. Among the possible approaches, we present two short indications.

A.2 Step Length Adjusting in the Active Set Method

The same approaches described in Chapter 5 can be adapted to consider minimum and maximum batch sizes. One of the heuristics which can be easily adapted is the Batch-Feasibility Heuristic. The other heuristic discussed in Chapter 5 is the Adapted Active-Set Method. We described in Section 5.3.2.1 that in case where full-batches can be made, the constraint becomes active as the variable equals its upper bound [START_REF] Gill | Practical Optimization[END_REF]. The same reasoning holds for the minimum batch size. Minimum batch size acts like a lower bound. So, when the step length equals the lower bound, the corresponding constraint becomes active. Now the question is how to define the 1 if recipe r is to be qualified on machine m

Constraint A.1 guarantees that the sum of the production volume of each recipe of type r in each production run k is equal to the total amount of production volume of recipe r. Constraint A.2 states that each production run consists of only one one recipe and no empty runs are authorized. In order to allow empty runs and at the same time oblige each production run to only consist of one recipe, Constraint A.2 must be changed to:

Constraint A.3 and A.4 state respectively that the number of wafers of each recipe at each production run must lie between minBS r and maxBS r . Y r,k is binary, and X r,k is continuous. The product of a binary and a continuous variable creates a semi-continuous variable. Constraint A.1 is nonlinear.

A semi-continuous variable can take the value zero or any continuous value between its lower and upper bounds. The lower bound is any positive value while the upper bound can be infinite. Semi-continuous variables are often used in real-world modeling, such as production planning, transportation, electrical power generation, portfolio selection, etc. Our case is similar to production planning in which either no products are fabricated or if fabricated, the production must be higher than a min-Chapter A. Batching: Minimum and Maximum Batch Sizes Y r,k X r,k is substituted by the semi-continuous variable Z r,k .

In the Flexibility, Variability and Deviation Ratio Measures, W IP r,m is to be replaced by k Z r,k,m .

Remark

In the above model, the minimum production volume of each recipe (wip r ) must be equal to its minimum batch size (minBS r ). It is possible to define

A.3 Mathematical Model using Semi-Continuous Variables

supplementary variables to cover the case where wip r ≤ minBS r .

A.3.3 Case of Multiple Qualifications

A.3.4 Best Qualification Selection: Alternative Criterion

Flexibility, Variability and Deviation Ratio Measures try to balance the best the workload on the toolset. These measures can be used in the precedent model as the objective function. However, while scheduling with minimum and maximum batch size, it would be preferable to choose a qualification which maximizes the total number of maximum and minimum batch size runs on the toolset. As perspective, it is interesting to make the model choose between minimum and maximum batch size by favoring bigger batch sizes, for instance with a nonlinear piece-wise function A.1.

Minimum Batch Size

Maximum Batch Size

SOI-Refresh Lines Modeling

Consider the refresh and SOI production processes as a whole system. This system has different states, such as Fresh, Refresh 1, Negative 1x, Scrap, etc. The models described until now does not consider all of the refresh process line charachteristics in detail. The system is simplified and reduced to its essential elements ( of the system evolve at discrete points in time, for instance a Fresh is used to make SOI wafers (and there are some scraps), the generated Negative 1x is refreshed to give some Refresh 1 wafers and some scraps, and so on. Let X t represent the characteristics of the system at time t. X t is a random variable as its value is not known certainly before time t. A description of the relationship between the random variables X t ∀t is a discrete-time stochastic process.

Here, we consider a special type of discrete-time stochastic processes called a Markov Chain. In a Markov chain, the probability distribution of the state at time t + 1 depends only and only of the probability distribution of the state at time t and not other states through which the chain has passed. Lets take a look at our problem. Once a Negative 1x is generated, it only depends upon its precedent state, i.e. SOI production and not two states before, i.e. Fresh state; Or production of

Thus, about 94.09% of the Fresh wafers which enter the SOI-and-Refresh Lines are fully refreshed, i.e. until R3, about 3.94% can be used as monitor wafers and 1.97% are to be scraped. In reality, the refresh process is not always successful and some The value of monitor wafers, also called NPW (Non-productive Wafers) is not considered in our models. One of the advantages of considering the refresh process line in its details is a better management of NPWs.

Acronyms