Alexis Tsoukiàs
email: tsoukias@lamsade.dauphine.fr

Herimandimbiniaina Ralijaona
email: herimandimbiniaina@agetipa.mg

Rural Road Maintenance in Madagascar The GENIS project

published or not. The documents may come

Introduction

This is a paper on a real case study concerning decision support issues arising from the problem of improving the rural road maintenance activity in Madagascar. This is one among the regular activities of AGETIPA and the occasion to focus the attention on it was given within the GENIS project.

AGETIPA (Agence d'Execution des Travaux d'Intérêt Public) is a non profit Agency established in 1993 by the Government of Madagascar and the consulting and executive companies working in the sector of the public work in Madagascar. Since 1994 it is responsible for the executive follow up of most of the public works taking place in Madagascar, the development projects as well as in charge of capacity building in this sector. It is a member of AFRICATIP the African Association of Public Work Agencies (there are 18 such agencies over the continent). Besides its activities in executive support AGETIPA provides technical support to other Institutions and Government bodies, training for the staff of the consulting and construction companies and recently develops also Decision Support Activities.

The GENIS project (Gestion des Routes par Niveau de Service) has been proposed by the Transportation Minister in collaboration with the World Bank in order to improve rural road maintenance. A pilot study area has been selected, the greater Antananarivo urban and peri-urban area. The principal aim of the project is to contribute, to the improvement of accessibility from rural areas to markets, social services and economic activities, thus increasing income for rural population, improving their life conditions and ultimately reducing poverty. The implementation of the GENIS project should: -satisfy the needs of the users in terms of necessary physical conditions of the roads to be maintained; -the service level of the roads to be considered should be maintained or improved. The principal decision problem within the GENIS project is to establish which roads should be considered first in order to maximise the positive economic, social and environmental impact.

While trying to formalise the problem on how to make any decisions concerning priorities in rural road maintenance the CEO of AGETIPA realised that:

1. there were no internal resources (at that time) able to follow a decision support problem (not only the specific one);

2. there was a clear necessity to re-think the whole evaluation model as far as public works financing was concerned, at least in the case of rural roads in Madagascar.

Although evaluating public projects in economical mature societies can be considered as a typical cost-benefit analysis (the reader should note that this is an approach strongly argued in the literature; see [START_REF] Boardman | Cost benefit analysis: Concepts and practices[END_REF], [START_REF] Bouyssou | Evaluation and decision models: a critical perspective[END_REF], [START_REF] Dasgupta | Economic Theory and Exhaustible Resources[END_REF], [START_REF] Dinwiddy | Principles of cost-benefit analysis for developing countries[END_REF], [START_REF] Keeney | Value-Focused Thinking. A Path to Creative Decision Making[END_REF], [START_REF] Kulkarni | Needbased project prioritization: Alternative to cost-benefit analysis[END_REF], [START_REF] Liesiö | Preference programming for robust portfolio modeling and project selection[END_REF], [START_REF] Mild | Combining a multiattribute value function with an optimization model: An application to dynamic resource allocation for infrastructure maintenance[END_REF], [START_REF] Roy | L'analyse coûts-avantages, outil de concertation et de légitimation ?[END_REF]), this is much more difficult in the case of countries under development. Deciding to fund a rural road maintenance project can be of vital importance for a local community and is basically a political decision which needs to be legitimated, explained and possibly negotiated. Behind such decisions there is a whole formal or informal process involving the participation of the local community and thus influencing also the future of this young democracy.

On that basis AGETIPA considered the hypothesis of involving a Decision Analysis expert with the aim: on the one hand to help conducting the GENIS project and on the other hand to introduce and enhance within AGETIPA a Decision Support capacity. This is how the first author has been involved in the project.

The presentation of this case study will follow the scheme presented in [START_REF] Tsoukiàs | On the concept of decision aiding process[END_REF] in order to emphasise in a structured way the different cognitive artifacts produced in the decision aiding process. More precisely we will try to describe the: • evaluation model; what are the elements composing such a model (set of alternatives, attributes and their scales, criteria, uncertainties and procedures to be used)?

•
There will be no final recommendation to talk about, since the study concerned the construction of a method to apply in several different problem situations repeatedly. There has been a pilot study though for which recommendations have been formulated, but this was basically a validation activity.

The objectives of this study can be thus summarised.

• provide a methodology for handling the rural road network maintenance problem;

• develop a decision aiding methodology for similar problems;

• create decision support capacity within AGETIPA with respect both to "internal" and "external" clients.

The paper is thus organised as follows. In Section 2 we present the Problem Situation. In Section 3 we present the Problem Formulation, while in Section 4 we present the Evaluation Model(s) used in order to handle the problem as formulated in Section 3. In Section 5 we present the pilot study conducted in order to validate the method suggested. In Section 6 we present the feedback obtained after two years of using this method in rural road maintenance in Madagascar. Section 7 concludes the paper.

Problem Situation

The maintenance and improvement of the rural road network in Madagascar is a vital project for the population living in these areas both for economic reasons (improve accessibility to land and markets) and social ones (improve accessibility to schools and health centres as well as improve communication among remotely located communities).

The result is an extremely high demand for rural road maintenance. In front of that the availability of funds to dedicate to this purpose is rather limited and largely depending from external donors and financial institutions such as the World Bank. The basic problem arises from this scarce availability of funds: given the annual spending of the national government for rural road maintenance where should be given the priority?

The demand for maintenance generates locally from communities and their local government which for simplicity we are going to represent by the local mayor. At the national level the management of the process is delegated by the government to AGETIPA who is expected to establish within the GENIS project the priorities and thus decide every year which are the projects that can be satisfied. However, in managing this problem AGETIPA is expected to pursue a precise policy: -on the one hand maximise the positive impact of the policy in terms of benefiting population, economic activities impacted and general improvement of the whole road network of the island in terms of "level of service" guaranteed; -on the other hand promote the participation and active implication of the local communities and their representatives as well as of possibly existing economic actors both to the decision and to the implementation process. Under such a perspective, the accountability of the decision process is extremely important. It is important to explain why a project has been rejected AND under which conditions would have been accepted (or could actually be accepted).

The traditional approach used until the time of this study aiming at calculating the overall economic return of each project (seen as a public investment) seems inadequate with respect to the above policy for essentially two reasons: -the difficulty to take into account qualitative information in order to establish an economic return (not all impacts of such projects have a clear economic dimension); -the impossibility to establish a negotiation table with the local actors: indeed the accountability of decisions based on purely economic reasons is extremely difficult (at least with respect to precisely the local actors which are expected to be implied). It should also be emphasised that the "client" (the government) considers very important to allow transparent negotiations with the local actors where the reasons for which projects submitted were rejected should be clear. A rural road can be of vital importance for a local community and rejecting its maintenance project can induce severe consequences.

We can summarise the problem situation as follows.

• The actors involved in the decision process (the stakeholders) are the National Government, AGETIPA, the local actors (economic, social and political), the population itself, fund raising institutions, international donors and financial institutions.

• The objects/concerns of such actors include: the rural roads network and the specific maintenance projects for given segments of the network, the funds available for maintenance at the National level, any local funds available, the decision process itself, the negotiation capacity, the local actors implication and commitment, the long term evolution of the whole road network of the island, the long term impact of this process on regional planning at the National level.

• The resources committed in the decision process include the maintenance funds, the local knowledge of the rural road network and the social and economic needs of the local communities, the satellite images of the rural areas, other information (statistics) about the areas crossed by the rural roads network, the technical knowledge about planning and road maintenance.

Decision Aiding is thus expected to: -convince AGETIPA that the projects chosen will have the maximum positive impact; -be inserted within a long term effort of improving regional planning and more precisely road maintenance; -be accountable with respect to the National government, the local actors and the international donors and financial institutions; -allow a transparent decision process enabling negotiations with the local actors.

Problem Formulation

The problem situation as previously described has been discussed within AGETIPA together with their advisor (the first author). Several hypotheses have been considered. In order to establish a precise problem formulation the following points have been fixed: -the results of the model have to be easily explainable and give an immediate intuitive idea of the decision; -the results of the model have to be consistent with the long term government policy in road network improvement; -the model has to take into account the local actors' commitment. On this basis the following problem formulation has been adopted: Classify the submitted projects into three classes: accepted, rejected, negotiable, taking into account the social-economic impact of the projects, the local actors' commitment and the cost. More precisely the problem formulation contains the following elements.

A The actions to be considered by the decision aiding process should be the precise rural road maintenance projects expressed and submitted by the local communities. More precisely the projects are established through a consultation process organised by FIFTAMA (the Association of the mayors of the towns surrounding Antananarivo). Some basic conditions are expected to be met in order to be eligible to apply:

-the roads to be maintained need to be already inserted within the Provincial Transportation Plan established in 2004; -the roads need to already exist in "acceptable" physical conditions and to mobilise local resources; -the roads need to be part of transportation axes linking the communities between them, axes identified in local development plans and have to intersect the national road network.

V There are basically three points of view to take into account for assessing the various projects:

-the first one concerns the "commitment" of the local actors to the maintenance effort. The idea here is that the local actors should consider the rural road maintenance as one of their major commitments and that they should feel responsible for such a resource; -the second one concerns the "social-economic" impact of each project for the local community. As already mentioned the rural road maintenance is considered as a key element of an overall policy aiming to sustain local development, reduce isolation, increase the economic, social and cultural exchanges and integrate the whole road network of the island. Proposed projects therefore need to demonstrate that they improve both internal accessibility (to key areas of the community) and external accessibility (to the national road network, both in quantity and in quality. Besides, priority should be given to areas with high population and/or economic density (priority expressed by the national government).

-the third one is the "cost", including both short term and long term financial costs as well as negative environmental and social impacts.

Π The problem statement adopted in this case is to classify the projects in three ordered categories:

accepted, the ones that meet all requirements and result at the top of all priorities in "all" dimensions (the ones for which the positive decision appears fully legitimate); -rejected, the ones that definitely cannot be accepted resulting at the bottom of the priorities in "all" dimensions (the ones for which the negative decision appears fully legitimate); -negotiable, the ones who could be accepted in case one or more dimensions could be improved.

Evaluation Model

The evaluation model has been conceived in a straightforward implementation of the problem formulation. It has been constructed in two steps. The first has been a generic model suggested by AGETIPA which has been refined after discussion with the advisor in order to fulfill basic conditions of coherence of the model (for a discussion about the coherence of a model the reader can see [START_REF] Bouyssou | Evaluation and decision models: a critical perspective[END_REF], [START_REF] Ph | Multicriteria Decision-Aid[END_REF]). The second has been a more precise version adapted to the precise issues raised by the GENIS project. The presentation which follows concerns the model as conceived before the first round of applications (there is no mention to uncertainty issues since it was considered that there was no such problem presently). In the feedback section we present how the model has been modified.

Alternatives

The alternatives to be evaluated are considered to be the projects submitted by the local communities. Such projects concern precise segments of the rural roads network which can normally be identified on the satellite images of the whole island (data to which AGETIPA has access). Such projects concern already existing segments of the network and not new connections to build. It is assumed that for each project presented it is possible to identify the local community submitting it (a single administration or a group of administrations).

Dimensions and Measurement Scales

The structure of dimensions results from the analysis of the three points of view introduced in the problem formulation.

1. The "Local Commitment" dimension decomposes to two attributes: public and private participation and is measured considering the level of financial involvement of the local community and/or the private actors to both the immediate costs of the project as well as the subsequent maintenance costs.

Practically we get two figures expressed in percentages.

2. The "social-economic impact" dimension decomposes to four attributes:

• Internal accessibility. On its turn this attribute results taking into account four sub-attributes:

-the number of social, cultural and administrative centres served by the project (schools, churches, townhalls etc.); -the agricultural area served by the project (measured in ha); -the number of "economic activities centres" served (such as markets, industries etc.); -the "level of service" presently offered by the road under consideration. The "level of service" is established considering three types of data:

-seasonal availability; -maximum speed; -traveling comfort;

• External accessibility. On its turn this attribute results taking into account four sub-attributes:

-number of intersections of the proposed road maintenance project with the national road network; -state of the above intersections; -volume of traffic on the road (including the transit traffic); -existence of alternatives and/or shortcuts in case of major impediments.

• Population density. Measured by number of people living in the area by square km.

• Economic density. Measured by the average taxes perceived in the area by the number of people living in the area.

3. The "cost" dimension is represented by a single attribute representing the cost of the project submitted taken into account both the immediate costs and the subsequent maintenance costs. No further environmental or social costs have been considered at this level of the study.

The information concerning the above attributes have been collected from the satellite images to whom AGETIPA has access, from the census information of Madagascar and finally from direct visits in the interested area in order to double check doubtful information.

Criteria

The set of criteria results in an hierarchy as can be shown in figure 1. Leaves have simple preference models resulting from the client's values. Intermediate "parent" nodes have preference models resulting from aggregating the preferences expressed on the "son" nodes.

In the following we are going to present the preference models starting from the leaves of the criteria tree (the ones which have no "son" nodes).

1. Public Participation will be represented through a simple model, such that x y iff pu(x) ≥ pu(y), where pu(x) is the percentage of funds contributed by the local community to the project.

Private Participation will be represented through a simple model, such that

x y iff pr(x) ≥ pr(y), where pr(x) is the percentage of funds contributed by private actors, possibly interested within the local community, to the project.

3 6. Economic Density will be represented through a simple model, such that x y iff e(x) ≥ e(y), where e(x) is the average taxes perceived for each inhabitant in the area interested by the project.

7. Cost will be represented through a simple model such that x y iff c(x) ≤ c(y) where c(x) is the cost (immediate and subsequent maintenance) of the project.

In order to present the model of criterion 4 we need to present the preference models of nodes 4.1 to 4.4.

Number of Intersections simply takes into account how many intersections

the project has with the national road network. There are three possible states (1, 2 or 3 intersections) and the more intersections the better it is (the case of no intersection is no considered since it is a reason of exclusion).

State of

Intersections is a qualitative judgement: 0 for a bad condition, 1 for acceptable condition and 2 for good condition.

4.3 Traffic measures the volume of traffic as resulting from the official statistics (considering a market day and using the usual coefficients for "homogeneous vehicles", HV). Three levels are considered: less than 500 HV daily, less than 1000 HV daily and more than 1000 HV daily. The more trafic registered the more important is the project.

4.4 Alternatives and/or Shortcuts takes into account the presence of such routes. Three possible values are considered: no alternatives and shortcuts, either an alternative or a shortcut, both alternatives and shortcuts.

Criterion 4 is thus established on a discrete scale with values 1 (bad), 2 (acceptable), 3 (good). The procedure for assigning a value to a precise alternative is based on an ordered classification procedure of the type ELECTRE TRI (a simplified version based only on simple majority rules, for more details see for instance [START_REF] João | An approach to support negotiation processes with imprecise information multicriteria additive models[END_REF], [START_REF] Mousseau | A user-oriented implementation of the ELECTRE TRI method integrating preference elicitation support[END_REF]). The model is presented in figure 2. Projects whose profile is (by majority) on the left of the frontier 1, 1, 500, N N get the value 1, projects whose profile is (by majority) on the right of the frontier 2, 2, 1000, Y Y get the value 3, the rest get the value 2. In order to establish the model of criterion 3 we first need to establish the model of node 3.4. This is shown in figure 3. Projects are noted on a four valued scale (1-4) resulting from an explicit enumeration of the 48 combinations of the 11 values on the nodes 3.4.3 (comfort I: unacceptable, A: acceptable, M: medium, B: good), 3.4.2 (maximum speed, less than 10km/h, 10-30km/h, 30-50km/h, more than 50km/h), 3.4.1 (seasonal availability: less than 6 months annually, from 6 to 9 months annually, more than 9 months annually). 3.1 There are three possible values: less than 5 social centres served, 5 to 10 and more than 10 social centres served. In counting such social centres we count principally schools, hospitals (and other health centres) churches and administration centres.

3.2

We take into account the land used (or potentially used) for agricultural purposes. This is measured in hectares.

3.3 There are three possible values: less than 5 economic centres served, 5 to 10 and more than 10 economic centres served. In counting such centres we count principally markets and any other economic activity which may attract mobility within the community.

Criterion 3 is thus established on a discrete scale with values M (bad), L (borderline), A (acceptable), B (good). The procedure for assigning a value to a precise alternative is based on an ordered classification procedure of the type ELECTRE TRI (a simplified version based only on simple majority rules, as for criterion 4). The model is presented in figure 4. Projects whose profile is (by majority) on the left of the frontier < 5, 20, < 5, 3 get the value M, projects whose profile is (by majority) on the right of the frontier > 10, 100, > 10, 1 get the value B. Projects who are on the right of the lowest profile, but lower than the profile 5 -10, 50, 5 -10, 2 get the value L, the rest get the value A.

Aggregation Procedure

The overall aggregation procedure was aimed at assigning each project to one among the following classes: accepted, negotiable, rejected. The procedure once again is a simplified version of the ELECTRE TRI method. The final model can be seen in figure 8. A brief description of the method is provided in Appendix A. In the pilot study a "pessimistic" assignment procedure has been adopted. Generally speaking we adopted a very simple ordinal sorting procedure mostly in order to ease the explanations for each assignment decision and thus the discussion which followed.

Pilot Study

The proposed method has been tested in a pilot study concerning an area at the Nord East of the Great Antananarivo Metropolitan Area. This area can be seen in figure 5.

Data have been collected from satellite images (to which AGETIPA has access), from the National Statistics office and from direct visits in the area under study. A data base with all relevant information has been thus established. A typical record of this data base can be seen in figure 6.

Four maintenance projects have been selected in order to be evaluated as a test for the method. These were at the moment of the study under discussion, but a basic consensus was already reached, although not using a formal evaluation tool. For this reason they seemed to be an interesting benchmark. The values of the four Figure 5: Map of the pilot study area: Nord East of Antananarivo alternatives together with the chosen profiles for the three categories (to finance, to negotiate, to reject) can be seen in figure 7.

The classification model specifying the shape of the three categories can be seen in figure 8 In order to complete the model we need to establish the "importance parameters", that is the importance of each criterion for establishing the class of each alternative. While in the intermediate nodes of the hierarchy a simple majority rule has been adopted, the final model needs to take more precisely into account the importance that the decision maker desires to associate to each criterion (as a manifestation of a certain policy). Since the comparison between alternatives and profiles is based on a "weighted majority" rule the procedure for constructing such "weights" is the following (considering a set H of m criteria): 1. We first check if any majority of m-1, m-2 ... criteria is sufficient to establish a preference relation (winning coalitions). 2. If all coalitions with m/2 + 1 criteria are "winning" then we stop and consider that all criteria have the same importance and the rule is a simple majority rule. 3. If it is not the case we identify the first k such that not all coalitions with m -k criteria are winning (obviously k < m/2) and we ask which among these ones are winning and which not. 4. For each winning coalition W ⊂ H we can write j∈W w j ≥ δ (δ being the majority threshold, unknown). 5. We repeat the same procedure for k + 1, k + 2 until for some k + l all coalitions are not winning. 6. At this point we have a set of linear inequalities, the variables being the different w j and δ. 7. We fix an arbitrary value K such that j∈H w j = K.

Tableau de performance We solve a linear programme minimising or maximising δ and we submit the result to the decision maker. In reality any feasible solution of the linear programme will fit, but the use of extreme solutions helps the decision maker to better discriminate among the values. For instance maximising δ we obtain a result which appears "highly confident" due to the qualified majority thus imposed. The reader should note that the arbitrary value K is irrelevant for the model and only allows to find a numerical representation. 9. In case the decision maker is able to establish a partial order among the criteria this will translate in some further inequalities in the linear programme. It is also interesting to note that in case the decision maker makes any inconsistent statements these will result in an unfeasible linear programme.

The four alternatives have been compared to the two profiles. The first three have been classified as "to be financed", while the fourth one as "to be negotiated". The result has been discussed and validated with the public authorities concerned. The successful application of the procedure to the pilot study has been thus followed by introducing the method in the regular business conducted by AGETIPA. and 2) the necessity to have clear the reasons for which a project is not chosen (rejected or to negotiate) in order to pursue further participation of the local communities. It is interesting to notice however, the lack of information and training in evaluation models within an institution such as the Word Bank.

The engineers experts who applied the method in further experiences raised a number of questions concerning the definitions and the construction and the criteria. More precisely it has been discussed: -The necessity to fix veto thresholds not in absolute value, but as a percentage of the criterion scale.

-The necessity to use attribute's information only once: in certain cases costs have been calculated as a cost by individual in the community concerned in order to compare a relative cost instead of an absolute one. However, this is an information already considered in the population density criterion. In order to avoid this problem it has been clarified to avoid such as a double use and prefer to establish a relative cost based on the length of the road to be maintained.

-The re-definition of the two criteria representing the commitment of the local community and the private actors. Most of the times it is unlike that any private actor can effectively contribute to the project. The result is that although in principle it is interesting to emphasise the presence of any such private actor, the criterion itself almost always never discriminates the alternatives. The suggestion (adopted) has been to consider the private actor's financial application as part of the whole involvement of the community and not separately. Instead it has been suggested, within the group, to separate the financial involvement of the local community to the short term direct costs of the project, from the further medium term maintenance costs.

A further element the discussion emphasised, was the necessity to establish a formal procedure, not only in order to apply, but also in order to further negotiate the application. The method as it is established today requires a procedure in order to formulate an application. Once the applicant knows that the proposal has been "approved" or "rejected" or that it is "negotiable", it is necessary to fix a procedure describing how these decisions are going to be implemented. For instance it is necessary to fix how the commitment of the local community to participate to the costs will be maintained after the decision to start the project has been taken (and the national financial commitment has been decided).

The overall assessment of the method has been positive. The projects handled using this approach show a better understanding, a major commitment of the local communities, while introducing accountability and elements of rationality in handling the road maintenance priorities.

Conclusions

In the paper we present a real world decision support case: aiding AGETIPA (the operational agency of the Minister of Public Works in Madagascar) to handle the problem of "improving the rural road maintenance in Madagascar". As often happens behind the verbal presentation of the "problem" there exist multiple problem situations and problem formulations. In the document we present the ones fixed in this study and the reasons for doing so: -take into account the qualitative information without necessarily transforming it in monetary consequences (not always feasible or accepted); -allow and help to conduct negotiations with the local actors given the political nature of such decisions.

Technically the problem can be seen as an "ordinal measurement" problem where to each object to be measured is associated a label from an ordinal scale (in other terms a class among a set of ordered classes; in this case accepted, negotiable, rejected). A further specificity results from the existence of a hierarchy of similar evaluation models where once again the objects to be evaluated must be classified in ordered classes (for instance level of service 1, 2, 3). The overall assessment resulting from the hierarchical aggregation of the values (assignment to ordered classes) obtained on the nodes of the evaluation tree.

The model has been first applied with a pilot case study (in the Metropolitan Area of the Great Antananarivo, reported in this document) and then used for the last two years. We briefly report the feedback received from this experience. AGETIPA and the teams of engineers working in this area are satisfied with the methodology. Presently AGETIPA is considering the extension of the use of decision aiding methodology in other case studies including the location of a new port, the construction of new schools and the renovation of the "pedestrian path network" within Antananarivo. 2.1 pessimistic assignment a i is iteratively compared with p t • • • p 1 , -as soon as s(a i , p h ) is established, assign a i to category c h .

2.2 optimistic assignment a i is iteratively compared with p 1 • • • p t , -as soon as is established s(p h , a i )∧¬s(a i , p h ) then assign a i to category c h-1 .

The pessimistic procedure finds the highest profile for which the element is not worse. The optimistic procedure finds the lowest profile against which the element is surely worse. If the optimistic and pessimistic assignments coincide, then no uncertainty exists for the assignment. Otherwise, an uncertainty exists and should be considered by the user.

  problem situation; who are the actors involved, what are their concerns and what are their commitments? • problem formulation; on what the decision process focusses, under which points of view (available and/or requested knowledge) and for what precise purpose?

Figure 1 :

 1 Figure 1: The set of criteria

Figure 2 :

 2 Figure 2: Model of criterion 4

Figure 3 :

 3 Figure 3: Model of node 3.4

Figure 4 :

 4 Figure 4: Model of criterion 3

Figure 6 :

 6 Figure 6: Extract of the data base information

Figure 7 :

 7 Figure 7: Data concerning the four alternatives and the profiles

1 .

 1 Establish the outranking relation on the basis of the following rule: s(x, y) ⇔ C(x, y) and not D(x, y) where ∀x ∈ A, y ∈ P : C(x, y) ⇔ j∈G ± w j ≥ c and (j∈G + w j ≥ j∈G - w j ) ∀y ∈ A, x ∈ P : C(x, y) ⇔ ( j∈G ± w j ≥ c and j∈G + w j ≥ j∈G - w j ) or ( j∈G + w j > j∈G - w j ) ∀(x, y) ∈ (A × P) ∪ (P × A) : not D(x, y) ⇔ j∈G - w j ≤ d and ∀g j not v j (x, y) where -G + = {g j ∈ G : P j (x, y)} -G -= {g j ∈ G : P j (y, x)} -G = = {g j ∈ G : I j (x, y)} -G ± = G + ∪ G = -c: the concordance threshold c ∈ [0.5, 1] -d: the discordance threshold d ∈ [0, 1] -v j (x,y): veto, expressed on criterion g j , of y on x 2. When the relation S is established, assign any element a i on the basis of the following rules.

The method suggested in this document has been conceived in September 2006 and since then applied by AGETIPA together with the "engineering" teams working in rural road maintenance and other public works under the supervision of the World Bank and the government. In September 2008 there has been a follow up meeting involving expert engineers, the Public Work Minister and the World Bank.

An issue raised by the World Bank concerned the legitimation of the precise method adopted. Basically two questions have been raised: -are we sure about the "correctness" of the results? -why not using a "weighted sum" as any other does? On the first question we offered the necessary literature references (such as [START_REF] Bouyssou | Evaluation and decision models: a critical perspective[END_REF], [START_REF] Keeney | Decisions with multiple objectives: Preferences and value tradeoffs[END_REF], [START_REF] Maystre | Méthodes multicritères ELECTRE. Description, conseils pratiques et cas d'application à la gestion environnementale[END_REF], [START_REF] Pictet | Adjuger un marché au mieux-disant. Analyse multicritère, pratique et droit des marchés publics[END_REF]). On the second question we emphasised the reasons for which the choice of establishing a single value function has been rejected: 1) the presence of information for which is not immediate establishing a trade-off among the criteria

Appendix A

The basic concepts adopted in the procedure used (based on ELECTRE TRI) are the following.

) is attributed to each criterion g j .

• Each criterion g j is equipped with an ordinal scale E j with degrees e l j , l = 1 • • • k.

such that if e h j belongs to profile p h , e h+1 j cannot belong to profile p h-1 .

• A set C of categories c λ , λ = 1 • • • t + 1, such that the profile p h is the upper bound of category c h and the lower bound of category c h+1 .

• An outranking relation S ⊂ (A × P) ∪ (P × A), where s(x, y) should be read as "x is at least as good as y".

• A set of preference relations P j , I j for each criterion g j such that:

-∀x ∈ A P j (x, e h j ) ⇔ g j (x) e h j -∀x ∈ A P j (e h j , x) ⇔ g j (x) ≺ e h j -∀x ∈ A I j (x, e h j ) ⇔ g j (x) ≈ e h j ≺, ≈ induced by the ordinal scale associated to criterion g j .

The procedure works in two basic steps.


Öztürk Meltem

Guerrand Sylvie

Meltem Öztürk

On the use of a multicriteria decision aiding tool for the evaluation of comfort

2. Global approach for assessing the comfort components and their interaction for long range trains

Problem definition

In this study we are interested in defining and evaluating the coach comfort for the French national railways, SNCF. More precisely, we conducted a feasibility study about the use of a multicriteria decision aiding tool within the ACONIT project 2 assigning comfort in high speed passenger trains. The main idea of this project consisted in considering the comfort as a judgment based on a set of complex elements by taking into account the traveller's point of view. This project was focused on two key points : 1. the comfort is a notion integrating different concepts which may be objective or subjective ; 2. the perception of the comfort by the passengers may be different from the definition of the comfort given by the SNCF experts. The first part of the ACONIT project focused on the comprehension of this complex notion of comfort and tried to define it from the passengers point of view. Before the ACONIT project, comfort was defined only by the experts of SNCF ; for instance seating comfort was defined using robots and technical informations such as resonance, vibration, vision, etc. The ACONIT project was interested in a similar subject but under a different perspective : "how is comfort appreciated by passengers ?". During the first part of the ACONIT project, different surveys have been realized with the SNCF passengers, experts and a supporting team of psycholinguists in order to understand how passengers perceive and define comfort.

The study presented in this article has been conducted at the last step of the ACONIT project and was focused on structuring the knowledge acquired in the preliminary steps. It was a three months feasibility analysis where two specialists of multicriteria decision making (MCDM) worked in collaboration with the SNCF Comfort department. The role of the MCDM experts was to guide the Comfort department to put together different results and provide a global vision of the comfort ; to analyze existing methods and to propose an adequate method for the evaluation of the comfort. Because of confidentiality reasons the ancient methods can not be presented in this article. We just want to mention that these were basically based on weighted sums with some flexibility scales. Results of the previous steps of the ACONIT project have shown that there are three main types of comfort :

-Quality of the services proposed by the SNCF (on-line booking, service for disabled persons, service on the train (restaurants, controllers, etc.), quality of train connections, etc.) ; -comfort on the train (seats, sensorial aspects like noise, vibrations etc., place for the baggages, etc.) ; -comfort of the train stations. The SNCF decided to focus their efforts firstly on" the comfort on the train". The issue was how to put together different aspects of rolling materials in order to build an overall assessment of comfort compatible with the passengers perception. Our contribution to the ACONIT project was exactly on that issue. Some previous studies in the literature have shown the complexity of the task of "how to define the confort" ( [START_REF] Florès | Le confort du passager de véhicule ferroviaire : problématique et méthode d'approche[END_REF] for confort in general, [START_REF] Lavaud | L'évolution des intérieurs répondant aux soucis de confort d'agrément et de sécurité des utilisateurs[END_REF] for confort in a car, [START_REF] Goff | L'invention du confort[END_REF] from the railway point of view).

Before describing the model we have to fix the perspectives under which the model was expected to be used : why is evaluating comfort a problem and for whom and for what purpose ? Naturally the comfort on passengers trains is strongly related to the quality and the description of the rolling stock. The purchase of such stock is done through call for tenders. For the purchase or renovation of a coach the acquisition department prepares a call for tenders by specifying some needs (dimensions of the seats, material of the seats, distance between foot-rest and the seat, etc.) and evaluates them from especially two points of view : the cost and the number of specifications satisfied by the supplier. The final aggregation is done using a weighted sum.

Before our study, even if some specifications were directly related to the notion of comfort, the global evaluation of offers did not provide a view of the comfort level of each offer. On the other hand, the specifications within the call for tenders related to the comfort were defined by the acquisition department above while the ACONIT project has shown that the vision of comfort of this department could be different from the vision of some experts and from the passengers perception. For all these reasons the comfort department decided to develop a tool in order to propose an evaluation of suppliers' offers from the comfort point of view where the notion of comfort would be in coherence with the perception of the acquisition department but also with the experts and the passengers point of view. Since there are a number of different components determining the comfort, it has been decided to do a feasibility analysis for the use of a multicriteria decision aiding tool where different components of comfort would play the role of criteria. The evaluation of offers from the comfort point of view would be in term of a classification, i.e. pre-established classes would present different comfort level. This choice was related to some semantical and mathematical reasons that we will present in the rest of this article.

The feasibility analysis has been done in four steps :

-Definition of comfort integrating the passengers perception through the previous steps of the ACONIT project ; -Construction of a hierarchy of the comfort components ; -Definition of value scales for the evaluation of the comfort components ; -Proposition of a classification model. We present in Section 2 different components of comfort and we propose a hierarchy of such components in Section 3. The fact that comfort components are of different nature imposes a detailed study of their value scales, Section 4 gives some details on this subject. Section 5 shows basic notions related to the classification method that we chose, Section 6 provides decision parameters related to comfort components and chosen classification methods and finally Section 7 presents some examples.

Comfort Components

In this first step we tried to give a definition of the comfort from the passengers point of view. This construction step and the proposition of the model were the most important part of our study. As it has been already mentioned in the literature by a number of authors structuring a MCDA problem is very crucial ( [START_REF] Ec Correa | Decision support systems in action : integrated application in a multicriteria decision aid process[END_REF], [START_REF] Antunes | Structuring an mcda model using ssm : A case study in energy efficiency[END_REF], [START_REF] Keeney | Value-Focused Thinking. A Path to Creative Decision Making[END_REF]).

The data which we dealt with in this part arised from surveys done within the framework of a thesis on psycholinguistic ( [START_REF] Delepaut | Contribution de la linguistique cognitive à l'identification du confort[END_REF]) conducted on the survey used by the SNCF. The aim of this PhD thesis was to define comfort on the train and analyze it from the linguistic point of view. Methodologies used in this thesis were based on the relation between the cognition and the linguistic and different surveys were designed in order to understand how passengers "feel" the comfort. A first survey, called exploratory, was used in order to build the main inquiry by enabling to clarify the shape of more relevant and more productive questions. The main survey was formed from fifteen open written questions and 240 passengers answered to the questions. The answers of the passengers were analyzed from the cognitive linguistic point of view which is more convenient in the case of open questions. Hence different linguistic technics are used during the analysis. Syntactical and morphological tools led to the naming of various semantic categories defining the global comfort. For instance, each syntagm (syntactical unit) was studied in order to identify different nomination ways of the comfort and the implication of the use of substantives, verbs, adverbs, adjectives, concession or opposition pointers, etc. Seventy-seven categories were defined from the answers of the questioned passengers. They were then classified in twelve meta-categories. Table 1 shows these categories.

Our commitment being to implement a model of comfort to use in rolling stocks acquisition call for tenders, we analyzed more in detail these semantic categories in order to be able to select those that seemed relevant for the preparation of call for tenders. This analysis was done by the two MCDM experts who proposed to the Comfort department of the SNCF a list of components defining comfort on the train. After a discussion little modifications have been asked by the SNCF. We present in the following the final version of this list.

It is also necessary to clarify that even if the surveys were mainly related to the comfort on the train, passengers mentioned also issues related to other types of comfort (bookings, strikes, connections, etc.). For that reason among such categories there were some which were not relevant for our study. Hence we began our analysis by a detailed study of these categories and subcategories. At this step, the components of comfort were analyzed one by one from a team formed by perception experts of SNCF, experts from the comfort and the acquisition departments and decision experts. Such an analysis led to dropping off some meta-categories or categories. We first present some examples of different dropping off reasons and then list the abandoned meta-categories.

-Dropping off of meta-categories which were not related to the comfort on the train : The meta-category called comfort before and after the train which contained categories like booking facilities, access to the train, location of train stations, etc. did not directly relate to the comfort on the train, hence it was abandoned. However the elimination of each category was discussed by the team participants (SNCF experts, and decision aiding experts) ; such discussion provided a better understanding of categories and allowed some modifications or addition of new categories. For instance, the category reservation facilities had a number of sub-categories : the possibility to choose the direction of the seat (in the same direction of the train or not) was one of them. With the new technologies its is possible to have seats with a convertible back (this option is generally not chosen because of its cost). Even if this notion was really related to reservation, it showed that people were sensitive to the direction of their seat. For that reason, a new category was added into the meta-category seating comfort showing if the direction of seats is convertible or not. Similarly, the category access to the train related to the equipment for disabled people or people having luggage was moved to the meta-category service with a restriction of its contents, naturally the presence of elevators etc. was eliminated. -Dropping off of some categories which were irrelevant to the rolling stock quality even if they were related to the comfort in the train : After a discussion it was decided to abandon the categories landscape, maintenance, punctuality, trip time, speed, efficiency of the SNCF staff, ticket cost, presence of non real passengers (beggars, etc.), restoration cost. Such a decision was confirmed by the fact that such categories have been identified asuncharacteristic components of comfort in previous steps of the ACONIT project. -Some other categories were also ignored for other reasons :

-they were very general (for instance condition of transportation), -they were redundant with other categories (for instance personal characteristics and the categories of disabled people, smokers, etc.) -they were blurred and/or not frequently expressed (for instance being soft, train dimension, functionality). Such eliminations were also justified by the weak frequencies of such items in the answers directly related to the comfort in the train.

After our analysis, seven meta-categories were dropped off, five were retained and a new one was added. Some more comments are given in the following for specific reasons of abandoning each category :

-the meta-category before-after the trip : as it is mentioned before this meta-category was not directly related to the comfort on the train. Only the category access to the train was retained and moved to the meta-category stand up comfort -the meta-category generic aspect : all its categories, except the activity, being too general were redundant with other categories. Answers to surveys and other studies done by the SNCF have shown that the practice of an activity (reading, writing, working with a laptop, etc.) becomes more and more important for passengers. For that reason it was decided to create a new meta-category named activity.

-the meta-category relational : some of its components being directly related to the personality of people were abandoned. The others, number of passengers, shifting of passengers and intimacy became subcategories of the category atmosphere Sensorial meta-category. -the meta-category train : some categories were rebuild : compartmentalization, localisation, number of places, door and luggage places. The compartmentalization (coaches for different types of people and their activities) was divided into two sub-categories : disabled people, pregnant women. These categories were then placed into the meta-category Services. The number of passengers, influencing the perception of ambiance is placed into the category atmosphere. The door has appeared in the majority of answers related to the moving on the train and noise, hence it was placed in two different meta-categories, sensorial and stand up comfort. Finally, the luggage place was placed into the meta-category service.

-the meta-category temporal aspect : under the hypothesis that the speed of the train does not depend on the suppliers (but depends on the railroad company) all the components of this meta-category were abandoned. -the meta-category financial aspect : its components being not directly related to the comfort in the train were abandoned. -the meta-category emotional feeling : its components pleasure and experience, the first one being redundant with other categories and the second one being not related to our framework were abandoned.

We give now some details about the five retained meta-categories.

-the meta-category Sensorial : it was divided into five categories sound, visual, air-conditioning, atmosphere and security feeling. An analysis of dependency showed that a majority of such categories were perceived simultaneously.

-the meta-category Seating comfort : it was divided into ten categories arm-rest, head-rest, back, leg comfort, net, garbage, foot-rest, direction, train movement and table.

-the meta-category Stand up comfort : it was divided into five categories corridor, access to the train, door and movement of the train.

-the meta-category Activity : it was divided into eight categories multimedia, socket, internet connection, light, table, visibility, ambiance and seating comfort.

-the meta-category Service : it was divided into nine categories restaurant, information, toilet, nursery phone, luggage, disabled people, pregnant women and animals.

Model

This second stage consisted in the constructing a hierarchy of the comfort components.

The procedure that we undertook in this stage was the following : the MCDM experts proposed a hierarchical model to the Comfort department of the SNCF. The SNCF approved the schema of the model.

The nature of the comfort components and the results of the previous steps of the ACONIT project were the main reasons for choosing a hierarchical model. Some previous studies regrouping some of the comfort components in different classes have been already done from a psycholinguistic point of view by M. Mzali ([10]). Our model completed this one.

A hierarchical model was defined as various levels, the highest level represented the global purpose. Within our framework, the highest level represents the comfort in the train which was decomposed in different parts named metacategories which were decomposed in smaller parts, entitled categories, and so on... Such a model can be represented in a graphic way by a tree where the root represents the global purpose, nodes are the main components of the global purpose and leaves (nodes having no branches) are the components of the lowest level.

The choice of a hierarchy had certain advantages, (the interested reader may find some examples of such models in [START_REF] Bouyssou | Evaluation and decision models : a critical perspective[END_REF], [START_REF] Bouyssou | Evaluation and decision models with multiple criteria : Stepping stones for the analyst[END_REF]) : We obtained a simple representation of our model, the decomposition at several levels facilitates the aggregation procedures The construction of the hierarchy of comfort was done using information from previous steps of the ACONIT project :

-We used basically the hierarchy determined by the PhD thesis -During the PhD thesis a dependency analysis between different comfort components has been done. Some "father and son" and "brothers" dependencies are found during this analysis, which helped us to define the relations between some categories and meta-categories. -Answers in inquiries to some questions which were directly related to one of the meta-categories showed also "father and son" relations.

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!"#$%&'() The basic levels of the hierarchical model of the comfort components proposed in this study are presented in Table 2. Five meta-categories are retained : sensorial comfort, seating comfort, stand up comfort, activity and services. These meta-categories have a number of categories which are also shown in Table 2. Some of these categories are divided in sub-categories, etc. Because of lack of space we only show the sub-categories related to the seating comfort in Table 2 since in the rest of this paper we are going to develop only the aggregation of seating comfort components.

We were interested in a feasibility analysis showing the interest of a multicriteria decision aiding tool summarizing passengers evaluation of the comfort for rolling stock procurement. This evaluation aimed at allocating suppliers' offers to different categories representing the level of comfort of their proposition. Under such a perspective and for this specific purpose our model showed that global comfort could be divided into five meta-categories which were then divided into categories which were divided into sub-categories. In the rest of this paper, we are going to present and discuss the evaluation of one of the meta-categories, seating comfort ; the evaluation of the others being similar. This choice is related to the importance and the complexity of seating comfort . It had many categories with different nature and these were generally divided into sub-categories combining experts' opinion, surveys' results and the team's point of view.

Value scales for seating comfort

The components of seating comfort were of different nature and both qualitative or quantitative data could be used. Such a diversity imposed the use of different types of scale for different components.

The construction of such scales was done by the MCDM experts, one comfort engineer and one perception expert. In fact the SNCF has an Equipment Department which is specialized in the definition of a "good" ("acceptable" or "comfortable" ) equipment (for instance the minimum distance between the seat and the foot-rest must be 850 mm). A number of perception experts working at this department defined physical indicators through some physical experiences. The majority of the values presented in this part came from these indicators.

The construction of such scales being an important step of our analysis, we devote this section to this issue.

Before giving more details about different types of scales used in our analysis, we present different categories (and their sub-categories) of the seating comfort :

arm-rest : its evaluation was done by perception experts who classified it among five options : bad, not bad, normal, good and very good. head-rest : its evaluation was the result of an aggregation of assessments on two sub-categories named angle of head-rest and sensorial of headrest (perception of head-rest). Such an aggregation was used in order to class an object into three ordered categories : bad head-rest comfort, normal head-rest comfort and good head-rest comfort. i. angle : its evaluation was done by binary data : 1 when passengers can change the angle of the head-rest and 0 otherwise. ii. sensorial : its evaluation (integrating maximal acceptable pression in the contact area, the repartition of soft material and resistance to damages) was done by perception experts who gave scores between 0 and 10 (10 being the best score). back comfort : its evaluation was the result of an aggregation of assessments of five sub-categories named width, length, hardness, uniformity and angle. Such an aggregation was used in order to have five ordered categories for back comfort (bad, not bad, normal, good, very good). i. width : this was the width while seated ; its value was presented in mm and the minimum acceptable value for SNCF was 450mm. ii. length : this was the width while seated en charge, its value was presented in mm. iii. hardness : this was the hardness of the dossier and fessier. It was evaluated by perception experts and graded between 10 and 0 (10 being the best score). iv. uniformity : it showed the presence or absence of hard parts. Its was evaluated by perception experts and graded between 10 and 0 (10 being the best score). v. angle : as the angle of head-rest, its evaluation was done by binary data : 1 when passengers could change the angle of the back of the seat and 0 otherwise leg comfort : its evaluation was the distance between the seat and footrest (in mm). net : its evaluation was done by perception experts who gave notes between 0 and 10 (10 being the best score). garbage : its evaluation was the result of an aggregation of the evaluation of its two sub-categories named size and ergonomy. Such an aggregation was used in order to have three ordered categories for gar-bage.

i. size : its evaluation was qualitative but ordinal with three levels :

small, normal, big. ii. ergonomy : its evaluation was done by perception experts who gave notes between 0 and 10 (10 being the best score). foot-rest : its evaluation was the result of an aggregation of the evaluation of its three sub-categories named width, slipperiness and distance.

Such an aggregation was used in order to have three ordered categories for foot-rest. i. width : its evaluation was in mm and the minimum acceptable value for SNCF was 300mm. ii. slipperiness : Its showed the quality of having a non skid surface.

The experts evaluated it by three levels : good, normal, bad. iii. distance : this was the distance (in mm) between the surface of foot-rest and the sole of the seat. direction : it showed the presence or absence of double numbering of seats (in this case the numbers of seats could be changed by the SNCF, hence for a given travel the SNCF could know which seats would be in the same direction as the train movement). It had a binary evaluation, 1 (resp. 0) representing the presence (resp. absence). movement of train : its evaluation was done by perception experts who gave notes between 0 and 10 (10 being the best score). table : its evaluation (score between 0 and 10) was the result of an aggregation of the evaluation of its two sub-categories named bulk and ergonomy. i. bulk : its evaluation was done by perception experts who gave notes between 0 and 10 (10 being the best score). ii. ergonomy : its evaluation was done by perception experts who gave notes between 0 and 10 (10 being the best score).

Table 3 summarizes the scales of the categories and the sub-categories of the seating comfort meta-category.

In order to perform the measures on parent roods of the hierarchy above presented, we could use different procedures, not necessarily the same. The last level aggregation has been done using the ELECTRE TRI method (we explain this choice in the following section). Other procedures have also been tested and used for simple aggregation issues. 5 The Electre TRI method as the evaluation tool of our study

Category

We were looking for the intrinsic evaluation of the offers and not for just a ranking. The reason was that within a ranking approach the comparison of each pair of objects did not provide information about the quality level of the objects. For instance, if a ranking approach provides the following ranking from the best to the worst : Offer 1 Offer 2 Offer 4 Offer 3, we know that Offer 1 is at least as good as Offer 2 but no one can guarantee that Offer 1 has a "good comfort". The intrinsic evaluation had to be performed despite the presence of quantitative information. This practically amounts in classifying the offers in pre-defined ordered classes ; such a problem statement is called sorting in the literature.

We chose to work with five categories representing very bad, bad, intermediate, comfortable, very comfortable levels. Note that a categorization is an ordered one by nature and the number of categories could be changed if one wants more or less detailed results. Since such categories are ordered (category comfortable is better than category intermediate etc.) one can separate them with some frontiers. This is what the ELECTRE TRI method is doing for the sorting problem situation.

Why Electre TRI ?

The ELECTRE TRI method is a multicriteria decision aiding tool designed for sorting problems. Sorting problems consist in analyzing the intrinsic value of each object to be classified in order to propose an appropriate recommendation for each one. The ELECTRE TRI consists in allocating each object into one class. Classes are ordered (good, intermediate, not good, etc.) and are defined by the decision maker. The characterization of each class is done by defining frontiers between classes. Such frontiers are called the limit profiles.

The assignment of an object into a class is done by the help of the comparison of this object with the limit profiles. Objects to be classified are not compared to each other, hence the assignment of one object to a class is completely independent from the evaluation of another object. The comparison between an object and a limit profile is done by a binary relation called outranking relation.

The use of the outranking relation which is based on majority principal, contrary to what happens with the methods based on the weighted sum principle, does not allow compensation between different performances of the object according to different criteria (for instance a very good evaluation for seating comfort can not compensate for a bad evaluation on stand up Moreover, compensatory methods need to impose commensurability of scales of the different attributes on which the criteria are established. In our framework different types of scales with different domains are used, hence such an approach would need supplementary studies in order to define such a common scale for all the components. Such a study needs a very strong interaction between different agents of the problem (decision experts, perception experts, people from comfort and acquisition department, etc.), can take a long time and may be difficult (for instance we have to be capable to say how many centimeters we have to increase the distance between the foot-rest and the seat if the garbage looses 8 mm 3 of its volume). As in the previous example the compensation may be relatively difficult to be expressed in some cases.

Briefly, the definition of our problem (allocation of offers into ordered categories), the nature of the comfort components (presence of qualitative data, different value scales and different value domains), our preference for a non compensatory method and our will to have a method where the assignment of objects are independent from each other were the main reasons for choosing the ELECTRE TRI for our problem. However as it will be explained in the rest of the paper, the evaluation of offers according to some sub-categories may be done by other methods.

The interested reader can find more details about the ELECTRE TRI in Appendix.

Decision parameters

Let us remind that in this section we are only interested in the seating comfort selected as an example.

Importance parameters

The construction of the importance parameters for the seating comfort was based on an analysis done before our study. This analysis was done by the SNCF experts and a PhD student on psycholinguistic. In their study, the SNCF people calculated the frequency of answers of passengers to some questions. Such frequencies were presented in their study in two parts : the ones related to a positive evaluation of the comfort and the others related to a negative evaluation. We used the sum of these frequencies in order to get an idea of the importance of different categories.

Thresholds

Concerning the majority threshold, we chose to make use of the default value of the ELECTRE TRI, 0.76. This value means that in order to say that object x is at least as good as a limit profile, at least 76% of criteria must be in concordance with this affirmation (naturally after the weighting).

Veto thresholds helped us to give up an assignment into a class if the offer was not strong enough according to one or more important criteria even if the majority (more than 76% of criteria) was for this assignment. We decided to define three veto thresholds for the most important criteria, back comfort, leg comfort and train movement. We set these thresholds to 1 for back comfort, to 3 for train movement and to 30 for leg comfort. A veto threshold fixed to 1 represented a very strong demand for the evaluation of its criterion. For instance, concerning the seating comfort, if the limit profile between the classes very good and good required a very good evaluation for back comfort, an offer having a normal evaluation for back comfort, it could never be classed into the very good seating comfort class. Note that the smaller is the veto threshold, the stronger is its power.

The introduction of indifference thresholds was not judged necessary because of the small number of levels of scales.

Limit

The limit profiles are imaginary objects representing the limit between two consecutive classes. Figure 1 shows a graphical representation of classes where g i represents the ith criterion, C k the kth class and a j the limit profile between the classes C j-1 and C j . The limit profiles a 0 and a m were omitted in our study.

✲ ✲ ✲ ✲ g 4 g 3 g 2 g 1 ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❅ ❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ g 4 (a 1 ) g 4 (a 2 ) g 2 (a 1 ) g 2 (a 2 ) g 1 (a 1 ) C 1 g 1 (a 2 ) C 2 C 3 g 3 (a 1 ) g 3 (a 2 ) Figure 1 -General graphical representation of classes
The definition of classes and the characterization of limit profiles were done by MCDM experts in collaboration with the Comfort department. The definition of the limit profiles was done by the help of some intuitive allocation examples. For instance the MCDM experts showed a fictitious offer (for instance very bad in almost all criteria except back comfort) and asked to comfort engineers in which category they wanted to put it. We defined five ordered classes for seating comfort and called them very bad, bad, normal, good and very good. Four limit profiles, a 1 , a 2 , a 3 and a 4 were defined in order to separate these classes. The profile a 1 separated the class very bad from the class bad, the profile a 2 separated the class bad from the class normal, etc. The evaluation of profiles for seating comfort components is showed in Table 5. Naturally such evaluations depended on the value scales defined in Section 4. A graphical representation of such classes can be found in Figure 2.

Frontier Table 5 -Presentation of limit profiles for seating comfort

Aggregation of sub-categories

Some of the seating comfort components had sub-components. The evaluation of such components could be done by different aggregation methods on their sub-components. We present in the following such aggregation procedures.

Head-rest comfort

There were two sub-categories : angle and sensorial. These sub-categories were defined with different types of scales (binary data for angle meaning that

NB N G VG ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ❆ ❆ ❆ ❆ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ❅ ❅ ❅ ❅ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❇ ❇ ❇ ❇ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ◗ ◗ ◗ ◗ ◗ ◗ Figure 2 -Seating comfort classes
the angle of the head-rest could be changed and scores gave by experts for sensorial, score between 0 and 10) and the final recommendation on head-rest comfort was a classification in three ordered classes (bad, normal and good ). The classification was done with simple rules :

-if angle is one and sensorial note is more than 7 then the evaluation is good, -if angle is one and sensorial note is between 4 and 7 then the evaluation is normal, -otherwise the evaluation is bad.

Back comfort

There were five sub-categories called width, length, hardness, uniformity and angle. As in the case of head-rest comfort these sub-categories were defined on different types of scales (interval scales, binary data or ratio scales). In order to deal with this scale's diversity and to not allow compensation, the ELECTRE TRI method was again used. The final recommendation was about the assignment of offers into five ordered classes very bad, bad, normal, good and very good back comfort.

Foot-rest and garbage comforts

There exist two sub-categories, size and ergonomy, for the component garbage and three sub-categories, width, slipperiness and distance, for the component foot-rest. We used a rule based method for the evaluation of garbage comfort : if size is big and the note of ergonomy is grather than 7 then the garbage comfort is good, otherwise if size is normal or big and the note of ergonomy is grather than 4 then the garbage comfort is normal and in all the other cases the garbage comfort is bad. Again for scale and non compensation reasons and the final recommandation being a classification, the ELECTRE TRI method was used for foot-rest comfort. Figure 7 presents the limit profiles of classes.

a

Table comfort

There were two sub-categories, volume and ergonomy, both of them defined on a ratio scale ({0, . . . , 10}) representing the scores given by experts. The aggregation method could provide a score for table comfort as it is shown in Table 3. For this aggregation a weighted sum was used with the substitution rates presented in Table 8. The score of table comfort of an offer x, denoted for instance by g(x) is calculated by g(x) = i w i g i (x) where g i (x) presented the score of x for the sub-component i and w i presents the substitution rate of the sub-component i.

Table

Substitution rate (w i ) volume 0.5 ergonomy 0.5

Examples

In this section we present three imaginary offers and analyze their assignment into different classes. Such examples were prepared in order to illustrate and explain the principles of the the ELECTRE TRI method (veto threshold, incomparability, optimistic and pessimistic procedures, etc.) . Table 9 represents the assessments of these three offers for the seating comfort components and sub-components. Such evaluations were chosen by MCDM experts.

In the following we will focus on the aggregation of the components.

Assignment of Offer 1 to the class normal seating comfort

We analyzed the outranking relations between Offer 1 and the limit profiles for both the "optimistic" and "pessimistic" procedure :

-Pessimistic procedure : The procedure compared first of all the offer to the limit profile a 4 and then to a 3 , a 2 etc. and stopped when the offer outranked a limit profile. Offer 1 did not outrank profiles a 4 and a 3 and was indifferent to profile a 2 (indifference being a part of the outranking relation -x outranked y if and only if x was preferred to y or x and y were indifferent-Offer 1 outranked a 2 ). Hence the procedure stopped and Offer 1 was affected to normal seating comfort class. -Optimistic procedure : The procedure began by comparing Offer 1 to the limit profile a 1 (profile separating the lowest class C 1 from C 2 ), if limit profile as not preferred to Offer 1, the comparison procedure continued, Offer 1 was compared to a 2 , a 3 , etc. limit profile a 1 was outranked by Offer 1, a 2 and Offer 1 were indifferent and Offer 1 was outranked by a 3 ; hence the procedure stopped and Offer 1 was affected to the normal seating comfort class. Table 9 presents the evaluation of Offer 1 for the seating comfort components. This shows that the evaluation of Offer 1 for the back comfort, train movement and leg comfort attributes are higher than the ones of the limit profile existing between classes not bad and normal. Hence we concluded that if the supplier wanted to improve his offer he must improve his offer from the table point of view, table being one of the important seating comfort components.

Assignment of Offer 2 to the class good seating comfort

Table 9 presents the evaluation of Offer 2 for the seating comfort components. It is easy to remark that the veto threshold on leg comfort had a strong The evaluations of Offer 2 were at least as good as (better than or preferred to) all the evaluations of the limit profile a 4 except the one of the leg comfort. Thank to these good evaluations Offer 2 obtained a majority coalition in his favor against a 4 however the category leg comfort introduced a veto to the outranking of a 4 by Offer2. Hence Offer 2, instead of being classified into the very good seating comfort class was classified into the good seating comfort class only. This example showed the importance of the veto thresholds for imposing very good evaluations. In this example, if the supplier wanted to improve his offer he must improve his offer from leg comfort point of view.

Assignment of Offer 3 to two different classes not bad seating comfort and good seating comfort

We analyzed the outranking relations between Offer 3 and the limit profiles for optimistic and pessimistic procedure since these two procedures did not provide the same assignment. Table 9 presents the evaluation of Offer 3 for seating comfort components.

-Pessimistic procedure : The procedure compared first of all the offer to the limit profile a 4 , Offer3 did not outrank a 4 . Then for the comparison of Offer3 and a 3 and a 2 we got incomparabilities. The procedure continued, Offer3 outranked a 1 , hence the procedure stopped and Offer3 was affected to not bad seating comfort class. -Optimistic procedure : The procedure began by comparing Offer 1 to the limit profile a 1 which did not outrank Offer3. Limit profiles a 2 and a 3 did not outrank Offer 3 because of the incomparabilities. The profile a 4 was preferred to Offer 3, hence the procedure stopped and Offer3 was affected to good seating comfort class. This example showed the role of incomparabilities in the assignment difference.

Conclusion

In this paper we presented a real word application of muticriteria decision aiding : Evaluating suppliers offers to call for tenders in rolling stock procurement from the comfort point of view. Our study was used as a feasibility analysis for the introduction of multicriteria tools in the SNCF. The results found in this study were jugged to be interesting by the SNCF who wants now to use similar approaches for the evaluation of other comfort aspect and/or other components of offers to call for tenders. A PhD student began to work on this subject since april 2010.

The complex nature of comfort was presented here using a hierarchical model. In the paper we showed how and why to use such a model. The presence of different types of data, -qualitative, quantitative, binary, etc.-was handled using different methods on different nodes of the comfort hierarchy, with a special interest on outranking methods. The overall assessment resulting from the hierarchical aggregation of the values helped the SNCF to classify the suppliers offers into five comfort categories. Such a classification gave information about the quality of offers and allowed also negotiation with suppliers.

Our contribution being a feasibility analysis focused specially on the test of the use of data coming from the PhD thesis on psycholinguistic, some aspects of MCDA have been ommited, such as sensibility analysis, management of diverging opinions, etc. These aspects will be naturally taken into account in the final application of the MCDA tool.

The comparison between two elements x and y (x may represent an object and y a limit profile or the inverse) is done by an outranking relation denoted by S. The affirmation xSy (or S(x, y)) means that "the element x is at least as good as the element y" and is calculated using two indices, the Concordance and the Discordance index. One can find different, more or less refined, definition of such indices in the literature but all of them are based on the same following idea :

-Concordance index : shows if there is a sufficiently strong majority of criteria in favor of the outranking relation ; -Discordance index : shows if there is at least one criterion "strongly opposed" to the outranking relation (in such a case we say that the criterion has a veto for the outranking relation).

In the following we note C(S(x, y)) (resp. D(S(x, y))) in order to say that there is concordance (resp. discordance) for the outranking S(x, y). Hence the relation S(x, y) is verified if there is concordance but not discordance :

xSy if and only if C(S(x, y)) and not D(S(x, y))

We will present in the following a classical formule of concordance and discordance indices. The interested reader can find more detailed explanations on the subject in [START_REF] Roy | Multicriteria Methodology for Decision Aiding[END_REF], [START_REF] Roy | Aide Multicritère à la Décision : Méthodes et Cas[END_REF]. The computation of these indices makes use of different parameters such as the importance of a criterion, the indifference threshold, the veto threshold and the majority threshold.

C(x, y) ⇐⇒

j∈Jxy w j j w j ≥ γ,

D(x, y) ⇐⇒ ∃j : g j (y)g j (x) > v j

where :

g j is a real valued function representing the evaluation of alternatives with respect to the criterion c j (to be maximized) ; w j is a non negative coefficient which represents the importance of the criterion c j ; -J xy represents the set of criteria for which x is at least as good as y ; more precisely, J xy = {j : g j (y)-g j (x) ≤ q j } where q j is the indifference threshold associated to criterion c j ; γ is a majority threshold ; v j is a veto threshold on criterion c j ;

The majority threshold represents the minimum percentage of criteria (weighted according to their importance) needed in order to have a concordance. The veto threshold is used for the discordance index and represents for each criteria the threshold for which a difference of evaluation on this criterion becomes problematic for the construction of the outranking relation. The indifference threshold represents the maximum tolerated difference between evaluations of two objects x and y in order to say that x and y are indifferent. In what follows, we will assume, without any loss of generality, that preferences increase with the value on each criterion.

It is easy to see that comparing two objects x and y, four situations may appear :

-xSy and not ySx : we say that "x is preferred to y" ; -not xSy and ySx : we say that "y is preferred to x" ; -xSy and ySx : we say that "x and y are indifferent" ; -not xSy and not ySx : we say that "x and y" can not be compared ;

The last case shows that the outranking relation is not necessary a complete relation.

After the construction of all comparisons between alternatives and profiles, the exploitation procedure begins. The role of this procedure is to analyse now which an alternative x compares to subsequent profiles in order to determine the class to which x should be assigned. The ELECTRE TRI proposes two different assignment procedures :

the pessimistic assignment procedure : i. compare x successively to limit profiles a i , for i ∈ {p, p -1, . . . , 0}, ii. a h being the first profile such that xSa h , assign x to class C h+1 .

If a h-1 and a h denote the lower and upper profile of the category C h , the pessimistic procedure assigns alternative a to the highest class C h such that x outranks a h-1 , i.e., xSa h-1 . the optimistic assignment procedure : i. compare x successively to a i , for i ∈ {1, 2, . . . , p}, ii. a h being the first profile such that a h Sx and not xSa h (i.e. x is preferred to a h ), assign x to class C h .

The optimistic procedure assigns x to the lowest class C h for which the upper profile a h is preferred to x.

The ideas that ground the two assignment procedures being different, these assignment procedures might assign some alternatives to different classes. The difference is basically related to the fact that the outranking relation is not complete, more precisely :

-when the evaluation of an alternative is between the two profiles of a class on each criterion, then both procedure assign this alternative to this class, -a divergence exists among the results of the two assignment procedures only when an alternative is incomparable to one or several profiles ; in such a case the pessimistic assignment rule assigns the alternative to a lower class than the optimistic one.

Table 1 -

 1 Comfort model given by the SNCF

	Comfort before and after the train access to the train, station, reservation,
		nombre of people waiting for the train
	General aspect	activity
	Sensorial	air, atmosphere, noise, train movement,
		color, space, light, material, carpet,
		landscape, cleanliness, toilet cleanliness,
		security, temperature, air-condition,
		visibility, smell
	Seating comfort	arm-rest, head-rest, back comfort,
		leg comfort, internet connexion, garbage,
		net, passengers movements,
		functionality, socket, foot-rest,
		train direction, table
	Stand up comfort	corridor , door
	Relational	crowd, other passengers, civicism,,
		shifting of passengers,
		efficiency of SNCF staff, intimacy,
		presence of non real passengers
	Services	bar, material disadvantages, information
		multi-media, personnalization
		catering, phone, toilets
		animals, nursery
	Train	being practical, compartmentalization,
		number of places, location, functionality,
		condition of transportation, door,
		luggage places, maintenance, classes,
		material,train dimension, being soft
	SNCF image	modernity
	Temporal aspect	trip time, speed, punctuality
	Emotional aspect	passengers emotion, train level, experience,
		pleasure
	Financial aspect	restoration cost, ticket cost

Table 2 -

 2 Hierarchy of comfort on the train

Table 3 -

 3 Scales of seating comfort components and sub-components

		Category Scale	Sub-category	Sub-category Scale
	Arm-rest	bad, not bad, normal,	-	-
		good, very good
	Head-rest	bad, normal, good	angle sensorial width	{0, 1} {0, . . . , 10} [400, 650]
		bad, not bad, normal,	length	[500, 700]
	Back comfort Leg comfort	good, very good [850, 980],	hardness uniformity angle -	{0, . . . , 10} {0, . . . , 10} {0, 1} -
	Net Garbage	{0, . . . , 10} bad, normal, good	-size	-bad, normal, good
			ergonomy width	{0, . . . , 10} [200, 600]
	Foot-rest	bad, normal, good	slipperiness	bad, normal, good
			distance	[80, 200]
	Direction Train Movement Table	{0, 1} {0, . . . , 10} {0, . . . , 10}	--volume ergonomy	--{0, . . . , 10} {0, . . . , 10}

Table 4 -

 4 Table 4 shows this evaluation. Frequency of answers and weights

		Positive Negative	Total	Weight
		answers	answers	answers
	Arm-rest	1	2	3	3/176
	Head-rest	0	5	5	5/176
	Back comfort	37	25	62	62/176
	Leg comfort	2	42	44	44/176
	Net	0	3	3	3/176
	Garbage	0	5	5	5/176
	Foot-rest	3	2	5	5/176
	Direction	1	7	8	8/176
	Train movem.	4	27	31	31/176
	Table	4	6	10	10/176
	TOTAL	52	124	176	1

  a 1 Frontier a 2

				Frontier a 3	Frontier a 4
		bad-	not bad-	normal-	good-
		not bad	normal	good	very good
	Arm-rest	not bad	normal	good	very good
	Head-rest	normal	normal	good	good
	Back comfort	not bad	normal	good	very good
	Leg comfort	850	920	950	970
	Net	4	6	7	9
	Garbage	normal	normal	good	good
	Foot-rest	normal	normal	good	good
	Direction	0	0	1	1
	Train movem.	4	5	6	8
	Table	4	5	6	8

Table 6 -

 6 Table 6 presents limit profiles of this component. No veto or indifference thresholds were defined and the majority threshold was fixed to 0.76. Presentation of limit profiles for back comfort

		a 1	a 2	a 3	a 4
	Back Comfort Frontier Frontier	Frontier	Frontier
		bad	not bad	normal	good
		not bad	normal	good	very good
	Widht	450	460	480	550
	Lenght	530	550	550	550
	Hardness	0	1	1	1
	Uniformity	4	6	8	9
	Angle	8	8	10	10

Table 7 -

 7 Presentation of limit profiles for foot-rest

		1	a 2
	Foot-rest	Frontier	Frontier
		bad/normal normal/good
	Width (mm)	300	400
	slipperiness	normal	good
	Distance (mm)	130	130

Table 8 -

 8 Substitution rates for table

Table 9 -

 9 Examplesinfluence on the assignment of Offer 2 to the not bad seating comfort class.

Appendix

The general procedure of the ELECTRE TRI has two consecutive steps : -construction of a binary relation establishing how alternatives are compared to the boundaries of classes, -exploitation (through assignment procedures) of the binary relation in order to assign each alternative to a specific class.

We present first of all the first step consisting in comparing alternatives to profiles representing the frontiers between ordered categories. We will note by X the set of objects to be classified (for instance suppliers' offers), X = {x 1 , x 2 , . . . , x n }, and by A = {a 0 , a 2 , . . . , a m } the set of limit profiles. Let us denote by C = {C 1 , C 2 , . . . , C m } the set of classes, the class C 1 being the worst one and C m the best one etc. If we have m classes, we will have m + 1 limit profiles where a 0 (resp. a m ) represents a fictive profile having the worst (resp. the best) evaluation on each criterion while a limit profile a i , i ∈ {1, 2, . . . , m -1}, represents the frontier between the classes C i and C i+1 .


Fadi El Haddad

Lionel Fourment
email: lionel.fourment@mines-paristech.fr

David Ryckelynck

François Bay

Application of P.O.D. and D.E.P.O.D. model reduction methods to semithick sheet metal forming

Model reduction allows transforming a high-dimensional model into a very low-dimensional approximation, which makes it possible to significantly decrease the computational time while preserving the physics of the problem. In this paper, it is applied to a metal forming process, the semi-thick sheet bulging test, where computing time can be quite significant when using the associated inverse analysis. The problem is characterized by large deformations, development of instability, implicit velocity / pressure formulation and unstructured meshes of tetrahedrons. A Proper Orthogonal Decomposition (POD) is implemented in the commercial software Forge®. The proposed reduced order model (ROM) involves a reduced basis for both velocity and pressure fields to account for the incompressibility constraint. The ROM by POD allows reducing the number of unknowns to less than 12 while keeping the results accuracy and precisely predicting the sheet instability development. The ROM remains accurate for a wide range of material parameters variations. The DEPOD approach allows further extending the ROM validity range, through the information of additional snapshots in the vicinity of the initial ones.

Introduction

Simulation techniques have brought huge contributions to engineering problems. Despite major progress in numerical methods and computers capacities, accurate solutions require refined space discretizations, which consequently increase computational time. Model Order Reduction (MOR) seeks to overcome this issue by approximating the high dimensional problem with a lower one, following several techniques as proposed in literature [START_REF] Krysl | Dimensional model reduction in non-linear finite element dynamics of solids and structures[END_REF][START_REF] Nguyen | Certified Real-Time Solution of Parametrized Partial Differential Equations, Handbook of Materials Modeling[END_REF]. This work is based on the Proper Orthogonal Decomposition (POD); it does not aim to develop a new variant of the method but instead to study its capacity to treat metal forming problems including high non-linearities, large deformations and unstructured meshes.

Bulging Rheological Test

The considered metal forming problem is the bulged test (see Figure 1 and 2) often used to identify the material behaviour through inverse analysis. An oil pressure is enforced on the metal sheet while the sides of the sheet are maintained by a blank holder. The inverse analysis requires carrying out several simulations of the forming problem with different material parameters, so resulting into large computational time. The circular sheet is 1 mm thick with a 50 mm radius. The material, formed under hot conditions, is assumed to follow the Norton-Hoff law [START_REF] Nguyen | Certified Real-Time Solution of Parametrized Partial Differential Equations, Handbook of Materials Modeling[END_REF]. Its consistency K is equal MATEC Web of Conferences to 202 MPa.s and its strain rate sensitivity m is equal to 0.15. Two oil pressure cycles are considered from the initial value of 0.3 MPa at t = 0 s. 1. Stable cycle: linear increase to 3 MPa at t = 20 s followed by a linear increase to 4.125 MPa at t = 50 s 2. Unstable cycle: same linear increase to 3 MPa at t = 20 s followed by a linear increase to 6 MPa at t = 100 s. Figure 3 shows the deformed bulged sheets at the end of the process in both cases, computed with an unstructured finite element mesh of tetrahedra with 700 nodes. Figure 4 shows the sudden increase of velocity at the central node of the sheet with the "unstable" cycle and that is triggered by the thickening of the sheet before tearing.

Metal Forming Problem Equations

Inertia and gravity forces can be neglected in the problem equations (1.a). Elasticity is also neglected during large material deformations under hot conditions, so that the material is fully incompressible (1.b), where  is the stress tensor and v the material velocity.

. 0 over . 0 over

         v  (1)
Norton-Hoff constitutive equation is given by ( 2) where () v  is the strain rate tensor,  the equivalent strain and p the hydrostatic pressure. Boundary conditions are summarized in equation ( 3), and the resulting weak form of the mixed velocity / pressure problem is presented in equation ( 4).

 

1 2 3 ( )   m K v pI    (2) 0 . 0 over . over 0 over            free pressure blankholder n np v   (3)           1 , 2 3 : div 0 , q div 0                       m w K v w d p w d q v d       (4)
The computational domain  is discretized using an unstructured mesh made of tetrahedra and equation ( 4) by the Mini-Element P1+/P1 formulation [START_REF] Coupez | From a direct solver to a parallel iterative solver in 3-D forming simulation[END_REF]. The resulting system of  

, hh vp equations is nonlinear: R denotes the residual of the equations and J its Jacobian. The matrix formulation of the linear system solved at each iteration of the nonlinear resolution algorithm can be written as:

, with , and

           vv vp t vp pp JJ v J q R J q p JJ    (5)

Reduced Order Model (ROM)

Proper Orthogonal Decomposition (POD) methods generally aim at finding a reduced equivalent system to (5) while preserving the accuracy of the solution. It consists in building a reduced orthonormal basis

  1, r rm N 
with m < N (number of dof) that minimizes the sum of the squared projection error over the chosen snapshots, following the snapshot POD approach introduced by Sirovich in [START_REF] Sirovich | Turbulence and the dynamics of coherent structures part I: coherent structures[END_REF]. At every snapshot time

  1, n i it
t  , the velocity and pressure fields are stored respectively into two matrices v Q and p Q :

1 1 1 1 ( , ) ( , ) ( , ) ( , ) n v n v x t v x t Q v x t v x t       NN (6) 1 1 1 /3 1 /3 ( , ) ( , ) ( , ) ( ,

   

,,   

t i i i i Q U V i v p (8)
Selecting the m first vectors for velocity and pressure provides the A reduction matrix (9). The solution in the reduced space writes  q Aa (10) and the reduced matrix formulation of the problem is expressed by (11):

0 0     v p U A U (9) , with ,           v p a v q Aa
q and a a p (10)

 tt A J A a A R  (11)
In order to extend the validity range of the ROM in respect to a specific material parameter coefficient such as m, computing snapshots for different m values is necessary. In the DEPOD approach [START_REF] Schmidt | Derivative-Extended POD Reduced-Order Modeling for Parameter Estimation[END_REF], the new m value is obtained from a slight modification of the initial value (12), and the new combined snapshot matrix    

, i i v p Q 
includes results from both simulations (13) where is a numerical coefficient:

21  m m m  (12)     1 1 2 1 21 , ,,        i i i i i ii Q i Q Q Q Q QQ vp  (13)
4 Applications

Modes Selection

Snapshots are extracted at every time step, Figure 4 shows the relative L 2 error on the velocity field at the end of the process, according to the number of velocity and pressure modes. The error decreases very rapidly and the minimum value is obtained with 12 modes. This number is then selected for following computations.

Accuracy of the Reduced Order Model

The ROM constructed in this way is now used with different values of the m coefficient of the Norton-Hoff constitutive equation (2). Figure 5 shows that for m ranging between 0.08 and 0.18 the relative L 2 error is less than 1%, which is more than acceptable. This domain can be further increased by the DEPOD approach. A new simulation is run with m=0.16 in order to compute new snapshots and to extract new modes. The errors for m=0.05 and m=0.2 are then reduced to less than 1%.

In the unstable case, Figure 6 shows that the moment for which the instability occurs depends significantly on the m value and is quite well predicted by the ROM.

Figure 1 .

 1 Figure 1. Boundary conditions for the bulging test.

Figure 2 .

 2 Figure 2. Velocity field at the end of the bulged test (stable case).

Figure 3 .

 3 Figure 3. Isovalues of the equivalent cumulated strain at the end of the process. Left: stable case. Right: unstable case.

Figure 4 .

 4 Figure 4. Time evolution of the vertical velocity at the centre of the sheet in the unstable case.

( 7 )

 7 Short title of the conferenceThe reduced basis is derived from the Singular Value Decomposition (SVD) of     ,

Figure 4 .

 4 Figure 4. Approximation error of the ROM versus the number of velocity/pressure modes.

Figure 5 .

 5 Figure 5. Approximation error of the ROM according to the strain rate sensitivity coefficient value of the material.

Figure 6 .

 6 Figure 6. Time evolution of the vertical velocity at the centre of the sheet in the unstable case for different values of the material strain rate sensitivity coefficient with the ROM and full model.

  )

	Q	p	p x t  p x t n   
			n   NN p x t p x t