F Garcia

B Grabot

Enterprise Web portals for Supply Chain coordination: a case study

Keywords: Supplier web portal, supply chain, coordination

The performance of the supply chains is highly linked with the quality of relationship between partners. Web portals are more and more used for facilitating this relationship, but the consequences of their implementation on real cases are still seldom analysed. Such analysis is provided in this communication, highlighting the interest but also the problems of supplier portals thanks to a questionnaire filled by more than 130 suppliers of a large company.

Introduction

The use of the Information and Communication Technologies (ICT) is now quite mature inside companies, with the generalisation of the ERPs (Enterprise Resource Planning) systems), and in Supply Chains (SCs), with EDI (Electronic Data Interchange) and Internet as privileged communication tools [START_REF] Williamson | Information Systems Development within Supply Chain Management[END_REF]. The study of the influence of these tools on the relationships and quality of cooperation between customers and suppliers is as a consequence a field of growing interest. In this context, this communication focuses on the use of supplier web portals in a multi-site company of the high-tech sector, characterised by long cycle times, short series and high need of reactivity, in which each factory has to synchronize many suppliers. In that case, the diversity of size and culture of the partners, leading to quite different maturity levels towards ICT, considerably complicates the coordination between partners [START_REF] Ming | A Typology of the Situations of Cooperation in Supply Chains[END_REF].

After a short survey on the role of ICT in integration and coordination of Supply Chains, we investigate the case of Inter-Organizational information Systems Infrastructures (IOSI), and more specifically supplier web portals. As a second step, a case study is presented: the use of the supplier web portal of a French large company, "Electra" (fictive name). The methodology of the study and its main results are then discussed.

There is an interesting chronological correlation between the development of the concept of SC, appeared in the 80's, developed in the 90's and generalised in the 2000's, the evolution of ICTs and the context of globalisation. These three phenomena are indeed intrinsically linked: the strategy of the companies aiming at developing their SC has relied on the new possibilities of information systems, especially linked to Internet, in order to benefit from the ever increasing globalisation of economy. In their survey, Arshinder at al. [START_REF] Arshinder | Supply Chain Coordination: Perspectives, Empirical Studies and Research Directions[END_REF] suggest a classification of the themes attached to the coordination of the SC: the Information and Communication Technologies are one of the considered mechanisms for coordinating a SC, information systems and contracts allowing information sharing and collaborative decision-making. The integration of functions inside a company is usually performed thanks to the large functional coverage of an ERP, while with an external point of view, the EDIs have improved the quality and the quantity of information exchanged by the partners of the SC. A study realised on an important panel of French companies [START_REF] De Corbière | De l'intégration interne du système d'information à l'intégration du système d'information de la chaîne logistique[END_REF] shows that the companies using an ERP have a better external degree of integration with their suppliers and service providers (transporters for instance), using tools like EDIs, internet, web portals, than those who do not internally use an integrated information system. In that context, an IOSI is a collection of information technology resources, which includes communication network, hardware, IT applications, standards for data transmission, and human IT skills and experiences. IOSI helps organisations to establish and maintain interfirm relationships such as alliances, partnerships, and buyersupplier relationship [START_REF] Haiwook | The effects of interorganisational information systems infrastructure on electronic cooperation: An investigation of the ''move to the middle[END_REF]. Fours stages of evolution of IOSI may be distinguished [START_REF] Williamson | Information Systems Development within Supply Chain Management[END_REF]: -stage 1: IOSI using paper supports (order forms, delivery orders...). We suggest to extend this stage to the use of Excel or Word files.

-stage 2: IOSI using EDIs.

-stage 3: IOSI using an ERP.

-stage 4: IOSI using internet portals or platforms. We shall see that these four stages coexist in the IOSI used by Electra and his suppliers. Finally, it is possible to characterise IOSI according to the inter-enterprise coordination mechanisms [START_REF] Corbière | SIIO et amélioration de la qualité des données : analyse de la synchronisation des données par catalogues électroniques[END_REF], by distinguishing those based on bilateral relationship, e.g. an EDI between two companies; those centred on a partner who is in relation with multiple customers or suppliers (this is the case for the considered portal) and finally IOSI putting in relation multiple customers with multiple suppliers (like electronic marketplaces, but also supplier portals like Boostaerospace1 , providing a unique portal for customers of the aeronautical industry and thousand of their suppliers).

The supplier web portal: an inter-organisational technological tool allowing coordination of customer and supplier

Under different names ("B2B portal", "Enterprise Information Portal", "Corporate Information Portal", or "Corporate Portal") the concept of enterprise web portal appeared in the end of the 90's with the development of Internet [START_REF] Dias | Corporate portals: a literature review of a new concept in Information Management[END_REF]. According to [START_REF] Shilakes | Enterprise information portals[END_REF], "Enterprise Information Portals are applications that enable companies to unlock internally and externally stored information, and provide users a single gateway to personalized information needed to make informed business decisions". Enterprise portals can have various contents, aims, technologies or targeted users. It is possible to classify them according to [START_REF] Clarke | Web-based B2B portals[END_REF]:

-the content provided by the portal: transactional (order entry, facture payment) vs. informational (technical documentations techniques, price catalogue), -the targeted portal users: wide target and generic purpose portal (horizontal) or more specific target with deep technical content (vertical), -a public-private dimension, with portals open to a wide audience or restricted to given users.

In the case of Electra, the portal is a transactional, vertical and private one.

In addition, three distinct orientations of B2B portals may be distinguished in SCs: customer oriented portals, supplier oriented portals, and digital marketplaces where customers and suppliers can meet [START_REF] Chan | A framework to develop an enterprise information portal for contract manufacturing[END_REF]. Most of the existing studies deal with customer portals in the domain of Customer Relationship Management or e-commerce, or with digital marketplaces providing new opportunities to the company through a new marketing channel [START_REF] Helmy | An Integrated ERP with Web Portal[END_REF]. In contrast, little research has been conducted on supplier portals with a technical point of view [START_REF] Boyson | The e-supply chain portal: a core business model[END_REF]. These supplier portals may be the forefront of an evolution of the customer-supplier relationship and of the integration of the SC information systems, since they allow to streamline and integrate the information flows between partners of the SC. Some examples of integration of the information flows by means of supplier portals have been reported, e.g. in the US Air Force [START_REF] Boyson | The e-supply chain portal: a core business model[END_REF] in a Taiwanese company [START_REF] Chan | A framework to develop an enterprise information portal for contract manufacturing[END_REF], or in the supply chain of a company of the automotive sector [START_REF] Chang | Enterprise Information Portals in Support of Business Process, Design Teams and Collaborative Commerce Performance[END_REF].

Often imposed by the customers, supplier web portals may create an asymmetry in the customer-supplier relationship, since the supplier is committed to use the customer's tool (the web portal), follow the rules of utilisation of the portal, train on the tool, and provide the information requested by the customer, in the required format.

The supplier hopes in return to consolidate his relationship and obtain a better quality of information, that he will be able to integrate in his internal information system. Moreover, the necessity to be integrated with the customer through a supplier portal creates a "barrier" for new suppliers, which protects as a consequence the "old" ones.

In practice, it can be noticed that this barrier increases the feeling of security of the supplier, together with his trust in the customer. After a study on Ferrari's supplier portal, [START_REF] Baglieri | Exploring the Impact of a Supplier Portal on the Buyer-supplier Relationship. The Case of Ferrari Auto[END_REF] even concludes that the use of a portal more impacts the improvement of the communication and relationships with the customer than the logistic performance or the purchasing process. Nevertheless, Robey et al. [START_REF] Robey | Theoretical foundations of empirical research on interorganizational systems: assessing past contributions and guiding future directions[END_REF] analyse the interorganisational impact of IOSIs, focusing on EDIs, and conclude that the use of EDIs reinforces the power of the customer at the expense of their suppliers, while the quality of the customer-supplier relationship influences the good use of the EDIs, and as a consequence the integration and coordination mechanisms.

Case study

The inputs of the literature summarized in section 3 have been confronted to a real case: the Electra company and its relationships with its suppliers through its supplier portal.

Context of the company

Electra manufactures high tech products on several production sites. Its activity is based on contracts on several years; its cycle times are long; the company has longterm relationships with many highly specialized suppliers. The company uses the SAP ERP for many years, and has recently implemented a supplier portal integrated in SAP. Within this portal, the exchanges between the customer and the suppliers are performed according to the process summarized in Fig. 1.

Figure 1. Information exchange between Electra and its supplier through the portal

The project of implementation of the portal at Electra's was launched in the beginning of 2010, with a progressive deployment since 2011 on pilot sites and pilot suppliers, until the integration of all the sites and a majority of suppliers in 2013. In 2014, all the industrial sites of Electra, and about 450 suppliers, were using the portal. Our case study concerns the post-implementation phase of the project, and studies the appropriation of the portal by the suppliers, from their perspective.

Methodology

One of the authors has followed the implementation of the web portal in a pilot site during two years, and has had meetings with the project leader on the site, with the supply chain leader and with the project leader in the head office. An investigation has then been decided with two main objectives: for Electra, in order to better understand the modalities of use of the portal by the suppliers and the linked problems; for the authors, in order to analyse how this portal influences the relationship between customer and supplier.

A questionnaire with 45 items has been created using a methodology of investigation based on questionnaires used in social sciences [START_REF] Singly | L'enquête et ses méthodes -Le questionnaire[END_REF]: 38 questions are "closed" while 7 are "open" in order to allow comments and reactions to the closed questions. It has been decided with Electra to keep the questionnaire anonymous but to give to the suppliers the possibility to provide their name if they want a feedback from Electra.

The questionnaire was sent to the supplier user (mainly supply chain user or sales administration user) and addresses four main topics: general information on the supplier and his information system; use of the portal; integration of the information coming from the portal in the IS of the suppliers; evolution of the collaboration with Electra since the implementation of the portal. The questionnaire has been created using a dedicated web tool, the access link being sent to 450 suppliers by Electra. The questionnaire has been available during five weeks, a reminder being sent after three weeks. 138 questionnaires have been collected, 134 of them being considered as exploitable. This number so that the quality of the answers (nearly all the exploited questionnaires were complete, and included argued answers to open questions), allows to consider that the sample is good, which was confirmed by Electra. This is also shown by the distribution according to company size (25% having less than 50 employees, 27% between 50 and 200 employees, 26% between 200 and 1000 employees and 12% more than 1000 employees) so that by country (46% of French suppliers, 54% from Europe and Asia).

Results

Even if the answer could be anonymous, 59 suppliers have given their name. Most of the answering suppliers have stable relationships with Electra: 86% work for Electra for more than 5 years. In the supply domain managed by the portal, the relationships are frequent and intense: the supplier work in average for 5 sites of Electra. Two to three persons use regularly the portal at the supplier's, and log on 14,6 times per month in average. Usage is then regular, almost daily in 33% of the cases. A majority of the suppliers also use this type of tool with other customers (57% use other portals, and 32% EDIs).

Most users benefited from training on the portal. Nevertheless, 18.7% have not been trained, mostly because they got their position of web portal user after the training period. How new users should be trained is so an important question.

The opinions on the improvement of the relationship with Electra, so that on the quality of the exchanged information, are dispersed: 50% of the respondents see an improvement while 32% do not see any change, 18% estimating that the situation worsened. This may be set in relation with 69% of the users expecting an improvement of the portal, and 59% providing their expectations as free comments.

Globally, the users know and use the simple functions of the portal (delivery programs, confirmations, delivery notices), but much less the advanced functions. They criticize:

-the slowness of the system (26%) (due to poor web connections), -its poor user-friendliness (in 44% of the answers), which does not facilitate its appropriation (complex menus, many manipulations required, missing lines on the screen), -a vocabulary not always understandable or well translated (12%), -non optimized processes: redundancies, necessity of useless back and forth exchanges with Electra (21%), -the poor quality of the data, not always reliable (18%), -the difficulties for integrating the data in their own system (18%), -a deviation of the use of the portal by its users at Electra (11%), that can be indirectly related with the number of suppliers receiving additional Excel files from their contacts at Electra (45%), -marginally, an insufficient training (9%).

Almost all the respondents use production management tools (70%). These tools are mainly integrated tools (ERPs) but sometimes specialized ones. Nevertheless, Excel is still considered as a basic tool for forecasting. 72% of the suppliers integrate the forecasts provided by the web portal, and 64% the orders. Between a quarter and a third of the respondents do not integrate them at all. This integration, when done, is mainly performed manually (47%), often after correction of the data using Excel files (46%), which is accepted by the portal. This correction of the data can be explained by free comments underlining in 18% of the cases a poor reliability of the data (presence of null quantities for instance). The suppliers clearly suffer from difficulties for performing a more systematic and automated integration of the data provided by the portal. The EDI solution can provide an improvement, but only 13% of the present users ask for such a solution.

45% of the suppliers receive Excel files from their contacts at Electra, in parallel with the information provided by the portal. These files may concern forecasts, orders, and deliveries, all theoretically provided by the portal. The suppliers complain about redundant, and sometimes conflicting information, the data provided on Excel files being considered as more reliable than the ones provided by the portal. On the other hand, the reaction of the suppliers when facing a problem also shows unexpected behaviours: 40% call their usual contact at Electra and 15% call a colleague, instead of referring to the user manuals (18%) or contacting the key users (23%) as recommended. A familiar human relationship if therefore preferred through phone or email, even if it leads to neglect the nominal procedure.

A very positive point of the investigation is that the suppliers do not reject the portal but ask for improvements. 56% of the suppliers consider themselves as "members of the users' community". The analysis of the correlation between answers shows that this feeling is correlated to a more positive opinion on the portal, and to a better internal integration of the forecasts and orders.

Although the sizes of the companies and sectors of activity and country are diverse, no correlation can be established between the use of the portal, the satisfaction felt and the type of company (especially its size). This may set into question the usual consideration that small companies are more reluctant than large ones to use ICTs.

Technically, it is clear that the portal would need some improvements: better connexion to the portal from some places and better ergonomics (especially in the structuration of the screens) for instance.

In terms of training, tools or training modules should be systematically suggested to the new users, the supported process being relatively complex. An extensive training of all the users on all the functions of the portal could be of immediate benefit. A better communication with the suppliers on the resources and competences that could support them should be done. The animation of a real "users' community" using social tools (Facebook or dedicated tools [START_REF] Grabot | ERP 2.0, what for and how[END_REF]) could be considered.

The reasons for the use of Excel files should be investigated more thoroughly, and solutions inside the web portal should be found. The suppliers should be helped for finding practical solutions in order to integrate the data provided by the web portal in their information system.

Conclusion

Supplier web portals are becoming a key tool of the collaboration between customer and supplier. Through a questionnaire, we have investigated the first results of a recent experiment of such portal in a high tech industry, showing that in spite of high potential, many practical (and sometimes anecdotal) problems may set into question the efficiency of such tools. This investigation has intrinsic limits that could provide new perspectives for this topic of research. Firstly, the domain of activity of Electra concerns industrial products of high technology. The results can clearly not be generalized to other domains like the automotive sector or mass-market products. Moreover, the study addresses the appropriation of the portal by the users at Electra's suppliers, but does not analyze its appropriation by Electra's users. Nevertheless, our study shows that such investigation would deserve some interest. Finally, it would be interesting to deepen the conclusions by meeting some suppliers. In spite of these limitations, we hope to have shown the great potential of these tools for a better integration of the partner's processes and information systems in nowadays supply chains.

http://www.boostaerospace.com/


Sylvain Sorin
email: sylvain.sorin@imj-prg.fr

Guillaume Vigeral
email: vigeral@ceremade.dauphine.fr

LIMIT OPTIMAL TRAJECTORIES IN ZERO-SUM STOCHASTIC GAMES

We consider zero-sum stochastic games. For every discount factor λ, a time normalization allows to represent the discounted game as being played during the interval [0, 1]. We introduce the trajectories of cumulated expected payoff and of cumulated occupation measure on the state space up to time t ∈ [0, 1], under ε-optimal strategies. A limit optimal trajectory is defined as an accumulation point as (λ, ε) tend to 0. We study existence, uniqueness and characterization of these limit optimal trajectories for compact absorbing games.

Introduction

The analysis of two-person zero-sum repeated games in discrete time may be performed along two lines:

1) asymptotic approach: to each probability distribution θ on the set of stages (m = 1, 2, ...) one associates the game G θ where the evaluation of the stream of stage payoffs {g m } is +∞ m=1 θ m g m , and one denotes its value by v θ . Given a preordered family {Θ, } of probability distributions, one studies whether v θ converges as θ ∈ Θ "goes to ∞" according to . Typical exemples correspond to n-stage games (θ m = 1 n I m≤n , n → ∞), λ-discounted games (θ m = λ(1 -λ) m-1 , λ → 0), or more generally decreasing evaluations (θ m ≥ θ m+1 ) with θ 1 → 0. The game has an asymptotic value v * when these limits exist and coincide.

2) uniform approach: for each strategy of player 1, one evaluates the amount that can be obtained against any strategy of the opponent in every sufficiently long game for {Θ, }. This allows to define a min max and a max min and the game has a uniform value v ∞ when min max = max min.

The second approach is stronger than the first one ( existence of v ∞ implies existence of v * and their equality), but there are games with asymptotic value and no uniform value (repeated games with incomplete information on both sides, Aumann and Maschler [START_REF] Aumann | Repeated Games with Incomplete Information[END_REF], Mertens and Zamir [START_REF] Mertens | The value of two-person zero-sum repeated games with lack of information on both sides[END_REF]; stochastic games with signals on the moves, see e.g. [START_REF] Sorin | A First Course on Zero-Sum Repeated Games[END_REF]). The first approach deals only with families of values while the second explicitly consider strategies. The main difference is that in the first case ε-optimal strategies of the players may depend on the evaluation represented by θ.

We focus here on a class of games where this dependence has a smooth representation and allows for a more precise analysis. Basically in addition to the asymptotic properties of the value, one studies the asymptotic behavior along the play induced by ε-optimal strategies. Date: October 13, 2019. Some of the results of this paper were presented in "Atelier Franco-Chilien: Dynamiques, optimisation et apprentissage" Valparaiso, November 2010 and a preliminary version of this paper was given at the Game Theory Conference in Stony Brook, July 2012. This research was supported by grant PGMO 0294-01 (France).

In order to do this, we first normalize the duration of the game using the evaluation θ. We thus consider each game G θ as being played on [0, 1], stage n lasting from time t n-1 = m<n θ m to time t n = t n-1 + θ n (with t 0 = 0).

Note that here time t corresponds to the fraction t of the total duration of the game, as evaluated trough θ. In particular given ε-optimal strategies in G θ the stream of expected stage payoffs generates a bounded measurable trajectory on [0, 1] and one will consider its asymptotic behavior.

The next section introduces the basic definitions and concepts that allow to describe our results. The main proofs are in Section 3. Further examples are in Sections 4 and 5.

To end this quick overview let us recall that there are games without asymptotic value: stochastic games with compact action spaces, Vigeral [START_REF] Vigeral | A zero-sum stochastic game with compact action sets and no asymptotic value[END_REF]; finite stochastic games with signals on the state, Ziliotto [START_REF] Ziliotto | Zero-sum repeated games: counterexamples to the existence of the asymptotic value and the conjecture maxmin = lim vn[END_REF]; or in a more general framework, Sorin and Vigeral [START_REF] Sorin | Reversibility and oscillations in zero-sum discounted stochastic games[END_REF].

Limit optimal trajectories

Let Γ be a two-person zero-sum stochastic game with state space Ω, action spaces I and J, stage payoff g and transition ρ from Ω × I × J to IR (resp. ∆(Ω)). We assume that Ω is finite, I and J are compact metric sets, g and ρ continuous functions. We keep the same notations for the multilinear extensions to X = ∆(I) and Y = ∆(J), where as usual ∆(A) denotes the set of probabilities on A.

For any pair of stationary strategies (x, y) ∈ X Ω ×Y Ω = X×Y, any state ω ∈ Ω and any stage n, denote by c ω,x,y n the expected payoff at stage n under these stationary strategies, given the initial state ω, and by q ω,x,y n ∈ ∆(Ω) the corresponding distribution of the state at stage n. Hence c ω,x,y n = q ω,x,y n , ḡ(x, y) where ḡ(x, y) stands for the vector payoff with component in state ζ ∈ Ω given by g(ζ; x(ζ), y(ζ)).

Definition 1.

For any (x, y) ∈ X Ω × Y Ω , any discount factor λ ∈ (0, 1], and any starting state ω, define the function

l ω,x,y λ : [0, 1] → R by l ω,x,y λ (t n ) = λ n i=1 (1 -λ) i-1 c ω,x,y i for t n = λ n i=1 (1 -λ) i-1
and a linear interpolation between these dates {t n }.

Thus l ω,x,y λ (t n ) corresponds to the expectation of the accumulated payoff for the n first stages, or equivalently up to time t n , and l ω,x,y λ (t) to the same at the fraction t of the game, both under x and y in the λ-discounted game starting from ω.

Let M (Ω) denote the set of positive measures on Ω. We introduce similarly the expected accumulated occupation measure at time t under x and y in the λ-discounted game starting from ω as follows:

Definition 2.
For any (x, y) ∈ X Ω × Y Ω , any discount factor λ, and any starting state ω, define the function

Q ω,x,y λ : [0, 1] → M (Ω) by Q ω,x,y λ (t n ) = λ n i=1 (1 -λ) i-1 q ω,x,y i
and by a linear interpolation between these dates {t n }.

Note that for any

t ∈ [0, 1], Q ω,x,y λ (t) ∈ t ∆(Ω).
Denote by l x,y λ and Q x,y λ the Ω-vectors of functions l ω,x,y λ (•) and Q ω,x,y λ (•) respectively.

Limit trajectories for the payoff and occupation measures will be defined as accumulation points of l x λ ,y λ λ and Q x λ ,y λ λ under ε-optimal strategies (ε ≥ 0) x λ and y λ in the λ-discounted game as (ε, λ) tend to 0. More precisely, denote by X ε λ (resp. Y ε λ ) the set of ε-optimal stationary strategies in the λ-discounted game Γ λ (with value v λ ) for Player 1 (resp. for Player 2).

Then we introduce:

Definition 3. l = (l ω : [0, 1] → R)
ω∈Ω is a limit optimal trajectory for the expected accumulated payoff

(LOT P ) if : ∀ε > 0, ∃λ 0 > 0, ∀λ < λ 0 , ∃x λ ∈ X ε λ , ∃y λ ∈ Y ε λ , ∀ω ∈ Ω, ∀t ∈ [0, 1], |l ω (t) -l ω,x λ ,y λ λ (t)| ≤ ε. Q = (Q ω : [0, 1] → M (Ω)
) ω∈Ω is a limit optimal trajectory for the expected accumulated occupation measure (LOT M ) if :

∀ε > 0, ∃λ 0 > 0, ∀λ < λ 0 , ∃x λ ∈ X ε λ , ∃y λ ∈ Y ε λ , ∀ω ∈ Ω, ∀t ∈ [0, 1], Q ω (t) -Q ω,x λ ,y λ λ (t) ≤ ε.
Alternative weaker and stronger definitions are as follows:

Definition 4. l = (l ω : [0, 1] → R) ω∈Ω is a weak limit optimal trajectory for the expected accumulated payoff (W LOT P ) if :

∀ε > 0, ∃λ ≤ ε, ∃x λ ∈ X ε λ , ∃y λ ∈ Y ε λ , ∀ω ∈ Ω, ∀t ∈ [0, 1], |l ω (t) -l ω,x λ ,y λ λ (t)| ≤ ε. l = (l ω : [0, 1] → R)
ω∈Ω is a strong limit optimal trajectory for the expected accumulated payoff

(SLOT P ) if : ∀ε > 0, ∃λ 0 > 0, ∃ε 0 > 0, ∀λ < λ 0 , ∀x λ ∈ X ε0 λ , ∀y λ ∈ Y ε0 λ , ∀ω ∈ Ω, ∀t ∈ [0, 1], |l ω (t) -l ω,x λ ,y λ λ (t)| ≤ ε.
and similar definitions for weak (resp. strong) limit optimal trajectory for the expected accumulated occupation measure, (W LOT M ) (resp. (SLOT M )).

Remark 5. Thus LOTP (resp. LOTM) basically are limit points of the functions t → l ω λ (t) (resp.

t → Q ω λ (t))
as lambda goes to 0 and players play almost optimally in the λ-discounted game. The weak definitions are only accumulation points of the same trajectories, while the strong ones also ask for some robustness with respect to the almost optimal strategies. It is readily checked that -W LOT P (and W LOT M ) always exists by standard arguments of equicontinuity.

-if a LOT P l exists, v λ converges to l(1).

-if a SLOT P exists, it is unique -no SLOT M exists in general (just consider a game where payoff is always 0).

Remark 6.

If the game has a uniform value v and both players use ε-optimal strategies the average expected payoff is essentially constant along the play.

Hence (t) = tv is a LOT P .

A first approach to this topic concerns one player games (or games where one player controls the transitions), where there is no finiteness assumption on Ω. In such games, if v λ converges uniformly then there exists a SLOT P and it is linear w.r.t. t, which means that the expected payoff is constant along the trajectory (Sorin, Venel and Vigeral [START_REF] Sorin | Asymptotic properties of optimal trajectories in dynamic programming[END_REF]). The same article provides an example of a two player game with finite action and countable state spaces, where LOT P is not unique.

Let us also mention recent results of Oliu-Barton and Ziliotto [START_REF] Oliu-Barton | Constant payoff in zero-sum stochastic games[END_REF] establishing the existence of linear SLOT P for finite stochastic games and optimal strategies: the class of games is larger and they allow for behavioral 0-optimal strategies (hence not only stationary). Our results deal with compact action spaces and ε-optimal stationary strategies.

More precisely, the main contributions of the current paper are: a) For absorbing games, existence of a linear LOT P , and existence of a "geometric" algebraic LOT M . b) For finite absorbing games, existence of a SLOT P .

c) An example of a finite game where LOT M is not semialgebraic.

d) An example of compact absorbing game with non uniqueness of LOT P .

As a final comment, let us underline the fact that the previous concepts and definitions can be extended to any repeated game, for any evaluation and any type of strategies.

Absorbing games

An absorbing game Γ is defined by two sets of actions I and J, two stage payoff functions g, g * from (i) the payoff at stage t is g (i t , j t );

I × J to [-1,
(ii) with probability p * (i t , j t ) absorption is reached and the payoff in all future stages s > t is g * (i t , j t );

(iii) with probability p (i t , j t ) := 1 -p * (i t , j t ) the game is repeated at stage t + 1.

Recall that the asymptotic analysis for these games is due to Kohlberg [START_REF] Kohlberg | Repeated games with absorbing states[END_REF] in the case where I and J are finite and Rosenberg and Sorin [START_REF] Rosenberg | An operator approach to zero-sum repeated games[END_REF] in the current framework. In either case the value v λ of the discounted game Γ λ converges to some v as λ goes to 0. This does not require any assumption on the information of the players. In case of full observation of the actions -or even only of the stage payoff -, a uniform value exists, see Mertens and Neyman [START_REF] Mertens | Stochastic games[END_REF] in the finite case and Mertens, Neyman and Rosenberg [START_REF] Mertens | Absorbing games with compact action spaces[END_REF] for compact actions.

Recall that X = ∆(I) and Y = ∆(J) are the sets of probabilities on I and J. The functions g, p and

p * are bilinearly extended to X × Y . Let G * (x, y) := p * (x, y)g * (x, y) := I×J p * (i, j)g * (i, j)x(di)y(dj).
g * (x, y) is thus the expected absorbing payoff conditionally to absorption (and is thus only defined for p * (x, y) = 0).

An auxiliary game.

Consider the two-person zero-sum (one-shot) game A, with actions (x, x , a) ∈ S = X 2 ×R + and (y, y , b) ∈ T = Y 2 × R + , defined by the payoff function

A(x, x , a, y, y , b) = g(x, y) + a G * (x , y) + b G * (x, y ) 1 + a p * (x , y) + b p * (x, y ) . (1)
3.1.1. General properties.

The following proposition extends to the compact case results due to Laraki [START_REF] Laraki | Explicit formulas for repeated games with absorbing states[END_REF] (Theorem 3 and Corollary 4) in the finite case (which were later on simplified by Cardaliaguet, Laraki and Sorin [START_REF] Cardaliaguet | A continuous time approach for the asymptotic value in two-person zero-sum repeated games[END_REF]).

Proposition 7.

1)

The game A has a value, which is v = lim v λ .

More precisely

v = max x∈X sup (x ,a)∈X×R + inf (y,y ,b)∈T A(x, x , a, y, y , b) = min y∈Y inf (y ,b)∈Y ×R + sup (x,x ,a)∈S A(x, x , a, y, y , b).
2) Moreover, if (x, x , a) is ε-optimal in the game A then for any λ small enough the stationary strategy

xλ := x+λax 1+λa is 2ε-optimal in Γ λ .
Proof.

1) Consider an accumulation point w of the family {v λ } and let λ n → 0 such that v λn converges to w.

We will show that w ≤ sup

(x,x ,a)∈S inf (y,y ,b)∈T g(x, y) + a G * (x , y) + b G * (x, y ) 1 + a p * (x , y) + b p * (x, y ) (2)
A dual argument proves at the same time that the family {v λ } converges and that the auxiliary game A has a value.

Let r λ (x, y) be the payoff in the game Γ λ , induced by a pair of stationary strategies (x, y) ∈ X × Y . It satisfies

r λ (x, y) = λg(x, y) + (1 -λ)[(1 -p * (x, y))r λ (x, y) + G * (x, y)] (3)
hence

r λ (x, y) = λg(x, y) + (1 -λ)G * (x, y) λ + (1 -λ)p * (x, y) . (4)
In particular for any x λ ∈ X optimal for Player 1 in Γ λ , one obtains

v λ ≤ λg(x λ , y) + (1 -λ)G * (x λ , y) λ + (1 -λ)p * (x λ , y) , ∀y ∈ Y, (5)
that one can write

v λ ≤ g(x λ , y) + (1-λ) λ G * (x λ , y) 1 + (1-λ) λ p * (x λ , y) , ∀y ∈ Y. (6)
Let x ∈ X be an accumulation point of {x λn } and given ε > 0 let λ in the sequence {λ n } such that

|g(x, y) -g(x λ , y)| ≤ ε, ∀y ∈ Y
(we use the fact that g is uniformly continuous on X × Y ) and

|v λ -w| ≤ ε.
Then with a = (1-λ) λ and x = x λ , (6) implies

w -ε ≤ g(x, y) + aG * (x , y) 1 + ap * (x , y) + ε, ∀y ∈ Y. (7)
On the other hand, going to the limit in (5) leads to

w p * (x, y ) ≤ G * (x, y ), ∀y ∈ Y. (8)
We multiply [START_REF] Mertens | The value of two-person zero-sum repeated games with lack of information on both sides[END_REF] by the denominator 1 + ap * (x , y) and we add to (8) multiplied by b ∈ R + to obtain the property:

∀ε > 0, ∃ x, x ∈ X and a ∈ R + such that w ≤ g(x, y) + a G * (x , y) + b G * (x, y ) 1 + a p * (x , y) + b p * (x, y ) + 2ε, ∀y, y ∈ Y, b ∈ R + (9)
which implies [START_REF] Cardaliaguet | A continuous time approach for the asymptotic value in two-person zero-sum repeated games[END_REF]. Note moreover that ( * ) x is independent of ε, which allows to replace sup by max hence the result.

2) Let (x, x , a) be ε-optimal in the game A and xλ := x + λax 1 + λa . Using (4) one obtains

r λ (x λ , y) = λ[g(x, y) + λag(x , y)] + (1 -λ)[G * (x, y) + λaG * (x , y)] λ(1 + λa) + (1 -λ)(p * (x, y) + λap * (x , y)) .

Note that

A(x, x , a, y, y,

1 -λ λ ) = λg(x, y) + λa G * (x , y) + (1 -λ) G * (x, y) λ + λa p * (x , y) + (1 -λ) p * (x, y) .
Thus

|r λ (x λ , y) -A(x, x , a, y, y, 1 -λ λ )| ≤ 4Cλa
where C is a bound on the payoffs. Hence, for any

y ∈ Y v -r λ (x λ , y) ≤ ε + 4Cλa ≤ 2ε
for λ small enough.

A is an auxiliary limit game in the sense that:

i) There is a map φ 1 from S × (0, 1] to X (that associates to a strategy of player 1 in A and a discount factor a stationary strategy of player 1 in Γ).

ii) There is a map ψ 1 from Y × (0, 1] to T (that associates to a stationary strategy of player 2 in Γ and a discount factor a stationary strategy of player 2 in A).

iii)

r λ (φ 1 (λ, s), y) ≥ A(s, ψ 1 (λ, y)) -o(1)
, ∀s ∈ S, ∀y ∈ Y iv) A dual property holds with some φ 2 and ψ 2 .

These properties imply: lim v λ exists and equals v(A).

We then recover Corollary 3.2 in Sorin and Vigeral [START_REF] Sorin | Existence of the limit value of two person zero-sum discounted repeated games via comparison theorems[END_REF], with a new proof that will be useful in the sequel. {g * (x", y)} ; inf

y"|p * (x,y")>0 {g * (x, y")} ( 10
)
where med is the median of three numbers, and with the usual convention that sup x"∈∅ = -∞; inf y"∈∅ = +∞. Moreover if (x, x , a) (resp (y, y , ε)) is ε-optimal in A then x (resp. y) is ε-optimal in (10).

Proof.

For any ε > 0 fix a triplet (y, y ε , b ε ) ∈ T of the second player ε-optimal in A, where we can assume that y does not depend on ε by the previous proof ( * ). Then for any x" ∈ X such that p * (x", y) > 0, one has

v + ε ≥ lim a→+∞ A(x, x", a, y, y ε , b ε ) = g * (x", y) (11)
thus v + ε ≥ h + (y) := sup x"|p * (x",y)>0 {g * (x", y)}. Denote similarly h -(x) = inf y"|p * (x,y")>0 {g * (x, y")}.
On the other hand, for any

x v + ε ≥ A(x, x", 0, y, y ε , b ε ) = g(x, y) + b ε G * (x, y ε ) 1 + b ε p * (x, y ε ) Now if p * (x, y ε ) > 0, g(x, y) + b ε G * (x, y ε ) 1 + b ε p * (x, y ε ) ≥ min{g(x, y), g * (x, y ε )}, hence in all cases v + ε ≥ A(x, x", 0, y, y ε , b ε ) ≥ min{g(x, y), h -(x)}. Thus for any x ∈ X v + ε ≥ med g(x, y); h + (y); h -(x) (12)
Letting ε go to 0 and using the dual inequality establish the results.

Further properties of optimal strategies.

We establish here more precise results concerning the decomposition of the payoff induced by ε-optimal strategies in the game A.

Proposition 9. Let (x, x , a) and (y, y , b) be ε-optimal in the game A.

a) If p * (x, y) > 0 then |g * (x, y) -v| ≤ ε b) |g(x, y) -v| ≤ 2(1 + ap * (x , y) + bp * (x, y ))ε c) If ap * (x , y) + bp * (x, y ) > 0 then aG * (x , y) + bG * (x, y ) ap * (x , y) + bp * (x, y ) -v ≤ 3 1 + ap * (x , y) + bp * (x, y ) ap * (x , y) + bp * (x, y ) ε.
Proof. a) This is exactly equation [START_REF] Sorin | A First Course on Zero-Sum Repeated Games[END_REF] and its dual.

b) From v + ε ≥ A(x, x , 0, y, y , b) we get g(x, y) + bG * (x, y ) 1 + bp * (x, y ) ≤ v + ε.
On the other hand, v -ε ≤ lim b→∞ A(x, x , a, y, y , b) hence G * (x, y ) ≥ (v -ε)p * (x, y ). Combining both inequalities yields

g(x, y) ≤ (v + ε)(1 + bp * (x, y )) -b(v -ε)p * (x, y ) ≤ v + ε(1 + 2bp * (x, y )) ≤ v + 2ε(1 + ap * (x , y) + bp * (x, y )) (13)
and the dual inequality is similar.

c) Since A(x, x , a, y, y , b) ≥ v -ε, one has (v -ε)(1 + ap * (x , y) + bp * (x, y )) ≤ g(x, y) + aG * (x , y) + bG * (x, y ) ≤ v + 2ε(1 + ap * (x , y) + bp * (x, y )) + aG * (x , y) + bG * (x, y ) by (13) hence v(ap * (x , y) + bp * (x, y )) -aG * (x , y) -bG * (x, y ) ≤ 3ε(1 + ap * (x , y) + bp * (x, y ))
and the dual inequality is similar.

Asymptotics properties in

Γ λ .
Since the game is absorbing, we write simply Q x,y λ (t) for Q ω0,x,y λ (t)(ω 0 ), where ω 0 is the nonabsorbing state.

Lemma 10. Let x λ and y λ be two families of (non necessarily optimal) stationary strategies of Player 1

and Player 2 respectively. Assume that p * (x λ , y λ ) λ converges to some γ in [0, +∞] as λ goes to 0. Then

Q x λ ,y λ λ (t) converges, uniformly in t, to 1 -(1 -t) 1+γ
1 + γ , as λ goes to 0, with the natural convention that

1 -(1 -t) 1+γ 1 + γ = 0 for γ = +∞.
Proof. By definition 2, for any λ and

t n = λ n i=1 (1 -λ) i-1 = 1 -(1 -λ) n , Q x λ ,y λ λ (t n ) = λ n i=1 (1 -λ) i-1 (1 -p * (x λ , y λ )) i-1 = 1 -((1 -λ)(1 -p * (x λ , y λ ))) n 1 + p * (x λ ,y λ ) λ -p * (x λ , y λ )
with linear interpolation between these dates.

Remark first that this implies that

Q x λ ,y λ λ (t) ≤ [1 + p * (x λ ,y λ ) λ
] -1 for all t and λ, which gives at the limit the desired result if γ = +∞.

Assume now that γ ∈ [0, +∞[, and thus that p * (x λ , y λ ) tends to 0 as λ goes to 0. Fix t and λ, and let n be the integer part of ln(1-t) ln(1-λ) so that

t n ≤ t ≤ t n+1 . Since Q x λ ,y λ λ (t n ) is decreasing in n, Q x λ ,y λ λ (t) ≤ Q x λ ,y λ λ (t n ) = 1 -((1 -λ)(1 -p * (x λ , y λ ))) n 1 + p * (x λ ,y λ ) λ -p * (x λ , y λ ) ≤ 1 -((1 -λ)(1 -p * (x λ , y λ ))) ln(1-t) ln(1-λ) -1 1 + p * (x λ ,y λ ) λ -p * (x λ , y λ ) = 1 -(1 -t) 1+ ln(1-p * (x λ ,y λ )) ln(1-λ) -ln(1-λ) 1 + p * (x λ ,y λ ) λ -p * (x λ , y λ ) (14)
Similarly,

Q x λ ,y λ λ (t) ≥ Q x λ ,y λ λ (t n+1 ) = 1 -((1 -λ)(1 -p * (x λ , y λ ))) n+1 1 + p * (x λ ,y λ ) λ -p * (x λ , y λ ) ≥ 1 -((1 -λ)(1 -p * (x λ , y λ ))) ln(1-t) ln(1-λ) +1 1 + p * (x λ ,y λ ) λ -p * (x λ , y λ ) = 1 -(1 -t) 1+ ln(1-p * (x λ ,y λ )) ln(1-λ) +ln(1-λ) 1 + p * (x λ ,y λ ) λ -p * (x λ , y λ ) (15)
Letting λ go to 0 in ( 14) and ( 15) yields the result.

(t) = 1-(1-t) 1+γ 1+γ
for some γ ∈ [0, +∞], and a LOT P l(t) = tv.

Proof. For every n let (x, x n , a n ) and (y, y n , b n ) be 1 n -optimal strategies for each player in A (recall that x and y can be chosen independently of n). Up to extraction γ n := γ(x, x n , a n , y, y n , b n ) converges to some 1+λbn are ε-optimal in Γ λ for λ small enough. Corollary 11 and equation ( 16) imply that

γ in [0, +∞]. Fix ε > 0 and let n ≥ 2 ε such that 1 -(1 -t) 1+γn 1 + γ n - 1 -(1 -t) 1+γ 1 + γ ≤ ε 2 (16)
Q xn λ ,ŷ n λ λ (t) - 1 -(1 -t) 1+γ 1 + γ ≤ ε (17)
for all λ small enough and t ∈ [0, 1]. This answers the first part of the Proposition.

Clearly

l xn λ ,ŷ n λ λ (t) = Q xn λ ,ŷ n λ (t)g(x n λ , ŷn λ ) + t -Q xn λ ,ŷ n λ (t) g * (x n λ , ŷn λ ). ( 18
)
Recall that the payoff function is assumed bounded by 1. Then equations ( 17) and (18) imply that for λ small enough and every t,

l xn λ ,ŷ n λ λ (t) - 1 -(1 -t) 1+γ 1 + γ g(x n λ , ŷn λ ) -t - 1 -(1 -t) 1+γ 1 + γ g * (x n λ , ŷn λ ) ≤ 2ε
Since x n λ and y n λ converge to x and y, we then have for λ small enough

l xn λ ,ŷ n λ λ (t) - -(1 -t) 1+γ 1 + γ g(x, y) -t - 1 -(1 -t) 1+γ 1 + γ g * (x n λ , ŷn λ ) ≤ 3ε (19)
.

We (t) -tv| ≤ 9ε for all λ small enough, uniformly in t.

As claimed, in every case we see that l(t) = tv.

Remark 13. Recall that Q(•) and l(•) represent expected cumulated occupation measure and payoff. By deriving these quantities with respect to t we get that the asymptotic probability q(t) of still being in the non absorbing state at time t is (1 -t) γ , and that asymptotically the current payoff is v at any time.

Remark 14. Let us give a simple heuristic behind the form (1-t) γ for q(t). Assuming that this quantity is well defined and smooth, note that at time t the remaining game has a length 1-t and weight q(t) = 1-r(t)

hence by renormalization (see figure below). (where a * denotes an absorbing payoff ) with asymptotic value 1. Let x = 1/2T + 1/2B. Then (x, x, n) is 1/n-optimal in A for Player 1, while any (y, y , b) is optimal for Player 2. Since p * (x, y) > 0 for all y, case 1 (see proof of Proposition 12) occurs for any choice of (y, y , b), hence the corresponding γ is +∞ and Q(t) = 0 for all t.

r (t) 1 - (1 -t) = r (0) so that - r (t) 1 -r(t) = - r (0) 1 -t which leads, with r(0) = 0 to r(t) = 1 -(1 -t) γ for some γ.
Notice that in Γ λ the only optimal stationary strategy is (1/2, 1/2) for each player, leading to the same asymptotic trajectory Q(.) = 0. Notice that in Γ λ the only optimal stationary strategy is

x λ = y λ = ( √ λ 1+ √ λ , 1 1+ √ λ ) for each player. Since p * (x λ , y λ ) = λ (1+ √
λ) 2 ∼ λ, Lemma 10 implies that the asymptotic trajectory associated to optimal strategies is Q

(t) = t -t 2
2 . Moreover for any γ ≥ 0 the strategy of Player 2 with asymptotic value 1/2. Let y = 1/2L+1/2R. Then (B, T, 1) and (y, y, 0) are optimal in A, with γ = 1.

z λ = ( γ √ λ 1+ √ λ , 1 -γ √ λ 1+ √ λ ) is ε-optimal in Γ λ for λ small enough, and p * (x λ , z λ ) ∼ γλ hence any Q(t) of the form 1-(1-t) 1+γ

Hence case 3 holds, and the corresponding

Q(t) is t -t 2
2 . The optimal strategies in Γ λ are ( λ 1+λ , 1 1+λ ) and (1/2, 1/2) respectively, leading to the same Q(t) = t -t 2 2 .

Finite case.

We now prove that when the game is finite, the limit payoff trajectory is linear for every couple of nearly optimal stationary strategies, not only those given by Proposition 7. That is, l(t) = tv is a SLOT P .

Proposition 18. Let Γ be a finite absorbing game with asymptotic value v, x λ and y λ families of ε(λ)optimal stationary strategies in Γ λ , with ε(λ) going to 0 as λ goes to 0. Then for every t ∈ [0, 1], l x λ ,y λ λ (t) converges to tv as λ goes to 0.

We will use in the proof of this proposition the following elementary lemma given without proof. Proof of Proposition 18. The result is clear for t = 0 or 1, assume by contradiction that it is false for some t ∈]0, 1[. Hence there is a sequence λ n going to 0 and optimal strategies x λn and y λn such that l

x λn ,y λn λn (t) converges to tw with w = v. Up to extraction of subsequences, x λn and y λn converge to x and y respectively. Also up to extraction, all the following limits exist in [0, +∞]:

α(i) = lim n→∞ x λn (i) λn , β(j) = lim n→∞ y λn (j)
λn , and γ = lim n→∞ p * (x λn ,y λn ) λn

. If γ = 0 (and hence p * (x λn , y λn ) > 0 for n large enough), denote g * (x, y) := lim

G * (x λn ,y λn )
p * (x λn ,y λn ) , which also exists up to extraction. Recall formula (4):

r λn (x λn , y λn ) = λ n g(x λn , y λn ) + (1 -λ n )G * (x λn , y λn ) λ n + (1 -λ n )p * (x λn , y λn ) (20)
and since x λ and y λ are families of ε(λ)-optimal strategies in Γ λ , r λn (x λn , y λn ) converges to v as n tends to infinity.

Recall that by Lemma 10, at the limit

l(t) = 1 -(1 -t) 1+γ 1 + γ g(x, y) + t - 1 -(1 -t) 1+γ 1 + γ g * (x, y).
We first claim that γ ∈]0, +∞[. If γ = 0, l(t) = tg(x, y), and by near optimality of x λ and y λ v = l(1) = g(x, y), hence l(t) = tv a contradiction.

If γ = +∞, l(t) = tg * (x, y), and by near optimality of x λ and y λ v = l(1) = g * (x, y), hence l(t) = tv a contradiction.

Hence γ ∈]0, +∞[, and w is a nontrivial convex combination of g(x, y) and g * (x, y). Since v = l(1) is also a convex combination of g(x, y) and g * (x, y), the assumption that v = w implies g(x, y) = g * (x, y).

Assume without loss of generality g(x, y) < v < g * (x, y).

We classify the actions i of Player 1 in 4 categories I 1 to I 4 :

• i ∈ I 1 if x(i) > 0, • i ∈ I 2 if x(i) = 0 and α(i) = +∞, • i ∈ I 3 if x(i) = 0 and α(i) ∈]0, +∞[, • i ∈ I 4 if x(i) = 0 and α(i) = 0.
Hence actions of category 1 are of order 1, actions of category 3 are of order λ, actions of category 4 are of order o(λ), and actions of category 2 are played with probability going to 0 but large with respect to λ.

Define categories J 1 to J 4 of player 2 in a similar way. By definition,

p * (x λn , y λn ) λ n = I×J p * (i, j)x λn (i)y λn (j) λ n (21)
Recall that the left hand side converges to γ ∈]0, +∞[, hence up to extraction p * (i,j)x λn (i)y λn (j) λn converge in [0, +∞[ for any i and j, denote by δ ij the limit. If i and j are of category k and l with k + l > 4 then

δ ij = 0. If k + l < 4 then
x λn (i)y λn (j) λn diverges to +∞ which implies that p * (i, j) = 0 = δ ij .

Hence going to the limit in (21) we get that γ = γ 1,3 +γ 2,2 +γ 3,1 where where γ 1,3 := I1×J3 p * (i, j)x(i)β(j), γ 2,2 := I2×J2 δ ij , and γ 3,1 := I3×J1 p * (i, j)α(i)y(j). Recall that γ > 0 thus at least one of γ 1,3 , γ 2,2 or γ 3,1 is positive as well.

Similarly, passing to the limit in the definition of G * (x λn , y λn ) := I×J G * (i, j)x λn (i)y λn (j) yields µ := lim

G * kl (x λn ,y λn ) λn = µ 1,3 + µ 2,2 + µ 3,1 where µ 1,3 := I1×J3 x(i)β(j)G * (i, j), µ 2,2 := I2×J2 δ ij g * (i, j),
and µ 3,1 := I3×J1 α(i)y(j)G * (i, j). Note that if γ k,4-k = 0 for some k then µ k,4-k = 0 as well.

Finally, going to the limit in equation ( 20) yields

v = g(x, y) + µ 1,3 + µ 2,2 + µ 3,1 1 + γ 1,3 + γ 2,2 + γ 3,1
and similarly going to the limit in the definition of g * (x, y) := lim

G * (x λn ,y λn ) p * (x λn ,y λn ) yields g * (x, y) = µ 1,3 + µ 2,2 + µ 3,1 γ 1,3 + γ 2,2 + γ 3,1
Recall that we assumed g * (x, y) > v. By Lemma 19, there exists k such that γ k,4-k > 0 and

µ k,4-k γ k,4-k ≥ g * (x, y) > v.
Assume first that k = 1. Consider now the following strategy y λn : y λn (j) = 0 for j ∈ J 3 , and y λn (j) = y λn (j) for all other j except for an arbitrary j 0 ∈ J 1 for which y λn (j 0 ) = y λn (j 0 ) + j∈J3 y λn (j).

The only effect of this deviation is that now γ 1,3 = µ 1,3 = 0. Hence lim n→∞ r λn (x λn , y λn ) = g(x, y)

+ µ 2,2 + µ 3,1 1 + γ 2,2 + γ 3,1
which is strictly less than v by Lemma 19 since v < µ1,3 γ1,3 ; this contradicts the (λ n )-optimality of x λn . Assume next that k = 2. Consider now the following strategy y λn : y λn (j) = 0 for j ∈ J 2 , and y λn (j) = y λn (j) for all other j except for an arbitrary j 0 ∈ J 1 for which y λn (j 0 ) = y λn (j 0 ) + j∈J2 y λn (j).

The only effect of this deviation is that now γ 2,2 = µ 2,2 = 0. Hence lim n→∞ r λn (x λn , y λn ) = g(x, y)

+ µ 1,3 + µ 3,1 1 + γ 2,2 + γ 3,1
which is strictly less than v by Lemma 19 since v < µ2,2 γ2,2 ; this again contradicts the (λ n )-optimality of x λn .

Finally assume that k = 3. Consider now the following strategy x λn : x λn (i) = 2x λn (i) for i ∈ I 3 and x λn (i) = x λn (i) for all other i except for an arbitrary i 0 ∈ I 1 for which x λn (i 0 ) = x λn (i 0 ) -j∈J3 x λn (j) (which is nonnegative for n large enough). The only effect of this deviation is that now γ 3,1 = 2γ 3,1 and µ 3,1 = 2µ

L R U 1* -1* D -1 1
Since v λ (s 2 ) = 0 for all λ and since there is no return once the play has entered state s 2 , it implies that the optimal play in state s 1 is the same as in the Big Match, in which γ = 1. So in both states the optimal strategies in Γ λ are D + λU for Player 1 and 1/2L + 1/2R for Player 2. By a scaling of time, the preceding section tells us that at the limit game, the probability of being in state 2 at time t, given that there were transition from s 1 to s 2 at time z, is 1-t 1-z . Since (also from the preceding section) the time of transition from s 1 to another state (which is s 2 with probability 1/2) has a uniform law on [0, 1], the probability of being in s 2 at time t is

p(t) = 1 2 t 0 1 -t 1 -z dz = - (1 -t) ln(1 -t) 2 .
Notice that this not an algebraic function of t as it was always the case in the preceding section. Similarly the probability of absorption before time

t is 1 -(1 -t)(1 -ln(1-t) 2
) and is also non algebraic.

4.2.

No ε-optimal strategies of the form x + aλx .

In the following game with two non absorbing states the payoff is always 1 in state a and -1 in state b with It is easy to see that the asymptotic value is 0 and that optimal strategies in the λ-discounted game put a weight ∼ √ λ on D and R in both states, hence the absorbing probability is of the order of λ per stage in each state.

We show that strategies of the form (x a + C a λx a , x b + C b λx b ) cannot guarantee more than -1 to player 1, as λ goes to 0.

-If x a (D)x b (D) > 0, player 2 plays L in a and R in b inducing an absorbing payoff of -1.

From a we reach eventually b were -1 has a positive probability.

-If x a (D) = 0, x b (D) > 0, player plays R in both games inducing an absorbing payoff of -1.

The payoff will be absorbing with high probability in finite time and the relative probability of 1 * vanishes with λ.

-If x a (D) > 0, x b (D) = 0, player plays L in both games inducing a non absorbing payoff of -1.

For λ small, most of the time the state is b.

- Proposition 20. For any discount factor λ ∈]0, 1], 0 (resp. 1) is optimal for Player 1 (resp. Player 2)

in the λ-discounted game, and v λ = λ.

The corresponding payoff trajectory is: l(t) = 0 on [0, 1].

Proof. Action 0 of Player 1 ensures that there will never be absorption to state -1 * , and thus that the stage payoff from stage 2 on is nonnegative. Action 1 of Player 2 ensures that there will be absorption with probability 1 at the end of stage 1, and thus that the stage payoff from stage 2 on is nonpositive.

Since the payoff in stage 1 is 1 irrespective of player's actions, the proposition is established.

Notice that the play under this couple of optimal strategies is simple: there is immediate absorption to 0 * , and in particular the limit payoff trajectory is linear and equals 0 for every time t.

We now prove that there are other ε-optimal strategies, with a different limit payoff trajectory. Denote Proposition 21. For any discount factor λ ∈]0, 1], {λ} is λ-optimal for Player 1 and √ λ-optimal for Player 2 in the λ-discounted game.

The corresponding payoff trajectory is: l(t) = t -t 2 .

Proof. If both players play {λ}, the payoff in the λ-discounted game is, according to formula (4),

r λ ({λ}, {λ}) = λ -(1 -λ){λ} λ + (1 -λ){λ}
which is nonnegative since {λ} < λ 1-λ . On the other hand, since λ ≤ {λ}, one gets r λ ({λ}, {λ}) ≤ λ. If Player 1 plays {λ} while Player 2 plays y = {λ}, there is no absorption to -1 * hence r λ ({λ}, y) ≥ 0.

Thus {λ} is λ-optimal for Player 1.

If If Player 2 plays {λ} while Player 1 plays x = {λ}, then, according once again to formula (4),

r λ (x, {λ}) = λ λ + (1 -λ) {λ} ≤ λ λ + (1 -λ) √ λ ≤ √ λ.
Thus {λ} is √ λ-optimal for Player 2.

Notice that while the limit value is 0 and ({λ}, {λ}) is a couple of near optimal strategies, along the induced play the nonabsorbing payoff is 1 and the absorbing payoff is -1. One can compute that the associated γ is 1, hence under these strategies Q(t) = t -t 2 2 . So that the accumulated limit payoff up to time t is t -t 2 , which is non linear and positive for every t ∈]0, 1[. Basically the players use a jointly controlled procedure either to follow ({λ}, {λ}) or to get at most (resp. at least) 0.

Concluding comments

A first series of interesting open questions is directly related to the results presented here such as:

-extension of Proposition 18 to non stationary strategies, or more global analysis in the framework of -arbitrary (not discounted) evaluations where optimal strategies are not stationary, -general stochastic games.

It is also natural to consider other families of repeated games: a first class that is of interest is games with incomplete information. The natural equivalent of LOT M in this framework is the way information is transmitted during the game.

  x, y); sup x"|p * (x",y)>0 {g * (x", y)} ; inf y"|p * (x,y")>0 {g * (x, y")} = max x∈X min y∈Y med g(x, y); supx"|p * (x",y)>0

ForProposition 12 .

 12 any (x, x , a) ∈ X 2 × R + and (y, y , b) ∈ Y 2 × R + define γ(x, x , a, y, y , b) if p * (x, y) > 0 ap * (x , y) + bp * (x, y ) if p * (x, y) = 0 An immediate consequence of the previous lemma is Corollary 11. Let (x, x , a) ∈ X 2 ×R + and (y, y , b) ∈ Y 2 ×R + and denote xλ := x+λax 1+λa and ŷλ := y+λby 1+λb .Then Q xλ ,ŷ λ λ (t) converges uniformly in t to 1 -(1 -t) 1+γ(x,x ,a,y,y ,b) 1 + γ(x, x , a, y, y , b) as λ goes to 0. Any absorbing game has a LOT M Q

Example 15 .

 15 now the four cases in the preceding proof by giving examples. Consider the absorbing game L R T 1 * 0 B 0 1 *

Example 16 .T 1 * 0 B 0 1 with asymptotic value 1 .

 161011 Consider the absorbing game L R Then (B, T, n) is 1/n-optimal in A, while any (y, y , b) is optimal. The associated γ is ny(L), hence either y(L) = 0 and case 2 holds with γ = 0 and Q(t) = t, or y(L) > 0 and case 4 occurs with γ = +∞ and Q(t) = 0.

Lemma 19 .

 19 Let a, b, c, d be real numbers with c and d positive. Then min( a c , b d ) ≤ a+b c+d ≤ max( a c , b d ) with equality if and only if a c = b d

5 .y

 5 If x a (D) = 0, x b (D) = 0, player plays R in game a and L in game b. The event "absorbing payoff of 1" occurs at stage n if ω n = a and i n = D. Hence ω n-1 = b and i n-1 = D. Now this event "i n = D and i n-1 = D" has probability of order λ 2 . Then the absorbing component of the λ discounted payoffs converges to 0 with λ. Moreover the non absorbing payoff is mainly -1. An absorbing game with compact action sets and non linear LOT P We consider the following absorbing game with compact actions sets. There are three states, two absorbing 0 * and -1 * , and the non absorbing state ω, in which the payoff is 1 whatever the actions taken. The sets of action are X = Y = {0} ∪ {1/n, n ∈ N * } with the usual distance. The probabilities of absorption are given by : ρ(0 * |x, y) = if x = y 0 if x = y It is easily checked that both functions ρ(0 * | • •) and ρ(-1 * | • •) are (jointly) continuous.

{λ} := 1 [ 1

 11 /λ] where [•] is the integer part ; hence λ ≤ {λ} < λ 1-λ and 1/{λ} ∈ N * for all λ ∈]0, 1].

  Then Proposition 9 b) implies |g(x, y) -v| ≤ 2(2 + γ)ε. Moreover, p * (x, y) = 0 implies that g )ε for all λ small enough, uniformly in t.Case 4 : p * (x, y) = 0 and γ = +∞, hence γ n ≥ 1 (up to chosing a larger n). Then g * (x n λ , ŷn

	Hence equation (19) yields |l	xn λ ,ŷ n λ λ	(t) -tv| ≤ (7 + 2γ + 3 1+γ/2 γ/2 -v ≤ 6ε. Thus
	xn λ ,ŷ n λ λ equation (19) yields |l

now consider four separate cases. Basically either γ = 0 or +∞ and equation (19) implies that l(t) is linear, and hence equals tv since l(1) = v by near optimality of the strategies xn λ and ŷn λ in Γ λ ; or γ ∈]0, +∞[ and then both g(x, y) and g * (x n λ , ŷn λ ) are close to v by Proposition 9, which once again implies l(t) = tv. Case 1 : p * (x, y) > 0. Then g * (x n λ , ŷn λ ) converges to g * (x, y) as λ go to 0, and by Proposition 9 a) |g * (x, y) -v| ≤ ε. Since γ = +∞ in that case, equation (19) yields |l xn λ ,ŷ n λ λ (t) -tv| ≤ 5ε for all λ small enough, uniformly in t. Case 2 : p * (x, y) = 0 and γ = 0, hence γ n ≤ 1 (up to chosing a larger n). Then Proposition 9 b) implies |g(x, y) -v| ≤ 4ε, and equation (19) yields |l xn λ ,ŷ n λ λ (t) -tv| ≤ 7ε for all λ small enough, uniformly in t. Case 3 : p * (x, y) = 0 and γ ∈]0, +∞[, hence γ 2 ≤ γ n ≤ 1 + γ (up to chosing a larger n). * (x n λ , ŷn λ ) converges to aG * (x n , y) + bG * (x, y n ) ap * (x n , y) + bp * (x, y n ) as λ go to 0, and by Proposition 9 c) aG * (x n , y) + bG * (x, y n ) ap * (x n , y) + bp * (x, y n ) -v ≤ 3 1+γ/2 γ/2 ε. λ ) converges to aG * (x n , y) + bG * (x, y n ) ap * (x n , y) + bp * (x, y n ) as λ go to 0, and by Proposition 9 c) aG * (x n , y) + bG * (x, y n ) ap * (x n , y) + bp * (x, y n )

  Consider the following zero-sum stochastic game with two non absorbing states and two actions for each player. In the first state s 1 (which is the starting state) the payoff and transitions are as follows:

	3,1 . Hence
	lim n→∞	r λn (x λn , y λn ) =	g(x, y) + µ 1,3 + µ 2,2 + 2µ 3,1 1 + γ 1,3 + γ 2,2 + 2γ 3,1
	which is strictly more than v by Lemma 19 since v <	µ3,1 γ3,1 ; this contradicts the ε(λ n )-optimality of y λn .
		4. Finite stochastic games
	4.1. Non algebraic limit trajectories.
			L R
		U 1* 0+
		D 0	1

where * denotes absorption and + that there is a deterministic transition to state 2. Starting from the second state s 2 the game is a linear variation of the Big Match:


Rihab Khemiri

Bernard Grabot

Khaoula Elbedoui-Maktouf

Belhassen Zouari

Integrating fuzzy TOPSIS and goal programming for multiple objective integrated procurement-production planning

Keywords: Integrated Procurement-Production, Fuzzy TOPSIS, performance, risk, Possibilistic mixed integer linear programming, goal programming

In this paper, a four-phase approach for Integrated Procurement-Production (IPP) tactical planning in a multiechelon, multi-product and multi-period Supply Chain (SC) network is proposed. To account for ambiguity and vagueness in some real-world data and preferences, in the first phase of the approach, the Fuzzy Technique for Order Preference by Similarity to Ideal Solution (fuzzy TOPSIS) method is used to obtain the overall performance and risk ratings of the suppliers with regard to a set of qualitative and quantitative criteria. In the second phase, we introduce a novel multi-objective possibilistic mixed integer linear programming model (MOPMILP) for solving an IPP planning considering conflicting goals simultaneously: maximization of the overall performance and minimization of the overall risk. Then, after converting this MOPMILP model into an equivalent crisp multi-objective mixed integer linear programming (MOMILP) model, we use the Goal Programming (GP) approach to solve this MOMILP model in order to find an efficient compromise solution (i.e. an efficient procurement production plan) for the whole SC. The proposed approach and solution methodology are validated through a numerical example.

I.! INTRODUCTION

Supply Chain Planning (SCP) problem is one of the important issues that face Supply Chain (SC) managers. Traditionally, these problems involve three levels of decision: strategic, tactical and operational. The strategic decisions are related to the SC design, ranging from 5 to 10 years. The major task of the tactical decisions is to determine an optimal use of the various resources on a medium term horizon (from 1 to 2 years). Operational planning decisions search to address an exact scheduling on a short-term horizon (from 1 to 2 weeks). The tactical planning problem is the focus of this study.

Traditionally, production and procurement activities were conducted separately, that may lead to a very poor overall performance [START_REF] Goyal | Integrated procurement-production systems: a review[END_REF]. However, in the design of an Integrated Procurement-Production (IPP) system involving resources belonging to different actors within the SC, the decision maker (DM) will be confronted with three important characteristics: (i) Conflicting goals that must be considered simultaneously (e.g., increase performance and at the same time minimize risk). (ii) Presence of different sources of uncertainty such as market demands and available capacities. (iii) Some critical information and preferences are captured through human judgments, and can hardly be expressed using deterministic or probabilistic formulation.

This study suggests to design an IPP planning system for a multi-echelon SC consisting of multiple suppliers, multiple parallel manufacturing plants and multiple subcontractors. To do this, we propose a four-phase approach, which starts with calculating performance and risk ratings of the various actors based on a Multi-Criteria Decision Making (MCDM) approach under a fuzzy environment. In the second phase, the problem is formulated using a Multi-Objective Possibilistic Mixed Integer Linear Programming Model (MOPMILP). Then, in the third phase, an appropriate strategy is applied for converting the MOPMILP into an equivalent crisp Multi-Objective Mixed Integer Linear Programming (MOMILP). Finally, using goal programming (GP) approach, we solve the proposed MOMILP to obtain an efficient compromise solution (i.e. an efficient procurement production plan) for the whole SC.

The rest of the paper is organized as follows. In the next section, we review related works. The proposed four-phase approach is described in section 3. In Section 4, an illustrative example is presented to show the applicability of the proposed approach. Conclusions and further research directions are discussed in the last section.

II.! LITERATURE REVIEW

In the scientific literature, several models for tactical SC planning under uncertainty are proposed. Most of them are based on stochastic programming models [START_REF] Kazemi Zanjani | A multi-stage stochastic programming approach for production planning with uncertainty in the quality of raw materials and demand[END_REF], which are generally derived from statistical data. However, in many practical situations, such historical data are often not available and so some imprecise information can, more suitably, be obtained using human subjective judgments. In this case, the Fuzzy Set Theory [START_REF] Zadeh | Fuzzy sets[END_REF] and the Possibility Theory [START_REF] Zadeh | Fuzzy sets as a basis for a theory of possibility[END_REF][5] can be used as an efficient way to model uncertainties and subjectivity. Nevertheless, few studies address the IPP planning problem in a fuzzy environment. For example, the authors of [START_REF] Chen | A mathematical programming approach to supply chain models with fuzzy parameters[END_REF] propose a multi-echelon, multi-products and multi-periods SC model to deal with the fuzziness of the market demand, the unit costs of raw materials and the unit transportation costs when the objective is to minimize the total cost. A method is also proposed, based on the α-cut representation and Zadeh's extension principle [START_REF] Zadeh | Fuzzy sets as a basis for a theory of possibility[END_REF][7] [START_REF] Zimmermann | Fuzzy Set Theory and its Applications[END_REF], to transform the proposed fuzzy SC model into a family of crisp models. Torabi and Hassini [START_REF] Torabi | An interactive possibilistic programming approach for multiple objective supply chain master planning[END_REF] proposed a MOPMILP model for solving an integrated production, procurement and distribution planning problem considering the imprecise nature of some input data as well as two conflicting objective functions: the total value of purchasing and the total cost of logistics. Even if through the first objective, it is proposed to consider the impact of subjective factors in purchasing decisions (such as technical capacities of the suppliers, after sale services and business structure), these criteria are quantitatively evaluated. On the basis of Lai and Hwang's approach [START_REF] Lai | A new approach to some possibilistic linear programming problems[END_REF] [START_REF] Lai | Fuzzy Multiple Objective Decision Making, Methods and Applications[END_REF] and the weighted average method [START_REF] Lai | A new approach to some possibilistic linear programming problems[END_REF][12] [START_REF] Wang | Applying possibilistic linear programming to aggregate production planning[END_REF], the original fuzzy model is converted into an equivalent crisp MOMILP model. Finally, a new interactive fuzzy approach is introduced to find a compromise solution. This study was extended in [START_REF] Torabi | Multi-site production planning integrating procurement and distribution plans in multi-echelon supply chains: an interactive fuzzy goal programming approach[END_REF] by developing an interactive fuzzy GP approach which incorporates four objective functions: maximizing the total value of purchasing, minimizing the total cost of logistics, minimizing the defective items and minimizing the late deliveries. As in [START_REF] Torabi | An interactive possibilistic programming approach for multiple objective supply chain master planning[END_REF], the qualitative factors are quantitatively evaluated. In [START_REF] Peidro | A fuzzy linear programming based approach for tactical supply chain planning in an uncertainty environment[END_REF], the authors proposed a fuzzy linear programming based approach to deal with the integrated production, procurement and distribution planning problem. This approach considers the various sources of uncertainty in a SC (process, demand and supply uncertainties) in an integrated manner. The aim of the proposed approach is the best utilization of the available resources so that customer demands are satisfied at minimum cost. Then, the authors introduce a solution procedure for converting the fuzzy model into an equivalent crisp one. This procedure allows the participation of the decision makers, who can express their opinions in linguistic terms. Recently, [START_REF] Gholamian | Comprehensive fuzzy multi-objective multi-product multi-site aggregate production planning decisions in a supply chain under uncertainty[END_REF] presented a fuzzy multi-objective mixed-integer nonlinear programming model dealing with a multi-echelon SC. The main objectives of the proposed model are: minimizing the total cost, minimizing the rate of changes of the workforce, improving the customer satisfaction as well as maximizing the total value of purchasing. The fuzzy original model is transformed into an auxiliary MOMILP model through a threephase approach. The authors extended then their work in [START_REF] Gholamian | Multiobjective multi-product multi-site aggregate production planning in a supply chain under uncertainty: fuzzy multi-objective optimisation[END_REF] by developing a multi-objective mixed-integer non-linear programming model integrating tactical and operational planning, solved by fuzzy optimization.

We can see that in most of the aforementioned studies, the authors limit their objective to minimize total cost/maximize total profit while satisfying customer demand. In our opinion, these research works fail to address the factors related to risk, which become critical given the rapid market variations. To the best of our knowledge, the only work including a risk management process in an IPP planning model is [START_REF] Khemiri | A fuzzy multi-criteria decision making approach for managing performance and risk in integrated procurement-production planning[END_REF], but this latter does not take into account the lack of knowledge in some critical parameters such as customer demand and partners' capacities.

Moreover, the survey shows that the majority of existing approaches does not include subjective criteria, or transform them into quantitative ones, as in [START_REF] Torabi | An interactive possibilistic programming approach for multiple objective supply chain master planning[END_REF] or [START_REF] Torabi | Multi-site production planning integrating procurement and distribution plans in multi-echelon supply chains: an interactive fuzzy goal programming approach[END_REF].

To overcome these limitations, we introduce in this paper an integrated fuzzy!multiple criteria decision making (MCDM) technique and a multi-objective programming approach to deal with an IPP planning system. The MCDM technique is chosen because it is a powerful tool for simultaneously accommodate qualitative and quantitative criteria, whereas using an analytical model is the only way to take into account the various restrictive assumptions of the planning problem.

According to our knowledge, it is the first paper that integrates fuzzy TOPSIS, MOPMILP and GP to solve a multiechelon, multi-product, multi-period IPP problem including a risk management process. In the next section is summarized the proposed approach.

III.! PROPOSED APPROACH

In this section, we present our four-phase approach dealing with IPP planning, consisting of multiple manufacturing plants, multiple suppliers and multiple subcontractors, considering various sources of uncertainty. We assume that a set of predefined qualified suppliers is provided. The supplier selection problem is indeed not our objective in this work: we focus here on determining the orders to be allocated to each actor (supplier, manufacturing plan and subcontractor). Moreover, since the DM cannot provide exact data for some critical parameters in a mid-term horizon, fuzzy sets and possibility theory is used to handle this vagueness.

In the first phase, the various partners are evaluated using the fuzzy TOPSIS method based on two classes of criteria: performance-based and risk-based decision criteria. In the second phase, the overall performance is computed and the overall risk scores calculated in the first phase serve as input data in the MOPMILP model suggested to determine an optimal procurement-production plan. This possibilistic model is then converted into an equivalent crisp MOMILP using an appropriate strategy. Finally, a multi-echelon, multi-period and multi-product GP method is adopted to solve this MOMILP model and finding a preferred procurement-production plan.

The main steps of the proposed approach are summarized in Fig. 1; more details are given in the following sub-sections.

A.! Phase I: Evaluating partners 1)! Selecting decision criteria

Two classes of decision criteria are defined. For each class, we enumerate the most important criteria. The decision maker can select the most appropriate ones according to his strategy. a)! Class I: Performance-based decision criteria •! Cost: The unit cost related to each partner.

•! Capacity: Partners' production capacity.

•! Quality: The capacity of the partners to deliver the orders with the required quality of packaging.

•! Reliability: The ability of the partners to deliver the orders on time.

b)! Class II: Risk-based decision criteria

•! Flexibility: The ability of the partners to take into consideration the changes in product characteristics proposed by the customer while the order is in process.

•! Responsiveness: The ability of the partners to take into account the changes imposed by the customer in the due dates of orders. •! Robustness: The insensitivity of the partners to disturbances.

•! Resilience: The ability of the partners to return to a satisfactory state after disruption.

For modelling uncertainty , we assume that the cost and the capacity criteria are given by a triangular fuzzy number, whereas the other criteria are qualitatively assessed by the decision maker using linguistic labels.

2)! Evaluating the performance of each partner

The fuzzy TOPSIS method is used according to the selected Class I criteria for providing an overall rating of each partner. Note that this overall rating, also called the closeness coefficient, is computed according to the distance to both a "negative" (worst) and "positive" (best) ideal solution. Therefore, the ranking of the partners can be determined according to the descending order of the closeness coefficient.

The overall performance rating of a partner j is denoted PI j

3)! Evaluating the risk of each partner

The fuzzy TOPSIS method is also invoked in this step according to the Class II selected criteria. The closeness coefficient related to the partner j is note CCII j . The overall risk rating of the partner j, equal to 1-CCII j is denoted RI j . -! &'(). /$#$% : Maximum capacity of subcontractor b for product p during period t.

B.! Phase

-! &'()0" 1$2$% : Maximum procurement capacity of supplier s for raw material r during period t. 3 456 OV_PI t" " """"""""""" """"""""""""" "" (1)" Such that: OV_PI t =" "

•! Objective 2: Minimizing the overall risk measurement Min F R = " 3 456 OV_RI t" " """"""""""" """"""""""""" "" (3)" Such that: OV_RI t =" " 7 856 " 9":"; < " =":"> < ?":"@ A Constraint ( 5) ensures the satisfaction of the customers' demand at each period. The amount of raw material to be supplied in each period is determined using constraint [START_REF] Chen | A mathematical programming approach to supply chain models with fuzzy parameters[END_REF]. Constraints ( 7) and ( 8) are respectively the regular and overtime production capacity limitations. Constraints ( 9) and ( 10) indicate the limited capacities for each supplier and subcontractor, respectively. Finally, constraint ( 11) is a nonnegativity constraint of the various decision variables.

2)! Model the uncertain input data with triangular possibility distribution

In this study, we adopt the pattern of triangular possibility distribution to represent the uncertain/imprecise data. Fig. 2 shows the triangular possibility distribution of fuzzy number ' (a pp , a mm , a oo ) where a pp , a oo and a mm are the pessimistic, optimistic and most possible value of '.

The imprecise input data for the proposed model can thus be represented using triangular possibility distributions as follows:

E 8$4 = (E 8$
LMN)G H$=$4 T"CLMN)G H$=$4 88 ,LMN)G H$=$4 ?? , LMN)G H$=$4 SS ) """B"s,r,t (15)
C.! Phase III: Strategy for processing the fuzzy constraints Let us consider constraints ( 5) and ( 7)- [START_REF] Lai | A new approach to some possibilistic linear programming problems[END_REF] in which the fuzzy left-hand sides are compared to the crisp right-hand sides. A usual method for dealing with such situation is the defuzzification process, which consists in approximating the fuzzy parameters by crisp numbers.

In this work, we use the well-known weighted average method originally proposed in [START_REF] Lai | A new approach to some possibilistic linear programming problems[END_REF] and successfully applied to several problems [START_REF] Torabi | An interactive possibilistic programming approach for multiple objective supply chain master planning[END_REF], [START_REF] Liang | Distribution planning decisions using interactive fuzzy multi-objective linear programming[END_REF], [START_REF] Wang | Applying possibilistic linear programming to aggregate production planning[END_REF], [START_REF] Torabi | Multi-site production planning integrating procurement and distribution plans in multi-echelon supply chains: an interactive fuzzy goal programming approach[END_REF] to convert the fuzzy constraints ( 5) and ( 7)- [START_REF] Lai | A new approach to some possibilistic linear programming problems[END_REF].

The popularity of this method is due to its simplicity and its reliability of defuzzification. To do this, we first need to establish the minimum acceptable possibility level of occurrence for the corresponding fuzzy data,"β. Then the equivalent crisp constraints can be stated as follows:

C ?":"@ A x_r m,p,t + x_ov m,p,t ) + " 9":"; A x_b b,p,t = w 1 *"E 8$4$U 88 "V w 2 *"E 8$4$U ?? "V w 3 *"E 8$4$U SS """ " Bp, t (16)
Where, w 1 + w 2 + w 3 = 1 and w 1 , w 2 and w 3 represent respectively the weights of the most pessimistic, the weights of a pp a mm a oo the most possible and the weights of the most optimistic value of the fuzzy data. In practice, these weights as well as the minimum acceptable possibility level are determined subjectively based on the DM's experience.

In this study, we adopt the concept of the most likely values [START_REF] Lai | A new approach to some possibilistic linear programming problems[END_REF], assuming that w 1 = w 2 = 1/6, w 2 =4/6 and β = 0.5. The reason for considering the above weighted values is that the most possible value is frequently the most important one and is associated as a consequence to the higher weight [START_REF] Lai | A new approach to some possibilistic linear programming problems[END_REF].

D.! Phase IV: Goal programming-based solution approach

In previous sections, the problem was originally formulated as a MOPMILP model, then converted into an equivalent auxiliary crisp MOMILP model.

Goal Programming (GP) [START_REF] Charnes | Management models and industrial applications of linear programming[END_REF] is the most widely used approach to deal with such multi-criteria and multi-objective decision-making problems [START_REF] Chang | Binary fuzzy goal programming[END_REF]. GP allows the DM to specify an "aspiration level" for the various goals and to reduce the original problem into a single-objective formulation that seeks to minimize the deviations between the realized results and the aspiration goals. The main advantages of the GP approach are its robustness, its mathematical flexibility and its accuracy, i.e. the possibility to introduce several system constraints [START_REF] Dhahri | Nonlinear goal programming models quantifying the bullwhip effect in supply chain based on ARIMA parameters[END_REF].

We have used the Weighted Programming (WGP) model to convert the proposed MOMILP model into an equivalent ordinary linear programming model. Consequently, our problem can be reformulated as follow: -! F P * is the performance goal calculated using the mathematical model with objective function (1) subject to constraints (6), ( 16)- [START_REF] Chang | Binary fuzzy goal programming[END_REF].

Min F = w p * Y Z [ + w R * Y \ ] (21)
-! F R * is the risk goal calculated using the mathematical model with objective function (3) subject to constraints (6), ( 16)-( 20).

-! ^7 [ `G"abc"Negative deviation from the target value of performance goal F 1 * . -! ^> ] is the Positive deviation from the target value of Risk goal F 2 * .

-! d 7 "and"d > represent respectively the importance weights of the performance goal and the risk goal. These parameters are generally determined by the decision makers such that d 7 "+ d > = 1.

IV.! NUMERICAL EXAMPLE

To demonstrate the feasibility of the proposed approach, we consider a SC involving three manufacturing plants m1, m2 and m3 and three subcontractors b1, b2 and b3 who provide the finished good p using three common purchased items (one unit of r1 and two units of r2) which can be supplied from two suppliers (S1 and S2).

The planning horizon is one year decomposed into six monthly periods. These periods correspond to six different forecasted demands with triangular distributions, summarized in Table I. We consider that the DM selects the following criteria: -! Class I: cost (CI.1), quality (CI.2), reliability (CI.3) and capacity (CI.4). -! Class II: flexibility (CII.1), reliability (CII.2), resilience (CII.3) and robustness (CII.4).

The classical linguistic variables presented in Table II [22] are used to assess the importance weights and ratings of actors. The importance of each decision criteria, as well as the actor's ratings, are given in Table III -Table VII.

We assume that the DM does not have sufficient information to determine the weights"e Z and e \ of each goal in the objective function F GP . We thus offer the DM the possibility of exploring a wider range of potential solutions. To do this, we vary each value of the weights"e Z and e \ between 0 and 1 by increasing one and decreasing the other simultaneously, such that the sum is equal to 1. The GP model is solved using the LINGO optimization package. The order quantities assigned to each actor in each period are shown in Table VIII.

Two solutions representing two extreme situations can be identified. In the first one, only objective F P is considered (e Z =1; e \ =0) and in the second one, only objective F R is considered. By decreasing by one step the weight of the performance objective and increasing on the other hand the weight of the risk objective, we increase the load of the least risky and least performing actors (i.e. the subcontractors) and we reduce the load of the riskiest and the most performing actors (i.e. the suppliers and the manufacturing plants).

The deviations from the target values for goals F P * and F R * are summarized in Table IX. The optimum performance measure F P * and the worst risk measure F R* are obtained in the first extreme situation (e Z =1; e \ =0). In the same way, the second extreme situation (e Z =0; e \ =1) provides the optimum risk measure F R * and the worst performance measure F P* .

Moreover, by reducing the weight of the performance objective by two steps from its optimal value, we degrade very slightly the performance measure F P and we improve very slightly the risk measure F R .

Solution (e Z = 0.6; e \ = 0.4) marks a turning point in the evaluation of performance and risk measures. Indeed, the value of the first objective function F P passes abruptly from a deviation equal to 0.101% to a deviation equal to 84.460% whereas the risk measure is significantly improved (from a deviation equal to 99.648 % to a deviation equal to 13.894%).

For the weights (e Z = 0.5; e \ = 0.5), the performance measure is slightly degraded (from a deviation equal to 84.460 % to a deviation equal to 99.782 %) and the value of F R becomes very close to the optimum value F R* .

We also notice that we have obtained very different solutions by varying the weights (e Z ; e \ ). It is then possible to orientate the calculation on a particular solution expressing the compromise required by the DM. However, the compromises achieved are not balanced, given that the performance and risk objectives are not correlated.

V.! CONCLUSION Nowadays, global SC networks are affected by various sources of uncertainty and imprecision. Consequently, an effective risk management process becomes mandatory for the success of the organization. However, there is still a gap in the scientific literature in providing risk-based factors when designing a SC tactical plan. To deal with this drawback, we present in this paper a four-phase approach integrating Fuzzy MCDM, MOPMILP and GP to deal with a multi-echelon, multi-product and multi-period IPP planning problem under a fuzzy environment.

In the first phase, based on the DM's judgement and using the fuzzy TOPSIS method, we calculate the overall performance score and the overall risk score of the various partners. Next, the overall scores of the partners are incorporated into the MOPMILP model in which two important objectives are taken into account: maximizing performance and minimizing risk. In the third phase, the possibilistic programming model is transformed into an equivalent crisp MOMILP model by applying the Lai and Hwang's approach [START_REF] Lai | A new approach to some possibilistic linear programming problems[END_REF]. Then, a GP approach is applied to determine an efficient compromise solution.

A numerical example is given to demonstrate the importance of the proposed approach.

Future research may focus on the case where several distribution centers are taken into account in the considered SC network to deal with integrated procurement-productiondistribution systems.

Fig 1 .

 1 Fig 1. Framework of the proposed approach.

  II: Proposed Multi-objective possibilistic mixed integer linear programming model 1)! Formulation of the model a)! Notation •! Set of indices: -! T: Set of time periods (t = 1,…,T) -! S: Set of suppliers (s = 1,…, S ) -! M: Set of manufacturing plants (m = 1,…, M ) -! B: Set of subcontractors (b = 1,…, B) -! R: Set of raw materials (r = 1,…, R) -! P: Set of finished goods (p= 1,…, P) •! Certain parameters: -! S r : Set of qualified suppliers offering raw material r (i.e. S r ⊆ {1,…,S}). -! M p : Set of plants producing finished good p (i.e. M p ⊆ {1,…, M}). -! B p : Set of subcontractors producing finished good p (i.e. B p ⊆ {1,…, B}). -! α p,r : Quantity of item r to produce a unit of finished good p. -! PI j : Performance index for partner j (j∈ {S}U{M_r}U {M_ov} U {B}) computed using Fuzzy TOPSIS during the first stage, where {M_r} (respectively {M_ov}) represents the set of manufacturing plants producing in regular time (respectively in overtime). -! RI j : Risk index for the partner j (j∈ {S} U {M_r} U {B}) computed using Fuzzy TOPSIS during the first stage. •! Fuzzy parameters: -! !" #$% : Demand of final product p in period t. -! &'()* +$#$% : Production capacity of manufacturing plant m for product p during period t. -! &'(),-+$#$% : Overtime capacity of manufacturing plant m for product p during period t.

  x_r m,p,t ≤ w 1 *"LMN)O ?$8r,t ≤ w 1 *"LMN)G H$=$4$U 88 + w 2 *"LMN)G H$=$4$U ?? + w 3 *"LMN)G H$=$4$U SS Bs,r, t(19)x_b b,p,t ≤w 1 *"LMN)R 9$8$4$U 88 + w 2 *"LMN)R 9$8$4$U ?? + w 3 *"LMN)R 9$8$4$U SS Bb,p, t

Fig 2 .

 2 Fig 2. The triangular possibility distribution of"W X.

  Decision variables: -! x_r m,p,t : Production amount in regular time for product p at manufacturing plant m in period t. -! x_ov m,p,t : Production amount in overtime for product p at manufacturing plant m in period t. -! x_b b,p,t : Subcontracting amount at subcontractor b for product p in period t. -! x_s s,r,t : Purchase amount from supplier s for raw material r in period t. -! OV_PI t : Overall performance measurement in period t. -! OV_RI t : Overall risk measurement in period t.

	b)!Objective functions
	•! Objective 1: Maximizing the overall performance
	measurement
	Max F P =	"

•!

  ( x_s s,r,t * PI s ) + ( x_r m,p,t * PI m_r ) + ( x_ov m,p,t * PI m_ov ) + ( x_b b,p,t * PI b ) Bt

	7 856	?":"@ A	9":"; <	"	=":"> <	"

TABLE I .

 I FUZZY DEMAND QUANTITIES FOR EACH PERIOD

			Periods		Demand
			1			(110;120;130)
			2			(120;135;140)
			3			(125;135;145)
			4			(145;150;160)
			5			(150;160;175)
		TABLE II. LINGUISTIC VARIABLES AND FUZZY VALUES [22]
		Linguistic			Value
		Very Low (VL)			( 0 ; 0 ; 1 )
			Low (L)			( 0 ; 1 ; 3 )
		Medium Low (ML)		( 1 ; 3 ; 5 )
		Medium (M)			( 3 ; 5 ; 7 )
		Medium High (MH)		( 5 ; 7 ; 9 )
		High (H)			( 7 ; 9 ; 10 )
		Very High (VH)			( 9 ; 10 ; 10 )
		TABLE III. IMPORTANCE WEIGHTS OF CLASS I CRITERIA
		Cost	Quality		Reliability	Capacity
		VH		M		ML		MH
		TABLE IV. IMPORTANCE WEIGHTS OF CLASS II CRITERIA
	Flexibility	Reliability		Resilience	Robustness
		VH	MH		ML		VL
	TABLE V. RATINGS OF THE ACTORS USING SELECTED CRITERIA OF CLASS II
	Criteria	S1	S2	M1	Actors M2 M3	B1	B2	B3
	CII.1	VL	M	VL	MH	H	VH	VH	VH
	CII.2	VL	VL	L	M	M	MH	MH	VH
	CII.3	VL	MH	VL	M	MH	H	VH	VH
	CII.4	VL	MH	VL	VH	VH	MH	VH	VH

TABLE VI .

 VI RATINGS OF THE ACTORS USING SELECTED CRITERIA OF CLASS I

	Critera	S1	S2	M1	M2	Actors	M3	B1	B2	B3
	C.I1	(1;2;3)	(3;3;4)	R: (8;9;9) Ov:(25;26;26)	R: (10;12;14) Ov:(30;31;33)	R: (25;27;28) Ov:(35;36;36)	(29;30;31)	(68;69;70)	(72;73;75)
	CI.2	VH	MH	VH	MH		M	L	L	VL
	CI.3	VH	M	MH	M		M	L	VL	VL
	CI.4	(95;100;110)	(60;70;90)	R (61;65;80) Ov: (10;15;20)	R (8;15;20) Ov: (10;15;20)	R (8;10;15) Ov: (10;15;20)	(55;65;70)	(36;40;50)	(15;20;22)
									NB:R: Regular Time; Ov: Overtime

TABLE VII .

 VII PERFORMANCE AND RISK MEASURES OF ACTORS

	Global						Actors
	indicators	S1	S2	M1_r	M1_ov	M2_r	M2_ov	M3_r	M3_ov	B1	B2	B3
	PI	0.6172	0.4475	0.4090	0.2806	0.2459	0.2300	0.1927	0.1986	0.1992	0.1324	0.0551
	RI	0.9437	0.6639	0.9024	0.4901	0.4185	0.3275	0.3212	0.2652

TABLE VIII .

 VIII PERIODIC ORDER ALLOCATIONS OBTAINED WITH THE GP METHOD

TABLE IX .

 IX DEVIATIONS FROM THE TARGET VALUES.

	Poids	FP	FR	f g [ (%)	f h ] (%)
	(1,0) (0.9, 0.1)	1660.556 *	2721.815*	00.00	100
	(0.

8, 0.2) (0.7, 0.3)

		1659.267	2714.557	0.101	99.648
	(0.6, 0.4)	580.0491	945.9591	84.460	13.894
	(0.5, 0.5)	384.0375	659.6424	99.782	0.012
	(0.

4, 0.6) (0.3, 0.7) (0.2, 0.8) (0.1, 0.9)

		383.7282	659.4185	99.80652	0.0012
	(0 , 1)	381.2529*	659.3935*	100	00.00


State of Research

qEvaluation of some Vibration-based damage identification methods thanks to numerical model of a concrete beam and a 3-storey building. Damage is introduced as a simple local reduction of the Young's modulus. § Various methods to assess the damage based on vibration measures have been used and compared such as:

-Frequency changes -Modal assurance criterion (MAC) -Mode shape changes -Mode shape curvature (MSCM) -Curvature damage factor (CDF) -Flexibility change q Elaboration of an algorithm by combining some methods in order to detect, locate and quantify accurately the damage.

Next Steps

qNumerical evaluation of the detection algorithm on a numerical model of a damaged reinforced-concrete-building as a result of a seismic excitation. qExperimental evaluation with a dummy building model realized with 3D printing technology. qDefining signal processing techniques with real sensors. qExperimental evaluation of the detection algorithm. qExtension of the research to other structure types. a Laboratoire Génie de Production INP-ENIT, University of Toulouse, France, ffrigui@enit.fr, jean-pierre.faye@enit.fr, carmen.martin-loren@enit.fr, olivier.dalverny@enit.fr, francois.peres@enit.fr b STANEO SAS, 90 rue des 36 ponts, 31400 Toulouse, sebastien.judenherc@staneo.fr

Acknowledgments and

Research Objective j

Defining an appropriate methodology for detecting damages of a reinforced concrete structure.

It's a fact that France is a country with moderate seismic activity, but it's also well known that it went through some devastating earthquakes in the past. Therefore, damage structure detection is crucial to ensure citizen safety. Currently detection methods allowing preventive maintenance or enabling the definition of repair actions after the seismic event, do not exist. The aim of this work is to detect damages by analyzing the dynamic behaviour of the structure following three steps :

1. Sensor-based instrumentation building 2. Signal processing. 3. Developing an accurate methodology to detect damages.

  an accurate methodology of detection and location of structural damages. qStudying sensors capabilities and optimizing their numbers and locations. qOptimizing maintenance cost and reducing the risk of collapse. qDemocratization to a large potential users of the instrumentation means.

  P r o d u c t i o n n , 4 7 a v e n u e d ' A z e r e i x , 6 5 0 1 6 Ta r b e s C e d e x -F r a n c e L a b o r a t o i 9 0 r u e d e s t r e n t e -s i x p o n t s -3 1 4 0 0 To u l o u s e -F r a n c e !," , J-P.Faye ! , C.Martin ! , O.Dalverny ! , F.Peres ! , S.Judenherc "