Eric Danan
email: eric.danan@u-cergy.fr

Thibault Gajdos
email: thibault.gajdos@univ-amu.fr

Brian Hill
email: hill@hec.fr

Jean-Marc Tallon
email: jean-marc.tallon@univ-paris1.fr

Marc Fleurbaey

Tzachi Gilboa

Peter Klibanoff

Fabio Maccheroni

Philippe Mongin

Sujoy Mukerji

Klaus Nehring

Efe Ok

Marcus Pivato

Xiangyu Qu

David Schmeidler

Peter Wakker

Stéphane Zuber

Robust Social Decisions *

Keywords: Unambiguous preferences, Pareto dominance, Preference aggregation, Social choice, Uncertainty JEL Classification. D71, D81

We propose and operationalize normative principles to guide social decisions when individuals potentially have imprecise and heterogeneous beliefs, in addition to conflicting tastes or interests. To do so we adapt the standard Pareto principle to those preference comparisons that are robust to belief imprecision and characterize social preferences that respect this robust principle. We also characterize a suitable restriction of this principle. The former principle provides stronger guidance when it can be satisfied; when it cannot, the latter always provides minimal guidance.

A major issue is whether there will be significant global warming -for example, of 4 • C or more (relative to preindustrial levels) -which would have wide-ranging, and unevenly distributed, consequences on economic activity, human settlement and health around the world (IPCC, 2014). This depends on future concentrations of greenhouse gases (GHG), which themselves depend on climate policy. Both of these dependencies involve considerable uncertainty. On the one hand, climate sensitivity to GHG concentrations is imperfectly understood and cannot as yet be accurately described, even probabilistically, with full precision. Rather, a range of probabilistic models are considered plausible by climate scientists (IPCC, 2013, Section 10.8).

On the other hand, the effect of a given policy on GHG concentrations depends, among other factors, on technological evolutions that are highly unpredictable and for which any prediction is essentially subjective [START_REF] Stern | The Structure of Economic Modeling of the Potential Impacts of Climate Change: Grafting Gross Underestimation of Risk onto Already Narrow Science Models[END_REF][START_REF] Pindyck | Climate Change Policy: What Do the Models Tell Us?[END_REF]. So different actors evaluating a given policy -say the French and British governments evaluating a European climate policy -may rely on different predictions and, hence, end up using different plausible ranges for the probability of global warming reaching 4 • C under this policy -say 10% to 50% and 40% to 60%, respectively.

In such a situation, how should the policy be evaluated at the European level?

This paper aims at providing guidance for such policy decisions. Situations of this sort involve a "social" decision maker (the European Commission) who must choose a policy whose outcome is uncertain and affects several "individual" actors (the French and British governments).

Individuals may have different utility functions -or have heterogeneous "tastes" -and consider different probabilistic models to be plausible -or have heterogeneous "beliefs". Moreover, a given individual may also consider more than one model to be plausible -or have an imprecise belief.

For such an individual, which of two policies yields the highest expected utility may depend on the model considered. When a policy yields a higher expected utility than another one for all plausible models, we say that the individual unambiguously prefers the former policy to the latter. Unambiguous preferences are thus robust to belief imprecision. 2The Pareto principle is a natural guide for such decisions. We propose a robust version of this principle, requiring that if all individuals unambiguously prefer a policy to another one then so should the policy maker. We show that this unambiguous Pareto principle prescribes that the policy maker must only rely on probabilistic models that are considered plausible by all individuals. In the example above, this means that in order to guarantee that the implemented policy is unambiguously Pareto optimal, the European Commission must restrict attention to probabilities of global warming reaching 4 • C that belong to both the French and British ranges -between 40% and 50%.

As this example illustrates, the policy maker can respect unambiguous Pareto dominance even when individuals have heterogeneous beliefs, as long as these beliefs are compatible -at least one model is unanimously considered plausible. Heterogeneous yet compatible beliefs arise naturally in some contexts. 3 But they are ruled out by the standard assumption that all individuals have precise beliefs -each individual considers a single probabilistic model plausible. Under this particular assumption, we recover the well-known result that the standard Pareto principle can only be respected when all individuals have identical beliefs [START_REF] Hylland | The Impossibility of Bayesian Group Decision Making with Separate Aggregation of Beliefs and Values[END_REF][START_REF] Mongin | Consistent Bayesian Aggregation[END_REF]Mongin, , 1998)).

When individuals have incompatible beliefs -no probabilistic model is unanimously considered plausible -the unambiguous Pareto principle yields no prescription: whatever probabilistic models the policy maker takes as plausible, she may end up implementing an unambiguously Pareto dominated policy. We therefore propose restricting this principle to policies that only involve outcomes on which individual tastes are homogeneous. We show that this common-taste unambiguous Pareto principle prescribes that the policy maker must only rely on probabilistic models that are weighted averages of models considered plausible by at least some individuals. Thus this common-taste restriction provides weaker guidance than the unambiguous Pareto principle when individual beliefs are compatible -in the example above, it prescribes that the European Commission must rely on probabilities between 10% and 60%. On the other hand, it still provides guidance when beliefs are incompatible -it yields the same prescription if the French range were narrowed to between 10% and 30%.

Except in a few special cases, neither the unambiguous Pareto principle nor its common-taste restriction constrain the policy maker to rely on a single probabilistic model. She may do so if she wishes, but she could also rely on a range of models.4 A wider range of models results in a larger set of unambiguously optimal policies and, consequently, allows the policy maker more flexibility in selecting the policy to implement within this set. As we demonstrate, the set of unambiguously optimal policies, however large, can be computed very simply. Moreover, any policy selected within this set reflects a more or less cautious -or conservative -attitude.

Section 1 introduces the formal setup for our analysis. Section 2 contains the main results: characterizations of the unambiguous Pareto principle and its common-taste restriction. Section 3 presents additional results on computing the set of unambiguous optima and making a selection within this set. Section 4 discusses related literature. Proofs are gathered in the Appendix.

Setup

Social decisions

Consider a society made of a finite number n of individuals. Let S be a finite set of states of the world and X be a set of outcomes. Society (the social decision maker) has to choose an act (a policy) f whose outcome f (s) ∈ X depends on which state s ∈ S will occur. Let F denote the set of all acts, that is all functions f : S → X. We identify an outcome x ∈ X with the constant act yielding outcome x no matter which state occurs, thus viewing X as a subset of F .

An element of X specifies an outcome for all individuals in society. We assume that X is a convex subset of some Euclidean space. One particular case is the classical setting of [START_REF] Anscombe | A Definition of Subjective Probability[END_REF] where X is the set of lotteries over some finite set of prizes. Another is when X is a convex subset of the set R n of monetary allocations or, more generally, of the set R dn of allocations of a finite number d of commodities. Since X is convex, given any two acts f, g ∈ F and any coefficient λ ∈ [0, 1] there exists a "mixed" act λf + (1 -λ)g ∈ F yielding outcome λf (s) + (1 -λ)g(s) in each state s ∈ S.

Unambiguous preferences

Each individual i = 1, . . . , n has preferences over the acts in F , described by a binary relation i on F . That is to say, we write f i g when individual i weakly prefers act f to act g. As usual we use i to indicate strict preference and ∼ i for indifference. Society also has preferences described by a binary relation 0 on F . We assume that all these relations are unambiguous preference relations in the following sense (we use the generic notation when the subscript i can be omitted). where E p (u(f )) = s∈S p(s)u(f (s)) for all f ∈ F and p ∈ P .

P is interpreted as the set of all probability distributions (probabilistic models) the individual (or society) considers a plausible description of the uncertainty about the state of the world.

When P contains a single probability distribution, the agent has standard subjective expected utility (SEU) preferences and prefers the act yielding the highest expected utility under this probability distribution. When P contains multiple probability distributions, the agent only has an unambiguous preference between two acts when one act yields a higher expected utility than the other under every distribution. If the act yielding the highest expected utility depends on which distribution in P is used, then the individual has no unambiguous preference between the two acts.5

Unambiguous preferences were introduced by [START_REF] Bewley | Knightian decision theory. Part I[END_REF]. They satisfy all the properties characterizing SEU preferences, except the completeness property. The belief P is uniquely pinned down by the preference relation , whereas the utility function u is cardinally unique (i.e. unique up to a positive affine transformation).

Taste heterogeneity

We focus on situations where individuals' tastes or interests, as captured by their respective utility functions, are not perfectly aligned. More precisely, we shall assume that for each individual one can find two constant acts between which this individual is the only one to have a strict preference (all other individuals being indifferent). The profile ( i ) n i=1 of individuals' unambiguous preference relations is said to satisfy c-diversity if for all i = 1, . . . , n, there exist

x, y ∈ X such that x i y whereas x ∼ j y for all j = 1, . . . , n, j = i. 6 C-diversity is known to be equivalent to the individuals' utility functions being linearly independent (note that this is only possible if X is at least n-dimensional; [START_REF] Weymark | Harsanyi's social aggregation theorem and the weak Pareto principle[END_REF].

Thus individual tastes cannot be in full agreement, but neither can they be in full disagreement.

In fact c-diversity implies that the profile ( i ) n i=1 satisfies the following c-minimal agreement property: there exist two constant acts x, y ∈ X such that x i y for all i = 1, . . . , n.

Robust Pareto principles

This section contains the main results of the paper. We first state a robust version of the standard Pareto principle and characterize its implications for social preferences. We then consider a weakening of this robust principle to a particular subset of acts, yielding a more general characterization.

Unambiguous Pareto dominance

The following is the most straightforward application of the Pareto principle in our context. It simply states that if all individuals unambiguously prefer f to g, then so should society.

Definition 2. The social unambiguous preference relation 0 satisfies unambiguous Pareto dominance with respect to the profile ( i ) n i=1 of individual unambiguous preference relations if for all acts f, g ∈ F , f 0 g whenever f i g for all i = 1, . . . , n.

The following characterization result shows that the unambiguous Pareto principle provides guidance as to which beliefs society may adopt, provided individuals' beliefs are not too heterogeneous.

Theorem 1. Let i be an unambiguous preference relation on F with representation (u i , P i ) for all i = 0, . . . , n. Assume ( i ) n i=1 satisfies c-diversity. Then 0 satisfies unambiguous Pareto dominance with respect to ( i ) n i=1 if and only if there exist a vector of weights θ ∈ R n + , θ = 0, and a constant γ ∈ R such that

u 0 = n i=1 θ i u i + γ and P 0 ⊆ n i=1 θ i >0 P i .
Theorem This in particular implies that society has a more precise belief than all the individuals. The condition that the intersection of individuals' beliefs is non-empty is not new in the literature;

it appears for instance in [START_REF] Rigotti | Uncertainty and Risk in Financial Markets[END_REF], where it is needed to prove that, absent any aggregate risk, the set of Pareto optima coincides with the set of full insurance allocations.

If individual beliefs are incompatible -or have an empty intersection -then some individuals have to be "excluded" as it were, i.e. given zero weight in the social utility function. For instance, if all individuals have distinct precise beliefs, then the only way for society to satisfy unambiguous Pareto dominance is that its preferences coincide with those of a particular individual, who then acts as a dictator. More generally, SEU individuals are either given zero weight or are dictators:

any individual with SEU preferences and a non-zero weight forces society to have SEU preferences with her prior, in a way forcing her "certitude" on the society.7

Common-taste unambiguous Pareto dominance

When individuals have incompatible beliefs and society does not wish to exclude some of them, the unambiguous Pareto principle yields no prescription for society. To recover some guidance in these situations, we now restrict this principle to acts that are "consensual" in a particular sense.

Let us start with a situation where our notion of consensus takes a particularly simple form.

Consider two constant acts x, y ∈ X such that x i y for all i = 1, . . . , n (such acts exist by c-minimal agreement) and two acts f, g ∈ F that never yield an outcome different from x or y in any state. Such acts are consensual in the sense that all individuals agree state by state on the ranking of their respective outcomes: for all s ∈ S, f (s) is either unanimously "good" -if it is x -or unanimously "bad" -if it is y -and similarly for g. Now if f unambiguously Pareto dominates g then all individuals, notwithstanding their incompatible and potentially imprecise beliefs, further agree that f is more likely than g to yield the "good" outcome. Put differently, they would continue to unanimously prefer f to g if they agreed to "pool" their beliefs -each of them incorporating the others' beliefs into her own.

More generally, we say that two acts are "common-taste" acts if all individuals have the same cardinal preferences over their possible outcomes.8 Formally, given an act f , let f (S) = {f (s) : s ∈ S} denote the image of f , i.e. the set of all possible outcomes of f . Given a set Y of outcomes, let conv(Y ) denote the convex hull of Y , i.e. the set of all convex combinations (or weighted averages) of outcomes in Y . Two acts f and g are common-taste acts if x i y is equivalent to x j y for all x, y ∈ conv(f (S) ∪ g(S)) and i, j = 1, . . . , n. Equivalently, f and g are common-taste acts if all individual utility functions, once restricted to the set of all possible outcomes of these two acts, are identical up to positive affine transformations.

Definition 3. The social unambiguous preference relation 0 satisfies common-taste unambiguous Pareto dominance with respect to the profile ( i ) n i=1 of individual unambiguous preference relations if for all common-taste acts f, g ∈ F , f 0 g whenever f i g for all i = 1, . . . , n.

Theorem 2. Let i be an unambiguous preference relation on F with representation (u i , P i ) for all i = 0, . . . , n. Assume ( i ) n i=1 satisfies c-minimal agreement. Then 0 satisfies common-taste unambiguous Pareto dominance with respect to ( i ) n i=1 if and only if there exist a vector of weights θ ∈ R n + , θ = 0, and a constant γ ∈ R such that

u 0 = n i=1 θ i u i + γ

Social optima and social choice

This section turns to the problem of choosing a socially optimal act among a given set of feasible acts. We provide results helping society to compute the set of optimal acts and make a further selection among them.

A feasible act is optimal if no other feasible act is strictly preferred to it. When society has a precise belief p 0 , the socially optimal acts are thus simply those that maximize expected social utility under p 0 . When society has an imprecise belief P 0 , on the other hand, the set of socially optimal acts cannot be computed by maximizing a single function, reflecting the incompleteness of the social unambiguous preference relation. However, we show that maximizing expected social utility under each "interior" distribution in P 0 separately always yields a lower bound for -that is, a subset of -this set. Moreover, when the feasible set is convex, doing so under all "boundary" distributions in P 0 as well yields an upper bound for this set. Finally, when the feasible set is polyhedral -determined by a finite system of weak linear inequalities -the lower bound is actually an exact characterization of the set of social optima.9

Proposition 1. Let be an unambiguous preference relation with representation (u, P ) and G be a subset of F . Then any act maximizing E p (u(f )) in G for some relatively interior p ∈ P is optimal for in G. Conversely, if G is convex then any optimal act for in G maximizes E p (u(f )) in G for some p ∈ P , and if G is polyhedral then any optimal act for in G maximizes E p (u(f )) in G for some relatively interior p ∈ P . 10

Once the socially optimal acts are identified, society may wish to select among them by "completing" the social unambiguous ranking in a consistent way rather than picking an act arbitrarily. Formally, we say that a binary relation on F is a completion of an unambiguous preference relation on F if (i) is complete, (ii) f g implies f g for all f, g ∈ F , and (iii)

x y implies x y for all x, y ∈ X. As we next show, virtually any consistent completion can be interpreted as evaluating the different acts with varying degrees of "caution" in the following sense. such that for all acts f, g ∈ F ,

f g if and only if V (f ) ≥ V (g),
where

V (f ) = α(f ) min p∈P E p (u(f )) + (1 -α(f )) max p∈P E p (u(f )) for all f ∈ F .
The coefficient α(f ) is interpreted as the degree of caution with which act f is evaluated. It is unique whenever the minimal and maximal expected utilities of f do not coincide (otherwise it is irrelevant), and independent of (u, P ). The most cautious rule corresponds to α(f ) = 1 for all f ∈ F [START_REF] Gilboa | Objective and Subjective Rationality in a Multiple Prior Model[END_REF]. It is akin to the precautionary principle, each act being evaluated by its minimal expected utility. The least cautious rule corresponds to α(f ) = 0 for all f ∈ F . More generally, taking α constant corresponds to the Hurwicz (1951) "optimism-pessimism" criterion.

Letting α vary with the act allows for more general rules. For instance, choosing a distribution p ∈ P and taking α(f ) =

max p∈P Ep(u(f ))-E p (u(f ))
max p∈P Ep(u(f ))-min p∈P Ep(u(f )) corresponds to a SEU rule.

Proposition 2. If a binary relation on F is a transitive, c-Archimedean completion of an unambiguous preference relation on F then it is a variable caution choice rule for . 11

Transitivity requires the completion to rank acts in a consistent way. The c-Archimedean property, on the other hand, is a mild continuity requirement. When these two requirements are met, selecting among socially optimal acts thus amounts to adopting a more or less cautious attitude towards social belief imprecision. The degree of social caution may depend on the act under consideration.

Remark 3. The converse of Proposition 2 does not hold: some variable caution choice rulesor α functions -reverse some unambiguous rankings and hence are not completions of it. The converse holds, however, for all the particular cases discussed above. It holds, more generally, if 10 A distribution p ∈ P is relatively interior if for every distribution q ∈ P , there exist a distribution r ∈ P and a coefficient λ ∈ (0, 1) such that p = λq + (1 -λ)r.

11 A binary relation on F is c-Archimedean if for all f ∈ F and x, y ∈ X such that x f y, there exist λ, µ ∈ (0, 1) such that λx

+ (1 -λ)y f µx + (1 -µ)y.
the definition of a variable caution choice rule is strengthened to further require that V (f ) ≥ V (g) whenever E p (u(f )) ≥ E p (u(g)) for all p ∈ P .

Remark 4. Our definition of a variable caution choice rule is identical to Cerreia-Vioglio et al.

(2011)'s definition of a "generalized Hurwicz representation", except that they require to be derived from in a specific way whereas we more generally allow to be any unambiguous preference relation admitting as a completion. [START_REF] Cerreia-Vioglio | Rational preferences under ambiguity[END_REF] show that any "monotonic Bernoullian Archimedean" (MBA) preference relation admits such a representation. 12 Any MBA preference relation is a transitive, c-Archimedean completion of some unambiguous preference relation, but the converse is not true as MBA preferences satisfy a stronger Archimedean property.

4 Discussion and related literature

In this section, we further discuss the relationship between our main results and the existing literature on social decisions. Whereas most of the literature assumes SEU preferences, a recent strand of papers considers ambiguity-sensitive-or non-expected utility -preferences.

Social decisions with SEU preferences

When individuals and society have SEU preferences and individual tastes are heterogeneous, respecting Pareto dominance is impossible unless all individuals (with non-zero weight) have identical beliefs [START_REF] Hylland | The Impossibility of Bayesian Group Decision Making with Separate Aggregation of Beliefs and Values[END_REF][START_REF] Mongin | Consistent Bayesian Aggregation[END_REF]Mongin, , 1998Mongin, , 2014)). Theorem 1 generalizes this result to unambiguous preferences. This generalization is partly a possibility result:

the unambiguous Pareto principle can accommodate simultaneous heterogeneity in tastes and beliefs, as long as beliefs are compatible. In the particular case where all individuals have SEU preferences, it also yields the following corollary, showing that the assumption that society has a precise belief is not necessary for the impossibility result.

Corollary 1. Let 0 be an unambiguous preference relation on F and i be a SEU preference relation on F for all i = 1, . . . , n. Assume ( i ) n i=1 satisfies c-diversity. If 0 satisfies unambiguous Pareto dominance with respect to ( i ) n i=1 then 0 is a SEU preference relation. [START_REF] Gilboa | Utilitarian Aggregation of Beliefs and Tastes[END_REF] restrict the Pareto principle to "common-belief" acts, i.e. acts whose outcome only depends on events to which all individuals assign the same probability. In the setting of [START_REF] Savage | The Foundations of Statistics[END_REF], they show that this restriction allows aggregation of SEU preferences with heterogeneous tastes and beliefs, and requires the social belief to be a weighted average of the individuals'. In an Anscombe-Aumann setting, [START_REF] Qu | Separate Aggregation of Beliefs and Values under Ambiguity[END_REF] obtains the same characterization by restricting the Pareto principle to common-taste acts. These two restrictions have the same flavor of allowing society to ignore "spurious" unanimities (that is, cases where individuals agree for opposite reasons), which are the source of the impossibility. The Savage setting features a rich set of states, making the common-belief restriction stronger, whereas the Anscombe-Aumann 12 The MBA class includes virtually all popular ambiguity models, such as maxmin expected utility (MEU; [START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF], Choquet expected utility (CEU; [START_REF] Schmeidler | Subjective Probability and Expected Utility without Additivity[END_REF], smooth ambiguity [START_REF] Klibanoff | A Smooth Model of Decision Making under Ambiguity[END_REF], variational [START_REF] Maccheroni | Ambiguity Aversion, Robustness, and the Variational Representation of Preferences[END_REF], and multiplier [START_REF] Hansen | Robust Control and Model Uncertainty[END_REF] preferences.

setting features a rich set of outcomes, making the common-taste restriction stronger. Theorem 2 generalizes Qu (2015)'s result to unambiguous preferences. 13 Gilboa et al. (2014) say that an act f no-betting Pareto dominates an act g if f Pareto dominates g and there exists a probability distribution p on S such that E p (u i (f )) ≥ E p (u i (g))

for every involved individual i (their definition requires strict inequality, but this weak version is more directly comparable to ours). Their definition can be generalized to unambiguous preferences by requiring unambiguous Pareto dominance instead of Pareto dominance. [START_REF] Gayer | Pareto Efficiency with Different Beliefs[END_REF] say that an act f unanimity Pareto dominates an act g if E p j (u i (f )) ≥ E p j (u i (g)) for all involved individuals i, j. Their definition (which again requires strict inequality) can also be generalized to unambiguous preferences by requiring that E p j (u i (f )) ≥ E p j (u i (g)) for all individuals i, j and all probability distributions p j ∈ P j . Common-taste unambiguous Pareto dominance then implies unanimity unambiguous Pareto dominance, which itself implies no-betting unambiguous Pareto dominance and in turn unambiguous Pareto dominance. Moreover, the last two are equivalent when individual beliefs are compatible. Finally, Theorem 2 implies that it is equivalent for a social unambiguous preference relation to respect either one of the first two when c-minimal agreement holds. [START_REF] Brunnermeier | A Welfare Criterion for Models with Distorted Beliefs[END_REF] propose a belief neutral social welfare criterion that essentially consists in a social unambiguous ranking whose belief is the convex hull of the individuals'. This corresponds to the particular case of Theorem 2 where individuals have SEU preferences and society has the least complete unambiguous preferences satisfying common-taste unambiguous Pareto dominance. The common-taste unambiguous Pareto principle thus provides foundations for a generalization of their criterion allowing, on the one hand, for more precise social beliefsor more complete social preferences -and, on the other hand, for imprecise individual beliefs.

Social decisions with ambiguity-sensitive preferences

When individuals and society have ambiguity-sensitive preferences and individual tastes are heterogeneous, respecting Pareto dominance becomes impossible even when all individuals have identical beliefs. This has been shown in various settings covering in particular the class of MBA preferences [START_REF] Gajdos | Representation and aggregation of preferences under uncertainty[END_REF][START_REF] Herzberg | Aggregation of Monotonic Bernoullian Archimedean preferences: Arrovian impossibility results[END_REF][START_REF] Chambers | Preference aggregation with incomplete information[END_REF][START_REF] Mongin | Ranking Multidimensional Alternatives and Uncertain Prospects[END_REF][START_REF] Zuber | Harsanyi's theorem without the sure-thing principle: On the consistent aggregation of Monotonic Bernoullian and Archimedean preferences[END_REF]. In contrast, Theorems 1 and 2 show that unambiguous preferences allow the aggregation of imprecise beliefs. Moreover, our results can be used to obtain positive aggregation results for ambiguitysensitive preferences as well. Indeed, an ambiguity-sensitive preference relation naturally induces a "revealed unambiguous preference" relation, capturing the part of the preference ranking that is not affected by the ambiguity the individual perceives [START_REF] Ghirardato | Differentiating ambiguity and ambiguity attitude[END_REF][START_REF] Nehring | Bernoulli Without Bayes: A Theory of Utility-Sophisticated Preferences under Ambiguity[END_REF][START_REF] Klibanoff | Perceived Ambiguity and Relevant Measures[END_REF]. For a MBA preference relation , this revealed unambiguous preference relation * is an unambiguous preference relation in the sense of Definition 1, and is a variable caution choice rule for * . can therefore be represented by a triple (u, P, α) where (u, P ) is as in Definition 1 and α is as in Definition 4. 14 The function α is then interpreted as reflecting the individual's attitude towards the ambiguity she perceives.

We may therefore restrict the Pareto principle as follows: say that the social preference relation 0 satisfies revealed unambiguous Pareto dominance with respect to the profile ( i

) n i=1 of individual preference relations if the social revealed unambiguous preference relation * 0 satisfies unambiguous Pareto dominance with respect to the profile ( * i ) n i=1 of individual revealed unambiguous preference relations. This principle, and its restriction to common-taste acts, are then characterized as in Theorems 1 and 2, respectively. Note that these characterizations do not involve the functions α i and thus relate the individuals' and society's beliefs independently of their ambiguity attitudes. We explicitly state the latter result.

Corollary 2. Let i be a MBA preference relation on F with representation (u i , P i , α i ) for all i = 0, . . . , n. Assume ( i ) n i=1 satisfies c-minimal agreement. Then 0 satisfies common-taste revealed unambiguous Pareto dominance with respect to ( i ) n i=1 if and only if there exist a vector of weights θ ∈ R n + , θ = 0, and a constant γ ∈ R such that

u 0 = n i=1 θ i u i + γ and P 0 ⊆ conv n i=1 P i .
Several particular specifications of this general characterization have been studied within various subclasses of MBA preferences. [START_REF] Crès | Aggregation of multiple prior opinions[END_REF][START_REF] Nascimento | The ex ante aggregation of opinions under uncertainty[END_REF], [START_REF] Hill | Unanimity and the aggregation of multiple prior opinions[END_REF], to show, as in the proof of Theorem 1, that for all p 0 ∈ P 0 , there exist

(p i ) n i=1 ∈ n i=1 P i , θ ∈ R n + , and c ∈ C such that p 0 (s)u 0 (z) = n i=1 θ i p i (s)u i (z) + c (s) (6)
for all s ∈ S and z ∈ conv({x, y}). Summing over S and using (5) yields

u 0 (z) = n i=1 θ i u i (z) + s∈S c (s) = n i=1 θ i a i u 0 (z) + n i=1 θ i b i + s∈S c (s)
for all z ∈ conv({x, y}), so that n i=1 θ i a i = 1 and n i=1 θ i b i = -s∈S c (s) since u 0 is nonconstant on conv({x, y}). Hence (6) implies that

p 0 (s)(u 0 (x) -u 0 (y)) = n i=1 θ i p i (s)(u i (x) -u i (y)) = n i=1 θ i p i (s)a i (u 0 (x) -u 0 (y))
and, hence, that

p 0 (s) = n i=1 θ i a i p i (s) for all s ∈ S, so that p 0 = n i=1 θ i a i p i . Let λ = (θ i a i ) n i=1 ∈ R n . Since θ i ≥ 0 and a i > 0 for all i = 1, . . . , n and n i=1 θ i a i = 1, we have λ ∈ ∆ n .
Proof of Proposition 1. First, let G be any subset of X and let g ∈ arg max f ∈G E p (u(f ))

for some relatively interior p ∈ P . We show that g is optimal for in G. Suppose not, i.e.

there exists f ∈ G such that f g. It must then be that E p (u(f )) = E p (u(g)) whereas E q (u(f )) > E q (u(g)) for some q ∈ P . Moreover, since p is relatively interior in P , there exist r ∈ P and λ ∈ (0, 1) such that p

= λq + (1 -λ)r, i.e. r = 1 1-λ p -λ 1-λ q. It follows that E r (u(f )) = 1 1 -λ E p (u(f )) - λ 1 -λ E q (u(f )) < 1 1 -λ E p (u(g)) - λ 1 -λ E q (u(g)) = E r (u(g)), contradicting f g.
Next, assume G is convex and g ∈ G is optimal for in G, i.e. there exists no f ∈ G such that f g. We show that g ∈ arg max f ∈G E p (u(f )) for some p ∈ P . Let

A = v ∈ R S : there exists f ∈ G such that v(s) = u(f (s)) -u(g(s)) for all s ∈ S , B = v ∈ R S : E q (v) > 0 for all q ∈ P .
Then A is convex since G is convex and u is affine, and B is a convex cone whose dual cone is cone(P ). Moreover, since g is optimal for in G, A and B must be disjoint by Definition 1.

Hence by a separation argument (Rockafellar, 1970, Theorem 11.3), there exists p ∈ R S , p = 0, for all a ∈ A and b ∈ B. The former inequality implies that p ∈ cone(P ), so we can assume without loss of generality that p ∈ P . The latter inequality then implies that E p (u(g)) ≥

E p (u(f )) for all f ∈ G.

Finally, assume G is polyhedral and g ∈ G is optimal for in G, i.e. there exists no f ∈ G such that f g. We show that g ∈ arg max f ∈G E p (u(f )) for some relatively interior p ∈ P .

Define A as above and let

B = v ∈ R S : E q (v) ≥ 0 for all q ∈ P .
Then A is polyhedral since G is polyhedral and u is affine, and B is a closed, convex cone whose dual cone is cone(P ). Since A is polyhedral and 0 ∈ A, cone(A) is a closed, convex cone (Rockafellar, 1970, Corollary 19.7.1). We also have B = B 1 + B 2 where B 1 is the lineality space of B and B 2 is a pointed, closed, convex cone orthogonal to B 1 . Since B 2 is pointed, there exists a compact, convex set D ⊂ B 2 , 0 / ∈ D, such that cone(D) = B 2 and, hence, cone(B 1 + D) = B .

Moreover, since g is optimal for in G, we have A ∩ B ⊆ B 1 by Definition 1 and, hence, A and B 1 + D must be disjoint since 0 / ∈ D. Hence by a separation argument (Rockafellar, 1970, Corollary 20.3.1), there exists q ∈ R S , q = 0, and ε ∈ R such that for all a ∈ A and b ∈ B . The former inequality implies that Q ⊂ cone(P ), so we can assume without loss of generality that q ∈ P . By definition, Q must then contain a relatively interior p ∈ P . The latter inequality then implies that E p (u(g)) ≥ E p (u(f )) for all f ∈ G.

Proof of Proposition 2. Assume is a transitive, c-Archimedean completion of an unambiguous preference relation on F with representation (u, P ). First note that since is a completion of and by Definition 1, we have

x y if and only if x y if and only if u(x) ≥ u(y)

for all x, y ∈ X. For all f ∈ F , let x f ∈ arg max s∈S u(f (s)) and y f ∈ arg min s∈S u(f (s)). This is well-defined since S is finite. We then have

u(x f ) ≥ max p∈P E p (u(f )) ≥ min p∈P E p (u(f )) ≥ u(y f ).
Since X is convex and u is affine, there then exist x f , y f ∈ conv({x f , y f }) such that u(x f ) = max p∈P E p (u(f )) and u(y f ) = min p∈P E p (u(f )). It follows that x f f y f by Definition 1 and, hence, x f f y f since is a completion of .

We now show that there exist α(f ) ∈ [0, 1] such that f ∼ α(f )y f + (1 -α(f ))x f .

If f ∼ x f or f ∼ y f then we are done, so assume x f f y f . By (7), we then have u(x f ) > u(y f ). Let L = {λ ∈ [0, 1] :

λy f + (1 -λ)x f f }, M = {µ ∈ [0, 1] : f µy f + (1 -µ)x f }.
We then have L ∪ M = [0, 1] since is complete. Moreover, for all λ ∈ L and µ ∈ M , we have λy f + (1 -λ)x f µy f + (1 -µ)x f since is transitive and, hence, u(λy f + (1 -λ)x f ) ≥ u(µy f + (1 -µ)x f ) by ( 7). Since u is affine and u(x f ) > u(y f ), this is only possible if λ ≤ µ.

z f = α(f )y f + (1 -α(f ))x f , we have f ∼ z f .
Finally, let V (f ) = u(z f ). Since u is affine, we then have

V (f ) = α(f )u(y f ) + (1 -α(f ))u(x f ) = α(f ) min p∈P E p u(f )) + (1 -α(f )) max p∈P E p (u(f )).
Moreover, since is transitive and by (7), we have f g if and only if z f z g if and only if V (f ) ≥ V (g) for all acts f, g ∈ F , which completes the proof.

  1 provides a way of aggregating individuals' tastes and beliefs. The social utility function is a utilitarian -or linear -aggregation of individuals' utility functions. This simply comes from applying the unambiguous Pareto principle to the constant acts, where it reduces to the standard Pareto principle since beliefs do not matter for the evaluation of these acts. It is thus a direct extension of Harsanyi (1955)'s aggregation theorem. More interesting is the way the social belief is constrained by individuals'. When individual beliefs are compatible in the sense of having a non-empty intersection, the social belief must lie inside this intersection. The unambiguous Pareto principle thus yields a strong but intuitive prescription: society must only use probability distributions that all individuals consider plausible.

  The common-taste unambiguous Pareto principle thus allows aggregation of unambiguous preferences even with incompatible beliefs. As in Theorem 1, society can have SEU preferences even if all individuals have imprecise beliefs. The opposite case is now also possible: society can have imprecise beliefs even if all individuals have SEU preferences, in which case social belief imprecision results from individual belief heterogeneity. Although more permissive than Theorem 1 in the way society's beliefs could be related to individuals', this result always provides guidance for the construction of these beliefs.Remark 1. We would obtain the same characterization if we strengthened the common-taste unambiguous Pareto principle by focusing on the "involved" individuals, in the spirit of[START_REF] Gilboa | No-Betting-Pareto Dominance[END_REF]: individual i is involved in the comparison between f and g if f (s) i g(s) for some s ∈ S. That is, we would now say, more generally, that f and g are common-taste acts if x i y is equivalent to x j y for all x, y ∈ conv(f (S) ∪ g(S)) and all individuals i, j that are involved in f and g. Equivalently, f and g are common-taste acts if all individual utility functions, once restricted to the set of all possible outcomes of these two acts, are either identical up to positive affine transformations or constant.Remark 2. Unlike Theorem 1, Theorem 2 does not require individual preferences to satisfy c-diversity but only c-minimal agreement. It is therefore applicable to the particular case where all individuals have identical tastes.

Definition 4 .

 4 A binary relation on F is a variable caution choice rule for an unambiguous preference relation on F with representation (u, P ) if there exists a function α : F → [0, 1]

and

  [START_REF] Gajdos | Decisions with conflicting and imprecise information[END_REF] assume that individuals have identical tastes. Allowing for taste heterogeneity,[START_REF] Qu | Separate Aggregation of Beliefs and Values under Ambiguity[END_REF] characterizes a strengthening of the common-taste Pareto principle within the MEU and CEU classes.[START_REF] Alon | Utilitarian Preferences with Multiple Priors[END_REF] assume that individuals have SEU preferences whereas society has MEU preferences and characterize the unanimity Pareto principle.

  a ∈ A and b ∈ B 1 + D. It follows that there exists an open neighborhood Q of q such that, for all r ∈ Q, s∈S r(s)b(s) ≥ 0 ≥ s∈S r(s)a(s).

  It follows that sup L = inf M . Finally, L and M are closed since is c-Archimedean and, hence, sup L = max L and inf M = min M . Hence, letting α(f ) = max L = min M and

Such preferences are also called[START_REF] Bewley | Knightian decision theory. Part I[END_REF] preferences. They are incomplete expected utility preferences and are thus distinct from "robust" preferences in the sense of[START_REF] Hansen | Robust Control and Model Uncertainty[END_REF] Sargent (2001, 2008), which are complete non-expected utility preferences.

For instance, if individuals' beliefs originate from partial and distinct but mutually consistent pieces of evidence, or from a common "baseline" probabilistic model that they do not fully trust.

For instance, the common-taste unambiguous Pareto principle allows the policy maker to consider plausible all models that at least one individual considers plausible, as recently proposed by[START_REF] Brunnermeier | A Welfare Criterion for Models with Distorted Beliefs[END_REF].

The individual may still come up with an overall preference judgment or reveal a behavioral disposition for one of the two acts; such a preference would simply not be unambiguous. See Section 3.

This property, which is standard in the preference aggregation literature, is often named "independent prospects".

A similar pattern was experimentally observed by[START_REF] Baillon | Aggregating imprecise or conflicting beliefs: An experimental investigation using modern ambiguity theories[END_REF].

In the two-outcome situations discussed above there is no distinction between ordinal and cardinal preferences. This is no longer true with more than two outcomes and requiring identical cardinal preferences turns out to provide the relevant notion of common-taste acts for our purposes.

[START_REF] Aumann | Utility theory without the completeness axiom[END_REF]Aumann ( , 1964) ) and[START_REF] Evren | Scalarization methods and expected multi-utility representations[END_REF] prove similar "scalarization" results in different settings.

Note that[START_REF] Qu | Separate Aggregation of Beliefs and Values under Ambiguity[END_REF] also defines common-taste acts more narrowly as those yielding only convex combinations of two exogenously fixed outcomes between which individuals have a unanimous strict preference. Our more general definition yields the same characterization while retaining a stronger Pareto principle. See also[START_REF] Billot | Utilitarianism with Prior Heterogeneity[END_REF] for a different resolution of the impossibility through an extension of the state space.

Note that in this approach is the only primitive relation whereas * is derived from . Note also that the definition of * -or, equivalently, of P -by[START_REF] Ghirardato | Differentiating ambiguity and ambiguity attitude[END_REF] and[START_REF] Nehring | Bernoulli Without Bayes: A Theory of Utility-Sophisticated Preferences under Ambiguity[END_REF] does not necessarily coincide with that by[START_REF] Klibanoff | Perceived Ambiguity and Relevant Measures[END_REF], the former being generally more complete.

for useful comments and discussions. Danan thanks support from the Labex MME-DII program (ANR-11-LBX-0023-01). Gajdos thanks support from the A MIDEX project (ANR-11-IDEX-0001-02) funded by the Investissements d'Avenir program. Hill thanks support from the ANR DUSUCA (ANR-14-CE29-0003-01) and the Investissements d'Avenir program (ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047). Tallon thanks support from ANR grant AmGames (ANR-12-FRAL-0008-01) and from the Investissements d'Avenir program (ANR-10-LABX-93).

Appendix: proofs

Given a utility function u : X → R and a probability distribution p on S, define the "statedependent utility" function w u,p : X × S → R by w u,p (x, s) = p(s)u(x). Given a set P of probability distributions on S, let W u,P = {w u,p : p ∈ P }. Let C = c ∈ R X×S : c(x, s) = c(y, s) for all x, y ∈ X and s ∈ S denote the set of "state-dependent constant" functions. Let cone(•) denote conic hull.

Lemma 1. Let i be an unambiguous preference relation on F with representation (u i , P i ) for all i = 0, . . . , n. Assume ( i ) n i=1 satisfies c-minimal agreement. Then 0 satisfies unambiguous Pareto dominance with respect to ( i ) n i=1 if and only if

Proof. X has a finite affine basis X since it is a subset of a Euclidean space. Given a statedependent utility function w : X × S → R, denote by ŵ its restriction to X × S. Given a set W of such functions, denote by Ŵ the set of corresponding restrictions. Then (1) is equivalent to

It follows from a straightforward generalization of [START_REF] Danan | Harsanyi's Aggregation Theorem with Incomplete Preferences[END_REF]'s aggregation theorem that 0 satisfies unambiguous Pareto dominance with respect to ( i ) n i=1 if and only if W u 0 ,P 0 is included in the closure of the right hand side of (2). Hence it suffices to prove that this right hand side is closed. We first show that cone( Ŵu i ,P i ) + Ĉ is a closed, convex cone for all i = 1, . . . , n.

That it is a convex cone is easily checked. For closedness, note that 0 / ∈ Ŵu i ,P i since u i is nonconstant and, hence, cone( Ŵu i ,P i ) is closed since Ŵu i ,P i is compact and convex (Rockafellar, 1970, Corollary 9.6.1). Moreover, cone( Ŵu i ,P i ) ∩ Ĉ = {0} and, hence, cone( Ŵu i ,P i ) + Ĉ is closed as well (Rockafellar, 1970, Corollary 9.1.3).

It remains to show that the sum of these closed, convex cones is itself closed. As explained in [START_REF] Danan | Harsanyi's Aggregation Theorem with Incomplete Preferences[END_REF], this will be the case if there exist two acts f, g ∈ F such that, for all i = 1, . . . , n and all g i ∈ F such that f i g i , there exist g i ∈ F and λ i ∈ (0, 1) such that f i g i and g = λ i g i + (1 -λ i )g i . To establish this property, recall that by c-minimal agreement, there exist x, y ∈ X such that x i y for all i = 1. . . . , n. Hence, for all i = 1, . . . , n, there exists an open neighborhood Y i of y in X such that x i z for all z ∈ Y i . Let g i ∈ F such that x i g i and, given a coefficient λ ∈ (0, 1), let

Moreover, since S is finite, there exists λ ∈ (0, 1) small enough so that g i ∈ Y S ⊂ F and, hence, x i g i (s) for all s ∈ S. It follows that x i g i .

Proof of Theorem 1. The "if" part is easily checked. For the "only if" part, assume 0 satisfies unambiguous Pareto dominance with respect to ( i ) n i=1 . Restricting attention to the constant acts, unambiguous Pareto dominance reduces to standard Pareto dominance. We can therefore apply Harsanyi (1955)'s aggregation theorem (for a rigorous proof in our setting, see de Meyer and [START_REF] Mongin | Consistent Bayesian Aggregation[END_REF] to obtain θ ∈ R n + and γ ∈ R such that u 0 = n i=1 θ i u i + γ. θ and γ are unique by c-diversity. Moreover, θ = 0 since u 0 is non-constant.

It remains to prove that for all p 0 ∈ P 0 and all i = 1, . . . , n such that θ i > 0, p 0 ∈ P i . To this end, note that by Lemma 1, there exist (p i ) n i=1 ∈ n i=1 P i , θ ∈ R n + , and c ∈ C such that

It follows that

for all s ∈ S and x ∈ X, where c (s) stands for c (x, s) since the latter is independent of x.

Summing over S yields

for all x ∈ X, so that θ = θ and γ = s∈S c (s). Hence (3) implies that

and, hence, that

for all s ∈ S and x, y ∈ X. Fix an individual i such that θ i > 0. By c-diversity, there exist

x, y ∈ X such that u i (x) > u i (y) whereas u j (x) = u j (y) for all j = 1, . . . , n, j = i. By (4), it follows that p 0 (s) = p i (s) for all s ∈ S, so that p 0 = p i ∈ P i .

Proof of Theorem 2. The "if" part is easily checked. For the "only if" part, assume 0 satisfies common-taste unambiguous Pareto dominance with respect to ( i ) n i=1 . As in the proof of Theorem 1, we first restrict attention to the constant acts to obtain θ ∈ R n + , θ = 0, and γ ∈ R such that u 0 = n i=1 θ i u i + γ. It remains to prove that for all p 0 ∈ P 0 , there exist (p i ) n i=1 ∈ n i=1 P i and λ ∈ ∆ n such that p 0 = n i=1 λ i p i . By c-minimal agreement, there exist x, y ∈ X such that x i y for all i = 1, . . . , n. Hence all acts in conv({x, y}) S are common-taste acts. It follows that x 0 y, so that individual and social preferences all agree on conv({x, y}). Hence for all i = 1, . . . , n, there exist a i ∈ R + , a i > 0, and b i ∈ R such that u i (z) = a i u 0 (z) + b i

(5) for all z ∈ conv({x, y}). We can therefore use the common-taste unambiguous Pareto principle


Emmanuel Zimmermann
email: emmanuel.zimmermann@univ-lorraine.fr

Mélanie Noyel
email: melanie.noyel@univ-lorraine.fr

Philippe Thomas
email: philippe.thomas@univ-lorraine.fr

Hind BRIL E L Haouzi

André Thomas
email: andre.thomas@univ-lorraine.fr

K-means Clustering Based Method for Production Flow Management Facing Rework Disturbances

Keywords: reworks, flow disturbance, indicators, clustering, design of experiments

Manufacturing companies that suffer from a high rework rate on their production have mostly problems with their production scheduling flow, too. Author's previous work have highlighted a panel of Key Performance Indicators (KPI) and their interactions. A simulation model has been made to obtain a cartography representing the evolution of the amount of delayed products depending of two indicators highlighting four areas useful to decision making. The main goal of this work is to propose an automated approach to determine these areas and the relating production flow management decisions. The determination of these areas is performed by using clustering approach, when the production flow management rules associated to each area are determined by performing experimental design. This work refers to a bigger project that tries to develop a production flow monitoring and control system based on the Product Driven System (PDS).

Introduction

The manufacturing companies producing very high quality products have sometimes to deal with heavy rework rate. Acta-Mobilier company, which offers very high quality lacquered furniture, is a good example. Thus, the company has to deal with a reworks rate higher than 30% and able to jump up to 80% for some product ranges. This is resulting from the gap between the growing customer requirements and the speed by which the company implements the new processes. The purpose of the study presented in this paper is to give those companies support tools or methods in order to understand their flows behavior in reaction to the non-quality rate variations and by the same way prevent the logistic problems. The flows disturbances resulting from the non-quality could not be easily observed precisely but they highlight behaviors that could be associated to snowball effect (Forester effect) making them hard to evaluate and to control. In the previous works [START_REF] Noyel | Contrôle intégré du pilotage et de la qualité des produits, application à la société Acta-Mobilier[END_REF], ARENA software was used to find out the non-quality indicator families to see with a better accuracy the workshop state. A combination of different indicators allows establishing a cartography of the perturbation impact on the workshop which highlights different behavior areas depending on the predefined KPIs. The approach proposed in this paper flows two stages: First, the clustering algorithm is used to define automatically the borders of these areas. In a second step an experimental design is performed to determine the best control method to apply for each of the cluster.

In the next section, we will make an assessment of the state-of-the-art about flows control, the KPIs and clustering algorithms. Then, a discussion about the proposed clustering algorithm and its results will be presented. The third section presents the approach used to determine the best production flow management rules for each cluster. In the end, a conclusion will be drawn and perspectives of evolution will be stated.

State-of-the-art

Flow monitoring disturbed by non-quality

Even if the non-quality became a well-known notion, as reflected by the countless works about it. The bound between rework and flow disturbance may not always be well highlighted. The COQ (Cost of Quality) [START_REF] Feigenbaum | Quality control[END_REF] method with the use of the ghost factory, a parallel workshop whose mission is to repair defaults generated in the official one, introduced the term of non-quality already in the 50 th . This workshop represents 40% of the official production plant capability. The first norm on this topic emerged in France in 1986 (X 50-126). It aims was to evaluate the non-quality costs [START_REF] Abouzahir | Pilotage de l'amélioration des processus par les coûts de non-qualité[END_REF]. A statistical analysis in relation to a problem of workstation highlights a vibrational phenomenon [START_REF] Skidmann | Performance evaluation of a flexible manufacturing cell with random multiproduct feedback flow[END_REF] mentioned in other works whose aim is to evaluate a multiproduct workshop performance [START_REF] Skidmann | Performance evaluation of a flexible manufacturing cell with random multiproduct feedback flow[END_REF]. The flow disturbance of the production by reworks is especially significant if reworks occur on the downstream of the production process [START_REF] Love | Rework: a symptom of a dysfunctional supply-chain[END_REF]. Therefore, on production monitoring subject, it is necessary to follow and maintain a low rework rate to be certain to simplify the initial monitoring problem [START_REF] Noyel | Implantation of an on-line quality process monitoring[END_REF]. However, even if the reworks rate is reduced and stabilized, the resulting disturbance on the product flow is still constant and complicates the scheduling function. In order to evaluate the reworks rate, the most used indicator is relative to the amount of defects. But the more the reworks rate increases, the higher is the probability to repair the same product many times. The challenging issue is how to quantify the impact of these repaired products several times? In this situation, it becomes hard to establish a direct link between reworks rate and flow disturbance.

In order to evaluate both the non-quality and the flow disturbance, a lot of indicators were used. Like all of the KPIs, these ones are formulized by the triplet (purpose, unit of measure, variable), they represent the system efficiency related to a pre-established norm. In [START_REF] Noyel | Contrôle intégré du pilotage et de la qualité des produits, application à la société Acta-Mobilier[END_REF], we highlights some of them, as, for example, the number of defect Ndefect, or the time to finish all the production traditionally called Cmax. Each indicator highlights only one sight of the problem. In order to outline a more global and realistic view, [START_REF] Noyel | Contrôle intégré du pilotage et de la qualité des produits, application à la société Acta-Mobilier[END_REF] has demonstrated the pertinence to work with a combination of these indicators. In other word, combining two or more indicators allows to sketch-up different views and makes a more complete assessment about the production workshop efficiency. For example, Ndefect evaluates the number of pieces which have at least presented one defect. It is frequently used in factories because it comes directly from quality control methods. This indicator is connected to the ability of the workshop to make a product well from the first time. But, alone, it does not allow understanding and prevised the effect on the flow disturbance. Thus combined to another flow indicator, it might become possible to identify one or several thresholds and then determine areas in which flow disturbance becomes uncontrollable. As a result, cartography of the workshop state can be established according to the values of the different indicators.

The goal is to develop a synthetic system of indicators usable to determine the best control method to apply for the considered situation. The figure 1 shows the combination between Nwip and Ndefect. In this chart, four behavior areas seem to appear and our goal was to determine the boundaries of these areas. ILS Conference 2016, June 1 -4, Bordeaux, France

The bound determination by using clustering algorithms

One of the most powerful approaches to determine the membership of data to an area and to set up the limits of each of them is the clustering [START_REF] Jain | Data clustering: a review[END_REF]. Many clustering algorithms exist and some of them are recapitulated in table 1 [START_REF] Hastie | The elements of statistical learning, data mining, inference and prediction[END_REF]. In our work, the K-means algorithm is chosen for two raisons: in our case study the number of the needed clusters and the algorithm is well known as simple to implement and gives a quick result. And on the other hand, the K-means algorithm aim is to gather N points of observation in K clusters with a constraint: each of the N points has to belong to the cluster for which the distance to the mean is the smallest. The idea is to use clustering algorithm on indicators values (Nwip, Ndefect, Cmax, Nlate…) on the considered process during production to determine the different functioning conditions. In a second step, the goal is to determine the best control rule to apply for each of these functioning conditions.

Control rules

The piloting rule has a huge impact on the flow of products in manufacturing. For example, whereas SPT (Shortest Processing Time) privileges the shortest tasks which consume few time, EDD (Earliest Due Date) focuses on the products with the nearest due date and that without regards on the required time to achieve them. As soon as the repair rate becomes important, the most suitable control rules aren't necessarily the same regarding to the situations. There are at least 150 different control rules. FIFO (First In First Out), SPT, EDD and CR (Critical Ratio) are really frequently used rules in workshop and are considered as benchmark for simulating research [START_REF] Noyel | Contrôle intégré du pilotage et de la qualité des produits, application à la société Acta-Mobilier[END_REF] where the repair rate varies. Lots of work in the semi-conductor sector show that when there are huge quantity of production and repairs a specific rule like FSVCT ( Fluctuation Smoothing of Variance of Cycle Time) can provide better results [START_REF] Mittler | Comparison of dispatching rules for semiconductor manufacturing using large facility models[END_REF]. However, the factor "production volume" and the repairs rate inferred are really different from our context. It seems that generally, in workshops with many repairs, SPT, EDD and FIFO rules are the most frequent, even if SPT seems to be the most efficient. Although, FIFO and EDD are principally used by Acta-Mobilier because of their simplicity to be understood and applied.

The areas determination

The data used to establish the cartography, were produced by simulation with the software ARENA which gave us the possibility to record as many evaluation points as we needed. Another point is that it allows to test the system in the critical conditions of use. The proposed model is based on a system with 2 workstations which have the same defect rate. In figure 2: Np represents the number of produced pieces and the formula on each arrow represents the probability to reach the next step. The system variability is that the evolution of the rate of defectuous pieces producted and the outstanding rate are completely random. These ones are represented by sinusoides with high periodicity but different so that, after some temporal periods, we are certain to evaluate the most possibles cases. Scenarii were launched for 3 different values of Ndefaut, Nwip and monitoring rules: EDD, FIFO and SPT. This system will allow obtaining a global sight of the workshop state in real-time, the reworks impact and/or the non-quality of the flow. We studied the behavior depending on the progressive rise of the reworks rate. But flow disturbance stays something difficult to measure.

The K-means algorithm has been used in order to well define the behaviors areas. Data processing was made with SCILAB which contains a standard version of the K-means in the functions library of the software [11]. The chosen settings are: 4 clusters and the Euclidian distance.

The figure 3 shows the result obtained on the data by the use of K-means algorithm with the settings described before. The bounds between the different areas is also drawn. This figure shows that the four areas are well identified and the bound between these areas may be determined. The second steps consist to determine the best rule to use for each identified operating points (each areas). An experiment design involves that several physical or numeric experiments have to be made. Settings have to be chosen, the values they can take have also to be set up and all of the possible settings combinations have to be tested, then results have to be analyzed. The chart shows each of the settings and highlights the selected values for the experiments and a curb for every interaction (an interaction is the representation of the combined effects of 2 setting). After that, with a simple read of the chart, the best solution to adopt can be deduced.

The definition of levels for each factors is described in Table 2. The produced simulations take into account all of the combinations of the 3 values of Nwip, Ndefect and the control rules. The obtained results about the number of overdue on the production chain allowed to establish a design of experiments and (Figure 4 shows the factor effect graphs), this one allows to visualize which monitoring rule is the best to apply for a given configuration of Nwip and Ndefect. This analysis is the key to finally associate the « best » or the less bad possible monitoring rule to every cluster.

To associate control rules to the map established previously, we made an experiments plan with the challenge to identify the influence of variables Ndefect, Nwip and the monitoring rule on the system. The first 3 curves show the impact of three factors individually. Analyzed alone, they confirm that the best case for not having delays use SPT rule in conditions of Work in Process low and with few defects. However, when looking at interactions (3 right curves), we realize that in conditions of Work in Process and reparations rate imposed, the most suitable management flow rule is not always the same. We can construct the table 3 having the most appropriate rule to a position given by the value of the two indicators considered. The corresponding limit for the changeover between EDD and SPT is the central green curve in Figure 3.

The other two curves are only state of saturation but does not justify the passage of a control rule to another one. It may instead be associated with common sense warnings or recommendations as shown in Table 4.

From an operator point of view, the most appropriate rule for the operation in the workshop is the FIFO rule with a batch processing in the order of their arrival, without any additional information. The EDD rule also applies easily since the departure dates are clearly marked on organized pallet so that it can choose any. However, the SPT rule is more difficult to implement because it depends on the working time which must be known before choosing the next work to be performed. This information which is available in the ERP may not be displayed directly on the follower sheet items because there is one per workstation / phase range. The implementation of the latter rule therefore requires the presence of an efficient information system. One can possibly imagine the presence of this information directly on the product itself, thanks to Auto-ID technologies, which will return it at the right time by consensus or with neighboring products.

Define batches to prioritize

The recommendation of the Area 2 aims to quickly reduce the number of ongoing repairs and thus move the point representing the state saturation of the workshop to the left to return in Area 1. Once the item returns to Area 1, one can return to a normal capacity for the repair station. The recommendation of Area 3 has the objective of act more on the number of delays. The combined action on delays and repairs will tend to move the point simultaneously down and to the left with the objective to reach the Area 2 rapidly. The recommendation of Area 4 rather palliative as the situation is serious. It tries to "save" lots of great importance in treatment of temporarily abandoning the others. This area, materialized in red on the map, is ideally never reached.

Conclusions

This paper addresses a challenging issue for factories suffering from a high rework rate. The proposed approach has as objective to follow and to give a help decision tool in order to decrease the impact of rework on the factory performances. The main contribution is to combine two methods Clustering and Experiment plan. Based on actual data from Acta-Mobilier and an ARENA simulation model, four clusters of data that correspond to four workshop saturation zones were identified. The use of an experiment plan performed on the same data, leads to identify two best rules for different situations and distribute them on four areas. The main advantage of this method is its simplicity and the visual aspect suitable for intuitive use on the shop floor. It is clearly oriented towards the easy and fast application, and to communication and empowerment of workers. Based on such approach and the use of PDS concept, an auto-control system and dynamical scheduling flows could be created. Indeed, this work refers to a bigger project trying to develop a production flow monitoring helping system based on the Product Driven System concept and real-time visuals indicators combination. This way, the product and the map based on the indicators described before will become an agent of the global system. Together they will impact on the scheduling and the dynamical re-scheduling by influencing the management flow rule to use and the scheduling horizon.

Figure 1 :

 1 Figure 1: evolution of Nwip (number of pieces in outstanding) depending on Ndefect

Figure 2 :

 2 Figure 2: Model of the system used for the tests

Figure 3 :

 3 Figure 3: Result obtained with a 4-Means algorithm and a Euclidian distance

Figure 4 :

 4 Figure 4: Experimental Design with management flow rules, Nwip, Ndefect as factors and their interactions.

Table 1 :

 1 Strengths & weaknesses of some unsupervised learning clustering algorithms[START_REF] Jain | Data clustering: a review[END_REF]

	Algorithms	Strengths & weaknesses
	K-means / K-methods	+ Most popular
		+ Draw cluster's border
		+ Already implemented in calculating software
		+ Give good results
		-Number of clusters has to be fixed before
		-Changing the number of cluster can arbitrary modify the
		owning
		-Risk to fall in a local minimum
		-Dependent of the initialization parameters and the chosen
		distance
	Agglomerative	+ Monotonicity property
	hierarchical	-Unappropriated for a lot of data
	Divisive hierarchical	-Rarely used

Table 2 :

 2 Levels for factors in the experimental design factors

	Factors

Table 3 :

 3 Best control rule regarding to the imposed indicator values

				NDefect
			30%	45%	60%
	Wip	High	SPT	SPT	SPT
		Average	EDD	SPT	SPT
		Low	EDD	EDD	SPT

Table 4 :

 4 Work recommendation depending on workshop saturation areas

	Area 1	Area 2	Area 3	Area 4