91 research outputs found

    Solving multi-criteria decision problems under possibilistic uncertainty using optimistic and pessimistic utilities

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    International audienceThis paper proposes a qualitative approach to solve multi-criteria decision making problems under possibilistic uncertainty. De-pending on the decision maker attitude with respect to uncertainty (i.e. optimistic or pessimistic) and on her attitude with respect to criteria (i.e. conjunctive or disjunctive), four ex-ante and four ex-post decision rules are dened and investigated. In particular, their coherence w.r.t. the principle of monotonicity, that allows Dynamic Programming is studied

    Lexicographic refinements in possibilistic decision trees and finite-horizon Markov decision processes

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    Possibilistic decision theory has been proposed twenty years ago and has had several extensions since then. Even though ap-pealing for its ability to handle qualitative decision problems, possibilisticdecision theory suffers from an important drawback. Qualitative possibilistic utility criteria compare acts through min and max operators, which leads to a drowning effect. To over-come this lack of decision power of the theory, several refinements have been proposed. Lexicographic refinements are particularly appealing since they allow to benefit from the Expected Utility background, while remaining qualitative. This article aims at extend-ing lexicographic refinements to sequential decision problems i.e., to possibilistic decision trees and possibilistic Markov decision processes, when the horizon is finite. We present two criteria that refine qualitative possibilistic utilities and provide dynamic programming algorithms for calculating lexicographically optimal policies

    Possibilistic decision theory: from theoretical foundations to influence diagrams methodology

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    Le domaine de prise de décision est un domaine multidisciplinaire en relation avec plusieurs disciplines telles que l'économie, la recherche opérationnelle, etc. La théorie de l'utilité espérée a été proposée pour modéliser et résoudre les problèmes de décision. Ces théories ont été mises en cause par plusieurs paradoxes (Allais, Ellsberg) qui ont montré les limites de son applicabilité. Par ailleurs, le cadre probabiliste utilisé dans ces théories s'avère non approprié dans certaines situations particulières (ignorance totale, incertitude qualitative). Pour pallier ces limites, plusieurs travaux ont été élaborés concernant l'utilisation des intégrales de Choquet et de Sugeno comme critères de décision d'une part et l'utilisation d'une théorie d'incertitude autre que la théorie des probabilités pour la modélisation de l'incertitude d'une autre part. Notre idée principale est de profiter de ces deux directions de recherche afin de développer, dans le cadre de la décision séquentielle, des modèles de décision qui se basent sur les intégrales de Choquet comme critères de décision et sur la théorie des possibilités pour la représentation de l'incertitude. Notre objectif est de développer des modèles graphiques décisionnels, qui représentent des modèles compacts et simples pour la prise de décision dans un contexte possibiliste. Nous nous intéressons en particulier aux arbres de décision et aux diagrammes d'influence possibilistes et à leurs algorithmes d'évaluation.The field of decision making is a multidisciplinary field in relation with several disciplines such as economics, operations research, etc. Theory of expected utility has been proposed to model and solve decision problems. These theories have been questioned by several paradoxes (Allais, Ellsberg) who have shown the limits of its applicability. Moreover, the probabilistic framework used in these theories is not appropriate in particular situations (total ignorance, qualitative uncertainty). To overcome these limitations, several studies have been developed basing on the use of Choquet and Sugeno integrals as decision criteria and a non classical theory to model uncertainty. Our main idea is to use these two lines of research to develop, within the framework of sequential decision making, decision models based on Choquet integrals as decision criteria and possibility theory to represent uncertainty. Our goal is to develop graphical decision models that represent compact models for decision making when uncertainty is represented using possibility theory. We are particularly interested by possibilistic decision trees and influence diagrams and their evaluation algorithms

    Lexicographic refinements in possibilistic sequential decision-making models

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    Ce travail contribue à la théorie de la décision possibiliste et plus précisément à la prise de décision séquentielle dans le cadre de la théorie des possibilités, à la fois au niveau théorique et pratique. Bien qu'attrayante pour sa capacité à résoudre les problèmes de décision qualitatifs, la théorie de la décision possibiliste souffre d'un inconvénient important : les critères d'utilité qualitatives possibilistes comparent les actions avec les opérateurs min et max, ce qui entraîne un effet de noyade. Pour surmonter ce manque de pouvoir décisionnel, plusieurs raffinements ont été proposés dans la littérature. Les raffinements lexicographiques sont particulièrement intéressants puisqu'ils permettent de bénéficier de l'arrière-plan de l'utilité espérée, tout en restant "qualitatifs". Cependant, ces raffinements ne sont définis que pour les problèmes de décision non séquentiels. Dans cette thèse, nous présentons des résultats sur l'extension des raffinements lexicographiques aux problèmes de décision séquentiels, en particulier aux Arbres de Décision et aux Processus Décisionnels de Markov possibilistes. Cela aboutit à des nouveaux algorithmes de planification plus "décisifs" que leurs contreparties possibilistes. Dans un premier temps, nous présentons des relations de préférence lexicographiques optimistes et pessimistes entre les politiques avec et sans utilités intermédiaires, qui raffinent respectivement les utilités possibilistes optimistes et pessimistes. Nous prouvons que les critères proposés satisfont le principe de l'efficacité de Pareto ainsi que la propriété de monotonie stricte. Cette dernière garantit la possibilité d'application d'un algorithme de programmation dynamique pour calculer des politiques optimales. Nous étudions tout d'abord l'optimisation lexicographique des politiques dans les Arbres de Décision possibilistes et les Processus Décisionnels de Markov à horizon fini. Nous fournissons des adaptations de l'algorithme de programmation dynamique qui calculent une politique optimale en temps polynomial. Ces algorithmes sont basés sur la comparaison lexicographique des matrices de trajectoires associées aux sous-politiques. Ce travail algorithmique est complété par une étude expérimentale qui montre la faisabilité et l'intérêt de l'approche proposée. Ensuite, nous prouvons que les critères lexicographiques bénéficient toujours d'une fondation en termes d'utilité espérée, et qu'ils peuvent être capturés par des utilités espérées infinitésimales. La dernière partie de notre travail est consacrée à l'optimisation des politiques dans les Processus Décisionnels de Markov (éventuellement infinis) stationnaires. Nous proposons un algorithme d'itération de la valeur pour le calcul des politiques optimales lexicographiques. De plus, nous étendons ces résultats au cas de l'horizon infini. La taille des matrices augmentant exponentiellement (ce qui est particulièrement problématique dans le cas de l'horizon infini), nous proposons un algorithme d'approximation qui se limite à la partie la plus intéressante de chaque matrice de trajectoires, à savoir les premières lignes et colonnes. Enfin, nous rapportons des résultats expérimentaux qui prouvent l'efficacité des algorithmes basés sur la troncation des matrices.This work contributes to possibilistic decision theory and more specifically to sequential decision-making under possibilistic uncertainty, at both the theoretical and practical levels. Even though appealing for its ability to handle qualitative decision problems, possibilisitic decision theory suffers from an important drawback: qualitative possibilistic utility criteria compare acts through min and max operators, which leads to a drowning effect. To overcome this lack of decision power, several refinements have been proposed in the literature. Lexicographic refinements are particularly appealing since they allow to benefit from the expected utility background, while remaining "qualitative". However, these refinements are defined for the non-sequential decision problems only. In this thesis, we present results on the extension of the lexicographic preference relations to sequential decision problems, in particular, to possibilistic Decision trees and Markov Decision Processes. This leads to new planning algorithms that are more "decisive" than their original possibilistic counterparts. We first present optimistic and pessimistic lexicographic preference relations between policies with and without intermediate utilities that refine the optimistic and pessimistic qualitative utilities respectively. We prove that these new proposed criteria satisfy the principle of Pareto efficiency as well as the property of strict monotonicity. This latter guarantees that dynamic programming algorithm can be used for calculating lexicographic optimal policies. Considering the problem of policy optimization in possibilistic decision trees and finite-horizon Markov decision processes, we provide adaptations of dynamic programming algorithm that calculate lexicographic optimal policy in polynomial time. These algorithms are based on the lexicographic comparison of the matrices of trajectories associated to the sub-policies. This algorithmic work is completed with an experimental study that shows the feasibility and the interest of the proposed approach. Then we prove that the lexicographic criteria still benefit from an Expected Utility grounding, and can be represented by infinitesimal expected utilities. The last part of our work is devoted to policy optimization in (possibly infinite) stationary Markov Decision Processes. We propose a value iteration algorithm for the computation of lexicographic optimal policies. We extend these results to the infinite-horizon case. Since the size of the matrices increases exponentially (which is especially problematic in the infinite-horizon case), we thus propose an approximation algorithm which keeps the most interesting part of each matrix of trajectories, namely the first lines and columns. Finally, we reports experimental results that show the effectiveness of the algorithms based on the cutting of the matrices

    Collective decision making under qualitative possibilistic uncertainty: principles and characterization

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    Cette Thèse pose la question de la décision collective sous incertitude possibiliste. On propose différents règles de décision collective qualitative et on montre que dans un contexte possibiliste, l'utilisation d'une fonction d'agrégation collective pessimiste égalitariste ne souffre pas du problème du Timing Effect. On étend ensuite les travaux de Dubois et Prade (1995, 1998) relatifs à l'axiomatisation des règles de décision qualitatives (l'utilité pessimiste) au cadre de décision collective et montre que si la décision collective comme les décisions individuelles satisfont les axiomes de Dubois et Prade ainsi que certains axiomes relatifs à la décision collective, particulièrement l'axiome de Pareto unanimité, alors l'agrégation collective égalitariste s'impose. Le tableau est ensuite complété par une axiomatisation d'un pendant optimiste de cette règle de décision collective. Le système axiomatique que nous avons développé peut être vu comme un pendant ordinal du théorème de Harsanyi (1955). Ce résultat á été démontré selon un formalisme qui et basé sur le modèle de de Von NeuMann and Morgenstern (1948) et permet de comparer des loteries possibilistes. Par ailleurs, on propose une première tentative pour la caractérisation des règles de décision collectives qualitatives selon le formalisme de Savage (1972) qui offre une représentation des décisions par des actes au lieu des loteries. De point de vue algorithmique, on considère l'optimisation des stratégies dans les arbres de décision possibilistes en utilisant les critères de décision caractérisés dans la première partie de ce travail. On offre une adaptation de l'algorithme de Programmation Dynamique pour les critères monotones et on propose un algorithme de Programmation Multi-dynamique et un algorithme de Branch and Bound pour les critères qui ne satisfont pas la monotonie. Finalement, on établit une comparaison empirique des différents algorithmes proposés. On mesure les CPU temps d'exécution qui augmentent linéairement en fonction de la taille de l'arbre mais restent abordable même pour des grands arbres. Ensuite, nous étudions le pourcentage d'exactitude de l'approximation des algorithmes exacts par Programmation Dynamique: Il apparaît que pour le critère U-max ante l'approximation de l'algorithme de Programmation Multi-dynamique n'est pas bonne. Mais, ceci n'est pas si dramatique puisque cet algorithme est polynomial (et efficace dans la pratique). Cependant, pour la règle U+min ante l'approximation par Programmation Dynamique est bonne et on peut dire qu'il devrait être possible d'éviter une énumération complète par Branch and Bound pour obtenir les stratégies optimales.This Thesis raises the question of collective decision making under possibilistic uncertainty. We propose several collective qualitative decision rules and show that in the context of a possibilistic representation of uncertainty, the use of an egalitarian pessimistic collective utility function allows us to get rid of the Timing Effect. Making a step further, we prove that if both the agents' preferences and the collective ranking of the decisions satisfy Dubois and Prade's axioms (1995, 1998) and some additional axioms relative to collective choice, in particular Pareto unanimity, then the egalitarian collective aggregation is compulsory. The picture is then completed by the proposition and the characterization of an optimistic counterpart of this pessimistic decision rule. Our axiomatic system can be seen as an ordinal counterpart of Harsanyi's theorem (1955). We prove this result in a formalism that is based on Von NeuMann and Morgenstern framework (1948) and compares possibilisitc lotteries. Besides, we propose a first attempt to provide a characterization of collective qualitative decision rules in Savage's formalism; where decisions are represented by acts rather than by lotteries. From an algorithmic standpoint, we consider strategy optimization in possibilistic decision trees using the decision rules characterized in the first part of this work. So, we provide an adaptation of the Dynamic Programming algorithm for criteria that satisfy the property of monotonicity and propose a Multi-Dynamic programming and a Branch and Bound algorithm for those that are not monotonic. Finally, we provide an empirical comparison of the different algorithms proposed. We measure the execution CPU times that increases linearly according to the size of the tree and it remains affordable in average even for big trees. Then, we study the accuracy percentage of the approximation of the pertinent exact algorithms by Dynamic Programming: It appears that for U-max ante criterion the approximation of Multi-dynamic programming is not so good. Yet, this is not so dramatic since this algorithm is polynomial (and efficient in practice). However, for U+min ante decision rule the approximation by Dynamic Programming is good and we can say that it should be possible to avoid a full Branch and Bound enumeration to find optimal strategies

    Anytime Algorithms for Solving Possibilistic MDPs and Hybrid MDPs

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    The ability of an agent to make quick, rational decisions in an uncertain environment is paramount for its applicability in realistic settings. Markov Decision Processes (MDP) provide such a framework, but can only model uncertainty that can be expressed as probabilities. Possibilistic counterparts of MDPs allow to model imprecise beliefs, yet they cannot accurately represent probabilistic sources of uncertainty and they lack the efficient online solvers found in the probabilistic MDP community. In this paper we advance the state of the art in three important ways. Firstly, we propose the first online planner for possibilistic MDP by adapting the Monte-Carlo Tree Search (MCTS) algorithm. A key component is the development of efficient search structures to sample possibility distributions based on the DPY transformation as introduced by Dubois, Prade, and Yager. Secondly, we introduce a hybrid MDP model that allows us to express both possibilistic and probabilistic uncertainty, where the hybrid model is a proper extension of both probabilistic and possibilistic MDPs. Thirdly, we demonstrate that MCTS algorithms can readily be applied to solve such hybrid models. © Springer International Publishing Switzerland 2016.This work is partially funded by EPSRC PACES project (Ref: EP/J012149/1).Peer Reviewe

    Possibilistic sequential decision making

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    International audienceWhen the information about uncertainty cannot be quantified in a simple, probabilistic way, the topic of possibilistic decision theory is often a natural one to consider. The development of possibilistic decision theory has lead to the proposition a series of possibilistic criteria, namely: optimistic and pessimistic possibilistic qualitative criteria [7], possibilistic likely dominance [2] and [9], binary possibilistic utility [11] and possibilistic Choquet integrals [24]. This paper focuses on sequential decision making in possibilistic decision trees. It proposes a theoretical study on the complexity of the problem of finding an optimal strategy depending on the monotonicity property of the optimization criteria – when the criterion is transitive, this property indeed allows a polytime solving of the problem by Dynamic Programming. We show that most possibilistic decision criteria, but possibilistic Choquet integrals, satisfy monotonicity and that the corresponding optimization problems can be solved in polynomial time by Dynamic Programming. Concerning the possibilistic likely dominance criteria which is quasi-transitive but not fully transitive, we propose an extended version of Dynamic Programming which remains polynomial in the size of the decision tree. We also show that for the particular case of possibilistic Choquet integrals, the problem of finding an optimal strategy is NP-hard. It can be solved by a Branch and Bound algorithm. Experiments show that even not necessarily optimal, the strategies built by Dynamic Programming are generally very good

    New Graphical Model for Computing Optimistic Decisions in Possibility Theory Framework

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    This paper first proposes a new graphical model for decision making under uncertainty based on min-based possibilistic networks. A decision problem under uncertainty is described by means of two distinct min-based possibilistic networks: the first one expresses agent's knowledge while the second one encodes agent's preferences representing a qualitative utility. We then propose an efficient algorithm for computing optimistic optimal decisions using our new model for representing possibilistic decision making under uncertainty. We show that the computation of optimal decisions comes down to compute a normalization degree of the junction tree associated with the graph resulting from the fusion of agent's beliefs and preferences. This paper also proposes an alternative way for computing optimal optimistic decisions. The idea is to transform the two possibilistic networks into two equivalent possibilistic logic knowledge bases, one representing agent's knowledge and the other represents agent's preferences. We show that computing an optimal optimistic decision comes down to compute the inconsistency degree of the union of the two possibilistic bases augmented with a given decision

    Optimization Models Using Fuzzy Sets and Possibility Theory

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    Optimization is of central concern to a number of disciplines. Operations Research and Decision Theory are often considered to be identical with optimization. But also in other areas such as engineering design, regional policy, logistics and many others, the search for optimal solutions is one of the prime goals. The methods and models which have been used over the last decades in these areas have primarily been "hard" or "crisp", i.e. the solutions were considered to be either feasible or unfeasible, either above a certain aspiration level or below. This dichotomous structure of methods very often forced the modeler to approximate real problem situations of the more-or-less type by yes-or-no-type models, the solutions of which might turn out not to be the solutions to the real problems. This is particularly true if the problem under consideration includes vaguely defined relationships, human evaluations, uncertainty due to inconsistent or incomplete evidence, if natural language has to be modeled or if state variables can only be described approximately. Until recently, everything which was not known with certainty, i.e. which was not known to be either true or false or which was not known to either happen with certainty or to be impossible to occur, was modeled by means of probabilities. This holds in particular for uncertainties concerning the occurrence of events. probability theory was used irrespective of whether its axioms (such as, for instance, the law of large numbers) were satisfied or not, or whether the "events" could really be described unequivocally and crisply. In the meantime one has become aware of the fact that uncertainties concerning the occurrence as well as concerning the description of events ought to be modeled in a much more differentiated way. New concepts and theories have been developed to do this: the theory of evidence, possibility theory, the theory of fuzzy sets have been advanced to a stage of remarkable maturity and have already been applied successfully in numerous cases and in many areas. Unluckily, the progress in these areas has been so fast in the last years that it has not been documented in a way which makes these results easily accessible and understandable for newcomers to these areas: text-books have not been able to keep up with the speed of new developments; edited volumes have been published which are very useful for specialists in these areas, but which are of very little use to nonspecialists because they assume too much of a background in fuzzy set theory. To a certain degree the same is true for the existing professional journals in the area of fuzzy set theory. Altogether this volume is a very important and appreciable contribution to the literature on fuzzy set theory
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