Possibilistic sequential decision making
When 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: optimist...
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| Vydáno v: | International journal of approximate reasoning Ročník 55; číslo 5; s. 1269 - 1300 |
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| Jazyk: | angličtina |
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01.07.2014
Elsevier |
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| ISSN: | 0888-613X, 1873-4731 |
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| Abstract | When 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 [2,9], 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.
•We study the sequential decision making problem in possibilistic decision trees.•We propose a theoretical study on the complexity of finding an optimal strategy w.r.t. possibilistic decision criteria.•We propose a resolution algorithm (Dynamic programming (standard-extended) or Branch and Bound) for each criterion.•We show that for possibilistic Choquet integrals the problem of finding an optimal strategy is NP-hard. |
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| AbstractList | When 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 [2,9], 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.
•We study the sequential decision making problem in possibilistic decision trees.•We propose a theoretical study on the complexity of finding an optimal strategy w.r.t. possibilistic decision criteria.•We propose a resolution algorithm (Dynamic programming (standard-extended) or Branch and Bound) for each criterion.•We show that for possibilistic Choquet integrals the problem of finding an optimal strategy is NP-hard. |
| Author | Ben Amor, Nahla Fargier, Hélène Guezguez, Wided |
| Author_xml | – sequence: 1 givenname: Nahla surname: Ben Amor fullname: Ben Amor, Nahla email: nahla.benamor@gmx.fr organization: LARODEC, Institut Supérieur de Gestion Tunis, Université de Tunis, Tunisia – sequence: 2 givenname: Hélène surname: Fargier fullname: Fargier, Hélène email: fargier@irit.fr organization: IRIT, University of Paul Sabatier, Toulouse, France – sequence: 3 givenname: Wided surname: Guezguez fullname: Guezguez, Wided email: widedguezguez@gmail.com organization: LARODEC, Institut Supérieur de Gestion Tunis, Université de Tunis, Tunisia |
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| Keywords | Possibility theory Choquet integrals Decision analysis Dynamic Programming Decision tree Branch and Bound Process dynamics Multicriteria analysis Quantization Sequential decision Modeling Optimization Optimal strategy Uncertain system Dynamic programming Mathematical programming Probabilistic approach Branch and bound method Decision making Dominance Polynomial time Choquet expected utility Measure theory Decision theory Implicit enumeration method NP hard problem Problem solving Matrix polynomial Monotonicity |
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Res. doi: 10.1016/j.ejor.2003.05.004 – volume: 195 start-page: 223 issue: 1 year: 2009 ident: 10.1016/j.ijar.2013.11.005_br0140 article-title: Qualitative possibilistic influence diagrams based on qualitative possibilistic utilities publication-title: Eur. J. Oper. Res. doi: 10.1016/j.ejor.2008.01.051 – volume: 57 start-page: 517 year: 1989 ident: 10.1016/j.ijar.2013.11.005_br0280 article-title: Subjective probability and expected utility without additivity publication-title: Econometrica doi: 10.2307/1911053 – volume: 19 start-page: 441 issue: 3–4 year: 1998 ident: 10.1016/j.ijar.2013.11.005_br0270 article-title: Towards qualitative approaches to multi-stage decision making publication-title: Int. J. Approx. Reason. doi: 10.1016/S0888-613X(98)10019-1 – volume: 49 start-page: 455 year: 2002 ident: 10.1016/j.ijar.2013.11.005_br0030 article-title: Qualitative decision theory: from Savage's axioms to nonmonotonic reasoning publication-title: J. ACM doi: 10.1145/581771.581772 |
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| SubjectTerms | Applied sciences Branch and Bound Choquet integrals Decision analysis Decision theory. Utility theory Decision tree Dynamic Programming Exact sciences and technology Mathematical analysis Mathematical programming Mathematics Measure and integration Operational research and scientific management Operational research. Management science Possibility theory Sciences and techniques of general use |
| Title | Possibilistic sequential decision making |
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