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: 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.     

Possibilistic nested logic programs and strong equivalence

In this paper, the class of possibilistic nested logic programs is introduced. These possibilistic logic programs allow us to use nested expressions in the bodies and heads of their rules. By considering a possibilistic nested logic program as a possibilistic theory, a construction of a possibilistic logic programing semantics based on answer sets for nested logic programs and the proof theory of possibilistic logic is defined. In order to define a general method for computing the possibilistic answer sets of a possibilistic nested program, the idea of equivalence between possibilistic nested programs is explored. By considering properties of equivalence between possibilistic programs, a process of transforming a possibilistic nested logic program into a possibilistic disjunctive logic program is defined. Given that our approach is an extension of answer set programming, we also explore the concept of strong equivalence between possibilistic nested logic programs. To this end, we introduce the concept of poss SE-models. Therefore, we show that two possibilistic nested logic programs are strong equivalents whenever they have the same poss SE-models.The expressiveness of the possibilistic nested logic programs is illustrated by a scenario from the medical domain. In particular, we exemplify how possibilistic nested logic programs are expressive enough for capturing medical guidelines which are pervaded by vagueness and qualitative information.     

Qualitative possibilistic influence diagrams based on qualitative possibilistic utilities

This paper proposes a new approach for decision making under uncertainty based on influence diagrams and possibility theory. The so-called qualitative possibilistic influence diagrams extend standard influence diagrams in order to avoid difficulties attached to the specification of both probability distributions relative to chance nodes and utilities relative to value nodes. In fact, generally, it is easier for experts to quantify dependencies between chance nodes qualitatively via possibility distributions and to provide a preferential relation between different consequences. In such a case, the possibility theory offers a suitable modeling framework. Different combinations of the quantification between chance and utility nodes offer several kinds of possibilistic influence diagrams. This paper focuses on qualitative ones and proposes an indirect evaluation method based on their transformation into possibilistic networks. The proposed approach is implemented via a possibilistic influence diagram toolbox (PIDT).     

Approaches to multistage one-shot decision making

In this research, multistage one-shot decision making under uncertainty is studied. In such a decision problem, a decision maker has one and only one chance to make a decision at each stage with possibilistic information. Based on the one-shot decision theory, approaches to multistage one-shot decision making are proposed. In the proposed approach, a decision maker chooses one state amongst all the states according to his/her attitude about satisfaction and possibility at each stage. The payoff at each stage is associated with the focus points at the succeeding stages. Based on the selected states (focus points), the sequence of optimal decisions is determined by dynamic programming. The proposed method is a fundamental alternative for multistage decision making under uncertainty because it is scenario-based instead of lottery-based as in the other existing methods. The one-shot optimal stopping problem is analyzed where a decision maker has only one chance to determine stopping or continuing at each stage. The theoretical results have been obtained.     

On the transformation between possibilistic logic bases and possibilistic causal networks

Possibilistic logic bases and possibilistic graphs are two different frameworks of interest for representing knowledge. The former ranks the pieces of knowledge (expressed by logical formulas) according to their level of certainty, while the latter exhibits relationships between variables. The two types of representation are semantically equivalent when they lead to the same possibility distribution (which rank-orders the possible interpretations). A possibility distribution can be decomposed using a chain rule which may be based on two different kinds of conditioning that exist in possibility theory (one based on the product in a numerical setting, one based on the minimum operation in a qualitative setting). These two types of conditioning induce two kinds of possibilistic graphs. This article deals with the links between the logical and the graphical frameworks in both numerical and quantitative settings. In both cases, a translation of these graphs into possibilistic bases is provided. The converse translation from a possibilistic knowledge base into a min-based graph is also described.     

Solving Large-Scale Fuzzy and Possibilistic Optimization Problems

Fuzzy and possibilistic optimization methods are demonstrated to be effective tools in solving large-scale problems. In particular, an optimization problem in radiation therapy with various orders of complexity from 1000 to 62,250 constraints for fuzzy and possibilistic linear and nonlinear programming implementations possessing (1) fuzzy or soft inequalities, (2) fuzzy right-hand side values, and (3) possibilistic right-hand side is used to demonstrate that fuzzy and possibilistic optimization methods are tractable and useful. We focus on the uncertainty in the right side of constraints which arises, in the context of the radiation therapy problem, from the fact that minimal and maximal radiation tolerances are ranges of values, with preferences within the range whose values are based on research results, empirical findings, and expert knowledge, rather than fixed real numbers. The results indicate that fuzzy/possibilistic optimization is a natural and effective way to model various types of optimization under uncertainty problems and that large fuzzy and possibilistic optimization problems can be solved efficiently.     

Group decision making methods based on intuitionistic fuzzy soft matrices

Soft set theory was originally proposed by Molodtsov as a general mathematical tool for dealing with uncertainty in 1999. Recently, researches of decision making based on soft sets have got some progress, but few people consider multi-experts situation. As such, this paper discusses multi-experts group decision making problems. Firstly, we give a concept of intuitionistic fuzzy soft matrix (IFSM) and prove some relevant properties of IFSM. Then, an adjustable approach is presented by means of median level soft set and p-quantile level soft set for dealing with decision making problems based on IFSM. Thirdly, we study aggregation methods of IFSM, give two kinds of aggregation operators and methods that how to determine experts’ weights under different situation with programming models, four corresponding algorithms have been proposed, too. Finally, a practical example has been demonstrated the reasonability and efficiency of these new algorithms.     

On the Evaluation of Uncertain Courses of Action

We consider the problem of decision making under uncertainty. The fuzzy measure is introduced as a general way of representing available information about the uncertainty. It is noted that generally in uncertain environments the problem of comparing alternative courses of action is difficult because of the multiplicity of possible outcomes for any action. One approach is to convert this multiplicity of possible of outcomes associated with an alternative into a single value using a valuation function. We describe various ways of providing a valuation function when the uncertainty is represented using a fuzzy measure. We then specialize these valuation functions to the cases of probabilistic and possibilistic uncertainty.     

Utility efficiency and its approximation

Summary We assume a multi-attribute decision making problem under uncertainty with partial information on the decision maker's preferences, by a vector utility function with two components and imprecision over their scaling constants. We propose an approximation set whose determination may be easier than the one of the utility efficient set and we consider an interactive procedure which uses such approximation to decision aid. We study some nesting and convergence properties based on the interactive reduction of the approximation set. Finally, we illustrate the procedure with a numerical example.     

A hybrid model for selection of an outsourcing provider

Outsourcing is a good strategy for firms that need to reduce operating costs and improve competitiveness and it is important that firms scientifically select appropriate outsourcing providers. Some efforts have been made to find systematic ways to deal with outsourcing problems, but these efforts incorrectly assumed that the criteria used in the decision process are independent, which is not true in the real world. In this study, we propose a new hybrid multiple criteria decision-making (MCDM) model, which addresses the dependent relationships between the various criteria. The relations-structure among the criteria is built with the aid of the decision-making trial and evaluation laboratory (DEMATEL) method. Decision-makers tend to hold diverse opinions about their preferences due to incomplete information and knowledge, or inherent conflict between various departments. We further used the fuzzy preference programming and the analytic network process (ANP) to form a model for the selection of partners for outsourcing providers. The proposed model can help practitioners improve their decision making process, especially when criteria are numerous and inter-related. The method is demonstrated using data from a Taiwanese airline.     

Decision-Making with Sugeno Integrals

This paper clarifies the connection between multiple criteria decision-making and decision under uncertainty in a qualitative setting relying on a finite value scale. While their mathematical formulations are very similar, the underlying assumptions differ and the latter problem turns out to be a special case of the former. Sugeno integrals are very general aggregation operations that can represent preference relations between uncertain acts or between multifactorial alternatives where attributes share the same totally ordered domain. This paper proposes a generalized form of the Sugeno integral that can cope with attributes having distinct domains via the use of qualitative utility functions. It is shown that in the case of decision under uncertainty, this model corresponds to state-dependent preferences on consequences of acts. Axiomatizations of the corresponding preference functionals are proposed in the cases where uncertainty is represented by possibility measures, by necessity measures, and by general order-preserving set-functions, respectively. This is achieved by weakening previously proposed axiom systems for Sugeno integrals.     

Kernel-based learning methods for preference aggregation

The mathematical representation of human preferences has been a subject of study for researchers in different fields. In multi-criteria decision making (MCDM) and fuzzy modeling, preference models are typically constructed by interacting with the human decision maker (DM). However, it is known that a DM often has difficulties to specify precise values for certain parameters of the model. He/she instead feels more comfortable to give holistic judgements for some of the alternatives. Inference and elicitation procedures then assist the DM to find a satisfactory model and to assess unjudged alternatives. In a related but more statistical way, machine learning algorithms can also infer preference models with similar setups and purposes, but here less interaction with the DM is required/allowed. In this article we discuss the main differences between both types of inference and, in particular, we present a hybrid approach that combines the best of both worlds. This approach consists of a very general kernel-based framework for constructing and inferring preference models. Additive models, for which interpretability is preserved, and utility models can be considered as special cases. Besides generality, important benefits of this approach are its robustness to noise and good scalability. We show in detail how this framework can be utilized to aggregate single-criterion outranking relations, resulting in a flexible class of preference models for which domain knowledge can be specified by a DM.      

International portfolio management with affine policies

While dynamic decision making has traditionally been represented as scenario trees, these may become severely intractable and difficult to compute with an increasing number of time periods. We present an alternative tractable approach to multiperiod international portfolio optimization based on an affine dependence between the decision variables and the past returns. Because local asset and currency returns are modeled separately, the original model is non-linear and non-convex. With the aid of robust optimization techniques, however, we develop a tractable semidefinite programming formulation of our model, where the uncertain returns are contained in an ellipsoidal uncertainty set. We add to our formulation the minimization of the worst case value-at-risk and show the close relationship with robust optimization. Numerical results demonstrate the potential gains from considering a dynamic multiperiod setting relative to a single stage approach.     

Eliciting and mapping qualitative preferences to numeric rankings in group decision making

Group work is becoming the norm in organizations. From strategy planning committees to quality management teams, organizational members are collaborating on problem solving. One area of team support that is often desired is the scoring and ranking of decision alternatives on qualitative/subjective domains, and the aggregation of individual preferences into group preferences. In this paper we present a new conceptual approach to qualitative preference elicitation and aggregation. This approach is based on well established decision analysis techniques. It significantly advances the state of the art of group decision making by addressing four common limitations: (1) the inability to deal with vagueness of human decision makers in articulating preferences; (2) difficulties in mapping qualitative evaluation to numeric estimates; (3) problems in aggregating individual preferences into meaningful group preference; and (4) the lack of simple user friendly techniques for dealing with a large number of decision alternatives. Our approach is easy to implement in stand alone personal computers and groupware. We illustrate this with a real-world problem.     

Dominance-based fuzzy rough set analysis of uncertain and possibilistic data tables

In this paper, we propose a dominance-based fuzzy rough set approach for the decision analysis of a preference-ordered uncertain or possibilistic data table, which is comprised of a finite set of objects described by a finite set of criteria. The domains of the criteria may have ordinal properties that express preference scales. In the proposed approach, we first compute the degree of dominance between any two objects based on their imprecise evaluations with respect to each criterion. This results in a valued dominance relation on the universe. Then, we define the degree of adherence to the dominance principle by every pair of objects and the degree of consistency of each object. The consistency degrees of all objects are aggregated to derive the quality of the classification, which we use to define the reducts of a data table. In addition, the upward and downward unions of decision classes are fuzzy subsets of the universe. Thus, the lower and upper approximations of the decision classes based on the valued dominance relation are fuzzy rough sets. By using the lower approximations of the decision classes, we can derive two types of decision rules that can be applied to new decision cases.     

A two-stage fuzzy robust integer programming approach for capacity planning of environmental management systems

In this study, a two-stage fuzzy robust integer programming (TFRIP) method has been developed for planning environmental management systems under uncertainty. This approach integrates techniques of robust programming and two-stage stochastic programming within a mixed integer linear programming framework. It can facilitate dynamic analysis of capacity-expansion planning for waste management facilities within a multi-stage context. In the modeling formulation, uncertainties can be presented in terms of both possibilistic and probabilistic distributions, such that robustness of the optimization process could be enhanced. In its solution process, the fuzzy decision space is delimited into a more robust one by specifying the uncertainties through dimensional enlargement of the original fuzzy constraints. The TFRIP method is applied to a case study of long-term waste-management planning under uncertainty. The generated solutions for continuous and binary variables can provide desired waste-flow-allocation and capacity-expansion plans with a minimized system cost and a maximized system feasibility.     

Modeling preferences and conditional preferences on resource consumption and production in ASP

In this paper, we extend our previous work on Resourced ASP, or for short RASP, where we have introduced the possibility of defining and using resources in ASP. In RASP, one can define resources with their amounts, where available resources can be used for producing other resources and the remaining amount, if any, can be used in a different way. In this paper, we introduce P-RASP (RASP with Preferences) where it is possible to express preferences about which resources should be either consumed or produced. Moreover, conditional preferences, of different forms, allow one to express preferences according to certain conditions, that are to be evaluated “dynamically”, namely, with respect to the specific answer set at hand. The semantic of conditional preferences is given in terms of (non-conditional) preferences, though the translation is not straightforward and thus the new features are not syntactic sugar. Complexity of P-RASP is also discussed.     

基于双边犹豫模糊非均衡语言优先集成算子的应急响应预案评估方法

在应用多属性决策理论求解应急响应预案评估问题时,问题结构的复杂性往往使决策者评价信息存在高度不确定性且属性相对重要性仅能以优先级关系来表征。为此,本文首先提出了双边犹豫模糊非均衡语言集这种新型信息形式以使决策者能够灵活有效的表征复杂评价信息,并定义了运算法则、熵和距离测度;其次,基于熵测度开发了双边犹豫模糊非均衡语言优先加权集成算子,并构建了能够考虑属性优先关系的多属性决策方法;进一步针对属性相对重要性不能由定性分析获得的情况,设计了客观权重确定方法,并构建了另一种更具实际灵活性的VIKOR决策方法;最后,实例研究表明了方法的有效性与优势。     

Genetic algorithms in logic tree decision modeling

An important approach to decision modeling is the induction of knowledge structures—such as rules, trees, and graphs—from empirical data describing previous conditions and the resulting decisions. We examine here a specific knowledge structure, a logic tree, in which the conditions are leaves, the decision is the root, and the intermediate nodes are logical operators. We then use genetic algorithms (GAs) to construct logic trees that best represent the correspondence between conditions and decisions described by the empirical data. We also investigate an important characteristic of the GA search, the fitness distance correlation. Finally, we comment on the usefulness of GAs in knowledge modeling.