Confidence structures in decision making

Decision making is defined in terms of four elements: the set of decisions, the set of outcomes for each decision, a set-valued criterion function, and the decision maker's value judgment for each outcome. Various confidence structures are defined, which give the decision maker's confidence of a given decision leading to a particular outcome. The relation of certain confidence structures to Bayesian decision making and to membership functions in fuzzy set theory is established. A number of schemes are discussed for arriving atbest decisions, and some new types of domination structures are introduced.This research was partly supported by Project No. NR-047-021, ONR Contract No. N-00014-75-C-0569 with the Center for Cybernetic Studies, The University of Texas, Austin, Texas, and by ONR Contract No. N-00014-69-A-0200-1012 with the University of California, Berkeley, California.     

Multiple criteria decision problems with fuzzy domination structures

The concepts of domination structures and nondominated solutions in multiple criteria decision problems, which were introduced by Yu, enable us to tackle general situations in which there exists information concerning the decision maker's preferences.In many of the multiple criteria decision problems the underlying domination structures are not known precisely but only fuzzily determined. Yu primarily works with the case where the domination structure at each point is a convex cone. As a result, there exists a sharp borderline dividing all solutions into nondominated solutions and the others.This paper fuzzifies the concepts of domination structures and nondominated solutions to allow them to be applied to a larger class of the multiple criteria decision problems mentioned above. Introducing the concepts of fuzzy convex cones and fuzzy polar cones, it is shown how some of the main results obtained by Yu are extended.     

Stability of nondominated solutions in multicriteria decision-making

Decision-making problems with multiple noncommensurable objectives are specified by two factors, i.e., the set of all feasible solutions and the domination structure. The solutions are characterized as nondominated points. Hence, in these problems, there may exist two parameter vectors, according to which the above two factors change. The stability of the solution set for perturbations of these parameters is investigated in this paper. The analysis is guided by using the concept of continuity of the solution map defined on the two parameter spaces.     

Fuzzy set based models and methods of multicriteria group decision making

In the paper, the term consensus scheme is utilized to denote a dynamic and iterative process where the experts involved discuss a multicriteria decision problem. This discussion process is conducted by a human or artificial moderator, with the purpose of minimizing the discrepancy between the individual opinions.During the process of decision making, each expert involved must provide preference information. The information format and the circumstances where it must be given play a critical role in the decision process. This paper analyses a generic consensus scheme, which considers many different preference input formats, several possible interventions of the moderator, as well as admitting several stop conditions for interrupting the discussion process. In addition, a new consensus scheme is proposed with the intention of eliminating some difficulties met when the traditional consensus schemes are utilized in real applications. It preserves the experts’ integrity through the intervention of an external person, to supervise and mediate the conflicting situations. The human moderator is supposed to interfere in the discussion process by adjusting some parameters of the mathematical model or by inviting an expert to update his opinion. The usefulness of this consensus scheme is demonstrated by its use to solve a multicriteria group decision problem, generated applying the Balanced Scorecard methodology for enterprise strategy planning. In the illustrating problem, the experts are allowed to give their preferences in different input formats. But the information provided is made uniform on the basis of fuzzy preference relations through the use of adequate transformation functions, before being analyzed. The advantage of using fuzzy set theory for solving multiperson multicriteria decision problems lies in the fact that it can provide the flexibility needed to adequately deal with the uncertain factors intrinsic to such problems.     

Necessary and sufficient conditions for local and global nondominated solutions in decision problems with multi-objectives

In this paper, the decision problem with multi-objectives is considered, and the nondominated solutions associated with a polyhedral domination cone are discussed. The necessary and sufficient conditions for the solutions are given in the decision space rather than the objective space. The similarity of the solution conditions obtained in this article to the Kuhn-Tucker condition of a convex programming problem is examined. As an application of the solution condition, an algorithm to locate the set of all nondominated solutions is shown for the linear multi-objective decision problem.The author would like to thank the reviewer for his helpful comments.     

Necessary conditions of nondominated solutions in multicriteria decision making

A partial integrodifferential equation is studied in which the derivatives of highest order also contain a discrete and a distributed delay. By means of abstract regularity results, global existence and uniqueness of a strict solution are obtained; moreover a characterization of the infinitesimal generator of the solution operator is given.     

Modeling subjective evaluation for fuzzy group multicriteria decision making

This paper presents a new fuzzy multicriteria decision making (MCDM) approach for evaluating decision alternatives involving subjective judgements made by a group of decision makers. A pairwise comparison process is used to help individual decision makers make comparative judgements, and a linguistic rating method is used for making absolute judgements. A hierarchical weighting method is developed to assess the weights of a large number of evaluation criteria by pairwise comparisons. To reflect the inherent imprecision of subjective judgements, individual assessments are aggregated as a group assessment using triangular fuzzy numbers. To obtain a cardinal preference value for each decision alternative, a new fuzzy MCDM algorithm is developed by extending the concept of the degree of optimality to incorporate criteria weights in the distance measurement. An empirical study of aircraft selection is presented to illustrate the effectiveness of the approach.     

Quasiconcavity and nondominated solutions in multiobjective programming

In the multiobjective programming literature, the concavity of the objectives is usually assumed to be a sufficient condition in seeking Pareto-optimal solutions. This paper investigates nondominated solutions associated with dominance cones via the assumption of the quasiconcavity of the objectives. Necessary as well as sufficient conditions for such quasiconcave multiobjective programming problems are obtained.The author is indebted to one of the referees for detailed constructive comments and suggestions. Thanks are also due to the late Professor Abraham Charnes, University of Texas at Austin, and Professor Zhimin Huang, Adelphi University.     

Orderings based on the banks set: Some new scoring methods for multicriteria decision making

This article introduces new methods for ranking alternatives in multicriteria decision making situations. Each is based on the normative position that the strength of an alternative is inversely related to the number of alternatives that could prevent it from being chosen. The scores discriminate among elements of the Banks set (Banks, Soc Choice Welfare, 1985, 1, 295–306). The new scoring methods are compared to traditional scoring methods and related to the amount of intransitivity (specifically, the size of the top‐cycle) of aggregated preference. The new scores are shown to measure important aspects of alternatives not captured by extant scoring methods and are illustrated in collective choice settings. © 2014 Wiley Periodicals, Inc. Complexity 20: 63–76, 2015     

Theory of convex cones in multicriteria decision making

We develop the theory of convex polyhedral cones in the objective-function space of a multicriteria decision problem. The convex cones are obtained from the decision-maker's pairwise judgments of decision alternatives and are applicable to any quasiconcave utility function. Therefore, the cones can be used in any progressively articulated solution procedure that employs pairwise comparisons. The cones represent convex sets of solutions that are inferior to known solutions to a multicriteria problem. Therefore, these convex sets can be eliminated from consideration while solving the problem. We develop the underlying theory and a framework for representing knowledge about the decision-maker's preference structure using convex cones. This framework can be adopted in the interactive solution of any multicriteria problem after taking into account the characteristics of the problem and the solution procedure. Our computational experience with different multicriteria problems shows that this approach is both viable and efficient in solving practical problems of moderate size.     

Indefinite preference structures and decision analysis

It always takes time for people to digest information and make judgments. The decision maker's preference is not always clear and stable when decision analysis and decision making are performed. In this paper, we introduce a generalized preference structure to cope with indefinite preferences. We describe its general properties, its implication on value function representation, its solution concepts, and methods for obtaining the solutions.This research has been partially supported by NSF Grant No. IST-84-18863. The authors are grateful to Dr. D. J. White for his helpful comments on a previous draft.     

Decision making in multiobjective optimization aided by the multicriteria tournament decision method

This paper proposes a new method for multicriteria analysis, named Multicriteria Tournament Decision (MTD). It provides the ranking of alternatives from best to worst, according to the preferences of a human decision-maker (DM). It has some positive aspects such as: it has a simple algorithm with intuitive appeal; it involves few input parameters (just the importance weight of each criterion).The helpfulness of MTD is demonstrated by using it to select the final solution of multiobjective optimization problems in an a posteriori decision making approach. Having at hand a discrete approximation of the Pareto front (provided by a multiobjective evolutionary search algorithm), the choice of the preferred Pareto-optimal solution is performed using MTD.A simple method, named Gain Analysis method (GAM), for verifying the existence of a better solution (a solution associated to higher marginal rates of return) than the one originally chosen by the DM, is also introduced here. The usefulness of MTD and GAM methods is confirmed by the suitable results shown in this paper.     

Energy planning: a multi-level and multicriteria decision making structure proposal

Energy planning is a complex issue because of its multi-scale and multicriteria issues. In this contribution, a prospective analysis on the development of a multi-level and multicriteria decision-making structure dedicated to energy planning is developed. This analysis is based on an examination of the specificity of the energy supply chain as well as a state of the art aimed at classifying the issues solved according to the time horizon and size of the geographical area under consideration (models developed, the degree of accuracy of the information used, criteria taken into account).     

Connectedness of the set of nondominated outcomes in multicriteria optimization

Sufficient conditions for the set of nondominated outcomes of a multicriteria optimization problem to be connected are given.The author would like to thank Professors G. Leitmann and P. L. Yu for many critical remarks.The contents of this article are taken from the author's doctoral dissertation written at the University of California at Berkeley under the supervision of Professor G. Leitmann.     

Strong equality of domination parameters in trees

We study the concept of strong equality of domination parameters. Let P₁ and P₂ be properties of vertex subsets of a graph, and assume that every subset of V(G) with property P₂ also has property P₁. Let ψ₁(G) and ψ₂(G), respectively, denote the minimum cardinalities of sets with properties P₁ and P₂, respectively. Then ψ₁(G)ψ₂(G). If ψ₁(G)=ψ₂(G) and every ψ₁(G)-set is also a ψ₂(G)-set, then we say ψ₁(G) strongly equals ψ₂(G), written ψ₁(G)≡ψ₂(G). We provide a constructive characterization of the trees T such that γ(T)≡i(T), where γ(T) and i(T) are the domination and independent domination numbers, respectively. A constructive characterization of the trees T for which γ(T)=γ_t(T), where γ_t(T) denotes the total domination number of T, is also presented.     

Characterizations of trees with equal paired and double domination numbers

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, and a double dominating set is a dominating set that dominates every vertex of G at least twice. We show that for trees, the paired-domination number is less than or equal to the double domination number, solving a conjecture of Chellali and Haynes. Then we characterize the trees having equal paired and double domination numbers.     

An intelligent-agent-based fuzzy group decision making model for financial multicriteria decision support: The case of credit scoring

Credit risk analysis is an active research area in financial risk management and credit scoring is one of the key analytical techniques in credit risk evaluation. In this study, a novel intelligent-agent-based fuzzy group decision making (GDM) model is proposed as an effective multicriteria decision analysis (MCDA) tool for credit risk evaluation. In this proposed model, some artificial intelligent techniques, which are used as intelligent agents, are first used to analyze and evaluate the risk levels of credit applicants over a set of pre-defined criteria. Then these evaluation results, generated by different intelligent agents, are fuzzified into some fuzzy opinions on credit risk level of applicants. Finally, these fuzzification opinions are aggregated into a group consensus and meantime the fuzzy aggregated consensus is defuzzified into a crisp aggregated value to support final decision for decision-makers of credit-granting institutions. For illustration and verification purposes, a simple numerical example and three real-world credit application approval datasets are presented.     

Second-order optimality conditions for nondominated solutions of multiobjective programming with C 1,1 data

We examine new second-order necessary conditions and sufficient conditions which characterize nondominated solutions of a generalized constrained multiobjective programming problem. The vector-valued criterion function as well as constraint functions are supposed to be from the class C ^1,1. Second-order optimality conditions for local Pareto solutions are derived as a special case.     

Using interdependent information on the relative importance of criteria in decision making

A multicriteria choice problem is considered. The Edgeworth-Pareto principle is established under the assumption that certain axioms hold true. Quantitative interdependent information on the relative importance of two groups of criteria is used to derive upper bounds for the unknown set of selected vectors.