Variable preference modeling with ideal-symmetric convex cones

Based on the concept of general domination structures, this paper presents an approach to model variable preferences for multicriteria optimization and decision making problems. The preference assumptions for using a constant convex cone are given, and, in remedy of some immanent model limitations, a new set of assumptions is presented. The underlying preference model is derived as a variable domination structure that is defined by a collection of ideal-symmetric convex cones. Necessary and sufficient conditions for nondominance are established, and the problem of finding corresponding nondominated solutions is addressed and solved on examples.     

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).     

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.     

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.     

Methods of support cones and simplices in convex programming and their applications to physicochemical systems

A variant of the embedding technique proposed earlier by the second author is suggested in which the sets to be embedded are support cones. Replacing the cones by simplices gives a modification with a polynomial convergence rate.     

Multicriteria analysis in decision making under information uncertainty

This paper presents results of research related to multicriteria decision making under information uncertainty. The Bellman–Zadeh approach to decision making in a fuzzy environment is utilized for analyzing multicriteria optimization models (X,M models) under deterministic information. Its application conforms to the principle of guaranteed result and provides constructive lines in obtaining harmonious solutions on the basis of analyzing associated maxmin problems. This circumstance permits one to generalize the classic approach to considering the uncertainty of quantitative information (based on constructing and analyzing payoff matrices reflecting effects which can be obtained for different combinations of solution alternatives and the so-called states of nature) in monocriteria decision making to multicriteria problems. Considering that the uncertainty of information can produce considerable decision uncertainty regions, the resolving capacity of this generalization does not always permit one to obtain unique solutions. Taking this into account, a proposed general scheme of multicriteria decision making under information uncertainty also includes the construction and analysis of the so-called X,R models (which contain fuzzy preference relations as criteria of optimality) as a means for the subsequent contraction of the decision uncertainty regions. The paper results are of a universal character and are illustrated by a simple example.     

A uniform boundedness theorem for locally convex cones

We prove a uniform boundedness theorem for families of linear operators on ordered cones. Using the concept of locally convex cones we introduce the notions of barreled cones and of weak cone-completeness. Our main result, though no straightforward generalization of the classical case, implies the Uniform Boundedness Theorem for Fréchet spaces.

     

MCS—A new algorithm for multicriteria optimisation in constraint programming

In this paper we propose a new algorithm called MCS for the search for solutions to multicriteria combinatorial optimisation problems. To quickly produce a solution that offers a good trade-off between criteria, the MCS algorithm alternates several Branch & Bound searches following diversified search strategies. It is implemented in CP in a dedicated framework and can be specialised for either complete or partial search.     

Developments in fuzzy multicriteria analysis

Selecting or ranking available alternatives (observations/objects) with respect to multiple, often conflicting criteria in a fuzzy environment usually referred to as fuzzy multicriteria analysis is a problem of a major interest in information and engineering. Methodologies for addressing this problem have been developed from a variety of research disciplines, including statistics, econometrics, artificial intelligent, and operations research. This paper presents an overview of the developments in fuzzy multicriteria analysis. It discusses the complexity of fuzzy multicriteria analysis and analyses the existing approaches from four different perspectives for facilitating a better understanding of the recent development in this domain. Finally, the paper elaborates on the future research areas in fuzzy multicriteria analysis.     

Value functions,domination cones and proper efficiency in multicriteria optimization

In the absence of a clear objective value function, it is still possible in many cases to construct a domination cone according to which efficient (nondominated) solutions can be found. The relations between value functions and domination cones and between efficiency and optimality are analyzed here. We show that such cones must be convex, strictly supported and, frequently, closed as well. Furthermore, in most applications potential optimal solutions are equivalent to properly efficient points. These solutions can often be produced by maximizing with respect to a class of concave functions or, under convexity conditions, a class of affine functions.     

Locally convex quotient lattice cones

Walter Roth has investigated certain equivalence relations on locally convex cones in [W. Roth, Locally convex quotient cones, J. Convex Anal. 18, No. 4, 903–913 (2011)] which give rise to the definition of a locally convex quotient cone. In this paper, we investigate some special equivalence relations on a locally convex lattice cone by which the locally convex quotient cone becomes a lattice. In the case of a locally convex solid Riesz space, this reduces to the known concept of locally convex solid quotient Riesz space. We prove that the strict inductive limit of locally convex lattice cones is a locally convex lattice cone. We also study the concept of locally convex complete quotient lattice cones.     

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.     

Aggregation of fuzzy preference relations to multicriteria decision making

Weighted aggregation of fuzzy preference relations on the set of alternatives by several criteria in decision-making problems is considered. Pairwise comparisons with respect to importance of the criteria are given in fuzzy preference relation as well. The aggregation procedure uses the composition between each two relations of the alternatives. The membership function of the newly constructed fuzzy preference relation includes t-norms and t-conorms to take into account the relation between the criteria importance. Properties of the composition and new relation, giving a possibility to make a consistent choice or to rank the alternatives, are proved. An illustrative numerical study and comparative examples are presented.     

Bases of convex cones and Borwein's proper efficiency

In this note, we establish some interesting relationships between the existence of Borwein's proper efficient points and the existence of bases for convex ordering cones in normed linear spaces. We show that, if the closed unit ball in a smooth normed space ordered by a convex cone possesses a proper efficient point in the sense of Borwein, then the ordering cone is based. In particular, a convex ordering cone in a reflexive space is based if the closed unit ball possesses a proper efficient point. Conversely, we show that, in any ordered normed space, if the ordering cone has a base, then every weakly compact set possesses a proper efficient point.The research was conducted while the author was working on his PhD Degree under the supervision of Professor J. M. Borwein, whose guidance and valuable suggestions are gratefully appreciated. The author would like to thank two anonymous referees for their constructive comments and suggestions. This research was supported by an NSERC grant and a Mount Saint Vincent University Research Grant.     

Generalization of domination structures and nondominated solutions in multicriteria decision making

The concepts of domination structures and nondominated solutions are important in tackling multicriteria decision problems. We relax Yu's requirement that the domination structure at each point of the criteria space be a convex cone (Ref. 1) and give results concerning the set of nondominated solutions for the case where the domination structure at each point is a convex set. A practical necessity for such a generalization is discussed. We also present conditions under which a locally nondominated solution is also a globally nondominated solution.     

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.     

Onn-person noncooperative multicriteria games described in strategic form

This paper undertakes the problem of multicriteria decision support in conflict situations described as a noncooperative game. Construction of such a decision support requires the development of the noncooperative game theory to be generalized for the multicriteria case. New theoretical results in this case are presented. Features of the multicriteria noncooperative games are shown with use of a duopoly model in case of two goods and two criteria of each player. Concepts of the decision support are discussed.     

Fuzzy preference orderings in group decision making

In this paper, some use of fuzzy preference orderings in group decision making is discussed. First, fuzzy preference orderings are defined as fuzzy binary relations satisfying reciprocity and max-min transitivity. Then, particularly in the case where individual preferences are represented by utility functions (utility values), group fuzzy preference orderings of which fuzziness is caused by differences or diversity of individual opinions are defined. Those orderings might be useful for proceeding the group decision making process smoothly, in the same manner as the extended contributive rule method.