Matrices.- Solvability of systems of interval linear equations and inequalities.- Interval linear programming.- Linear programming with set coefficients.- Fuzzy linear optimization.- Interval linear systems and optimization problems over max-algebras.

Linear Optimization Problems with Inexact Data

Soft Computing (SC) represents a significant paradigm shift in the aims of computing, which reflects the fact that the human mind, unlike present day computers, possesses a remarkable ability to store and process information which is pervasively imprecise, uncertain and lacking in categoricity. At this juncture, the principal constituents of Soft Computing (SC) are: Fuzzy Systems (FS), including Fuzzy Logic (FL); Evolutionary Computation (EC), including Genetic Algorithms (GA); Neural Networks (NN), including Neural Computing (NC); Machine Learning (ML); and Probabilistic Reasoning (PR). In this work, we focus on fuzzy methodologies and fuzzy systems, as they bring basic ideas to other SC methodologies. The other constituents of SC are also briefly surveyed here but for details we refer to the existing vast literature. In Part 1 we present an overview of developments in the individual parts of SC. For each constituent of SC we overview its background, main problems, methodologies and recent developments. We focus mainly on Fuzzy Systems, for which the main literature, main professional journals and other relevant information is also supplied. The other constituencies of SC are reviewed shortly. In Part 2 we investigate some fuzzy optimization systems. First, we investigate Fuzzy Sets we define fuzzy sets within the classical set theory by nested families of sets, and discuss how this concept is related to the usual definition by membership functions. Further, we will bring some important applications of the theory based on generalizations of concave functions. We study a decision problem, i.e. the problem to find a ”best” decision in the set of feasible alternatives with respect to several (i.e. more than one) criteria functions. Within the framework of such a decision situation, we deal with the existence and mutual relationships of three kinds of ”optimal decisions”: Weak Pareto-Maximizers, Pareto-Maximizers and Strong Pareto-Maximizers particular alternatives satisfying some natural and rational conditions. We also study the compromise decisions maximizing some aggregation of the criteria. The criteria considered here will be functions defined on the set of feasible alternatives with the values in the unit interval. In Fuzzy mathematical programming problems (FMP) the values of the objective function describe effects from choices of the alternatives. Among others we show that the class of all MP problems with (crisp) parameters can be naturally embedded into the class of FMP problems with fuzzy parameters. Finally, we deal with a class of fuzzy linear programming problems. We show that the class of crisp (classical) LP problems can be embedded into the class of FLP ones. Moreover, for FLP problems we define the concept of duality and prove the weak and strong duality theorems. Further, we investigate special classes of FLP interval LP problems, flexible LP problems, LP problems with interactive coefficients and LP problems with centered coefficients. We present here an original mathematically oriented and unified approach.

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Soft Computing: Overview and Recent Developments in Fuzzy Optimization

A fuzzy preference framework is developed within the paradigm of the graph model for conflict resolution. This framework takes into account both certain and uncertain information about the preferences of decision makers (DMs) involved in a strategic conflict. The graph model is a solution methodology for conflict decision making that begins with a model of the problem and suggests possible resolutions through a number of stability definitions. Four basic fuzzy stability definitions are introduced for a two-DM graph model to analyze conflict behavior and identify possible resolutions even when preferences are fuzzy. Fuzzy stability definitions describe varied human behavior in a conflict model; a state is fuzzy stable for a DM according to a specific fuzzy stability definition if a move to any other state, evaluated according to that definition, does not meet the DM's fuzzy satisficing threshold. A state that is fuzzy stable for all DMs under a specific fuzzy stability definition constitutes a fuzzy equilibrium under that definition, and is interpreted as a possible resolution of the conflict. Fuzzy stability definitions include fuzzy Nash stability, fuzzy general metarationality, fuzzy symmetric metarationality, and fuzzy sequential stability.

Fuzzy preferences in a two-decision maker graph model

A signed network analysis-based Consensus Reaching Process in Group Decision Making

In this work we present a construction method for interval-valued fuzzy preference relations from a fuzzy preference relation and the representation of the lack of knowledge or ignorance that experts suffer when they define the membership values of the elements of that fuzzy preference relationWe also prove that, with this construction method, we obtain membership intervals for an element which length is equal to the ignorance associated with that element We then propose a generalization of Orlovsky’s non dominance method to solve decision making problems using interval-valued fuzzy preference relations

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Construction of Interval-Valued Fuzzy Preference Relations Using Ignorance Functions: Interval-Valued Non Dominance Criterion

A general decision-making problem is considered in which a set of alternatives and a non-strict preference relation specified in this set are of fuzzy nature. The corresponding fuzzy equivalence and strict preference relations are defined which allow to introduce a fuzzy set of non-dominated solutions. It is shown that under some weak topological conditions unfuzzy non-dominated solutions to the problem exist. The questions of equivalence of these solutions are studied and a method of calculating these solutions is suggested.

S. A. Orlovsky

Papers

Decision-making With a Fuzzy Preference Relation in a Fuzzy Set of Alternatives