Fuzzy Mathematical Programming Problems Research Articles

Optimization under fuzzy if–then rules

The aim of this paper is to introduce a novel statement of fuzzy mathematical programming problems and to provide a method for finding a fair solution to these problems. Suppose we are given a mathematical programming problem in which the functional relationship between the decision variables and the objective function is not completely known. Our knowledge-base consists of a block of fuzzy if–then rules, where the antecedent part of the rules contains some linguistic values of the decision variables, and the consequence part consists of a linguistic value of the objective function. We suggest the use of Tsukamoto's fuzzy reasoning method to determine the crisp functional relationship between the objective function and the decision variables, and solve the resulting (usually nonlinear) programming problem to find a fair optimal solution to the original fuzzy problem.

Multiobjective linguistic optimization

Generalizing our earlier results on optimization with linguistic variables we introduce a novel statement of fuzzy multiobjective mathematical programming problems and provide a method for finding a fair solution to these problems. Suppose we are given a multiobjective mathematical programming problem in which the functional relationship between the decision variables and the objective functions is not completely known. Our knowledge-base consists of a block of fuzzy if–then rules, where the antecedent part of the rules contains some linguistic values of the decision variables, and the consequence part consists of a linguistic value of the objective functions. We suggest the use of Tsukamoto's fuzzy reasoning method to determine the crisp functional relationship between the decision variables and objective functions. We model the anding of the objective functions by a t-norm and solve the resulting (usually nonlinear) programming problem to find a fair optimal solution to the original fuzzy multiobjective problem.

Using ranking functions in multiobjective fuzzy linear programming

Multiobjective mathematical programming problems, in particular the vector optimization problems, define a well known and studied area because of its relevance to numerous practical applications. In this paper vector optimization problems with a fuzzy nature are considered. In these problems usually it is assumed that all the objective functions involved come from the same decision maker. The problem considered here assumes, however, that the objective functions can be defined by different decision makers, and that the coefficients in each of these objective functions are fuzzy numbers. Hence, solution methodologies for these multiobjective fuzzy mathematical programming problems, using different ordering methods ranking fuzzy numbers, are proposed. As an illustration a bi-objective model for land use is presented.

User oriented ranking criteria and its application to fuzzy mathematical programming problems

In this paper, we propose two kinds of ranking criteria for fuzzy numbers, and a simple and systematical method for producing these criteria. We show that any ranking proposed by Dubois is one of the rankings of each kind. That is, we show that the proposed rankings of each kind are generalization of the rankings proposed by Dubois. We also show the application of that to a fuzzy mathematical programming problem (FMP) with imprecise resources and technological coefficients. In the proposed method, a DM can easily select ranking criteria which can reflect the DM's subtle request. A DM also can have clearly the information about selected ranking criteria although so far in many cases a DM was difficult to know the meaning of using ranking criteria. Hence, when applied to an FMP with imprecise resources and technological coefficients, a DM can easily set the way of solving the FMP approximately, and also have clearly the information how the FMP is solved approximately. This information is very useful especially when a DM makes decision by using the solution of the FMP.

Fuzzy mathematical programming based on some new inequality relations

This paper deals with inequality relations in fuzzy mathematical programming problem (FMP) not necessarily linear. Moreover, fuzzy parameters may have nonlinear membership functions. A new approach for comparing fuzzy sets is proposed, which is more general than the well known proposals in the literature.

On fuzzy stochastic optimization

This paper extends the applicability of optimization models to situations where both fuzzy and random data are in the state of affairs. The major mathematical tool used to this end is the concept of fuzzy random variables, which provides a synergistic way for dealing with the two kinds of imprecision. The extension principle is canonically extended to fuzzy random variables and exploited to derive a way for handling robust fuzzy stochastic mathematical programming problems. Finally, we provide some possible avenues for further fruitful developments in this field.

Flexible generator maintenance scheduling based on subjective relaxation of constraints

AbstractThis paper presents a fuzzy‐based method for determining a flexible generator maintenance scheduling by means of subjective relaxation of constraints imposed on the maintenance scheduling problem. The constraints are divided into hard (crisp) constraint set and soft (fuzzy) constraint set according to reflecting conditions which surround power systems. The problem is formulated as a fuzzy mathematical programming problem and solved with the fuzzy branch‐and‐bound method using Bellman‐Zadeh maximizing decision. The proposed approach provides not only a new flexible concept of planning problems in power systems, but also natural expansion of conventional approaches based on crisp set theory. The effectiveness and feasibility of the proposed approach are demonstrated on two typical power system models which consist of 15 generators and 60 generators, respectively.

A cooperative fuzzy game theoretic approach to multiple objective design optimization

The utility and applicability of cooperative game theory in an engineering design process is examined. It is shown how game theory may be used as a tool for solving multiple objective optimization (MOO) problems. The concepts in cooperative game theory and fuzzy set theory are combined to yield a new optimization method referred to herein as cooperative fuzzy games. The concept of cooperative fuzzy games can be applied to solve not only well- and ill-structured single and MOO problems, but also preliminary decision making and design problems where only a feasible solution is sought and no objective functions are specified. A completely general formulation capable of solving decision making problems with partly crisp and partly fuzzy objective functions, as well as partly crisp and partly fuzzy constraints is presented. It is shown that existing techniques for solving crisp and fuzzy mathematical programming problems are special cases of this general formulation. The computational procedure is illustrated via an application to a MOO problem dealing with the design of high speed mechanisms.

制約の主観的な緩和に基づく柔軟な電源補修計画

This paper presents a fuzzy-based method for determining a flexible generator maintenance scheduling by means of subjective relaxation of constraints imposed on the maintenance scheduling problem. The constraints are divided into hard (crisp) constraint set and soft (fuzzy) constraint set according to reflecting conditions which surround power systems. The problem is formulated as a fuzzy mathematical programming problem and solved with the fuzzy branch-and-bound method using Bellman-Zadeh maximizing decision. The proposed approach provides not only a new flexible concept of planning problems in power systems, but also natural expansion of conventional approaches based on crisp set theory. The effectiveness and feasibility of the proposed approach are demonstrated on two typical power system models which consist of 15 generators and 60 generators, respectively.

A generalized approach to fuzzy optimization

A new approach to fuzzy optimization based on the generalization of Bellman-Zadeh's (BZ) concept is proposed in this article. It consists of a parametric generalization of intersection of fuzzy sets and a generalized defuzzification method. This approach allows the solving of a fuzzy mathematical programming (FMP) problem without transformation to a crisp one. It takes into account all possible fuzzy decisions and allows the degree of conjunction of criteria and constraints to vary. BZ method can be considered a special case of the approach proposed here. A simple algorithm for noniterative solving FMP problem is proposed whereas well-known Zimmermann's approach uses numerical methods. an illustrative example is presented. © 1994 John Wiley & Sons, Inc.

Fuzzy reasoning for solving fuzzy mathematical programming problems

We interpret fuzzy linear programming (FLP) problems with fuzzy coefficients and fuzzy inequality relations as multiple fuzzy reasoning schemes (MFR), where the antecedents of the scheme correspond to the constraints of the FLP problem and the fact of the scheme is the objective of the FLP problem. Then the solution process consists of two steps: first, for every decision variable x ϵ R n , we compute the (fuzzy) value of the objective function, MAX( x), via sup-min convolution of the antecedents/constraints and the fact/objective. Then an (optimal) solution to FLP problem is any point which produces a maximal element of the set {MAX(x)|x ϵ R n} (in the sense of the given inequality relation). We show that our solution process for a classical (crisp) LP problem results in a solution in the classical sense, and (under well-chosen inequality relations and objective function) coincides with those suggested by Buckley [Fuzzy Sets and Systems 31 (1989) 329–341], Delgado et al. [Control and Cybernetics 16 (1987) 114–121, Fuzzy sets and Systems 26 (1988) 49–62], Negoita [Fuzzy Sets and Systems 6 (1981) 261–269], Ramik and Rimanek [Fuzzy Sets and Systems 16 (1985) 123–138], Verdegay [Fuzzy Information and Decision Processes (North-Holland, Amsterdam, 1982), Control and Cybernetics 13 (1984) 230–239] and Zimmermann [Int. J. General Systems 2 (1976) 209–215]. Furthermore, we show how to extend the proposed solution principle to non-linear programming problems with fuzzy coefficients. We illustrate our approach by some simple examples.

Optimality for fuzzified mathematical programming problems: A parametric approach

From a conventional mathematical programming model, and in accordance with which fuzzification is used, several models of fuzzy mathematical programming problems can be obtained. This paper deals with the study of the optimality concept for ( g, p)-fuzzified mathematical programming problems. An auxiliary parametric mathematical programming problem is presented which allows the above model to be solved in a straightforward way. In addition, some results about the ( g, p)-fuzzified mathematical programming problem are obtained using the parametric mathematical programming problem.

Modality constrained programming problems: A unified approach to fuzzy mathematical programming problems in the setting of possibility theory

In this paper, fuzzy mathematical programming problems are formulated based on the idea analogous with the chance constrained programming problem. The difference in meaning between the ambiguity of the coefficients and that of the decision maker's preference is emphasized. The constraints with fuzzy coefficients are treated as the restriction that should be satisfied properly rather than perfectly. The objective functions with fuzzy coefficients are treated variously depending on the interpretations, i.e., the optimization of the modalities, the optimization of the fractile, the minimization of the ambiguity, and so forth. The deterministic equivalent constraints and the deterministic equivalent problems are shown when the constraints and the objective functions are linear. A numerical example is given to illustrate the proposed formulations.

Relationships between modality constrained programming problems and various fuzzy mathematical programming problems

Abstract In this paper, we focus on the relationships between modality constrained programming problems and various fuzzy mathematical programming problems, i.e., flexible programming problem, fuzzy goal programming problem and fuzzy mathematical programming problems proposed by Orlovski, Tanaka et al., Inuiguchi et al., Dubois and Sakawa et al. Fuzzy mathematical programming problems discussed in this paper can be interpreted in the framework of modality constrained programming problems. Thus, modality constrained programming problems can be regarded as a base model for unifying various fuzzy mathematical programming problems.

Modality Constrained Programming Problems Introduced Gödel Implication and Various Fuzzy Mathematical Programming Problems

The modality constrained programming (MCP) problem is previously proposed as the base model for unifying various fuzzy mathematical programming (FMP) problems. The relationships between MCP problems and six kinds of FMP problems have been discussed. However, some FMP problems, such as robust programming problem, FMP problem using fuzzy max, etc., did not have the relationships to MCP problems.In this paper, introducing the Gödel implication into modality measures, a fuzzy relation between elements is extended to eight kinds of relations between possibility distributions. MCP problems are reformulated using these eight kinds of relations. The relationships between reformulated MCP problems and four kinds of FMP problems, i. e., robust programming problem, FMP problem using fuzzy max, FMP problem proposed by Luhandjula and modality goal programming problem, are investigated. Itis emphasized that most of FMP problems are interpreted in the framework of MCP problems and formulated based on the modal concept.

Modality Constrained Programming Problems and Various Fuzzy Mathematical Programming Problems

The modality constrained programming (MCP) problems were previously proposed by the authors. MCP problems are a sort of fuzzy mathematical programming (FMP) problems and have some analogy with the chance constrained programming (CCP) problems. In MCP problems, the probability in CCP problems is replaced by the possibility or the necessity.In this paper, we focus on the relationships between MCP problems and the other various FMP problems, i.e., flexible programming problem, fuzzy goal programming problem, FMP problem of Orlovsky, that of Tanaka et al., that of Inuiguchi et al. and that of Dubois. FMP problems discussed in this paper can be interpreted in the framework of MCP problems. MCP problems are the basic models for unifying FMP problems.

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A Modalistic Formalization of Fuzzy Mathematical Programming Problems

本研究では, 機会制約条件計画問題と同様な考え方に基づき, 係数のあいまいさと意思決定者の選好態度のあいまいさの意味の相違を踏まえて, ファジィ数理計画問題を構成する.また, 制約条件式および目的関数が線形である場合の等価確定制約条件および等価問題を示す.本研究の方法を通して, 従来のファジィ数理計画問題の多くを統一的に解釈することができる.