Fuzzy reasoning for solving fuzzy mathematical programming problems

R. Fullér, Hans Jürgen Zimmermann

Research output: Contribution to journalArticle

50 Citations (Scopus)

Abstract

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{lunate} Rn, 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{lunate} Rn} (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.

Original language English 121-133 13 Fuzzy Sets and Systems 60 2 https://doi.org/10.1016/0165-0114(93)90341-E Published - Dec 10 1993

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Fuzzy Mathematical Programming
Fuzzy Reasoning
Mathematical programming
Fuzzy systems
Fuzzy sets
Linear programming
Fuzzy Linear Programming
Fuzzy Systems
Cybernetics
Fuzzy Sets
Nonlinear programming
Objective function
Convolution
Maximal Element
Fuzzy Decision
Fuzzy Information
Coefficient
Nonlinear Programming
Optimal Solution
Fuzzy linear programming

Keywords

• Compositional rule of inference
• fuzzy mathematical programming
• multiple fuzzy reasoning
• possibilistic mathematical programming

ASJC Scopus subject areas

• Artificial Intelligence
• Computer Science Applications
• Computer Vision and Pattern Recognition
• Information Systems and Management
• Statistics, Probability and Uncertainty
• Electrical and Electronic Engineering
• Statistics and Probability

Cite this

Fuzzy reasoning for solving fuzzy mathematical programming problems. / Fullér, R.; Zimmermann, Hans Jürgen.

In: Fuzzy Sets and Systems, Vol. 60, No. 2, 10.12.1993, p. 121-133.

Research output: Contribution to journalArticle

Fullér, R. ; Zimmermann, Hans Jürgen. / Fuzzy reasoning for solving fuzzy mathematical programming problems. In: Fuzzy Sets and Systems. 1993 ; Vol. 60, No. 2. pp. 121-133.
title = "Fuzzy reasoning for solving fuzzy mathematical programming problems",
abstract = "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{lunate} Rn, 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{lunate} Rn} (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.",
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