### 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} 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{lunate} 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.

Original language | English |
---|---|

Pages (from-to) | 121-133 |

Number of pages | 13 |

Journal | Fuzzy Sets and Systems |

Volume | 60 |

Issue number | 2 |

DOIs | |

Publication status | Published - Dec 10 1993 |

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### 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 Sets and Systems*,

*60*(2), 121-133. https://doi.org/10.1016/0165-0114(93)90341-E

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

Research output: Contribution to journal › Article

*Fuzzy Sets and Systems*, vol. 60, no. 2, pp. 121-133. https://doi.org/10.1016/0165-0114(93)90341-E

}

TY - JOUR

T1 - Fuzzy reasoning for solving fuzzy mathematical programming problems

AU - Fullér, R.

AU - Zimmermann, Hans Jürgen

PY - 1993/12/10

Y1 - 1993/12/10

N2 - 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.

AB - 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.

KW - Compositional rule of inference

KW - fuzzy mathematical programming

KW - multiple fuzzy reasoning

KW - possibilistic mathematical programming

UR - http://www.scopus.com/inward/record.url?scp=38249000273&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38249000273&partnerID=8YFLogxK

U2 - 10.1016/0165-0114(93)90341-E

DO - 10.1016/0165-0114(93)90341-E

M3 - Article

VL - 60

SP - 121

EP - 133

JO - Fuzzy Sets and Systems

JF - Fuzzy Sets and Systems

SN - 0165-0114

IS - 2

ER -