Possible and necessary efficiency in possibilistic multiobjective linear programming problems and possible efficiency test

Possible and necessary efficiency in possibilistic multiobjective linear programming problems and possible efficiency test

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Article ID: iaor19961305
Country: Netherlands
Volume: 78
Issue: 2
Start Page Number: 231
End Page Number: 241
Publication Date: Mar 1996
Journal: Fuzzy Sets and Systems
Authors: ,
Keywords: programming: linear, programming: multiple criteria
Abstract:

In this paper, the concept of efficient solutions to the conventional multiobjective linear programming problems is extended to the fuzzy (possibilistic) coefficients case. Two kinds of efficient solution sets, i.e., a set of possible efficient solutions and a set of necessarily efficient solutions, are defined as fuzzy sets whose membership grades represent the possibility and necessity degrees to which the solution is efficient. A test to check the possible efficiency is discussed when a feasible solution is given. To do this, the authors first consider the interval case, where all fuzzy (possibilistic) coefficients degenerate into interval coefficients. In this case, a set of possibly efficient solutions degenerates into a usual (crisp) set. A necessary and sufficient condition of the possible efficiency for the interval case is presented. This condition shows that the possible efficiency is checked by solving a system of linear inequalities. Extending this result to the fuzzy (possibilistic) case, the degree of possibility efficiency is obtained by solving a nonlinear programming problem. The nonlinear programming problem is solved by the simplex and bisection methods.

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