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

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 possibly 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, we 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.