Multiobjective possibilistic linear programming (Q911995)

The paper explores multiobjective linear programming with fuzzy variables expressed by possibility distributions and formulated accordingly: \[ \max Z(C_ 1x,C_ 2x,...,C_ Kx)\quad subject\quad to\quad A_ ix * B_ i,\quad i=1,2,...,m,\quad x\geq 0, \] where \(C_ 1,C_ 2,...,C_ K\), \(A_ i\), \(B_ i\), \(i=1,2,...,m\) denote fuzzy numbers and \(x=[x_ 1x_ 2...x_ n]^ T\) stands for a vector of (non-fuzzy) decision variables. The operation ``*'' stands for \(<\), \(\leq\), \(=\), \(\geq\), \(>\) for each i. Two solutions to the above problem are discussed. The first one finds the possibility distribution of the objective function Z. The resulting solution is obtained by calculating undominated vectors of maximum possibility and translating it back to decision variables. The second approach can be considered as a sort of reverse method to the previous one. Namely, one first finds the possibility distribution of the undominated sets. In the sequel, a solution consists of those values of the decision variables which are undominated with maximum possibility and simultaneously produce the largest (undominated) values of Z. The main theorem of the paper states conditions under which the two solutions are identical.