ELSEVIER Fuzzy Sets and Systems 78 (1996) 231-241 sets and systems Possible and necessary efficiency in possibilistic multiobjective linear programming problems and possible efficiency test Masahiro Inuiguchi*, Masatoshi Sakawa Department of lndustrial and Systems Engineering, Faculty of Engineering, Hiroshima University, 4-1, Kagamivama l-chome, Higashi-Hiroshima, 724, Japan 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 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. Keywords: Multiobjective linear programming; Efficiency; Possibility; Necessity; Simplex method 1. Introduction We usually face some difficulties when a real world problem is formulated to a mathematical programming problem. One of the difficulties is caused by the uncertainty in knowledge, informa- tion and/or decision maker's preference; for example, it is difficult to identify the coefficient under the ambiguous information such as "the * Corresponding author. profit rate will be almost 10 ($/min)" and to fix the target value under the vague aspiration such as "we want the profit substantially larger than 10 mil- lion". If such parameters are determined as real numbers appropriately, the formulated problem of- ten has no feasible solution or yields an improper solution. In these cases, we have to investigate the causes, change the determined real numbers, solve the reformulated problem and repeat this proced- ure until we obtain the satisficing solution. Thus, this requires a formidable effort and plenty of time. Fuzzy programming is proposed in order to deal with such uncertainties [2,4,5, 10, 12, 14, 15]. In 0165-0114/96/$15.00 © 1996 - Elsevier Science B.V. All rights reserved SSDI 0165-01 14(95)00169-7