Nondominated solutions in a fully fuzzy linear programming problem

Abstract

In this work, a new method is presented for locating fuzzy optimal (nondominated) solutions of a fully fuzzy linear programming problem with inequality constraints and triangular fuzzy numbers, not necessarily symmetric, without ranking functions, by means of solving a multiobjective linear problem. An equivalence is proved between the set of nondominated solutions of the fully fuzzy linear programming problem and the set of weakly efficient solutions of the considered and related multiobjective linear problem. An algorithm is provided to generate nondominated solutions of a fully fuzzy linear programming problem, as well as an example to illustrate it.

An epsilon‐constraint method for fully fuzzy multiobjective linear programming

“…To handle fuzzy inequality constraints, some authors suggest to transform them into equalities by introducing nonnegative fuzzy slack and surplus variables 7,11,29,30 ; this is, however, incorrect as shown by Bhardwaj and Kumar 31 and Gupta et al 32 Other authors, for example, 6,8,14,26,33 use a partial order to handle the fuzzy inequality constraints and a linear ranking function or lexicographic ranking criterion with the objective function. Thus, it can be observed that, for both approaches in the literature, two different order relations are used for the inequality of fuzzy numbers in the same problem.…”

“…Sharma and Aggarwal 7 used the nearest interval approximation of LR fuzzy numbers to transform the FFMOLP problem into a multiobjective interval linear programming problem, which was further transformed into a crisp single-objective one by means of a weighted sum of the centre and width of the intervalvalued objective functions. Arana-Jiménez 8 extended a previous result from Reference 14 and used a partial order relation to define fuzzy inequality constraints on the set of triangular fuzzy numbers; the author developed a weighted sum-based procedure to obtain a set of nondominated solutions of FFMOLP problems, without explicitly setting a fuzzy number ranking criterion beforehand. However, a decision-maker still has to select a solution from this set; therefore, the use of a ranking criterion to reach a final conclusion cannot be avoided.…”

On the fuzzy maximal covering location problem

“…On the contrary, it is well-known that the general multiplication operator (3) is not suitable for fuzzy numbers with a FRS, and, in particular, for TFNs. Different multiplication rules have been proposed for TFNs (see [1,23,24,26]). In the case of nonnegative TFNs all of them coincide in the following simple expression, which we will use through this paper:…”

Solving fuzzy linear fractional set covering problem by a goal programming based solution approach

“…For example, in the case of facility location problem, the set covering problem can be applied to find the locations for a new facility to serve all of the existing facilities. To be more close to real situations, the engineering and non-engineering problems are modeled with fuzzy parameters in the literature (see [4,5,7,16,17,24,25,33]). Furthermore, the studies of Niroomand et al [22], Mahmoodirad et al [19], Mahmoodirad and Niroomand [17] are some recent studies that applied fuzziness and uncertainty in optimization problems of supply chain network design.…”

“…• For the case of problem instances with high number of variables, the method becomes inefficient. 4. Proposed solution approach.…”