This paper models a crisp Linear Programming (LP) as a Compromise Fuzzy Multi-Objective LP (CFMOLP). The application of CFMOLP has been focused on an industrial engineering problem that seeks profit maximisation by optimising product-mix. Imprecision of the large volume of industrial data and the conglomerated decision from all levels of management lead fuzzication of the identified constraints and the objective functions as well. The crisp model described is in the form of crisp-Multi-Objective Linear Programming (MOLP) with objective functions, functional constraints and non-negativity constraints. This model is formulated as a fuzzy-MOLP and subsequently converted into an equivalent compromise-MOLP model. The paper describes the development process for the CFMOLP model and its application along with appropriate interpretation.
This paper, among many other objectives, specifically aims at developing the CFMOLP model capable of determining the product-mix decision problem. The contribution enables one to locate the feasible optimal space of the product-mix decision. The fuzzy multi-objective LP model for the product-mix problem having four objective functions is represented by optimisation problem with twelve objective fuzzy MFs which are (μu1,μu2,μu3,μv1,μv2,μv3,μy1,μy2,μy3,μz1,μz2,μz3)(μu1,μu2,μu3,μv1,μv2,μv3,μy1,μy2,μy3,μz1,μz2,μz3). The values of these functions at the optimal solution represent the levels-of-satisfaction achieved by the functions u1, u2, u3, v1, v2, v3, y1, y2, y3 and z1, z2, z3, respectively. It is observed from the discussion laid down in Section 4 that the solution offered guarantees a minimum value of level-of-satisfaction as 50% for the objective functions. Therefore, this minimum level-of-satisfaction value clearly indicates the acceptability of the proposed CFMOLP model.
It is worthwhile to mention that the level-of-satisfaction elucidated in this paper is meant for the optimal solution space during a search for the feasible optimal space using the CFMOLP model. Therefore, the computation of the level-of-satisfaction in the proposed CFMOLP model is one of the essential components for both solution and validation of the product-mix decision problem.
The findings and validation of the presented heuristic lead the authors to conclude that the strong background of the proposed CFMOLP model generates a better solution than that of the earlier reported fuzzy MOLP formulations in finding out feasible optimal product-mix solution. Hence the proposed CFMOLP model is capable of tackling large-scale LP problems having imprecision in the raw product-mix decision datasets.
Customised versions of the devised CFMOLP model find specific applications in econometrics, business management, decision sciences and engineering where candidate alternatives are to be selected with a pre-determined level-of-satisfaction of the decision makers. Further, triangular membership functions are the approximations of real-life curves. Thus, to take into account the real-world scenario, triangular MF may be replaced with Gaussian, Sigmoid and bell type MFs as an extension of the proposed CFMOLP.
