رویکرد مبتنی بر فضای وزن به برنامه ریزی با اهداف چندگانه خطی فازی

In this paper, the effects of uncertainty on multiple-objective linear programming models are studied using the concepts of fuzzy set theory. The proposed interactive decision support system is based on the interactive exploration of the weight space. The comparative analysis of indifference regions on the various weight spaces (which vary according to intervals of values of the satisfaction degree of objective functions and constraints) enables to study the stability and evolution of the basis that correspond to the calculated efficient solutions with changes of some model parameters.

Most of realistic decision-making problems, essentially those stemming from complex and ill-structured situations, are characterized by the existence of multiple, conflicting and incommensurate objectives and are subject to the unavoidable influence of distinct sources of uncertainty. Therefore, models must take into account vague information, imprecise requirements, modifications of the original input data, imprecision stemming from the modeling phase, needed simplifications, unexpected occurrence of important events and the subjective and evolutive nature of human preference structures whenever multiple objectives and trade-offs are at stake. Interactive techniques based on the weight space decomposition have been developed and computationally implemented as the core of a decision support system (DSS) to deal with uncertainty in multiple-objective linear programming (MOLP) models by using fuzzy set theory concepts. The decision maker (DM) has the possibility of interactively changing the membership functions associated with the mathematical constraint relations and the objective functions optimization. It is then possible to evaluate the effects of changing the model parameters and to study alternative scenarios without having to reformulate the problem. The comparative analysis of the weight spaces corresponding to distinct satisfaction degrees is a valuable tool to study the fuzzy efficient solution set. Among these fuzzy solutions, the DM may choose a satisfactory compromise one according to his/her preference structure which may change as more knowledge about the problem is acquired throughout the interactive decision aid process. This paper is organized in five sections. The introduction of the main concepts of fuzzy multiple-objective linear optimization problems is made in Section 2. The conceptual aspects of the proposed DSS are presented in Section 3. The example presented in Section 4 aims at illustrating the concepts presented. Some conclusions about the potentialities of this approach are drawn in Section 5.

Decisions to be made in complex contexts, characterized by the presence of multiple evaluation aspects, are normally affected by uncertainty, which is essentially due to the insufficient and/or imprecise nature of input data as well as the subjective and evolutive preferences of the decision maker. An interactive approach, based on the search of the weight space, to deal with FMOLP problems has been proposed and implemented as a DSS. Linear fuzzy objective functions and fuzzy constraints have been considered. The analysis is based on the weight space which enables to show graphical information interactively to the DM in a way that promotes to gain new insights into the problem and the trade-offs to be made in order to select a satisfactory compromise solution. Special attention has been paid to the computational simplicity and graphical interactivity, in order to visualize dynamically the behavior of the efficient solutions according to changes in the initial model coefficients, by displaying the indifference regions on the weight space. The comparative study of distinct weight space decomposition, which changes according to the range of the parameter α, shows the evolution of the indifference regions corresponding to the calculated efficient solutions, in a way that enables to understand the shape of the fuzzy efficient feasible region and the nature of the trade-offs to be made in selecting a final satisfactory compromise solution. The interactive computer environment contributes to stimulate the DM to take a more active role in the decision process by exploring the problem and his/her convictions, criticizing the obtained results and carefully considering distinct situations that can arise (regarding objective functions values, used resources, intervals of values of the objective functions' and constraints' satisfaction degree, etc.). The membership functions can also be interactively changed, thus, allowing to further study the fuzzy efficient solution set. Despite the fact that uncertainty elements in the coefficients of the objective functions have not been incorporated, this seems very easy to integrate in the proposed approach both methodologically and computationally. In this situation the shape and the size of the indifference regions on the weight spaces would change dynamically as the value of the objective functions and constraints satisfaction degree varies. Research is currently underway to extend this DSS based on the weight space to incorporate uncertainty elements in the coefficients of the objective functions.