یک روش دو قطبی در برنامه ریزی خطی چندهدفه فازی

The traditional frameworks for fuzzy linear optimization problems are inspired by the max–min model proposed by Zimmermann using the Bellman–Zadeh extension principle to aggregate all the fuzzy sets representing flexible (fuzzy) constraints and objective functions together. In this paper, we propose an alternative approach to model fuzzy multi-objective linear programming problems (FMOLPPs) from a perspective of bipolar view in preference modeling. Bipolarity allows us to distinguish between the negative and the positive preferences. Negative preferences denote what is unacceptable while positive preferences are less restrictive and express what is desirable. This framework facilitate a natural fusion of bipolarity in FMOLPPs. The flexible constraints in a fuzzy multi-objective linear programming problem (FMOLPP) are viewed as negative preferences for describing what is somewhat tolerable while the objective functions of the problem are viewed as positive preferences for depicting satisfaction to what is desirable. This approach enables us to handle fuzzy sets representing constraints and objective functions separately and combine them in distinct ways. After aggregating these fuzzy sets separately, coherence (or consistency) condition is used to define the fuzzy decision set.

One of the early models of fuzzy linear programming was proposed by Zimmermann [46] and [47]. In the model he studied, an aspiration level for the objective function had been assumed to be prescribed along with certain tolerances, and the inequalities in linear constraints were made flexible. Thereafter, the objective functions and the flexible constraints received the same treatment leading to a symmetric model. The classical Bellman–Zadeh principle [4] had been used to define a fuzzy decision set of solutions by aggregating all fuzzy inequalities using the ‘min’ aggregation. An optimal solution of the linear program was the one for which this minimum (aggregated function) is maximal. The core ideas of Zimmerman and Bellman–Zadeh paradigms have been adopted in the years to follow so much so that several other fuzzy optimization models ultimately boil down to max–min optimization models. A large literature is available in this context; just to cite some, please refer to [2], [3], [25], [27], [28], [29], [33] and [49] and references therein. In [6], [20], [21], [30], [36], [37], [45] and [47], to name a few, researchers had used various approaches to model FMOLPPs. In particular [6], [30], [36] and [47] used the max–min approach to solve FMOLPPs. There are certain non-symmetric models for FMOLPP, for instance [7] and [38]. However, ultimately these models reduce to symmetric model where in the fuzzy sets representing objective functions and the constraints are aggregated using ‘min’ aggregation operator. The main advantage of the max–min framework is that it enables a better discrimination between good and less good solutions to a set described by linear inequalities representing both the fuzzy objective function and the flexible constraints. However, an interpretation of a decision as the intersection of fuzzy sets, computed by applying ‘min’ operator to the membership functions of the fuzzy sets of objectives and constraints, implies that there is no compensation between low and high degree of membership. However, an aggregation of subjective categories in the framework of human decisions almost always shows some degree of compensation [48], hence the max–min approach does not found favor in certain situations in decision making problems. With this motivation, Luhandjula [23], Sommer and Pollatschek [31], Tsai et al. [34], Süer et al. [32], Werner [39], proposed and used various compensatory aggregation operators to solve FMOLPPs. Any approach which aggregates all fuzzy sets representing the objective functions and the constraints together abolishes the distinction between the two. Physically, a constraint is something which should be satisfied, at least to some extent for a flexible constraint. In other words, constraints are something to be respected. On the other hand, there is no idea of requirement associated with the objective functions. The objective functions aspiration levels in FMOLPP only depict the desire or wish of the decision maker (DM) and hence non-compulsory. It makes sense that if a solution of FMOLPP has positive membership degrees in not all but some fuzzy sets representing the objective functions then it should continue to have a positive membership degree on their aggregation. What we observe that the ‘min’ aggregation operator, being non-compensatory, does not support this property. At this point we would like to cite Benferhat et al. [5] and Dubois and Prade [12] and [13], who suggested the concept of negative preferences and positive preferences to respectively discriminate between what is unacceptable and what is really satisfactory for DM. The negative preferences act as constraints discarding the unacceptable solutions and the positive preferences lead to support appealing or desired solutions. Thus, an FMOLPP can be modeled for computing the best solutions after merging the negative and the positive preferences separately. But this amalgamate of two different kinds of information may create inconsistency. The latter can be enforced by restricting what is desirable to what is tolerated using coherence or consistency condition [5]. The above ideas naturally provide a bipolar view of preferences in fuzzy optimization. Motivated by these thoughts, in this paper, we attempt to study FMOLPPs with afore described bipolar approach. We shall be using separate aggregation schemes for aggregating the flexible constraints with predefined admissible tolerances and the objective functions with preset desired aspiration levels by the DM. The paper is structured as follows. In Section 2, we present a set of concepts from bipolarity and ordered weighted averaging (OWA) operator which facilitate the subsequent discussion. In Section 3, a general framework for FMOLPPs in a bipolar setting is explained. Section 4 discusses the crisp formulations of FMOLPP using OWA operator and some other compensatory aggregation operators. Section 5 presents a numerical illustration while the paper concludes in Section 6 with some remarks and future directions.

In this paper, we make an attempt to visualize an FMOLPP with a heterogeneous bipolar view which refers to opposition between constraints and objectives. The bipolarity framework allows to distinguish between two fundamental concepts involved in an FMOLPP formulation: one representing constraints whose violation is unacceptable, and the other is objectives whose achievement generate satisfaction. We used an OWA operator, fuzzy-and, and fuzzy-or operators to aggregate the fuzzy objectives; while ‘min’ operator is used to aggregate the constraints. The solution methodologies given by Yager [43] and Werner [39] to solve the aggregated problems are used to solve FMOLPP. Yager's [43] approach formulates an OWA optimization problem as a mixed integer linear programming problem. In [26], Ogryczak and Śliwiński showed that an OWA optimization problem with monotonic weights can be formulated as a standard linear programming problem in higher dimension. Therefore, for an OWA aggregation operator F with monotonic weights, the models proposed in [26] can also be used to formulate the deterministic equivalent problem of (FMOLP). Our approach in this paper differs in viewpoint from the other studies on similar class of fuzzy optimization problems that are inspired mainly by the Zimmermann modeling ideas. We highlight an important difference between the bipolar approach and the max–min approach to model FMOLPP. In our proposed model (using bipolar approach) for FMOLPP, the obtained form of an overall decision function on aggregation of individual objectives most nearly corresponds to the situation in which we get as much satisfaction to objectives as possible unlike the other studies (using max–min approach) where the aim is to achieve positive satisfaction in all objectives simultaneously. The bipolar approach appears to be promising in fuzzy mathematical programming. Its further exploration can indeed improvise the modeling aspect for such a class of optimization problems to take account of more practical issues related with them. This paper attempts to manifest bipolarity in fuzzy linear programming, and it opens the gateway to explore other related application areas. Certainly one can also initiate designing new aggregation operators specific to the need of the problem. Another interesting direction in similar context is suggested by Felix [14], Grabisch [18], and many others to model an interaction between objective functions using fuzzy relations which require no preference ordering. In contrast to the approach adopted in the present work, where we have assumed that the target values and the tolerances in them are priori provided for each of the p objectives, the interactive structure of goals for each decision situation are calculated explicitly based on fuzzy types of interaction between the objective functions. The aggregation of objective functions based on what kind of fuzzy relations they share among themselves (see, [15] for explicit meaning), instead of simple OWA operators, can be studied within a bipolar framework of FMOLPP in future research.