A fuzzy goal programming approach for solving the decentralized bi-level optimization problem with imprecise cooperation relations

Abstract

This paper studies the decentralized bi-level multiobjective programming (DBLMOP) problem with one decisionmaker (DM) at the higher level and more than one DM at the lower level. The number of objective functions to be optimized by each DM may be different. The fuzzy relation technique is first incorporated into fuzzy goal programming (FGP) to solve the hierarchical optimization problem. After characterizing the fuzzy goals of the objective functions and the higher-level DM's decision vector by the corresponding membership functions, the concept of fuzzy binary relation is introduced to define the imprecise cooperation relations between the two DMs located at the different levels. An FGP model is formulated to reach the satisfactory values of all the fuzzy goals as close to the optimums as possible by minimizing their deviation variables and thereby finding a candidate solution. The proposed solution algorithm involving an interactive procedure with two orientations enables the higher-level DM to adjust the decision powers of the DMs through changing the cooperation relations, if the solution needs to be re-evaluated. A numerical example is presented to illustrate and demonstrate the proposed method.

A two-phase fuzzy approach for solving multi-level decision-making problems

“…; v i g for each higher-level DM i ; i 2 I h , to describe the level of the relative satisfaction of the lower-level DM iþ1 compared to their own. The relative satisfaction between DM iþ1 and DM i is measured by the ratio of l f ðiþ1Þ to l f ðiÞ , i.e., p i ¼ l f ðiþ1Þ =l f ðiÞ ; i 2 I h [8,9,11,21]. Each linguistic term is characterized by a membership function with the base variable of p i and the most likely value pẼ ði;sÞ ; s 2 S i .…”

“…. ; k À 1g, based on the concept presented by Chen and Chen [8,9]. The fuzzy variableẼ i includes a set of linguistic terms S i ¼ f1; 2; .…”

“…However, these methods [3,17] may produce a higher satisfaction degree for the fuzzy goal of a lower-level DM than that for a higher-level one when the former's goal is relatively easy to achieve. To deal with this problem, Chen and Chen [8,9] applied a fuzzy variable of relative satisfactions between lower-and higher-level DMs to solve the decentralized bi-level optimization problem, ensuring that the integrated satisfaction of the higherlevel DM is not less than that of the lower-level DM.…”

A fuzzy approach with required minimum decision tolerances for multi-level multi-objective decision-making problems

“…Such problems are usually formulated as multi-level mathematical programming models [26] with a continuous feasible solution set. Several features are considered, such as sequential decision processes from the top down, interactive decisionmaking, and the inclusion of the leader-follower relationship between higher-and lower-level DMs [9,10,15]. For example, in a company, departments and their associated branches usually make their decisions in order to optimize their own objectives.…”

“…However, Baky's approach [4] very likely produces multiple final solutions (alternative optima) so that the optimal solutions that contribute to the satisfaction degrees of fuzzy goals cannot be easily found for large-scale problems. In addition, the application of FGP [21] may violate the inherent leader-follower relationship in terms of the DMs' integrated satisfactions of the respective fuzzy goals [9,10]. Particularly, most existing fuzzy approaches [9,10,15,[21][22][23][24][25] assume that their solution models can always find satisfactory solutions, ignoring the possibility of high conflicts of membership domains between the higher-level DMs' decision variables, because the tolerances of decision variables are individually and subjectively defined.…”