Yager ranking index in fuzzy bilevel optimization

Bilevel programming problems are challenging problems of mathematical optimization, which are interesting from the theoretical point-of-view (since it is a special case in nonsmooth optimization) and have a variety of applications. They are hierarchical problems of two decision makers, in which one – the so-called leader – has the first choice and the other one – the so-called follower – reacts optimally on the leader’s selection. The formulation of the bilevel programming problem for crisp (i.e. with exactly known) data can be found [1]. Problems with a predominantly hierarchical structure as a decision-making process are extremely practical to such decentralized systems as agriculture, government policy, economic systems, finance, warfare and are especially suitable for conflict resolutions [1-4].

Considering the inherently difficult nature of bilevel problems due to their nonconvexity, nonsmoothness and implicitly determined feasible set, it is difficult to design convergent algorithms, and the few algorithms that converge appear to be slow most of the time. Even in the simplest case, i.e. when the upper and lower level problems are crisp and linear, the bilevel programming problem has been shown to be NP-hard [5, 6].

Moreover, in practical situations data are often not known exactly, i.e. only (subjective) estimates are provided. One commonly used approach to deal with these problems is to model them as fuzzy optimization problems [7]. This approach proved very useful in many applied sciences, such as economics, physics [8-11].

As it was shown [12], the Yager ranking indices approach can be very useful in solving (single level) fuzzy optimization problems. In the present paper each fuzzy optimization problem is solved by its reformulation into the crisp optimization problem using the Yager ranking indices [13]. According to Liu and Kao [12], an optimal solution for a fixed parameter is then taken as an optimal solution of the initial fuzzy optimization problem.

The above concepts are combined, if the data involved in the bilevel optimization problem are known only approximately. A number of fuzzy bilevel programming problems can be found in Dempe et al. [14], Dempe and Starostina [15] and references therein. While some convergent algorithms for crisp bilevel problems exist in the literature (see e.g. Dempe [1], Bard and Moore [16], Ishizuka and Aiyoshi [17]), solution strategies for fuzzy bilevel programming problems are an emerging new field with a wide range of practical applicability. One of the main problems of a large amount of strategies developed for fuzzy bilevel optimization is a lack of ability to solve bilevel optimization problem. Some of the authors solve this problem as biobjective optimization problems [18]. This approach was criticized in Dempe [19] as one, that does not lead to a satisfactory solution.

In the present paper we introduce a sensible attempt to solve fuzzy bilevel optimization problems, that is organized as follows.

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(Author: Alina Ruzíyeva, S. Dempe

Published by Sciedu Press)