A choice of bilevel linear programming solving parameters: factoraggregation approach

Abstract

Our paper deals with the problem of choosing correct parameters for the bilevel linear program- ming solving algorithm proposed by M. Sakawa and I. Nishizaki. We suggest an approach based on fac- toraggregation, which is a specially designed general aggregation operator. The idea of factoraggregation arises from factorization by the equivalence relation generated by the upper level objective function. We prove several important properties of the factorag- gregation result regarding the analysis of param- eters in order to find an optimal solution for the problem. We illustrate the proposed method with some numerical and graphical examples, in particu- lar we consider a modification of the mixed produc- tion planning problem. operator. The idea of factoraggregation is based on factorization by an equivalence relation. We show that all properties of the definition of a general ag- gregation operator such as the boundary conditions and the monotonicity hold for the factoraggrega- tion operator. In the third section we show how factoraggregation can be applied for solving BLPP. This section is based on the interactive method of solution of bilevel linear programming problems in- troduced by M. Sakawa and I. Nishizaki (7),(8) and involving some parameters for the upper and lower level objectives. We illustrate with some numeri- cal and graphical examples showing how factorag- gregation is applied to the analysis of the choice of the parameters for solving BLPP. Several impor- tant properties of the result of factoraggregation are proved, these properties help us in the process of choosing the solving parameters. In the final section we illustrate the factoraggregation approach with the analysis of solving parameters for one particu- lar problem called the mixed production planning problem. We modify the problem described in (3) by considering new objective functions: we maxi- mize the profit of a production company with the higher priority and minimize the volume of environ- mentally damaging products and the dependence of external suppliers. We give numerical values for the parameters of the problem and describe how the analysis of solving parameters could be performed. 2. BLPP fuzzy solution approach