An Efficient Genetic Algorithm for Interval Linear Bilevel Programming Problems

This paper deals with a class of interval linear bilevel programming problems, in which some or all of the leader's and follower's objective function coefficients are specified in terms of intervals. The focus of solving this class of problems is on determining the optimal value range when different coefficients of objectives are taken in intervals given. In order to obtain the best and the worst optimal solutions to this class of problems, an efficient genetic algorithm is developed. Firstly, the objective coefficients of the lower level are encoded as individuals using real coding scheme, and the relative intervals are taken as the search space of the genetic algorithm. Secondly, for each encoded individual, a simplified interval linear bilevel program is obtained, in which interval coefficients are simply in the upper level objective function. Finally, the simplified problem is further divided into two linear bilevel programs without interval coefficients and solved by using the optimality theory of linear programming. The optimal values are taken as fitness values, by which the best and the worst optimal solutions can be obtained. In order to illustrate the efficiency of the proposed algorithm, two examples are solved and the results show that the algorithm is feasible and robust.