Abstract
Problem statement: We deal with the bi-level linear programming problem. A bi-level programming problem is formulated for a problem in which two Decision-Makers (DMs) make decisions successively. Approach: In this research we studied and designs a Genetic Algorithm (GA) of Bi-Level Linear Programming Problems (BLPP) by constructing the fitness function of the upper-level programming problems based on the definition of the feasible degree. This GA avoids the use of penalty function to deal with the constraints, by changing the randomly generated initial population into an initial population satisfying the constraints in order to improve the ability of the GA to deal with the constraints. Also we designed software to solve this problem. A comparative study between proposed method and previous methods through numerical results of some examples. Finally, parametric information of the GA was introduced. Results: Results of the study showed that the proposed method is feasible and more efficient to solve (BLPP), also there exist package to solve (BLPP) problem. Conclusion: This GA avoids the use of penalty function to deal with the constraints, by changing the randomly generated initial population into an initial population satisfying the constraints in order to improve the ability of the GA to deal with the constraints.
Highlights
Multi-level programming techniques are developed to solve decentralized planning problems with multiple decision makers in a hierarchal organization[1]
Numerical example: In order to compare the performance of Genetic Algorithm (GA) which introduced in this study with the other existing methods, proposed three examples solved by proposed method in this study and compare our results with the results in the previous methods[14,23]
This study designs the GA for solving Bi-Level Linear Programming Problems (BLPP) which the optimal solution of the lower-level problem is dependent on the upper-level problem
Summary
Multi-level programming techniques are developed to solve decentralized planning problems with multiple decision makers in a hierarchal organization[1]. The BiLevel Programming (BLP) problem is a special case of multilevel programming problems with a two-level structure. This problem is an important case in nonconvex optimization and a leader-follower game in which play is sequential and cooperation is not permitted[2]. Many instances of decision problems can be fined, which are formulated as two-level programming problems and concerning the above mentioned hierarchical decision problem in the decentralized firm, it is natural that decision makers behave cooperatively rather than non-cooperatively. Lai and Lee [14] have proposed a solution concept, which is different from the concept of the Stackelberg solution, for multi level linear programming problems such that decisions of DMs in both levels are sequential and all of the DMs essentially cooperate with each other[15,16]
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