Integrating goal programming, Kuhn–Tucker conditions, and penalty function approaches to solve linear bi-level programming problems

Bi-level programming, a tool for modeling decentralized decisions, consists of the objective(s) of the leader at its first level and the follower(s) at the second level. Three level programming results when the second level itself is a bi-level programming which helps us develop the idea of bi-level programming to multi-level programs with any number of levels.There are many approaches for encountering with bi(multi)-level problems. The method introduced by Chenggen Shi et al. [Chenggen Shi, Jie Lu, Guangquan Zhang, An extended Kuhn–Tucker approach for linear bi-level programming, Applied Mathematics and Computation 162 (2005) 51–63] uses extended Kuhn–Tucker conditions to solve bi-level programming problems. In this paper, we integrate goal programming (GP), Kuhn–Tucker conditions (KKT) and Penalty Function approaches to solve linear bi-level programming problems.By using a numerical example we illustrate efficiency of our methodology.