Taylor Approach for Solving Non-Linear Bi-level Programming Problem Eghbal Hosseini 1 , Isa Nakhai Kamalabadi 2 1 Department of Mathematics, Payamenur University of Tehran, Tehran, Iran eghbal_math@yahoo.com 2 Department of Industrial Engineering, University of Kurdistan, Sanandaj. Iran nakhai.isa@gmail.com Abstract In recent years the bi-level programming problem (BLPP) is interested by many researchers and it is known as an appropriate tool to solve the real problems in several areas such as computer science, engineering, economic, traffic, finance, management and so on. Also it has been proved that the general BLPP is an NP-hard problem. The literature shows a few attempts for using approximate methods. In this paper we attempt to develop an effective approach based on Taylor theorem to obtain an approximate solution for the non-linear BLPP. In this approach using the Karush-Kuhn–Tucker, the BLPP has been converted to a non-smooth single problem, and then it is smoothed by the Fischer – Burmeister function. Finally the smoothed problem is solved using an approach based on Taylor theorem. The presented approach achieves an efficient and feasible solution in an appropriate time which has been is evaluated by comparing to references and test problems. Keywords: Non-linear bi-level programming problem, Taylor theorem, Karush-Kuhn–Tucker conditions, smoothing methods. 1. Introduction The bi-level programming problem (BLPP) is a nested optimization problem, which has two levels in hierarchy. The first level is called leader and the second one is called follower. They have their own objective functions and constraints. The leader actions first, and the follower reacts to the leader decision. The follower should optimize its objective function according to the leader decision and delivered answers of the leader. In fact, the leader inflicts his decision on and obtains reaction of the follower. It has been proved that the BLPP is an NP- Hard problem even to seek for the locally optimal solutions [1, 2]. Nonetheless the BLPP is an applicable problem and a practical tool to solve decision making problems. It is used in several areas such as transportation, finance and so on. Therefore finding the optimal solution has a special importance to researchers. Several algorithms have been presented for solving the BLPP [3, 4, 11, 12, 13, 21, 25]. These algorithms are divided into the following classes: Transformation methods [3, 4, 22, 23, 36], Fuzzy methods [5, 6, 7, 8, 24, 35], Global techniques [9, 10, 11, 12, 38, 39], Primal–dual interior methods [13], Enumeration methods [14], Meta heuristic approaches [15, 16, 17, 18, 19, 25, 37, 40, 41]. The purpose of this paper is to develop two efficient approaches for solving linear bi-level programming problems (LBPP). We mainly concentrate on LBPP, in which both the upper level objective function and the lower level objective function are convex functions. In the present work, first, different from all previous works, we use a new proposed function to smoothen the problem. Then, an approximate approach is proposed which provides an efficient solution requiring much less times as compared to already available methods. The remainder of the paper is structured as follows: in Section 2, basic concepts of the non-linear BLPP and a smooth method to BLPP are introduced. Main theoretical results and steps of proposed algorithm are presented in Section 3. Computational results are presented for in Section 4. Finally, the paper is finished in Section 5 by presenting the concluding remarks. 2. Non-Linear BLPP and Smoothing Method The BLPP is used frequently by problems with decentralized planning structure. It is defined as [20]: (1) ACSIJ Advances in Computer Science: an International Journal, Vol. 3, Issue 5, No.11 , September 2014 ISSN : 2322-5157 www.ACSIJ.org 91 Copyright (c) 2014 Advances in Computer Science: an International Journal. All Rights Reserved.