[01] Eghbal Hosseini, Payamenur University of Tehran, Department of Mathematics, Tehran, Iran. [02] Isa Nakhai Kamalabadi, University of Kurdistan, Department of Industy, Sanandaj, Iran.

The multi-level programming problems are attractive for many researchers because of their application in several areas such as economic, traffic, finance, management, computer science, informatics, transportation and so on. Among these, the quad-level programming problem (QLPP) is an appropriate tool to model these real problems because many real problems have four levels. It has been proven that even the general bi-level programming problem is an NP-hard problem, so the multi-level problem are practical and complicated problems therefore solving these problem would be significant. The literature shows several algorithms to solve different forms of the bi-level programming problems (BLPP). Not only there is no any algorithm for solving QLPP, but also it has not been studied by any researcher. The most important part in this paper is presentation and studying of a new model (QLPP) of multi-level problems. Also we attempt for developing two applied problems which would be modeled to the linear QLPP, then we attempt for developing an effective approach based on analyze theorems for solving the linear QLPP. In this approach, by using the heuristic method the QLPP is converted to a linear single problem. Finally, the single level problem is solved using the enumeration algorithm. The presented approach achieves an efficient and feasible solution in an appropriate time which has been evaluated by solving test problems.

Linear Quad-Level Programming Problem, Heuristic Method, Enumeration Algorithm

[01] J.F. Bard, Some properties of the bi-level linear programming, Journal of Optimization Theory and Applications (1991) 68 371–378. [02] L. Vicente, G. Savard, J. Judice, Descent approaches for quadratic bi-level programming, Journal of Optimization Theory and Applications (1994) 81 379–399. [03] Lv. Yibing, Hu. Tiesong, Wang. Guangmin , A penalty function method Based on Kuhn–Tucker condition for solving linear bilevel programming, Applied Mathematics and Computation (2007) 1 88 808–813. [04] G. B. Allende, G. Still, Solving bi-level programs with the KKT-approach, Springer and Mathematical Programming Society (2012) 1 31:37 – 48. [05] M. Sakava, I. Nishizaki, Y. Uemura, Interactive fuzzy programming for multilevel linear programming problem, Computers & Mathematics with Applications (1997) 36 71–86. [06] S Sinha, Fuzzy programming approach to multi-level programming problems, Fuzzy Sets And Systems (2003) 136 189–202. [07] S. Pramanik, T.K. Ro, Fuzzy goal programming approach to multilevel programming problems, European Journal of Operational Research (2009) 194 368–376. [08] S.R. Arora, R. Gupta, Interactive fuzzy goal programming approach for bi-level programming problem, European Journal of Operational Research (2007) 176 1151–1166. [09] J. Nocedal, S.J. Wright, 2005 Numerical Optimization, Springer-Verlag, New York. [10] A.AL Khayyal, Minimizing a Quasi-concave Function Over a Convex Set: A Case Solvable by Lagrangian Duality, proceedings, I.E.E.E. International Conference on Systems, Man,and Cybemeties, Tucson AZ (1985) 661-663. [11] R. Mathieu, L. Pittard, G. Anandalingam, Genetic algorithm based approach to bi-level Linear Programming, Operations Research (1994) 28 1–21. [12] G. Wang, B. Jiang, K. Zhu, (2010) Global convergent algorithm for the bi-level linear fractional-linear programming based on modified convex simplex method, Journal of Systems Engineering and Electronics 239–243. [13] W. T. Wend, U. P. Wen, (2000) A primal-dual interior point algorithm for solving bi-level programming problems, Asia-Pacific J. of Operational Research, 17. [14] N. V. Thoai, Y. Yamamoto, A. Yoshise, (2002) Global optimization method for solving mathematical programs with linear complementary constraints, Institute of Policy and Planning Sciences, University of Tsukuba, Japan 978. [15] S.R. Hejazi, A. Memariani, G. Jahanshahloo, (2002) Linear bi-level programming solution by genetic algorithm, Computers & Operations Research 29 1913–1925. [16] G. Z. Wang, Wan, X. Wang, Y.Lv, Genetic algorithm based on simplex method for solving Linear-quadratic bi-level programming problem, Computers and Mathematics with Applications (2008) 56 2550–2555. [17] T. X. Hu, Guo, X. Fu, Y. Lv, (2010) A neural network approach for solving linear bi-level programming problem, Knowledge-Based Systems 23 239–242. [18] B. Baran Pal, D .Chakraborti , P. Biswas, (2010) A Genetic Algorithm Approach to Fuzzy Quadratic Bi-level Programming, Second International Conference on Computing, Communication and Networking Technologies. [19] Z. G.Wan, Wang, B. Sun, ( 2012) A hybrid intelligent algorithm by combining particle Swarm optimization with chaos searching technique for solving nonlinear bi-level programming Problems, Swarm and Evolutionary Computation. [20] J.F. Bard, Practical bi-level optimization: Algorithms and applications, Kluwer Academic Publishers, Dordrecht, 1998. [21] J.F. Bard, Some properties of the bi-level linear programming, Journal of Optimization Theory and Applications 68 (1991) 371–378. [22] S Hosseini, E & I.Nakhai Kamalabadi. Line Search and Genetic Approaches for Solving Linear Tri-level Programming Problem International Journal of Management, Accounting and Economics Vol. 1, No. 4, 2014. [23] S Hosseini, E & I.Nakhai Kamalabadi. aylor Approach for Solving Non-Linear Bi-level Programming Problem ACSIJ Advances in Computer Science: an International Journal, Vol. 3, Issue 5, No.11 , September. 5. [24] S Hosseini, E & I.Nakhai Kamalabadi. Two Approaches for Solving Non-linear Bi-level Programming Problem , Advances in Research Vol. 4, No.3 , 2015. ISSN: 2348-0394 [25] J. Yan, Xuyong.L, Chongchao.H, Xianing.W, Application of particle swarm optimization based on CHKS smoothing function for solving nonlinear bi-level programming problem, Applied Mathematics and Computation 219 (2013) 4332–4339. [26] Xu, P, & L. Wang. An exact algorithm for the bilevel mixed integer linear programming problem under three simplifying assumptions, Computers & Operations Research, Volume 41, January, Pages 309-318 (2014). [27] Wan, Z, L. Mao, & G. Wang. Estimation of distribution algorithm for a class of nonlinear bilevel programming problems, Information Sciences, Volume 256, 20 January, Pages 184-196(2014). [28] Zheng, Y, J. Liu, & Z. Wan. Interactive fuzzy decision making method for solving bi-level programming problem, Applied Mathematical Modelling, Volume 38, Issue 13, 1 July, Pages 3136-3141(2014). [29] Zhang , G, J. Lu , J. Montero , & Y. Zeng , Model. solution concept, and Kth-best algorithm for linear tri-level, programming Information Sciences 180 481–492 (2010) . [30] A. Silverman. Richard, Calculus with analytic geometry, ISBN:978-964-311-008-6, 2000. [31] Y. Zheng, J. Liu, Z. Wan, Interactive fuzzy decision making method for solving bi-level programming problem, Applied Mathematical Modelling, Volume 38, Issue 13, 1 July 2014, Pages 3136-3141. [32] Y. Jiang, X. Li, C. Huang, X. Wu, An augmented Lagrangian multiplier method based on a CHKS smoothing function for solving nonlinear bi-level programming problems, Knowledge-Based Systems, Volume 55, January 2014, Pages 9-14. [33] X. He, C. Li, T. Huang, C. Li, Neural network for solving convex quadratic bilevel programming problems, Neural Networks, Volume 51, March 2014, Pages 17-25. [34] Z. Wan, L. Mao, G. Wang, Estimation of distribution algorithm for a class of nonlinear bilevel programming problems, Information Sciences, Volume 256, 20 January 2014, Pages 184-196. [35] P. Xu, L. Wang, An exact algorithm for the bilevel mixed integer linear programming problem under three simplifying assumptions, Computers & Operations Research, Volume 41, January 2014, Pages 309-318. [36] E. Hosseini, I.Nakhai Kamalabadi, A Genetic Approach for Solving Bi-Level Programming Problems, Advanced Modeling and Optimization, Volume 15, Number 3, 2013. [37] E. Hosseini, I.Nakhai Kamalabadi, Solving Linear-Quadratic Bi-Level Programming and Linear-Fractional Bi-Level Programming Problems Using Genetic Based Algorithm, Applied Mathematics and Computational Intelligence, Volume 2, 2013. [38] E. Hosseini, I.Nakhai Kamalabadi, Taylor Approach for Solving Non-Linear Bi-level Programming Problem ACSIJ Advances in Computer Science: an International Journal, Vol. 3, Issue 5, No.11, September 2014. [39] E. Hosseini, I.Nakhai Kamalabadi, Solving Linear Bi-level Programming Problem Using Two New Approaches Based on Line Search International Journal of Management sciences and Education, Vol. 2, Issue 6, 2014, 243-252. [40] E. Hosseini, I.Nakhai Kamalabadi, Two Approaches for Solving Non-linear Bi-level Programming Problem, Advances in Research, 3(5), 2015.