Fuzzy programming approach to Bi-level linear programming problems

Document Type : Research Paper

Authors

Department of Mathematics, Wollega University, Nekemte, Ethiopia.

Abstract

In this study, we discussed a fuzzy programming approach to bi-level linear programming problems and their application. Bi-level linear programming is characterized as mathematical programming to solve decentralized problems with two decision-makers in the hierarchal organization. They become more important for the contemporary decentralized organization where each unit seeks to optimize its own objective. In addition to this, we have considered Bi-Level Linear Programming (BLPP) and applied the Fuzzy Mathematical Programming (FMP) approach to get the solution of the system. We have suggested the FMP method for the minimization of the objectives in terms of the linear membership functions. FMP is a supervised search procedure (supervised by the upper Decision Maker (DM)). The upper-level decision-maker provides the preferred values of decision variables under his control (to enable the lower level DM to search for his optimum in a wider feasible space) and the bounds of his objective function (to direct the lower level DM to search for his solutions in the right direction).

Keywords

Main Subjects


[1]     Shih, H. S., Lai, Y. J., & Lee, E. S. (1996). Fuzzy approach for multi-level programming problems. Computers & operations research23(1), 73-91.
[2]     Bialas, W. F., & Karwan, M. H. (1984). Two-level linear programming. Management science30(8), 1004-1020.
[3]     Wen, U. P., & Hsu, S. T. (1991). Linear bi-level programming problems—a review. Journal of the operational research society42(2), 125-133.
[4]     Shih, H. S., Lai, Y. J., & Lee, E. S. (1996). Fuzzy approach for multi-level programming problems. Computers and operations research23(1), 73-91.
[5]     Sakawa, M., & Nishizaki, I. (2009). Cooperative and noncooperative multi-level programming (Vol. 48). Springer Science & Business Media.
[6]     Lai, Y. J. (1996). Hierarchical optimization: a satisfactory solution. Fuzzy sets and systems77(3), 321-335.
[7]     Sinha, S. (2003). Fuzzy programming approach to multi-level programming problems. Fuzzy sets and systems136(2), 189-202.
[8]     Zimmermann, H. J. (1978). Fuzzy programming and linear programming with several objective functions. Fuzzy sets and systems1(1), 45-55.
[9]     Shamooshaki, M. M., Hosseinzadeh, A., & Edalatpanah, S. A. (2015). A new method for solving fully fuzzy linear programming problems by using the lexicography method. Applied and computational mathematics1, 53-55.
[10] Pérez-Cañedo, B., Concepción-Morales, E. R., & Edalatpanah, S. A. (2020). A revised version of a lexicographical-based method for solving fully fuzzy linear programming problems with inequality constraints. Fuzzy information and engineering, 1-20.
[11] Hosseinzadeh, A., & Edalatpanah, S. A. (2016). A new approach for solving fully fuzzy linear programming by using the lexicography method. Advances in fuzzy systems. https://doi.org/10.1155/2016/1538496 
[12] Das, S. K., Edalatpanah, S. A., & Mandal, T. (2018). A proposed model for solving fuzzy linear fractional programming problem: numerical point of view. Journal of computational science25, 367-375.
[13] Elsisy, M. A., & El Sayed, M. A. (2019). Fuzzy rough bi-level multi-objective nonlinear programming problems. Alexandria engineering journal58(4), 1471-1482.
[14] Edalatpanah, S. A., & Shahabi, S. (2012). A new two-phase method for the fuzzy primal simplex algorithm. International review of pure and applied mathematics8(2), 157-164.
[15] Najafi, H. S., Edalatpanah, S. A., & Dutta, H. (2016). A nonlinear model for fully fuzzy linear programming with fully unrestricted variables and parameters. Alexandria engineering journal55(3), 2589-2595.
[16] Kamal, M., Gupta, S., Chatterjee, P., Pamucar, D., & Stevic, Z. (2019). Bi-Level multi-objective production planning problem with multi-choice parameters: a fuzzy goal programming algorithm. Algorithms12(7), 143.
[17] Maiti, S. K., & Roy, S. K. (2020). Analysing interval and multi-choice bi-level programming for Stackelberg game using intuitionistic fuzzy programming. International journal of mathematics in operational research16(3), 354-375.
[18] Yue, Q., Wang, Y., Liu, L., Niu, J., Guo, P., & Li, P. (2020). Type-2 fuzzy mixed-integer bi-level programming approach for multi-source multi-user water allocation under future climate change. Journal of hydrology591, 125332.
[19] Liu, J., Xue, W., Pang, G., & Guo, Y. (2019, December). Application of gray discrete bi-level linear programming model in the double emergency management network. 2019 12th international symposium on computational intelligence and design (ISCID) (Vol. 1, pp. 232-235). IEEE.
[20] Rizk-Allah, R. M., & Abo-Sinna, M. A. (2020). A comparative study of two optimization approaches for solving bi-level multi-objective linear fractional programming problem. OPSEARCH, 1-29.
[21] Han, J., Liu, Y., Luo, L., & Mao, M. (2020). Integrated production planning and scheduling under uncertainty: A fuzzy bi-level decision-making approach. Knowledge-based systems, 106056.
[22] Bellman, R. E., & Zadeh, L. A. (1970). Decision-making in a fuzzy environment. Management science17(4), B-141.