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A class of expected value bilevel programming problems with random coefficients based on rough approximation and its application to a production-inventory system. (English) Zbl 1278.90286
Summary: This paper focuses on the development of a bilevel optimization model with random coefficients for a production-inventory system. The expected value operator technique is used to deal with the objective function, and rough approximation is applied to convert the stochastic constraint into a crisp constraint. Then an interactive programming method and genetic algorithm are utilized to solve the crisp model. Finally, an application is given to show the efficiency of the proposed model and approaches in solving the problem.
90C15Stochastic programming
90B05Inventory, storage, reservoirs
90C59Approximation methods and heuristics
Full Text: DOI
[1] O. E. Emam, “A fuzzy approach for bi-level integer non-linear programming problem,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 62-71, 2006. · Zbl 1169.90408 · doi:10.1016/j.amc.2005.01.149
[2] G. B. Dantzig and A. Madansky, “On the solution of two-stage linear programs under uncertainty,” in Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Statistical Laboratory, University of California, June 1960. · Zbl 0104.14401
[3] C. Liu, Y. Fan, and F. Ordóñez, “A two-stage stochastic programming model for transportation network protection,” Computers & Operations Research, vol. 36, no. 5, pp. 1582-1590, 2009. · Zbl 1179.90246 · doi:10.1016/j.cor.2008.03.001
[4] J. Y. Jung, G. Blau, J. F. Pekny, G. V. Reklaitis, and D. Eversdyk, “Integrated safety stock management for multi-stage supply chains under production capacity constraints,” Computers and Chemical Engineering, vol. 32, no. 11, pp. 2570-2581, 2008. · doi:10.1016/j.compchemeng.2008.04.003
[5] T. Paksoy and C.-T. Chang, “Revised multi-choice goal programming for multi-period, multi-stage inventory controlled supply chain model with popup stores in Guerrilla marketing,” Applied Mathematical Modelling, vol. 34, no. 11, pp. 3586-3598, 2010. · Zbl 1201.90015 · doi:10.1016/j.apm.2010.03.008
[6] H. Sun, Z. Gao, and J. Wu, “A bi-level programming model and solution algorithm for the location of logistics distribution centers,” Applied Mathematical Modelling, vol. 32, no. 4, pp. 610-616, 2008. · Zbl 1171.90409 · doi:10.1016/j.apm.2007.02.007
[7] E. Roghanian, S. J. Sadjadi, and M. B. Aryanezhad, “A probabilistic bi-level linear multi-objective programming problem to supply chain planning,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 786-800, 2007. · Zbl 1137.90659 · doi:10.1016/j.amc.2006.10.032
[8] X. Ji and Z. Shao, “Model and algorithm for bilevel newsboy problem with fuzzy demands and discounts,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 163-174, 2006. · Zbl 1169.90496 · doi:10.1016/j.amc.2005.01.139
[9] A. Charnes and W. W. Cooper, “Deterministic equivalents for optimizing and satisficing under chance constraints,” Operations Research, vol. 11, pp. 18-39, 1963. · Zbl 0117.15403 · doi:10.1287/opre.11.1.18
[10] J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, Springer, New York, NY, USA, 1997. · Zbl 0892.90142
[11] Z. Pawlak, “Rough sets,” International Journal of Computer and Information Sciences, vol. 11, no. 5, pp. 341-356, 1982. · Zbl 0525.04005 · doi:10.1007/BF01001956
[12] Z. Pawlak and R. Sowinski, “Rough set approach to multi-attribute decision analysis,” European Journal of Operational Research, vol. 72, no. 3, pp. 443-459, 1994. · Zbl 0805.90069 · doi:10.1016/0377-2217(94)90415-4
[13] J. Xu and L. Yao, “A class of multiobjective linear programming models with random rough coefficients,” Mathematical and Computer Modelling, vol. 49, no. 1-2, pp. 189-206, 2009. · Zbl 1165.90614 · doi:10.1016/j.mcm.2008.01.003
[14] E. A. Youness, “Characterizing solutions of rough programming problems,” European Journal of Operational Research, vol. 168, no. 3, pp. 1019-1029, 2006. · Zbl 1077.90085 · doi:10.1016/j.ejor.2004.05.019
[15] Y. Shi, L. Yao, and J. Xu, “A probability maximization model based on rough approximation and its application to the inventory problem,” International Journal of Approximate Reasoning, vol. 52, no. 2, pp. 261-280, 2011. · Zbl 1222.90056 · doi:10.1016/j.ijar.2010.08.012
[16] L. N. Vicente and P. H. Calamai, “Bilevel and multilevel programming: a bibliography review,” Journal of Global Optimization, vol. 5, no. 3, pp. 291-306, 1994. · Zbl 0822.90127 · doi:10.1007/BF01096458
[17] J. F. Bard and J. T. Moore, “A branch and bound algorithm for the bilevel programming problem,” Society for Industrial and Applied Mathematics, vol. 11, no. 2, pp. 281-292, 1990. · Zbl 0702.65060 · doi:10.1137/0911017
[18] O. Ben-Ayed and C. E. Blair, “Computational difficulties of Bi-level Linear Programming,” Operations Research, vol. 38, no. 3, pp. 556-560, 1990. · Zbl 0708.90052 · doi:10.1287/opre.38.3.556
[19] W. F. Bialas and M. H. Karwan, “Two-level linear programming,” Management Science, vol. 30, no. 8, pp. 1004-1020, 1984. · Zbl 0559.90053 · doi:10.1287/mnsc.30.8.1004
[20] J. Fortuny-Amat and B. McCarl, “A representation and economic interpretation of a two-level programming problem,” The Journal of the Operational Research Society, vol. 32, no. 9, pp. 783-792, 1981. · Zbl 0459.90067 · doi:10.2307/2581394
[21] S. Sinha and S. B. Sinha, “KKT transformation approach for multi-objective multi-level linear programming problems,” European Journal of Operational Research, vol. 143, no. 1, pp. 19-31, 2002. · Zbl 1073.90552 · doi:10.1016/S0377-2217(01)00323-X
[22] M. Sakawa, I. Nishizaki, and Y. Uemura, “Interactive fuzzy programming for multilevel linear programming problems,” Computers & Mathematics with Applications, vol. 36, no. 2, pp. 71-86, 1998. · Zbl 0937.90123 · doi:10.1016/S0898-1221(98)00118-7
[23] M. Sakawa, “Interactive fuzzy goal programming for nonlinear programming problems and its applications to water quality management,” Control and Cybernetics, vol. 13, pp. 217-228, 1984. · Zbl 0551.90089
[24] S. Pramanik and T. K. Roy, “Fuzzy goal programming approach to multilevel programming problems,” European Journal of Operational Research, vol. 176, no. 2, pp. 1151-1166, 2007. · Zbl 1110.90084 · doi:10.1016/j.ejor.2005.08.024
[25] S. R. Hejazi, A. Memariani, G. Jahanshahloo, and M. M. Sepehri, “Linear bilevel programming solution by genetic algorithm,” Computers and Operations Research, vol. 29, no. 13, pp. 1913-1925, 2002. · Zbl 1259.90120 · doi:10.1016/S0305-0548(01)00066-1
[26] K. H. Sahin and A. R. Ciric, “A dual temperature simulated annealing approach for solving bilevel programming problems,” Computers and Chemical Engineering, vol. 23, no. 1, pp. 11-25, 1998. · doi:10.1016/S0098-1354(98)00267-1
[27] M. Gendreau, P. Marcotte, and G. Savard, “A hybrid tabu-ascent algorithm for the linear bilevel programming problem,” Journal of Global Optimization, vol. 8, no. 3, pp. 217-233, 1996. · Zbl 0859.90097 · doi:10.1007/BF00121266
[28] S. F. Woon, V. Rehbock, and R. C. Loxton, “Global optimization method for continuous-time sensor scheduling,” Nonlinear Dynamics and Systems Theory, vol. 10, no. 2, pp. 175-188, 2010. · Zbl 1225.37120
[29] S. F. Woon, V. Rehbock, and R. Loxton, “Towards global solutions of optimal discrete-valued control problems,” Optimal Control Applications & Methods, vol. 33, no. 5, pp. 576-594, 2012. · Zbl 1275.49057 · doi:10.1002/oca.1015
[30] D. D. Wu, “BiLevel programming data envelopment analysis with constrained resource,” European Journal of Operational Research, vol. 207, no. 2, pp. 856-864, 2010. · Zbl 1205.90165 · doi:10.1016/j.ejor.2010.05.008
[31] R. Slowinski and D. Vanderpooten, “A generalized definition of rough approximations based on similarity,” IEEE Transactions on Knowledge and Data Engineering, vol. 12, no. 2, pp. 331-336, 2000.
[32] Y. Yao, “Probabilistic rough set approximations,” International Journal of Approximate Reasoning, vol. 49, no. 2, pp. 255-271, 2008. · Zbl 1191.68702 · doi:10.1016/j.ijar.2007.05.019
[33] M. Sakawa, I. Nishizaki, and Y. Uemura, “Interactive fuzzy programming for multi-level linear programming problems with fuzzy parameters,” Fuzzy Sets and Systems, vol. 109, no. 1, pp. 3-19, 2000. · Zbl 0956.90063 · doi:10.1016/S0165-0114(98)00130-4
[34] J. H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan, Ann Arbor, Mich, USA, 1975. · Zbl 0317.68006
[35] D. B. Fogel, Evolution Computation: Toward a New Philosophy of Machine Intelligence, IEEE Press, Piscataway, NJ, USA, 1995.
[36] J. R. Koza, Genetic Programming, The MIT Press, Cambridge, Mass, USA, 1992. · Zbl 0850.68161
[37] C. Fonseca and P. Fleming, “An overview of evolutionary algorithms in multiobjective optimization,” Evolutionary Computation, vol. 3, pp. 1-16, 1995.
[38] Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, Springer, New York, NY, USA, 1994. · Zbl 0818.68017
[39] D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, New York, NY, USA, 1989. · Zbl 0721.68056
[40] M. Gen and R. Cheng, Gennetic Algorithms and Engineering Design, Wiley, New York, NY, USA, 1997.
[41] J. Gao and B. Liu, “Fuzzy multilevel programming with a hybrid intelligent algorithm,” Computers & Mathematics with Applications, vol. 49, no. 9-10, pp. 1539-1548, 2005. · Zbl 1138.90508 · doi:10.1016/j.camwa.2004.07.027