zbMATH — the first resource for mathematics

Linear optimization with bipolar max-min constraints. (English) Zbl 1284.90104
Summary: We consider a generalization of the linear optimization problem with fuzzy relational (in)equality constraints by allowing for bipolar max-min constraints, i.e. constraints in which not only the independent variables but also their negations occur. A necessary condition to have a non-empty feasible domain is given. The feasible domain, if not empty, is algebraically characterized. A simple procedure is described to generate all maximizers of the linear optimization problem considered and is applied to various illustrative example problems.

90C70 Fuzzy and other nonstochastic uncertainty mathematical programming
90C08 Special problems of linear programming (transportation, multi-index, data envelopment analysis, etc.)
Full Text: DOI
[1] Abbasbandy, S.; Babolian, E.; Allame, M., Numerical solution of fuzzy MAX-MIN systems, Applied Mathematics and Computation, 174, 1321-1328, (2006) · Zbl 1094.65035
[2] Allame, M.; Vatankhahan, B., Iteration algorithm for solving ax=b in MAX-MIN algebra, Applied Mathematics and Computation, 175, 269-276, (2006) · Zbl 1088.65027
[3] Czogała, E.; Drewniak, J.; Pedrycz, W., Fuzzy relation equations on a finite set, Fuzzy Sets and Systems, 7, 89-101, (1982) · Zbl 0483.04001
[4] De Baets, B., Analytical solution methods for fuzzy relational equations, (Dubois, D.; Prade, H., Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series, (2000), Kluwer Academic Publishers Dordrecht), 291-340 · Zbl 0970.03044
[5] De Baets, B.; Kerre, E., A primer on solving fuzzy relational equations on the unit interval, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2, 205-225, (1994) · Zbl 1232.03039
[6] Di Nola, A.; Sessa, S.; Pedrycz, W.; Sanchez, E., Fuzzy relation equations and their applications to knowledge engineering, (Theory and Decision Library, Series D. System Theory, Knowledge Engineering and Problem Solving, (1989), Kluwer Academic Publishers) · Zbl 0694.94025
[7] Drewniak, J.; Matusiewicz, Z., Properties of \(\max -^\ast\) fuzzy relation equations, Soft Computing, 14, 1037-1041, (2010) · Zbl 1206.03047
[8] Fang, S.-C.; Li, G., Solving fuzzy relation equations with a linear objective function, Fuzzy Sets and Systems, 103, 107-113, (1999) · Zbl 0933.90069
[9] Gavalec, M., Solvability and unique solvability of MAX-MIN fuzzy equations, Fuzzy Sets and Systems, 124, 385-393, (2001) · Zbl 0994.03047
[10] Gavalec, M.; Zimmermann, K., Solving systems of two-sided (MAX,MIN)-linear equations, Kybernetika, 46, 405-414, (2010) · Zbl 1195.65037
[11] Ghodousian, A.; Khorram, E., Solving a linear programming problem with the convex combination of the MAX-MIN and the MAX-average fuzzy relation equations, Applied Mathematics and Computation, 180, 411-418, (2006) · Zbl 1102.90036
[12] Ghodousian, A.; Khorram, E., Fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with MAX-MIN composition, Information Sciences, 178, 501-519, (2008) · Zbl 1149.90189
[13] Guu, S.-M.; Wu, Y.-K., Minimizing a linear objective function with fuzzy relation equation constraints, Fuzzy Optimization and Decision Making, 347-360, (2002) · Zbl 1055.90094
[14] Higashi, M.; Klir, G. J., Resolution of finite fuzzy relation equations, Fuzzy Sets and Systems, 13, 65-82, (1984) · Zbl 0553.04006
[15] Islam, S.; Roy, T. K., Fuzzy multi-item economic production quantity model under space constraint: a geometric programming approach, Applied Mathematics and Computation, 184, 326-335, (2007) · Zbl 1149.90011
[16] Kabalek, P.; Pozdilkova, A., Maximal solutions of two-sided linear systems in MAX-MIN algebra, Kybernetika, 46, 501-512, (2010) · Zbl 1204.15008
[17] Kantorovich, L. V., Mathematical methods of organizing and planning production, Management Science, 6, 366-422, (1960) · Zbl 0995.90532
[18] Khorram, E.; Ghodousian, A.; Molai, A. A., Solving linear optimization problems with MAX-star composition equation constraints, Applied Mathematics and Computation, 178, 654-661, (2006) · Zbl 1103.65067
[19] Li, J.-X., A new algorithm for minimizing a linear objective function with fuzzy relation equation constraints, Fuzzy Sets and Systems, 159, 2278-2298, (2008) · Zbl 1180.90382
[20] Lichun, C.; Boxing, P., The fuzzy relation equation with union or intersection preserving operator, Fuzzy Sets and Systems, 25, 191-204, (1988) · Zbl 0651.04005
[21] Loetamonphong, J.; Fang, S.-C., An efficient solution procedure for fuzzy relation equations with MAX-product composition, IEEE Transactions on Fuzzy Systems, 7, 441-445, (1999)
[22] Loetamonphong, J.; Fang, S.-C., Optimization of fuzzy relation equations with MAX-product composition, Fuzzy Sets and Systems, 118, 509-517, (2001) · Zbl 1044.90533
[23] Loetamonphong, J.; Fang, S.-C.; Young, R. E., Multi-objective optimization problems with fuzzy relation equation constraints, Fuzzy Sets and Systems, 127, 141-164, (2002) · Zbl 0994.90130
[24] Lu, J.; Fang, S.-C., Solving nonlinear optimization problems with fuzzy relation equation constraints, Fuzzy Sets and Systems, 119, 1-20, (2001)
[25] Luoh, L.; Wang, W.-J.; Liaw, Y.-K., New algorithms for solving fuzzy relation equations, Mathematics and Computers in Simulation, 59, 329-333, (2002) · Zbl 0999.03513
[26] Nitica, V., The structure of MAX-MIN hyperplanes, Linear Algebra and its Applications, 432, 402-429, (2010) · Zbl 1180.52005
[27] Peeva, K.; Kyosev, Y., Fuzzy relational calculus: theory, applications and software, (Advances in Fuzzy Systems, Applications and Theory, vol. 22, (2004), World Scientific Publishing) · Zbl 1083.03048
[28] Peeva, K., Universal algorithm for solving fuzzy relational equations, Italian Journal of Pure and Applied Mathematics, 19, 169-188, (2006) · Zbl 1137.03030
[29] Qu, X.-B.; Wang, X.-P., Minimization of linear objective functions under the constraints expressed by a system of fuzzy relation equations, Information Sciences, 178, 3482-3490, (2008) · Zbl 1190.90300
[30] Sanchez, E., Resolution of composite fuzzy relation equations, Information and Control, 30, 38-48, (1976) · Zbl 0326.02048
[31] Shieh, B.-S., Solutions of fuzzy relation equations based on continuous t-norms, Information Sciences, 177, 4208-4215, (2007) · Zbl 1122.03054
[32] Wang, P. Z.; Zhang, D. Z.; Sanchez, E.; Lee, E. S., Latticized linear programming and fuzzy relation inequalities, Journal of Mathematical Analysis and Applications, 159, 72-87, (1991) · Zbl 0746.90081
[33] Wu, Y.-K., Optimization of fuzzy relational equations with MAX-av composition, Information Sciences, 177, 4216-4229, (2007) · Zbl 1140.90523
[34] Wu, Y.-K.; Guu, S.-M., A note on fuzzy relation programming problems with MAX-strict-t-norm composition, Fuzzy Optimization and Decision Making, 3, 271-278, (2004) · Zbl 1091.90087
[35] Wu, Y.-K.; Guu, S.-M., Minimizing a linear function under a fuzzy MAX-MIN relational equation constraint, Fuzzy Sets and Systems, 150, 147-162, (2005) · Zbl 1074.90057
[36] Wu, Y.-K.; Guu, S.-M.; Liu, J. Y.-C., An accelerated approach for solving fuzzy relation equations with a linear objective function, IEEE Transactions on Fuzzy Systems, 10, 552-558, (2002)
[37] Yeh, C.-T., On the minimal solutions of MAX-MIN fuzzy relational equations, Fuzzy Sets and Systems, 159, 23-39, (2008) · Zbl 1176.03040
[38] Yu, G., MIN-MAX optimization of several classical discrete optimization problems, Journal of Optimization Theory and Applications, 98, 221-242, (1998) · Zbl 0908.90220
[39] Zimmerman, K., Disjunctive optimization, MAX-separable problems and extremal algebras, Theoretical Computer Science, 293, 45-54, (2003) · Zbl 1031.90046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.