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Solving a class of fuzzy linear programs by using semi-infinite programming techniques. (English) Zbl 1061.90122
Summary: This paper deals with a class of Fuzzy Linear Programming problems characterized by the fact that the coefficients in the constraints are modeled as LR-fuzzy numbers with different shapes. Solving such problems is usually more complicated than finding a solution when all the fuzzy coefficients have the same shape. We propose a primal semi-infinite algorithm as a valuable tool for solving this class of Fuzzy Linear programs and, we illustrate it by means of several examples.

##### MSC:
 90C70 Fuzzy programming 90C34 Semi-infinite programming
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##### References:
 [1] Anderson, E. J.; Lewis, A. S.: An extension of the simplex algorithm for semi-infinite linear programming. Math. programming 44, 247-269 (1989) · Zbl 0682.90058 [2] Carlsson, C.; Korhonen, P.: A parametric approach to fuzzy linear programming. Fuzzy sets and systems 20, 17-30 (1986) · Zbl 0603.90093 [3] Chanas, S.: On the interval approximation of a fuzzy number. Fuzzy sets and systems 122, 353-359 (2001) · Zbl 1010.03523 [4] Chang, P. T.; Lee, E. S.: Fuzzy arithmetics and comparison of fuzzy numbers. Fuzzy optimization: recent advances, 69-82 (1994) · Zbl 0925.90384 [5] M. Delgado, J. Kacprzyk, J.-L. Verdegay, M.A. Vila (Eds.), Fuzzy Optimization: Recent Advances, Physica-Verlag, Wursburg, 1994. · Zbl 0812.00011 [6] Dubois, D.; Kerre, E.; Mesiar, R.; Prade, H.: Fuzzy interval analysis. Fundamentals of fuzzy sets, 483-581 (2000) · Zbl 0988.26020 [7] Dubois, D.; Prade, H.: Systems of linear fuzzy constraints. Fuzzy sets and systems 3, 37-48 (1980) · Zbl 0425.94029 [8] D. Dubois, H. Prade, Fuzzy numbers: an overview, in: J. Bezdek (Ed.), Analysis of Fuzzy Information, vol. 2, CRC-Press, Boca Raton, FL, 1988, pp. 3--39. · Zbl 0663.94028 [9] D. Dubois, H. Prade (Eds.), Fundamentals of Fuzzy Sets, Kluwer Academic, Dordrecht, 2000. · Zbl 1018.03045 [10] Fang, S. C.; Hu, C. F.; Wang, H. F.; Wu, S. Y.: Linear programming with fuzzy coefficients in constraints. Comput. math. Appl. 37, 63-76 (1999) · Zbl 0931.90069 [11] Goberna, M. A.; Lopez, M. A.: Linear semi-infinite optimization. (1998) · Zbl 0635.90059 [12] Hettich, R.; Kortanek, K. O.: Semi-infinite programmingtheory, methods and applications. SIAM rev. 35, 380-429 (1993) · Zbl 0784.90090 [13] Inuiguchi, M.; Ichihashi, H.; Tanaka, H.: Fuzzy programming--a survey of recent developments. Stochastic versus fuzzy approaches to multi objective mathematical programming under uncertainty, 45-68 (1990) · Zbl 0728.90091 [14] Klir, G. J.; Folger, T. A.: Fuzzy sets, uncertainty and information. (1988) · Zbl 0675.94025 [15] Kolesárová, A.: Triangular norm-based additions of similar fuzzy numbers and preserving of similarity. Busefal 69, 43-54 (1997) · Zbl 0920.04009 [16] Lai, Y. J.; Hwang, C. L.: Fuzzy mathematical programming: theory and applications. (1992) · Zbl 0793.90094 [17] León, T.; Sanmatı\acute{}as, S.; Vercher, E.: A multi-local optimization algorithm. Top 6, 1-18 (1998) · Zbl 0910.90258 [18] León, T.; Sanmatı\acute{}as, S.; Vercher, E.: On the numerical treatment of linearly constrained semi-infinite problems. Eur. J. Oper. res. 121, 78-91 (2000) [19] T. León, E. Vercher, New descent rules for solving the linear semi-infinite programming problem, Oper. Res. Lett. (1994) 105--114. · Zbl 0810.90123 [20] Pedrycz, W.: Why triangular membership functions?. Fuzzy sets and systems 64, 21-30 (1994) [21] Ramik, J.; Rimanek, J.: Inequality relation between fuzzy numbers and its use in fuzzy optimization. Fuzzy sets and systems 16, 123-138 (1985) [22] Reemtsen, R.; Görner, S.: Numerical methods for semi-infinite programming: a survey. Semi-infinite programming, 195-275 (1998) · Zbl 0908.90255 [23] Rommelfanger, H.; Slowinski, R.: Fuzzy linear programming with single or multiple objective functions. Fuzzy sets in decision analysis, operations research and statistics, 179-213 (1998) [24] Romelfanger, H.: Inequality relations in fuzzy constraints and its use in linear fuzzy optimization. The interface between artificial intelligence and operational research in fuzzy environment, 195-211 (1989) [25] Sakawa, M.: Fuzzy nonlinear programming with single or single or multiple objective functions. Fuzzy sets in decision analysis, operations research and statistics, 215-248 (1998) · Zbl 0927.90106 [26] Tanaka, H.; Guo, P.: Possibilistic data analysis for operations research. (1999) · Zbl 0931.91011 [27] Tanaka, H.; Ichihashi, H.; Asai, K.: A formulation of fuzzy linear programming problems based on comparison of fuzzy numbers. Control cybernet. 13, 186-194 (1984) · Zbl 0551.90062 [28] Zimmermann, H. -J.: Fuzzy set theory and its applications. (1996) · Zbl 0845.04006