Applications of fuzzy set theory to mathematical programming. (English) Zbl 0578.90095

Summary: Mathematical programming is one of the areas to which fuzzy set theory has been applied extensively. Primarily based on Bellman and Zadeh’s model of decision in fuzzy environments, models have been suggested which allow flexibility in constraints and fuzziness in the objective function in traditional linear and nonlinear programming, in integer and fractional programming, and in dynamic programming. These models in turn have been used to offer computationally efficient approaches for solving vector maximum problems. This paper surveys major models and theories in this area and offers some indication on future developments which can be expected.


90C99 Mathematical programming
03E72 Theory of fuzzy sets, etc.
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
Full Text: DOI


[1] Behringer, F. A., Lexicographic quasiconcave multiobjective programming, Z. Oper. Res., 21, 103-116 (1977) · Zbl 0362.90101
[2] Behringer, F. A., A simplex based algorithm for the lexicographically extended linear maxmin problem, European J. Oper. Res., 7, 274-283 (1981) · Zbl 0455.90053
[3] Bellman, R. E.; Zadeh, L. A., Decision-making in a fuzzy environment, Management Sci., 17, B141-B164 (1970) · Zbl 0224.90032
[4] Chanas, S., The use of parametric programming in fuzzy linear programming, Fuzzy Sets and Systems, 11, 243-251 (1983) · Zbl 0534.90056
[5] Chang, L. L., Interpretation and execution of fuzzy programs, (Zadeh, L. A.; etal. (1975)), 191-218 · Zbl 0322.68057
[6] Fabian, L.; Stoica, M., Fuzzy integer programming, (Zimmermann, H.-J.; etal. (1984)), 123-132, in [34] · Zbl 0546.90067
[7] Hamacher, H., Über logische Aggregationen nicht binär expliziter Entscheidungskriterien (1978), Frankfurt/Main
[8] Hamacher, H.; Leberling, H.; Zimmermann, H.-J., Sensitivity analysis in fuzzy linear programming, Fuzzy Sets and Systems, 1, 269-281 (1978) · Zbl 0408.90051
[9] Hannan, E. L., Linear programming with multiple fuzzy goals, Fuzzy Sets and Systems, 6, 235-248 (1981) · Zbl 0465.90080
[10] Kuhn, H. W.; Tucker, A. W., Nonlinear programming, (Neyman, J., Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability (1951)) · Zbl 0044.05903
[11] Leberling, H., On finding compromise solutions in multicriteria problems using the fuzzy min-operator, Fuzzy Sets and Systems, 6, 105-118 (1981) · Zbl 0465.90081
[12] Leberling, H., Entscheidungsfindung bei divergierenden Faktorinteressen und relaxierten Kapazitätsrestriktionen mittels eines unscharfen Lösungsansatzes, Z. Betriebsw. Forsch., 35, 398-419 (1983)
[13] Luhandjula, M. K., Linear programming under randomness and fuzziness, Fuzzy Sets and Systems, 10, 45-55 (1983) · Zbl 0514.90067
[14] Luhandjula, M. K., Fuzzy approaches for multiple objective linear fractional optimization, Fuzzy Sets and Systems, 13, 11-24 (1984) · Zbl 0546.90094
[15] K. Nakamura, Some extension of fuzzy linear programming, Fuzzy Sets and Systems; K. Nakamura, Some extension of fuzzy linear programming, Fuzzy Sets and Systems
[16] Orlovsky, S. A., On programming with fuzzy constraint sets, Kybernetes, 6, 197-201 (1977) · Zbl 0369.90129
[17] Ostasiewicz, W., A new approach to fuzzy programming, Fuzzy Sets and Systems, 7, 139-152 (1982) · Zbl 0474.68007
[18] Rödder, W.; Zimmermann, H.-J., Analyse, Beschreibung und Optimierung von unscharf formulierten Problemen, Z. Oper. Res., 21, 1-18 (1977) · Zbl 0397.90099
[19] Rödder, W.; Zimmerman, H.-J., Duality in fuzzy linear programming, (Fiacco, A. V.; Kortanek, K. O., Extremal Methods and Systems Analyses (1980)), 415-429, New York · Zbl 0428.90035
[20] Rubin, P. A.; Narasimhan, R., Fuzzy goal programming with rested priorities, Fuzzy Sets and Systems, 14, 115-130 (1984) · Zbl 0546.90092
[21] Tanaka, H.; Asai, K., Fuzzy linear programming problems with fuzzy numbers, Fuzzy Sets and Systems, 13, 1-10 (1984) · Zbl 0546.90062
[22] Tanaka, K.; Mizumoto, M., Fuzzy programs and their execution, (A. Zadeh, L.; etal. (1975)), 41-76, in [28] · Zbl 0314.68008
[23] Tanaka, H.; Okuda, T.; Asai, K., On fuzzy mathematical programming, J. Cybern., 3, 37-46 (1974) · Zbl 0297.90098
[24] Thole, U.; Zimmermann, H.-J.; Zysno, P., On the suitability of minimum and product operators for the intersection of fuzzy sets, Fuzzy Sets and Systems, 2, 167-180 (1979) · Zbl 0408.94030
[25] Verdegay, J. L., A dual approach to solve the fuzzy linear programming problem, Fuzzy Sets and Systems, 14, 131-141 (1984) · Zbl 0549.90064
[26] Werners, B., Interaktive Entscheidungsunterstützung durch ein flexibles mathematisches Programmierungssystem (1984), München
[27] Zadeh, L. A., On fuzzy algorithms, (Memo ERL-M325 (1972), Univ. of California: Univ. of California Berkeley) · Zbl 0182.33301
[28] (Zadeh, L. A.; Fu, K. S.; Tanaka, K.; Shimura, M., Fuzzy Sets and Their Applications to Cognitive and Decision Processes (1975)), New York · Zbl 0307.00008
[29] Zangwill, W. I., Nonlinear Programming (1969), Prentice-Hall · Zbl 0191.49101
[30] Zimmermann, H.-J., Description and optimization of fuzzy systems, Internat. J. Gen. Systems, 2, 209-215 (1976) · Zbl 0338.90055
[31] Zimmermann, H.-J., Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems, 1, 45-55 (1978) · Zbl 0364.90065
[32] Zimmermann, H.-J.; Zysno, P., Latent connectives in human decision making, Fuzzy Sets and Systems, 4, 37-51 (1980) · Zbl 0435.90009
[33] Zimmermann, H.-J.; Zysno, P., Decisions and evaluations by hierarchical aggregation of information, Fuzzy Sets and Systems, 10, 243-266 (1983) · Zbl 0519.90049
[34] (Zimmermann, H.-J.; Zadeh, L. A.; Gaines, B. R., Fuzzy Sets and Decision Analysis (1984)), New York · Zbl 0534.00023
[35] Karmarkar, N., A new polynomial-time algorithm in linear programming, (Proceedings of the · Zbl 0557.90065
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