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Fuzzy optimization: An appraisal. (English) Zbl 0677.90088

Summary: This paper takes a general look at core ideas that make up the burgeoning body of fuzzy mathematical programming emphasizing the methodological view. Although fuzzy mathematical programming has enjoyed a rapidly increasing acceptance within the scientific community, some technical hurdles exist to hinder unanimity. Reasons for this as well as possible ways for improvement are also discussed.

MSC:

90C99 Mathematical programming
90B50 Management decision making, including multiple objectives
03E72 Theory of fuzzy sets, etc.
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[1] Afgoun, A., Solving a production problem in the presence of ill known data (in French) (1985), Institut d’Informatique de Tizi-Ouzou
[2] Baldwin, J. F., Support logic programming, (Proceedings of the Nato Advanced Study Institute on Fuzzy Sets (1985)) · Zbl 0606.68089
[3] Baptistella, A., A fuzzy approach to solven on convex multicriteria optimization, (Workshop on Mathematical Programming with Multiple Objectives. Workshop on Mathematical Programming with Multiple Objectives, Brussels (1980))
[4] Baptistella, A.; Ollero, A., Fuzzy methodologies for interactive multicriteria optimization, IEEE Trans. System Man Cybernet., 10, 355-365 (1980) · Zbl 0442.90088
[5] Bellman, R.; Zadeh, L. A., Decision making in a fuzzy environment, Management Sci., 17, 141-164 (1970) · Zbl 0224.90032
[6] Blankenship, J. W.; Falk, J. E., Infinitely constrained optimisation problems, J. Optim. Theory Appl., 19, 261-281 (1976) · Zbl 0307.90071
[7] Buckley, J. J., Fuzzy programming and the Pareto optimal set, Fuzzy Sets and Systems, 10, 57-63 (1983) · Zbl 0517.90073
[8] Buckley, J. J., Possibilistic programming, Fuzzy Sets and Systems, 26, 135-138 (1988) · Zbl 0644.90059
[9] Buckley, J. J., Possibility and necessity in optimization, Fuzzy Sets and Systems, 25, 1-13 (1988) · Zbl 0638.90078
[10] Carlsson, C., Tackling an MCDM-problem with the help of some results from fuzzy sets theory, European J. Oper. Res., 10, 270-281 (1982) · Zbl 0481.90050
[11] Carlsson, C., An approach to handle fuzzy problems structures, Cybernet. and Systems, 14, 33-54 (1983) · Zbl 0541.93002
[12] Carlsson, C.; Korhonen, P., A parametric approach to fuzzy linear programming, Fuzzy Sets and Systems, 20, 17-30 (1986) · Zbl 0603.90093
[13] Chanas, S., Parametric programming in fuzzy linear programming, Fuzzy Sets and Systems, 11, 243-251 (1983) · Zbl 0534.90056
[14] Chanas, S., Fuzzy programming in multiple objective linear programming: A parametrical approach, Fuzzy Sets and Systems, 29, 303-313 (1989) · Zbl 0676.90077
[15] Chanas, S.; Kulej, M., A fuzzy linear programming problem with equality constraints, Control and Cybernet., 13, 195-201 (1984) · Zbl 0551.90060
[16] Czogala, E., On distribution function description of probabilistic set and its applications in decision making, Fuzzy Sets and Systems, 10, 21-29 (1983) · Zbl 0535.62011
[17] Delgado, J. L.; Verdegay, J. L.; Vila, M. A., Mathematical programming problems with fuzzy costs, (Proceedings of the First IFSA Congress. Proceedings of the First IFSA Congress, Palma, Spain (1985)) · Zbl 0637.90063
[18] Dubois, D., Linear programming with fuzzy data, (Presented at the International Conference on Fuzzy Information Processing. Presented at the International Conference on Fuzzy Information Processing, Kauai, Hawaiian Islands (1984)) · Zbl 0657.90064
[19] Dubois, D.; Prade, H., Systems of fuzzy linear constraints, Fuzzy Sets and Systems, 3, 37-48 (1978) · Zbl 0425.94029
[20] Dubois, D.; Prade, H., Fuzzy Sets and Systems: Theory and Applications (1980), Academic Press: Academic Press New York · Zbl 0444.94049
[21] Dumitru, V.; Luban, F., On some optimization problems under uncertainty, Fuzzy Sets and Systems, 18, 257-272 (1986) · Zbl 0629.90066
[22] Dyson, R. G., Maximin programming, fuzzy linear programming and multicriteria decision making, J. Oper. Res. Soc., 31, 263-267 (1980) · Zbl 0436.90106
[23] Ernst, E., Fahrplanerstellung und Umlaufdisposition in Containerschiffsverkehr, Diss. Aachen (1982)
[24] Fabian, C.; Ciobanu, G.; Stoica, M., O metoda generala pentru rezolvarea problemelor multicriteriale, Econom. Comput. Econom. Cybernet. Stud. Res., 8, 47-55 (1983)
[25] Feng, Y. J., A method using fuzzy mathematical programming to solve the vector maximum problem, Fuzzy Sets and Systems, 9, 129-136 (1983)
[26] Flachs, J.; Pollatcheck, M. A., Further results on fuzzy mathematical programming, Inform. and Control, 38, 241-257 (1979) · Zbl 0412.90079
[27] Freeling, A. N.S., Fuzzy sets and decision analysis, IEEE Trans. Systems Man Cybernet., 10, 254-341 (1980)
[28] Glashoff, G.; Gustafson, S. A., Linear Optimization and Approximation (1983), Springer: Springer Berlin-New York
[29] Hamacher, H.; Leberling, H.; Zimmermann, H.-J., Sensitivity analysis in fuzzy linear programming, Fuzzy Sets and Systems, 1, 269-281 (1978) · Zbl 0408.90051
[30] Hannan, E. L., On the efficiency of the product operator in fuzzy programming with multiple objectives, Fuzzy Sets and Systems, 2, 259-262 (1979) · Zbl 0413.90070
[31] Hannan, E. L., Linear programming with multiple fuzzy goals, Fuzzy Sets and Systems, 6, 235-248 (1981) · Zbl 0465.90080
[32] Hettich, R., Semi-infinite Programming (1979), Springer: Springer Berlin-New York · Zbl 0406.90063
[33] Hirota, K., Concepts of probabilistics sets, Fuzzy Sets and Systems, 5, 31-46 (1981)
[34] Hogna, A. J.; Morris, J. G.; Thompson, H. E., Decision problems under risk and chance constrained programming: Dilemma in the transition, Management Sci., 27, 698-716 (1981)
[35] Holtz, M.; Desonki, D., Fuzzy Modell für Instandhaltung, (Unscharfe Modellbildung und Steuerung LV (1981), Karl-Marx-Stadt)
[36] Ignizio, J. P.; Daniels, S. D., Fuzzy multicriteria integer programming via fuzzy generalized networks, Fuzzy Sets and Systems, 11, 1-10 (1983)
[37] Kacprzyk, J., Multistage decision making under fuzziness (1983), Verlag TUV Rheinland: Verlag TUV Rheinland Köln · Zbl 0507.90023
[38] Kacprzyk, J., Towards ‘human-consistent’ multistage decision making and control models using fuzzy sets and fuzzy logic, Fuzzy Sets and Systems, 18, 299-314 (1986) · Zbl 0613.90057
[39] Kall, P., Stochastic Linear Programming (1976), Springer: Springer Berlin-New York · Zbl 0317.90042
[40] Kickert, W. J.M., Fuzzy Theories on Decision Making (1978), Martinus Nijhoff: Martinus Nijhoff Dordrecht · Zbl 0364.93022
[41] Kuz’min, V. B., A parametric approach to description of linguistic values of variables and hedges, Fuzzy Sets and Systems, 6, 27-41 (1981)
[42] Kwakernaak, H., Fuzzy random variables, Inform. Sci., 1, 1-29 (1978) · Zbl 0438.60004
[43] Lebbah, H., A fuzzy mathematical programming approach for a diet problem (1985), Institut d’Informatique de Tizi-Ouzou
[44] Leberling, H., On finding compromise solutions in multicriteria problems using the min operator, Fuzzy Sets and Systems, 6, 105-118 (1981) · Zbl 0465.90081
[45] Little, J. D.C., Model and Managers, Management Sci., 16, 446-458 (1970)
[46] Luhandjula, M. K., Compensatory operators in fuzzy linear programming with multiple objectives, Fuzzy Sets and Systems, 8, 245-252 (1982) · Zbl 0492.90076
[47] Luhandjula, M. K., Linear programming under randomnes and fuzziness, Fuzzy Sets and Systems, 10, 45-55 (1983) · Zbl 0514.90067
[48] Luhandjula, M. K., Fuzzy approaches for multiple objective linear fractional optimization, Fuzzy Sets and Systems, 13, 11-24 (1984) · Zbl 0546.90094
[49] Luhandjula, M. K., On possibilistic linear programming, Fuzzy Sets and Systems, 18, 15-30 (1986) · Zbl 0616.90038
[50] Luhandjula, M. K., Multiple objective programming problems with possibilistic coefficients, Fuzzy Sets and Systems, 21, 135-145 (1987) · Zbl 0621.90085
[51] Luhandjula, M. K., Linear programming with a possibilistic objective function, European J. Oper. Res., 31, 110-117 (1987) · Zbl 0635.90057
[52] Luhandjula, M. K., Satisfying solutions for possibilistic linear programming, Inform. Sci., 40, 247-265 (1987) · Zbl 0623.90091
[53] Mekaouche, A.; Rezzik, C., User friendly software for fuzzy optimization (in French) (1986), Institut d’Informatique de Tizi-Ouzou
[54] Nakamura, K., Some extensions of fuzzy linear programming, (Working Paper RWTH Aahen (1978)) · Zbl 0552.90063
[55] Narasimhan, R., Goal programming in a fuzzy environment, Decision Sci., 11, 325-336 (1980)
[56] Negoita, C. V., Management Applications of System Theory (1975), Birkhauser: Birkhauser Basel-Boston · Zbl 0326.94002
[57] Negoita, C. V., The current interest in fuzzy optimization, Fuzzy Sets and Systems, 6, 261-269 (1981) · Zbl 0465.90091
[58] Negoita, C. V.; Sularia, M., On fuzzy mathematical programming and tolerances in planning, Econom. Comput. Econom. Cybernet Stud. Res., 1, 3-15 (1976) · Zbl 0336.90060
[59] ÓhÉigeartaigh, A fuzzy transportation algorithm, Fuzzy Sets and Systems, 8, 235-243 (1982) · Zbl 0493.90058
[60] Orlovski, S. A., On programming with fuzzy constraints sets, Kybernetes, 6, 197-201 (1977)
[61] Orlovski, S. A., Decision making with a fuzzy preference relation, Fuzzy Sets and Systems, 1, 155-167 (1978) · Zbl 0396.90004
[62] Orlovski, S. A., On formalization of a general mathematical programming problem, Fuzzy Sets and Systems, 3, 311-321 (1980) · Zbl 0435.90008
[63] Orlovski, S. A., Mathematical programming problems with fuzzy parameters, (Working Paper IIASA (1984)) · Zbl 0624.90105
[64] Orlovski, S. A., Two approaches to multiple objective programming problems with fuzzy parameters, (Working Paper IIASA (1984)) · Zbl 0624.90105
[65] Pearman, A. D.; Kmietowicz, Z. W., Stochastic dominance with linear partial information, European J. Oper. Res., 23, 57-63 (1986) · Zbl 0581.90041
[66] Philip, P., Algorithms for the vector maximization problem, Math. Prog., 2, 229-297 (1972) · Zbl 0288.90052
[67] Ramik, J., Extension principle and fuzzy mathematical programming, Kybernetica, 19, 516-523 (1983) · Zbl 0528.90059
[68] Ramik, J.; Rimanek, J., Inequality relation between fuzzy numbers and its use in fuzzy optimization, Fuzzy Sets and Systems, 16, 123-138 (1985) · Zbl 0574.04005
[69] Rinks, D. B., A heuristic approach to aggregate production scheduling using linguistic variables, (Yager, R., Fuzzy Sets and Possibility Theory. Recent Developments (1982), Pergamon Press: Pergamon Press New York), 582-588
[70] Rödder, W.; Zimmermann, H.-J., Duality in fuzzy linear programming, (Working Paper, RWTH Aachen (1977)) · Zbl 0585.90056
[71] Rommelfanger, H., Concave membership functions and their application in fuzzy mathematical programming, (Presented at the Euro Working Group on Fuzzy Sets. Presented at the Euro Working Group on Fuzzy Sets, Brussels (1984))
[72] Rommelfanger, H.; Hanuschek, R.; Wolf, J., Linear programming with fuzzy objective functions, (Presented at the first IFSA Congress. Presented at the first IFSA Congress, Palma de Mallorca (1985))
[73] Roubens, M.; Teghem, J., Comparison between Multiple objective fuzzy linear programming and multiple objective stochastic linear programming, (Working Paper, Mons (1986)) · Zbl 0655.90072
[74] Roy, B., Problems and methods with multiple objective functions, Math. Prog., 1, 239-266 (1971) · Zbl 0254.90061
[75] Rudin, P. A.; Narasimhan, R., Fuzzy goal programming with nested priorities, Fuzzy Sets and Systems, 14, 115-130 (1984) · Zbl 0546.90092
[76] Sakawa, M.; Yumine, T., Interactive fuzzy decision making for multiple objective linear programming problems, Large Scale Systems, 5, 105-114 (1983) · Zbl 0533.90085
[77] Sakawa, L.; Yumine, T.; Nango, Y., Interactive fuzzy decision making for multiple objective nonlinear programming problems, Trans. Inst. Elec. Commun. Eng. Japan, 1243-1250 (1983)
[78] Sakawa, L.; Yumine, T., Interactive fuzzy decision making for multiple objective nonlinear programming using augmented minimax problems, Fuzzy Sets and Systems, 20, 31-43 (1986) · Zbl 0629.90054
[79] Sher, A. P., Solution of the mathematical programming problem with a linear objective function and fuzzy constraints, Automat. i Telemeh., 41, 137-143 (1980) · Zbl 0453.90059
[80] Simon, H. A., Administrative Behavior (1945), Free press: Free press New York
[81] Slowinski, R., A multicriteria fuzzy linear programming method for water supply system development planning, Fuzzy Sets and Systems, 19, 217-237 (1986) · Zbl 0626.90085
[82] Slowinski, R.; Teghem, J., Comparison des approches floues et stochastiques en programmation linéaire multicritère en avenir incertain, LAMSADE, Document No. 35 (1986)
[83] Sommer, G.; Pollatschek, M. A., A fuzzy programming approach to an air pollution regulation problem, (Working Paper, RWTH Aachen (1976)) · Zbl 0422.90030
[84] Soyster, A. L., Convex programming with set inclusive constraints Application to inexact programming, Oper. Res., 21, 1154-1157 (1973) · Zbl 0266.90046
[85] Steuer, R., Algorithms for linear programming problems with interval objective function coefficients, Math. Oper. Res., 6, 333-348 (1981) · Zbl 0491.90059
[86] Stoica, M.; Fabian, C., Fuzzy integer programming, (Zimmermann, H.-J.; Zadeh, L. A.; Gaines, B. R., Fuzzy Sets and Decision Analysis (1984), North-Holland: North-Holland Amsterdam), 123-132 · Zbl 0546.90067
[87] Sugeno, M., An introductory survey of fuzzy control, Inform. Sci., 36, 59-83 (1985) · Zbl 0586.93053
[88] Tanaka, H.; Asai, J., Fuzzy linear programming based on fuzzy functions, Bull. Univ. Osaka Prefecture, 29, 113-125 (1980) · Zbl 0525.90065
[89] Tanaka, H.; Asai, K., Fuzzy linear programming with fuzzy numbers, Fuzzy Sets and Systems, 13, 1-10 (1984) · Zbl 0546.90062
[90] Tanaka, H.; Ichihachi, H.; Asai, K., A formulation of fuzzy linear programming based on comparison of fuzzy numbers, Control and Cybernet., 13, 185-194 (1984) · Zbl 0551.90062
[91] Tanaka, H.; Ichihashi, H.; Asai, K., Fuzzy decision in linear programming with trapezoid fuzzy parameters, (Kackprzyk, J.; Yager, R., Decision Systems Using Fuzzy Sets and Possibility Theory (1985), Verlag TÜV Rheinland: Verlag TÜV Rheinland Köln) · Zbl 0657.90063
[92] Tanaka, H.; Ichihashi, H.; Asai, K., A value of information in fuzzy linear programming problems via sensitivity analysis, Fuzzy Sets and Systems, 18, 119-129 (1986) · Zbl 0601.90098
[93] Tanaka, H.; Okuda, T.; Asai, K., On fuzzy mathematical programming, Cybernet., 3, 37-46 (1974) · Zbl 0297.90098
[94] Takeda, D.; Nishida, T., Multiple criteria decision problems with domination structures, Fuzzy Sets and Systems, 3, 123-136 (1980) · Zbl 0429.90073
[95] Timsi, B.; Kerri, D., Programmation semi-infinite: Analyse réalisation et applications, (Diss. Institu d’Informatique de Tizi-Ouzou (1987))
[96] Tiwari, R. N.; Dhamar, S.; Rao, J. R., Priority structure in fuzzy goal programming, Fuzzy Sets and Systems, 19, 251-259 (1986) · Zbl 0602.90078
[97] Vaja, S., Probabilistic Programming (1972), Academic Press: Academic Press New York
[98] Verdegay, J. L., Fuzzy mathematical programming, (Gupta, M. M.; Sanchez, E., Fuzzy Information and Decision Processes (1982), North-Holland: North-Holland Amsterdam), 231-237 · Zbl 0504.90056
[99] Verdegay, J. L., A dual approach to solve the fuzzy linear programming problems, Fuzzy Sets and Systems, 14, 131-141 (1984) · Zbl 0549.90064
[100] Werners, B., Interaktive Entscheidungsunterstützung durch ein flexibles mathematisches Programmierungsystem, (Diss. Aachen (1984))
[101] Wiedey, G.; Zimmermann, H.-J., Media selection and fuzzy linear programming, J. Oper. Res. Soc., 29, 1071-1084 (1978) · Zbl 0388.90087
[102] Yager, R. R., A foundation of theory of possibility, J. Cybernet., 10, 177-204 (1980) · Zbl 0438.94042
[103] Yazenin, A. V., Fuzzy and stochastic programming, Fuzzy Sets and Systems, 22, 171-180 (1987) · Zbl 0623.90058
[104] Zadeh, L. A., Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1, 3-28 (1978) · Zbl 0377.04002
[105] Zimmermann, H.-J., Description and optimization of fuzzy systems, Internat. J. General Systems, 2, 209-215 (1976) · Zbl 0338.90055
[106] Zimmermann, H.-J., Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems, 1, 46-55 (1978) · Zbl 0364.90065
[107] Zimmermann, H.-J., Applications of fuzzy set theory to mathematical programming, Inform. Sci., 36, 29-58 (1985) · Zbl 0578.90095
[108] Zimmermann, H.-J., Fuzzy Sets and Applications (1985), Kluwer Nijhoff: Kluwer Nijhoff Dordrecht
[109] Zimmermann, H.-J.; Pollatschek, M. A., Fuzzy 0-1 linear programming, (Zimmermann, H.-J.; Zadeh, L. A.; Gaines, B., Fuzzy Sets and Decision Analysis (1984), North-Holland: North-Holland Amsterdam) · Zbl 0545.90070
[110] Zimmermann, H.-J.; Zysno, P., Latent connectives in human decision making, Fuzzy Sets and Systems, 4, 37-51 (1980) · Zbl 0435.90009
[111] Zimmermann, H.-J.; Zysno, P., Decisions and evaluations by hierarchical aggregation of information, Fuzzy Sets and Systems, 10, 243-260 (1983) · Zbl 0519.90049
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