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Fuzzy linear programming and applications. (English) Zbl 0914.90265
Summary: This paper presents a survey on methods for solving fuzzy linear programs. First LP models with soft constraints are discussed. Then LP problems in which coefficients of constraints and/or of the objective function may be fuzzy are outlined. Pivotal questions are the interpretation of the inequality relation in fuzzy constraints and the meaning of fuzzy objectives. In addition to the commonly applied extended addition, based on the min-operator and used for the aggregation of the left-hand sides of fuzzy constraints and fuzzy objectives, a more flexible procedure, based on Yager’s parametrized $$t$$-norm $$T_p$$, is presented. Finally, practical applications of fuzzy linear programs are listed.

##### MSC:
 90C70 Fuzzy and other nonstochastic uncertainty mathematical programming 90C10 Integer programming
##### Keywords:
survey; fuzzy linear programs; fuzzy constraints
FULPAL
Full Text:
##### References:
 [1] Bellman, R.E.; Zadeh, L.A., Decision-making in a fuzzy environment, Management science, 17, 149-156, (1973) [2] Bortolan, G.; Degani, R., Ranking fuzzy subsets, Fuzzy sets and systems, 15, 1-19, (1985) · Zbl 0567.90056 [3] Buckley, J.J., Possibilistic linear programming with triangular fuzzy numbers, Fuzzy sets and systems, 26, 135-138, (1988) · Zbl 0788.90076 [4] Buckley, J.J., Solving possibilistic linear programming problems, Fuzzy sets and systems, 31, 329-341, (1989) · Zbl 0671.90049 [5] Carlsson, C.; Korhonen, P., A parametric approach to fuzzy linear programming, Fuzzy sets and systems, 20, 17-30, (1986) · Zbl 0603.90093 [6] Chanas, S., Fuzzy programming in multiobjective linear programming — A parametric approach, Fuzzy sets and systems, 29, 303-313, (1989) · Zbl 0676.90077 [7] Czyzak, P., Application of the ‘FLIP’ method to farm structure optimization under uncertainty, (), 263-278 [8] Darzentas, J., On fuzzy location models, (), 328-341 [9] Delgado, M.; Verdegay, J.L.; Vila, M.A., A general model for fuzzy linear programming, Fuzzy sets and systems, 29, 21-30, (1989) · Zbl 0662.90049 [10] Dubois, D.; Prade, H., Fuzzy sets and systems: theory and applications, (1980), Academic Press New York · Zbl 0444.94049 [11] Dubois, D.; Prade, H., Ranking of fuzzy numbers in the setting of possibility theory, Information sciences, 30, 183-224, (1989) · Zbl 0569.94031 [12] Hannan, E.L., Linear programming with multiple fuzzy goals, Fuzzy sets and systems, 6, 235-248, (1981) · Zbl 0465.90080 [13] Hanuscheck, R., Investitionsplanung auf der grundlage vager daten, (1986), Schulz-Kirchner Verlag Idstein [14] Kacprzyk, J., A branch and bound algorithm for the multistage control of a fuzzy system in a fuzzy environment, Kybernetes, 8, 139-147, (1979) · Zbl 0401.49026 [15] Kacprzyk, J., Multistage decision-making under fuzzyness, (1983), TUV Rheinland Köln [16] () [17] Kivijärvi, J.; Korhonen, P.; Wallenius, J., Operations research and its practice in Finland, Interfaces, 16, 53-59, (1986) [18] Lai, Y.-J.; Hwang, Ch.-L., Fuzzy mathematical programming: methods and applications, (1992), Springer-Verlag Heidelberg [19] Lai, Y.-J.; Hwang, Ch.-L., Fuzzy multiple objective decision making, methods and applications, (1994), Springer-Verlag Heidelberg [20] 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 [21] Leberling, H., Entscheidungsfindung bei divergierenden faktorinteressen und relaxierten kapazitätsrestriktionen mittels eines unscharfen Lösungsansatzes, Zeitschrift für betriebswirtschaftliche forschung, 35, 398-419, (1983) [22] Leung, Y.; Leung, Y., Interregional equilibrium and fuzzy linear programming, Environment and planning A, Environment and planning A, 20, 219-230, (1988) [23] Lilien, G., MS/OR: A mid-life crisis, Interfaces, 17, 53-59, (1987) [24] Luhandjula, M.K., Multiple objective programming with possibilistic coefficients, Fuzzy sets and systems, 21, 135-146, (1987) · Zbl 0621.90085 [25] Meyer zu Selhausen, H., Repositioning OR’s products in the market, Interfaces, 19, 79-87, (1989) [26] Mjelde, K.M., Fuzzy resource allocation, Fuzzy sets and systems, 19, 239-250, (1986) · Zbl 0603.90083 [27] Nakamura, K., Some extensions of fuzzy linear programming, Fuzzy sets and systems, 14, 211-219, (1984) · Zbl 0552.90063 [28] Negoita, C.V.; Ralescu, D., Simulation, knowledge-based computing, and fuzzy statistics, (1987), Van Nostrand Reinhold New York · Zbl 0683.68097 [29] Negoita, C.V.; Sularia, M., On fuzzy mathematical programming and tolerances in planning, Economic computation and economic cybernetics studies and research, 3, 3-15, (1976) · Zbl 0336.90060 [30] Negoita, C.V.; Minoiu, S.; Stan, E., On considering imprecision in dynamic linear programming, Economic computation and economic cybernetics studies and research, 3, 83-95, (1976) · Zbl 0364.90114 [31] Oder, C.; Rentz, O., Entwicklung eines auf der theorie unscharfer mengen basierenden energie-emissions-modells, (), 111-118 [32] Orlovski, S.A., Mathematical programming problems with fuzzy parameters, (), 136-145 [33] Ostermark, R., Profit apportionment in concerns with mutual ownership - an application of fuzzy inequalities, Fuzzy sets and systems, 26, 283-297, (1988) [34] Ostermark, R., Fuzzy linear constraints in the capital asset pricing model, Fuzzy sets and systems, 30, 93-102, (1989) · Zbl 0664.90008 [35] Owsinski, J.W.; Zadrozny, S.; Kacprzyk, J., Analysis of water use and needs in agriculture through a fuzzy programming model, (), 377-395 [36] Ramik, J., A unified approach to fuzzy optimization, (), 128-130 [37] Ramik, J.; Rimanek, J., Inequality between fuzzy numbers and its use in fuzzy optimization, Fuzzy sets and systems, 16, 123-138, (1985) · Zbl 0574.04005 [38] Ramik, J.; Rimanek, J., Fuzzy parameters in optimal allocation of resources, (), 359-374 [39] Ramik, J.; Rommelfanger, H., A single- and a multi-valued order on fuzzy numbers and its use in linear programming with fuzzy coefficients, Fuzzy sets and systems, 57, 203-208, (1993) · Zbl 0787.90107 [40] Rommelfanger, H., Concave membership functions and their application in fuzzy mathematical programming, (), 88-101 [41] Rommelfanger, H., Rangordnungsverfahren für unscharfe mengen, OR spektrum, 8, 219-228, (1986) · Zbl 0604.90006 [42] Rommelfanger, H.; Rommelfanger, H., Entscheiden bei unschärfe — fuzzy decision support-systeme, (1994), Springer-Verlag Berlin/Heidelberg · Zbl 0795.90001 [43] Rommelfanger, H., Inequality relations in fuzzy constraints and its use in linear fuzzy optimization, (), 195-211 [44] Rommelfanger, H., FULPAL — an interactive method for solving multiobjective fuzzy linear programming problems, (), 279-299 · Zbl 0734.90120 [45] Rommelfanger, H., FULP — A PC-supported procedure for solving multicriteria linear programming problems with fuzzy data, (), 154-167 · Zbl 0782.90100 [46] Rommelfanger, H., Some problems of fuzzy optimization with t-norm based addition, (), 158-168 · Zbl 0826.90132 [47] Rommelfanger, H.; Keresztfalvi, T., Multicriteria fuzzy optimization based on Yager’s parametrized t-norm, Foundations of computing and decision sciences, 16, 99-110, (1993) · Zbl 0814.90130 [48] Rommelfanger, H.; Keresztfalvi, T., Fuzzy-optimierungsmodelle auf der basis der yagerschen t-norm tp-theoretische und empirische überprüfung des einflusses des parameters p auf die Lösungsmengen und entwicklung geeigneter Lösungsalgorithmen, () [49] Rommelfanger, H.; Hanuscheck, R.; Wolf, J., Linear programming with fuzzy objectives, Fuzzy sets and systems, 29, 31-48, (1989) · Zbl 0662.90045 [50] Roubens, M.; Teghem, J., Comparison of methodologies for multicriteria feasibility constraint fuzzy and multiobjective stochastic linear programming, (), 240-265 [51] Sakawa, M., Interactive computer programs for fuzzy linear programming with multiple objectives, International journal of man-machine studies, 18, 489-503, (1983) · Zbl 0513.90079 [52] Sakawa, M.; Yano, H., Interactive fuzzy satisficing method for multiobjective nonlinear programming problems with fuzzy parameters, Fuzzy sets and systems, 30, 221-238, (1989) · Zbl 0676.90078 [53] Sakawa, M.; Yano, H., Interactive decision making for multiobjective programming problems with fuzzy parameters, (), 191-228 · Zbl 0738.90089 [54] Slowinski, and J. Teghem (ed.), Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming Under Uncertainty, Reidel, Dordrecht, 191-228. · Zbl 0724.00033 [55] Schwab, K.-D., Ein auf dem konzept der unscharfen mengen basierendes entscheidungsmodell bei mehrfachener zielsetzung, (1983), Lang-Verlag Frankfurt [56] 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 [57] Slowinski, R., An interactive method for multiobjective linear programming with fuzzy parameters and its application to water supply planning, (), 396-414 [58] Sommer, G., Lineare ersatzprogramme für unscharfe entscheidungsprobleme — zur optimierung bei unscharfer problembeschreibung, Zeitschrift für operations research, 22, B1-B24, (1978) · Zbl 0372.90081 [59] Sommer, G.; Pollatschek, M.A., A fuzzy programming approach to an air pollution regulation problem, (), 303-313 · Zbl 0422.90030 [60] Spengler, T., Fuzzy-entscheidungsmodelle fur die planung der personalbereitstellung, (), 501-508 [61] Tanaka, H.; Asai, K., Fuzzy linear programming with fuzzy numbers, Fuzzy sets and systems, 13, 1-10, (1984) · Zbl 0546.90062 [62] Tanaka, H.; Ichihashi, H.; Asai, K., A formulation of linear programming problems based on comparison of fuzzy numbers, Control and cybernetics, 13, 185-194, (1984) · Zbl 0551.90062 [63] Tingley, G.A., Can MS/OR Sell itself well enough?, Interfaces, 17, 41-52, (1987) [64] Trappey, J.F.C.; Liu, C.R.; Chang, T.C., Fuzzy non-linear programming: theory and application in manufacturing, International journal of production research, 26, 957-985, (1988) [65] Verdegay, J.L., Application of fuzzy optimization in operational research, Control and cybernetics, 13, 229-239, (1984) · Zbl 0551.90059 [66] Wagenknecht, M.; Hartmann, K., Fuzzy evaluation of Pareto points and its application to hydrocracking processes, (), 415-431 [67] Werners, B., Interaktive entscheidungsunterstützung durch ein flexibles mathematisches programmierungssystem, (1984), Minerwa Publikation München [68] Wolf, J., Lineare fuzzy-modelle zur unterstützung der investitionsentscheidung, (1988), Lang Verlag Frankfurt [69] Yazenin, A.V., Fuzzy and stochastic programming, Fuzzy sets and systems, 22, 171-180, (1987) · Zbl 0623.90058 [70] Zeleny, M., Optimal system design with multiple criteria: de novo programming approach, Engineering cost and production economics, 10, 89-94, (1986) [71] Zimmermann, H.-J., Optimale entscheidungen bei unscharfen problembeschreibungen, Zeitschrift fur betriebswirtschaftliche forschung, 27, 785-795, (1975) [72] Zimmermann, H.-J., Description and optimization of fuzzy systems, International journal of general systems, 2, 209-216, (1976) · Zbl 0338.90055 [73] Zimmermann, H.-J., Fuzzy programming and linear programming with several objective functions, Fuzzy sets and systems, 1, 45-55, (1978) · Zbl 0364.90065 [74] Zimmermann, H.-J., Fuzzy sets, decision making and expert systems, (1987), Kluwer Academic Publishers Boston, MA [75] Zimmermann, H.-J.; Zysno, P., Latent connectives in human decision making, Fuzzy sets and systems, 3, 37-51, (1979) · Zbl 0435.90009 [76] Zimmermann, H.-J.; Zysno, P., Zugehörigkeitsfunktion: modellierung, empirische bestimmung und verwendung in entscheidungsmodellen, Arbeitsbericht 1. teil des DFG-projektes zi, 104, 15-1, (1982), 1982
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