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A Survey of Fuzzy Optimization and Mathematical Programming

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Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 368))

Abstract

A brief survey of various concepts, problem classes, issues, etc. related to fuzzy optimization and fuzzy mathematical programming is provided. Emphasis is on various approaches to fuzzy linear programming as the most important from both the practical point of view and the purpose and scope fo the volume.

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Literature

  • Bellman R.E. and Zadeh L.A. (1970), Decision Making in a Fuzzy Environment. Man. Sci. 17 (B), 4, 141–164.

    Google Scholar 

  • Buckley J.J. (1983), Fuzzy Programming and the Pareto Optial Set; Fuzzy Sets and Systems 10, 57–63.

    Google Scholar 

  • Buckley J.J. (1988), Possibility and Necessity in Optimization. Fuzzy Sets and Systems 25, 1–13.

    Article  Google Scholar 

  • Campos L. (1989), Fuzzy Linear Programming Models to Solve Fuzzy Matrix Games. Fuzzy Sets and Systems 32, 275–289.

    Article  Google Scholar 

  • Campos L. and Verdegay J.L. (1986), On fuzzy Games. Proc. of Fall International Seminar on Applied Logic. Palma de Mallorca (Spain).

    Google Scholar 

  • Campos L. and Verdegay J.L. (1989), Linear Programming Problems and Ranking of Fuzzy Numbers. Fuzzy Sets and Systems 32, 1–11.

    Article  Google Scholar 

  • Campos L., M. Delgado and Vila M.A. (1990), Solving Fuzzy Matrix Games by Using a Direct Approach. In Verdegay and Delgado (1990), 69-84.

    Google Scholar 

  • Carls Son Ch. and Korhonen P. (1986), A parametric Approach to Fuzzy Linear Programming. Fuzzy Sets and Systems 20, 17–30.

    Article  Google Scholar 

  • Chanas S. (1983), Parametric Programming in Fuzzy Linear Programming. Fuzzy Sets and Systems, 11, 243–251.

    Article  Google Scholar 

  • Chanas S. (1989), Fuzzy Programming in Multiobjective Linear Programming — A Parametric Approach. Fuzzy Sets and Systems 29, 303–313.

    Article  Google Scholar 

  • Chanas S., Kolodziejczyk W. and Machaj (1984), A Fuzzy Approach to the Transportation Problem. Fuzzy Sets and Systems 13, 211–222.

    Article  Google Scholar 

  • Chanas S. and Kulej M. (1984), A Fuzzy Linear Programming Problem with equality constraints. In Kacprzyk (1984). 195-202.

    Google Scholar 

  • Delgado M. (1983), A Resolution Method for Multiobjective Problems. Europ. J. of Op. Res. 13, 165–172.

    Article  Google Scholar 

  • Delgado M., Verdegay J.L. and Vila M.A. (1985a), Solving the Biobjective Linear Programming Problem: A Fuzzy Approach. In Approximate Reasoning in Expert Systems (M.M. Gupta et al. Eds.), (1985) 317-322. North-Holland.

    Google Scholar 

  • Delgado M., Verdegay J.L. and Vila M.A. (1985b), Fuzzy Vectormaximum Problem and Parametric Programming. (In Spanish). Trabajos de Estadistica y de Investigacion Operativa 36, 2, 126–137.

    Article  Google Scholar 

  • Delgado M., Verdegay J.L. and Vila M.A. (1987a), Imprecise Costs in Mathematical Programming Problems. Control and Cybernetics 16, 113–121.

    Google Scholar 

  • Delgado M., Verdegay J.L. and Vila M.A. (1987b), On Fuzzy Linear Programming Models. Preprints of II IFSA Congress. Tokyo, 715-718.

    Google Scholar 

  • Delgado M., Verdegay J.L. and Vila M.A. (1987c), Fuzzy Transporation Problems: A General Analysis. In Kacprzyk and Orlovski (1987a), 342-358.

    Google Scholar 

  • Delgado M., Verdegay J.L. and Vila M.A. (1989a), A General Model for Fuzzy Linear Programming. Fuzzy Sets and Systems 29, 21–29.

    Article  Google Scholar 

  • Delgado M., Verdegay J.L. and Vila M.A. (1989b), Using Linguistic Labels in Games (In Spanish). Proc. of the III Technical Meeting of the Spanish Association for Artificial Intelligence, 349-354.

    Google Scholar 

  • Delgado M., Verdegay J.L. and Vila M.A. (1990), Relating Different Approaches to solve Linear Programming Problems with Imprecise Costs. Fuzzy Sets and Systems 37, 33–42.

    Article  Google Scholar 

  • Dubois D. and Prade H. (1980a): Systems of Linear Fuzzy Constraints. Fuzzy Sets and Systems 3, 37–48.

    Article  Google Scholar 

  • Dubois D. and Prade H. (1980), Fuzzy Sets and Systems. Theory and Applications. Academic Press, New York.

    Google Scholar 

  • Dyson R.G. (1980), Maxmin Programming, Fuzzy Linear Programming and Multicriteria Decision Making. J. of Oper. Res. Soc. 31, 263–267.

    Google Scholar 

  • Garcia-Aguado M.C. (1990), Sensitivity of Membership Functions in Fuzzy Linear Programming (In Spanish). Ph. D. University of Granada.

    Google Scholar 

  • Hamacher H., Leberling H. and Zimmermann H.J. (1978), Sensitivity Analysis in Fuzzy Linear Programming. Fuzzy Sets and Systems 1, 269–281.

    Article  Google Scholar 

  • Hannan E.L. (1979), On the Efficiency of the Product Operator in Fuzzy Programming with Multiple Objectives. Fuzzy Sets and Systems 2, 259–262.

    Article  Google Scholar 

  • Hannan E.L. (1981a), Linear Programming with Multiple Fuzzy Goals. Fuzzy Sets and Systems 6, 235–248.

    Article  Google Scholar 

  • Hannan E.L. (1981b), On Fuzzy Goal Programming. Decision Sciences 12, 522–531.

    Article  Google Scholar 

  • Hannan E.L. (1981b), Contrasting Fuzzy Goal Programming and «Fuzzy» Multicriteria Programming. Decision Sciences 13, 337–339.

    Article  Google Scholar 

  • Ignizio J.P. (1982), On the (re) Discovery of Fuzzy Goal Programming Decision Sciences 13, 331–336.

    Google Scholar 

  • Kabbara G. (1982), New Utilization of Fuzzy Optimization Method. In M.M. Gupta and E. Sanchez (Eds), Fuzzy Information and Decision Processes, North-Holland, Amsterdam, 239–246.

    Google Scholar 

  • Kacprzyk J. (1983), Multistage Decision — Making under Fuzziness. ISR Series, Verlag TÜV Rheinalnd, Cologne.

    Google Scholar 

  • Kacprzyk J. (1984), Guest Ed.: Special Issue on Fuzzy Sets and Possibility Theory in Optimization Models. Control and Cybernetics 4, No. 3.

    Google Scholar 

  • Kacprzyk J. and Orlovski S.A., Eds (1987a), Opimization Models Using Fuzzy Sets and Possibility Theory. Reidel, Dordrecht.

    Google Scholar 

  • Kacprzyk J. and Orlovski S.A.(1987b), Fuzzy Optimization and Mathematical Programming: A Brief Introduction and Survey. In Kacprzyk and Orlovski (1987a).

    Google Scholar 

  • Kacprzyk J. and Yager R.R. (1984A), «Softer» Optimization and Control Models via Fuzzy Linguistic Quantifiers. Information Sciences 34, 157–178.

    Article  Google Scholar 

  • Kacprzyk J. and Yager R.R. (1984b), Linguistic Quantifiers and Belief Qualification in Fuzzy Multicriteria and Multistage Decision Making. In Kacprzyk (1984), 155-174.

    Google Scholar 

  • Lamata M.T., Moral S. and Verdegay J.L. (1990), Transforming Fuzzy Measures. In Verdegay and Delgado (1990), 146-158.

    Google Scholar 

  • Leberling H. (1981), On finding Compromise Solutions in MultiCriteria Problems Using the Fuzzy Min-Operator. Fuzzy Sets and Systems 6, 105–118.

    Article  Google Scholar 

  • Leung Y. (1982), Multicriteria Conflict Resolution Through a Theory of Displaced Fuzzy Ideal. In M.M. Gupta and E. Sanchez (Eds), Approximate Reasoning in Decision Analysis. North-Holland. 381-390.

    Google Scholar 

  • Llena J. (1985), On Fuzzy Linear Programming. Eur. J. Op. Res. 22, 216–223.

    Article  Google Scholar 

  • Luhandjula M.K. (1982), Compensatory Operators in Fuzzy Linear Programming with Multiple Objectives. Fuzzy Sets and Systems 8, 245–252.

    Article  Google Scholar 

  • Luhandjula M.K. (1983), Linear Programming under Randomness and Fuzziness. Fuzzy Sets and Systems 10, 57–63.

    Article  Google Scholar 

  • Luhandjula M.K. (1984), Fuzzy Approaches for Multiple Objective Linear Fractional Optimization. Fuzzy Sets and Systems 13, 11–23.

    Article  Google Scholar 

  • Luhandjula M.K. (1986), Satisfying Solutions for a Possibilistic Linear Program. Information Sciences 40, 247–265.

    Article  Google Scholar 

  • Luhandjula M.K. (1987a), Multiple Objective Programming Problems with Possibilistic Coefficients. Fuzzy Sets and Systems 21, 135–146.

    Article  Google Scholar 

  • Luhandjula M.K. (1987), Linear Programming with a Possibilistic Objective function. Eur. J. of Op. Res. 31, 110–117.

    Article  Google Scholar 

  • Luhandjula M.K. (1989), Fuzzy Optimization: an appraisal. Fuzzy Sets and Systems 30, 257–282.

    Article  Google Scholar 

  • Nakamura K. (1984), Some Extension of Fuzzy Linear Programming. Fuzzy Sets and Systems 14, 211–229.

    Article  Google Scholar 

  • Narasimhan R. (1980), Goal Programming in a Fuzzy Environment. Decision Sci. 11, 325–336.

    Article  Google Scholar 

  • Narasimhan R. (1981), On Fuzzy Goal Programming — Some Comments. Decision Sci. 12, 532–538.

    Article  Google Scholar 

  • Negoita C.V. (1981), The Current Interest in Fuzzy Optimization. Fuzzy Sets and Systems 6, 261–269.

    Article  Google Scholar 

  • Negoita C.V. (1984), Structure and Logic in Optimization. In Kacprzyk (1984), 121-128.

    Google Scholar 

  • Negoita C.V., Flondor P. and Sularia M. (1977), On Fuzzy Environment in Optimization Problems. In Modern Trends in Cybernetic and Systems. J. Rose and C. Bilciu (Eds). Springer-Verlag.

    Google Scholar 

  • Negoita C.V., S. Minoiu and Stan E. (1976), In Considering Imprecision in Dynamic Linear Programming. Economic Computation and Economic Cybernetics Studies and Research 3, 83–95.

    Google Scholar 

  • Negoita C.V. and Ralescu D. (1975), Applications of Fuzzy Sets to Systems Analysis. Birkhauser-Verlag.

    Google Scholar 

  • Negoita C.V. and Ralescu D. (1977), On Fuzzy Optimization. Kybernetes, 6, 193–195.

    Article  Google Scholar 

  • Negoita C.V. and Stefanescu A.C. (1982), On Fuzzy Optimization. In M.M. Gupta and E. Sanchez (Eds), Fuzzy Information and Decision Processes. North-Holland, Amsterdam, 247-250.

    Google Scholar 

  • Negoita C.V. and Sularia M. (1976): On Fuzzy Programming and Tolerances in Planning. Econom. Comp. Econom. Cybernet. Stud. Res., 1, 3–15.

    Google Scholar 

  • Oh’Eigeartaigh M. (1982), A Fuzzy Transportation Algorithm. Fuzzy Sets and Systems, 8, 235–245.

    Article  Google Scholar 

  • Orlovski S.A. (1977), On Programming with Fuzzy Constraint Sets. Kybernetes 6, 197–201.

    Article  Google Scholar 

  • Orlovski S.A. (1978), Decision Making with a Fuzzy Preference Relation. Fuzzy Sets and Systems 1, 155–167.

    Article  Google Scholar 

  • Orlovski S.A. (1980), On Formulization of a General Fuzzy Mathematical Problem. Fuzzy Sets and Systems 3, 311–321.

    Article  Google Scholar 

  • Orlovski S.A. (1982), Effective Alternatives for Multiple Fuzzy Preference Relations. In R. Trappl (Ed.), Cybvernetics and Systems Research. North-Holland, Amsterdam.

    Google Scholar 

  • Orlovski S.A. (1984), Multiobjective Programming Problems with Fuzzy Parameters. In Kacprzyk (1984), 175-184.

    Google Scholar 

  • Ralescu D. (1984), Optimization in a Fuzzy Environment. In M.M. Gupta and E. Sanchez (Eds), Fuzzy Information, Knowledge Representation and Decision ANalysis. Pergamon Press, Oxford.

    Google Scholar 

  • Ramk J. (1983), Extension Principle and Fuzzy Mathematical Programming. Kybernetika 19, 516–525.

    Google Scholar 

  • Ramik J. (1986), Extension Principle in Fuzzy Optimization. Fuzzy Sets and Systems 19, 29–36.

    Article  Google Scholar 

  • Rödder W. and H.J. Zimmermann (1980): Duality in Fuzzy Linear Programming. In A.V. Fiacco and K.O. Kortanek (Eds), Extremal Methods and Systems Analyses. Springer-Verlag. Berlin, 415–429.

    Google Scholar 

  • Rommelfanger, H., R. Hanuscheck and Wolf J. (1985), Linear Programming with Fuzzy objectives. Fuzzy Sets and Systems 29, 31–48.

    Article  Google Scholar 

  • Rubin P.A. and Narasimhan R. (1984), Fuzzy Goal Programming with Nested Priorities. Fuzzy Sets and Systems 14, 115–129.

    Article  Google Scholar 

  • Slowinski R. (1986), A Multicriteria Fuzzy Linear Programming Method for Water Supply System Development Planning. Fuzzy Sets and Systems 19, 217–238.

    Article  Google Scholar 

  • Sakawa M. (1983), Interactive Computer Programs for Fuzzy Linear Programming with Multiple Objectives. Int J. of Man-Machine Studies 18, 489–503.

    Article  Google Scholar 

  • Sakawa M. (1984a), Interactive Fuzzy Goal Programming for Multiobjective Nonlinear Problems and its Application to Water Quality Management. In Kacprzyk (1984), 217-228.

    Google Scholar 

  • Sakawa M. (1984b), Interactive Multiobjective Decision Making by the Fuzzy Sequential Proxy Optimization Technique: FSPOT. In Zimmermann, Zadeh and Gaines (1984), 241-260.

    Google Scholar 

  • Sakawa M. and F. Seo (1983), Interactive Multiobjective Decision Making in Environmental Systems using the Fuzzy Sequential Proxy Optimization Technique. Large Scale Systems 4, 223–243.

    Google Scholar 

  • Sakawa M. and Yano H. (1985), Interactive Decision Making for Multiobjective Linear Fractional Programming Problems with Fuzzy Parameters. Cybernetics and Systems 16, 377–394.

    Article  Google Scholar 

  • Sakawa M., Yano H. and Yumine T. (1987), An Interactive Fuzzy Satisficing Method for Multiobjective Linear Programming Problems and its Application. IEEE Transactions on Systems, Man and Cybernetics SMC-17, 654-651.

    Google Scholar 

  • Sakawa M., Yumine T. (1983), Interactive Fuzzy Decision Making for Multiobjective Linear Fractional Programming Problems. Large Scale Systems 5, 105–114.

    Google Scholar 

  • Sher A.P. (1980), Solving Mathematical Programming Problem with a Linear Goal Function in Fuzzy Constraints (in Russian). Automation and Remote Control 40, 137–143.

    Google Scholar 

  • Sommer G. and Pollatschek M.A. (1978), A Fuzzy Programming Approach to an Air Pollution Regulation Problem. Eut. J. Op. Res. 10, 303–313.

    Google Scholar 

  • Takeda E. and Nishida N.T. (1980), Multiple Criteria Decision Making with Fuzzy Domination Structures. Fuzzy Sets and Systems 3, 123–136.

    Article  Google Scholar 

  • Tanaka H. and Asai K, (1981), Fuzzy Linear Programming Based on Fuzzy Functions. Proc. 8th EFAC World Congress (Kyoto, Japan), Pergamon Press, Oxford.

    Google Scholar 

  • Tanaka H. and Asai K. (1984a), Fuzzy Linear Programming Problems with Fuzzy Numbers. Fuzzy Sets and Systems 13, 1–10.

    Article  Google Scholar 

  • Tanaka H. and Asai K. (1984b): Fuzzy Solution in Fuzzy Linear Programming Problems. IEEE Trans. on Systems, Man and Cybernetics SMC-14, 285-288.

    Google Scholar 

  • Tanaka H., Ichihashi H. and Asai K. (1984), A Formulation of Fuzzy Linear Programming Problems based on Comparison of Fuzzy Numbers. In Kacprzyk (1984), 185-194.

    Google Scholar 

  • Tanaka H., Okuda T. and Asai K. (1974), On Fuzzy Mathematical Programming. Journal of Cybernetics, 3, 37–46.

    Article  Google Scholar 

  • Verdegay J.L. (1982): Fuzzy Mathematical Programming. In Fuzzy Information and Decision Processes (M.M. Gupta and E. Sanchez, Eds.). North-Holland, 231-237.

    Google Scholar 

  • Verdegay J.L. (1983), Transportation Problems with Fuzzy Parameter. (In Spanish). Rev. Acad. Ciencias Mat. Fis. Quim y Nat. De Granada, 2, 47–56.

    Google Scholar 

  • Verdegay J.L. (1984a), A Dual Approach to Solve the Fuzzy Linear Programming Problem. Fuzzy Sets and Systems 14, 131–141.

    Article  Google Scholar 

  • Verdegay J.L. (1984b), Application of Fuzzy Optimization in Operational Research. In Kacprzyk (1984) 229-239.

    Google Scholar 

  • Verdegay J.L. (1987), Fuzzy Mathematical Programming Problem: Resolution. In M.G. Singh (Ed.), Systems and Control Encyclopedia. Theory, Technology, Applications. Pergamon Press, 1816-1819.

    Google Scholar 

  • Verdegay J.L. and Delgado M. (1989), The Interface between Artificial Intelligence and Operations Research in Fuzzy Environment. Verlag TÜV Rheinland, ISR Series no. 95.

    Google Scholar 

  • Verdegay J.L. and Delgado M. (1990), Approximate Reasoning Tools for Artificial Intelligence. Verlag TÜV Theinland, ISR Series no. 96.

    Google Scholar 

  • Werners B. (1987), Interactive Multiple Objective Programming Subject to Flexible Coefficients. Europ. J. of Op. Res. 31, 342–349.

    Article  Google Scholar 

  • Wiedey G. and Zimmermann H.J. (1978), Media Selection and Fuzzy Linear Programming. J. Op. Res. Soc. 29, 1071–1084.

    Google Scholar 

  • Wierzchon S.T. (1987), Linear Programming with Fuzzy Sets: A General Approach. Mathematical Modelling 9, 447–460.

    Article  Google Scholar 

  • Yager R.R. (1979), Mathematical Programming with Fuzzy Constraints and a Preference on the Objective. Kybernetes 9, 109–114.

    Article  Google Scholar 

  • Zimmermann H.J. (1975), Description and Optimization of Fuzzy Systems. Int. J. of General Systems 2, 209–215.

    Article  Google Scholar 

  • Zimmermann H.J. (1978), Fuzzy Programming and Linear Programming with Several Objective Functions. Fuzzy Sets and Systems 1, 45–55.

    Article  Google Scholar 

  • Zimmermann H.J. (1983), Using Fuzzy Sets in Operational Research. European Journal of Operational Research 13, 201–216.

    Article  Google Scholar 

  • Zimmermann H.J. (1985), Applications of Fuzzy Sets Theory to Mathematical Programming. Information Sciences 36, 29–58.

    Article  Google Scholar 

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Fedrizzi, M., Kacprzyk, J., Verdegay, J.L. (1991). A Survey of Fuzzy Optimization and Mathematical Programming. In: Fedrizzi, M., Kacprzyk, J., Roubens, M. (eds) Interactive Fuzzy Optimization. Lecture Notes in Economics and Mathematical Systems, vol 368. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45700-5_2

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  • DOI: https://doi.org/10.1007/978-3-642-45700-5_2

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