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Fuzzy Optimization

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

Fuzzy optimization is normative and as a mathematical model deals with transitional uncertainty and information deficiency uncertainty. Someliterature calls these uncertainties vagueness and ambiguity respectively.Transitional uncertainty is the domain of fuzzy set theory while uncertainty resulting from information deficiency isthe domain of possibility theory. Suppose one is told by one's employer to go to the airport and meet a tallfemale visitor at the baggage claim of the airport. The notion of “tall” is transitional uncertainty. As the passengers arrive at the baggageclaim, one compares each female passenger to the ascribed characteristic (tall woman) as determined by one's evaluation of what the boss' function is for“tall woman” and all the information one might have regarding “tall women”. The resulting function is used to obtaina possibility value for each female that appears at the baggage claim. This uncertainty arises from information deficiency.

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Abbreviations

Fuzzy set:

A set whose membership is characterized by gradualness and uniquely described by a membership function which measures the degree of membership that domain values possess with respect to belonging to the set in question.

Fuzzy set theory:

The theory of uncertainty associated with sets characterized by gradual membership.

Possibility theory:

The theory of uncertainty associated with deficiency of information.

Fuzzy number:

A fuzzy set whose membership function is upper semi‐continuous with a unique modal value and whose domain is the set of real numbers.

Possibilistic number:

A variable described by a possibilistic distribution whose domain is the set of real numbers.

Optimization:

The mathematical field that studies normative processes.

Bibliography

  1. Asai K, Tanaka H (1973) On the fuzzy – mathematical programming,Identification and System Estimation Proceedings of the 3rd IFAC Symposium, The Hague, 12–15 June 1973,pp 1050–1051

    Google Scholar 

  2. Audin J-P, Frankkowska H (1990) Set‐Valued Analysis. Birkhäuser,Boston

    Google Scholar 

  3. Baudrit C, Dubois D, Fargier H (2005) Propagation of uncertainty involvingimprecision and randomness. ISIPTA 2005, Pittsburgh, pp 31–40

    Google Scholar 

  4. Bector CR, Chandra S (2005) Fuzzy Mathematical Programming and Fuzzy MatrixGames. Springer, Berlin

    MATH  Google Scholar 

  5. Bellman RE, Zadeh LA (1970) Decision‐Making in a Fuzzy Environment. ManagSci B 17:141–164

    MathSciNet  Google Scholar 

  6. Brooke A, Kendrick D, Meeraus A (2002) GAMS: A User's Guide. Scientific Press,San Francisco (the latest updates can be obtained from http://www.gams.com/)

    Google Scholar 

  7. Buckley JJ (1988) Possibility and necessity in optimization. Fuzzy Sets Syst25(1):1–13

    MathSciNet  MATH  Google Scholar 

  8. Buckley JJ (1988) Possibilistic linear programming with triangular fuzzynumbers. Fuzzy Sets Syst 26(1):135–138

    Google Scholar 

  9. Buckley JJ (1989) Solving possibilistic linear programming problems. Fuzzy SetsSyst 31(3):329–341

    MathSciNet  MATH  Google Scholar 

  10. Buckley JJ (1989) A generalized extension principle. Fuzzy Sets Syst33:241–242

    MathSciNet  MATH  Google Scholar 

  11. Delgado M, Verdegay JL, Vila MA (1989) A general model for fuzzy linearprogramming. Fuzzy Sets Syst 29:21–29

    MathSciNet  MATH  Google Scholar 

  12. Delgado M, Kacprzyk J, Verdegay J-L, Vila MA (eds) (1994) Fuzzy Optimization:Recent Advances. Physica, Heidelberg

    MATH  Google Scholar 

  13. Demster AP (1967) Upper and lower probabilities induced by multivaluedmapping. Ann Math Stat 38:325–339

    Google Scholar 

  14. Dempster MAH (1969) Distributions in interval and linear programming. In:Hansen ER (ed) Topics in Interval Analysis. Oxford Press, Oxford, pp 107–127

    Google Scholar 

  15. Diamond P (1991) Congruence classes of fuzzy sets form a Banach space. J MathAnal Appl 162:144–151

    MathSciNet  MATH  Google Scholar 

  16. Diamond P, Kloeden P (1994) Metric Spaces of Fuzzy Sets. World Scientific,Singapore

    MATH  Google Scholar 

  17. Diamond P, Kloeden P (1994) Robust Kuhn–Tucker conditions andoptimization under imprecision. In: Delgado M, Kacprzyk J, Verdegay J-L, Vila MA (eds) Fuzzy Optimization: Recent Advances. Physica, Heidelberg,pp 61–66

    Google Scholar 

  18. Dubois D (1987) Linear programming with fuzzy data. In: Bezdek JC (ed)Analysis of Fuzzy Information, vol III: Applications in Engineering and Science. CRC Press, Boca Raton, pp 241–263

    Google Scholar 

  19. Dubois D, Prade H (1980) Fuzzy Sets and Systems: Theory andApplications. Academic Press, New York

    MATH  Google Scholar 

  20. Dubois D, Prade H (1980) Systems of linear fuzzy constraints. Fuzzy Sets Syst3:37–48

    MathSciNet  MATH  Google Scholar 

  21. Dubois D, Prade H (1981) Additions of interactive fuzzy numbers. IEEE TransAutom Control 26(4):926–936

    MathSciNet  Google Scholar 

  22. Dubois D, Prade H (1983) Ranking fuzzy numbers in the setting of possibilitytheory. Inf Sci 30:183–224

    MathSciNet  MATH  Google Scholar 

  23. Dubois D, Prade H (1986) New results about properties and semantics of fuzzyset‐theoretical operators. In: Wang PP, Chang SK (eds) Fuzzy Sets. Plenum Press, New York, pp 59–75

    Google Scholar 

  24. Dubois D, Prade H (1987) Fuzzy numbers: An overview. Tech. Rep. no 219, (LSI,Univ. Paul Sabatier, Toulouse, France). Mathematics and Logic. In: James Bezdek C (ed) Analysis of Fuzzy Information, vol 1, chap 1. CRC Press, BocaRaton, pp 3–39

    Google Scholar 

  25. Dubois D, Prade H (1988) Possibility Theory an Approach to ComputerizedProcessing of Uncertainty. Plenum Press, New York

    MATH  Google Scholar 

  26. Dubois D, Prade H (2005) Fuzzy elements in a fuzzy set. Proceedings ofthe 11th International Fuzzy System Association World Congress, IFSA 2005, Beijing, July 2005, pp 55–60

    Google Scholar 

  27. Dubois D, Moral S, Prade H (1997) Semantics for possibility theory based onlikelihoods. J Math Anal Appl 205:359–380

    MathSciNet  MATH  Google Scholar 

  28. Ertuğrul I, Tuş A (2007) Interactive fuzzy linear programming and anapplication sample at a textile firm. Fuzzy Optim Decis Making 6:29–49

    Google Scholar 

  29. Fortin J, Dubois D, Fargier H (2008) Gradual numbers and their application tofuzzy interval analysis. IEEE Trans Fuzzy Syst 16:2, pp 388–402

    Google Scholar 

  30. Fourer R, Gay DM, Kerninghan BW (1993) AMPL: A Modeling Language forMathematical Programming. Scientific Press, San Francisco, CA (the latest updates can be obtained fromhttp://www.ampl.com/)

    Google Scholar 

  31. Fullér R, Keresztfalvi T (1990) On generalization of Nguyen's theorem. FuzzySets Syst 41:371–374

    Google Scholar 

  32. Ghassan K (1982) New utilization of fuzzy optimization method. In: Gupta MM,Sanchez E (eds) Fuzzy Information and Decision Processes. North‐Holland, Netherlands, pp 239–246

    Google Scholar 

  33. Gladish B, Parra M, Terol A, Uría M (2005) Management of surgical waitinglists through a possibilistic linear multiobjective programming problem. Appl Math Comput 167:477–495

    Google Scholar 

  34. Greenberg H (1992) MODLER: Modeling by Object‐Driven Linear ElementalRelations. Ann Oper Res 38:239–280

    Google Scholar 

  35. Hanss M (2005) Applied Fuzzy Arithmetic. Springer,Berlin

    MATH  Google Scholar 

  36. Hsu H, Wang W (2001) Possibilistic programming in production planning ofassemble-to-order environments. Fuzzy Sets Syst 119:59–70

    MathSciNet  Google Scholar 

  37. Inuiguchi M (1992) Stochastic Programming Problems Versus Fuzzy MathematicalProgramming Problems. Jpn J Fuzzy Theory Syst 4(1):97–109

    MathSciNet  MATH  Google Scholar 

  38. Inuiguchi M (1997) Fuzzy linear programming: what, why and how? Tatra Mt MathPubl 13:123–167

    Google Scholar 

  39. Inuiguchi M (2007) Necessity measure optimization in linear programmingproblems with fuzzy polytopes. Fuzzy Sets Syst 158:1882–1891

    MathSciNet  MATH  Google Scholar 

  40. Inuiguchi M (2007) On possibility/fuzzy optimization. In: Melin P, CastilloO, Aguilar LT, Kacprzyk J, Pedrycz W (eds) Foundations of Fuzzy Logic and Soft Computing: 12th International Fuzzy System Association World Congress, IFSA2007, Cancun, June 2007, Proceedings. Springer, Berlin, pp 351–360

    Google Scholar 

  41. Inuiguchi M, Ramik J (2000)Possibilistic linear programming: A brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolioselection problem. Fuzzy Sets Syst 111:97–110

    MathSciNet  Google Scholar 

  42. Inuiguchi M, Sakawa M (1997) An achievement rate approach to linearprogramming problems with an interval objective function. J Oper Res Soc 48:25–33

    MATH  Google Scholar 

  43. Inuiguchi M, Tanino T (2004) Fuzzy linear programming with interactiveuncertain parameters. Reliab Comput 10(5):512–527

    MathSciNet  Google Scholar 

  44. Inuiguchi M, Ichihashi H, Tanaka H (1990) Fuzzy programming: A survey ofrecent developments. In: Slowinski R, Teghem J (eds) Stochastic versus Fuzzy Approaches to Multiobjective Mathematical Programming UnderUncertainty. Kluwer, Netherlands, pp 45–68

    Google Scholar 

  45. Inuiguchi M, Ichihashi H, Kume Y (1992) Relationships Between ModalityConstrained Programming Problems and Various Fuzzy Mathematical Programming Problems. Fuzzy Sets Syst 49:243–259

    MathSciNet  MATH  Google Scholar 

  46. Inuiguchi M, Ichihashi H, Tanaka H (1992) Fuzzy Programming: A Survey ofRecent Developments. In: Slowinski R, Teghem J (eds) Stochastic versus Fuzzy Approaches to Multiobjective Mathematical Programming underUncertainty.Springer, Berlin, pp 45–68

    Google Scholar 

  47. Inuiguchi M, Sakawa M, Kume Y (1994) The usefulness of possibilisticprogramming in production planning problems. Int J Prod Econ 33:49–52

    Google Scholar 

  48. Jamison KD (1998) Modeling Uncertainty Using Probabilistic Based PossibilityTheory with Applications to Optimization. Ph D Thesis, University of Colorado Denver, Department of Mathematical Sciences.http://www-math.cudenver.edu/graduate/thesis/jamison.pdf

  49. Jamison KD (2000) Possibilities as cumulative subjective probabilities anda norm on the space of congruence classes of fuzzy numbers motivated by an expected utility functional. Fuzzy Sets Syst111:331–339

    MathSciNet  MATH  Google Scholar 

  50. Jamison KD, Lodwick WA (2001) Fuzzy linear programming using penaltymethod. Fuzzy Sets Syst 119:97–110

    MathSciNet  MATH  Google Scholar 

  51. Jamison KD, Lodwick WA (2002) The construction of consistent possibility andnecessity measures. Fuzzy Sets Syst 132(1):1–10

    MathSciNet  MATH  Google Scholar 

  52. Jamison KD, Lodwick WA (2004) Interval‐valued probabilitymeasures. UCD/CCM Report No. 213, March 2004

    Google Scholar 

  53. Joubert JW, Luhandjula MK, Ncube O, le Roux G, de Wet F (2007) An optimizationmodel for the management of South African game ranch. Agric Syst 92:223–239

    Google Scholar 

  54. Kacprzyk J, Orlovski SA (eds) (1987) Optimization Models Using Fuzzy Sets andPossibility Theory. D Reidel, Dordrecht

    MATH  Google Scholar 

  55. Kacprzyk J, Orlovski SA (1987) Fuzzy optimization and mathematicalprogramming: A brief introduction and survey. In: Kacprzyk J, Orlovski SA (eds) Optimization Models Using Fuzzy Sets and Possibility Theory. DReidel, Dordrecht, pp 50–72

    Google Scholar 

  56. Kasperski A, Zeilinski P (2007) Using gradual numbers for solvingfuzzy‐valued combinatorial optimization problems. In: Melin P, Castillo O, Aguilar LT, Kacprzyk J, Pedrycz W (eds) Foundations of Fuzzy Logic andSoft Computing: 12th International Fuzzy System Association World Congress, IFSA 2007, Cancun, June 2007, Proceedings. Springer, Berlin,pp 656–665

    Google Scholar 

  57. Kaufmann A, Gupta MM (1985) Introduction to Fuzzy Arithmetic –Theory and Applications. Van Nostrand Reinhold, New York

    MATH  Google Scholar 

  58. Kaymak U, Sousa JM (2003) Weighting of Constraints in FuzzyOptimization. Constraints 8:61–78 (also in the 2001 Proceedings of IEEE Fuzzy Systems Conference)

    MathSciNet  MATH  Google Scholar 

  59. Klir GJ, Yuan B (1995) Fuzzy Sets and Fuzzy Logic. Prentice Hall, Upper SaddleRiver

    MATH  Google Scholar 

  60. Lai Y, Hwang C (1992) Fuzzy MathematicalProgramming. Springer, Berlin

    MATH  Google Scholar 

  61. Lin TY (2005) A function theoretic view of fuzzy sets: New extensionprinciple. In: Filev D, Ying H (eds) Proceedings of NAFIPS05

    Google Scholar 

  62. Liu B (1999) Uncertainty Programming. Wiley, NewYork

    Google Scholar 

  63. Liu B (2000) Dependent‐chance programming in fuzzy environments. FuzzySets Syst 109:97–106

    MATH  Google Scholar 

  64. Liu B (2001) Fuzzy random chance‐constrained programming. IEEE Trans Fuzzy Syst 9:713–720

    Google Scholar 

  65. Liu B (2002) Theory and Practice of Uncertainty Programming. Physica,Heidelberg

    Google Scholar 

  66. Liu B, Iwamura K (2001) Fuzzy programming with fuzzy decisions and fuzzysimulation‐based genetic algorithm. Fuzzy Sets Syst 122:253–262

    MathSciNet  MATH  Google Scholar 

  67. Lodwick WA (1990) Analysis of structure in fuzzy linear programs. Fuzzy SetsSyst 38:15–26

    MathSciNet  MATH  Google Scholar 

  68. Lodwick WA (1990) A generalized convex stochastic dominancealgorithm. IMA J Math Appl Bus Ind 2:225–246

    MathSciNet  MATH  Google Scholar 

  69. Lodwick WA (1999) Constrained Interval Arithmetic. CCM Report 138,Feb. 1999. CCM, Denver

    Google Scholar 

  70. Lodwick WA (ed) (2004) Special Issue on Linkages Between Interval Analysis andFuzzy Set Theory. Reliable Comput 10

    Google Scholar 

  71. Lodwick WA (2007) Interval and fuzzy analysis: A unified approach. In:Advances in Imaging and Electronic Physics, vol 148. Elsevier, San Diego, pp 75–192

    Google Scholar 

  72. Lodwick WA, Bachman KA (2005) Solving large-scale fuzzy and possibilisticoptimization problems: Theory, algorithms and applications. Fuzzy Optim Decis Making 4(4):257–278. (also UCD/CCM Report No. 216, June 2004)

    MathSciNet  MATH  Google Scholar 

  73. Lodwick WA, Inuiguchi M (eds) (2007) Special Issue on Optimization Under Fuzzyand Possibilistic Uncertainty. Fuzzy Sets Syst 158:17; 1 Sept 2007

    Google Scholar 

  74. Lodwick WA, Jamison KD (1997) A computational method for fuzzyoptimization. In: Bilal A, Madan G (eds) Uncertainty Analysis in Engineering and Sciences: Fuzzy Logic, Statistics, and Neural Network Approach, chap19. Kluwer, Norwell

    Google Scholar 

  75. Lodwick WA, Jamison KD (eds) (2003) Special Issue: Interfaces Fuzzy Set TheoryInterval Anal 135:1; April 1, 2003

    Google Scholar 

  76. Lodwick WA, Jamison KD (2003) Estimating and Validating the CumulativeDistribution of a Function of Random Variables: Toward the Development of Distribution Arithmetic. Reliab Comput9:127–141

    MathSciNet  MATH  Google Scholar 

  77. Lodwick WA, Jamison KD (2005) Theory and semantics for fuzzy and possibilisticoptimization, Proceedings of the 11th International Fuzzy System Association World Congress. IFSA 2005, Beijing, July2005

    Google Scholar 

  78. Lodwick WA, Jamison KD (2006) Interval‐valued probability in theanalysis of problems that contain a mixture of fuzzy, possibilistic and interval uncertainty. In: Demirli K, Akgunduz A (eds) 2006 Conference of theNorth American Fuzzy Information Processing Society, 3–6 June 2006, Montréal, Canada, paper 327137

    Google Scholar 

  79. Lodwick WA, Jamison KD (2008)Interval‐Valued Probability in the Analysis of Problems Containing a Mixture of Fuzzy, Possibilistic, Probabilistic and Interval Uncertainty. FuzzySets and Systems 2008

    Google Scholar 

  80. Lodwick WA, Jamison KD (2007) The use of interval‐valued probabilitymeasures in optimization under uncertainty for problems containing a mixture of fuzzy, possibilistic, and interval uncertainty. In: Melin P, CastilloO, Aguilar LT, Kacprzyk J, Pedrycz W (eds) Foundations of Fuzzy Logic and Soft Computing: 12 th International Fuzzy System Association World Congress,IFSA 2007, Cancun, Mexico, June 2007, Proceedings. Springer, Berlin, pp 361–370

    Google Scholar 

  81. Lodwick WA, Jamison KD (2007) Theoretical and semantic distinctions of fuzzy,possibilistic, and mixed fuzzy/possibilistic optimization. Fuzzy Sets Syst 158(17):1861–1872

    MathSciNet  MATH  Google Scholar 

  82. Lodwick WA, McCourt S, Newman F, Humpheries S (1999) Optimization Methods forRadiation Therapy Plans. In: Borgers C, Natterer F (eds) IMA Series in Applied Mathematics – Computational Radiology and Imaging: Therapy andDiagnosis. Springer, New York, pp 229–250

    Google Scholar 

  83. Lodwick WA, Neumaier A, Newman F (2001) Optimization under uncertainty:methods and applications in radiation therapy. Proc 10th IEEE Int Conf Fuzzy Syst 2001 3:1219–1222

    Google Scholar 

  84. Luhandjula MK (1986) On possibilistic linear programming. Fuzzy Sets Syst18:15–30

    MathSciNet  MATH  Google Scholar 

  85. Luhandjula MK (1989) Fuzzy optimization: an appraisal. Fuzzy Sets Syst30:257–282

    MathSciNet  MATH  Google Scholar 

  86. Luhandjula MK (2004) Optimisation under hybrid uncertainty. Fuzzy Sets Syst146:187–203

    MathSciNet  MATH  Google Scholar 

  87. Luhandjula MK (2006) Fuzzy stochastic linear programming: Survey and futureresearch directions. Eur J Oper Res 174:1353–1367

    MathSciNet  MATH  Google Scholar 

  88. Luhandjula MK, Ichihashi H, Inuiguchi M (1992) Fuzzy and semi‐infinitemathematical programming. Inf Sci 61:233–250

    MathSciNet  MATH  Google Scholar 

  89. Markowitz H (1952) Portfolio selection. J Finance7:77–91

    Google Scholar 

  90. Moore RE (1979) Methods and Applications of Interval Analysis. SIAM,Philadelphia

    MATH  Google Scholar 

  91. Negoita CV (1981) The current interest in fuzzy optimization. Fuzzy Sets Syst6:261–269

    MathSciNet  MATH  Google Scholar 

  92. Negoita CV, Ralescu DA (1975) Applications of Fuzzy Sets to SystemsAnalysis. Birkhäuser, Boston

    MATH  Google Scholar 

  93. Negoita CV, Sularia M (1976) On fuzzy mathematical programming and tolerancesin planning. Econ Comput Cybern Stud Res 3(31):3–14

    MathSciNet  Google Scholar 

  94. Neumaier A (2003) Fuzzy modeling in terms of surprise. Fuzzy Sets Syst135(1):21–38

    MathSciNet  MATH  Google Scholar 

  95. Neumaier A (2004) Clouds, fuzzy sets and probability intervals. Reliab Comput10:249–272. Springer

    MathSciNet  MATH  Google Scholar 

  96. Neumaier A (2005) Structure of clouds. (submitted – downloadablehttp://www.mat.univie.ac.at/%7Eneum/papers.html)

  97. Ogryczak W, Ruszczynski A (1999) From stochastic dominance to mean-riskmodels: Semideviations as risk measures. Eur J Oper Res 116:33–50

    MATH  Google Scholar 

  98. Nguyen HT (1978) A note on the extension principle for fuzzy sets. J MathAnal Appl 64:369–380

    MathSciNet  MATH  Google Scholar 

  99. Ralescu D (1977) Inexact solutions for large-scale control problems. In:Proceedings of the 1st International Congress on Mathematics at the Service of Man, Barcelona

    Google Scholar 

  100. Ramik J (1986) Extension principle in fuzzy optimization. Fuzzy Sets Syst19:29–35

    MathSciNet  MATH  Google Scholar 

  101. Ramik J, Rimanek J (1985) Inequality relation between fuzzy numbers and itsuse in fuzzy optimization. Fuzzy Sets Syst 16:123–138

    MathSciNet  MATH  Google Scholar 

  102. Ramik J, Vlach M (2002) Fuzzy mathematical programming: A unifiedapproach based on fuzzy relations. Fuzzy Optim Decis Making 1:335–346

    MathSciNet  MATH  Google Scholar 

  103. Ramik J, Vlach M (2002) Generalized Concavity in Fuzzy Optimization andDecision Analysis. Kluwer, Boston

    MATH  Google Scholar 

  104. Riverol C, Pilipovik MV (2007) Optimization of the pyrolysis of ethane usingfuzzy programming. Chem Eng J 133:133–137

    Google Scholar 

  105. Riverol C, Pilipovik MV, Carosi C (2007) Assessing the water requirements inrefineries using possibilistic programming. Chem Eng Process 45:533–537

    Google Scholar 

  106. Rommelfanger HJ (1994) Some problems of fuzzy optimization with T-norm basedextended addition. In: Delgado M, Kacprzyk J, Verdegay J-L, Vila MA (eds) Fuzzy Optimization: Recent Advances. Physica, Heidelberg,pp 158–168

    Google Scholar 

  107. Rommelfanger HJ (1996) Fuzzy linear programming and applications. Eur J OperRes 92:512–527

    MATH  Google Scholar 

  108. Rommelfanger HJ (2004) The advantages of fuzzy optimization models inpractical use. Fuzzy Optim Decis Making 3:293–309

    Google Scholar 

  109. Rommelfanger HJ, Slowinski R (1998) Fuzzy linear programming with single ormultiple objective functions. In: Slowinski R (ed) Fuzzy Sets in Decision Analysis, Operations Research and Statistics. The Handbooks of FuzzySets. Kluwer, Netherlands, pp 179–213

    Google Scholar 

  110. Roubens M (1990) Inequality constraints between fuzzy numbers and their usein mathematical programming. In: Slowinski R, Teghem J (eds) Stochastic versus Fuzzy Approaches to Multiobjective Mathematical Programming UnderUncertainty. Kluwer, Netherlands, pp 321–330

    Google Scholar 

  111. Russell B (1924) Vagueness. Aust J Philos1:84–92

    Google Scholar 

  112. Sahinidis N (2004) Optimization under uncertainty: State-of-the-art andopportunities. Comput Chem Eng 28:971–983

    Google Scholar 

  113. Saito S, Ishii H (1998) Existence criteria for fuzzy optimizationproblems. In: Takahashi W, Tanaka T (eds) Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis, Niigata, Japan,28–31 July 1998. World Scientific Press, Singapore, pp 321–325

    Google Scholar 

  114. Shafer G (1976) Mathematical A Theory of Evidence. Princeton UniversityPress, Princeton

    MATH  Google Scholar 

  115. Shafer G (1987) Belief functions and possibility measures. In: James BezdekC (ed) Analysis of Fuzzy Information. Mathematics and Logic, vol 1. CRC Press, Boca Raton, pp 51–84

    Google Scholar 

  116. Sousa JM, Kaymak U (2002) Fuzzy Decision Making in Modeling andControl. World Scientific Press, Singapore

    MATH  Google Scholar 

  117. Steinbach M (2001) Markowitz revisited:Mean‐variance models in financial portfolio analysis. SIAM Rev 43(1):31–85

    MathSciNet  ADS  MATH  Google Scholar 

  118. Tanaka H, Asai K (1984) Fuzzy linear programming problems with fuzzynumbers. Fuzzy Sets Syst 13:1–10

    MathSciNet  MATH  Google Scholar 

  119. Tanaka H, Okuda T, Asai K (1973) Fuzzy mathematical programming. Trans Soc Instrum Control Eng 9(5):607–613; (in Japanese)

    Google Scholar 

  120. Tanaka H, Okuda T, Asai K (1974) On fuzzy mathematical programming. J Cybern3(4):37–46

    MathSciNet  Google Scholar 

  121. Tanaka H, Ichihasi H, Asai K (1984) A formulation of fuzzy linearprogramming problems based on comparison of fuzzy numbers. Control Cybern 13(3):185–194

    MATH  Google Scholar 

  122. Tanaka H, Ichihasi H, Asai K (1985) Fuzzy decisions in linear programmingwith trapezoidal fuzzy parameters. In: Kacprzyk J, Yager R (eds) Management Decision Support Systems Using Fuzzy Sets and Possibility Theory. Springer, Heidelberg,pp 146–159

    Google Scholar 

  123. Tang J, Wang D, Fung R (2001) Formulation of general possibilistic linearprogramming problems for complex industrial systems. Fuzzy Sets Syst 119:41–48

    MathSciNet  Google Scholar 

  124. Thipwiwatpotjani P (2007) An algorithm for solving optimization problemswith interval‐valued probability measures. CCM Report, No. 259 (December 2007). CCM, Denver

    Google Scholar 

  125. Untiedt E (2006) Fuzzy and possibilistic programming techniques in theradiation therapy problem: An implementation‐based analysis. Masters Thesis, University of Colorado Denver, Department of Mathematical Sciences(July 5, 2006)

    Google Scholar 

  126. Untiedt E (2007) A robust model for mixed fuzzy and possibilisticprogramming. Project report for Math 7593: Advanced Linear Programming. Spring, Denver

    Google Scholar 

  127. Untiedt E (2007) Using gradual numbers to analyze non‐monotonicfunctions of fuzzy intervals. CCM Report No 258 (December 2007). CCM, Denver

    Google Scholar 

  128. Untiedt E, Lodwick WA (2007) On selecting an algorithm for fuzzyoptimization. In: Melin P, Castillo O, Aguilar LT, Kacprzyk J, Pedrycz W (eds) Foundations of Fuzzy Logic and Soft Computing: 12th International FuzzySystem Association World Congress, IFSA 2007, Cancun, Mexico, June 2007, Proceedings. Springer, Berlin, pp 371–380

    Google Scholar 

  129. Vasant PM, Barsoum NN, Bhattacharya A (2007) Possibilistic optimization inplanning decision of construction industry. Int J Prod Econ (to appear)

    Google Scholar 

  130. Verdegay JL (1982) Fuzzy mathematical programming. In: Gupta MM, Sanchez E(eds) Fuzzy Information and Decision Processes. North‐Holland, Amsterdam, pp 231–237

    Google Scholar 

  131. Vila MA, Delgado M, Verdegay JL (1989) A general model for fuzzy linearprogramming. Fuzzy Sets Syst 30:21–29

    MathSciNet  Google Scholar 

  132. Wang R, Liang T (2005) Applying possibilistic linear programming toaggregate production planning. Int J Prod Econ 98:328–341

    Google Scholar 

  133. Wang S, Zhu S (2002) On fuzzy portfolio selection problems. Fuzzy OptimDecis Making 1:361–377

    MATH  Google Scholar 

  134. Wang Z, Klir GJ (1992) Fuzzy Measure Theory. Plenum Press, NewYork

    MATH  Google Scholar 

  135. Weichselberger K (2000) The theory of interval‐probability asa unifying concept for uncertainty. Int J Approx Reason 24:149–170

    MathSciNet  MATH  Google Scholar 

  136. Werners B (1988) Aggregation models in mathematical programming. In: Mitra G(ed) Mathematical Models for Decision Support, NATO ASI Series vol F48. Springer, Berlin

    Google Scholar 

  137. Werners B (1995) Fuzzy linear programming – Algorithms andapplications. Addendum to the Proceedings of ISUMA‐NAPIPS'95, College Park, Maryland, 17–20 Sept 1995, pp A7-A12

    Google Scholar 

  138. Whitmore GA, Findlay MC (1978) Stochastic Dominance: An Approach toDecisionDecision‐Making Under Risk. Lexington Books, Lexington

    Google Scholar 

  139. Yager RR (1980) On choosing between fuzzy subsets. Kybernetes9:151–154

    MATH  Google Scholar 

  140. Yager RR (1981) A procedure for ordering fuzzy subsets of the unitinterval. Inf Sci 24:143–161

    MathSciNet  MATH  Google Scholar 

  141. Yager RR (1986) A characterization of the extension principle. FuzzySets Syst 18:205–217

    MathSciNet  MATH  Google Scholar 

  142. Zadeh LA (1965) Fuzzy Sets. Inf Control8:338–353

    MathSciNet  MATH  Google Scholar 

  143. Zadeh LA (1968) Probability measures of fuzzy events. J Math Anal Appl23:421–427

    MathSciNet  MATH  Google Scholar 

  144. Zadeh LA (1975) The concept ofa linguistic variable and its application to approximate reasoning. Inf Sci, Part I: 8:199–249; Part II: 8:301–357; Part III:9:43–80

    MathSciNet  MATH  Google Scholar 

  145. Zadeh LA (1978) Fuzzy sets as a basis for a theory ofpossibility. Fuzzy Sets Syst 1:3–28

    MathSciNet  MATH  Google Scholar 

  146. Zeleny M (1994) Fuzziness, knowledge and optimization: New optimalityconcepts. In: Delgado M, Kacprzyk J, Verdegay J-L, M Vila A (eds) Fuzzy Optimization: Recent Advances. Physica, Heidelberg,pp 3–20

    Google Scholar 

  147. Zimmermann HJ (1974) Optimization in fuzzy environment. Paper presented atthe International XXI TIMS and 46th ORSA Conference, Puerto Rico Meeting, San Juan, Porto Rico, Oct 1974

    Google Scholar 

  148. Zimmermann HJ (1976) Description and optimization of fuzzy systems. Int JGeneral Syst 2(4):209–216

    MATH  Google Scholar 

  149. Zimmermann HJ (1978) Fuzzy programming and linear programming with severalobjective functions. Fuzzy Sets Syst 1:45–55

    MATH  Google Scholar 

  150. Zimmermann HJ (1983) Fuzzy mathematical programming. Comput Oper Res10:291–298

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Colorado Denver, Denver, USA

    Weldon A. Lodwick & Elizabeth A. Untiedt

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  1. Weldon A. Lodwick
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  2. Elizabeth A. Untiedt
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Lodwick, W.A., Untiedt, E.A. (2009). Fuzzy Optimization. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_236

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