Advertisement

Fuzziness, Rationality, Optimality and Equilibrium in Decision and Economic Theories

  • Kofi Kissi Dompere
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 254)

Abstract

This essay presents structural categories of theories of optimization. It begins with the classical system leading to the establishment of an entry point of fuzzy optimization from the logic of the classical optimization. Four categories of optimization problems are identified. They are exact (non-fuzzy) and nonstochastic, exact (non-fuzzy) and stochastic categories that are associated with classical laws of thought and mathematics. The other categories are fuzzy and non-stochastic, and fuzzy-stochastic problems that are associated with fuzzy laws of thought and fuzzy mathematics. The canonical structures of the problems and their solutions are presented.

From these structures, similarities and differences in the problem structures and corresponding solutions are abstracted and discussed. The similarities and differences in the problem-solution structures of different categories are attributed to properties of exactness and completeness about information-knowledge structures in which the optimization problems are formulated and solved. The assumed degrees of exactness and completeness establish defective informationknowledge structure that generates uncertainties and produces inter-category differences in the optimization problem. The specific differences of intra-category algorithms are attributed to differences in the assumed functional relationships of the variables that establish the objective and constraint sets. The essay is concluded with taxonomy of solution structures and discussions on future research directions.

Keywords

Membership Function Fuzzy Variable Fuzzy Optimization Fuzzy Program Fuzzy Linear Programming 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Black, Max, The Nature of Mathematics, Totowa, Littlefield, Adams and Co. (1965)Google Scholar
  2. 2.
    Boolos, G.S., Jeffrey, R.C.: Computability and Logic. Cambridge University Press, New York (1989)zbMATHGoogle Scholar
  3. 3.
    Bose, R.K., Sahani, D.: Fuzzy Mappings and Fixed Point Theorems. Fuzzy Sets and Systems 21, 53–58 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Buckley, J.J.: Fuzzy Programming And the Pareto Optimal Set. Fuzzy Set and Systems 10, 57–63 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Butnariu, D.: Fixed Points for Fuzzy Mappings. Fuzzy Sets and Systems 7, 191–207 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Carlsson, G.: Solving Ill-Structured Problems Through Well Structured Fuzzy Programming. In: Brans, J.P. (ed.) Operation Research, vol. 81, pp. 467–477. North-Holland, Amsterdam (1981)Google Scholar
  7. 7.
    Carlsson, C.: Tackling an AMDM - Problem with the Help of Some Results From Fuzzy Set Theory. European Journal of Operational Research 10(3), 270–281 (1982)zbMATHCrossRefGoogle Scholar
  8. 8.
    Cerny, M.: Fuzzy Approach to Vector Optimization. Intern. Jour. of General Systems 20(1), 23–29Google Scholar
  9. 9.
    Chang, S.S.: Fixed Point Theorems for Fuzzy Mappings. Fuzzy Sets and Systems 17, 181–187 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Chang, S.S.L.: Fuzzy Dynamic Programming and the Decision Making Process. In: Proc. 3rd Princeton Conference on Information Science and Systems, Princeton, pp. 200–203 (1969)Google Scholar
  11. 11.
    Churchman, C.W.: Prediction and Optimal Decision. Pretice-Hall, Englewood Cliffs (1961)Google Scholar
  12. 12.
    Cohn, D.L.: Measure Theory. Birkhauser, Boston (1980)zbMATHGoogle Scholar
  13. 13.
    Delgado, M., et al.: On the Concept of Possibility-Probability Consistency. Fuzzy Sets and Systems 21(3), 311–318 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    DiNola, A., et al. (eds.): The Mathematics of Fuzzy Systems. Verlag TUV Rheinland, Koln (1986)Google Scholar
  15. 15.
    Dompere, K.K.: Epistemic Foundations of Fuzziness: Unified Theories on Decision-Choice Processes. Studies in Fuzziness and Soft Computing, vol. 236. Springer, New York (2009)zbMATHGoogle Scholar
  16. 16.
    Dompere, K.K.: Fuzziness and Approximate Reasoning: Epistemics on Uncertainty, Expectations and Risk in Rational Behavior. Studies in Fuzziness and Soft Computing, vol. 237. Springer, New York (2009)Google Scholar
  17. 17.
    Dompere, K.K.: Fuzzy Rationality: Methodological Critique and Unity of Classical, Bounded and Other Rationalities. Studies in Fuzziness and Soft Computing, vol. 235. Springer, New York (2009)zbMATHGoogle Scholar
  18. 18.
    Dompere, K.K.: Cost-Benefit Analysis and the Theory of Fuzzy Decision: Identification and Measurement Theory. Springer, New York (2004)zbMATHGoogle Scholar
  19. 19.
    Dompere, K.K.: Cost-Benefit Analysis and the Theory of Fuzzy Decision: Fuzzy Value Theory. Springer, New York (2004)zbMATHGoogle Scholar
  20. 20.
    Delbaen, F.: Stochastic Preferences and General Equilibrium Theory. In: Dreze, J.H. (ed.) Allocation Under Uncertainty: Equilibrium and Optimality. Wiley, New York (1974)Google Scholar
  21. 21.
    Dubois, D., et al.: Systems of Linear Fuzzy Constraints. Fuzzy Sets and Systems 3(1), 37–48 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Dubois, D.: An Application of Fuzzy Arithmetic to the Optimization of Industrial Machining Processes. Mathematical Modelling 9(6), 461–475 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Dumitrescu, D.: Entropy of a Fuzzy Process. Fuzzy Sets and Systems 55(2), 169–177 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Eaves, B.C.: Computing Kakutani Fixed Points. Journal of Applied Mathematics 21, 236–244 (1971)zbMATHMathSciNetGoogle Scholar
  25. 25.
    El Rayes, A.B., et al.: Generalized Possibility Measures. Information Sciences 79, 201–222 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Ernst, G.C., et al.: Principles of Structural Equilibrium, a Study of Equilibrium Conditions by Graphic, Force Moment and Virtual Displacement (virtual work). University of Nebraska Press, Lincoln Na. (1962)Google Scholar
  27. 27.
    Feng, Y.J.: A Method Using Fuzzy Mathematics to Solve the Vector Maxim Problem. Fuzzy Set and Systems 9(2), 129–136 (1983)zbMATHGoogle Scholar
  28. 28.
    Fischer, R.B., Peters, G.: Chemical Equilibrium. Saunders Pub., Philadelphia (1970)Google Scholar
  29. 29.
    Fisher, F.M.: Disequilibrium Foundations of Equilibrium Economics. Cambridge University Press, New York (1983)zbMATHGoogle Scholar
  30. 30.
    Foster, D.H.: Fuzzy Topological Groups. Journal of Math. Analysis and Applications 67, 549–564 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Goetschel Jr., R., et al.: Topological Properties of Fuzzy Number. Fuzzy Sets and Systems 10(1), 87–99 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Fourgeaud, C., Gouriéroux, C.: Learning Procedures and Convergence to Rationality. Econometrica 54, 845–868 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Freeman, A., Carchedi, G. (eds.): Marx and Non-Equilibrium Economics, Cheltenham, UK, Edward Elgar (1996)Google Scholar
  34. 34.
    Gaines, B.R.: Fuzzy and Probability Uncertainty logics. Information and Control 38(2), 154–169 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Newton, G., Hare, P.H. (eds.): Naturalism and Rationality. Prometheus Books, Buffalo (1986)Google Scholar
  36. 36.
    Ginsburgh, V.: Activity Analysis and General Equilibrium Modelling. North-Holland, New York (1981)zbMATHGoogle Scholar
  37. 37.
    Grabish, M., et al.: Fundamentals of Uncertainty Calculi with Application to Fuzzy Systems. Kluwer, Boston (1994)Google Scholar
  38. 38.
    Hahn, F.: Equilibrium and Macroeconomics. MIT Press, Cambridge (1984)Google Scholar
  39. 39.
    Hamacher, H., et al.: Sensitivity Analysis in Fuzzy Linear Programming. Fuzzy Sets and Systems 1, 269–281 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Hannan, E.L.: On the Efficiency of the Product Operator in Fuzzy Programming with Multiple Objectives. Fuzzy Sets and Systems 2(3), 259–262 (1979)zbMATHCrossRefGoogle Scholar
  41. 41.
    Hannan, E.L.: Linear Programming with Multiple Fuzzy Goals. Fuzzy Sets and Systems 6(3), 235–248 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Hansen, B.: A Survey of General Equilibrium Systems. McGraw-Hill, New York (1970)Google Scholar
  43. 43.
    Heilpern, S.: Fuzzy Mappings and Fixed Point Theorem. Journal of Mathematical Analysis and Applications 83, 566–569 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Hestenes, M.R.: Optimization Theory: The Finite Dimensional Case, New York (1975)Google Scholar
  45. 45.
    Heyman, D.P., Sobel, M.J.: Stochastic Models in Operations Research. Stochastic Optimization, vol. II. Mc Graw-Hill, New York (1984)zbMATHGoogle Scholar
  46. 46.
    Ulrich, H.: A Mathematical Theory of Uncertainty. In: Yager, R.R. (ed.) Fuzzy Set and Possibility Theory: Recent Developments, pp. 344–355. Pergamon, New York (1982)Google Scholar
  47. 47.
    Ignizio, J.P., et al.: Fuzzy Multicriteria Integer Programming via Fuzzy Generalized Networks. Fuzzy Sets and Systems 10(3), 261–270 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    Jarvis, R.A.: Optimization Strategies in Adaptive Control: A Selective Survey. IEEE Trans. Syst. Man. Cybernetics SMC-5, 83–94 (1975)MathSciNetGoogle Scholar
  49. 49.
    Kabbara, G.: New Utilization of Fuzzy Optimization Method. In: Gupta, M.M., et al. (eds.) Approximate Reasoning In Decision Analysis, pp. 239–246. North Holland, New York (1982)Google Scholar
  50. 50.
    Kacprzyk, J., Fedrizzi, M. (eds.): Combining Fuzzy Imprecision with Probabilistic Uncertainty in Decision Making. Plenum Press, New York (1992)Google Scholar
  51. 51.
    Kacprzyk, J., et al. (eds.): Optimization Models Using Fuzzy Sets and Possibility Theory. D. Reidel, Boston (1987)zbMATHGoogle Scholar
  52. 52.
    Kakutani, S.: A Generalization of Brouwer’s Fixed Point Theorem. Duke Mathematical Journal 8, 416–427 (1941)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Kaleva, O.: A Note on Fixed Points for Fuzzy Mappings. Fuzzy Sets and Systems 15, 99–100 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  54. 54.
    Kandel, A.: On Minimization of Fuzzy Functions. IEEE Trans. Comp. C-22, 826–832 (1973)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Kandel, A.: Comments on Minimization of Fuzzy Functions. IEEE Trans. Comp. C-22, 217 (1973)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Kandel, A.: On the Minimization of Incompletely Specified Fuzzy Functions. Information, and Control 26, 141–153 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  57. 57.
    Kaufmann, A., Gupta, M.M.: Introduction to fuzzy arithmetic: Theory and applications. Van Nostrand Rheinhold, New York (1991)zbMATHGoogle Scholar
  58. 58.
    Kaufmann, A.: Introduction to the Theory of Fuzzy Subsets, vol. 1. Academic Press, New York (1975)zbMATHGoogle Scholar
  59. 59.
    Kaufmann, A.: The Science of Decision-Making: Introduction to Praxeology. McGraw-Hill, New York (1968)Google Scholar
  60. 60.
    Kickert, W.J.M.: Organization of Decision-Making: A Systems-Theoretical Approach. North-Holland, Amsterdam (1980)Google Scholar
  61. 61.
    Klir, G.J., et al.: Fuzzy Sets, Uncertainty and Information. Prentice Hall, Englewood Cliffs (1988)zbMATHGoogle Scholar
  62. 62.
    Klir, G.J., et al.: Probability-Possibility Transformations: A Comparison. Intern. Jour. of General Systems 21(3), 291–310 (1992)zbMATHCrossRefGoogle Scholar
  63. 63.
    Klir, G.J.: Where Do we Stand on Measures of Uncertainty. Ambignity, Fuzziness and the like? Fuzzy Sets and Systems 24(2), 141–160 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  64. 64.
    Kosko, B.: Fuzziness vs Probability. Intern. Jour. of General Systems 17(1-3), 211–240 (1990)zbMATHCrossRefGoogle Scholar
  65. 65.
    Lai, Y., et al.: Fuzzy Mathematical Programming. Springer, New York (1992)zbMATHGoogle Scholar
  66. 66.
    Lee, E.S., et al.: Fuzzy Multiple Objective Programming and Compromise Programming with Pareto Optimum. Fuzzy Sets and Systems 53(3), 275–288 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  67. 67.
    Lowen, R.: Connex Fuzzy Sets. Fuzzy Sets and Systems 3, 291–310 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  68. 68.
    Luhandjula, M.K.: Compensatory Operators in Fuzzy Linear Programming with Multiple Objectives. Fuzzy Sets and Systems 8(3), 245–252 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  69. 69.
    Luhandjula, M.K.: Linear Programming Under Randomness and Fuzziness. Fuzzy Sets and Systems 10(1), 45–54 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  70. 70.
    McKenzie, L.W.: Classical General Equilibrium Theory. MIT Press, Cambridge (2002)Google Scholar
  71. 71.
    Negoita, C.V., et al.: Fuzzy Linear Programming and Tolerances in Planning. Econ. Group Cybernetic Studies 1, 3–15 (1976)MathSciNetGoogle Scholar
  72. 72.
    Negoita, C.V., Stefanescu, A.C.: On Fuzzy Optimization. In: Gupta, M.M., et al. (eds.) Approximate Reasoning In Decision Analysis, pp. 247–250. North Holland, New York (1982)Google Scholar
  73. 73.
    Negoita, C.V.: The Current Interest in Fuzzy Optimization. Fuzzy Sets and Systems 6(3), 261–270 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  74. 74.
    Negoita, C.V., et al.: On Fuzzy Environment in Optimization Problems. In: Rose, J., et al. (eds.) Modern Trends in Cybernetics and Systems, pp. 13–24. Springer, Berlin (1977)Google Scholar
  75. 75.
    Nguyen, H.T.: Random Sets and Belief Functions. Jour. of Math. Analysis and Applications 65(3), 531–542 (1978)zbMATHCrossRefGoogle Scholar
  76. 76.
    Orlovsky, S.A.: On Programming with Fuzzy Constraint Sets. Kybernetes 6, 197–201 (1977)zbMATHCrossRefGoogle Scholar
  77. 77.
    Orlovsky, S.A.: On Formulation of General Fuzzy Mathematical Problem. Fuzzy Sets and Systems 3, 311–321 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  78. 78.
    Ostasiewicz, W.: A New Approach to Fuzzy Programming. Fuzzy Sets and Systems 7(2), 139–152 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  79. 79.
    Patty Wayne, C.: Foundations of Topology. PWS Pub. Co., Boston (1993)zbMATHGoogle Scholar
  80. 80.
    Parthasarath, K.R.: Probability Measure on Metric Spaces. Academic Press, New York (1967)Google Scholar
  81. 81.
    Ponsard, G.: Partial Spatial Equilibra With Fuzzy Constraints. Journal of Regional Science 22(2), 159–175 (1982)CrossRefGoogle Scholar
  82. 82.
    Prade, H., et al.: Representation and Combination of Uncertainty with belief Functions and Possibility Measures. Comput. Intell. 4, 244–264 (1988)CrossRefGoogle Scholar
  83. 83.
    Prade, M.: Operations Research with Fuzzy Data. In: Want, P.P., et al. (eds.) Fuzzy Sets, pp. 155–170. Plenum, New York (1980)Google Scholar
  84. 84.
    Puri, M.L., et al.: Fuzzy Random Variables. Jour. of Mathematical Analysis and Applications 114(2), 409–422 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  85. 85.
    Ralescu, D.: Optimization in a Fuzzy Environment. In: Gupta, M.M., et al. (eds.) Advances in Fuzzy Set Theory and Applications, pp. 77–91. North-Holland, New York (1979)Google Scholar
  86. 86.
    Ralescu, D.A.: Orderings, Preferences and Fuzzy Optimization. In: Rose, J. (ed.) Current Topics in Cybernetics and Systems. Springer, Berlin (1978)Google Scholar
  87. 87.
    Rao, M.B., et al.: Some Comments on Fuzzy Variables. Fuzzy Sets and Systems 6(2), 285–292 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  88. 88.
    Rao, N.B., Rashed, A.: Some Comments on Fuzzy Random Variables. Fuzzy Sets and Systems 6(3), 285–292 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  89. 89.
    Ralescu, D.: Toward a General Theory of Fuzzy Variables. Jour. of Math. Analysis and Applications 86(1), 176–193 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  90. 90.
    Stein, N.E., Talaki, K.: Convex Fuzzy Random Variables. Fuzzy Sets and Systems 6(3), 271–284 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  91. 91.
    Rodabaugh, S.E.: Fuzzy Arithmetic and Fuzzy Topology. In: Lasker, G.E. (ed.) Applied Systems and Cybernetics. Fuzzy Sets and Systems, vol. VI, pp. 2803–2807. Pergamon Press, New York (1981)Google Scholar
  92. 92.
    Rodabaugh, S., et al. (eds.): Application of Category Theory to Fuzzy Subsets. Kluwer, Boston (1992)Google Scholar
  93. 93.
    Russell, B.: Vagueness. Australian Journal of Philosophy 1, 84–92 (1923)CrossRefGoogle Scholar
  94. 94.
    Russell, B.: Logic and Knowledge: Essays 1901-1950. Capricorn Books, New York (1971)Google Scholar
  95. 95.
    Sakawa, M.: Fuzzy Sets and Interactive Multiobjective Optimization. Plenum Press, New York (1993)zbMATHGoogle Scholar
  96. 96.
    Sakawa, M., et al.: Feasibility and Pareto Optimality for Multi-objective Nonlinear Programming Problems with Fuzzy Parameters. Fuzzy Sets and Systems 43(1), 1–15Google Scholar
  97. 97.
    Tanaka, K., et al.: Fuzzy Programs and Their Execution. In: Zadeh, L.A., et al. (eds.) Fuzzy Sets and Their Applications to Cognitive and Decision Processes, pp. 41–76 (1974)Google Scholar
  98. 98.
    Tanaka, K., et al.: Fuzzy Mathematical Programming. Transactions of SICE, 109–115 (1973)Google Scholar
  99. 99.
    Tanaka, H., et al.: On Fuzzy-Mathematical Programming. Journal of Cybernetics 3(4), 37–46 (1974)CrossRefGoogle Scholar
  100. 100.
    Tanaka, H., et al.: Fuzzy Linear Programming, Based on Fuzzy Functions. Bulletin of Univ. of Osaka Prefecture, Series A 29(2), 113–125 (1980)zbMATHGoogle Scholar
  101. 101.
    Tomovic, R., Vukobratovic, M.: General Sensitivity Theory. American Elsevier, New York (1972)zbMATHGoogle Scholar
  102. 102.
    Torr, C.: Equilibrium, Expectations, and Information: A Study of General Theory and Modern Classical Economics. Westview Press, Boulder Colo (1988)Google Scholar
  103. 103.
    Tsypkin, Y.Z.: Foundations of the Theory of Learning Systems. Academic Press, New York (1973)zbMATHGoogle Scholar
  104. 104.
    Valentinuzzi, M.: The Organs of Equilibrium and Orientation as a Control System. Hardwood Academic Pub., New York (1980)Google Scholar
  105. 105.
    Vira, J.: Fuzzy Expectation Values in Multistage Optimization Problems. Fuzzy Sets and Systems 6(2), 161–168 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  106. 106.
    Verdegay, J.L.: Fuzzy Mathematical Programming. In: Gupta, M.M., et al. (eds.) Fuzzy Information and Decision Processes, pp. 231–238. North-Holland, New York (1982)Google Scholar
  107. 107.
    Walsh, V.C., Gram, H.: Classical and Neoclassical Theories of General Equilibrium: Historical Origins and Mathematical Structure. Oxford University Press, New York (1980)Google Scholar
  108. 108.
    Wang, G.Y., et al.: The Theory of Fuzzy Stochastic Processes. Fuzzy Sets and Systems 51(2), 161–178 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  109. 109.
    Wang, X., et al.: Fuzzy Linear Regression Analysis of Fuzzy Valued Variable. Fuzzy Sets and Systems 36(1), 19Google Scholar
  110. 110.
    Warren, R.H.: Optimality in Fuzzy Topological Polysystems. Jour. Math. Anal. 54, 309–315 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  111. 111.
    Roy, W.E.: General Equilibrium Analysis: Studies in Appraisal. Cambridge University Press, Cambridge (1985)Google Scholar
  112. 112.
    Roy, W.E.: Microfoundations: The Compatibility of Microeconomics and Macroeconomics. Cambridge University Press, Cambridge (1980)Google Scholar
  113. 113.
    Weiss, M.D.: Fixed Points, Separation and Induced Topologies for Fuzzy Sets. Jour. Math. Anal. and Appln. 50, 142–150 (1975)zbMATHCrossRefGoogle Scholar
  114. 114.
    Whittle, P.: Systems in Stochastic Equilibrium. Wiley, New York (1986)zbMATHGoogle Scholar
  115. 115.
    Wiedey, G., Zimmermann, H.J.: Media Selection and Fuzzy Linear Programming. Journal Oper. Res. Society 29, 1071–1084 (1978)zbMATHGoogle Scholar
  116. 116.
    Wong, C.K.: Fuzzy Points and Local Properties of Fuzzy Topology. Jour. Math. Anal. and Appln. 46, 316–328 (1987)CrossRefGoogle Scholar
  117. 117.
    Yager, R.R.: Mathematical Programming with Fuzzy Constraints and Preference on the Objective. Kybernetes 8, 285–291 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  118. 118.
    Zadeh, L.A.: The Role of Fuzzy Logic in the Management of Uncertainty in expert Systems. Fuzzy Sets and Systems 11, 199–227 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  119. 119.
    Zadeh, L.A.: Probability Measure of Fuzzy Event. Jour. of Math. Analysis and Applications 23, 421–427 (1968)zbMATHMathSciNetCrossRefGoogle Scholar
  120. 120.
    Zimmerman, H.-J.: Description and Optimization of Fuzzy Systems. Intern. Jour. Gen. Syst. 2(4), 209–215 (1975)CrossRefGoogle Scholar
  121. 121.
    Zimmerman, H.-J.: Fuzzy Programming and Linear Programming with Several Objective Functions. Fuzzy Sets and Systems 1(1), 45–56 (1978)MathSciNetCrossRefGoogle Scholar
  122. 122.
    Zimmerman, H.J.: Applications of Fuzzy Set Theory to Mathematical Programming. Information Science 36(1), 29–58 (1985)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Kofi Kissi Dompere
    • 1
  1. 1.Department of EeconomicsHoward UniveristyWashingtonUSA

Personalised recommendations