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.
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References
Black, Max, The Nature of Mathematics, Totowa, Littlefield, Adams and Co. (1965)
Boolos, G.S., Jeffrey, R.C.: Computability and Logic. Cambridge University Press, New York (1989)
Bose, R.K., Sahani, D.: Fuzzy Mappings and Fixed Point Theorems. Fuzzy Sets and Systems 21, 53–58 (1987)
Buckley, J.J.: Fuzzy Programming And the Pareto Optimal Set. Fuzzy Set and Systems 10, 57–63 (1983)
Butnariu, D.: Fixed Points for Fuzzy Mappings. Fuzzy Sets and Systems 7, 191–207 (1982)
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)
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)
Cerny, M.: Fuzzy Approach to Vector Optimization. Intern. Jour. of General Systems 20(1), 23–29
Chang, S.S.: Fixed Point Theorems for Fuzzy Mappings. Fuzzy Sets and Systems 17, 181–187 (1985)
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)
Churchman, C.W.: Prediction and Optimal Decision. Pretice-Hall, Englewood Cliffs (1961)
Cohn, D.L.: Measure Theory. Birkhauser, Boston (1980)
Delgado, M., et al.: On the Concept of Possibility-Probability Consistency. Fuzzy Sets and Systems 21(3), 311–318 (1987)
DiNola, A., et al. (eds.): The Mathematics of Fuzzy Systems. Verlag TUV Rheinland, Koln (1986)
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)
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)
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)
Dompere, K.K.: Cost-Benefit Analysis and the Theory of Fuzzy Decision: Identification and Measurement Theory. Springer, New York (2004)
Dompere, K.K.: Cost-Benefit Analysis and the Theory of Fuzzy Decision: Fuzzy Value Theory. Springer, New York (2004)
Delbaen, F.: Stochastic Preferences and General Equilibrium Theory. In: Dreze, J.H. (ed.) Allocation Under Uncertainty: Equilibrium and Optimality. Wiley, New York (1974)
Dubois, D., et al.: Systems of Linear Fuzzy Constraints. Fuzzy Sets and Systems 3(1), 37–48 (1980)
Dubois, D.: An Application of Fuzzy Arithmetic to the Optimization of Industrial Machining Processes. Mathematical Modelling 9(6), 461–475 (1987)
Dumitrescu, D.: Entropy of a Fuzzy Process. Fuzzy Sets and Systems 55(2), 169–177 (1993)
Eaves, B.C.: Computing Kakutani Fixed Points. Journal of Applied Mathematics 21, 236–244 (1971)
El Rayes, A.B., et al.: Generalized Possibility Measures. Information Sciences 79, 201–222 (1994)
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)
Feng, Y.J.: A Method Using Fuzzy Mathematics to Solve the Vector Maxim Problem. Fuzzy Set and Systems 9(2), 129–136 (1983)
Fischer, R.B., Peters, G.: Chemical Equilibrium. Saunders Pub., Philadelphia (1970)
Fisher, F.M.: Disequilibrium Foundations of Equilibrium Economics. Cambridge University Press, New York (1983)
Foster, D.H.: Fuzzy Topological Groups. Journal of Math. Analysis and Applications 67, 549–564 (1979)
Goetschel Jr., R., et al.: Topological Properties of Fuzzy Number. Fuzzy Sets and Systems 10(1), 87–99 (1983)
Fourgeaud, C., Gouriéroux, C.: Learning Procedures and Convergence to Rationality. Econometrica 54, 845–868 (1986)
Freeman, A., Carchedi, G. (eds.): Marx and Non-Equilibrium Economics, Cheltenham, UK, Edward Elgar (1996)
Gaines, B.R.: Fuzzy and Probability Uncertainty logics. Information and Control 38(2), 154–169 (1978)
Newton, G., Hare, P.H. (eds.): Naturalism and Rationality. Prometheus Books, Buffalo (1986)
Ginsburgh, V.: Activity Analysis and General Equilibrium Modelling. North-Holland, New York (1981)
Grabish, M., et al.: Fundamentals of Uncertainty Calculi with Application to Fuzzy Systems. Kluwer, Boston (1994)
Hahn, F.: Equilibrium and Macroeconomics. MIT Press, Cambridge (1984)
Hamacher, H., et al.: Sensitivity Analysis in Fuzzy Linear Programming. Fuzzy Sets and Systems 1, 269–281 (1978)
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)
Hannan, E.L.: Linear Programming with Multiple Fuzzy Goals. Fuzzy Sets and Systems 6(3), 235–248 (1981)
Hansen, B.: A Survey of General Equilibrium Systems. McGraw-Hill, New York (1970)
Heilpern, S.: Fuzzy Mappings and Fixed Point Theorem. Journal of Mathematical Analysis and Applications 83, 566–569 (1981)
Hestenes, M.R.: Optimization Theory: The Finite Dimensional Case, New York (1975)
Heyman, D.P., Sobel, M.J.: Stochastic Models in Operations Research. Stochastic Optimization, vol. II. Mc Graw-Hill, New York (1984)
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)
Ignizio, J.P., et al.: Fuzzy Multicriteria Integer Programming via Fuzzy Generalized Networks. Fuzzy Sets and Systems 10(3), 261–270 (1983)
Jarvis, R.A.: Optimization Strategies in Adaptive Control: A Selective Survey. IEEE Trans. Syst. Man. Cybernetics SMC-5, 83–94 (1975)
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)
Kacprzyk, J., Fedrizzi, M. (eds.): Combining Fuzzy Imprecision with Probabilistic Uncertainty in Decision Making. Plenum Press, New York (1992)
Kacprzyk, J., et al. (eds.): Optimization Models Using Fuzzy Sets and Possibility Theory. D. Reidel, Boston (1987)
Kakutani, S.: A Generalization of Brouwer’s Fixed Point Theorem. Duke Mathematical Journal 8, 416–427 (1941)
Kaleva, O.: A Note on Fixed Points for Fuzzy Mappings. Fuzzy Sets and Systems 15, 99–100 (1985)
Kandel, A.: On Minimization of Fuzzy Functions. IEEE Trans. Comp. C-22, 826–832 (1973)
Kandel, A.: Comments on Minimization of Fuzzy Functions. IEEE Trans. Comp. C-22, 217 (1973)
Kandel, A.: On the Minimization of Incompletely Specified Fuzzy Functions. Information, and Control 26, 141–153 (1974)
Kaufmann, A., Gupta, M.M.: Introduction to fuzzy arithmetic: Theory and applications. Van Nostrand Rheinhold, New York (1991)
Kaufmann, A.: Introduction to the Theory of Fuzzy Subsets, vol. 1. Academic Press, New York (1975)
Kaufmann, A.: The Science of Decision-Making: Introduction to Praxeology. McGraw-Hill, New York (1968)
Kickert, W.J.M.: Organization of Decision-Making: A Systems-Theoretical Approach. North-Holland, Amsterdam (1980)
Klir, G.J., et al.: Fuzzy Sets, Uncertainty and Information. Prentice Hall, Englewood Cliffs (1988)
Klir, G.J., et al.: Probability-Possibility Transformations: A Comparison. Intern. Jour. of General Systems 21(3), 291–310 (1992)
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)
Kosko, B.: Fuzziness vs Probability. Intern. Jour. of General Systems 17(1-3), 211–240 (1990)
Lai, Y., et al.: Fuzzy Mathematical Programming. Springer, New York (1992)
Lee, E.S., et al.: Fuzzy Multiple Objective Programming and Compromise Programming with Pareto Optimum. Fuzzy Sets and Systems 53(3), 275–288 (1993)
Lowen, R.: Connex Fuzzy Sets. Fuzzy Sets and Systems 3, 291–310 (1980)
Luhandjula, M.K.: Compensatory Operators in Fuzzy Linear Programming with Multiple Objectives. Fuzzy Sets and Systems 8(3), 245–252 (1982)
Luhandjula, M.K.: Linear Programming Under Randomness and Fuzziness. Fuzzy Sets and Systems 10(1), 45–54 (1983)
McKenzie, L.W.: Classical General Equilibrium Theory. MIT Press, Cambridge (2002)
Negoita, C.V., et al.: Fuzzy Linear Programming and Tolerances in Planning. Econ. Group Cybernetic Studies 1, 3–15 (1976)
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)
Negoita, C.V.: The Current Interest in Fuzzy Optimization. Fuzzy Sets and Systems 6(3), 261–270 (1981)
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)
Nguyen, H.T.: Random Sets and Belief Functions. Jour. of Math. Analysis and Applications 65(3), 531–542 (1978)
Orlovsky, S.A.: On Programming with Fuzzy Constraint Sets. Kybernetes 6, 197–201 (1977)
Orlovsky, S.A.: On Formulation of General Fuzzy Mathematical Problem. Fuzzy Sets and Systems 3, 311–321 (1980)
Ostasiewicz, W.: A New Approach to Fuzzy Programming. Fuzzy Sets and Systems 7(2), 139–152 (1982)
Patty Wayne, C.: Foundations of Topology. PWS Pub. Co., Boston (1993)
Parthasarath, K.R.: Probability Measure on Metric Spaces. Academic Press, New York (1967)
Ponsard, G.: Partial Spatial Equilibra With Fuzzy Constraints. Journal of Regional Science 22(2), 159–175 (1982)
Prade, H., et al.: Representation and Combination of Uncertainty with belief Functions and Possibility Measures. Comput. Intell. 4, 244–264 (1988)
Prade, M.: Operations Research with Fuzzy Data. In: Want, P.P., et al. (eds.) Fuzzy Sets, pp. 155–170. Plenum, New York (1980)
Puri, M.L., et al.: Fuzzy Random Variables. Jour. of Mathematical Analysis and Applications 114(2), 409–422 (1986)
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)
Ralescu, D.A.: Orderings, Preferences and Fuzzy Optimization. In: Rose, J. (ed.) Current Topics in Cybernetics and Systems. Springer, Berlin (1978)
Rao, M.B., et al.: Some Comments on Fuzzy Variables. Fuzzy Sets and Systems 6(2), 285–292 (1981)
Rao, N.B., Rashed, A.: Some Comments on Fuzzy Random Variables. Fuzzy Sets and Systems 6(3), 285–292 (1981)
Ralescu, D.: Toward a General Theory of Fuzzy Variables. Jour. of Math. Analysis and Applications 86(1), 176–193 (1982)
Stein, N.E., Talaki, K.: Convex Fuzzy Random Variables. Fuzzy Sets and Systems 6(3), 271–284 (1981)
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)
Rodabaugh, S., et al. (eds.): Application of Category Theory to Fuzzy Subsets. Kluwer, Boston (1992)
Russell, B.: Vagueness. Australian Journal of Philosophy 1, 84–92 (1923)
Russell, B.: Logic and Knowledge: Essays 1901-1950. Capricorn Books, New York (1971)
Sakawa, M.: Fuzzy Sets and Interactive Multiobjective Optimization. Plenum Press, New York (1993)
Sakawa, M., et al.: Feasibility and Pareto Optimality for Multi-objective Nonlinear Programming Problems with Fuzzy Parameters. Fuzzy Sets and Systems 43(1), 1–15
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)
Tanaka, K., et al.: Fuzzy Mathematical Programming. Transactions of SICE, 109–115 (1973)
Tanaka, H., et al.: On Fuzzy-Mathematical Programming. Journal of Cybernetics 3(4), 37–46 (1974)
Tanaka, H., et al.: Fuzzy Linear Programming, Based on Fuzzy Functions. Bulletin of Univ. of Osaka Prefecture, Series A 29(2), 113–125 (1980)
Tomovic, R., Vukobratovic, M.: General Sensitivity Theory. American Elsevier, New York (1972)
Torr, C.: Equilibrium, Expectations, and Information: A Study of General Theory and Modern Classical Economics. Westview Press, Boulder Colo (1988)
Tsypkin, Y.Z.: Foundations of the Theory of Learning Systems. Academic Press, New York (1973)
Valentinuzzi, M.: The Organs of Equilibrium and Orientation as a Control System. Hardwood Academic Pub., New York (1980)
Vira, J.: Fuzzy Expectation Values in Multistage Optimization Problems. Fuzzy Sets and Systems 6(2), 161–168 (1981)
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)
Walsh, V.C., Gram, H.: Classical and Neoclassical Theories of General Equilibrium: Historical Origins and Mathematical Structure. Oxford University Press, New York (1980)
Wang, G.Y., et al.: The Theory of Fuzzy Stochastic Processes. Fuzzy Sets and Systems 51(2), 161–178 (1992)
Wang, X., et al.: Fuzzy Linear Regression Analysis of Fuzzy Valued Variable. Fuzzy Sets and Systems 36(1), 19
Warren, R.H.: Optimality in Fuzzy Topological Polysystems. Jour. Math. Anal. 54, 309–315 (1976)
Roy, W.E.: General Equilibrium Analysis: Studies in Appraisal. Cambridge University Press, Cambridge (1985)
Roy, W.E.: Microfoundations: The Compatibility of Microeconomics and Macroeconomics. Cambridge University Press, Cambridge (1980)
Weiss, M.D.: Fixed Points, Separation and Induced Topologies for Fuzzy Sets. Jour. Math. Anal. and Appln. 50, 142–150 (1975)
Whittle, P.: Systems in Stochastic Equilibrium. Wiley, New York (1986)
Wiedey, G., Zimmermann, H.J.: Media Selection and Fuzzy Linear Programming. Journal Oper. Res. Society 29, 1071–1084 (1978)
Wong, C.K.: Fuzzy Points and Local Properties of Fuzzy Topology. Jour. Math. Anal. and Appln. 46, 316–328 (1987)
Yager, R.R.: Mathematical Programming with Fuzzy Constraints and Preference on the Objective. Kybernetes 8, 285–291 (1979)
Zadeh, L.A.: The Role of Fuzzy Logic in the Management of Uncertainty in expert Systems. Fuzzy Sets and Systems 11, 199–227 (1983)
Zadeh, L.A.: Probability Measure of Fuzzy Event. Jour. of Math. Analysis and Applications 23, 421–427 (1968)
Zimmerman, H.-J.: Description and Optimization of Fuzzy Systems. Intern. Jour. Gen. Syst. 2(4), 209–215 (1975)
Zimmerman, H.-J.: Fuzzy Programming and Linear Programming with Several Objective Functions. Fuzzy Sets and Systems 1(1), 45–56 (1978)
Zimmerman, H.J.: Applications of Fuzzy Set Theory to Mathematical Programming. Information Science 36(1), 29–58 (1985)
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Dompere, K.K. (2010). Fuzziness, Rationality, Optimality and Equilibrium in Decision and Economic Theories. In: Lodwick, W.A., Kacprzyk, J. (eds) Fuzzy Optimization. Studies in Fuzziness and Soft Computing, vol 254. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13935-2_1
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