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Introduction to Fuzzy and Possibilistic Optimization

  • Weldon A. Lodwick
  • Elizabeth Untiedt
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 254)

Introduction

Deterministic optimization is a normative process which extracts the best from a set of options, usually under constraints. It is arguably true that optimization is one of the most used areas of mathematical applications. It is the thesis of this book that applied mathematical programming problems should be solved predominantly by using a fuzzy and possibilistic approaches. Rommelfanger ([42], p. 295), states that the only operations research methods that is widely applied is linear programming. He goes on to state that even though this is true, of the 167 production (linear) programming systems investigated and surveyed by Fandel [18], only 13 of these were ”purely” (my interpretation) linear programming systems. Thus, Rommelfanger concludes that even with this most highly used and applied operations research method, there is a discrepancy between classical linear programming and what is applied. Deterministic and stochastic optimization models require well-defined input parameters (coefficients, right-hand side values), relationships (inequalities, equalities), and preferences (real valued functions to maximize, minimize) either as real numbers or real valued distribution functions. Any large scale model requires significant data gathering efforts. If the model has projections of future values, it is clear that real numbers and real valued distributions are inadequate representations of parameters, even assuming that the model correctly captures the underlying system. It is also known from mathematical programming theory that only a few of the variables and constraints are necessary to describe an optimal solution (basic variables and active constraints), assuming a correct deterministic normative criterion (objective function). The ratio of variables that are basic and constraints that are active compared to the total becomes smaller, in general, as the model increases in size since in general large-scale models tend to become more sparse. Thus, only a few parameters need to be obtained precisely. Of course the problem is that it is not known a priori which variables will be basic and which constraints will be active.

Keywords

Membership Function Fuzzy Number Soft Constraint Aggregation Operator Fuzzy Measure 
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.

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References

  1. 1.
    Bellman, R.E., Zadeh, L.A.: Decision-making in a fuzzy environment. Management Science, Serial B 17, 141–164 (1970)MathSciNetGoogle Scholar
  2. 2.
    Birge, J.R., Louveux, F.: Introduction to Stochastic Programming. Springer, New York (1997)zbMATHGoogle Scholar
  3. 3.
    Buckley, J.J.: Solving possibilistic linear programming problems. Fuzzy Sets and Systems 31, 329–341 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    de Cooman, G.: Possibility theory I: The measure- and integral - theoretic groundwork. International Journal of General Systems 25(4), 291–323 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Delgado, M., Verdegay, J.L., Vila, M.A.: A general model for fuzzy linear programming. Fuzzy Sets and Systems 29, 21–29 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dempster, A.P.: Upper and lower probabilities induced by a multivalued mapping. Annals of Mathematical Statistics 38, 325–339 (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dubois, D.: Linear programming with fuzzy data. In: Bezdek, J. (ed.) Analysis of Fuzzy Information. Applications in Engineering and Science, vol. III, pp. 241–263. CRC Press, Boca Raton (1897)Google Scholar
  8. 8.
    Dubois, D.: Personal communications (2008, 2009)Google Scholar
  9. 9.
    Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York (1980)zbMATHGoogle Scholar
  10. 10.
    Dubois, D., Prade, H.: The mean value of a fuzzy number. Fuzzy Sets and Systems 24, 279–300 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dubois, D., Prade, H.: Possibility Theory. Plenum Press, New York (1988)zbMATHGoogle Scholar
  12. 12.
    Dubois, D., Prade, H.: The three semantics of fuzzy sets. Fuzzy Sets and Systems 90, 141–150 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Dubois, D., Prade, H. (eds.): Fundamentals of Fuzzy Sets. Kluwer Academic Press, Dordrecht (2000)zbMATHGoogle Scholar
  14. 14.
    Dubois, D., Prade, H.: Formal representations of uncertainty. In: IFSA 2009, Lisboa, Portugal (July 2009)Google Scholar
  15. 15.
    Dubois, D., Karre, E., Mesiar, R., Prade, H.: Fuzzy interval analysis. In: Dubois, D., Prade, H. (eds.) Fundamentals of Fuzzy Sets, ch. 10, pp. 483–581. Kluwer Academic Press, Dordrecht (2000)Google Scholar
  16. 16.
    Dubois, D., Nguyen, H., Prade, H.: Possibility theory, probability theory, and fuzzy sets: Misunderstanding, bridges and gaps. In: Dubois, D., Prade, H. (eds.) Fundamentals of Fuzzy Sets, ch. 7, pp. 343–438. Kluwer Academic Press, Dordrecht (2000)Google Scholar
  17. 17.
    Dubois, D., Prade, H., Sabbadin, R.: Decision-theoretic foundations of qualitative possibility theory. European Journal of Operational Research 128, 459–478 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Fandel, G.: PPS-Systeme: Grundlagen, Methoden, Software, Markanalyse. Springer, Heidelberg (1994)Google Scholar
  19. 19.
    Fishburn, P.C.: Stochastic dominance and moments of distributions. Mathematics of Operations Research 5, 94–100 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Gay, D.M.: Solving interval linear equations. SIAM Journal of Numerical Analysis 19(4), 858–870 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Greenberg, H.: Personal communication (2005)Google Scholar
  22. 22.
    Inuiguchi, M.: Stochastic programming problems versus fuzzy mathematical programming problems. Japanese Journal of Fuzzy Theory and Systems 4(1), 97–109 (1992)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Inuiguchi, M., Ichihashi, H., Kume, Y.: Relationships between modality constrained programming problems and various fuzzy mathematical programming problems. Fuzzy Sets and Systems 49, 243–259 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Inuiguchi, M., Sakawa, M., Kume, Y.: The usefulness of possibilistic programming in production planning problems. Int. J. of Production Economics 33, 49–52 (1994)Google Scholar
  25. 25.
    Inuiguchi, M., Tanino, T.: Two-stage linear recourse problems under non-probabilistic uncertainty. In: Yoshida, Y. (ed.) Dynamical Aspect in Fuzzy Decision Making, pp. 117–140. Physica-Verlag, Heidelberg (2001)Google Scholar
  26. 26.
    Jamison, K.D.: Modeling uncertainty using probability based possibility theory with applications to optimization. Ph.D. Thesis, UCD Department of Mathematics (1998)Google Scholar
  27. 27.
    Jamison, K.D., Lodwick, W.A.: Fuzzy linear programming using penalty method. Fuzzy Sets and Systems 119, 97–110 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Jamison, K.D., Lodwick, W.A.: The construction of consistent possibility and necessity measures. Fuzzy Sets and Systems 132, 1–10 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Kaymak, U., Sousa, J.M.: Weighting of constraints in fuzzy optimization. Constraints 8, 61–78 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, New Jersey (1995)zbMATHGoogle Scholar
  31. 31.
    Lodwick, W., Bachman, K.: Solving Large Scale Fuzzy Possibilistic Optimization Problems. Fuzzy Optimization and Decision Making 4(4), 257–278 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Lodwick, W.A., Jamison, K.D.: A computational method for fuzzy optimization. In: Ayyub, B., Gupta, M. (eds.) Uncertainty Analysis in Engineering and Sciences: Fuzzy Logic, Statistics, and Neural Network Approach, ch. 19. Kluwer Academic Publishers, Dordrecht (1997)Google Scholar
  33. 33.
    Lodwick, W.A., Jamison, K.D.: Estimating and validating the cumulative distribution of a function of random variables: Toward the development of distribution arithmetic. Reliable Computing 9, 127–141 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Lodwick, W.A., Jamison, K.D.: Theory and semantics for fuzzy and possibilistic optimization. In: Fuzzy Logic, Soft Computing and Computational Intelligence (Eleventh International Fuzzy Systems Association World Congress), Beijing, China, July 28-31, vol. III, pp. 1805–1810 (2005)Google Scholar
  35. 35.
    Lodwick, W.A., Jamison, K.D.: Theory and semantics for fuzzy and possibilistic optimization. Fuzzy Sets and Systems 158(7), 1861–1871 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Lodwick, W.A., Jamison, K.D.: The Use of interval-valued probability measure in optimization under uncertainty for problems containing a mixture of possibilistic, probabilistic and interval uncertainty. In: Melin, P., Castillo, O., Aguilar, L.T., Kacprzyk, J., Pedrycz, W. (eds.) IFSA 2007. LNCS (LNAI), vol. 4529, pp. 361–370. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  37. 37.
    Lodwick, W.A., Jamison, K.D.: Interval-Valued Probability in the Analysis of Problems Containing a Mixture of Possibilistic, Probabilistic, and Interval Uncertainty. Fuzzy Sets and Systems 159(1), 2845–2858 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Lodwick, W.A., Neumaier, A., Newman, F.D.: Optimization under uncertainty: methods and applications in radiation therapy. In: Proceedings 10th IEEE International Conference on Fuzzy Systems, vol. 3, pp. 1219–1222 (2001)Google Scholar
  39. 39.
    Luhandjula, M.K.: On possibilistic linear programming. Fuzzy Sets and Systems 18, 15–30 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Neumaier, A.: Fuzzy modeling in terms of surprise. Fuzzy Sets and Systems 135(1), 21–38 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Puri, M.L., Ralescu, D.: Fuzzy measures are not possibility measures. Fuzzy Sets and Systems 7, 311–313 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Rommelfanger, H.J.: The advantages of fuzzy optimization models in practical use. Fuzzy Optimization and Decision Making 3, 295–309 (2004)zbMATHCrossRefGoogle Scholar
  43. 43.
    Rommelfanger, H.J.: Personal communication (April 2009)Google Scholar
  44. 44.
    Russell, B.: Vagueness. Australasian Journal of Psychology and Philosophy 1, 84–92 (1923)CrossRefGoogle Scholar
  45. 45.
    Shafer, G.: Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)zbMATHGoogle Scholar
  46. 46.
    Simon, H.: The Sciences of the Artificial, 2nd edn. The MIT Press, Cambridge (1969/1981)Google Scholar
  47. 47.
    Tanaka, H., Asai, K.: Fuzzy linear programming with fuzzy numbers. Fuzzy Sets and Systems 13, 1–10 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Tanaka, H., Okuda, T., Asai, K.: On fuzzy mathematical programming. Transactions of the Society of Instrument and Control Engineers 9(5), 607–613 (1973) (in Japanese)Google Scholar
  49. 49.
    Tanaka, H., Okuda, T., Asai, K.: On fuzzy mathematical programming. J. of Cybernetics 3, 37–46 (1974)CrossRefMathSciNetGoogle Scholar
  50. 50.
    Tanaka, H., Ichihashi, H., Asai, K.: Fuzzy Decision in linear programming with trapezoid fuzzy parameters. In: Kacpryzk, J., Yager, R.R. (eds.) Management Decision Support Systems Using Fuzzy Sets and Possibility Theory, pp. 146–154. Verlag TUV, Koln (1985)Google Scholar
  51. 51.
    Verdegay, J.L.: Fuzzy mathematical programming. In: Gupta, M.M., Sanchez, E. (eds.) Fuzzy Information and Decision Processes, pp. 231–237. North Holland Company, Amsterdam (1982)Google Scholar
  52. 52.
    Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1947)Google Scholar
  53. 53.
    Yager, R.: On choosing between fuzzy subsets. Kybernetes 9, 151–154 (1980)zbMATHCrossRefGoogle Scholar
  54. 54.
    Yager, R.: A procedure for ordering fuzzy subsets of the unit interval. Information Sciences 24, 143–161 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1, 3–28 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Zimmermann, H.: Description and optimization of fuzzy systems. International J. of General Systems 2, 209–215 (1976)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Weldon A. Lodwick
  • Elizabeth Untiedt

There are no affiliations available

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