Linear Programming with Interval Type-2 Fuzzy Constraints

  • Juan C. Figueroa-GarcíaEmail author
  • Germán Hernández
Part of the Studies in Computational Intelligence book series (SCI, volume 539)


This chapter shows a method for solving Linear Programming (LP) problems that includes Interval Type-2 fuzzy constraints. A method is proposed for finding an optimal solution in these conditions, using convex optimization techniques. The entire method is presented and some interpretation issues are discussed. An introductory example is presented and solved using our proposal, and its results are explained and discussed.


Membership Function Fuzzy Number Fuzzy Linear Programming Fuzzy Relation Equation Fuzzy Constraint 
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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  1. 1.
    Ghodousiana, A., Khorram, E.: Solving a linear programming problem with the convex combination of the max-min and the max-average fuzzy relation equations. Applied Mathematics and Computation 180(1), 411–418 (2006)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Sy-Ming, G., Yan-Kuen, W.: Minimizing a linear objective function with fuzzy relation equation constraints. Fuzzy Optimization and Decision Making 1(4), 347 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Tanaka, H., Asai, K.: Fuzzy Solution in Fuzzy Linear Programming Problems. IEEE Transactions on Systems, Man and Cybernetics 14, 325–328 (1984)CrossRefzbMATHGoogle Scholar
  4. 4.
    Tanaka, H., Asai, K., Okuda, T.: On Fuzzy Mathematical Programming. Journal of Cybernetics 3, 37–46 (1974)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Inuiguchi, M., Sakawa, M.: A possibilistic linear program is equivalent to a stochastic linear program in a special case. Fuzzy Sets and Systems 76(1), 309–317 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Inuiguchi, M., Sakawa, M.: Possible and necessary optimality tests in possibilistic linear programming problems. Fuzzy Sets and Systems 67, 29–46 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Inuiguchi, M., Ramík, J.: Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem. Fuzzy Sets and Systems 111, 3–28 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Angelov, P.P.: Optimization in an intuitionistic fuzzy environment. Fuzzy Sets and Systems 86(3), 299–306 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Dubey, D., Chandra, S., Mehra, A.: Fuzzy linear programming under interval uncertainty based on ifs representation. Fuzzy Sets and Systems 188(1), 68–87 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Zimmermann, H.J.: Fuzzy programming and Linear Programming with several objective functions. Fuzzy Sets and Systems 1(1), 45–55 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Zimmermann, H.J., Fullér, R.: Fuzzy Reasoning for solving fuzzy Mathematical Programming Problems. Fuzzy Sets and Systems 60(1), 121–133 (1993)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Dantzig, G.: Linear Programming and Extensions. Princeton (1998)Google Scholar
  13. 13.
    Bazaraa, M.S., Jarvis, J.J., Sherali, H.D.: Linear Programming and Networks Flow. John Wiley and Sons (2009)Google Scholar
  14. 14.
    Bellman, R.E., Zadeh, L.A.: Decision-making in a fuzzy environment. Management Science 17(1), 141–164 (1970)MathSciNetGoogle Scholar
  15. 15.
    Kall, P., Mayer, J.: Stochastic Linear Programming: Models, Theory, and Computation. Springer (2010)Google Scholar
  16. 16.
    Mora, H.M.: Optimización no lineal y dinámica. Universidad Nacional de Colombia (2001)Google Scholar
  17. 17.
    Mendel, J.: Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions. Prentice Hall (2001)Google Scholar
  18. 18.
    Mendel, J., John, R.I., Liu, F.: Interval type-2 fuzzy logic systems made simple. IEEE Transactions on Fuzzy Systems 14(6), 808–821 (2006)CrossRefGoogle Scholar
  19. 19.
    Mendel, J., John, R.I.: Type-2 fuzzy sets made simple. IEEE Transactions on Fuzzy Systems 10(2), 117–127 (2002)CrossRefGoogle Scholar
  20. 20.
    Liang, Q., Mendel, J.: Interval type-2 fuzzy logic systems: Theory and design. IEEE Transactions on Fuzzy Systems 8(5), 535–550 (2000)CrossRefGoogle Scholar
  21. 21.
    Karnik, N.N., Mendel, J.: Operations on type-2 fuzzy sets. Fuzzy Sets and Systems 122, 327–348 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Karnik, N.N., Mendel, J., Liang, Q.: Type-2 fuzzy logic systems. Fuzzy Sets and Systems 17(10), 643–658 (1999)Google Scholar
  23. 23.
    Melgarejo, M.: A Fast Recursive Method to compute the Generalized Centroid of an Interval Type-2 Fuzzy Set. In: Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS), pp. 190–194. IEEE (2007)Google Scholar
  24. 24.
    Melgarejo, M.: Implementing Interval Type-2 Fuzzy processors. IEEE Computational Intelligence Magazine 2(1), 63–71 (2007)CrossRefGoogle Scholar
  25. 25.
    Figueroa, J.C.: Linear programming with interval type-2 fuzzy right hand side parameters. In: 2008 Annual Meeting of the IEEE North American Fuzzy Information Processing Society, NAFIPS (2008)Google Scholar
  26. 26.
    Figueroa, J.C.: Solving fuzzy linear programming problems with interval type-2 RHS. In: 2009 Conference on Systems, Man and Cybernetics, pp. 1–6. IEEE (2009)Google Scholar
  27. 27.
    Figueroa, J.C.: Interval type-2 fuzzy linear programming: Uncertain constraints. In: IEEE Symposium Series on Computational Intelligence, pp. 1–6. IEEE (2011)Google Scholar
  28. 28.
    Figueroa-García, J.C., Hernandez, G.: Computing optimal solutions of a linear programming problem with interval type-2 fuzzy constraints. In: Corchado, E., Snášel, V., Abraham, A., Woźniak, M., Graña, M., Cho, S.-B. (eds.) HAIS 2012, Part I. LNCS, vol. 7208, pp. 567–576. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  29. 29.
    Mendel, J.: Fuzzy sets for words: a new beginning. In: The IEEE International Conference on Fuzzy Systems, pp. 37–42 (2003)Google Scholar
  30. 30.
    Mendel, J.: Type-2 Fuzzy Sets: Some Questions and Answers. IEEE coNNectionS. A Publication of the IEEE Neural Networks Society (8), 10–13 (2003)Google Scholar
  31. 31.
    Mendel, J.M., Liu, F.: Super-exponential convergence of the Karnik-Mendel algorithms for computing the centroid of an interval type-2 fuzzy set. IEEE Transactions on Fuzzy Systems 15(2), 309–320 (2007)CrossRefGoogle Scholar
  32. 32.
    Kearfott, R.B., Kreinovich, V.: Beyond convex? global optimization is feasible only for convex objective functions: A theorem. Journal of Global Optimization 33(4), 617–624 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Niewiadomski, A.: On Type-2 fuzzy logic and linguistic summarization of databases. Bulletin of the Section of Logic 38(3), 215–227 (2009)MathSciNetGoogle Scholar
  34. 34.
    Niewiadomski, A.: Imprecision measures for Type-2 fuzzy sets: Applications to linguistic summarization of databases. In: Rutkowski, L., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2008. LNCS (LNAI), vol. 5097, pp. 285–294. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  35. 35.
    Wolsey, L.A.: Integer Programming. John Wiley and Sons (1998)Google Scholar
  36. 36.
    Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Dover Publications (1998)Google Scholar

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Universidad Nacional de ColombiaBogotaColombia

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