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Parametric Analysis in Fuzzy Number Linear Programming Problems

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Abstract

This paper describes parametric analysis in fuzzy number linear programming (FNLP) when the objective function coefficients and/or the right-hand-side constants are parameterized. Using a linear ranking function, we consider the problem variations. In fact, we find a range set of the parameters for which a given basis remains optimal for the FNLP problem. If the perturbation destroys optimality and/or feasibility of the optimal basis, we use of the fuzzy primal simplex method, the fuzzy dual simplex method and/or our proposed fuzzy primal-dual simplex method to find the new optimal basis. Finally, by numerical examples we demonstrate the computational procedure.

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References

  1. Tanaka, H., Asai, A.: Fuzzy linear programming problems with fuzzy numbers. Fuzzy Sets Syst. 13(1), 1–10 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bellman, R.E., Zadeh, L.A.: Decision making in a fuzzy environment. Manag. Sci. 17, 141–164 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  3. Tanaka, H., Okuda, T., Asai, K.: On fuzzy mathematical programming. J. Cybern. 3, 37–46 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  4. Yager, R.R.: A procedure for ordering fuzzy subsets of the unit interval. Inf. Sci. 24, 143–161 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Zimmermann, H.J.: Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1, 45–55 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bector, C.R., Chandra, S.: Fuzzy Mathematical Programming and Fuzzy Matrix Games. Springer, Berlin (2005)

    MATH  Google Scholar 

  7. Guu, S.S., Wu, Y.K.: Two phase approach for solving the fuzzy linear programming problems. Fuzzy Sets Syst. 107, 191–195 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lai, Y.J., Hwang, C.L.: Fuzzy Mathematical Programming: Methods and Applications. Springer-Verlag, Berlin (1992)

    Book  MATH  Google Scholar 

  9. Lodwick, W.A., Kacprzyk, J. (eds.): Fuzzy Optimization: Recent Advances and Applications. Springer-Verlag, Berlin (2010)

    MATH  Google Scholar 

  10. Sakawa, M., Yano, H., Nishizaki, I.: Linear and Multiobjective Programming with Fuzzy Stochastic Extensions. Springer, New York (2013)

    Book  MATH  Google Scholar 

  11. Ganesan, K., Veeramani, P.: Fuzzy linear programming with trapezoidal fuzzy numbers. Ann. Oper. Res. 143, 305–315 (2006)

    Article  MATH  Google Scholar 

  12. Maleki, H.R., Tata, M., Mashinchi, M.: Linear programming with fuzzy variables. Fuzzy Sets Syst. 109, 21–33 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Nasseri, S.H., Ebrahimnejad, A.: A fuzzy primal simplex algorithm and its application for solving flexible linear programming problems. Eur. J. Ind. Eng. 4(3), 372–389 (2010)

    Article  MathSciNet  Google Scholar 

  14. Nasseri, S.H., Ardil, E., Yazdani, A., Zaefarian, R.: Simplex method for solving linear programming problems with fuzzy numbers. Trans. Eng. Comput. Technol. 10, 184–288 (2005)

    Google Scholar 

  15. Mahdavi-Amiri, N., Nasseri, S.H., Yazdani, A.: Fuzzy primal simplex algorithms for solving fuzzy linear programming problems. Iran. J. Oper. Res. 1, 68–84 (2009)

    Google Scholar 

  16. Nasseri, S.H., Khabiri, B.: Revised fuzzy simplex algorithm for linear programming problems with fuzzy variables using linear ranking functions. Int. J. Math. Comput. 6, 44–55 (2010)

    MathSciNet  Google Scholar 

  17. Nasseri, S.H., Khabiri, B.: A revised simplex algorithm for fuzzy number linear programming problems using linear ranking functions. Int. J. Math. Comput. 8, 114–126 (2010)

    MathSciNet  Google Scholar 

  18. Mahdavi-Amiri, N., Nasseri, S.H.: Duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables. Fuzzy Sets Syst. 158, 1961–1978 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ebrahimnejad, A., Nasseri, S.H.: A dual simplex method for bounded linear programmes with fuzzy numbers. Int J Math Oper Res 2(6), 762–779 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mahdavi-Amiri, N., Nasseri, S.H.: Duality in fuzzy number linear programming by use of a certain linear ranking function. Appl. Math. Comput. 180, 206–216 (2006)

    MathSciNet  MATH  Google Scholar 

  21. Nasseri, S.H., Ebrahimnejad, A.: A fuzzy dual simplex method for fuzzy number linear programming problem. Adv Fuzzy Sets Syst. 5(2), 81–95 (2010)

    MathSciNet  MATH  Google Scholar 

  22. Nasseri, S.H., Ebrahimnejad, A., Mizuno, S.: Duality in fuzzy linear programming with symmetric trapezoidal numbers. Appl Appl Math 5(10), 1467–1482 (2010)

    MathSciNet  MATH  Google Scholar 

  23. Nasseri, S.H., Mahdavi-Amiri, N.: Some duality results on linear programming problems with symmetric fuzzy numbers. Fuzzy Inf. Eng. 1, 59–66 (2009)

    Article  MATH  Google Scholar 

  24. Ebrahimnejad, A., Nasseri, S.H., Lotfi, F.H., Soltanifar, M.: A primal-dual method for linear programming problems with fuzzy variables. Eur. J. Ind. Eng. 4(2), 189–209 (2010)

    Article  Google Scholar 

  25. Ebrahimnejad, A.: A primal-dual simplex algorithm for solving linear programming problems with symmetric trapezoidal fuzzy numbers. Appl. Math. 2, 676–684 (2011)

    Article  MathSciNet  Google Scholar 

  26. Ebrahimnejad, A., Nasseri, S.H.: Using complementary slackness property to solve linear programming with fuzzy parameters. Fuzzy Inf. Eng. 3, 233–245 (2009)

    Article  MATH  Google Scholar 

  27. Ebrahimnejad, A., Nasseri, S.H.: Linear programmes with trapezoidal fuzzy numbers: a duality approach. Int J Oper Res 13(1), 67–89 (2012)

    Article  MathSciNet  Google Scholar 

  28. Ebrahimnejad, A., Nasseri, S.H., Lotfi, F.H.: Bounded linear programs with trapezoidal fuzzy numbers. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 18(3), 269–286 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ebrahimnejad, A., Nasseri, S.H., Mansourzadeh, S.M.: Bounded primal simplex algorithm for bounded linear programming with fuzzy cost coefficients. Int. J. Oper. Res. Inf. Syst. 2(1), 100–124 (2011)

    Article  Google Scholar 

  30. Lotfi, F.H., Allahviranloo, T., Jondabeh, M.A., Alizadeh, L.: Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution. Appl. Math. Model. 33, 3151–3156 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mottaghi, A., Ezzati, R., Khorram, E.: A new method for solving fuzzy linear programming problems based on the fuzzy linear complementary problem (FLCP). Int. J. Fuzzy Syst. 17(2), 236–245 (2015)

    Article  MathSciNet  Google Scholar 

  32. Hamacher, H., Liberling, H., Zimmermann, H.J.: Sensitivity analysis in fuzzy linear programming. Fuzzy Sets Syst. 1, 269–281 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  33. Dutta, D., Rao, J.R., Tiwari, R.N.: Sensitivity analysis in fuzzy linear fractional programming problem. Fuzzy Sets Syst. 48, 211–216 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  34. Frahadian, B.: Sensitivity analysis in interval-valued trapezoidal fuzzy number linear programming problems. Appl. Math. Model. 38, 50–62 (2014)

    Article  MathSciNet  Google Scholar 

  35. Fuller, R.: On stability in fuzzy linear programming problems. Fuzzy Sets Syst. 30, 339–344 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tanaka, H., Ichihashi, H., Asai, K.: A value of information in FLP problems via sensitivity analysis. Fuzzy Sets Syst. 18, 119–186 (1986)

    Article  MATH  Google Scholar 

  37. Kheirfam, B., Hasani, F.: Sensitivity analysis for fuzzy linear programming problems with fuzzy variables. Adv. Model. Optim. 12, 257–272 (2010)

    MathSciNet  MATH  Google Scholar 

  38. Nasseri, S.H., Ebrahimnejad, A.: Sensitivity analysis on linear programming problems with trapezoidal fuzzy variables. Int. J. Oper. Res. Inf. Syst. 2, 23–29 (2011)

    Article  Google Scholar 

  39. Ebrahimnejad, A.: Sensitivity analysis in fuzzy number linear programming problems. Math. Comput. Model. 53, 1878–1888 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Bazaraa, M.S., Jarvis, J.J., Sherali, H.D.: Linear Programming and Network Flows, Forth edn. Wiley, New York (2009)

    Book  MATH  Google Scholar 

  41. Dantzig, G.B., Thapa, M.N.: Linear Programming, 2: Theory and Extensions. Springer-Verlag, New York (2003)

    MATH  Google Scholar 

  42. L.A. Zadeh, The concept of linguistic variable and its application to approximate reasoning I, II and III, Information Sciences 8, 199–249; 301–357; 9, 43–80 (1975)

  43. Bansal, A.: Trapezoidal fuzzy numbers (a, b, c, d): arithmetic behavior. Int. J. Phys. Math. Sci. 2, 39–44 (2011)

    Google Scholar 

  44. Liu, B.: Uncertainty Theory. Springer-Verlag, Berlin (2015)

    Book  MATH  Google Scholar 

  45. Yao, K.: Sine entropy of uncertain set and its applications. Appl. Soft Comput. 22, 432–442 (2014)

    Article  Google Scholar 

  46. Yao, K., Hua, K.: Entropy operator for membership function of uncertain set. Appl. Math. Comput. 242, 898–906 (2014)

    MathSciNet  MATH  Google Scholar 

  47. Bortolan, G., Degani, R.: A review of some methods for ranking fuzzy subsets. Fuzzy Sets Syst. 15, 1–19 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  48. Fortemps, P., Roubens, M.: Ranking and defuzzification methods based on area compensation. Fuzzy Sets Syst. 82, 319–330 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  49. Maleki, H.R.: Ranking functions and their applications to fuzzy linear programming. Far East J. Math. Sci. 4, 283–301 (2002)

    MathSciNet  MATH  Google Scholar 

  50. Wang, X., Kerre, E.: Reasonable properties for the ordering of fuzzy quantities (2 parts). Fuzzy Sets Syst. 118, 375–405 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  51. Baldwin, J., Guild, N.: Comparison of fuzzy sets on the same decision space. Fuzzy Sets Syst. 2, 213–233 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  52. Campos, L., Verdegay, J.L.: Linear programming problems and ranking of fuzzy numbers. Fuzzy Sets Syst. 32, 1–11 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  53. Arana-Jimnez, M., Rufin-Lizana, A., Chalco-Cano, Y., Romn-Flores, H.: Generalized convexity in fuzzy vector optimization through a linear ordering. Inf. Sci. 312, 13–24 (2015)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran

    M. Ghaznavi, F. Soleimani & N. Hoseinpoor

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  1. M. Ghaznavi
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  2. F. Soleimani
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  3. N. Hoseinpoor
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Correspondence to M. Ghaznavi.

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Ghaznavi, M., Soleimani, F. & Hoseinpoor, N. Parametric Analysis in Fuzzy Number Linear Programming Problems. Int. J. Fuzzy Syst. 18, 463–477 (2016). https://doi.org/10.1007/s40815-015-0123-3

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