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Fuzzy Optimal Solution of Fuzzy Number Linear Programming Problems

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Abstract

In this paper, by turning fuzzy number linear programming problem (FNLPP) into interval number linear programming problem, we give the conceptions of the \(r-\)best optimal solution and \(r-\)worst optimal solution of FNLPP. And from this, we finally put forward the definitions of the best fuzzy set and fuzzy number optimal solutions and the worst fuzzy set and fuzzy number optimal solutions of FNLPP. In order to solve fuzzy number linear programming problem in practice, we establish some relevant theories and introduce the definitions of the best fuzzy set and fuzzy number approximate optimal solutions and the worst fuzzy set and fuzzy number approximate optimal solutions of FNLPP. At last, we give two examples to show how to get the best (or worst) fuzzy number (or set) approximate optimal solution of FNLPP by using these established theories.

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References

  1. Buckley, J.J., Feuring, T.: Evolutionary algorithm solution to fuzzy problems: fuzzy linear programming. Fuzzy Sets Syst. 109(1), 35–53 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chang, S.S.L., Zadeh, L.A.: On fuzzy mappings and control. IEEE Trans. Syst. Man Cybernet 2, 30–34 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  3. Das, S.K., Mandal, T., Edalatpanah, S.A.: A mathematical model for solving fully fuzzy linear programming problem with trapezoidal fuzzy numbers. Appl. Intell. 46, 509–519 (2017)

    Article  Google Scholar 

  4. Diamond, P., Kloeden, P.: Metric Space of Fuzzy Sets. World Scientific, Singapore (1994)

    MATH  Google Scholar 

  5. Ebrahimnejada, A., Verdegayb, J.L.: A novel approach for sensitivity analysis in linear programs with trapezoidal fuzzy numbers. J. Intell. Fuzzy Syst. 27, 173–185 (2014)

    MathSciNet  Google Scholar 

  6. Ebrahimnejad, A.: Fuzzy linear programming approach for solving transportation problems with interval-valued trapezoidal fuzzy numbers. Sãdhanã 41(3), 299–316 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Ezzati, R., Khorram, E., Enayati, R.: A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem. Appl. Math. Model. 39, 3183–3193 (2015)

    Article  MathSciNet  Google Scholar 

  8. Fan, Y., Huang, G., Veawab, A.: A generalized fuzzy linear programming approach for environmental management problem under uncertainty. J. Air Waste Manag. Assoc. 62(1), 72C86 (2012)

    Article  Google Scholar 

  9. Fan, Y., Huang, G., Yang, A.: Generalized fuzzy linear programming for decision making under uncertainty: feasibility of fuzzy solutions and solving approach. Inf. Sci. 241, 12–27 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ganesan, K., Veeramani, P.: Fuzzy linear programs with trapezoidal fuzzy numbers. Ann. Oper. Res. 143(1), 305–315 (2006)

    Article  MATH  Google Scholar 

  11. Ghaznavi, M., Soleimani, F., Hoseinpoor, N.: Parametric analysis in fuzzy number linear programming problems. Int. J. Fuzzy Syst. 18(3), 463–477 (2016)

    Article  MathSciNet  Google Scholar 

  12. Inuiguchi, M., Ichihashi, H., Kume, Y.: A solution algorithm for fuzzy linear programming with piecewise linear membership functions. Fuzzy Sets Syst. 34(1), 15–31 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jimnez, M., Arenas, M., Bilbao, A., Rodriguez, M.V.: Linear programming with fuzzy parameters: an interactive method resolution. Eur. J. Oper. Res. 177(3), 1599–1609 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kumar, A., Appadoo, S.S., Bector, C.R.: A note on Generalized fuzzy linear programming for decision making under uncertainty: feasibility of fuzzy solutions and solving approach. Inf. Sci. 266, 226–227 (2014)

    Article  MATH  Google Scholar 

  15. Kumar, A., Kaur, J., Singh, P.: A new method for solving fully fuzzy linear programming problems. Appl. Math. Model. 35(2), 817–823 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kumar, A., Kaur, J.: Fuzzy optimal solution of fully fuzzy linear programming problems using ranking function. J. Intell. Fuzzy Syst. 26, 337–344 (2014)

    MathSciNet  MATH  Google Scholar 

  17. Li, W.: Linear Optimization and its Extension (in Chinese). National Defense Industry Press, Beijing (2011)

    Google Scholar 

  18. 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(7), 3151–3156 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Maleki, H.R., Tata, M., Mashinchi, M.: Linear programming with fuzzy variables. Fuzzy Sets Syst. 109(1), 21–33 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nehi, H.M., Maleki, H.R., Mashinchi, M.: Solving fuzzy number linear programming problem by lexicographic ranking function. Italian J. Pure Appl. Math. 15(1), 9–20 (2004)

    MathSciNet  MATH  Google Scholar 

  21. Rommelfanger, H.: Fuzzy linear programming and applications. Eur. J. Oper. Res. 92(3), 512–527 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Stanojević, B.: Extended procedure for computing the values of the membership function of a fuzzy solution to a class of fuzzy linear optimization problems. Fuzzy Sets Syst. 272, 47–59 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Stanojević, B., Stanojević, M.: Parametric computation of a fuzzy set solution to a class of fuzzy linear fractional optimization problems. Fuzzy Optim. Decis. Mak. 15, 435–455 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Tong, S.: Interval number and fuzzy number linear programmings. Fuzzy Sets Syst. 66(3), 301–306 (1994)

    Article  MathSciNet  Google Scholar 

  26. Wang, S., Huang, G.H., Lu, H.W., Li, Y.P.: An interval-valued fuzzy linear programming with infinite \(\alpha\)-cuts method for environmental management under uncertainty. Stoch. Environ. Res. Risk Assess. 25(2), 211–222 (2011)

    Article  Google Scholar 

  27. Wang, G., Li, J.: Approximations of fuzzy numbers by step type fuzzy numbers. Fuzzy Sets Syst. 310, 47–59 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, G., Li, Y.: The ranking of multi-dimensional uncertain information based on metrics on the fuzzy ellipsoid number space. IEEE Trans. Fuzzy Syst. 25, 614–626 (2017)

    Article  Google Scholar 

  29. Werners, B.: An interactive fuzzy programming system. Fuzzy Sets Syst. 23(1), 131–147 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yang, X., Cao, B., Zhou, X.: Solving fully fuzzy linear programming problems with flexible constraints based on a new order relation. J. Intell. Fuzzy Syst. 29, 1539–1550 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MATH  Google Scholar 

  32. Zhong, Y., Jiaa, Y., Zhaoa, J., Lina, X., Yanga, Y.: A revised simplex method of solving degenerate fuzzy number linear programming problems. J. Intell. Fuzzy Syst. 29, 2139–2146 (2015)

    Article  MathSciNet  Google Scholar 

  33. Zimmermann, H.J.: Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst. 11(1), 45–55 (1978)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is partially supported by the Nature Science Foundation of China (Nos. 61771174 and 61433001).

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Authors and Affiliations

  1. Institute of Operations Research and Cybernetics, Hangzhou Dianzi University, Hangzhou, 310018, Zhejiang, China

    Guixiang Wang & Jintao Peng

  2. Zhuoyue Honors College, Hangzhou Dianzi University, Hangzhou, 310018, Zhejiang, China

    Jintao Peng

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  1. Guixiang Wang
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  2. Jintao Peng
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Correspondence to Guixiang Wang.

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Wang, G., Peng, J. Fuzzy Optimal Solution of Fuzzy Number Linear Programming Problems. Int. J. Fuzzy Syst. 21, 865–881 (2019). https://doi.org/10.1007/s40815-018-0594-0

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  • DOI: https://doi.org/10.1007/s40815-018-0594-0

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