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A Survey on Models and Methods for Solving Fuzzy Linear Programming Problems

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Fuzzy Logic in Its 50th Year

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 341))

Abstract

Fuzzy Linear Programming (FLP), as one of the main branches of operation research, is concerned with the optimal allocation of limited resources to several competing activities on the basis of given criteria of optimality in fuzzy environment. Numerous researchers have studied various properties of FLP problems and proposed different approaches for solving them. This work presents a survey on models and methods for solving FLP problems. The solution approaches are divided into four areas: (1) Linear Programming (LP) problems with fuzzy inequalities and crisp objective function, (2) LP problems with crisp inequalities and fuzzy objective function, (3) LP problems with fuzzy inequalities and fuzzy objective function and (4) LP problems with fuzzy parameters. In the first area, the imprecise right-hand-side of the constraints is specified with a fuzzy set and a monotone membership function expressing the individual satisfaction of the decision maker. In the second area, a fuzzy goal and corresponding tolerance is defined for each coefficient of decision variables in the objective function. In the third area, both the coefficients of decision variables in the objective function and the right-hand-side of the constraints are specified by fuzzy goals and corresponding tolerances. In the fourth area, some or all parameters of the LP problem are represented in terms of fuzzy numbers. In this contribution, some of the most common models and procedures for solving FLP problems are analyzed. The solution approaches are illustrated with numerical examples.

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References

  1. Bector, C.R., Chandra, S.: Fuzzy mathematical programming and fuzzy matrix games. In: Studies in Fuzziness and Soft Computing, vol. 169. Springer, Berlin (2005)

    Google Scholar 

  2. Bellman, R.E., Zadeh, L.A.: Decision making in a fuzzy environment. Manage. Sci. 17(4), 141–164 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chanas, S.: The use of parametric programming in fuzzy linear programming problems. Fuzzy Sets Syst. 11, 243–251 (1983)

    Article  MATH  Google Scholar 

  4. Delgado, M., Verdegay, J.L., Vila, M.A.: Relating different approaches to solve linear programming problems with imprecise costs. Fuzzy Sets Syst. 37(1), 33–42 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dubois, D., Prade, H.: A review of fuzzy set aggregation connectives. Inf. Sci. 36(1–2), 85–121 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ebrahimnejad, A.: Sensitivity analysis in fuzzy number linear programming problems. Math. Comput. Model. 53(9–10), 1878–1888 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ebrahimnejad, A.: A primal–dual method for solving linear programming problems with fuzzy cost coefficients based on linear ranking functions and its applications. Int. J. Ind. Syst. Eng. 12(2), 119–140 (2012)

    Google Scholar 

  8. Ebrahimnejad, A.: A duality approach for solving bounded linear programming problems with fuzzy variables based on ranking functions and its application in bounded transportation problems. Int. J. Syst. Sci. 46(11), 2048–2060 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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 

  10. Ebrahimnejad, A., Tavana, M.: A novel method for solving linear programming problems with symmetric trapezoidal fuzzy numbers. Appl. Math. Model. 38(17–18), 4388–4395 (2014)

    Article  MathSciNet  Google Scholar 

  11. Ebrahimnejad, A., Verdegay, J.L.: A novel approach for sensitivity analysis in linear programs with trapezoidal fuzzy numbers. J. Intell. Fuzzy Syst. 27(1), 173–185 (2014)

    MathSciNet  MATH  Google Scholar 

  12. Ebrahimnejad, A., Verdegay, J.L.: On solving bounded fuzzy variable linear program and its applications. J. Intell. Fuzzy Syst. 27(5), 2265–2280 (2014)

    MathSciNet  Google Scholar 

  13. Ebrahimnejad, A., Nasseri, S.H., HosseinzadehLotfi, F., 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 

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

    Article  MATH  Google Scholar 

  15. Gupta, P., Mehlawat, M.K.: Bector-Chandra type duality in fuzzy linear programming with exponential membership functions. Fuzzy Sets Syst. 160(22), 3290–3308 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hatami-Marbini, A., Tavana, M.: An extension of the linear programming method with fuzzy parameters. Int. J. Math. Oper. Res. 3(1), 44–55 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Herrera, F., Kovacs, M., Verdegay, J.L.: Optimality for fuzzified mathematical programming problems: a parametric approach. Fuzzy Sets Syst. 54(3), 279–284 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hosseinzadeh Lotfi, F., Allahviranloo, T., Alimardani Jondabeh, M., 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. Kheirfam, B., Verdegay, J.L.: Optimization and reoptimization in fuzzy linear programming problems. In: The 8th Conference of the European Society for Fuzzy Logic and Technology, pp. 527–533 (2013)

    Google Scholar 

  20. Kheirfam, B., Verdegay, J.L.: Strict sensitivity analysis in fuzzy quadratic programming. Fuzzy Sets Syst. 198, 99–111 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kheirfam, B., Verdegay, J.L.: The dual simplex method and sensitivity analysis for fuzzy linear programming with symmetric trapezoidal numbers. Fuzzy Optim. Decis. Making 12(1), 171–189 (2013)

    Article  MathSciNet  Google Scholar 

  22. Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic, Theory and Applications. Prentice-Hall, PTR, Englewood Cliffs (1995)

    MATH  Google Scholar 

  23. Kumar, A., Kaur, J.: A new method for fuzzy linear programming programs with trapezoidal fuzzy numbers. J. Fuzzy Valued Syst. Anal. 1–12. Article ID jfsva-00102. doi:10.5899/2011/jfsva-00102

    Google Scholar 

  24. 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 

  25. Lai, Y.J., Hwang, C.L.: A new approach to some possibilistic linear programming problem. Fuzzy Sets Syst. 49(2), 121–133 (1992)

    Article  MathSciNet  Google Scholar 

  26. Lai, Y.J., Hwang, C.L.: Fuzzy Mathematical Programming. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  27. Mahdavi-Amiri, N., Nasseri, S.H.: Duality in fuzzy number linear programming by use of a certain linear ranking function. Appl. Math. Comput. 180(1), 206–216 (2006)

    MathSciNet  MATH  Google Scholar 

  28. 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(17), 1961–1978 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. 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 

  30. 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 

  31. 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 

  32. Ramik, J., Rimanek, J.: Inequality relation between fuzzy numbers and its use in fuzzy optimization. Fuzzy Sets Syst. 16(2), 123–138 (1085)

    Article  MathSciNet  MATH  Google Scholar 

  33. Rommelfanger, H., Hanuscheck, R., Wolf, J.: Linear programming with fuzzy objective. Fuzzy Sets Syst. 29(1), 31–48 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tanaka, H., Asai, K.: Fuzzy solution in fuzzy linear programming problems. IEEE Trans. Syst. Man Cybern. 14(2), 325–328 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tanaka, H., Okuda, T., Asai, K.: On fuzzy mathematical programming. J. Cybern. 3(4), 37–46 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tanaka, H., Ichihashi, H., Asai, K.: A formulation of fuzzy linear programming problems based on comparison of fuzzy numbers. Control Cybern. 13, 186–194 (1984)

    MathSciNet  MATH  Google Scholar 

  37. Verdegay, J.L.: Fuzzy mathematical programming. In: Gupta, M.M., Sanchez, E. (eds.) Fuzzy Information and Decision Processes. North Holland, Amsterdam (1982)

    Google Scholar 

  38. Verdegay, J.L.: A dual approach to solve the fuzzy linear programming problem. Fuzzy Sets Syst. 14(2), 131–141 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wan, S.P., Dong, J.Y.: Possibility linear programming with trapezoidal fuzzy numbers. Appl. Math. Model. 38, 1660–1672 (2014)

    Article  MathSciNet  Google Scholar 

  40. Werners, B.: Interactive multiple objective programming subject to flexible constraints. Eur. J. Oper. Res. 31, 342–349 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang, G., Wu, Y.H., Remias, M., Lu, J.: Formulation of fuzzy linear programming problems as four-objective constrained optimization problems. Appl. Math. Comput. 139(2–3), 383–399 (2003)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  44. Zimmermann, H.J.: Description and optimization of fuzzy Systems. Int. J. Gen. Syst. 2, 209–215 (1976)

    Article  MATH  Google Scholar 

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Acknowledgments

Research supported in part under projects P11-TIC-8001 (VASCAS) from the Andalusian Government and TIN2014-55024-P (MODAS-3D) from the Spanish Ministry of Economy and Competitiveness (both including FEDER funds).

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Authors and Affiliations

  1. Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran

    Ali Ebrahimnejad

  2. Department of Computer Science and Artificial Intelligence, University of Granada, 18014, Granada, Spain

    José L. Verdegay

Authors
  1. Ali Ebrahimnejad
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  2. José L. Verdegay
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Correspondence to Ali Ebrahimnejad .

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Editors and Affiliations

  1. Department of Industrial Engineering, Istanbul Technical University, Istanbul, Turkey

    Cengiz Kahraman

  2. School of Industrial Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

    Uzay Kaymak

  3. Department of Computer Engineering, Middle East Technical University (M, Ankara, Turkey

    Adnan Yazici

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Ebrahimnejad, A., Verdegay, J.L. (2016). A Survey on Models and Methods for Solving Fuzzy Linear Programming Problems. In: Kahraman, C., Kaymak, U., Yazici, A. (eds) Fuzzy Logic in Its 50th Year. Studies in Fuzziness and Soft Computing, vol 341. Springer, Cham. https://doi.org/10.1007/978-3-319-31093-0_15

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  • DOI: https://doi.org/10.1007/978-3-319-31093-0_15

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-31091-6

  • Online ISBN: 978-3-319-31093-0

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