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A New Method For Solving Fuzzy Linear Programming Problems Based on The Fuzzy Linear Complementary Problem (FLCP)

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Abstract

Linear programming (LP) is one of the most widely used methods in the area of optimization. Dealing with the formulation of LP problems, the parameters of objective function and constraints should be assigned by experts. In most cases, precise data have been used, but in most of the real-life situations, these parameters are imprecise and ambiguous. In order to deal with the problem of ambiguity and imprecision, fuzzy numbers can be appropriate. By replacing precise numbers with fuzzy numbers, LP problems change to fuzzy linear programming (FLP) problems. So FLPs can be considered as a broader category in comparison to LPs. Considering the above-mentioned points, FLP problems play an important rule in operational researches hence there is a need to investigate these problems. In this paper, a new method for solving the FLP problems is presented in which the coefficients of the objective function and the values of the right-hand side are represented by fuzzy numbers, while the elements of the coefficient matrix are represented by real numbers. To this end, we develop the Karush–Kuhn–Tucker (KKT) optimality conditions for FLP problems. Then, every FLP problem is converted to a fuzzy linear complementary problem (FLCP) by considering KKT conditions. In order to solve the FLCP problems, ranking functions and Lemke’s algorithm are used. Consequently, the solution to primal and dual problems of FLP is obtained. In addition to simplicity in calculations and feasibility, this method solves the primal and dual problems of FLP simultaneously. In order to illustrate the proposed method, some numerical examples are considered.

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References

  1. Allahviranloo, T., Shamsolkotabi, K.H., Kiani, N.A., Alizadeh, L.: Fuzzy integer linear programming problems. Int. J. Contemp. Math. Sci. 2, 167–181 (2007)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Bazaraa, M.S., Jarvis, J.J., Sherali, H.D.: Linear programming and network flows, pp. 221–223. Wiley, New York (1977)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Dehghan, M., Hashemi, B.: Solution of the fully fuzzy linear systems using the decomposition procedure. Appl. Math. Comput. 182–2, 1568–1580 (2007)

    MathSciNet  Google Scholar 

  7. Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. Syst. Sci. 9, 613–626 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dubois, D., Prade, H.: Fuzzy sets and system; theory and applications. Academic Press, New York (1980)

    Google Scholar 

  9. Ebrahimnejad, A., Nasseri, S.H.: Using complementary slackness property to solve linear programming with fuzzy parameters. Fuzzy Inf. Eng. 1, 233–245 (2009)

    Article  MATH  Google Scholar 

  10. 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, 189–209 (2010)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  13. Hashemi, S.M., Modarres, M., Nasrabadi, E., Nasrabadi, M.M.: Fully fuzzified linear programming, solution and duality. J. Intell. Fuzzy Syst. 17, 253–261 (2006)

    MATH  Google Scholar 

  14. 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  MATH  MathSciNet  Google Scholar 

  15. Jimenez, M., Arenas, M., Bilbao, A., Rodrguez, M.V.: Linear programming with fuzzy parameters: An interactive method resolution. Eur. J. Oper. Res. 177, 1599–1609 (2007)

    Article  MATH  Google Scholar 

  16. Lemke, C.E.: On complementary pivot theory. In: Dantzig, G.B., Veinott, A.F. (eds.) Mathematics of the Decision Sciences. American Mathematical Society, Providence (1968)

    Google Scholar 

  17. Liu, X.: Measuring the satisfaction of constraints in fuzzy linear programming. Fuzzy Sets Syst. 122, 263–275 (2001)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. 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  MATH  MathSciNet  Google Scholar 

  20. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  23. Nasseri, S.H.: A new method for solving fuzzy linear programming by solving linear programming. Appl. Math. Sciences 2, 2473–2480 (2008)

    MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  25. Ozelkan, E., Duckstein, L.: Primal fuzzy counterparts of scheduling rules. Eur. J. Oper. Res. 113, 593–609 (1999)

    Article  Google Scholar 

  26. Ramik, J.: Duality in fuzzy linear programming: Some new concepts and results. Fuzzy Optim. Decis. Mak. 4, 25–39 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  27. Rommelfanger, H.: A general concept for solving linear multicriteria programming problems with crisp, fuzzy or stochastic values. Fuzzy Sets Syst. 158, 1892–1904 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  28. Tanaka, H., Ichihashi, H., Asadi, K.: A formulation of fuzzy linear programming problem based on comparison of fuzzy numbers. Control Cybern. 13, 185–194 (1984)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  31. 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, 383–399 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  33. Zimmerman, H.J.: Fuzzy Set Theory and its Applications. Kluwer Academic Publishers, Boston (2001)

    Book  Google Scholar 

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Authors and Affiliations

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

    A. Mottaghi & R. Ezzati

  2. Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, 15914, Tehran, Iran

    E. Khorram

Authors
  1. A. Mottaghi
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  2. R. Ezzati
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  3. E. Khorram
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Correspondence to R. Ezzati.

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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, 236–245 (2015). https://doi.org/10.1007/s40815-015-0016-5

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  • DOI: https://doi.org/10.1007/s40815-015-0016-5

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