A method for solving linear programming with interval-valued trapezoidal fuzzy variables
RAIRO - Operations Research - Recherche Opérationnelle, Tome 52 (2018) no. 3, pp. 955-979.

An efficient method to handle the uncertain parameters of a linear programming (LP) problem is to express the uncertain parameters by fuzzy numbers which are more realistic, and create a conceptual and theoretical framework for dealing with imprecision and vagueness. The fuzzy LP (FLP) models in the literature generally either incorporate the imprecisions related to the coefficients of the objective function, the values of the right-hand side, and/or the elements of the coefficient matrix. The aim of this article is to introduce a formulation of FLP problems involving interval-valued trapezoidal fuzzy numbers for the decision variables and the right-hand-side of the constraints. We propose a new method for solving this kind of FLP problems based on comparison of interval-valued fuzzy numbers by the help of signed distance ranking. To do this, we first define an auxiliary problem, having only interval-valued trapezoidal fuzzy cost coefficients, and then study the relationships between these problems leading to a solution for the primary problem. It is demonstrated that study of LP problems with interval-valued trapezoidal fuzzy variables gives rise to the same expected results as those obtained for LP with trapezoidal fuzzy variables.

DOI : 10.1051/ro/2018007
Classification : 90Cxx, 90C05, 90C70
Mots-clés : Fuzzy linear programming, interval-valued trapezoidal fuzzy numbers, signed distance ranking
Ebrahimnejad, Ali 1

1 
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     title = {A method for solving linear programming with interval-valued trapezoidal fuzzy variables},
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     pages = {955--979},
     publisher = {EDP-Sciences},
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Ebrahimnejad, Ali. A method for solving linear programming with interval-valued trapezoidal fuzzy variables. RAIRO - Operations Research - Recherche Opérationnelle, Tome 52 (2018) no. 3, pp. 955-979. doi : 10.1051/ro/2018007. http://archive.numdam.org/articles/10.1051/ro/2018007/

[1] M.S. Bazaraa, J.J. Jarvis and H.D. Sherali, Linear Programming and Network Flows, 4th edn. John Wiley, New York (2010). | MR | Zbl

[2] J. Chiang, Fuzzy linear programming based on statistical confidence interval and interval-valued fuzzy set. Eur. J. Oper. Res. 129 (2001) 65–86. | DOI | MR | Zbl

[3] A. Ebrahimnejad, Sensitivity analysis in fuzzy number linear programming problems. Math. Comput. Model. 53 (2011) 1878–1888. | DOI | MR | Zbl

[4] A. Ebrahimnejad, A duality approach for solving bounded linear programming with fuzzy variables based on ranking functions. Int. J. Syst. Sci. 46 (2015) 2048–2060. | DOI | MR | Zbl

[5] A. Ebrahimnejad, Fuzzy linear programming approach for solving transportation problems with interval-valued trapezoidal fuzzy numbers. Sādhanā 41 (2016) 299–316. | MR | Zbl

[6] A. Ebrahimnejad and J.L. Verdegay, A novel approach for sensitivity analysis in linear programs with trapezoidal fuzzy numbers. J. Intell. Fuzzy Syst. 27 (2014) 173–185. | DOI | MR | Zbl

[7] A. Ebrahimnejad, S.H. Nasseri and F.H. Lotfi, Bounded linear programs with trapezoidal fuzzy numbers. Int. J. Uncertain. FuzzinessKnowl. Based Syst. 18 (2010) 269–286. | DOI | MR | Zbl

[8] A. Ebrahimnejad, S.H. Nasseri, F. Hosseinzadeh Lotfi and M. Soltanifar, A primal-dual method for linear programming problems with fuzzy variables. Eur. J. Ind. Eng. 4 (2010) 189–209. | DOI

[9] R. Ezzati, E. Khorram and R. Enayati, A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem. Appl. Math. Model. 39 (2015) 3183–3193. | DOI | MR | Zbl

[10] B. Farhadinia, Sensitivity analysis in interval-valued trapezoidal fuzzy number linear programming problems. Appl. Math. Model. 38 (2014) 40–62. | DOI | MR | Zbl

[11] K. Ganesan and P. Veeramani, Fuzzy linear programming with trapezoidal fuzzy numbers. Ann. Oper. Res. 143 (2006) 305–315. | DOI | Zbl

[12] A. Hatami-Marbini and M. Tavana, An extension of the linear programming method with fuzzy parameters. Int. J. Math. Oper. Res. 3 (2011) 44–55 | DOI | MR | Zbl

[13] A. Hatami-Marbini, P.J. Agrell, M. Tavana and A. Emrouznejad, A stepwise fuzzy linear programming model with possibility and necessity relations, J. Intell. Fuzzy Syst. 25 (2013) 81–93. | DOI | MR | Zbl

[14] B. Kheirfam and J.L. Verdegay, Strict sensitivity analysis in fuzzy quadratic programming. Fuzzy Sets Syst. 198 (2012) 99–111. | DOI | MR | Zbl

[15] S. Kumar Das, T. Mandal and S.A. Edalatpanah, A new approach for solving fully fuzzy linear fractional programming problems using the multi-objective linear programming. RAIRO: OR 51 (2017) 285–297. | DOI | Numdam | MR | Zbl

[16] Y.J. Lai and C.L. Hwang, A new approach to some possibilistic linear programming problem. Fuzzy Sets Syst. 49 (1992) 121–133. | DOI | MR

[17] T.-F. Liang, Application of fuzzy sets to multi-objective project management decisions. Int. J. Gen. Syst. 38 (2009) 311–330. | DOI | MR | Zbl

[18] T.-F. Liang, Applying fuzzy goal programming to project management decisions with multiple goals in uncertain environments. Expert Syst. Appl. 37 (2010) 8499–8507. | DOI

[19] F.H. Lotfi, T. Allahviranloo, M. Alimardani Jondabeh and L. Alizadeh, Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution. Appl. Math. Model. 33 (2009) 3151–3156. | DOI | MR | Zbl

[20] N. Mahdavi-Amiri and S.H. Nasseri, Duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables. Fuzzy Sets Syst. 158 (2007) 1961–1978. | DOI | MR | Zbl

[21] H.R. Maleki, M. Tata and M. Mashinchi, Linear programming with fuzzy variables. Fuzzy Sets Syst. 109 (2000) 21–33. | DOI | MR | Zbl

[22] S. Saati, A. Hatami-Marbini, M. Tavana and E. Hajiahkondi, A two-fold linear programming model with fuzzy data. Int. J. Fuzzy Syst. Appl. 2 (2012) 1–12.

[23] J.-S. Su, Fuzzy programming based on interval-valued fuzzy numbers and ranking. Int. J. Contemp. Math. Sci. 2 (2007) 393–410. | DOI | MR | Zbl

[24] H. Tanaka and K. Asai, Fuzzy solution in fuzzy linear programming problems. IEEE Trans. Syst. Man Cybern. 14 (1984) 325–328. | DOI | Zbl

[25] P. Veeramani and M. Sumathi, Fuzzy mathematical programming approach for solving fuzzy linear fractional programming problem. RAIRO: OR 48 (2014) 109–122. | DOI | Numdam | MR | Zbl

[26] J.L. Verdegay, A dual approach to solve the fuzzy linear programming problem. Fuzzy Sets Syst. 14 (1984) 131–141. | DOI | MR | Zbl

[27] S.H. Wei and S.M. Chen, Fuzzy risk analysis based on interval-valued fuzzy numbers. Expert Syst. Appl. 36 (2009) 2285–2299. | DOI

[28] R.R. Yager, A procedure for ordering fuzzy subsets of the unit interval. Inf. Sci. 24 (1981) 143–161. | DOI | MR | Zbl

[29] J.S. Yao and F.T. Lin, Constructing a fuzzy flow-shop sequencing model based on statistical data. Int. J. Approx. Reason. 29 (2002) 215–234. | DOI | MR | Zbl

[30] J. Ye, Multiple attribute group decision-making methods with unknown weights in intuitionistic fuzzy setting and interval-valued intuitionistic fuzzy setting. Int. J. Gen Syst. 42 (2013) 489–502. | DOI | MR | Zbl

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