Some new results in linear programs with trapezoidal fuzzy numbers: finite convergence of the Ganesan and Veeramani’s method and a fuzzy revised simplex method. (English) Zbl 1225.90165

Summary: In a recent paper, K. Ganesan and P. Veeramani [Ann. Oper. Res. 143, 305–315 (2006; Zbl 1101.90091)] considered a kind of linear programming involving symmetric trapezoidal fuzzy numbers without converting them to the crisp linear programming problems and then proved fuzzy analogues of some important theorems of linear programming that lead to a new method for solving fuzzy linear programming (FLP) problems. In this paper, we obtain some another new results for FLP problems. In fact, we show that if an FLP problem has a fuzzy feasible solution, it also has a fuzzy basic feasible solution and if an FLP problem has an optimal fuzzy solution, it has an optimal fuzzy basic solution too. We also prove that in the absence of degeneracy, the method proposed by Ganesan and Veermani stops in a finite number of iterations. Then, we propose a revised kind of their method that is more efficient and robust in practice. Finally, we give a new method to obtain an initial fuzzy basic feasible solution for solving FLP problems.


90C70 Fuzzy and other nonstochastic uncertainty mathematical programming
90C05 Linear programming
90C49 Extreme-point and pivoting methods


Zbl 1101.90091
Full Text: DOI


[1] Tanaka, H.; Okuda, T.; Asai, K., On fuzzy mathematical programming, J. cybernet., 3, 37-46, (1974) · Zbl 0297.90098
[2] Belman, R.E.; Zadeh, L.A., Decision making in a fuzzy environment, Manage. sci., 17, 141-164, (1970)
[3] Maleki, H.R.; Tata, M.; Mashinchi, M., Linear programming with fuzzy variables, Fuzzy sets syst., 109, 21-33, (2000) · Zbl 0956.90068
[4] 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) · Zbl 1102.90080
[5] 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) · Zbl 1135.90446
[6] 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) · Zbl 1203.90185
[7] 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, 96-120, (2011)
[8] Nasseri, S.H.; Ebrahimnejad, A., A fuzzy dual simplex method for fuzzy number linear programming problem, Adv. fuzzy set syst., 5, 2, 81-95, (2010) · Zbl 1197.90353
[9] Ebrahimnejad, A.; Nasseri, S.H.; Hosseinzadeh Lotfi, F.; Soltanifar, M., A primal – dual method for linear programming problems with fuzzy variables, Eur. J. ind. eng., 4, 2, 189-209, (2010)
[10] Ebrahimnejad, A.; Nasseri, S.H., Using complementary slackness property to solve linear programming with fuzzy parameters, Fuzzy inf. eng., 3, 233-245, (2009) · Zbl 1275.90131
[11] 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) · Zbl 1197.90353
[12] 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, 3151-3156, (2009) · Zbl 1205.90313
[13] Kumar, A.; Kaur, J.; Singh, P., A new method for solving fully fuzzy linear programming problems, Appl. math. model., 35, 817-823, (2011) · Zbl 1205.90310
[14] Ebrahimnejad, A., Sensitivity analysis in fuzzy number linear programming problems, Math. comput. model., 53, 1878-1888, (2011) · Zbl 1219.90198
[15] Ganesan, K.; Veeramani, P., Fuzzy linear programming with trapezoidal fuzzy numbers, Ann. oper. res., 143, 305-315, (2006) · Zbl 1101.90091
[16] Ebrahimnejad, A.; Nasseri, S.H.; Hosseinzadeh Lotfi, F., Bounded linear programs with trapezoidal fuzzy numbers, Int. J. uncertain. fuzziness knowledge-based systems, 18, 3, 269-286, (2010) · Zbl 1200.90173
[17] Nasseri, S.H.; Mahdavi-Amiri, N., Some duality results on linear programming problems with symmetric fuzzy numbers, Fuzzy inf. eng., 1, 56-59, (2009) · Zbl 1275.90132
[18] Nasseri, S.H.; Ebrahimnejad, A.; Mizuno, S., Duality in fuzzy linear programming with symmetric trapezoidal numbers, Appl. appl. math., 5, 10, 14671482, (2010) · Zbl 1205.90314
[19] Okada, S.; Soper, T., A shortest path problem on a network with fuzzy arc lengths, Fuzzy sets syst., 109, 129-140, (2000) · Zbl 0956.90070
[20] Cho, T.C.; Tsao, C.T., Ranking fuzzy numbers with an area between the centroid point and original point, Comput. math. appl., 4, 111-117, (2002) · Zbl 1113.62307
[21] Wang, Y.J.; Lee, H.S., The revised method of ranking fuzzy numbers with an area between the centroid and original points, Comput. math. appl., 55, 2033-2042, (2008) · Zbl 1137.62313
[22] Wu, D.; Mendel, J.M., A comparative study of ranking methods, similarity measures and uncertainty measures for interval type-2 fuzzy sets, Inf. sci., 179, 1169-1192, (2009)
[23] Wang, Z.X.; Liu, Y.J.; Fan, Z.P.; Feng, B., Ranking L-R fuzzy number based on deviation degree, Inf. sci., 179, 2070-2077, (2009) · Zbl 1166.90349
[24] Bazaraa, M.S.; Jarvis, J.J.; Sherali, H.D., Linear programming and network flows, (2005), John Wiley and Sons New York · Zbl 1061.90085
[25] Murty, G.H., Linear programming, (1983), John Wiley and Sons New York
[26] Mishra, S.; Ghosh, A., Interactive fuzzy programming approach to bi-level quadratic fractional programming problems, Ann. oper. res., 143, 251-263, (2006) · Zbl 1101.90093
[27] Mahdavi-Amiri, N.; Nasseri, S.H.; Yazdani, A., Fuzzy primal simplex algorithms for solving fuzzy linear programming problems, Iranian J. oper. res., 1, 68-84, (2009)
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