Multi-Objective Fuzzy Fully Linear Programming Transportation Problem using Ranking Function

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1 Volume 117 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Multi-Objective Fuzzy Fully Linear Programming Transportation Problem using Ranking Function P. Rajarajeswari 1 and M.Sangeetha 2 1 Department of mathematics, Chikkanna Govt. Arts College, Tirupur, Tamil Nadu, India, p.rajarajeswari29@gmail.com 2 Department of mathematics, Sri Ramalinga Sowdambigai College of science and commerce, Coimbatore ,Tamil Nadu, India. m.sangeethaphd@gmail.com Abstract The transportation model is a special case of linear programming models, widely used in the areas of inventory control, networks, employment scheduling aggregate planning, and personnal assignment, among others. This paper proposes a new approach, using ranking function to the fuzzy applied Linear Programming problem with transportation model(lptp). The algorithm of this approach is presented, and explained briefly as the regular multi objective LP transportation using ranking function and largest cost entry method. Finally, a numerical example is described in the paper to show its efficiency. Keywords: Fully fuzzy linear programming transportation problem, Ranking function, Multi objective fuzzy Transportation problem(moftp), Membership function fuzzy set. 1 Introduction Multi Objective linear programming transportation problem is an extension of linear programming. The optimal solution of MOFLPTP is solved using min operator chanas [3] proposed a fuzzy programming in multi objective linear programming transportation problem and it was developed by parametric approach. Zimmermann [12] proposed a multi criteria decision making set defined as the intersection of all fuzzy goals and constraints. In this section, we have proposed MOFLPTP 63

2 with mixed constraint is which right hand side and the constraint matrix are fuzzy numbers and it has been solved by min operator transportation Bablu and Tapan [8] applied transportation application of MOFLPP with a minimization objective whereas in this section the maximization objective with numerical example is proposed. 2 Fully fuzzy Linear Programming Transportation Problem (FFLPTP) According to Jayalakshmi and Pandian [10] the following FFLPTP with in fuzzy in equality, equality constraints and n fuzzy variables is formulated as follows Minimize Z = C T X Subject to à X(,, )ã, à X(,, ) b, X 0, where the cost vector C T = (C ij ) i n is a non-negative real fuzzy vector, the coefficient matrix à = (ã ij) m n is a non-negative real fuzzy matrix, ã = (ã i ) n l and b = ( bj ) m l are non-negative real fuzzy vectors such that ã i, x ij, b j, c ij F (R) for all 1 i n and for all 1 j m. 2.1 Fully Fuzzy LPTP using Robusts Ranking. Amit Kumar et. Al., [8] proposed a method for solving the FFLPTP by using fuzzy ranking function in the fuzzy objective function. Rangarajan and Solairaju [12] computed improved fuzzy optimal Hungarian assignment problems with fuzzy numbers by applying Robusts ranking techniques to transform the fuzzy assignment problem to a crisp one. 2.2 Ranking Function. Robusts ranking techniques satisfies compensation, linearity and provides results which are consistent with human institution. Given a converse fuzzy number ã the Robust ranking index is defined by R(ã) = (a r α, a u α)dx, Where (a l α, a u α) is the α-level cut of the fuzzy number ã. 3 Algorithm for the proposed method. Step 1. The problem is formulated as fuzzy LPTP. Step 2. on substitution of the values of x ij = (x ij, y ij, t ij ) and c ij = (p ij, q ij, r ij ), ã i = (a i, b i, c i ), b j = (b j, q j, h j ) in the fuzzy LPTP obtained in step 1. Step 3. By using the linearity property of ranking function and the Robusts ranking, all the fuzzy constraints and restrictions are converted into the crisp constraints. 64

3 Step 4. an optimal solutions is arrived from step 3 to the given (FFLPTP). 4 Numerical Example Minimize (R( Z)) = R(9, 10, 11) R(x 1 y 1 t 1 ) + R(8, 9, 10) R(x 2 y 2 t 2 ) Subject to; R( x 1 ) + R( x 2 ) R(3, 11, 22), R( x 3 ) + R( x 4 ) R(4, 12, 23), R( x 1 ) + R( x 3 ) R(5, 13, 24), R( x 2 ) + R( x 4 ) R(6, 14, 25), +R(6, 7, 8) R(x 3 y 3 t 3 ) + R(4, 5, 6) R(x 4 y 4 t 4 ) and x 1, x 2, x 3, x 4, y 1, y 2, y 3, y 4, t 1, t 2, t 3, t 4 0. The membership function of the triangular fuzzy number (9, 10, 11) x 9 if 9 x 10, 1 1 if x = 10, µãi = 11 x if 10 11, 1 0 if otherwise. The α cut of the fuzzy number (9, 10, 11) is (a α l, a α u ) = (α, 11 α) for which R(9, 10, 11) = 10, R(8, 9, 10) = 9, R(6, 7, 8) = 7, R(4, 5, 6) = 6, R(3, 11, 22) = 12.5, R(4, 12, 23) = 13.5, R(5, 13, 24) = 14.5, R(2, 14, 21) = Now the given LPTP becomes Minimize z = 10x 1 + 9x 2 + 7x 3 + 5x 4 Subject to x 1 + x ; x 3 + x ; x 1 + x ; x 2 + x ; and X 0 Source/Destination Destination I Destination II Supply Source Source Demand Total 16 By solving the above problem by transportation, we obtain the optimal solution z = 208, x 1 = 1, x 3 = 11.5, x 4 = Comparitive study of Largest cost fuzzy entry method and Robust s ranking method in the proposed method we obtain the optimal solution for the above transportation problem is z = 208, x 1 = 1, x 3 = 11.5, x 4 =

4 Method x 1 x 2 x 3 x 4 z FVAM, FNWCR,FLCM,Existing method Robust s ranking proposed method(flcem) A comparitive study on the two methods that is existing method and Robust s ranking has been done. 6 Conclusion In this continuation of the study, FFLPTP with ranking was worked out and the comparative results are tabulated as per the solution obtained from the existing and the proposed method. The main objective of the study is to check the effectiveness of the application of fuzzy sets in key areas of transportation, which has been substantially proved with numerical examples. References [1] Abbasbandy. S. and Hajjira, T., A new approach for ranking of trapezoidal numbers, Computer and mathematics with applications, 57 (2009), [2] Amit Kumar, Pushpinder Singh, Amarpreet Kaur and Parampreet Kaur, Ranking of generalized Trapezoidal Fuzzy numbers basedon Rank, mode, divergence and spread, TurkishJournal of Fuzzy systems, 1 (2010), [3] Bablu Jana and Tapan Kumar Roy, Multi objective Fuzzy linaer programming and its Application in Transportation model, Tamsui Oxford Journal of Mathematical Sciences, 21 (2005), [4] Chanas,D., Fuzzy Programming in multi objective linear programming- a parametric Approach, Fuzzy set and systems,29 (1989), [5] Dehghan, M., Hashemi, B., and Ghatee.M., Computational methods for solving fully fuzzy linear systems, Journal od applied mathematics and computing., 179 (2006), [6] Ganesan, K and Veeramani,.P., Fuzzy linear Programs with Trapezoidal Fuzzy Numbers, Annals of operation Researxch, 143 (2006), [7] Hu, C.F.., Teng, C.J. and Li, S. Y., A fuzzy goal programming approach to multiobjective optimization problem with priorities, European Journal of Operation Research, 176 (2007), [8] Kumar, P., Kaur, J. and Singh, Solving fully fuzzy linear problems with inequality constraints, International Journal of Physical and Mathematical Sciences, 1 (2010), [9] Verdegay, J.L., A dual approach to solve the fuzzy linear programming problem, Fuzzy sets and systems, 14 (1984),

5 [10] Jayalaksmi,M. and Pandian, P., A new Method for finding an Optimal Fuzzy Solution For fuzzy fully linear programming problems, International Journal of Engineering Research and Applications(IJERA) ISSN: , 2 (2012), [11] Zimmermann,H.J., Fuzzy programming and linear programming with several objective functions, Fuzzy sets and systems, 1 (1978), [12] Rangarajan, R. and Solairaju, A., Computing improved fuzzy optimal Hungarian assignment problems with fuzzy costs under robust ranking techniques. International journal of Computer Applications, 6 (2010),

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