Fuzzy Transportation Problem of Trapezoidal Numbers with Cut and Ranking Technique

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1 International Journal of Fuzzy Mathematics and Systems. ISSN Volume 2, Number 3 (2012), pp Research India Publications Fuzzy Transportation Problem of Trapezoidal Numbers with Cut and Ranking Technique 1 Shugani Poonam, 2 Abbas S.H. and 3 Gupta V.K. 1 Institute for Imsrastrure and Human Resource Development Vidisha, M. P, India 2 Saifia Science College, Bhopal M. P, India 3 UIT, RGPV, Bhopal, M.P, India punam.shugani.vds.2010@gmail.com Abstract The aim of Fuzzy transportation is to find the least transportation cost of some commodities through a capacitated network when the supply and demand of nodes and the capacity and cost of edges are represented as fuzzy numbers. Many previous papers [4, 5, 13, 14] have presented arithmetic operations, alpha level and simple ranking by operations. P. Pandian [7, 8] has presented some methods for fuzzy transportation problem. Finally all these research papers present the solutions of FTP by alpha level, simple ranking method and alpha level standard value. In this paper we are presenting a ranking technique with alpha optimal solution for solving fuzzy transportation problem, where fuzzy demand and supply all are in the form of trapezoidal fuzzy numbers. Keywords: Fuzzy Transportation Problem, Trapezoidal fuzzy numbers, optimal solution, Roubast Ranking Method Introduction The transportation problem is one of the earliest applications of linear programming problems. Transportation models have wide applications in logistics and supply chain for reducing the cost efficient algorithms have been developed for solving the transportation problem when the cost coefficients and the supply and demand quantities are known exactly. The occurrence of randomness and imprecision in the real world is inevitable owing to some unexpected situations. There are cases that the cost coefficients and the supply and demand quantities of a transportation problem may be uncertain due to some uncontrollable factors. To deal quantitatively with imprecise information in making decisions Bellman (1970) and zadeh (1978) introduced the notion of fuzziness.

2 264 Shugani Poonam et al A fuzzy transportation problem is a transportation problem in which the transportation cost, supply and demand quantities are fuzzy quantities. The objective of the fuzzy transportation problem is to determine the shipping schedule that minimizes the total fuzzy transportation cost while satisfying fuzzy supply and demand limits. In 1981 R.R. Yager [9] procedure for ordering fuzzy subsets of the unit interval, S.H. Chen [12] (1985) Ranking fuzzy numbers with maximizing set and minimizing set. S. Chanas, D. Kuchta [14] (1996) solved Fuzzy integer transportation problem. P. Fortemps and M. Roubens [2] (1996) work on Ranking and defuzzification methods based on area compensation. M. Sakawa, I. Nishizaki, Y. Uemura [6] (2001) Interactive with fuzzy programming for two level linear and linear fractional production and assignment problems. S.Abbasbandy and T.Hajjari [1], A new approach for ranking of trapezoidal Fuzzy numbers (2009). A.Nagoor Gani and K. Abdul Razak [10] (2006) have solved fuzzy transportation problem in two stages. Dongmei Huang, Junli Fu, Tao Xiao and Jing Zhou, Comparison between the Inductions Learning Algorithms of Fuzzy Number-Valued Decision Tree (2007). Then p. pandian and g. natrajan [7] (2010) has solved fuzzy transportation problem of trapezoidal numbers with algorithms and zero point method. They use simple operations for solving trapezoidal numbers and ranking also. In this paper we investigate more realistic problems, namely the transportation problem with fuzzy costs. Since the objective is to minimize the total cost or to maximize the total profit, subject to some fuzzy constraints, the objective function is also considered as a fuzzy number. The method is to rank the fuzzy objective values of the objective function by some ranking method for numbers to find the best alternative. On the basis of this idea the Roubast Ranking method [9] with the help of α solution has been adopted a transform the fuzzy transportation problem. The idea is to transform a problem with fuzzy parameters in the form of Linear programming problem and solve it by the Vogel Approximation Method. Trapezoidal fuzzy number For a trapezoidal number A(x), it can be represented by A (a,b,c,d;1) with membership function µ(x) given by, 1 µ(x) =, 0, Robust Ranking Technique Roubast ranking technique which satisfy compensation, linearity, and additivity properties and provides results which are consist human intuition. If ã is a fuzzy number then the Roubast Ranking is defined by R(ã) = 0.5, where is the level cut of the fuzzy number ã

3 Fuzzy Transportation Problem of Trapezoidal Numbers 265 In this paper we use this method for ranking the objective values. The Roubast ranking index R(ã) gives the representative value of fuzzy number ã. Numerical Example A company has four sources S 1, S 2, S 3 and S 4 and four destinations D 1, D 2, D 3 and D 4 ; the fuzzy transportation cost for unit quantity of the product from i th source to j th 1,2,3,41,3,5,69,11,12,145,7,8,11 destination is C ij where 0,1,2,41,0,1,25,6,7,80,1,2,3 3,5,6,85,8,9,1212,15,16,197,9,10,12 And fuzzy availability of the product at source are ( (1,6,7,12), (0,1,2,3), (5,10,12,17), ) and the fuzzy demand of the product at destinations are ((5,7,8,10), (1,5,6,10), (1,3,4,6) (1,2,3,4) ) respectively. Then the problem become as Table 1 FD1 FD2 FD3 FD4 Supply FS1 (1,2,3,4) (1,3,4,6) (9,11,12,14) (5,7,8,11) (1,6,7,12) FS2 (0,1,2,4) (-1,0,1,2) (5,6,7,8) (0,1,2,3) (0,1,2,3) FS3 (3,5,6,8) (5,8,9,12) (12,15,16,19) (7,9,10,12) (5,10,12,17) Demand (5,7,8,10) (1,5,6,10) (1,3,4,6) (1,2,3,4) Solution: In Conformation to model the fuzzy transportation problem can be formulated in the following mathematical programming form Min Z = R(1,2,3,4)x 11 + R(1,3,4,6)x 12 + R(9,11,12,14)x 13 + R(5,7,8,11)x 14 + R(0,1,2,4)x 21 + R(- 1,0,1,2)x 22 +R(5,6,78)x 23 +R(4,5,6,7)x 24 +R(3,5,6,8)x 31 +R(5,8,9,12)x 32 + R(12,15,16,19)x 33 + R(7,9,10,12)x 34 Where 0.5,,, 1,2,3, ,2,3, Similarly 1,2,3,4 2.5;1,3,5,6 3.75; 9,11,12, ;5,7,8, ,1,2,4 1.75;1,0,1,2 0.5;5,6,7,8 6.5;0,1,2, ,5,6,8 5.5;5,8,9,12 8.5;12,15,16, ;7,9,10,12 9.5

4 266 Shugani Poonam et al Rank of all supply 1,6,7,12 6.5;0,1,2,3 1.5;5,10,12,17 11 Rank of all demand 5,7,8,10 7.5;1,5,6,10 5.5;1,3,4,6 3.5; 1,2,3,4 2.5 Table after ranking Table 2 FD1 FD2 FD3 FD4 Supply FS FS FS Demand Table after applying Vogel approximation method Table 3 FD1 FD2 FD3 FD4 Supply FS FS FS Demand The transportation cost is (2.5)(1.0)+(3.5)(5.5)+(1.5)(1.5)+( 5.5)(6.5)+(15.5.5)(3.5)+(9.5)(1.0) = Conclusion In this paper, the transportation costs are considered as imprecise numbers described by fuzzy numbers which are more realistic and general in nature. Moreover, the fuzzy transportation problem of triangular numbers has been transformed into crisp transportation problem using Robust s ranking indices. Numerical examples show that by this method we can have the optimal solution as well as the crisp and fuzzy optimal total cost. By using Robust s [10] ranking method we have shown that the total cost obtained is optimal. Moreover, one can conclude that the solution of fuzzy problems can be obtained by Robust s ranking method effectively. This technique can also be used in solving other types of problems like, project schedules, assignment problems and network flow problems.

5 Fuzzy Transportation Problem of Trapezoidal Numbers 267 References [1] Abbasbandy, S. and Hajjari, T. (2009): A new approach for ranking computers and mathematics with application, 57 pp [2] Fortemps P. and Roubens M. (1996): Ranking and defuzzification methods based area compensation Fuzzy sets and system, vol. 82 pp [3] Gani, A. Nagoor and Rajak, K. Abdul (2006): Two stage fuzzy transportation problem, journal of physical science, vol. 10, [4] Lin, Feng_Tse and Tsai, Tzong_Ru (2009): A two stage genetic algorithm for solving the transportation problem with fuzzy demands and fuzzy supplies, Int. J. of innovative computing information and control. Vol.5.. [5] Liu, Shiang-Tai and Chiang, Kao (2004): Solving fuzzy transportation problemsbased on extension principle, European Journal of Operational Research, 153, pp [6] M.Sakawa, I.Nishizaki, Y.Uemura (2001): Interactive fuzzy programming for two level linear and linear fractional production and assignment problems a case study, European J. Oper.Res. 135 pp [7] Pandian, P. and Natrajan, G.(2010): An optimal More for less solution to fuzzy transportation problem with mixed constraints Applied mathematical sciences, vol.4, no.29, [8] Pandian, P. and Natrajan, G. (2010): A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problem, applied mathematical sciences, vol.4, No.2, [9] R. R. Yager (1981): A procedure for ordering fuzzy subsets of the unit interval, Information Sciences, 24, [10] R. Nagarajan and A. Solairaju (2010): Computing Improved Fuzzy Optimal Hungarian Assignment Problems with Fuzzy Costs under Robust Ranking Techniques international journal of computer application, vol 6, no. 4. [11] Ritha, W. and Vinotha, J. Merline (2009): Multi-objective two stage fuzzy transportation problem, journal of physical science, vol. 13, pp [12] S.H.Chen (1985), Ranking fuzzy numbers with maximizing set and minimizing set, Fuzzy Sets and System, 17 pp [13] S. Chanas, D. Kuchta (1996), A concept of the optimal solution of the transportation problem with fuzzy cost coefficients, Fuzzy Sets and Systems 82 pp [14] S. Chanas, D. Kuchta (1999): Fuzzy integer transportation problems, European J. operation research 118 pp [15] Sonia, and Malhotra, Rita (2003): A polynomial algorithm for a two stage time minimising transportation problem, OPSEARCH, 39, n0.5 and 6, pp [16] Zadeh, L. A (1965): Fuzzy sets, Information and Control, 8, pp

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