A NEW METHOD FOR SOLVING TWO VEHICLE COST VARYING FUZZY TRANSPORTATION PROBLEM

Transcription

1 ISSN: X CDEN: IJPTFI Available nline through esearch Article A NEW METHD F SLVING TW VEHICLE CST VAYING FUZZY TANSPTATIN PBLEM D.Kalpanapriya* and D.Anuradha Department of Mathematics School of Advanced Sciences VIT University Vellore-4 Tamilnadu India. dkalpanapriya@gmail.com eceived on Accepted on Abstract: This paper presents two vehicle cost varying transportation problem with imprecise costs. obust s ranking method is adopted for ranking the imprecise data. The two vehicle cost varying fuzzy transportation problem has been transformed into crisp one and solved by proposed algorithm. Numerical example is provided to illustrate the approach. Keywords: Cost varying fuzzy transportation problem anking method feasible solution.. Introduction Transportation problem (TP) is one of the most interesting linear programming problems concerning with the distribution of products or services. Whenever there is a physical movement of goods from the point of manufacturer to the final consumers through a variety of channels of distribution there is a need to minimize the cost of transportation so as to increase profit on sales. Efficient algorithms have been developed for solving transportation problems when the cost coefficients the demand and supply quantities are known precisely. In real life we frequently deal with vague or imprecise information. Vagueness is usually expressed by intervals or fuzzy numbers. Two vehicle cost varying FTP can arise when uncertainty exists in data problem. In fuzzy decision making the ranking of fuzzy number plays a vital role. anking of fuzzy numbers was first proposed by Jain [4]. Dominance of fuzzy numbers can be explained by many ranking methods of these obust s ranking method [] proposed four indices which may be employed for the purpose of ordering fuzzy quantities in [0]. Poonam et al. [9] discussed fuzzy transportation problem of trapezoidal numbers with ranking technique. Narayanamoorthy et al. [8] developed a new method for solving fuzzy transportation problems using fuzzy russell s method. Arpita and Bikash [] discussed two vehicle cost varying TP as a bi-level mathematical programming model. Sudha et al. [0] developed fuzzy linear programming techniques to transportation problems. IJPT Sep-06 Vol. 8 Issue No Page 654

2 Sharma et al. [] presented a new approach for solving fuzzy TP. Giancarlo [3] studied a new methodology to find initial solution for transportation problems with fuzzy parameters. Krishna prabha and Seerengasamy [5] discussed a new method for finding optimal solution to unbalanced FTP. Krishna Prabha imala [6] proposed a new technique for finding the maximum profit cost for fuzzy transportation Problem. Muruganandam and Srinivasan [7] proposed a new algorithm for solving a special type of fuzzy transportation problems. In this paper a heuristic algorithm for finding an initial basic feasible solution for two vehicle cost varying fuzzy transportation problem is proposed and the same is illustrated with the help of numerical example. This method can help the decision makers in the logistics related issues of real life problems.. Preliminaries We need the following definitions of fuzzy set fuzzy number and membership function which can be found in [].. Definition: Let A be a classical set and A (x) be a membership function from A to [0]. A fuzzy set A with the membership function A (x) is defined by A ( x ( x)) : x A and ( x) [0]. A A. Definition: A Fuzzy set A is called positive if its membership function is such that A( x ) 0 for all x 0..3 Definition: A Fuzzy set A defined on the set of real numbers is said to be a fuzzy number of its member ship function has the following conditions: (i) A( x ) : [0] is continuous. (ii) A( x ) 0 for all ( a ] [ c ) (iii) A( x ) is strictly increasing on [ab] and strictly decreasing on [bc]. (iv) A( x ) for all x b where ab c..4 Definition: A fuzzy number A is denoted as a triangular fuzzy number by ( a a a 3) and its membership function A ( x ) is given as: x a a a if a x a xa ( ) 3 A x a a3 if a x a3 0 otherwise IJPT Sep-06 Vol. 8 Issue No Page 655

3 .5 Definition: The -cut of a fuzzy number ( ) A( ) x/ ( x) [0] Ax is defined as.6 obust s ranking method: The obust s ranking is defined as L U where ( c c ) ( ) 0.5( L U c c c ) d 0 is the - level cut of the fuzzy number c. obust s ranking technique satisfies compensation linearity and additive property which provides results that are consistent with human intuition. 3. Two Vehicle Cost Varying Fuzzy Transportation Problem Suppose there are two types of vehicles V Vfrom each source to each destination. Let C and C (> C ) be the fuzzy capacities (in unit) of the vehicles V respectively. ( ) represent fuzzy transportation cost for each cell ( i j ) where is the fuzzy transportation cost from source i to the destination D j by the vehicle V and is the fuzzy transportation cost from source be represent in the following tabular form i to the destination D jby the vehicle V.So two vehicle cost varying fuzzy TP can D D. D n Stock. n.. n a n a n..... m m m m. m mn a mn m Demand b b. b n The proposed algorithm for two vehicle cost varying fuzzy transportation problem proceeds as follows: Step : Compute the obust s ranking index for each fuzzy cost and replace the fuzzy cost by their respective ranking indices. Step : Compute the difference between two least cost for each row and column. Identify the maximum interval cost difference and allocate the maximum possible units to the least interval cost cell. Step 3: epeat Step until the total available stock fully allocated to the cells as required. Then go to Step 4. IJPT Sep-06 Vol. 8 Issue No Page 656

4 Step 4: Compute the cost for each allocated cell using the algorithm []. Step 5: This allotment yields a basic feasible solution to the given two vehicle cost varying fuzzy transportation problem. The solution procedure of obtaining a basic feasible solution to a two vehicle cost varying fuzzy transportation problem using the proposed algorithm is illustrated by the following example. Example 3. Consider a fuzzy transportation problem with three origins three destinations and two types of vehicles. The fuzzy capacities of vehicles V respectively are C (0) and C (0). The basic aim is to find out a feasible solution. The following table I exhibit the cost which is in the form of fuzzy numbers. Table I: Two vehicle cost varying Fuzzy TP. D D D3 Stock (567) (0) (890)(34) (678)(90) (567) (678)(890) (34)(567) (567)(890) (34) 3 (456)(678) (890)(678) (567)(0) (345) Demand (0) (0) (0) Now using Step the obust s indices for the costs corresponding to the given two vehicle cost varying fuzzy transportation problem is given below: D D D3 Stock Demand The capacities of vehicles V respectively are C and C Now using step and Step 3 the basic cells to the above reduced problem is x 9 x3 7 x x and x Using step 4 the cost for each basic cells is c c3 c c and c33 which produces the total basic feasible transportation cost equal to 4. IJPT Sep-06 Vol. 8 Issue No Page 657

5 (890) Thus the basic cells and their corresponding cost to the given problem is x (890) c ; x3 (678) (890) (6 78) (6 78) (34) (567) c3 ; x (0) c ; x (3) c and x33 (345) c33 with the (6 78) (0) ( 3) (3 45) fuzzy feasible cost value equal to (37447). Example 3. Consider the following two vehicle cost varying fuzzy unbalanced TP with fuzzy capacities of vehicles V respectively are C (9 0 ) andc (39404). Table II: Two vehicle cost varying Fuzzy unbalanced TP. D D D3 Stock (456) (890) (567)(0) (0)(0) (484950) (34)(345) (897)(678) (678)(34) (555657) 3 (789)(456) (345)(678) (90)(890) (567) Demand (757677) (3033) (0) Now since m n ai bj the given problem is unbalanced. We introduce a dummy column D4 having all the values i j equal to zero to make the given problem as balanced. Now using Step the obust s indices for the costs corresponding to the given problem is given below: D D D3 D4 Stock Demand The capacities of vehicles V respectively are C 0 and C 40. Now using step to Step 4 the basic cells and their corresponding cost to the above reduced problem is x c ; x 5 c ; x3 c3 ; x4 3 c4 ; x 56 c and x3 6 c3 with feasible cost as 43. IJPT Sep-06 Vol. 8 Issue No Page 658

6 Thus the basic cells and their corresponding cost to the given unbalanced problem is (456) (567) (0 ) x (0 0 0) c ; x (555) c ; x3 (0) c3 ; (0 0 0) (555) (0 ) (000) (3 45) (6 78) x4 (333) c 4 ; x (555657) c ; x3 (567) c3 with the fuzzy feasible cost (333) (555657) (5 6 7) value equal to (384348). 4. Conclusion In this paper we consider the two vehicle cost varying balanced transportation problem and unbalanced transportation problem with uncertain data. The solution procedure of the proposed algorithm is illustrated with help of a real life example. The necessity of this problem arises when transportation cost depends on amount of transport quantity and capacity of vehicles. This method helps the decision-makers to choose an appropriate decision in real life situations. 5. eferences. Arpita Panda Chandan Bikash Das -Vehicle Cost Varying Transportation Problem Journal of Uncertain Systems Gaurav SharmaS H Abbas Vay Kumar Gupta A New Approach to solve Fuzzy Transportation Problem for Trapezoidal Number Scitech esearch organisation Giancarlo de França Aguiar (05) studied anew Methodology to Find Initial Solution for Transportation Problems with Fuzzy Parameters Applied Mathematical Sciences Jain Decision-making in the presence of fuzzy variables IEEE Transactions on Systems Man and Cybernetics S.Krishna prabha.seerengasamy Procedure for solving unbalanced fuzzy transportation problem for maximizing the profit International Journal of Computer and rganization Trends S. Krishna prabha S.Vimala An modified method for solving balanced fuzzy transportation problem for maximizing the profit International Journal of Pure and Applied Mathematics S. Muruganandam and. Srinivasan A New Algorithm for Solving Fuzzy Transportation Problems with Trapezoidal Fuzzy Numbers International Journal of ecent Trends in Engineering & esearch IJPT Sep-06 Vol. 8 Issue No Page 659

7 8. Narayanamoorthy S. Saranya S. Maheswari S. A Method for Solving Fuzzy Transportation Problems using Fuzzy ussell s Method. I.J. Intelligent Sys. and Applications Poonam S. Abbas S. H. Gupta V. K. Fuzzy Transportation Problem of Trapezoidal Numbers with -Cut and anking Technique International Journal of Fuzzy Mathematics and Systems Sudha S. Margaret A. M. Yuvarani. A New Approach for Fuzzy Transportation Problem International Journal of Innovative esearch and Studies Yager A procedure for ordering fuzzy subsets of the unit interval Information Sciences Zadeh L.A Fuzzy sets Information and control Corresponding Author: D.Kalpanapriya * dkalpanapriya@gmail.com IJPT Sep-06 Vol. 8 Issue No Page 660