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1. 1. M. Shanmugasundari, K. Ganesan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.1416-1424 A Novel Approach for the fuzzy optimal solution of Fuzzy Transportation Problem M. Shanmugasundari *, K. Ganesan ** *Department of Mathematics, Faculty of Science Humanities, SRM University, Kattankulathur, Chennai - 603203, India. ** Department of Mathematics, Faculty of Engineering and Technology, SRM University, Kattankulathur, Chennai - 603203, IndiaABSTRACT In this paper we propose a new method Bellman and Zadeh [3] proposed the concept offor the Fuzzy optimal solution to the decision making in Fuzzy environment. After thistransportation problem with Fuzzy parameters. pioneering work, several authors such asWe develop Fuzzy version of Vogels and MODI Shiang-Tai Liu and Chiang Kao[23], Chanas et alalgorithms for finding Fuzzy basic feasible and [4], Pandian et.al [19], Liu and Kao [17] etcfuzzy optimal solution of fuzzy transportation proposed different methods for the solution of Fuzzyproblems without converting them to classical transportation problems. Chanas and Kuchta [4]transportation problems. The proposed method proposed the concept of the optimal solution for theis easy to understand and to apply for finding Transportation with Fuzzy coefficient expressed asFuzzy optimal solution of Fuzzy transportation Fuzzy numbers. Chanas, Kolodziejckzy, Machaj[5]problem occurring in real world situation. To presented a Fuzzy linear programming model forillustrate the proposed method, numerical solving Transportation problem. Liu and Kao [17]examples are provided and the results obtained described a method to solve a Fuzzy Transportationare discussed. problem based on extension principle. Lin introduced a genetic algorithm to solveKeywords - Fuzzy sets, Fuzzy numbers, Fuzzy Transportation with Fuzzy objective functions.transportation problem, Fuzzy ranking, Fuzzy Nagoor Gani and Abdul Razak [12] obtained aarithmetic fuzzy solution for a two stage cost minimizing fuzzy transportation problem in which supplies andI. INTRODUCTION demands are trapezoidal fuzzy numbers. A. Nagoor The transportation problem is a special Gani, Edward Samuel and Anuradha [9] usedtype of linear programming problem which deals Arshamkhan’s Algorithm to solve a Fuzzywith the distribution of single product (raw or Transportation problem. Pandian and Natarajan [19]finished) from various sources of supply to various proposed a Fuzzy zero point method for finding adestination of demand in such a way that the total Fuzzy optimal solution for Fuzzy transportationtransportation cost is minimized. There are effective problem where all parameters are trapezoidal fuzzyalgorithms for solving the transportation problems numbers.when all the decision parameters, i. e the supply In general, most of the existing techniquesavailable at each source, the demand required at provide only crisp solutions for the fuzzyeach destination as well as the unit transportation transportation problem. In this paper we propose acosts are given in a precise way. But in real life, simple method, for the solution of fuzzythere are many diverse situations due to uncertainty transportation problems without converting them inin one or more decision parameters and hence they to classical transportation problems. The rest of thismay not be expressed in a precise way. This is due paper is organized as:to measurement inaccuracy, lack of evidence, In section II, we recall the basic conceptscomputational errors, high information cost, whether of Fuzzy numbers and related results. In section III,conditions etc. Hence we cannot apply the we define Fuzzy transportation problem and provetraditional classical methods to solve the the related theorems. In section IV, we proposetransportation problems successfully. Therefore the Fuzzy Version of Vogels Approximation Algorithmuse of Fuzzy transportation problems is more (FVAM) and fuzzy version of MODI methodappropriate to model and solve the real world (FMODI). In section V, numerical examples areproblems. A fuzzy transportation problem is a provided to illustrate the methods proposed in thistransportation problem in which the transportation paper for the fuzzy optimal solution of fuzzycosts, supply and demand are fuzzy quantities. transportation problems without converting them to classical transportation problems and the results obtained are discussed. 1416 | P a g e
2. 2. M. Shanmugasundari, K. Ganesan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.1416-1424II. PRELIMINARIES 2.3. Ranking of triangular fuzzy numbers The aim of this section is to present some Several approaches for the ranking of fuzzynotations, notions and results which are of useful in numbers have been proposed in the literature. Anour further consideration. efficient approach for comparing the fuzzy numbers is by the use of a ranking function based on their2.1 Fuzzy numbers graded means. That is, for every A fuzzy set Ã defined on the set of real A   a1 , a 2 , a 3   F  R  , the ranking function numbers R is said to be a fuzzy number if its : F(R)  R by graded mean is defined asmembership function  A : R  0,1    has the   a  4a 2  a 3 following characteristics  (A)   1 (i) Ã is normal. It means that there exists an xR  6 such that  A  x   1  For any two fuzzy triangular numbers  is convex. It means that for every x , x  R, A   a1 , a 2 , a 3  and B   b1 , b 2 , b3  in F(R), we   (ii) A 1 2 have the following comparison A  x1  1    x 2   min  A  x1  ,  A  x 2  ,   0,1    (iii)  A is upper semi-continuous.     (i). A  B if and only if  (A)  (B).  (iv) supp ( A ) is bounded in R.     (ii). A  B if and only if  (A)  (B).2.2. Triangular fuzzy numbers      (iii). A  B if and only if  (A)  (B). A fuzzy number A in R is said to be atriangular fuzzy number if its membership function      (iv). A  B  0 if and only if  (A)  (B)  0. A : R   0,1 has the following characteristics.   x  a1  a1  x  a 2 A triangular fuzzy number A   a1 , a 2 , a 3  in F  R    a 2  a1   1 is said to be positive if  (A)  0 and denoted  x  a2 A  x        by A  0. Also if  (A)  0, then   A  0 and if  a3  x a2  x  a3  a3  a2     (A)  0, then A  0. If  (A)   (B), then     0  otherwise the triangular numbers A and B are said to be   equivalent and is denoted by A  B.We denote this triangular fuzzy number byA   a1 , a 2 , a 3  . We use F(R) to denote the set of all 2.4. Arithmetic operations on triangular fuzzytriangular fuzzy numbers. Also if m  a 2 represents numbers Ming Ma et al. [9] have proposed a newthe modal value or midpoint,   a 2  a1 represents fuzzy arithmetic based upon both location index andthe left spread and   a 3  a 2 represents the right fuzziness index functions. The location indexspread of the triangular fuzzy number number is taken in the ordinary arithmetic, whereas A   a1 , a 2 , a 3  , then the triangular fuzzy number  the fuzziness index functions are considered to follow the lattice rule which is least upper bound in A can be represented by the triplet A   m, ,   .   the lattice L. That is for a, bL we defineThat is A   a1 , a 2 , a 3    m, ,   .  a  b  max a, b and a  b  min a, b . For arbitrary triangular fuzzy numbers   (a ,a ,a )   m ,  ,   , B  (b , b , b ) A  1 2 3 1 1 1 1 2 3   m 2 ,  2 , 2  and   , , ,  , the arithmetic operations on the triangular fuzzy numbers are defined by A  B   m 1 , 1 , 1    m 2 ,  2 , 2     (m 1  m 2 , max{1 ,  2 }, max{1 , 2 }  (m 1  m 2 , 1   2 , 1  2 ) 1417 | P a g e
3. 3. M. Shanmugasundari, K. Ganesan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.1416-1424In particular for any two triangular fuzzy numbers where c ij is the fuzzy unit transportation cost from ith  A  (a1 ,a 2 ,a 3 )   m 1 , 1 , 1  , B  (b1 , b2 , b3 )   source to the jth destination. The objective is to  m 2 ,  2 , 2  , we define: minimize the total fuzzy transportation cost, in this paper the fuzzy transportation problem is solved by(i). Addition fuzzy version of Vogel’s and MODI method. This A  B   a1 ,a 2 ,a 3    b1 ,b2 ,b3    fuzzy transportation problem is explicitly   m1 , 1 , 1    m 2 ,  2 , 2  represented by the following fuzzy transportation table.   m1  m 2 ,max  1 ,  2  ,max 1 , 2   Table 3.1 Fuzzy transportation table(ii). Subtraction A  B   a1 ,a2 ,a3    b1 , b2 , b3    Destination   m1 , 1 , 1    m 2 ,  2 , 2    m1  m 2 ,max 1 , 2  ,max 1 ,2   1 2 … n Supply(iii).Multiplication 1  c11  c12 …  c1n  a1 AB   a1 ,a 2 ,a 3  b1 ,b2 ,b3  Sources  2  c 21  c 22 ….  c 2n  a2   m1 , 1 , 1  m 2 ,  2 , 2    … … … … … …  m1m 2 ,max  1 ,  2  ,max 1 , 2 (iv). Division m  cm1  cm2 ….  cmn  an A  a1 ,a 2 ,a 3   m1        ,max( 2 , 1 ),max(1 , 2 )  ,  b ,b ,b   m Demand b1 b2 bm B 1 2 3  2  3.2. Basic Theorems if, B   b1 , b 2 , b 3  is non zero fuzzy number.  Theorem 3.1. (Existence of a fuzzy feasible solution)III. MAIN RESULTS The necessary and sufficient condition for the Consider a fuzzy transportation with m existence of a fuzzy feasible solution to the fuzzysources and n destinations with triangular fuzzy transportation problem (3.1) is,  m n  b numbers. Let a i  0 be the fuzzy availability at  ai   j (3.2)   source i and b , (b  0) be the fuzzy requirement i 1 j1 j j   at destinations j. Let cij (cij  0) be the unit (Total supply  Ttotal demand). .fuzzy transportation cost from source i to destination j. Let x ij denote the number of fuzzy units to be Proof: (Necessary condition) Let there exist a fuzzy feasible solution to the fuzzytransformed from source i to destination j. Now the transportation problemproblem is to find a feasible way of transporting the m n  Minimize Z    c x available amount at each source to satisfy the i 1 j1 ij ijdemand at each destination so that the total ntransportation cost is minimized. subject to  x ij  a i ,i  1, 2,3,..., m.   j1 (3.3) m3.1. Mathematical formulation of fuzzy    x ij  b j , j  1, 2,3,..., n.transportation problem i 1 The mathematical model of fuzzy   and x ij  0 for all i and j.transportation problem is as follows n From  x ij  a i , (i  1, 2,3,..., m), we have   m n j1 Minimize Z    cij x ij   m n m i 1 j1   x ij   a i    3.4  n i 1 j1 i 1 subject to  x ij  a i , i  1, 2,3,..., m   m j1   Also from  x ij  b j , ( j  1, 2,3,..., n), we have m i 1   x ij  b j , j  1, 2,3,..., n  (3.1) n m n i 1     x ij   b j (3.5) m n j1 i 1 j1   a i   b j , i  1, 2,3,..., m;  i 1 j1 From equations (3.4) and (3.5), we have   j  1, 2,3,..., n and x ij  0 for all i and j. m n  a  b  i j i 1 j1 1418 | P a g e
4. 4. M. Shanmugasundari, K. Ganesan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.1416-1424(Sufficient condition) will be atleast 2m and 2n respectively. Suppose if   Since all a and b are positive, xij must be all the total number of basic variables is D, then i j obviously D ≥ 2m , D ≥ 2n.positive. Therefore equation (3.2) yields a feasiblesolution. Case (i). If m < n, then m + n < n + n  m + n < 2n  m + n < 2n < D m + n < D .Theorem 3.2. The dimension of the basis of a fuzzytransportation are (m+n-1)(m+n-1).That a fuzzy Case (ii). If m  n, then m  n  m  ntransportation problem has only (m+n-1)  m  m  m  n  2m  m  nindependent structural constraints and its basic  m  n  2m  m  n  2m  Dfeasible solution has only (m+n-1) positivecomponents.  m  n  D.Proof: Consider a fuzzy transportation problem Case (iii). If m = n, then m + n = m + nwith m sources and n destinations,  m + m = m + n  2m = m + n m n  Minimize Z    c x    m + n = 2m  m + n = 2m < D ij ij i 1 j1 n  m + n < D. subject to  x ij  a i , i  1, 2,3,..., m.   Thus in all cases D ≥ m + n. But the number of j1 basic variables in fuzzy transportation problem is m    x ij  b j , j  1, 2,3,..., n. (m+n-1) which is a contradiction. i 1   and x ij  0 for all i and j. Therefore there is atleast one equation, either row or column having only one basic variable. Let us assume that a fuzzy transportation has m Let the rth equation have only one basic variable.rows (supply constraint equations) and n columns(demand constraint equations). Therefore there are Let x rs be the only basic variable in the rth row and totally (m+n) constraint equations. s column ,then x rs  a r . Eliminate r th row from the th   m n system of equation and substitute x rs  a r in sth  This is due to the condition that  i 1 ai   b  j1 j   column equation and replace b by b  b  a . s s   s rwhich is the last requirement constraint. Therefore After eliminating the r th row , the system has (m+n-one of (m+n) constraints can always be derived 2) linearly independent constraints. Hence thefrom the remaining (m+n-1). Thus there exists only number of basic variables is (m+n-2).(m+n-1) independent constraints and its basic Repeat the process and conclude that in thefeasible solution has only (m+n-1) positive reduced system of equations there is an equationcomponents. which has only one basic variable. But if this is in sth column equation, then it will have two basicTheorem 3.3 variables.The values of the fuzzy basic feasible solution are This concludes that in our original system all differences between the partial sum of a i and of equations, there is an equation which has atleast two basic variables. Continue the process to get the the partial sum of b . j theorem. m n   That is x ij    ri a i   s j b j , where ri ,s j are either   i 1 j1 IV. PROPOSED ALGORITHMS1  1,1,1 or 0 =  0, 0, 0  .  4.1 Fuzzy Version of Vogels Approximation Algorithm (FVAM)Theorem 3.4The fuzzy transportation problem has a triangular Step 1:basis. From the Fuzzy Transportation table, determine the penalty for each row or column. TheProof: We note that every equation has a basic penalties are calculated for each row column byvariable, otherwise, the equation cannot be satisfied substituting the lowest cost element in that row or    for a i  0, b j  0 . Suppose every row and column column from the next cost element in the same rowequations has atleast two basic variables, since there or column. Write down the penalties below andare m rows and n columns, the total number of basic aside of the rows and columns respectively of thevariables in row equations and column equations table. 1419 | P a g e
5. 5. M. Shanmugasundari, K. Ganesan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.1416-1424Step 2: Step 6: Identify the column or row with largest On the closed path, identify the corners with – θ.fuzzy penalty. In case of tie, break the tie arbitrarily. Select the smallest allocation among the cornersSelect a cell with minimum fuzzy cost in the with – θ which indicate the number of units that canselected column (or row, as the case may be) and be shifted to some other unoccupied cells. Add thisassign the maximum units possible by considering quantity to those corners marked with +θ andthe demand and supply position corresponding to subtract this quantity to those corners marked with –the selected cell. θ on the closed path and cheek whether the number of nonnegative allocations is (m+n-1) and repeatStep 3:   step 1 to step 7, till we reach  ij  0 for all i and j.Delete the column/row for which the supply anddemand requirements are met. 4.3 Unbalanced Fuzzy Transportation Problem.Step 4: Suppose the fuzzy transportation problem is Continue steps 1 to 3 for the resulting fuzzy  m n transportation table until the supply and demand of unbalanced one, i. e , if   i 1 ai       b j  , then all sources and destinations have been met.  j1  convert this into a balanced one by introducing a4.2 Fuzzy Version of MODI Algorithm (FMODI) dummy source or dummy destination with zero fuzzy unit transportation costs. Solve the resulting Step 1: balanced fuzzy transportation problem using aboveGiven an initial fuzzy basic feasible solution of a said algorithms.fuzzy transportation problem in the form ofallocated and unallocated cells of fuzzy V. NUMERICAL EXAMPLEStransportation table. Assign the auxiliary variables The following two numerical examples are u i , i=1,2,3,…,m and v j , j=1,2,3,…,n for rows and   taken from the paper “Simplex type algorithm for  solving fuzzy transportation problem by Edwardcolumns respectively. Compute the values of ui and Samuel et.al [9].v j using the relationship cij  ui  v j for all i, j for      all occupied cells. Assume either ui or v j as zero Example 5.1 A company has two factories O1, O2 andarbitrarily for the allocations in row/column. two retail stores D1, D2. The production quantities per month at O1, O2 are (150, 201, 246) and (50, 99,Step 2: 154) tons respectively. The demands per month forFor each unoccupied cell (i, j), compute the fuzzy D1 and D2 are (100,150,200) and (100,150,200) tonsopportunity cost ij using ij  cij  (ui  v j ).     respectively. The transportation cost per ton cij ,   i =1, 2; j=1, 2 are c11 =(15,19,29), c12 =(22,31,34),Step 3:   c21 =(8,10,12) and c22 =(30,39,54). (i) If all ij  0, then the current solution under the test is optimal Solution: Transportation table of the given fuzzy (ii) If at least one ij  0, then the current  transportation problem is Table 5.1 Fuzzy Transportation Problemsolution under the test is not optimal and proceeds tothe next step. D1 D2 Supply O1 (15,19,29) (22,31,34) (150,201,246)Step 4: O2 (8,10,12) (30,39,54) (50,99,154) Select an unoccupied cell (i, j) with most negative (100,150,200) (100,150,200) (200,300,400)opportunity cost among all unoccupied cells. DemandStep 5:Draw a closed path involving horizontal and verticallines for the unoccupied cells starting and ending atthe cell obtained in step 4 and having its othercorners at some allocated cells. Assign +θ and – θ To apply the proposed algorithms and the fuzzyalternately starting with +θ for the selected arithmetic, let us express all the triangular fuzzyunoccupied cells. numbers based upon both location index and fuzziness index functions. That is in the form of  m,  ,   given in table 5.2. 1420 | P a g e
6. 6. M. Shanmugasundari, K. Ganesan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.1416-1424Table 5.2 Balanced fuzzy transportation problem in The corresponding fuzzy optimal transportation costwhich all the triangular numbers are of the form is given by m,  ,    m n Minimize Z    c x   ij ij i 1 j1 D1 D2 Supply   6609,50  50r,55  55r  , where 0  r  1 can be suitably chosen by (19, 4-4r, 10-10r) the decision maker. (150, 50-50r, 50-50r) (39, 9-9r, 15-15r) (31, 9-9r, 3-3r)O1 (201, 51-51r, 45-45r) Case(i). When r=0, the fuzzy optimal transportation cost in terms of the form (m, α, β) is (6609, 50, 55). The corresponding fuzzy optimal transportation of the form  a1 , a 2 , a 3  is (6559, 6609, 6664) and its defuzzyfied transportation cost is 609.8. (150, 50-50r, 50-50r) (10, 2-2r, 2-2r) Case(ii). When r = 0.5, the fuzzy optimalO2 (99, 49-49r, 55-55r) transportation cost in terms of the form (m, α, β) is (6609, 25, 27.5). The corresponding fuzzy optimal transportation cost of the form  a1 , a 2 , a 3  is (6584, 6609, 6636.5) and its defuzzyfied transportation cost is 6609.4 Case(iii). When r=1, the fuzzy optimal transportation cost in terms of the form (m, α, β) is (6609,0,0). The corresponding fuzzy optimal transportation cost of the form  a1 , a 2 , a 3  is Demand (0, 6609, 0) and its defuzzyfied transportation cost is 4406. Optimum fuzzy transportation cost is (1150, 6609, 12882). Defuzzified fuzzyBy applying fuzzy version of Vogel’s transportation cost is 6745.Approximation method (FVAM)), the initial fuzzybasic feasible solution is given in table 5.3. Example 5.2 Find a fuzzy optimum solution for the unbalancedTable 5.3 Initial fuzzy basic feasible solution fuzzy transportation problem given below. D1 D2 Table 5.4. Unbalanced Fuzzy Transportation (19, 4-4r, 10-10r) (31, 9-9r, 3-3r) ProblemO1 (51, 50-50r, 55-55r) (150, 50-50r, 50-50r) B1 B2 Supply (10, 2-2r, 2-2r)O2 (39, 9-9r, 15-15r) (99, 49-49r, 55-55r) A1 (1,3,5) (3,5,13) (300,399,504)The corresponding initial fuzzy transportation cost A2 (2,3,10) (3,4,11) (250,301,346)is given by IFTC  19, 4  4r, 10  10r  51, 50  50r, 55  55r  A3 (5,6,13) (1,3,5) (300,399,504)   31, 9  9r, 3  3r 150, 50  50r, 50  50r   10, 2  2r, 2  2r  99, 49  49r, 55  55r  (400,448,508) (300,351,396)   6609,50  50r,55  55r  . DemandBy applying fuzzy version of MODI method(FMODI), it can be seen that the current initialfuzzy basic feasible solution is optimal.Hence the fuzzy optimal solution in terms oflocation index and fuzziness index is given by Solution: Introduce a dummy destination B3 withx11   51,50  50r,55  55r  ; (150,300,450) as its fuzzy demand and (0,0,0) as its fuzzy transportation costs per unit. Hence thex12  150,50  50r,50  50r  ; balanced fuzzy transportation with a dummyx 21   99, 49  49r,55  55r  , where 0  r  1. destination becomes 1421 | P a g e
7. 7. M. Shanmugasundari, K. Ganesan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.1416-1424Table 5.5 Balanced Fuzzy Transportation Problem By applying fuzzy version of Vogels Approximation method, the initial fuzzy basic feasible solution is B1 B2 B3 Supply given byA1 (1,3,5) (3,5,13) (0,0,0) (300,399,504) Table 5.7 Initial fuzzy basic feasible solutionA2 (2,3,10) (3,4,11) (0,0,0) (250,301,346)A3 (5,6,13) (1,3,5) (0,0,0) (300,399,504) B1 B2 B3 (400,448,508) (300,351,396) (150,300,450) (399,99-99r,105-105r)Demand (3,2-2r,2-2r) (5,2-2r,8-8r) (0,0,0) A1Now the balanced fuzzy transportation problem inwhich all the triangular numbers are of the form m,  ,   is given in table 5.6. (1,150-150r,150-50r)Table 5.6 Balanced Fuzzy Transportation Problemin which all the triangular numbers are of the form (3,1-r,7-7r) (4,1-r,7-7r)  m,  ,   (0,0,0) A2 B1 B2 B3 Supply (399,99-99r,105-105r) (48,99-99r,105-105r) (351,51-51r,45-45r) (3,2-2r,2-2r) (5,2-2r,8-8r) (0,0,0) (3,2-2r,2-2r) (6,1-r,7-7r) A1 (0,0,0) A3 (301,51-51r,45-45r) By applying fuzzy version of MODI method, it can be seen that the current initial fuzzy basic feasible (3,1-r,7-7r) (4,1-r,7-7r) (0,0,0) solution is optimal. The fuzzy optimal solution in A2 terms of location index and fuzziness index is given by x11   399, 99  99r, 105  105r  ;  x 21    49,150  150r, 150  150r  ; (399,99-99r,105-105r) x 23    252,150  150r, 150  150r  ; (3,2-2r,2-2r) (6,1-r,7-7r) x 32    351,51  51r, 45  45r  and (0,0,0) A3 x 33    48,99  99r,105  105r  , where 0  r  1. The corresponding fuzzy optimal transportation cost is given by m n  Minimize Z    c x   ij ij i 1 j1 (300,150-150r,150-150r)   2357,150  150r,150  150r  (448,48-48r,60-60r) (351,51-51r,45-45r) where 0 ≤ r ≤ 1 can be suitably chosen by the decision maker.Demand Case(i). When r = 0, the fuzzy optimal transportation cost in terms of the form (m, α, β) is (2397, 150,150). The corresponding fuzzy optimal 1422 | P a g e
8. 8. M. Shanmugasundari, K. Ganesan / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.1416-1424 transportation cost of the form A  (a1 ,a 2 ,a 3 ) is [6]. A. Charnes, W. W. Cooper, The stepping stone method for explaining linear(2217, 2397, 2547). Its defuzzyfied transportation programming calculation in transportation,cost is 2392. Management science, 1 (1954), 49-69.Case(ii). When r = 0.5, the fuzzy optimal [7]. G. B. Dantzig, M. N. Thapa, Springer:transportation cost in terms of the form (m, α, β) is L.P:2: Theory and Extensions, Princeton(2397, 75, 75). The corresponding fuzzy optimal university Press New Jersey, 1963. transportation cost of the form A  (a1 ,a 2 ,a 3 ) is [8]. D. Dubois and H. Prade, Fuzzy Sets and Systems, Theory and applications,(2322, 2397, 2472). Its defuzzyfied transportation Academic Press, New York,1980.cost is 2397. [9]. A. Edward Samuel and A. Nagoor Gani,Case(iii). When r =1, the fuzzy optimal Simplex type algorithm for solving fuzzytransportation cost in terms of the form (m, α, β) is transportation problem, Tamsui oxford(2397, 0, 0). The corresponding fuzzy optimal journal of information and mathematical  sciences, 27(1) (2011), 89-98.transportation cost of the form A  (a1 ,a 2 ,a 3 ) is [10]. Fang. S. C, Hu .C. F, Wang. H. F and(0, 2397, 0). The optimum fuzzy transportation cost Wu.S.Y, Linear Programming with fuzzy = (-548, 2397, 7120). Defuzzied fuzzy coefficients in constraints, Computers andtransportation cost is 2693. Mathematics with applications, 37 (1999), 63-76.VI. CONCLUSION [11]. Fegad. M. R, Jadhav. V. A and Muley. A. In this paper, the transportation costs are A, Finding an optimal solution ofconsidered as imprecise numbers described by transportation problem using interval andtriangular fuzzy numbers which are more realistic triangular membership functions, Europeanand general in nature. We proposed a fuzzy version Journal of Scientific Research, 60 (3)of VAM and MODI algorithms to solve fuzzy (2011), 415-421.transportation problem without converting them to [12]. A. Nagoor Gani, K. A. Razak, Two stageclassical transportation problems. Two numerical fuzzy transportation problem, Journal ofexamples are solved using the proposed algorithms Physical Sciences, 10 (2006), 63–69.and obtained results are better than the existing [13]. L. S. Gass, On solving the transportationresults. problem, Journal of operational research Society, 41 (1990), 291-297.AKNOWLEDGEMENTS [14]. E. Kaucher, Interval analysis in extended The authors are grateful to the anonymous interval space IR, Comput. Suppl, 2(1980),referees and the editors for their constructive 33-49.comments and suggestions. [15]. L. J. Krajewski, L. P. Ritzman and M. K. Malhotra, Operations management processREFERENCES and value chains, Upper Saddle River, NJ: [1]. Amarpreet Kau, Amit Kumar, A new Pearson / Prentice Hall, 2007. approach for solving fuzzy transportation [16]. Lious.T.S. and Wang.M.J, Ranking fuzzy problems using generalized trapezoidal numbers with integral value, Fuzzy sets fuzzy numbers, Applied Soft Computing, and systems, 50 (3) (1992), 247-255. 12 (2012) 1201–1213. [17]. S. T. Liu, C. Kao, Solving Fuzzy [2]. H. Arsham and A. B. Kahn, A simplex type transportation problem based on extension algorithm for general transportation principle, European Journal of Operations problems: An alternative to stepping- Research, 153 (2004), 661-674. stone, Journal of Operational Research [18]. T.Nirmala, D.Datta, H. S. Kushwaha and Society, 40 (1989), 581-590. K. Ganesan, Inverse Interval Matrix: A [3]. R. E. Bellman and L. A. Zadeh, Decision New Approach, Applied Mathematical making in a fuzzy environment, Sciences, 5(13) (2011) 607-624. Management science, 17(1970), 141-164. [19]. Pandian. P and Nagarajan. G, A new [4]. S. Chanas, D. Kuchta, A concept of algorithm for finding a fuzzy optimal optimal solution of the transportation with solution for fuzzy transportation problem, Fuzzy cost co efficient, Fuzzy sets and Applied Mathematics Sciences, 4 (2) systems, 82(9) (1996), 299-305. (2010),79-90 [5]. S. Chanas, W. Kolodziejczyk and A. [20]. G. Ramesh and K. Ganesan, Interval Linear Machaj, A fuzzy approach to the programming with generalized interval transportation problem, Fuzzy Sets and arithmetic, International Journal of Systems, 13(1984), 211-221. Scientific & Engineering Research, 2(11) (2011), 01-08. 1423 | P a g e
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