Applied Mathematical Sciences, Vol. 7, 2013, no. 54, 2661 - 2673 HIKARI Ltd, www.m-hikari.com
An Approach for Solving Fuzzy Transportation
Problem Using Octagonal Fuzzy Numbers
S. U. Malini
Research Scholar, Stella Maris College (Autonomous), Chennai
Felbin C. Kennedy
Research Guide, Associate Professor, Stella Maris College, Chennai.
Copyright © 2013 S. U. Malini and Felbin C. Kennedy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, a general fuzzy transportation problem model is discussed. There are several approaches by different authors to solve such a problem viz., [1,2,3,6,7,8]. We introduce octagonal fuzzy numbers using which we develop a new model to solve the problem. By defining a ranking to the octagonal fuzzy numbers, it is possible to compare them and using this we convert the fuzzy valued transportation problem (cost, supply and demand appearing as octagonal fuzzy numbers) to a crisp valued transportation problem, which then can be solved using the MODI Method. We have proved that the optimal value for a fuzzy transportation problem, when solved using octagonal fuzzy number gives a much more optimal value than when it is solved using trapezoidal fuzzy number as done by Basirzadeh [3] which is illustrated through a numerical example.
Mathematics Subject Classification: 03B52, 68T27, 68T37, 94D05
Keywords: Octagonal Fuzzy numbers, Fuzzy Transportation Problem.
1 Introduction
The transportation problem is a special case of linear programming problem, which enable us to determine the optimum shipping patterns between origins and 2662 S. U. Malini and Felbin C. Kennedy
destinations. Suppose that there are m origins and n destinations. The solution of the problem will enable us to determine the number of units to be transported from a particular origin to a particular destination so that the cost incurred is least or the time taken is least or the profit obtained is maximum. Let ai be the number of units of a product available at origin i, and bj be the number of units of the product required at destination j. Let cij be the cost of transporting one unit from origin i to destination j and let xij be the amount of quantity transported or shipped from origin i to destination j. A fuzzy transportation problem is a transportation problem in which the transportation costs, supply and demand quantities are fuzzy quantities. Michael [11] has proposed an algorithm for solving transportation problems with fuzzy constraints and has investigated the relationship between the algebraic structure of the optimum solution of the deterministic problem and its fuzzy equivalent. Chanas et al [4] deals with the transportation problem wherein fuzzy supply values of the deliverers and the fuzzy demand values of the receivers are analysed. For the solution of the problem the technique of parametric programming is used. Chanas and Kuchta [5] have given a definition for the optimal solution of a transportation problem and as also proposed an algorithm to determine the optimal solution. Shiang-Tai Liu and Chiang Kao[14] have given a procedure to derive the fuzzy objective value of the fuzzy transportation problem based on the extension principle. Two different types of the fuzzy transportation problem are discussed: one with inequality constraints and the other with equality constraints. Nagoor Gani and Abdul Razack [12] obtained a fuzzy solution for a two stage cost minimising fuzzy transportation problem in which supplies and demands are trapezoidal fuzzy numbers. Pandian et al [13] proposed a method namely fuzzy zero point method for finding fuzzy optimal solution for a fuzzy transportation problem where all parameters are trapezoidal fuzzy numbers. In a fuzzy transportation problem, all parameters are fuzzy numbers. Fuzzy numbers may be normal or abnormal, triangular or trapezoidal or it can also be octagonal. Thus, they cannot be compared directly. Several methods were introduced for ranking of fuzzy numbers, so that it will be helpful in comparing them. Basirzadeh et al [2] have also proposed a method for ranking fuzzy numbers using α – cuts in which he has given a ranking for triangular and trapezoidal fuzzy numbers. A ranking using α-cut is introduced on octagonal fuzzy numbers. Using this ranking the fuzzy transportation problem is converted to a crisp valued problem, which can be solved using VAM for initial solution and MODI for optimal solution. The optimal solution can be got either as a fuzzy number or as a crisp number.
2. Octagonal fuzzy numbers
Two relevant classes of fuzzy numbers, which are frequently used in practical purposes so far, are “triangular and trapezoidal fuzzy numbers”. In this paper we introduce octagonal fuzzy numbers which is much useful in solving
Solving fuzzy transportation problem 2663
fuzzy transportation problem (FTP).
Definition 2.1: An octagonal fuzzy number denoted by ω is defined to be the ordered quadruple ω , for , and t where (i) is a bounded left continuous non decreasing function over [0, ω1], [0 ω1 k] (ii) is a bounded left continuous non decreasing function over [k, 2], [k ω2 ] (iii) is bounded left continuous non increasing function over [k, ω2],
[k ω2 ] (iv) is bounded left continuous non increasing function over [0,ω1]. [0 ω1 k] Remark 2.1: If ω=1, then the above-defined number is called a normal octagonal fuzzy number.
The octagonal numbers we consider for our study is a subclass of the octagonal fuzzy numbers (Definition 2.1) defined as follows:
Definition 2.2: A fuzzy number is a normal octagonal fuzzy number denoted by (a1,a2,a3,a4,a5,a6,a7,a8) where a1, a2, a3, a4, a5, a6, a7, a8 are real numbers and its membership function (x) is given below
μÃ(x) =
where 0 < k < 1.
Remark 2.2: If k = 0, the octagonal fuzzy number reduces to the trapezoidal number (a3,a4,a5,a6) and if k=1, it reduces to the trapezoidal number (a1,a4,a5,a8).
Remark 2.3: According to the above mentioned definition, octagonal fuzzy number ω is the ordered quadruple , for , and t where
= , = ,
and
2664 S. U. Malini and Felbin C. Kennedy
Remark 2.4: Membership functions are continuous functions.
Remark 2.5: Here ω represents a fuzzy number in which “ω” is the maximum membership value that a fuzzy number takes on. Whenever a normal fuzzy number is meant, the fuzzy number is shown by , for convenience.
Definition 2.3: If ω be an octagonal fuzzy number, then the α-cut of ω is [ ω]α = ω α = α α α ω Remark 2.6: The octagonal fuzzy number is convex as their α-cuts are convex sets in the classical sense.
Remark 2.7: The collection of all octagonal fuzzy real numbers from R to I is denoted as Rω(I) and if ω=1, then the collection of normal octagonal fuzzy numbers is denoted by R(I).
Graphical representation of a normal octagonal fuzzy number for k=0.5 is 1.2
1
0.8 s1(t) s2(t) 0.6
0.4
l1(r) 0.2 l2(r)
0 a₁ a₂ a₃ a₄ a₅ a₆ a₇ a₈
Working Rule I:
Using interval arithmetic given by Kaufmann A, [10] we obtain α-cuts, α ∊ (0, 1], addition, subtraction and multiplication of two octagonal fuzzy numbers as follows: a) α-cut of an octagonal fuzzy number: The α-cut of a normal octagonal fuzzy number =n (a1, a2, a3, a4, a5, a6, a7, a8) given by Definition 2.3 (i.e. ), for α ∊ (0, 1] is:
∊
= – ∊
Solving fuzzy transportation problem 2665
b) Addition of octagonal fuzzy Numbers: Let =(a1, a2, a3, a4, a5, a6, a7, a8) and = (b1, b2, b3, b4, b5, b6, b7, b8) be two octagonal fuzzy numbers. To calculate addition of fuzzy numbers and we first add the α–cuts of and using interval arithmetic. [a (a a ), a (a a )][b (b b ), b (b b )] for [0,k] ~ ~ 1 2 1 8 8 7 1 2 1 8 8 7 A B k k k k k k k k [a3 (a4 a3), a6 (a6 a5 )][b3 (b4 b3), b6 (b6 b5 )] for (k,1] 1 k 1 k 1 k 1 k [a b (a a b b ), a b (a a b b )] for [0,k] ~ ~ 1 1 2 1 2 1 8 8 8 7 8 7 A B k k k k [a3 b3 (a4 a3 b4 b3), a6 b6 (a6 a5 b6 b5)] for (k,1] 1 k 1 k c) Subtraction of two octagonal fuzzy numbers: Let = (a1, a2, a3, a4, a5, a6, a7, a8) and = (b1, b2, b3, b4, b5, b6, b7, b8) be two octagonal fuzzy numbers. To calculate subtraction of fuzzy numbers and we first subtract the α–cuts of and using interval arithmetic.
= [ , where qL min{( aaabbbaaabbb ( )) ( ( )),( ( )) ( ( )), 1k 211 k 211 k 218 k 87 (a ( aab )) ( ( bba )),( ( aab )) ( ( bb ))} 8k 871 k 218 k 878 k 87 qR max{( aaabbbaaabbb ( )) ( ( )),( ( )) ( ( )), 1k 211 k 211 k 218 k 87 (a ( aab )) ( ( bba )),( ( aab )) ( ( bb ))} 8k 871 k 218 k 878 k 87 for [0, k ] and
L k k k k q min{( a3 ( aab 433 )) ( ( bba 433 )),( ( aab 436 )) ( ( bb 65 )), 1k 1 k 1 k 1 k k k k k (a6 ( a 6 a 5 ))( b 3 ()),( b 4 b 3 a 6 ( a 6 a 5 ))( b 6 (bb65 ))} 1k 1 k 1 k 1 k
R k k k k q max{( a3 ( aab 433 )) ( ( bba 433 )),( ( aab 436 )) ( ( bb 65 )), 1k 1 k 1 k 1 k k k k k (a6 ( a 6 a 5 ))( b 3 ()),( b 4 b 3 a 6 ( a 6 a 5 ))( b 6 (bb65 ))} 1k 1 k 1 k 1 k for ( k ,1] d) Multiplication of two octagonal fuzzy numbers: Let = (a1, a2, a3, a4, a5, a6, a7, a8) and = (b1, b2, b3, b4, b5, b6, b7, b8) be two octagonal fuzzy numbers. To calculate multiplication of fuzzy numbers and we first multiply the α–cuts of and using interval arithmetic.
= [ , where
2666 S. U. Malini and Felbin C. Kennedy
qL min{( a ( aab ))( ( bba )),( ( aab ))( ( bb )), 1k 211 k 211 k 218 k 87 (a ( aab ))( ( bba )),( ( aab ))( ( bb ))} 8k 871 k 218 k 878 k 87 qR max{( a ( aab ))( ( bba )),( ( aab ))( ( bb )), 1k 211 k 211 k 218 k 87 (a ( aab ))( ( bba )),( ( aab ))( ( bb ))} 8k 871 k 218 k 878 k 87 for [0, k ]
=[ , where
L k k k k q min{( a3 ( aab 433 ))( ( bba 433 )),( ( aab 436 ))( ( bb 65 )), 1k 1 k 1 k 1 k k k k k (a6 ( aab 653 ))( ( bba 436 )),( ( aab 656 ))( ( bb 6 5 ))} 1k 1 k 1 k 1 k
R k k k k q max{( a3 ( aab 433 ))( ( bba 433 )),( ( aab 436 ))( ( bb 65 )), 1k 1 k 1 k 1 k k k k k (a6 ( aab 653 ))( ( bba 436 )),( ( aab 656 ))( ( bb 6 5 ))} 1k 1 k 1 k 1 k for ( k ,1]
3 Ranking of octagonal fuzzy numbers [3]
The parametric methods of comparing fuzzy numbers, especially in fuzzy decision making theory are more efficient than non-parametric methods. Cheng’s centroid point method[6], Chu and Tsao’s method[7], Abbasbandy and Assady’s [1] sign-distance method was all non-parametric and was applicable only for normal fuzzy numbers. The non-parametric methods for comparing fuzzy numbers have some drawbacks in practice.
+ Definition 3.1: A measure of fuzzy number is a function Mα: Rω(I) → R which assigns a non-negative real number Mα( ) that expresses the measure of .
(Ãω) = + where 0
Definition 3.2: The measure of an octagonal fuzzy number is obtained by the average of the two fuzzy side areas, left side area and right side area, from membership function to α axis.
Definition 3.3: Let be a normal octagonal fuzzy number. The value , called the measure of is calculated as follows: Oct M0 ( ) = where 0 = [(a1+ a2+a7 + a8)k + (a3 +a4+a5+a6)(1-k)] ------(3.1) Solving fuzzy transportation problem 2667 Remark 3.1: Consider the trapezoidal number which is got from the above octagonal number by equating , for k=0.5 and ω=1 1.5 1 0.5 0 a₁ b a₄ a₅ c a₈ The measure of the normal fuzzy trapezoidal number is given by tra M0 ( ) = ------(3.2) Oct Remark 3.2: If k=0.5, M0 ( ) = (a1+ a2+ a3 +a4+a5+a6+a7 + a8) When a2 coincides with a3 and a6 coincides with a7 it reduces to trapezoidal fuzzy number, which is given by Equation (3.2). Remark 3.3: If a1+ a2+a7 + a8 = a3 +a4+a5+a6 ------(3.3) then the measure of an octagonal number is the same for any value of k (0 Remark 3.4: If ω=(a1 + (a2-a1), a3 + (a4-a3), a6 – (a6-a5), a8 - (a8-a7)) for , and t be an arbitrary octagonal fuzzy ct number at decision level higher than “α” and α, ω ∊ [0,1], the value α (A ω), assigned to A ω may be calculated as follows: Working Rule II: If ω>α, then (Ãω) = + ; = { [ + ] + + (ω+k-2 ω)] (ω-k)} Obviously if ω α then the above quantity will be zero. If ω = it becomes a normal octagonal number, then ct(A ) = + ; α ∊ α = { [ + ] + + (1-k)] (1-k)} α ∊ [0,1) 2668 S. U. Malini and Felbin C. Kennedy Remark 3.5: If ω and ω’ are two octagonal fuzzy numbers and ω, ω’ [0, 1], then we have: 1. ω ω’ ∀α [0, 1] ( ω) ( ω’) 2. ω = ω’ ∀α [0, 1] ( ω) = ( ω’) 3. ω ω’ ∀α [0, 1] ( ω) ( ω’) Remark 3.6: If α is close to one, the pertaining decision is called a “high level decision”, in which case only parts of the two fuzzy numbers, with membership values between “α” and “1”, will be compared. Likewise, if “α” is close to zero, the pertaining decision is referred to as a “low level decision”, since members with membership values lower than both the fuzzy numbers are involved in the comparison. 4 Mathematical formulation of a Fuzzy Transportation Problem Consider the following fuzzy transportation problem (FTP) having fuzzy costs, fuzzy sources and fuzzy demands, (FTP) Minimize z = Subject to ≈ ãi, for i= ,2,…m (4.1) ≈ , for j= , 2,…n (4.2) ≽ for i= , 2, …m and j= , 2,…n (4.3) where m = the number of supply points; n = the number of demand points; ≈ is the uncertain number of units shipped from supply point i to demand point j; ≈ is the uncertain cost of shipping one unit from supply point i to the demand point j; ãi ≈ is the uncertain supply at supply point i and ≈ is the uncertain demand at demand point j. The necessary and sufficient condition for the linear programming problem given above to have a solution is that ≈ The above problem can be put in table namely fuzzy transportation table given below: Supply ……………… ...... ……………… Demand ……………. Solving fuzzy transportation problem 2669 5. Procedure for Solving Fuzzy Transportation Problem We shall present a solution to fuzzy transportation problem involving shipping cost, customer demand and availability of products from the producers using octagonal fuzzy numbers. Step 1: First convert the cost, demand and supply values which are all octagonal fuzzy numbers into crisp values by using the measure defined by (Definition 3.3) in Section 3. Step 2: [9] we solve the transportation problem with crisp values by using the VAM procedure to get the initial solution and then the MODI Method to get the optimal solution and obtain the allotment table. Remark 5.1: A solution to any transportation problem will contain exactly (m+n-1) basic feasible solutions. The allotted value should be some positive integer or zero, but the solution obtained may be an integer or non-integer, because the original problem involves fuzzy numbers whose values are real numbers. If crisp solution is enough the solution is complete but if fuzzy solution is required go to next step. Step 3: Determine the locations of nonzero basic feasible solutions in transportation table. There must be atleast one basic cell in each row and one in each column of the transportation table. Also the m+n-1 basic cells should not contain a cycle. Therefore, there exist some rows and columns which have only one basic cell. By starting from these cells, we calculate the fuzzy basic solutions, and continue until (m+n-1) basic solutions are obtained. 6. Numerical Example Consider the following fuzzy transportation problem. According to the definition of an octagonal fuzzy number Ã, the measure of à is calculated as Oct M0 (A ) = where 0 k 1 2670 S. U. Malini and Felbin C. Kennedy = [(a1+a2+a7 + a8)k + (a3 +a4+a5+a6)(1-k)] where 0 k 1 Step 1: Convert the given fuzzy problem into a crisp value problem by using the measure given by Definition 3.3 in Section 3. This problem is done by taking the value of k as 0.4, we obtain the values of Oct Oct Oct M0 ( ), M0 ( ) and M0 ( ) as c Oct 2 (-1,0,1,2,3,4,5,6) M0 (c )= (-1+0+5+6)+ (1+2+3+4)] = 2.5 c Oct 2 2 (0,1,2,3,4,5,6,7) M0 (c )= (0+1+6+7)+ (2+3+4+5)] = 3.5 2 c Oct 2 (8,9,10, 11,12,13,14,15) M0 (c )= (8+9+14+15)+ (10+11+12+13)] = 11.5 c Oct 2 (4, 5,6,7,8,9,10,11) M0 (c )= [ (4+5+10+11)+ (6+7+8+9)] = 7.5 c Oct 2 2 (-2,-1,0,1,2,3,4,5) M0 (c )= (-2-1+4+5)+ (0+1+2+3)] = 1.5 2 c Oct 2 22 = (-3,-2,-1,0,1,2,3,4) M0 (c )= (-3-2+3+4)+ (-1+0+1+2)] = 0.5 22 c Oct 2 2 = (2,4,5,6,7,8,9,11) M0 (c )= (2+4+9+11)+ (5+6+7+8)] = 6.5 2 c Oct 2 2 = (-3,-1,0,1,2,4,5,6) M0 (c )= (-3-1+5+6)+ (0+1+2+4)] = 1.75 2 c Oct 2 (2,3,4,5,6,7,8,9) M0 (c )= (2+3+8+9)+ (4+5+6+7)] = 5.5 c Oct 2 2 (3,6,7,8,9,10,12,13) M0 (c )= (3+6+12+13)+ (7+8+9+10)] = 8.5 2 c Oct 2 = (11,12,14,15,16,17,18,21) M0 (c )= (11+12+18+21)+ (14+15+16+17)]=15.5 Oct 2 c (5,6,8,9,10,11,12,15) M0 (c )= (5+6+12+15)+ (8+9+10+11)] = 9.5 And the fuzzy supplies are Oct = (1,3,5,6,7,8,10,12) M0 ( ) = (1+3+10+12)+ (5+6+7+8)] = 6.5 Oct = (-2,-1,0,1,2,3,4,5) M0 ( ) = (-2-1+4+5)+ (0+1+2+3)] = 1.5 Oct = (5,6,8,10,12,13,15,17) M0 ( ) = (5+6+15+17)+ (8+10+12+13)] = 10.75 And the fuzzy demands are (4,5,6,7, 8,9,10,11) Oct M0 ( ) = [ (4+5+10+11)+ (6+7+8+9)] = 7.5 (1,2,3,5,6,7,8,10) Oct M0 ( ) = (1+2+8+10)+ (3+5+6+7)] = 5.25 Oct (0,1,2,3,4,5,6,7) M0 ( ) = (0+1+6+7)+ (2+3+4+5)] = 3.5 Oct (-1,0,1,2,3,4,5,6) M0 ( ) = (-1+0+5+6)+ (1+2+3+4)] = 2.5 Solving fuzzy transportation problem 2671 Remark 6.1: In the above problem since condition 3.3 (Equation 3.3) is satisfied by all the octagonal numbers (cost, supply and demand), for any value of k we will get the same table as below. 2.5 3.5 11.5 7.5 6.5 1.5 0.5 6.5 1.75 1.5 5.5 8.5 15.5 9.5 10.75 7.5 5.25 3.5 2.5 Step 2: Using VAM procedure we obtain the initial solution as 1.25 5.25 1.5 6.25 3.5 1 which is not an optimal solution. Hence by using the MODI method we shall improve the solution and get the optimal solution as 5.25 1.25 1.5 7.5 0.75 2.5 Step 3: Now using the allotment rules, the solution of the problem can be obtained in the form of octagonal fuzzy numbers DESTINATION SUPPLY (1,2,3,5,6,7,8,10) (-9,-5,-2,0,2,5,8,11) (1,3,5,6,7,8,10,12) (-2,-1,0,1,2,3,4,5) (-2,-1,0,1,2,3,4,5) (4,5,6,7,8,9,10,11) (-12,-9,-5,-1,3,6,10,14) (-1,0,1,2,3,4,5,6) (5,6,8,10,12,13,15,17) DEMAND (4,5,6,7,8,9,10,11) (1,2,3,5,6,7,8,10) (0,1,2,3,4,5,6,7) (-1,0,1,2,3,4,5,6) Therefore the fuzzy optimal solution for the given transportation problem is (1,2,3,5,6,7,8,10), = (-9,-5,-2,0,2,5,8,11) = (-2,-1,0,1,2,3,4,5), =(4,5,6,7,8,9,10,11), = (-12,-9,-5,-1,3,6,10,14), = (-1,0,1,2,3,4,5,6) and the fuzzy optimal value of z = (-416,-224,-73,58,188,333,516,773). And the crisp solution to the problem is Minimum cost = 119.125. Also for different values of k (0 k 1) we obtain the same solution. Hence the solution is 2672 S. U. Malini and Felbin C. Kennedy independent of k. Remark 6.2: When we convert the octagonal fuzzy transportation problem into trapezoidal fuzzy transportation problem we get Supply ( -1,2,3,6) (0,3,4,7) (8,11,12,15) (4,7,8,11) (1,6,7,12) (-2,1,2,5) (-3,0,1,4) (2,6,7,11) (-3,1,2,6) (2,5,6,9) (3,8,9,13) (11,15,16,21) (5,9,10,15) (-2,1,2,5) (5,10,12,17) Demand (4,7,8,11) (1,5,6,10) (0,3,4,7) (-1,2,3,6) When this problem is solved as in [13] we would get the fuzzy optimal solution for the given transportation problem as (1,5,6,10), = (-9,0,2,11) = (-2,-1,2,5), =(4,7,8,11), = (-12,-1,3,14), = (-1,2,3,6), the fuzzy optimal value of z as z = (-416,59,185,773) and the crisp value of the optimum fuzzy transportation cost for the problem is z = 140.83. If it is solved as in [3], we will get the same value for the variables but the optimal cost will be 121.375. On the other hand if the problem is solved using octagonal fuzzy numbers we get the optimum cost as 119.125 Remark 6.3: If the octagonal numbers are slightly modified so that the condition (Equation (3.3)) is not satisfied, i.e. a1+ a2+a7 + a8 ≠ a3 +a4+a5+a6, then for such a problem the optimal solution for different values of k (0 k 1) can be easily checked to lie in a finite interval. 7. Conclusion In this paper a simple method of solving fuzzy transportation problem (supply, demand, and cost are all octagonal fuzzy numbers) were introduced by using ranking of fuzzy numbers. The shipping cost, availability at the origins and requirements at the destinations are all octagonal fuzzy numbers and the solution to the problem is given both as a fuzzy number and also as a ranked fuzzy number. It also gives us the optimum cost which is much lower than, when it is done using trapezoidal fuzzy numbers. Acknowledgement. The authors wish to thank Professor M.S. Rangachari, Former Director and Head, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai and Professor P.V. Subramanyam, Department of Mathematics, IIT Madras, Chennai for their valuable suggestions in the preparation of this paper. Solving fuzzy transportation problem 2673 References [1]. S. Abbasbandy, B.Asady, Ranking of fuzzy numbers by sign distance, Information Science, 176 (2006) 2405-2416. [2]. H. Basirzadeh, R. Abbasi, A new approach for ranking fuzzy numbers based on α cuts, JAMI, Journal of Applied Mathematics & Informatic, 26(2008) 767-778. [3]. H. Bazirzadeh, An approach for solving fuzzy transportation problem, Applied Mathematical Sciences, 5(32)1549-1566. [4]. S.Chanas and W.Kolodziejczyk and A. Machaj, A fuzzy approach to the transportation problem, Fuzzy Sets and Systems, 13(1984) 211-221. [5]. S. Chanas and D. Kuchta, A concept of the optimal solution of the transportation problem with fuzzy cost coefficients, Fuzzy Sets and Systems, 82(1996), 299-305. [6]. C.H.Cheng, A new approach for ranking fuzzy numbers by distance method, Fuzzy sets and system s, 95 (1998) 307-317. [7]. T.C. Chu, C.T. Tsao, Ranking fuzzy numbers with an area between the centroid point, compute. Math. Appl., 43 (2002) 111-117. [8]. . Detyniecki, R.R. Yager, Ranking fuzzy numbers using α- weighted Valuations, International journal of Uncertainty, Fuzziness and knowledge-Based systems, 8(5) (2001) 573-592. [9]. Dr. S.P. Gupta, Dr. P.K. Gupta, Dr. Manmohan, Business Statistics and Operations Research, Sultan Chand & Sons Publications. [10]. A. Kaufmann and Madan M. Gupta, Introduction To Fuzzy Arithmetic, Theory And Applications, International Thomson Computer Press. [11]. Michéal ÓhÉigeartaigh, A fuzzy transportation algorithm, Fuzzy Sets Systems, 8(1982), 235–243. [12]. A.Nagoor Gani and K. Abdul Razak, Two stage fuzzy transportation problem, Journal of Physical Science, 10(2006), 63-69. [13]. P. Pandian and G. Natarajan, A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problem, Applied Mathematical Sciences, 4(2010), 79-90. [14]. Shiang-Tai Liu and Chiang Kao, Solving fuzzy transportation problems based on extension principle, European Journal of Operational Research, 153(2004), 661-674. Received: February 25, 2013