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 L q min{( aaabbbaaabbb1 ( 211 )) ( ( 211 )),( ( 218 )) ( ( 87 )), k k k k (a ( aab )) ( ( bba )),( ( aab )) ( ( bb ))} 8k 871 k 218 k 878 k 87 R q max{( aaabbbaaabbb1 ( 211 )) ( ( 211 )),( ( 218 )) ( ( 87 )), k k k k (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.