An Approach to Solve Unbalanced Intuitionisitic Fuzzy Transportation Problem Using Intuitionistic Fuzzy Numbers

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1 Volume 117 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu An Approach to Solve Unbalanced Intuitionisitic Fuzzy Transportation Problem Using Intuitionistic Fuzzy Numbers S. Narayanamoorthy 1 and S. Ranjitha 2 1 Department of Applied Mathematics, 2 Department of Mathematics, 1,2 Bharathiar University, Coimbatore , Tamilnadu, India. 1 snm phd@yahoo.co.in 2 ranjiswasti93@gmail.com Abstract Transportation problems have numerous application in logistics and supply chains for decreasing costs. In real life circumstances, all the parameters of Transportation Problems may not be well-known precisely because of uncontrollable factors. In this paper we are presenting a new method to find the exact fuzzy optimal solution of unbalanced intuitionistic fuzzy transportation problem by using Yager s Ranking function. AMS Subject Classification: 03F55, 03E72, 90B06. Key Words and Phrases: IFTP, Yager s Ranking Function, Fuzzy Numbers, LR Flat Trapezoidal Intuitionistic Fuzzy Numbers. 1 Introduction An important network-structured linear programming problem is the Transportation problem, that arises in various contexts and has deservedly accepted considerable attention in the literature. In general, transportation problems are solved with the assumption that the transportation costs, supply and demand are specified precisely. If the information is unclear, that is if it lacks precision, the similar coefficients defining the problem can be developed using fuzzy sets, giving rise to intuitionistic fuzzy transportation problem. 411

2 2 Preliminaries In this section the basic concepts of Intuitionistic Fuzzy Set, LR Fuzzy Number, LR Flat Fuzzy Number, α - cut LR Flat Fuzzy Number are recalled. Definition 1. (Intuitionistic Fuzzy set) Let X φ be a given set. An Intuitionistic fuzzy set in X is an object given by, à I = {(x, µãi(x), νãi(x)) : x X} where, µãi, νãi : X [0, 1] satisfy the condition, 0 µãi(x) + νãi(x) 1, for every x X. Here µãi(x) denotes the degree of membership and νãi(x) denotes the degree of non-membership of the element x X being in ÃI, h(x) = 1 µãi (x) ν à I(x) for all x X is called as a degree of hesitation for x X being in ÃI. Definition 2. (LR Fuzzy Number) A function L (or R) : [0, ) [0, 1] is said to be the reference function of fuzzy numbers if and only if, (i) L(0)= (R(0)= 1). (ii) L(or R) is non - increasing on [0, ). Definition 3. (LR Flat Fuzzy Number) A fuzzy number à = (a 1, a 2, a 3, a 4 ) LR, is an LR flat fuzzy number, if its membership function µã(x) is, ( ) a L 2 x a 2 a 1, a 1 x a 2, µã(x) = 1, ( a 2 x a 3, x a R 3, a 3 x a 4. a 4 a 3 ) The set of LR flat fuzzy number on the real line is denoted by lr(r). If L(x) = R(x) = max(0, 1 x), the LR fuzzy number à = (a 1, a 2, a 3, a 4 ) LR is denoted as à = (a 1, a 2, a 3, a 4 ) and called a trapezoidal fuzzy number with the following membership function: µã(x) = ( x a 1 a 2 a 1 ), a 1 x a 2, 1, ( a 2 x a 3, ) a 4 x a 4 a 3 a 3 x a 4, Definition 4. (Non - Negative LR Flat Fuzzy Number) An LR flat fuzzy number à = (a 1, a 2, a 3, a 4 ) LR is said to be non-negative LR flat fuzzy number if and only if a 1 0. lr(r) + is denoted by the set of all non-negative LR flat fuzzy number. 412

3 Definition 5. (α - cut LR Flat Fuzzy Number) Let à = (a 1, a 2, a 3, a 4 ) LR be an LR flat fuzzy number and α be a real number in the interval [0,1]. Then, the set Ãα = {x R, µã(x) α}. à α = [a 2 (a 2 a 1 )L 1 (α), a 3 + (a 4 a 3 )R 1 (α)] = [à α, Ã+ α ] is said to be the α cut of the fuzzy number Ã. 3 Intuitionistic Fuzzy Transportation Problem The special kind of linear programming problem is the transportation problem, where special scientific structure of limitations is utilized. Efficient algorithms have been established for solving the transportation problem where the cost coefficients, supply and demand quantities are known precisely. The occurrence of irregularity and imprecision in reality is inevitable owing to some sudden situations. These cases that the cost of coefficients, supply and demand quantities of a transportation problem might be uncertain due to some uncontrollable factors. 4 Yager s Ranking Approach A Yager s Ranking Approach have been proposed for the ranking of trapezoidal fuzzy numbers. Yager s Ranking Approach is easily understandable and simple computational ranking approach. This Ranking Approach obtained a ranking index R(Ã) for ordering fuzzy sets are calculated for the fuzzy number à = (a 1, a 2, a 3, a 4 ) LR from its λ cut A λ = [a 1 a 3 L 1 (λ), a 2 +a 4 R 1 (λ)] according the following formula, R(Ã) = 1( 1 (a 2 1 a 3 L 1 (λ))dλ + 0 à and B are the two fuzzy numbers such that, 1 0 (a 2 + a 4 R 1 (λ))dλ) (i) à B if R(Ã) > R( B), (ii) à B if R(Ã) = R( B), (iii) à B if R(Ã) R( B). To calculated R (A) from the extreme values of λ- cut of à rather than its membership functions. Using this approach, the value of R(Ã) for any parameter, represented by trapezoidal fuzzy number à = (a 1, a 2, a 3, a 4 ) LR with L(x) = R(x) = max{0, 1 x}, may be obtained by using the succeeding formula, R(Ã) = 1 2 (a 1 + a 2 a a 4 2 ). 413

4 4.1 Numerical Examples Unbalanced intuitionistic fuzzy transportation problem with LR flat intuitionistic fuzzy numbers, The fuzzy supply of the goods at the first and second origins are (70, 90, 110, 150; 80, 95, 100, 130) LR, and (40, 60, 100, 130; 50, 75, 90, 120) LR respectively, and the fuzzy demand for the product at the first, second and third destinations are(30, 40, 55, 75; 35, 45, 50, 65) LR, (20, 30, 45, 65; 25, 35, 40, 55) LR and (40, 50, 65, 80; 45, 55, 60, 75) LR respectively. On determining the fuzzy quantity of the commodity that should be transported from each origin to each destination, the total intuitionistic fuzzy transportation cost is minimized. The total fuzzy supply is, 2 s I i = (110, 150, 210, 280; 130, 170, 190, 250) LR i=1 and the fuzzy demand is, 3 j=1 d I j = (90, 120, 165, 220; 105, 135, 150, 195) LR. Since the fuzzy supply is not equal to the fuzzy demand, this is called as an unbalanced intuitionistic fuzzy transportation problem. By introducing an imaginary supply and imaginary demand the unbalanced problem can be changed into a balanced problem, 3 s I i = 4 i=1 Now, we are assigning the fuzzy costs for transporting a unit quantity of the commodity from all sources to the imaginary origin and from an imaginary origin to all destinations as zero LR flat intuitionistic fuzzy numbers. The obtained balanced intuitionistic fuzzy transportation problem can be developed. The balanced transportation table(3.1) for the considered problem is given below, j=1 d I j Source D 1 D 2 D 3 D n Supply O 1 (10, 20, 30, 40) LR (50, 60, 70, 90) LR (80, 90, 110, 120) LR (0, 0, 0, 0) LR (70, 90, 110, 150) LR (15, 25, 35, 45) LR (55, 65, 75, 95) LR (85, 95, 115, 125) LR (0, 0, 0, 0) LR (80, 95, 100, 130) LR O 2 (60, 70, 80, 90) LR (70, 80, 100, 120) LR (20, 30, 50, 60) LR (0, 0, 0, 0) LR (40, 60, 100, 130) LR (65, 75, 85, 95) LR (75, 85, 110, 125) LR (25, 35, 55, 65) LR (0, 0, 0, 0) LR (50, 75, 90, 120) LR O 2 (0, 0, 0, 0) LR (0, 0, 0, 0) LR (0, 0, 0, 0) LR (0, 0, 0, 0) LR (0, 0, 0, 0) LR (0, 0, 0, 0) LR (0, 0, 0, 0) LR (0, 0, 0, 0) LR (0, 0, 0, 0) LR (0, 0, 0, 0) LR O 2 (30, 40, 55, 75) LR (20, 30, 45, 65) LR (40, 50, 65, 80) LR (20, 30, 45, 60) LR (35, 45, 50, 65) LR (25, 35, 40, 55) LR (45, 55, 60, 75) LR (25, 35, 40, 55) LR 3 i=1 si i = 4 j=1 d I j Minimize (10, 20, 30, 40; 15, 25, 35, 45) LR x 11 (50, 60, 70, 90; 55, 65, 75, 95) LR x 12 (80, 90, 110, 120; 185, 95, 115, 125) LR x 13 (0, 0, 0, 0; 0, 0, 0, 0) LR x 14 (60, 70, 80, 90; 65, 75, 85, 95) LR x

5 (70, 80, 100, 120; 75, 85, 110, 125) LR x 22 (20, 30, 50, 60; 25, 35, 55, 65) LR x 23 (0, 0, 0, 0; 0, 0, 0, 0) LR x 24 (0, 0, 0, 0; 0, 0, 0, 0) LR x 31 (0, 0, 0, 0; 0, 0, 0, 0) LR x 32 (0, 0, 0, 0; 0, 0, 0, 0) LR x 33 (0, 0, 0, 0; 0, 0, 0, 0) LR x 34 Subject to x 11 x 12 x 13 x 14 = (70, 90, 110, 150; 80, 95, 100, 130) LR x 21 x 22 x 23 x 24 = (40, 60, 100, 130; 50, 75, 90, 120) LR x 31 x 32 x 33 x 34 = (0, 0, 0, 0; 0, 0, 0, 0) LR x 11 x 21 x 31 = (30, 40, 55, 75; 35, 45, 50, 65) LR x 12 x 22 x 32 = (20, 30, 45, 65; 25, 35, 40, 55) LR x 13 x 23 x 33 = (40, 50, 65, 80; 45, 55, 60, 75) LR x 14 x 24 x 34 = (20, 30, 45, 60; 25, 35, 40, 55) LR where x 11, x 12, x 13, x 14, x 21, x 22, x 23, x 24, x 31, x 32, x 33, x 34, are non-negative fuzzy numbers. 3 s I i = (110, 150, 210, 280; 130, 170, 190, 250) LR = 4 i=1 The unbalanced intuitionistic fuzzy transportation problem becomes an balanced intuitionistic fuzzy transportation problem when the total fuzzy supply is equal to the total fuzzy demand with applying an imaginary demand d 4 = (20, 30, 45, 60; 25, 35, 40, 55) LR. j=1 d I j Stage 3 Split the table into eight crisp transportation tables as shown below, O O Demand O O Demand Table 3.2 Table 3.3 O O Demand O O Demand Table 3.4 Table

6 O O Demand O O Demand Table: 3.6 Table: 3.7 O O Demand O O Demand Table 3.8 Table 3.9 stage 4 By substituting the values of x ij,1, x ij,2, x ij,3, x ij,4, x ij,5, x ij,6, x ij,7 and x ij,8 in x ij = (x ij,1, x ij,2, x ij,3, x ij,4, x ij,5, x ij,6, x ij,7, x ij,8) LR, the fuzzy optimal solution of the intuitionistic fuzzy transportation problem is obtained as follows, x 11 = (55, 65, 55, 75, 50, 65, 50, 65) LR, x 12 = (45, 55, 45, 65, 40, 45, 40, 55) LR, x 13 = (10, 10, 10, 10, 10, 10, 10, 10) LR, x 14 = (0, 0, 0, 0, 0, 0, 0, 0) LR, x 21 = (0, 0, 0, 0, 0, 0, 0, 0) LR, x 22 = (0, 0, 0, 0, 0, 5, 0, 0) LR, x 23 = (55, 65, 55, 70, 50, 60, 50, 65) LR, x 24 = (45, 55, 45, 60, 40, 50, 40, 55) LR, x 31 = (0, 0, 0, 0, 0, 0, 0, 0) LR, x 32 = (0, 0, 0, 0, 0, 0, 0, 0) LR, x 33 = (0, 0, 0, 0, 0, 0, 0, 0) LR, x 34 = (0, 0, 0, 0, 0, 0, 0, 0) LR. stage 5 The total intuitionistic fuzzy transportation cost with LR flat intuitionistic fuzzy numbers is determined by substituting the values of the fuzzy optimal solution of an intuitionistic fuzzy transportation problem in the objective function of an intuitionistic fuzzy transportation problem, as follows, 2 i=1 j=1 3 c ij x ij = (4700, 7450, 8650, 14250) LR (5050, 7850, 8650, 13625) LR. 416

7 Figure 1: Graphical Representation of Unbalanced IFTP with Membership grade Figure 2: Graphical Representation of Unbalanced IFTP with Non - membership grade 5 Conclusion We considered Intuitionistic Fuzzy Transportation Problem all the parameters were described by non-negative LR flat intuitionistic fuzzy numbers and solved by using the approach of classical transportation Method. Here, the results obtained by the proposed method is more efficient than that of the existing methods. To obtaining fuzzy optimal solution of unbalanced intuitionistic fuzzy transportation problems pretaining to real world applications. The advantages of the approach over existing methods are discussed in the context of examples. References [1] A. Ebrahimnejad, A simplified new approach for solving fuzzy transportation problems with generalized trapezoidal fuzzy numbers, Appl. Soft Comput., 19 (2014),

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