A Method for Solving Intuitionistic Fuzzy Transportation Problem using Intuitionistic Fuzzy Russell s Method

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1 nternational Journal of Pure and Applied Mathematics Volume 117 No , SSN: (printed version); SSN: (on-line version) url: Special ssue ijpam.eu A Method for Solving ntuitionistic Fuzzy Transportation Problem using ntuitionistic Fuzzy Russell s Method S. Narayanamoorthy 1 and A. Deepa 2 1 Department of Applied Mathematics, 2 Department of Mathematics, 1,2 Bharathiar University, Coimbatore , Tamilnadu, ndia. 1 snm phd@yahoo.co.in 2 deepa.sneha1993@gmail.com Abstract The fundamental transportation problem was originally developed by Hitchcock. n the literature several methods are proposed for solving intuitionistic fuzzy transportation problem. n this paper, we propose a new algorithm called intuitionistic fuzzy Russell s method for the initial basic feasible solution to a intuitionistic fuzzy transportation problem. A Numerical example is taken to illustrate the solution procedure. AMS Subject Classification: 03F55, 03E072, 90B06, 65K10. Key Words and Phrases: intuitionistic fuzzy transportation problem, triangular intuitionistic fuzzy numbers, fuzzy russell s method. 1 ntroduction The concept of intuitionistic fuzzy sets was presented by Atanassov which was established to be extremely useful to deal with vagueness. The intuitionistic fuzzy set seperates the degree of membership with degree of non membership of an element in the set. This is main advantage of the intuitionistic fuzzy set. Nagoorgani and Abbas had worked on intuitionistic fuzzy transportation probelems using a particular algorithm known as zero suffix algorithm and they introduced a new method for finding an optimal solution of intuitionistic fuzzy transportation problems based on ranking function, where the supply and demand are triangular fuzzy numbers. By 335

2 nternational Journal of Pure and Applied Mathematics Special ssue using a newly defined ranking function, intuitionistic fuzzy transportation problems was solved by Jahir hussain and Senthil kumar, and they also proposed operations for solving mixed intuitionistic fuzzy transportation problems. Shashi Aggarwal and Chaui Gupta introduced a new method in order to handle generalized trapezoidal intuitionistic fuzzy number and proposed a new method to deals with the generalized intuitionistic fuzzy transportation problems. 2 Preliminaries n this section the basic concepts of intuitionistic fuzzy set, intuitionistic fuzzy number, intuitionistic triangular fuzzy number are recalled. 2.1 ntuitionistic Fuzzy Set(FS) A in X is given by a set of ordered triples: à = {< x, µã(x), VÃ(x) > /x X} where µã, Và : X [0, 1] are function such that 0 µã + Và 1 for all x X. For each x the number µã and Và represent the degree of membership and degree of non membership of the element x X to A X respectively. 2.2 ntuitionistic Fuzzy Number(FN) An intuitionistic fuzzy number µã (i) An intuitionistic fuzzy subset of the real line. is: (ii) Normal. i.e., there is any x 1 R such that µã(x) = 1(sothatVÃ(X) = 0). (iii) Convex for the membership function µã(x) µã(x)(λx 1 + (1 λ)x 2 ) min(µã(x 1 ), µã(x 2 )), x 1, x 2 R, λ [0, 1]. (iv) Concave for the non membership function VÃ(X) VÃ(x)(λx 1 + (1 λ)x 2 ) min(vã(x 1 ), VÃ(x 2 )), x 1, x 2 R, λ [0, 1]. 2.3 ntuitionistic Triangular Fuzzy Number(TFN) µã is an intuitionistic fuzzy set in R with the following membership function µã(x) and the non membership function VÃ(X) : µã = x p 1 p 2 p 1 if p 1 x p 2. p 3 x p 3 p 2 if p 2 x p 3. 0 if otherwise. 336

3 nternational Journal of Pure and Applied Mathematics Special ssue and Và = p 2 x p 2 p 1 if p 1 x p 2. x p 2 p 3 p 2 if p 2 x p 3. 1 if otherwise. where p 1 p 1 p 2 p 3 p 3 and µã(x) + VÃ(X) 1 or µã(x) = VÃ(X) for all x R. This TFN is denoted by µã (T F N) (X) = {(p 1, p 2, p 3 ) : (p 1, p 2, p 3)} = (p 1, p 2, p 3 : p 1, p 2, p 3). 2.4 Ranking of an ntuitionistic Triangular Fuzzy Number Let P 1 = (p 1, p 2, p 3 : p 1, p 2, p 3) be a TFN. The ranking function for the membership function µ P is denoted by R(µ P ) and is defined by Let Then R(µ P ) = p 1 + 2p 2 + p 3 + p 1 + 2p 2 + p 3 8 P 1 = (p 1, p 2, p 3 : p 1, p 2, p 3) and Q 1 = (q 1, q 2, q 3 : q 1, q 2, q 3)be two TFNs. (i) P 1 Q 1 if R( P 1 ) R( Q 1). (ii) P 1 Q 1 if R( P 1 ) R( Q 1) (iii) P 1 = Q 1 if R( P 1 ) = R( Q 1) (iv)min( P 1, Q 1) = P 1 if P 1 Q 1 or Q 1 P 1 (v)max( P 1, Q 1) = P 1 if P 1 Q 1 or Q 1 P Mathematical Model for ntuitionistic Fuzzy Transportation Problem An intuitionistic fuzzy transportation problem in which a chief is questionable about the correct transportation costs, free market activity can be communicated as takes after: Minimize Z = x ij C ij Subject to x ij = p i, i = 1, 2,..., m x ij = q j, j = 1, 2,..., n 337

4 nternational Journal of Pure and Applied Mathematics Special ssue x ij 0 for all i and j. Where, m means the aggregate number of sources, n indicates the aggregate number of destinations, p i is the intuitionistic fuzzy supply of the item at the i th inception, q j is the intuitionistic fuzzy interest for the item at the j th goal, C ij is the intuitionistic fuzzy cost of transportation for a unit amount of the ware from the i th root to the j th goal and x ij is the intuitionistic fuzzy amount of the ware that ought to be transported from ith ause to the j th goal or intuitionistic fuzzychoice factors to limit the aggregate fuzzy transportation cost. Frame the transportation problem, the first m limitations stipulates that the intuitionistic fuzzy amount of the product sent from starting point levels with the intuitionistic fuzzy supply of the ware at cause. Also, the following n limitations stipulates that the intuitionistic fuzzy amount of the ware sent to goal j measures up to the intuitionistic fuzzy interest for the ware at goal j. At long last the target work demonstrates the aggregate intuitionistic fuzzy transportation cost. Thus,the intuitionistic fuzzy transportation problem is said to be adjusted, if The above FTP m p i = n q j i=1 j=1 Source G 1 G 2... G n Supply E 1 x 11 c 11 x 12 c x 1n c 1n p 1 E 2 x 21 c 21 x 22 c x 2n c 2n p E m x m1 c m1 x m2 c m2... x mn c mn p i m Demand q 1 q 2... q j p i = n i=1 ntuitionistic Fuzzy Transportation Table q j j=1 3 The Computational Procedure for ntuitionistic Fuzzy Russells Method n this section we proposes modified method called as ntuitionistic Fuzzy Russells method is used for finding initial basic feasible solution for ntuitionistic Fuzzy transportation problem. The solution procedure as follows, 3.1 Algorithm for ntuitionistic Fuzzy Russell s Method Step 1: Calculate the amounts u i, v j, ij by using u i = max {c ij } v j = max {c ij } for i= 1, 2, 3,..., m for j= 1, 2, 3,..., n 338

5 nternational Journal of Pure and Applied Mathematics Special ssue and ij = c ij u i v j for all i,j Step 2: choose the variables x ij (x 1 ij, x 2 ij, x 3 ij; x 1 ij) from the most negative value of ij. n the event that there are ties in the value of ij, choose x ij (x 1 ij, x 2 ij, x 3 ij; x 1 ij) with the smallest unit cost c ij (c 1 ij, c 2 ij, c 3 ij; c 1 ij, c 2 ij, c 3 ij). Again if there are ties in the value of c ij (c 1 ij, c 2 ij, c 3 ij; c 1 ij, c 2 ij, c 3 ij), choose x ij (x 1 ij, x 2 ij, x 3 ij; x 1 ij) with the highest amount of remaining source supply or destination demand. Step 3: Pick the activity level of x ij (x 1 ij, x 2 ij, x 3 ij; x 1 ij) equal to the smaller value between the source supply p i and the destination demand q j. Step 4: Subtract x ij (x 1 ij, x 2 ij, x 3 ij; x 1 ij) from p i and q j found in step 3. Eliminate from the transportation table the row or column that outcomes in a zero supply or destination demand after this subtraction. Stop if all p i (i = 1, 2, 3,..., m) and q j (j = 1, 2,..., n) are zero, or else go to step Numerical Example Consider the intuitionistic fuzzy transportation problem. Here the cost value, supplies and demands are intuitionistic triangular fuzzy numbers. G 1 G 2 G 3 G 4 p i E 1 (4, 7, 8; 3, 7, 9) (3, 6, 8; 2, 6, 8) (5, 7, 8; 4, 7, 9) (1, 2, 3; 1, 2, 4) (8, 11, 15; 6, 11, 18) E 2 (5, 6, 7; 4, 6, 8) (3, 5, 6; 2, 5, 7) (2, 5, 8; 1, 5, 9) (0, 1, 3; 1, 1, 4) (3, 8, 10; 2, 8, 13) E 3 (1, 4, 5; 1, 4, 5) (2, 3, 4; 1, 3, 5) (2, 5, 7; 1, 5, 7) (3, 4, 6; 2, 4, 7) (6, 10, 13; 4, 10, 15) q j (3, 6, 8; 2, 6, 10) (7, 12, 13; 6, 12, 15) (2, 4, 7; 1, 4, 9) (5, 7, 10; 3, 7, 12) (7, 29, 38; 12, 29, 46) Here p i and q j are intuitionistic fuzzy supply and intuitionistic fuzzy demand. ntuitionistic fuzzy Russell s method is used to finding the initial basic feasible solution. p i = (7, 29, 38; 12, 29, 46), q j = (7, 29, 38; 12, 29, 46) Since p i = qj, the problem is balanced intuitionistic fuzzy transportation problem. There exist a intuitionistic fuzzy initial basic feasible solution. By using ntuitionistic fuzzy Russell s method we have, 339

6 nternational Journal of Pure and Applied Mathematics Special ssue G 1 G 2 G 3 G 4 p i E 1 (4, 7, 8; 3, 7, 9) (3, 6, 8; 2, 6, 8) (5, 7, 8; 4, 7, 9) (1, 2, 3; 1, 2, 4) (8, 11, 15; 6, 11, 18) ( 3, 8, 15; 7, 8, 21) ( 3, 3, 14; 9, 3, 19) E 2 (5, 6, 7; 4, 6, 8) (3, 5, 6; 2, 5, 7) (2, 5, 8; 1, 5, 9) (0, 1, 3; 1, 1, 4) (3, 8, 10; 2, 8, 13) ( 7, 1, 5; 10, 1, 10) (5, 7, 10; 3, 7, 12) E 3 (1, 4, 5; 1, 4, 5) (2, 3, 4; 1, 3, 5) (2, 5, 7; 1, 5, 7) (3, 4, 6; 2, 4, 7) (6, 10, 13; 4, 10, 15) (3, 6, 8; 2, 6, 10) ( 2, 4, 10; 6, 4, 13) q j (3, 6, 8; 2, 6, 10) (7, 12, 13; 6, 12, 15) (2, 4, 7; 1, 4, 9) (5, 7, 10; 3, 7, 12) (7, 29, 38; 12, 29, 46) Hence the initial basic feasible solution is, [MinimumZ(Z 1, Z 2, Z 3 ; Z 1, Z 2, Z 3) = [( 3, 8, 15; 7, 8, 21)(3, 6, 8; 2, 6, 8)] +[( 3, 3, 14; 9, 3, 19)(5, 7, 8; 4, 7, 9)] +[( 7, 1, 5; 10, 1, 10)(2, 5, 8; 1, 5, 9)] +[(5, 7, 10; 3, 7, 12)(0, 1, 3; 1, 1, 4)] +[(3, 6, 8; 2, 6, 10)(1, 4, 5; 1, 4, 5)] +[( 2, 4, 10; 6, 4, 13)(2, 3, 4; 1, 3, 5)] Hence total intuitionistic fuzzy minimum transportation cost is, Minimum Z(Z 1, Z 2, Z 3, Z 1, Z 2, Z 3) = ( 39, 117, 382; 64, 117, 592) The crisp value of the ntuitionistic Fuzzy Transportation problem is Conclusion we proposed a new technique called as ntuitionistic Fuzzy Russell s method to find the initial basic feasible solution using ranking method with intuitionistic triangular fuzzy numbers to the intuitionistic fuzzy transportation problem. Numerical case demonstrates that by this method gives the base transportation cost. This method can be utilized for all kind of intuitionistic fuzzy numbers. References [1] K. T. Atanassov, (1986). ntuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, [2] S. Narayanamoorthy, S. Kalyani, Finding the nitial Basic Feasible Solution Problem of a Fuzzy Transportation Problem by a New Method, JPAM, (2015), Vol.101.No.5, pp:

7 nternational Journal of Pure and Applied Mathematics Special ssue [3] S. Narayanamoorthy, S. Saranya, S. Maheshwari, A Method for Solving Fuzzy Transportation Problem using Fuzzy Russell s Method, JSA, (2013), Vol.02, pp: [4] S. Narayanamoorthy, S. Kalyani, A Modified Concept of the Optimal Solution of the Transportation Problem in Fuzzy Environment, nternational Journal of Applied Engineering Research, (2014), Vol.9.No.11, pp: [5] V. L. G. Nayagam, G. Venkateshwari, and G. Sivaraman, (2008). Ranking of intuitionistic fuzzy numbers, Proceedings of EEE nternational Conference on Fuzzy Systems, Hong Kong, [6] H. M. Nehi, (2010). Nehi, HM (2010).A new ranking method for intuitionistic fuzzy numbers, nternational journal of fuzzy systems,12 (1), [7] P. Pandian and G. Natarajan, A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problem, Applied Mathematical Sciences, (2010), 4, pp: [8] Shashi Aggarwal and Chavi Gupta, A novel algorithm for solving intuitionistic fuzzy transportation problem via new ranking method, Annals of Fuzzy Mathematics and nformatics. [9] Shiang-Tai Liu and Chiang Kao, Solving fuzzy transportation problem based on extension principle, Journal of Physical Science, (2006), 10,pp: [10] D. Stephen Dinagar and K. Palanivel, The transportation problem in Fuzzy Environment. nternational Journal of Algorithms, Computing and Mathematics, August 2009, Vol 2, Number 3. [11] Tze-San lee, A complete Russells method for the Transportation Problem, (1986), Vol.28, No.4. [12] R. R. Yager, A procedure for ordering fuzzy subsets of the unit interval, nformation Sciences, (1981), 24, pp: [13] L. A. Zadeh, Fuzzy sets, nformation and Control, (1965), 8, pp:

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