A Strategy to Solve Mixed Intuitionistic Fuzzy Transportation Problems by BCM

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1 Middle-East Journal of Scientific Research 25 (2): , 207 SSN DOS Publications, 207 DO: /idosi.mesr A Strategy to Solve Mixed ntuitionistic Fuzzy Transportation Problems by BCM 2 S. Krishna Prabha and S. Vimala Ph.D Scholar, Moer Teresa Women's University, Kodaikannal, Assistant Professor, Department of Maematics, PSNA CET, Dindigul, ndia 2 Department of Maematics, Moer Teresa Women s University, Kodaikannal, Tamilnadu, ndia Abstract: Transportation Problem (TP) mainly deals wi e supply and demand of commodities transported from several sources to e different destinations. n is work a Mixed ntuitionistic Fuzzy Transportation Problem (MFTP) is considered for which e transportation costs are imprecise numbers described by fuzzy numbers and ntuitionistic fuzzy numbers which are more realistic and general in nature. An alternative meod called Best Candidate Meod (BCM) is used to solve e MFTP. The result is demonstrated by considering a numerical example. AMS Subect Classification: 90C08, 90C90, 90C29, 90C70, 90B06, 03E72, 03F55. Key words: Mixed ntuitionistic Fuzzy Transportation Problem Triangular intuitionistic fuzzy numbers and mixed constraints NTRODUCTON problems. f it is not possible to explain an imprecise concept by using e conventional fuzzy set e The challenge of how to supply e commodities to intustinistic fuzzy set can be used. The membership of an e customers in an more efficient way is tackled by an element to a fuzzy set is a single value wi e degree of efficient framework called Transportation model. The acceptance between zero and one. But in e case of problems on distribution and transportation of resources ntuitionistic fuzzy Set it is characterized by a membership from one place to anoer can be solved by function and a non-membership function so at e sum Transportation meods. They ensure e efficient of bo values is less an one. Optimization in movement and timely availability of raw materials and intuitionistic fuzzy environment was given by finished goods. The basic transportation problem was Angelov [3]. The idea of intuitionistic fuzzy set (FS) originally developed by Hitchcock [] in 94. n 963, introduced by Atanassov [4, 5] is e generalization of Dantzig [2] used e simplex meod to e Zadeh s [6] fuzzy set. transportation problems as e primal simplex n 996, Chanas et al. [7] presented a fuzzy approach transportation meod. Several researchers studied to e transportation problem. Nagoor Gani et al. [8] built extensively to solve cost minimizing transportation up a two stage cost minimizing fuzzy. Stephen Dinager problem in various ways. All e parameters of e et al. [9] inferred a fuzzy transportation problem wi e transportation problems may not be known precisely aid of trapezoidal fuzzy numbers. Pandian et al. [0] due to uncontrollable factors in real world applications. developed a new algorim for finding a fuzzy optimal This type of imprecise data is not always well represented solution for fuzzy transportation problem. An Algorimic by random variable selected from a probability approach for solving a mixed ntustinistic Fuzzy distribution. n day to day problems various calculations Assignment problem was presented by Senil Kumar should be solved wi uncertainty and inexactness. et al. []. A systematic approach for solving mixed Variations in measurement, deviation in accuracy, errors intustionistic fuzzy transportation problem by zero point in computation leads to uncertainty and in exactness. meod was proposed by Senil Kumar et al. [2]. n order to deal wi is uncertainty we use fuzzy Abdullah A. Hlayel, Mohammad A. Alia [3, 4] has transportation problems instead of classical assignment proposed e BCM for solving optimization problems. Corresponding Auor: S. Krishna Prabha, Department of Maematics, Moer Teresa Women s University, Kodaikannal, PSNA CET, Dindigul, ndia. 374

2 Middle-East J. Sci. Res., 25 (2): , 207 Annie Christi and Malini [5, 6] have solved e transportation problems wi hexagonal and octagonal fuzzy numbers using best candidate meod and centriod ranking techniques. S. Krishna Prabha and S. Vimala [7] have implemented BCM for Solving e Fuzzy Assignment Problem Wi Various Ranking Techniques. n is paper Best Candidate Meod (BCM) is introduced for solving a balanced mixed intustionistic fuzzy transportation problem. The mixed fuzzy quantities as e cost, coefficients, supply and demands are transformed into crisp quantities by e given ranking meods and e BCM is applied to obtain e solution. n section 2 elementary concepts have been reviewed. Maematical Model of MFTP and ranking meods of e various fuzzy numbers are introduced in section 3. n section 4, corresponding algorim called BCM have been proposed for solving MFTP. A numerical example is taken and e above ere meods are applied and tested numericallyin section 5. Section 6concludes e paper. Preliminaries Definition 2.: A Triangular ntuitionistic Fuzzy Number (à ) is an intuitionistic fuzzy set in R wi e following membership function µ A(x) and non membership function. v (x) ) A Case : f a'= a, a'= 3 a 3, en à represent Tringular Fuzzy Number (TrFN). t is denoted by A = (a, a 2, a 3). Case 2: f a '= a = a 2= a 3 = a 3' = m, en à represent a real number m. Definition 2.4.6: B Let A and B be two TrFNs. The ranking of à and by e R(.) on E, e set of TrFNs is defined as follows: Ranking Techniques: Yager s Ranking Technique: Yager s ranking technique which satisfies compensation, linearity, additivity properties and provides results which consists of human intuition. For a convex fuzzy number ã, e Robust s Ranking ndex is defined by, R (ã) (0.5)( L, U a a ) d () 0 where ( L U a, a ) = {( b a) + ac, ( c b) } which is e level cut of e fuzzy number ã. Ranking of Triangular ntuitionistic Fuzzy Numbers: The Ranking of a triangular intuitionistic fuzzy number à = (a, a 2, a 3) (a ', a 2, a') 3 is defined by R (à ) = The ranking technique is: fr (à ) R ( B ), en à = B i.e, min { Ã, B }= à (2) where ' 2 3 3' and µ A(x), v A(x) 0.5 for µ A(x) = v A (x), x R. This TrFN is denoted by à = (a, a 2, a 3) (a ', a 2, a') 3 Particular Cases Let à = (a, a 2, a 3) (a ', a 2, a') 3 be a TrFN. Then e following cases arise 375 Fuzzy Balanced and Unbalanced Transportation Problem [0, 8, 9]: The balanced fuzzy transportation problem, in which a decision maker is uncertain about e precise values of transportation cost, availability and demand, may be formulated as follows: p minimize c x q i= =

3 Middle-East J. Sci. Res., 25 (2): , 207 Subect to p x i = q x = p q a i = b i= = p a i = i q b = p q c i= = * x p q a i = b i= = = ã, i =, 2, 3,, p i = b, =, 2, 3,, q X is a non- negative trapezoidal fuzzy number, where p = total number of sources Q = total number of destinations a i = e fuzzy availability of e product at i source b = e fuzzy demand of e product at destination c = e fuzzy transportation cost for unit quantity of e product from i source to destination x = e fuzzy quantity of e product at should be transported from i source to destination to minimize e total fuzzy transportation cost. = total fuzzy availability of e product, = total fuzzy demand of e product = total fuzzy transportation cost. f en e fuzzy transportation problem is said to be balanced fuzzy transportation problem, oerwise it is called unbalanced fuzzy transportation problem. Consider transportation wi m fuzzy origins (rows) and n fuzzy destinations (Columns) () (2) (3) Let C =[C, C, C ] be e cost of transporting one unit of e product from i fuzzy origin to fuzzy () (2) (3) destination a=[a i i, a i, a i ] be e quantity of commodity () (2) (3) available at fuzzy origin i b = [b, b, b ] be e quantity of commodity requirement at fuzzy destination. 2 3 X =[X, X, X ] is quantity transported from i fuzzy origin to fuzzy destination. An unbalanced transportation problem is converted into a balanced transportation problem by introducing a dummy origin or dummy destinations which will provide for e excess availability or e requirement e cost of transporting a unit from is dummy origin (or dummy destination) to any place is taken to be zero. After converting e unbalanced problem into a balanced problem, we adopt e usual procedure for solving a balanced transportation problem. Algorim for Best Candidates Meod (BCM) has e Following Solution Steps: Step: From e matrix wi e Fuzzy Assignment Costs. Balance e unbalanced matrix and don't use e added row or column candidates in our solution procedure. Step2: The best candidates are selected by choosing minimum cost for minimization problems and maximum cost for maximization problems. Select e best two candidates in each row, if e candidate is repeated more an two times select it also. Check e columns at not have candidates and select one candidate for em, if e candidate is repeated more an one time select it also. Step3: The combinations are found by determining only one candidate for each row and column starting from e row at have least candidates and delete at row and column if ere is situation at has no candidate for some rows or columns, select directly e best available candidate. Repeat step 3 (, 2) by determining e next candidate in e row at started from. The total sum of candidates for each combination is computed and compared to determine e best combinations at give e optimal solution. Algorim to Solve MFTP wi BCM: Step: f e matrix is balanced go to step2, f not balance by adding dummy row/column as needed to make e supply equal to demand. Step2: Find e best combination by BCM to produce e lowest total weight of e cost, where is one candidate for each row or column. Step 3: Selected a row wi e smallest cost candidate from e choosen combination. Allocate e demand and supply as much as possible to e variable wi least unit cost in e selected row/column. Adust e supply and demand by crossing out e row /column. f e row/column is not assigned to zero en we check e selected row if it has an element wi lowest cost comparing to e determined element in e choosen combination en we elect it. Step 4: Find e next least cost from e choosen combination and repeat step3 until all columns and rows are exhausted. 376

4 Middle-East J. Sci. Res., 25 (2): , 207 Numerical Example: Consider e Mixed ntustionstic fuzzy transportation problem FD FD2 FD3 Fuzzy Available F0 (6, 7, 8:4, 7, 0) 3 (2, 4, 6) (, 2, 3) F02 (, 2, 3) (0,, 2) (, 3, 5:2, 3, 4) 3 FO3 (, 3, 5) (2, 4, 6:, 4, 7) 6 (3, 5, 7:, 5, 9) Fuzzy Requirement (2, 4, 6:, 4, 7) (3, 5, 7) By using e above ranking techniques for defuzzification, we get e following MFTP table FD FD2 FD3 Fuzzy Available F F FO Fuzzy Requirement 4 5 Determine e cost table from e given problem. Here total demand equals total demand. FD FD2 FD3 Fuzzy Available F F FO Fuzzy Requirement Applying e above new algorim for solving e transportation problem, we get e following allocations Select e best candidates from step 2. FD FD2 FD3 Fuzzy Available F0 7 2 F FOO3 6 5 Fuzzy Requirement Using BCM we select e best combination at will produce e lowest total weight of e costs, where ere is one candidate for each row and column. FD FD2 FD3 Fuzzy Available F F FO Fuzzy Requirement 4 5 Optimal Solution FD FD2 FD3 Fuzzy Available F \ 0 F \2\0 FO \ Fuzzy Requirement 4\0 \0 5\3\ The optimal solution is given by (6x)+(3x2)+(3x4)+(x)+(4x2) = =33 Total minimum cost will be Rs.33 CONCLUSON ranking techniques. The best combinations are selected wiout any complication. Optimal solution or n is paper, e transportation costs are closest to optimal solution can be obtained using BCM considered as mixed intustionistic fuzzy numbers. wi less computation time. This meod is very easy to Thus e MFTP has been transformed into crisp solve and it minimize e iterations and reduce e balanced transportation problem using various complexity. 377

5 Middle-East J. Sci. Res., 25 (2): , 207 REFERENCE 5. Annie Christi M.S. and D. Malini, 206. Solving Transportation Problems wi Hexagonal Fuzzy. Hitchcock, 94. F.L: The distribution of a product Numbers Using Best Candidates Meod and from several sources to numerous localities, Journal Different Ranking Techniques, Journal of of Maematical Physics, pp: Engineering Research and Applications, 6(2): Dantzig, G.B., 963. Linear Programming and 6. Annie Christi M.S. and D. Malini, 206. An Extensions, Princeton University Press, NJ. Approach To Solve Transportation Problems Wi 3. Angelov, P.P., 997. Optimization in an Octagonal Fuzzy Numbers Using Best Candidates intuitionistic fuzzy environment, Fuzzy Sets and Meod And Different Ranking Techniques, Systems, 86: nternational Journal of Computer Application ( Atanassov, K., 986. ntuitionistic fuzzy sets, 797), 6(), January- February 206 Fuzzy Sets and Systems, 20(): Krishna, Prabha S. and S. Vimala, Atanassov, K., 989. More on ntuitionistic fuzzy Optimal Solution for e ntuitionistic Fuzzy sets, Fuzzy Sets and Systems, 33: Assignment Problem via Three Meods- 6. Zadeh, L.A., 965. Fuzzy sets, nformation and FRMM, FOAM, FAM, Advances in Research, Control, 8: Science Domain nternational, 7(6): -8, SSN: 7. Chanas, S. and D. Kuchta, 996. A concept of e optimal solution of e transportation problem wi 8. smail Mohideen, S. and P. Senil Kumar, 200. A fuzzy cost coefficients Fuzzy Sets and Systems, Comparative Study on Transportation Problem in 82: fuzzy environment. nternational Journal of 8. Nagoor Gani A. and K. Abdul Razak, Two Maematics Research, 2(): stage fuzzy transportation problem, Journal of 9. Krishna Prabha, S. and S. Vimala, 206. Maximizing Physical Sciences, 0: The Profit For An Unbalanced Fuzzy Transportation 9. Stephen Dinager, D and K. Palanivel, The Problem Using Circumcenter Of Centroids Ranking Transportation problem in fuzzy environment, int. Technique, nternational Journal of Applied Journal of Algorim, Computing and Maematics, Engineering Research, SSN , (): (3). 20. Bellman, R.E. and L.A. Zadeh, 970. Decision making 0. Pandian, P. and G. Nataraan, 200. A new algorim in a fuzzy environment, Management Sci., 7: for finding a fuzzy optimal solution for fuzzy 2. Haari, T. and S. Abbasbandy, 20. A Transportation problems. Applied Maematics Promoter Operator for Defuzzification Meods, Sciences, 4(2): Australian Journal of Basic and Applied Sciences,. Senil Kumar, P. and R. Jahir Hussain, 204. New npress. Algorim for solving mixed ntuitionistic fuzzy 22. smail Mohideen, S. and P. Senil Kumar, 200. Assignment Problem, Elixir Applied Maematics, A Comparative Study on Transportation Problem in 73: fuzzy environment. nternational Journal of 2. Senil Kumar, P. and R. Jahir Hussain, 204. A Maematics Research, 2(): Systematic Approach for Solving Mixed ntuitionistic 23. Krishna Prabha, S. and S. Vimala, 206. Optimal Fuzzy Transportation Problems, nternational Solution for e ntuitionistic Fuzzy Assignment Journal of Pure and Applied Maematics, Problem via Three Meods-FRMM, FOAM, FAM, 92(2): Advances in Research, Science Domain nternational, 3. Abdullah A. Hlayel, Mohammad A. Alia, (6): -8. Solving Transportation Problems Using e Best 24. Nagaraan, R and A. Solairau, 200. Assignment Candidates Meod, Computer Science & Problems wi Fuzzy Costs under RobustRanking Engineering: An nternational Journal (CSEJ), Techniques, nternational Journal of Computer 2(5). Applications ( ), 6(4). 4. Hlayel Abdallah Ahma, 202. The Best Candidates 25. Nagoor Gani A. Abbas, ntuitionistic Fuzzy Meod For Solving Optimization Problems, Journal Transportation problem, Proceedings of e Heber of Computer Science, 8(5): nternational Conference, pp:

6 Middle-East J. Sci. Res., 25 (2): , Sagaya Roseline, Henry Amirara, 205. Meods to 27. Zimmermann, H.J., 934. Fuzzy set eory and its Find e Solution for e ntuitionistic Fuzzy applications, SBN Assignment Problem wi Ranking of ntuitionistic 26. Yager, R.R., 978. Ranking fuzzy subsets over e Fuzzy Numbers, nternational Journal of nnovative unit interval, in Proceedings of e EEE Conference Research in Science, Engineering and Technology, on Decision and Control (CDC 78), pp: , 4(7). Albuquerque, NM, USA, August