Realistic Method for Solving Fully Intuitionistic Fuzzy. Transportation Problems

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1 Applied Mthemticl Sciences, Vol 8, 201, no 11, 6-69 HKAR Ltd, wwwm-hikricom Relistic Method for Solving Fully ntuitionistic Fuzzy Trnsporttion Problems P Pndin Deprtment of Mthemtics, School of Advnced Sciences VT University, Vellore-1, Tmilndu, ndi Copyright 201 P Pndin This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited Abstrct A new method nmely, relistic method is proposed for finding n optiml intuitionistic fuzzy (F) solution to fully intuitionistic fuzzy trnsporttion (FT) problem in which rnking functions re not used The proposed method is bsed on the crisp trnsporttion lgorithm nd provides tht the optiml F solution nd the optiml oective F vlue of the fully FT problem do not contn ny negtive prt For illustrting, fully FT problem is solved by using the proposed method The proposed method is n pproprite method to pply for finding n optiml solution of FT problems occurring in rel life situtions Mthemtics Suect Clssifiction: 0E72, 0F, 90B06 Keywords: Tringulr intuitionistic fuzzy numbers, ntuitionistic fuzzy trnsporttion problem, Optiml ntuitionistic fuzzy solution, Relistic method 1 ntroduction The trnsporttion problem is specil clss of liner progrmming problem which hs pplictions in Science, Engineering nd Technology An uncertn trnsporttion problem is trnsporttion problem in which t lest one of the prmeters is uncertnty The theory of fuzzy set introduced by Zdeh [9] hs chieved successful pplictions in vrious fields including Mthemticl Progrmming n the literture, mny resercher hve developed vrious methods for solving different types of fuzzy trnsporttion (FT) problems Pndin nd Ntrjn [6] proposed the dynmic method for solving fully FT problems which provides tht optiml fuzzy decision vribles nd the optiml oective fuzzy vlue of the FT problems do not contn ny negtive prt Atnssov [1] introduced the concept of F

2 6 P Pndin sets which is found to be highly useful to del with vgueness The mjor dvntge of F set over fuzzy set is tht F set seprtes the degree of membership nd the degree of nonmembership of n element in the set Ngoor Gni nd Abbs [] solved FT problems using zero suffix lgorithm Ngoor Gni nd Abbs [] proposed new method bsed on rnking function for finding n optiml solution of FT problems where the supply nd demnd re tringulr F numbers Jhir Hussn nd Senthilkumr [2] solved FT problems using newly defined rnking function Shshi Aggrwl nd Chvi Gupt [7] hve introduced new rnking method for generlized trpezoidl F number nd proposed new method bsed on the new rnking method for solving generlized FT problems Senthilkumr nd Jhir Hussn [8] hve defined new multipliction opertions on set of tringulr F numbers nd proposed new method bsed on the newly proposed opertions for solving mixed FT problems n the existing methods [2,,,8], the optiml solution of some of the F decision vribles nd the optiml oective F vlue of the FT problem hve negtive prt which depicts tht quntity of the product nd trnsporttion cost my be negtive But the negtive quntity of the product nd negtive trnsporttion cost hve no physicl mening Therefore, the solution obtned in [2,,,8] for FT problems re not relistic nd not pplicble n this pper, we develop new method nmely, relistic method for obtning n F optiml solution to fully FT problem where ll prmeters re F tringulr numbers To overcome the shortcomings of the existing methods [2,,,8], the proposed method bsed on lgorithm of the crisp trnsporttion problem nd provides non-negtive optiml F solution nd non-negtive optiml oective F vlue of the fully FT problem Rnking functions nd F rithmetic opertions re not used By mens of numericl exmple, the proposed method of solving the fully FT problem is illustrted The relistic method is n pproprite method for solving FT problems of the rel life situtions nd provides n pplicble optiml solution 2 ntuitionistic Fuzzy Trnsporttion Problem We need the following mthemticl orientted definitions of F set, tringulr F number nd membership function nd non-membership function of F set/number which cn be found in [1,2,,,8] Definition 21 Let X denote universe of discourse nd A X Then, n F set of A in X, A ~ is defined s follows: ~ A {( x, ~ ( x), ~ ( x)); x X} where A( x), A( x)) : X [0,1] re functions such tht 0 ~ ( x) ~ ( x) 1 for ll x X For ech x in X, ~ ( x) nd ~ ( x) represent the membership the degree of nonmembership vlues of x in the set A X ~ ( x) nd ~ ( x)

3 Fully intuitionistic fuzzy trnsporttion problems 6 Definition 22: A fuzzy number ~ is tringulr F number denoted by ( 2, )( 1, ) where 1, 2,, nd re rel numbers such tht 1 2 nd its membership function ~ ( x) nd its non-membership function ( ) re given below ~ x x 2 2 : 2 x x ( x) : x ~ nd 0 : otherwise x 1 : 1 x x ( x) : x ~ 0 : otherwise Let F (R) be set of ll tringulr F numbers over R, set of rel numbers Bsed on ordering reltion in intervl theory/ fuzzy set theory, we define the following: Definition : Let ~ ~ ( 2, )( 1, ) nd b ( b2, b, b )( b1, b, b ) be in F (R) Then, () ~ nd b ~ re sd to be equl if bi, i =1,2,,, nd (b) ~ is sd to be less thn or equl b ~ if bi, i =1,2,,, Definition 2: Let ~ ( 2, )( 1, ) be in F (R) Then, ~ ~ ( 0 ) if 0 1 Definition 2: Let ~ (,, )(,, ) be in F (R) Then, i 0, i =1,2,,, re integers 2 1 ~ is sd to be positive ~ is sd to be integer if Consider the following fully FT problem in which ll prmeters, tht is, decision vribles, trnsporttion costs, supplies nd demnds, re tringulr F numbers: m n ~ (P) Minimize Z c ~ ~ x suect to n j1 ~ x ~ ~ i1 j 1, for i = 1,2,,m (1) m ~ x b, for j = 1,2,,n (2) i1 ~ x ~ 0, for i = 1,2,,m nd j = 1,2,,n nd integers ()

4 66 P Pndin where m = the number of supply points; n = the number of demnd points; 2 1 ~ 2 1 (,, )(,, c c c c c c ~ 2 1 (,, )(,, i ~ i 2 1 (,, )(,, j ( x, x, x )( x, x, x ) is the uncertn number of units shipped from supply point i to demnd point j; c ) is the uncertn cost of shipping one unit from supply point i to the demnd point j ; ) is the uncertn supply t supply point i nd b ) is the uncertn demnd t demnd point j ~ A set of tringulr F numbers { ~ 2 1 X (,, )(,, ), i 1,2,, m nd j ~ 1,2,, m} is sd to be fesible F solution to the problem (P) if { ~ 2 1 X x (,, )(,, ), i 1,2,, m nd j 1,2,, m} stisfies the conditions (1), (2) nd () ~ A fesible F solution { ~ 2 1 X (,, )(,, ), i 1,2,, m nd j 1,2,, m} of ~ ~ the problem (P) is sd to be n optiml F solution to the problem (P) if Z( X ) Z( U ) for ll fesible U ~ of the problem (P) x~ The Relistic Method We need the following theorem which is used in the proposed method, nmely relistic method for solving the fully FT problems Theorem 1: Let [ ] {, i 1,2,, m nd j 1,2,, n} be n optiml solution to the k k problem ( P ) nd [ ] {, i 1,2,, m nd j 1,2,, n} be n optiml solution to the problem ( P k ), k=,,2,1 where m n ( P ) Minimize Z c i1 j 1 suect to n j 1 m i1 b i j,for i = 1,2,,m, for j = 1,2,,n x 0, for i = 1,2,,m nd j = 1,2,,n nd integers, nd for k =,,, 2, m n k-1 k-1 ( Pk -1) Minimize Zk-1 c i1 j1

5 Fully intuitionistic fuzzy trnsporttion problems 67 suect to n k-1 k-1, for i = 1,2,,m j1 m k-1 k-1, for j = 1,2,,n i1 k-1 k x x, for i = 1,2,,m nd j = 1,2,,n k-1 x { ~ ] x ( x 0, for i = 1,2,,m nd j = 1,2,,n nd integers Then, [ ~ 2 1,, )(,, ), i 1,2,, m nd j 1,2,, m} is n optiml F solution to the given problem (P) 1 2 Proof: Now, since [ x ],[ ],[ ],[ ] nd [ x ] re fesible solutions of ( P 1),( P2 ),( P ) ( P ) nd ( P ) respectively, [ ~ x ] is fesible F solution to the problem (P) where [ ~ { ~ 2 1 ] (,, )(,, ), i 1,2,, m nd j 1,2,, m} Let [ ~ { ~ 2 1 y ] y ( y, y, y )( y, y, y ), i 1,2,, m nd j 1,2,, n} be fesible F 1 2 solution of the problem (P) Clerly, [ y ],[ y ],[ y ],[ y ] nd [ y ] re fesible solutions of ( P 1),( P2 ),( P ), ( P ) nd ( P ) respectively 1 2 Now, since [ x ],[ x ],[ x ],[ x ] nd [ x ] re optiml solutions of P ),( P ),( P ), ( ) nd ( P ) respectively, we hve Z ([ x ]) Z ([ y ]); Z ([ x ]) Z ([ y ]); ( 1 2 P Z([ ]) Z([ y ]) ; Z([ ]) Z([ y ]) nd Z([ ]) Z([ y ]) This implies tht, ([ ~ ]) ([ ~ Z Z y ]), for ll fesible [ ~ y ] of the problem (P) Thus, [ ~ { ~ 2 1 ] (,, )(,, ), i 1,2,, m nd j 1,2,, m} is n optiml F solution to the given problem (P) Hence the theorem is proved Remrk 1: The optiml F solution [ ~ { ~ 2 1 ] (,, )(,, ), i 1,2,, m nd j 1,2,, m} nd the totl minimum F trnsporttion cost ([ ~ 2 1 Z ]) ( Z2([ ]), Z([ ]), Z([ ]))( Z1([ ]), Z([ ]), Z([ ])) k re positive becuse for ech k=1,2,,,, x k nd Z k ([ x ]) re positive for ll i nd j We, now propose new method nmely, relistic method to obtn n optiml solution of fully FT problem bsed on the crisp trnsporttion lgorithm The relistic method is s follows:

6 68 P Pndin Algorithm: Step 1: Construct the problem ( P ) from the given FT problem, (P) nd solve it by the zero point method []/ clssicl trnsporttion lgorithm Let {, i 1,2,, m nd j 1,2,, m} be n optiml solution of the problem ( P ) x Step 2: Construct the problem ( Pk 1) from the given problem (P) nd solve it by the zero point method [] / clssicl trnsporttion lgorithm for ech k=,,,2 1 Let { x k, i 1,2,, m nd j 1,2,, n} be n optiml solution of ( P k 1), k=,,,2 Step : { ~ 2 1 (,, )(,, ), i 1,2,, m nd j 1,2,, n} is n optiml F solution to the given FT problem, (P) by the Theorem 1 Numericl Exmple The proposed method is illustrted by the following exmple Exmple 1: Consider the following fully FT problem D1 D2 D Supply S1 (,16,18)(1,16,20) (0,1,2)(0,1,) (,8,10)(1,8,) (2,,)(1,,6) S2 (,11,16)(,11,18) (2,,6)(1,,8) (,7,9)(1,7,10) (,6,8)(,6,10) S (2,8,10)(1,8,) (,1,16)(1,1,18) (,9,)(,9,18) (,7,)(2,7,1) Demnd (,,6)(2,,8) (2,,7)(1,,8) (,8,)(,8,1) (9,17,2)(6,17,0) The given FT problem is blnced one since totl F demnd = totl F supply =(9,17,2)(6,17,0) Now, by solving the problems ( P ), ( P ), ( P ), ( P 2 ) nd ( P 1 ) using the zero point method / ny one of the existing crisp trnsporttion lgorithms, we obtn the following optiml solutions of the crisp problems ( P ), ( P ), ( P ), ( P 2 ) nd ( P 1 ) respectively: ( P ) : x 6; x 2; x 8; x 8, x 6 nd the minimum trnsporttion cost is 8; ( P ) : x ; x 2 ; x 6 ; x 6, x 6 nd the minimum trnsporttion cost is 208; ( P ) : x ; x 1; x ; x, x nd the minimum trnsporttion cost is 102; ( P 2 ) : x 2; x 0 ; x ; x, x 0 nd the minimum trnsporttion cost is 18 nd ( P 1 ) : x 1; x 0 ; x ; x 2 nd x 0 nd the minimum trnsporttion cost is Thus, the optiml F solution of the given FT problem is ~ x (2,,)(1,,6 ) ; ~ x (0,1,2)(0,1,2) ; ~ x (,,6)(,,8 ) ; ~ x (,,6)(2,,8) nd ~ x (0,,6)(0,,6) nd the totl minimum FT cost is ( 18,102, 208)(,102, 8)

7 Fully intuitionistic fuzzy trnsporttion problems 69 Conclusion The mn dvntge of the relistic method is tht both the optiml F solution nd the optiml oective F vlue of the fully FT problem re non-negtive F numbers Since the proposed method is bsed on the clssicl trnsporttion lgorithm so it cn be esy to compute nd to pply Since the optiml solution obtned by the proposed method does not contn negtive prt, the solution is meningful nd cn be pplied in rel life situtions Thus, the relistic method provides n pplicble optiml solution which helps the decision mkers while they re hndling rel life trnsporttion problems hving F prmeters References [1] KTAtnssov, ntuitionistic fuzzy sets, Fuzzy Sets nd Systems, 20(1986),87-96 [2] R Jhir Hussn nd P Senthil Kumr, Algorithmic pproch for solving intuitionistic fuzzy trnsporttion problem, Applied Mthemticl Sciences, 6(20), [] A Ngoor Gni nd S Abbs, Solving intuitionistic fuzzy trnsporttion problem using zero suffix lgorithm, nterntionl J of Mth Sci & Eng Appliction, 6(20), 7-82 [] A Ngoor Gni nd S Abbs, A new method for solving intuitionistic fuzzy trnsporttion problem, Applied Mthemticl Sciences, 7(201), [] P Pndin nd GNtrjn, A new method for finding n optiml solution for trnsporttion problems, nterntionl J of Mth Sci & Engg Appls, (2010), 9-6 [6] P,Pndin nd GNtrjn, An pproprite method for rel life fuzzy trnsporttion problems, nterntionl Journl of nformtion Sciences nd Appliction, 2(2010), 7-82 [7] Shshi Aggrwl nd Chvi Gupt, A novel lgorithm for solving intuitionistic fuzzy trnsporttion problem vi new rnking method, Annls of Fuzzy Mthemtics nd nformtics (in press) [8] P Senthil Kumr nd R Jhir Hussn, A systemtic pproch for solving mixed intuitionistic fuzzy trnsporttion problems, nterntionl Journl of Pure nd Applied Mthemtics, 92(201), [9] LA Zdeh, Fuzzy sets, nformtion nd Control, 8 (196), 8- Received: July 9, 201