A Method for Solving Balanced Intuitionistic Fuzzy Assignment Problem

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1 P. Sethil Kumar et al t. Joural of Egieerig Research ad Applicatios SSN : , Vol. 4, ssue 3( Versio 1), March 2014, pp RESEARCH ARTCLE OPEN ACCESS A Method for Solvig Balaced tuitioistic Fuzzy Assigmet Problem P. Sethil Kumar *, R. Jahir Hussai # * Research Scholar, # Associate Professor, PG ad Research Departmet of Mathematics, Jamal Mohamed College, Tiruchirappalli dia. Abstract this paper, we ivestigate a assigmet problem i which cost coefficiets are triagular ituitioistic fuzzy umbers. covetioal assigmet problem, cost is always certai. This paper develops a approach to solve a ituitioistic fuzzy assigmet problem where cost is ot determiistic umbers but imprecise oes. Here, the elemets of the costs (profits) matrix of the assigmet problem are triagular ituitioistic fuzzy umbers. The its triagular shaped membership ad o-membership fuctios are defied. A ew rakig procedure which ca be foud i [4] ad is used to compare the ituitioistic fuzzy umbers so that a tuitioistic Fuzzy Hugaria method may be applied to solve the ituitioistic fuzzy assigmet problem. Numerical examples show that a ituitioistic fuzzy rakig method offers a effective tool for hadlig a ituitioistic fuzzy assigmet problem. Keywords: tuitioistic Fuzzy Set, Triagular Fuzzy Number, Triagular tuitioistic Fuzzy Number, tuitioistic Fuzzy Assigmet Problem, Optimal Solutio.. troductio Assigmet Problem(AP) is used worldwide i solvig real world problems. A assigmet problem plays a importat role i a assigig of persos to jobs, or classes to rooms, operators to machies, drivers to trucks, trucks to routes, or problems to research teams, etc. The assigmet problem is a special type of liear programmig problem (LPP) i which our objective is to assig umber of jobs to umber of machies (persos) at a miimum cost. To fid solutio to assigmet problems, various algorithm such as liear programmig [8,9,13,17], Hugaria algorithm [15], eural etwork [12], geetic algorithm [6] have bee developed. However, i real life situatios, the parameters of assigmet problem are imprecise umbers istead of fixed real umbers because time/cost for doig a job by a facility (machie/persio) might vary due to differet reasos. The theory of fuzzy set itroduced by Zadeh[21] i 1965 has achieved successful applicatios i various fields. 1970, Belma ad Zadeh itroduce the cocepts of fuzzy set theory ito the decisiomakig problems ivolvig ucertaity ad imprecisio. Amit Kumar et al ivestigated Assigmet ad Travellig Salesma Problems with cost coefficiets as LR fuzzy parameters[1], Fuzzy liear programmig approach for solvig fuzzy trasportatio problems with trasshipmet[2], Method for solvig fully fuzzy assigmet problems usig triagular fuzzy umbers[3]. [18], Sathi Mukherjee et al preseted a Applicatio of fuzzy rakig method for solvig assigmet problems with fuzzy costs. Li ad We [16] proposed a efficiet algorithm based a labelig method for solvig the liear fractioal programmig case. Y.L.P.Thorai ad N.Ravi Sakar did Fuzzy assigmet problem with geeralized fuzzy umbers [19].Differet kids of fuzzy assigmet problems are solved i the papers [1, 3, 10, 11, 12, 20]. The cocept of tuitioistic Fuzzy Sets (FSs) proposed by Ataassov[5] i 1986 is foud to be highly useful to deal with vagueess. [14], Jahir Hussia et all preseted A Optimal More-for-Less Solutio of Mixed Costrais tuitioistic Fuzzy Trasportatio Problems. Here we ivestigate a more realistic problem, amely ituitioistic fuzzy assigmet problem. Let cij be the ituitioistic fuzzy cost of assigig the j th job to the i th machie. We assume that oe machie ca be assiged exactly oe job; also each machie ca do at most oe job. The problem is to fid a optimal assigmet so that the total ituitioistic fuzzy cost of performig all jobs is miimum or the total ituitioistic fuzzy profit is maximum. this paper, rakig procedure of Aie Varghese ad Suy Kuriakose [4] is used to compare the ituitioistic fuzzy umbers. Fially a tuitioistic Fuzzy Hugaria method may be applied to solve a FAP. This paper is orgaized as follows: Sectio 2 deals with some basic termiology ad rakig of triagular ituitioistic fuzzy umbers, sectio 3, provides ot oly the defiitio of ituitioistic fuzzy assigmet problem but also its mathematical formulatio ad Fudametal Theorems of a tuitioistic Fuzzy 897 P a g e

2 P. Sethil Kumar et al t. Joural of Egieerig Research ad Applicatios SSN : , Vol. 4, ssue 3( Versio 1), March 2014, pp Assigmet Problem. Sectio 4 describes the solutio procedure of a ituitioistic fuzzy assigmet problem, sectio 5, to illustrate the proposed method a umerical example with results ad discussio is discussed ad followed by the coclusios are give i Sectio 6.. Prelimiaries Defiitio 2.1 Let A be a classical set, μ A (x) be a fuctio from A to [0,1]. A fuzzy set A with the membership fuctio μ A (x) is defied by A = x, μ A x ; x A ad μ A x 0,1. Defiitio 2.2 Let X be deote a uiverse of discourse, the a ituitioistic fuzzy set A i X is give by a set of ordered triples, A = {< x, μ A x, θ A x >; x X} Whereμ A, θ A : X [0,1], are fuctios such that 0 μ A x + θ A x 1, x X. For each x the membership μ A x ad θ A x represet the degree of membership ad the degree of o membership of the elemet x X to A X respectively. Defiitio 2.3 A fuzzy umber A is defied to be a triagular fuzzy umber if its membership fuctios μ A :R [0, 1] is equal to x a 1 if x [a a 2 a 1, a 2 ] 1 μ A x = a 3 x if x a a 3 a 2, a oterwise where a 1 a 2 a 3. This fuzzy umber is deoted by (a 1, a 2, a 3 ). Defiitio 2.4 A Triagular tuitioistic Fuzzy Number (A is a ituitioistic fuzzy set i R with the followig membership fuctio μ A x ad o membership fuctio θ A x : ) μ A x = 0 for x < a 1 x a 1 a 2 a 1 for a 1 x a 2 1 for x = a 2 a 3 x a 3 a 2 for a 2 x a 3 0 for x > a 3 θ A x = 1 for x < a 1 a 2 x a 2 a 1 for a 1 x a 2 0 for x = a 2 x a 2 a 3 a 2 for a 2 x a 3 1 for x > a 3 Where a 1 a 1 a 2 a 3 a 3 ad μ A x, θ A x 0.5 for μ A x = θ A x x R. This TrFN is deoted by A = a 1, a 2, a 3 ( a 1, a 2, a 3 ) Particular Cases Let A = a 1, a 2, a 3 ( a 1, a 2, a 3 ) be a TrFN. The the followig cases arise Case 1: f a 1 = a 1, a 3 = a 3, the A represet Trigular Fuzzy Number(TFN). t is deoted by A = a 1, a 2, a 3. Case 2: f a 1 = a 1 = a 2 = a 3 = a 3 = m, the A represet a real umber m. Defiitio 2.5 Let A ad B be two TrFNs. The rakig of A ad B by the (.) o E, the set of TrFNs is defied as follows: i. (A )> (B ) iff A B ii. (A )< (B ) iff A B iii. (A )= (B ) iff A B iv. (A + B )= (A )+ (B ) v. (A B ) = A (B ) Arithmetic Operatios 898 P a g e

3 P. Sethil Kumar et al t. Joural of Egieerig Research ad Applicatios SSN : , Vol. 4, ssue 3( Versio 1), March 2014, pp Let A = (a 1, a 2, a 3 )( a 1, a 2, a 3 ) ad B = (b 1, b 2, b 3 )( b 1, b 2, b 3 )be ay two operatios as follows: Additio: A B = a 1 + b 1, a 2 + b 2, a 3 + b 3 (a 1 + b 1, a 2 + b 2, a 3 +b 3 ) Subtractio: A B = a 1 b 3, a 2 b 2, a 3 b 1 (a 1 b 3, a 2 b 2, a 3 b 1 ) TrFNs the the arithmetic Rakig of triagular ituitioistic fuzzy umbers The Rakig of a triagular ituitioistic fuzzy umber A = (a 1, a 2, a 3 )( a 1, a 2, a 3 ) is defied by [4] R A = 1 (a 3 a 1 ) a 2 2a 3 2a 1 + a 3 a 1 a 1 + a 2 + a 3 + 3(a 2 3 a 2 1 ) 3 a 3 a 1 + a 3 a 1 The rakig techique [4] is: f (A ) (B ), the A B i.e., mi {A, B } = A Example: Let A = 8,10,12 (6,10,14) ad B = 3,5,8 (1,5,10) be ay two TrFN, the its rak is defied by A = 10, (B ) = 5.33 this implies A B. tuitioistic Fuzzy Assigmet Problem Cosider the situatio of assigig machies to jobs ad each machie is capable of doig ay job at differet costs. Let cij be a ituitioistic fuzzy cost of assigig the j th job to the i th machie. Let x ij be the decisio variable deotig the assigmet of the machie i to the job j. The objective is to miimize the total ituitioistic fuzzy cost of assigig all the jobs to the available machies (oe machie per job) at the least total cost. This situatio is kow as balaced ituitioistic fuzzy assigmet problem. (FAP)Miimize Z = i=i j =1 cij x ij Subject to, x ij = 1, for i = 1,2,, j =1 m i=1 x ij = 1, for j = 1,2,, (1) x ij 0,1 cij = (c 1 ij, c 2 ij, c 3 1 ij )(c ij, c 2 3 ij, c ij ) Were x ij = 1, if the ith machie is assiged to j th job 0, if i th machie is ot assiged to j th job 3.1 Fudametal Theorems of a tuitioistic Fuzzy Assigmet Problem The solutio of a ituitioistic fuzzy assigmet problem is fudametally based o the followig two theorems. Theorem 1: a ituitioistic fuzzy assigmet problem, if we add or subtract a ituitioistic fuzzy umber to every elemet of ay row(or colum) of the ituitioistic fuzzy cost matrix [cij ], the a assigmet that miimizes the total ituitioistic fuzzy cost o oe matrix also miimizes the total ituitioistic fuzzy cost o the other matrix. other words if x ij = x ij miimizes, Z = cij i=i j =1 x ij with i=1 x ij = 1, j =1 x ij = 1, x ij = 0 or 1 the x ij also miimizes Z = i=i j =1 cij x ij, where cij = cij u i v j for all i,j=1,2, ad u i, v j are some real triagular ituitioistic ituitioistic fuzzy umbers. i=i j =1 cij x ij = i=i j =1(cij = j =1 cij Proof: Now, Z = u i v j )x ij i=i x ij i=i u i j =1 x ij i=i x ij j =1 v j = Z i=i u i j =1 v j Sice i=1 x ij = j =1 x ij = 1 This shows that the miimizatio of the ew objective fuctio Z yields the same solutio as the miimizatio of origial objective fuctio Z because i=i u i ad j =1 v j are idepedet of x ij 899 P a g e

4 P. Sethil Kumar et al t. Joural of Egieerig Research ad Applicatios SSN : , Vol. 4, ssue 3( Versio 1), March 2014, pp Theorem 2: a ituitioistic fuzzy assigmet problem with cost [cij ], if all [cij ] 0 the a feasible solutio x ij which satisfies cij x ij = 0, is optimal for the problem. i=i j =1 Proof: Sice all [cij ] 0 ad all [x ij ] 0, The objective fuctio Z = i=i Z ca attai 0. Thus, ay feasible solutio [x ij ] that satisfies j =1 cij i=i x ij ca ot be egative. The miimum possible value that j =1 cij x ij = 0, will be a optimal. V. The Computatioal Procedure for tuitioistic Fuzzy Assigmet Problem. Step 1. the give ituitioistic fuzzy cost matrix, subtract the smallest elemet i each row from every elemet of that row by usig rakig procedure as metioed i sectio. Step 2. the reduced ituitioistic fuzzy cost matrix, subtract the smallest elemet i each colum from every elemet of that colum by usig rakig procedure as metioed i sectio. Step 3. Make the assigmet for the reduced ituitioistic fuzzy cost matrix obtaied from Step 2 i the followig way: i. Examie the rows successively util a row with exactly oe umarked ituitioistic fuzzy zero is foud. Eclose this ituitioistic fuzzy zero i a box ( ) as a assigmet will be made there ad cross ( ) all other ituitioistic fuzzy zeros appearig i the correspodig colum as they will ot be cosidered for further assigmet. Proceed i this way util all the rows have bee examied. ii. After examiig all the rows completely, examie the colums successively util a colum with exactly oe umarked ituitioistic fuzzy zero is foud. Make a assigmet to this sigle ituitioistic fuzzy zero by puttig a box ( ) ad cross out ( ) all other ituitioistic fuzzy zeros i the correspodig row. Proceed i this way util all colums have bee examied. iii. Repeat the operatio (i) ad (ii) util all the ituitioistic fuzzy zeros are either marked ( ) or crossed ( ). Step 4. f there is exactly oe assigmet i each row ad i each colum the the optimum assigmet policy for the give problem is obtaied. Otherwise go to Step-5. Step 5. Draw miimum umber of vertical ad horizotal lies ecessary to cover all the ituitioistic fuzzy zeros i the reduced ituitioistic fuzzy cost matrix obtaied from Step-3 by ispectio or by adoptig the followig procedure i. Mark ( ) all rows that do ot have assigmet ii. Mark ( ) all colums (ot already marked) which have ituitioistic fuzzy zeros i the marked rows iii. Mark ( ) all rows (ot already marked) that have assigmets i marked colums, iv. Repeat steps 5(ii) ad 5(iii) util o more rows or colums ca be marked. v. Draw straight lies through all umarked rows ad marked colums. Step 6. Select the smallest elemet amog all the ucovered elemets. Subtract this least elemet from all the ucovered elemets ad add it to the elemet which lies at the itersectio of ay two lies. Thus, we obtai the modified matrix. Go to Step 3 ad repeat the procedure. V. Numerical Examples: Example: Let us cosider a ituitioistic fuzzy assigmet problem with rows represetig 3 machies M 1, M 2, M 3 ad colums represetig the 3 jobs J 1, J 2, J 3. The cost matrix [c ] is give whose elemets are TrFN. The problem is to fid the optimal assigmet so that the total cost of job assigmet becomes miimum M 1 (7,21,29)(2,21,34) (7,20,57)(3,20,61) ( 12,25,56)(8,25,60) M 2 ( 8,9,16)(2,9,22) (4,12,35)(1,12,38) (6,14,28)(3,14,31) M 3 (5,9,22)(2,9,25) (10,15,20)(5,15,25) (4,16,19)(1,16,22) Solutio: The above ituitioistic fuzzy assigmet problem ca be formulated i the followig mathematical programmig form 900 P a g e

5 P. Sethil Kumar et al t. Joural of Egieerig Research ad Applicatios SSN : , Vol. 4, ssue 3( Versio 1), March 2014, pp Mi[(7,21,29)(2,21,34)x 11 +(7,20,57)(3,20,61) x 12 +(12,25,56)(8,25,60) x 13 +(8,9,16)(2,9,22) x 21 +(4,12,35)(1,12,38) x 22 +(6,14,28)(3,14,31) x 23 +(5,9,22)(2,9,25) x 31 +(10,15,20)(5,15,25 ) x 32 +(4,16,19)(1,16,22) x 33 Subject to x 11 + x 12 + x 13 = 1, x 11 + x 21 + x 31 = 1, x 21 + x 22 + x 23 = 1, x 12 + x 22 + x 32 = 1, x 31 + x 32 + x 33 = 1, x 13 + x 23 + x 33 = 1, x ij 0,1. Now, usig the Step 1 of the ituitioistic fuzzy Hugaria assigmet method, we have the followig reduced ituitioistic fuzzy assigmet table. M 1 (-22,0,22)(-32,0,32) (-22,-1,50)(-31,-1,59) (-17,4,49)(-26,4,58) M 2 (-8,0,8)(-20,0,20) (-12,3,27)(-21,3,36) (-10,5,20)(-19,5,29) M 3 (-17,0,17)(-23,0,23) (-12,6, 15)(-20,6,23) (-18,7,14)(-24,7,20) Now, usig the Step 2 of the ituitioistic fuzzy Hugaria assigmet method, we have the followig reduced ituitioistic fuzzy assigmet table. M 1 (-22,0,22)(-32,0,32) (-37,-7,62)(-54,-7,79) (-31,-3,67)(-46,-3,82) M 2 (-8,0,8)(-20,0,20) (-27,-3,39)(-44,-3,56) (-24,-2,38)(-39,-2,53) M 3 (-17,0,17)(-23,0,23) (-27,0,27)(-43,0,43) (-32,0,32)(-44,0,44) Now, usig the Step 3 to the Step 6 of the ituitioistic fuzzy Hugaria assigmet method, we have the followig reduced ituitioistic fuzzy assigmet table. M 1 (-22,0,22)(-32,0,32) (-76,-4,89)(-110,-4,123) (-70,0,94)(-102,0,126) M 2 (-8,0,8)(-20,0,20) (-66,0,66)(-100,0,100) (-63,1,65)(-95,1,97) M 3 (-27,-3,39)(-44,-3,56) (-27,0,27)(-43,0,43) (-32,0,32)(-44,0,44) The optimal solutio is x 11 = x 22 = x 33 = 1, x 12 = x 13 = x 21 = x 23 = x 31 = x 32 = 0, With the optimal objective value Z = 49 which represets the optimal total cost. other words the optimal assigmet is M 1 J 1, M 2 J 2, M 3 J 3 The ituitioistic fuzzy miimum total cost is calculated as c11 + c22 + c33 = (7,21,29)(2,21,34) + 4,12,35 1,12,38 + (4,16,19)(1,16,22)= 15,49,83 4,49,94 Also we fid that Z = 15,49,83 4,49,94 = Rs. 49 the above example it has bee show that the total optimal cost obtaied by our method remais same as that obtaied by covertig the total ituitioistic fuzzy cost by applyig the rakig method [4]. 901 P a g e

6 P. Sethil Kumar et al t. Joural of Egieerig Research ad Applicatios SSN : , Vol. 4, ssue 3( Versio 1), March 2014, pp Results ad discussio: The miimum total ituitioistic fuzzy assigmet cost is Z = (15, 49, 83) (4, 49, 94) (1) Figure1 Graphical Represetatio of FAC The result i (1) ca be explaied (Refer to figure1) as follows: (a) Assigmet cost lies i [15, 83]. (b) 100% expect are i favour that a assigmet cost is 49 as μ Z x = 1,x = 49. (c) Assumig that m is a membership value ad is a o-membership value at c. The 100m% experts are i favour ad 100% experts are opposig but m % are i cofusio that a assigmet cost is c. Values of μ Z c ad θ Z c at differet values of c ca be determied usig equatios give below. μ Z x = 0 for x < 15 x 15 for 15 x for x = x for 49 x for x > 83 θ Z x = 1 for x < 4 49 x 45 for 4 x 49 0 for x = 49 x for 49 x 94 1 for x > 94 V.Coclusio this paper, we discussed fidig a solutio of a assigmet problem i which cost coefficiets are triagular ituitioistic fuzzy umbers. The total optimal cost obtaied by our method remais same as that obtaied by covertig the total ituitioistic fuzzy cost by applyig the rakig method [4].Also the membership ad o-membership values of the ituitioistic fuzzy costs are derived. This techique ca also be used i solvig other types of problems like, project schedules, trasportatio problems ad etwork flow problems. Refereces [1] Amit Kumar ad Aila Gupta, Assigmet ad Travellig Salesma Problems with Coefficiets as LR Fuzzy Parameters, teratioal Joural of Applied Sciece ad Egieerig ,3: [2] Amit Kumar, Amarpreet Kaur, Aila Gupta, Fuzzy Liear Programmig Approach for Solvig Fuzzy Trasportatio probles with Trasshipmet, J Math Model Algor (2011) 10: [3] Amit Kumar, Aila Gupta ad Amarpreet Kumar,Method for Solvig Fully Fuzzy Assigmet Problems Usig Triagular Fuzzy Numbers, teratioal Joural of Computer ad formatio Egieerig 3: [4] Aie Varghese ad Suy Kuriakose, Notes o tuitioistic Fuzzy Sets Vol.18, 2012, No.1, [5] K.T.Ataassov, tuitioistic fuzzy sets, fuzzy sets ad systems, Vol.20, o.1.pp.87-96, [6] D.Avis, L.Devroye, A aalysis of a decompositio heuristic for the assigmet problem, Oper.Res.Lett., 3(6) (1985), P a g e

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