Assignment Problems with fuzzy costs using Ones Assignment Method

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1 IOSR Joural of Mathematics (IOSR-JM) e-issn: 8-8, p-issn: 9-6. Volume, Issue Ver. V (Sep. - Oct.06), PP Assigmet Problems with fuzzy costs usig Oes Assigmet Method S.Vimala, S.Krisha Prabha Assistat Professor, Departmet of Mathematics, Mother Teresa ome s Uiversity, Kodaikaal,Tamiladu Ph.D Scholar, Mother Teresa ome s Uiversity, Kodaikaal. Assistat Professor, Departmet of Mathematics, PSNA CET, Didigul, Tamiladu Abstract: I this paper we itroduce a ew approach to solve fuzzy assigmet problem amely, oes assigmet method. Cosiderig each fuzzy cost as a triagular fuzzy umbers the fuzzy assigmet problem has bee trasformed ito crisp values by usig liguistic variables ad rakig techiques ad the it is solved by oes assigmet method. The proposed method is a systematic procedure, easy to apply ad ca be utilized for all types of assigmet problem with maximize or miimize objective fuctios. At the ed, this method is illustrated with some umerical example Key words: rakig of fuzzy umbers, triagular fuzzy umbers, Fuzzy Assigmet problem, oes method Mathematics Subject Classificatio: 90C08, 90C0, 0E, I. Itroductio A assigmet problem (AP) is a particular type of trasportatio problem where tasks (jobs) are to be assiged to a equal umber of machies (workers) i oe to oe basis such that the assigmet cost (or profit) is miimum (or maximum).hece, it ca be cosidered as a balaced trasportatio problem i which all supplies ad demads are equal, ad the umber of rows ad colums i the matrix are idetical. Various rakig procedures have bee developed sice 9 where the theories of fuzzy sets first itroduced by adeh []. Rakig fuzzy umbers proposed by Jai [] for decisio makig i fuzzy situatios by represetig the ill-defied quatity as a fuzzy set. Sice the, various procedures to rak fuzzy quatities are proposed by various researchers. Bortola ad Degai [4] reviewed some of these rakig methods for rakig fuzzy subsets. Che [6] preseted rakig fuzzy umbers with maximizig set ad miimizig set. Dubois ad Prade preseted the mea value of fuzzy umber. Lee ad Li [6] preseted a compariso of fuzzy umbers based o the probability measure of fuzzy evets. Che ad Che [] derived a ew method o rakig geeralized trapezoidal fuzzy umbers based o cetroid poit ad stadard deviatios. Till date, several researchers studied extesively to solve fuzzy assigmet problems i various ways. The triagular fuzzy umbers are defuzzified by usig cetriod rakig formula.i sectio some elemetary cocepts ad operatios of fuzzy set theory have bee reviewed. I sectio, correspodig algorithms have bee proposed for Fuzzy assigmet problems. I sectio 4 the proposed method is illustrated by a umerical example II. Basic defiitios.fuzzy set: A fuzzy set is characterized by a membership fuctio mappig elemet of a domai, space or uiverse of discourse to the uit iterval [0, ] i.e. A = {(x, μ A (x); x }, Here μ A : [0,] is a mappig called the degree of membership fuctio of the fuzzy set A ad μ A (x) is called the membership value of x ε i the fuzzy set A. These membership grades are ofte represeted by real umbers ragig from [0,]..Fuzzy Number A real umber a is a fuzzy subset of the real umber R with membership fuctio μ a satisfyig the followig coditios, μ a is cotiuous from R to the closed iterval [0,] μ a is strictly icreasig ad cotiuous o [a,a ] μ a is strictly decreasig ad cotiuous o [a,a 4 ].. Triagular Fuzzy Number The fuzzy umbera is a triagular fuzzy umbers, deoted by a = (a,a,a ) its membership fuctio μ a is give below the figure. DOI: 0.990/ Page

2 F(R) represets the set all trapezoidal fuzzy umbers It R be ay rakig fuctio the, R (a) = (a +a +a ) / Let a = [a, a, a,] ad b = [b, b, b ] be two triagular fuzzy umbers the the arithmetic operatios o a ad b as follows..4 Properties of Triagular Fuzzy Number Additio: a + b = (a +b, a +b, +b ) Subtractio: a b = (a b, a b, a b ) Multiplicatio: a. b = a +b +b ), a +b +b ), a +b +b ) if R (a) >0 a. b = a +b +b ), a +b +b ), a +b +b ) if R (a) < 0. Liguistic Variable I more specific terms,a liguistic variable is characterized by a quituple (ν, T(ν),U,G,M) i which ν is the ame of the variable; T(ν) the term-set of ν, that is, the collectio of its liguistic values; U is a uiverse of discourse; G is a sytactic rule which geerates the terms i T(ν) ad M is a sematic rule which associates with each liguistic value its meaig, M(), where M() deotes a fuzzy subset of U. The meaig of a liguistic value is characterized by a compatibility fuctio, C: U [0, I], which associates with each u i U its compatibility with III. Assigmet Method The assigmet problem ca be stated i the form of x cost matrix [c ij ] of real umbers as give i the followig table Jobs Persos ---j--- C C C -- C j -- C C C C -- C j -- C I C i C i C i -- C ij -- C i - N C C C -- C j -- C Mathematically assigmet problem ca be stated as Miimize z = c i= j = ij x ij where i =,,., j =,,. Subject to i= x ij =, i =,,...() j = x ij = j =,,. x ij 0, where x ij = if the ith perso is assiged the j th job 0 otherwise is the decisio variable deotig the assigmet of the perso i to job j, Cij is the cost of assigig the j th job to the i th perso. The objective is to miimize the total cost of assigig all the jobs to the available persos. (Oe job to oe perso). he the costs cij are fuzzy umbers, the the fuzzy assigmet problem becomes DOI: 0.990/ Page

3 z = i= j = (cij ) x ij.() Subject to the same coditios () Assigmet Problems with fuzzy costs usig Oes Assigmet Method. Algorithm for the proposed method : ( Balaced /Ubalaced assigmet problem ) If the umber of rows is ot equal to the umber of colums the the problem is termed as ubalaced assigmet problem the this problem ito chage balaced assigmet problem as follows ecessary umber of dummy row (s) / colum(s) are added such that the cost matrix is a square matrix the values for the etries i the dummy row (s) / colum(s) are assumed to be zero. By defuzzifyig the fuzzy cost coefficiets ito crisp coefficiets by cetriod rakig formula we ca solve. This problem is obviously the crisp assigmet problem of the form () which ca be solved by Oes Assigmet Method. step. I a miimizatio (maximizatio) case, fid the miimum (maximum) elemet of each row i the assigmet matrix (say ai) ad write it o the right had side of the matrix. The divide each elemet of ith row of the matrix by ai. These operatios create at least oe oes i each rows. I term of oes for each row ad colum do assigmet, otherwise go to step. step. Fid the miimum (maximum) elemet of each colum i assigmet matrix (say bj ), ad write it below jth colum. The divide each elemet of jth colum of the matrix by bj.these operatios create at least oe oes i each colums. Make assigmet i terms of oes.if o feasible assigmet ca be achieved from step () ad () the go to step. Note: I a maximizatio case, the ed of step we have a fuzzy matrix,which all elemets are belog to [0, ], ad the greatest elemet is oe [4]. step. Draw the miimum umber of lies to cover all the oes of the matrix. If the umber of draw lies less tha, the the complete assigmet is ot possible, while if the umber of lies is exactly equal to, the the complete assigmet is obtaied step 4. If a complete assigmet program is ot possible i step, the select the smallest ( largest) elemet (say dij) out of those which do ot lie o ay of the lies i the above matrix. The divide by dij each elemet of the ucovered rows or colums, which dij lies o it. This operatio create some ew oes to this row or colum. If still a complete optimal assigmet is ot achieved i this ew matrix, the use step 4 ad iteratively. By repeatig the same procedure the optimal assigmet will be obtaied. Priority, plays a importat role i this method, he we wat to assig the oes. Priority rule. For maximizatio (miimizatio) assigmet problem, assig the oes o the rows which have greatest (smallest) elemet o the right had side, respectively. IV. Numerical example Cosider the followig fuzzy assigmet problem. Assig the four jobs to the four machies so as to miimize the total cost. 4 (9,0,) (4,,6) (,,4) (4,,6) (,,4) (8,9,0) (,8,9) (,,4) (9,0,) (6,,8) ((,,4) (,,) (4,,6) (0,,) (8,9,0) (6,,8) Defuzzyifyig usig the give rakig method. e have Fid the miimum (maximum) elemet of each row i the assigmet matrix (say ai) ad write it o the right had side of the matrix. The divide each elemet of ith row of the matrix by ai. DOI: 0.990/ Page

4 4 6 9 Fid the miimum( maximum) elemet of each colum i assigmet matrix (say bj ), ad write it below jth colum The divide each elemet of jth colum of the matrix by bj.these operatios create at least oe oes i each colums. Make assigmet i terms of oes. 4 () () 6 () () 8 The optimal assigmet schedule is, 4,,, the miimum cost is 6. V. Coclusios I this paper, the assigmet costs are cosidered as liguistic variables represeted by fuzzy umbers. Thus the fuzzy assigmet problem has bee trasformed ito crisp balaced ad ubalaced assigmet problem usig rakig method. Here we have show that the fuzzy assigmet problems of qualitative ature ca be solved i a effective way by oes assigmet method. This techique ca also be tried i solvig the problems like balaced ad ubalaced Trasportatio problems, Trasshipmet, project schedulig problems, etwork flow problems etc. This research work is uder process for hexagoal ad octagoal fuzzy umbers. Refereces []. Adamo J. M., Fuzzy decisio trees, Fuzzy Sets ad Systems, vol. 4, o., pp. 0 9, 980. []. Baas ad H. K. wakeraak, Ratig ad rakig of multiple-aspect alteratives usig fuzzy sets, Automatica, vol., o., pp. 4 8, 9. []. Baldwi J. F. ad Guild N. C. F., Compariso of fuzzy sets o the same decisio space, Fuzzy Sets ad Systems, vol., o., pp., 99. [4]. Bortola.G. ad Degai, R. A review of some methods for rakig fuzzy subsets, Fuzzy Sets ad Systems, vol., o.,pp. 9, 98. []. Chag., Rakig of fuzzy utilities with triagular membership fuctios, Proceedigs of Iteratioal Coferece o Policy Aalysis ad Systems, pp. 6, 98. [6]. Che S.H, Rakig fuzzy umbers with maximizig set ad miimizig set, Fuzzy Sets ad Systems, vol., o., pp. 9, 98. DOI: 0.990/ Page

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