An Algorithm to Solve Multi-Objective Assignment. Problem Using Interactive Fuzzy. Goal Programming Approach

Transcription

1 It. J. Cotemp. Math. Scieces, Vol. 6, 0, o. 34, A Algorm to Solve Multi-Objective Assigmet Problem Usig Iteractive Fuzzy Goal Programmig Approach P. K. De ad Bharti Yadav Departmet of Mathematics Natioal Istitute of Techology Silchar-78800, Assam, Idia pusde@redfmail.com Departmet of Mathematics Krisha Istitute of Egieerig ad Techology Ghaziabad- 006, UP, Idia bharti406@redfmail.com Abstract This paper proposes a algorm for solvig multi-objective assigmet problem (MOAP) through iteractive fuzzy goal programmig approach. A mathematical model has bee established to discuss about multi-objective assigmet problem which is characterized by o-liear(epoetial) membership fuctio. The approach emphasizes o optimal solutio of each objective fuctio by miimizig the worst upper boud, which is close to the best lower boud. To illustrate the algorm a umerical eample is preseted. Mathematics Subject Classicatio: 03E7, 90C9, 90B06, 90C70 Keywords: Assigmet, Multi-objective decisio maig, Fuzzy liear programmig, Goal programmig, No-liear membership fuctio (epoetial)

2 65 P. K. De ad B. Yadav. Itroductio The assigmet problem is oe of the most-studied, well-ow ad importat problem i mathematical programmig. The classical assigmet problem, that is, the umber of jobs ad the umber of the machies are equal, has bee well studied ad may algorms have bee produced to solve these type of problems. But due to the ucertaity of the real le, this problem turs ito a ucertai assigmet problem. The objective of assigmet problem is to assig a umber of jobs to a equal umber of machies so as to miimize the total assigmet cost or to miimize the total cosumed time for eecutio of all the jobs. I the multi-objective assigmet problem, the objectives aloe are cosidered as fuzzy. The classical assigmet problem refers to a special class of liear programmig problems. Liear programmig is oe of the most widely used decisio maig tool for solvig real world problems. I actual decisiomaig situatios, a major cocer is that most decisio problems ivolves multiple criteria (attributes or objectives). However, much of decisio maig i the real world taes place i a eviromet where the objectives, costraits or parameters are ot precise Liu[5].Therefore, a decisio is ofte made o the basis of vague iformatio or ucertai data. I 970, Zadeh & Bellma itroduced the cocept of fuzzy set theory ito the decisio-maig problem ivolvig ucertaity ad imprecisio. Accordig to fuzzy set theory, the fuzzy objectives ad costraits are represeted by associated membership fuctios. The Zimmerma [30] first applied suitable membership fuctios to solve liear programmig problem w several objective fuctios. He showed that solutios obtaied by fuzzy liear programmig are always efficiet. Leberlig [0] used a special type o-liear membership fuctios for the vector maimum liear programmig problem. Bit et al. [4] applied the fuzzy programmig techique w liear membership fuctio to solve the multi-objective trasportatio problem. Verma et al.[6] used the fuzzy programmig techique w some o-liear (hyperbolic & epoetial) membership fuctios to solve a multi-objective trasportatio problem. Li et al.[3] used a special type of o-liear (epoetial) membership fuctio for the multi-objective liear programmig problem. Belacela et al.[] studied a multi-criteria fuzzy assigmet problem. Geetha et al.[8] first epressed the cost-time miimizig assigmet as the multi-criteria problem. Yag et al.[9] desiged a tabu search algorm based o fuzzy simulatio to achieve a appropriate best solutio of fuzzy assigmet problem. Li et al.[4] cosidered a id of fuzzy assigmet problem ad desiged a labelig algorm for it. Wahed et al.[8] preseted a iteractive fuzzy goal programmig approach to determie the preferred compromise solutio for the multi-objective trasportatio problem w liear membership fuctio. Gao et al.[7] developed a two-phase fuzzy goal programmig techique for multiobjective trasportatio problem w liear ad o-liear parameters ad Pramai et al. [7] discussed the fuzzy goal programmig approach for multi-objective

3 Algorm to solve multi-objective assigmet problem 653 trasportatio, w crisp ad fuzzy coefficiets. Pramai et al.[8] preseted the priority based goal programmig approach for multi-objective trasportatio problem, w fuzzy parameters. Shaoyua et.al.[] proposed a satisfyig optimizatio method based o goal programmig for fuzzy multiple objective optimizatio problem. Tiwari et al.[5] formulated a additive model to solve fuzzy goal programmig. The liear iteractive ad discrete optimizatio [LINDO] [], geeral iteractive optimizer [GINO] [] ad TORA pacages [3]as well as may other commercial ad academic pacages are useful to fid the solutio of the assigmet problem. I this paper we are proposig a algorm for solvig multi-objective assigmet problem through iteractive fuzzy goal programmig approach. This paper is orgaized as follows: I sectio, mathematical model of multiobjective assigmet problem is described. Sectio 3 presets iteractive fuzzy goal programmig approach. The optimizatio algorm is provided i sectio 4. To illustrate the algorm a umerical eample is preseted i sectio 5. Sectio 6 gives few cocludig remars o the proposed algorm.. Establishmet of mathematical model of multi-objective assigmet problem Assume that there are jobs ad s. Jobs must be performed by s, where the costs deped o the specic assigmets. Each job must be assiged to oe ad oly oe ad each has to perform oe ad oly oe job. Let C be the cost the i th is assiged the j th job, the problem is to fid a assigmet (which job should be assiged to which ) so that the total cost for performig all jobs is miimum. Here mae a assumptio that j th job will be completed by i th, ad let = 0 Where is assiged is ot assiged deotes that j th job is to be assiged to the i th. The, the mathematical model of multi-objective assigmet problem is:

4 654 P. K. De ad B. Yadav Subject to i= j= =, j =,,..., (oly oe should be assiged the =, i =,,..., (oly oe job is doe by the ) ) (.) (.3) Where Z ( ) { Z ( ) Z ( ) Z ( ) } o Z ( ) ad superscript o fuctios ( =,,..., K ). =,,..., is a vector of K objective fuctios, the subscript c are used to idety the umber of objective 3. Iteractive Fuzzy goal programmig(ifgp) approach IFGP approach is the combiatio of three approaches: 3.. Iteractive approach, 3.. Fuzzy programmig approach, 3.3. Goal programmig approach, Let us briefly discuss about each approach: 3.. Iteractive approach Riguest et al.(987) ad climaco et al.(993) developed two iteractive approaches to determie the satisfactory solutio of multi-objective trasportatio problem. Sice the solutio maer is ivolved i the solutio procedure, the iteractive approaches play a importat role i derivig the best preferred compromise solutio. Iteractive approach facilitates efficiet solutio i large scale problems oce it is more effective ad suitable. 3.. Fuzzy programmig approach Fuzzy set theory is useful i solvig the iteractive multi-objective assigmet problem to improve ad stregthe the proposed solutio techiques. Ispite, this is a tool to treat the icomplete preferece iformatio of the decisio-maer. Bit et al.(99) developed a fuzzy approach to get the compromise solutio for multi-objective trasportatio problem. Also, Verma et al.(997) developed a fuzzy approach to solve multi-objective trasportatio problem w some o-liear membership fuctios. Wahed (00) developed a fuzzy approach to get the compromise solutio for multi-objective trasportatio problem.

5 Algorm to solve multi-objective assigmet problem Goal programmig approach The goal programmig approach is very useful tool for decisio-maer to discuss ad fid a set of suitable ad acceptable solutios to decisio problems. The term Goal programmig was itroduced by Chares et al.(96). Lee & Moore (973) ad Aeaida ad Kwa (994) applied goal programmig to fid a satisfactory solutio of multiobjective trasportatio problem. Goal programmig is a good decisio aid i modelig real world decisio problems which ivolves multiple objectives. Goal programmig requires the decisio-maer to set defiite aspiratio values for each objective that he wishes to achieve. The combiatio of the fuzzy set theory ad the goal programmig will defuse the coflict amog the objectives ad the aspiratio levels determiatio via goal programmig. At the same time the fuzzy set theory will tae care of ucertaity of the etracted iformatio from the decisio-maer. Thus the veture betwee two approaches is carried out through implemetatio of the aspiratio levels. Wahed ad Lee(006) combied three approaches, Iteractive approach, fuzzy programmig approach, ad goal programmig ad developed a Iteractive fuzzy goal programmig approach to determie the preferred compromise solutio for multiobjective trasportatio problem. Thus, combiatio of above three approaches produces a powerful method to solve liear multi-objective programmig problem. Now, the IFGP approach proposed by wahed ad Lee (006) is applied to solve MOAP. 4. Algorm Step : Solutio Represetatio Step.: Solve the multi-objective assigmet problem as a sigle objective assigmet problem K times by taig oe of the objectives at a time. Step.: Accordig to each solutio ad value for each objective, we ca fid a pay-off matri as follows: Z ( ) Z ( ).. Z ( X ) () X X X Z Z.. Z () Z Z.. Z : : : : : ( ) Z.. Z X X Z () () ( ) Where X, X,..., X are the isolated optimal solutios of the K dferet assigmet i problems for K dferet objective fuctios, Z = Z j ( X ),( i =,,..., K; j =,,..., K) be the i th row ad jth colum elemet of the pay-off matri.

6 656 P. K. De ad B. Yadav Step.3: From step., we fid for each objective fuctio the worst ( U ) ad the best ( L ) values correspodig to the set of solutios, where, U = Z ) = ma(z, Z,..., Z ) ad L = ( Z ) = mi(z, Z,...,Z ),,,..., K. ( ma mi = Step : Determiatio of membership fuctio (epoetial) for the th objective fuctio A epoetial membership fuctio for the th objective fuctio is defied as, Z ( ) L (4.) α( Z ) () L μ ( ( )) ep Z =, L < Z ( ) < U, L U 0, Z ( ) U ad α where α is a o - zero parameter, prescribed by the decisio maer. Step 3: Mathematical model structure By usig the epoetial membership fuctio as defied i (4.), ad followig the fuzzy decisio of Bellma ad Zadeh (970), the equivalet o-liear programmig model is: S : Ma λ (4.) Subject to α ( Z - L ) (4.3) λ ep, =,,..., K, L U =, j,,..., ; (4.4) i= = =, i =,,..., ; β 0 j = = 0 is assiged is ot assiged (4.5) (4.6) The above problem ca be trasformed ito the followig liear programmig model by substitutig β = lλ (4.7) Now we have: S3: Mi β (4.8) Subject to

7 Algorm to solve multi-objective assigmet problem 657 β i= j= α ( L Z L U ), =, j =,,..., ; = 0 =, i =,,..., ; =,,..., K, β 0 is assiged is ot assiged (4.9) (4.0) (4.) (4.) Step 4: Formulatio of goal programmig model To formulate model (S3) as a goal programmig model (Saawa,993), let us itroduce the followig positive ad egative deviatioal variables: + Z ( ) d + d = G, =,,..., K (4.3) Where G is the aspiratio level of the objective fuctio K. Usig the IFGP approach preseted by Wahed ad Lee (006), model (S3) ca be formulated as a mied iteger goal programmig as follows: S4 : Mi β (4.4) Subject to β i= j= α( L Z ), =,,..., K, L U =, j =,,..., ; =, i =,,..., ; + Z ( ) d + d = G = 0 +, d, d 0,, are itegers i, j. is assiged is ot assiged i =,,..., ; j =,,..., ; =.,..., K; 0 β Step 5: Determiatio of aspiratio level (4.5) (4.6) (4.7) (4.8) (4.9) (4.0)

8 658 P. K. De ad B. Yadav From step.3 we ow that ( Z ) mi Z ( Z ) ma. For the MOAP, we should get the optimal compromise solutio that is ear to the ideal solutio. This ca be obtaied by settig the aspiratio levels i model (S4) equal to the upper bouds ( Z ) ma, =,,..., K. let us solve model (S4) based o the above described algorm ad the correspodig solutio vector be X. If the decisio maer accepts this solutio, the go to step (6). Otherwise, model (S4) is modied as follows: Let Z, Z,..., Z are the objective fuctios vectors correspodig to the solutio vector X. Now compare each Z w the eistig upper boud U, =,,..., K. Now aspiratio level ca be updated by the followig steps: 5.. If Z < U (meas ew value of the objective fuctio is less tha the upper boud, cosider this as a ew upper boud, replace U by Z.Repeat this process K times ad go to step. 5.. If Z = U, o chage i aspiratio level ad algorm termiates, the go to et step. Step 6: Ed. 5. Implemetatio of the model I this sectio, we use umerical eample to illustrate the formulatio ad solutio procedure of the MOAP. For this model the proposed algorm gives the best optimal solutio. Numerical Eample Mi Z = Mi Z subject to 3 i= =, j =,,3; = 0 = j = =, i =,,3. is assiged is ot assiged As the first step, the solutio of each sigle objective assigmet problem is:

9 Algorm to solve multi-objective assigmet problem 659 X = (0,,0,0,0,,,0,0), X = (0,0,,,0,0,0,,0), The objective fuctio values are: Z ( X ) = 9, Z( X ) = 38, Z( X ) = 4, Z( X ) = 8. We ca write the pay-off matri as Z ( X ) Z ( X ) () X 9 4 () X 38 8 From the pay-off matri, the upper ad lower bouds of each objective fuctio ca be writte as follows: 9 Z 38 ad 8 Z developed as follows: Mi β subject to = β, d =, , d =, =, =, =, =, , d , d is assiged is ot assiged Now, model S3 w the parameterα =, is d d d + d = 38, + = 4, 0.6β, 0.5β, The problem is solved by usig the TORA pacage. The optimal solutio is preseted as follows: =, =, =, d = 8, d = 5, Therefore Z = 30 ad Z 4. = 37 w β =.8 ad λ =.8.

10 660 P. K. De ad B. Yadav Assume decisio-maer eeds more improvemet, the, go to step (5.) The ew upper ad lower bouds are: 9 Z 30 ad 8 Z 37. Accordig to algorm, the ew aspiratio levels of the two objective fuctios are 30 ad 37, respectively. Thus, the ew upper bouds i model S3 will be 30 & 37, respectively ad repeat step to step 6. Based o these modicatios, the above mathematical model is updated ad resolved. The, we fid that the ew upper bouds are equivalet to the earlier upper bouds so the algorm termiates. As the algorm termiates, the above solutio is accepted by the decisio maer ad validates as best optimal solutio. The set of solutios ( Z, Z ) ca be summarized i the followig way :( 38, 4), (30, 37), (9, 8) respectively. 6. Coclusios I this paper, we maily studied a fuzzy multi-objective assigmet problem. As a result, we used a iteractive fuzzy goal programmig (IFGP) approach to deal w it. The combiatio of goal programmig, fuzzy programmig ad iteractive programmig is a powerful method for solvig MOAP. I order to obtai the best solutio, a algorm w o-liear membership fuctio (epoetial) is desiged. The IFGP approach has the followig features:. The approach provides a optimal compromise solutio by updatig both upper bouds ad aspiratio level of each objective fuctio.. This is a powerful approach to obtai a appropriate aspiratio level of the objective fuctios. 3. The approach solves all types of MOAP, the vector miimum problem ad the vector maimum problem. 4. It is easy ad simple to use for the decisio- maer ad ca be easily implemeted to solve similar liear multi-objective programmig problems. 5. The approach solves a series of classical assigmet problem ad a liear iteger programmig problem, which ca be solved by ay available software. 6. This approach w epoetial membership fuctio is a suitable represetatio i may practical situatios. This feature maes this approach more practical tha the approach usig liear membership fuctio i solvig MOAP. Refereces. R.J.Aeaida,N.W.Kwa, A liear goal programmig for trasshipmet problems w fleible supply ad demad costraits, Joural of Operatioal Research Society, (994),5-4.

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