A General Approach for Solving Assignment Problems Involving with Fuzzy Cost Coefficients

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1 Moder Applied Siee Vol. 6, No. 3; Marh 202 A Geeral Approah for Solvig Assigmet Problems Ivolvig with Fuzzy ost oeffiiets P. K. De Departmet of Mathematis, Natioal Istitute of Tehology, Idia pusde@rediffmail.om Bharti Yadav Departmet of Mathematis, Krisha Istitute of Egieerig ad Tehology, Idia bharti406@rediffmail.om Reeived: Deember 6, 20 Aepted: Jauary 7, 202 Published: Marh, 202 doi:0.5539/mas.v63p2 URL: Abstrat Assigmet problem is oe of the most-studied, well kow ad importat problems i mathematial programmig. I this paper two differet type of assigmet problems are disussed: ovetioal ad fuzzy assigmet problem. I ovetioal assigmet problem, ost is always ertai. This paper develops a approah to solve the fuzzy assigmet problem where ost is ot determiisti umbers but impreise oes. Here, the elemets of the ost matri of the assigmet problem are triagular fuzzy umbers. Its triagular shaped membership futio is defied. The optimal solutio of fuzzy assigmet problem is obtaied suessfully by usig this approah. ompared with the result of ovetioal assigmet problem, the result obtaied by our approah is more advataged for deisio-makers. Fially, to show the effiiey of the proposed approah, the problem is demostrated by oe umerial eample. Keywords: Fuzzy sets, Fuzzy mathematial programmig, Assigmet problem, Triagular fuzzy umber. Itrodutio The assigmet problem (AP) is a speial type of liear programmig problem i whih our objetive is to assig a umber of jobs to a equal umber of persos, so as to miimize the total assigmet ost or to miimize the total osumed time for eeutio of all the jobs. However, muh of deisio-makig i the real world takes plae i a eviromet where the objetives, ostraits or parameters are ot preise. Therefore, a deisio is ofte made o the basis of vague iformatio or uertai data. I 970, Bellma ad Zadeh itrodued the oepts of fuzzy set theory ito the deisio-makig problems ivolvig uertaity ad impreisio. Fuzzy assigmet problems have reeived great attetio i reet years. Li ad We (2004) proposed a labelig algorithm for solvig fuzzy assigmet problems. Yaakob ad Watada (2009) proposed the fuzzy approah for solvig assigmet problem, i whih they preseted a worker s plaemet model apable of evaluatig worker s suitability for a speified task aordig their performae, soial ad metal fator. Yag ad Liu (2005) desiged a tabu searh algorithm based o fuzzy simulatio to ahieve a appropriate best solutio of fuzzy assigmet problem. he (985) proved some theorems ad proposed a fuzzy assigmet model whih did ot osider the differees of idividuals. Wag (987) solved a similar model by graph theory. Sakawa (200) dealt with atual problems o produtio ad work fore assigmet i a housig material maufaturer ad a subotrat firm ad formulated two kids of two level programmig problems. Applyig the iterative fuzzy programmig for two-level liear ad liear fratioal programmig problems, they desired satisfatory solutios to the problems ad therefore ompared the results. Liu ad Gao (2009) proposed a equilibrium optimizatio problem ad eteded the assigmet problem to the equilibrium multi-job assigmet problem, equilibrium multi-job quadrati assigmet problem ad used geeti algorithm to solve the proposed models. Majumdar ad Bhuia (2007) proposed a elitist geeti algorithm to solve the geeralized assigmet problem with impreise ost/time. Ye ad Xu (2008) proposed a effetive method o priority-based geeti algorithm to solve fuzzy vehile routig assigmet whe there is o geeti algorithm whih a give lear proedure of solvig it. The liear iterative ad disrete optimizatio [LINDO] 2 ISSN E-ISSN

2 Moder Applied Siee Vol. 6, No. 3; Marh 202 (984) geeral iterative optimizer [GINO] (986) ad TORA pakages (992) as well as may other ommerial ad aademi pakages are useful i fidig the solutio of the assigmet problems. I this paper, we are proposig a ew approah to fid the optimal solutio of fuzzy assigmet problems by represetig ost parameters as triagular fuzzy umbers. To illustrate the proposed approah a fuzzy assigmet problem is solved ad the obtaied results are disussed. This paper is orgaized as follows: I setio 2, some basi defiitios ad arithmeti operatios are reviewed. I setio 3, formulatio of fuzzy assigmet problem is desribed. I setio 4, a ew approah is proposed to fid the optimal solutio of fuzzy assigmet problem. Setio 5, presets a umerial eample to illustrate the proposed approah. The results are disussed i setio 6 ad setio 7 gives few oludig remarks o the proposed approah. 2. Fuzzy Prelimiaries The terms of epressio suh as very good, really good, ot bad ad rather lear are used very ofte i daily life, ommo that they are more or less taited with fuzziess. With differet daily deisio-makig problems of diverse itesity, the results a be misleadig if the fuzziess of huma deisio-makig is ot take ito aout. The theory of fuzzy set is based upo the ivestigatio reported by Bellma ad Zadeh (970), ivolves a mathematial desriptio of vague (ieat, fuzzy) elemets, with the vagueess of iformatio resultig ot from the stohasti harater of the system, but from the lak of uiqueess or seletivity that of. Aordigly, the aswer to the questio whether a elemet is assoiated with a fuzzy set will ot be i the form of a YES-OR-NO deisio but it will require arefully graded judgmet of its assoiatio. The degree of assoiatio of defied elemets is determied by a assoiatio futio that must ome withi the sope of partiular mathematial defiitios, aioms ad operatioal rules. Fuzzy set is a theory of graded oept, has a vague boudary set, as ompared to with risp set. This is also a powerful modelig laguage that a ope with a large fratio of uertaities of real life situatios. I this setio, some basi defiitios ad arithmeti operatios are reviewed. 2. Basi Defiitios Defiitio. A fuzzy set is a set whose boudary is ot lear, whose elemets are haraterized by a membership futio. Let X be a uiversal set. A fuzzy set A defie o X. A set of order pair of elemet whose first elemet X, seod elemet Α is the membership value of elemet i the set A. It is deoted by A or A, ad it defied by A, A X Where A K ad K [0, ] Defiitio 2. A fuzzy set A, defied o uiversal set of real umbers X, is said to be a fuzzy umber if (i) A is ove set i.e., 2 mi, 2,, 2 X, 0, ; A A A (ii) A is ormalized fuzzy set if there eists at least oe 0 X with ( 0) ; A (iii) it s membership futio ( ) is pieewise otiuous. A Defiitio 3. A fuzzy umber A, X is o-egative if ad oly if ( ) 0 for all 0. A A Defiitio 4. A fuzzy umber A ( a,b,) is said to be a triagular fuzzy umber, if its membership futio is give by A 0, a a l A, a b ba r A, b b 0, ad Where l A r A = left membership futio ad right membership futio of the fuzzy set A. Defiitio 5. A triagular fuzzy umber A ( a,b,) is said to be o-egative if ad oly if a 0. Defiitio 6. A triagular fuzzy umber A ( a,b,) is said to be zero triagular fuzzy umber if ad oly if a 0, b 0, 0. Published by aadia eter of Siee ad Eduatio 3

3 Moder Applied Siee Vol. 6, No. 3; Marh 202 Defiitio 7. Two triagular fuzzy umbers A ( a,a2,a3) ad B ( b, b2, b3 ) are said to be equal if ad oly if a b a b, a., b3 2.2 Arithmeti Operatios Let a, a2, a3 B as: A ad b, b, b 2 3 be two triagular fuzzy umbers, the arithmeti o them is defied Additio: A B a b, a2 b2, a3 b3 Subtratio: A ( ) B a b, a b a b 3 2 2, 3 ( a, a2, a3), if Salar multipliatio: A 0 Symmetri image: A - a3, a2,-a 3. Fuzzy Assigmet Problem We kow for every physial struture there is some mathematial pheomea ad for every mathematial pheomeo there may be or may ot be some physial struture. I this setio we will be desribig mathematial model of assigmet problems i the fuzzy eviromet. Assume that there are jobs ad persos. Jobs must be performed by persos, where the osts deped o the speifi assigmets. Eah job must be assiged to oe ad oly oe perso ad eah perso has to perform oe ad oly oe job. Let be the ost if the i th perso is assiged the j th job, the problem is to fid a assigmet (whih job should be assiged to whih perso) so that the total ost for performig all jobs is miimum. Here make a assumptio that j th job will be ompleted by i th perso, ad let 0 if if ith perso is assiged jth job ith perso is ot assiged jth job Where deotes that j th job is to be assiged to the i th perso. The, the mathematial model of assigmet problem i risp eviromet is: S: Mi i Z i j, j,2,..., j, i,2,..., () 0, for i, j,2,..., I above ovetioal assigmet problem, the variables, assigmet ost oeffiiets, are usually preise values. However, i real life situatios, the parameters of assigmet problem are impreise umbers beause time/ost for doig a job by a perso/mahie might vary due to differet reasos, suh as assigig me to offies, truks to delivery routes et. Espeially the assigmet ost, that whih is osidered as a ertai value is ot suitable, will be iflueed diretly by the above reasos. Therefore assigmet ost oeffiiets are usually uertai values ad will hage respetively i a frame. This paper osiders assigmet ost as a fuzzy umber deoted by ( / / ), i that represets the most possible assigmet ost, the most optimisti assigmet ost ad the most pessimisti assigmet ost. If ost oeffiiets are fuzzy umbers, the the total assigmet ost beomes fuzzy as well. Now the problem is how to ahieve a miimum total ost uder fuzzy ost. The, the ovetioal assigmet problem i () turs ito followig fuzzy assigmet problem. S2: Mi i j Z 4 ISSN E-ISSN

4 Moder Applied Siee Vol. 6, No. 3; Marh 202 i, j,2,..., j, i,2,..., (2) 0, for i, j,2,..., 4. Proposed Approah I this setio, a ew approah is proposed to fid the optimal solutio of fuzzy assigmet problems, ourrig i real life situatios, by represetig ost oeffiiets as triagular fuzzy umbers. The steps of proposed approah are as follows: 4. Mathematial Struture The fuzzy assigmet problem a be stated i the form of ost matri of real umbers as follows: 2 3 j N 2 3 j j 2 i i i 2 i 3 i is a o-egative triagular fuzzy umber. Where For the fuzzy assigmet problem, a triagular shaped membership futio for fuzzy ost oeffiiet is deoted by ad is defied as: 0, l l l, l m m l r (3) r, m r r m 0, r represet the left ad right had side of the triagular membership futio, Where l ad r N 2 3 j respetively. For fuzzy assigmet problem (S2), formulate the followig multi objetive liear programmig problem with fuzzy ost oeffiiets as: S3: Mi z ( ), z2( ),..., zk ( ) i, j,2,..., j, i,2,..., (4) 0, for i, j,2,..., Published by aadia eter of Siee ad Eduatio 5

5 Moder Applied Siee Vol. 6, No. 3; Marh 202 By osiderig the weightig fator, the multi objetive liear programmig problem is defied as: S4: w z ( ) w z ( )... w z ( ) Mi 2 2 k k k i.e. w z ( ) i m m m, j,2,..., j, i,2,..., (5) 0, for i, j,2,..., 4.2 Algorithm The algorithm for the solutio proedure of the proposed approah a be summarized i the followig steps: Step : Develop the fuzzy assigmet problem as desribed i (S2). Step 2: Write the elemets of the ost matri of the assigmet problem i the form of triagular fuzzy umbers. Step 3: Defie the triagular shaped membership futio of eah fuzzy ost oeffiiet as metioed i Eq. (3). Step 4: Formulate the multi objetive liear programmig problem with fuzzy ost oeffiiets for fuzzy assigmet problem (S3). Step 5: For differet weights develop problem (S4) to get a optimal solutio. Step 6: Fid the miimum total fuzzy ost by puttig the values of i i j. 5. Numerial Eample To illustrate the approah let us osider the followig 3 3 fuzzy assigmet problem. The osts are represeted by triagular fuzzy umbers ad are show i followig Table : where = (4.5,5,5.5), 2 =(8.,9,9.9), 3 =(2.7,3,3.3), 2 =(7.2,8,8.8), 22 =(6.3,7,7.7), 23 =(7.2,8,8.8) 3 =(5.4,6,6.6), 32 =(9,0,), 33 =(0.8,2,3.2) Fid the assigmet of persos to jobs that will miimize the total fuzzy ost. Solutio: The fuzzy optimal solutio of fuzzy assigmet problem by usig the proposed approah a be obtaied as follows: Step : The give fuzzy assigmet problem may be formulated i to the followig fuzzy liear programmig problem: Mi Z ((4.5,5, 5.5) (8., 9, 9.9) (2.7, 3, 3.3) (7.2,8,8.8) (6.3, 7, 7.7) (7.2,8,8.8) (5.4,6,6.6) (9,0,) (0.8,2,3.2) ) ,, i,2,3, j,2,3. Step 2: Usig step 3 to 5 of proposed approah, we trasform the fuzzy assigmet problem ito the followig multi-objetive liear programmig problem: (i dollars) 6 ISSN E-ISSN

6 Moder Applied Siee Vol. 6, No. 3; Marh 202 Miimize ( w w w ) , , 2 3, , , 0,, i,2,3, j,2,3. Solve the above problem for differet weights. For eample, w 0, w2, w3 Now above problem redues to Miimize , , , 2 3, , , 0,, i,2,3, j,2,3. The above problem is solved by usig the TORA pakage. The solutio is preseted as follows: 2 3, 22, 3, 2 2 Step 3: The fuzzy optimal total ost is alulated as: = (2.7, 3, 3.3) + (6.3, 7, 7.7) + (5.4, 6, 6.6) = (4.4, 6, 7.6) I other words the optimal assigmet is, 2 B, 3 A 6. Aalysis of the Results ad Disussios The obtaied result a be eplaied as follows: ) The total ost is greater tha 4.4 ad less tha 7.6 dollars. 2) Let T represets the total ost, the the peretage of the favouress for T is give by T T 00, where 0, T 4.4 T 4.4 TlT, 4.4 T 6.6 T T 7.6 T T rt, 6 T , T 7.6 Table 2 lists the solutio for above multi-objetive liear programmig problem for various weights ad it also shows that the solutios are idepedet of weights ( w m, m,2, 3 ) , 33 Published by aadia eter of Siee ad Eduatio 7

7 Moder Applied Siee Vol. 6, No. 3; Marh 202 Table 3 shows the total assigmet ost of the most possible, optimism ad pessimism value. So the fial result of fuzzy assigmet problem is show i Table 4. From Table 4 it a be see that total assigmet ost of ovetioal assigmet problem is just the most possible ost of fuzzy assigmet problem. The a olusio may be that the solutio of assigmet problem is oly a speial ase of fuzzy assigmet problem, ad that the paper osiders the assigmet ost as a fuzzy umber is more sigifiat ad atual. So the solutio obtaied by this approah is optimum. This shows the effiiey of our approah. 7. olusios I the proposed approah a assigmet problem with fuzzy ost oeffiiets has bee solved i order to defeat uertai eviromet i the real world situatio ad relevae to solve it. By the proposed approah fuzzy assigmet problem has bee trasformed ito multi-objetive liear programmig problem where the ost oeffiiets are fuzzy umbers ad also proved that the solutios are idepedet of weights. To illustrate the proposed approah a umerial eample is solved ad obtaied results are disussed. We have foud that i our omputatioal eperiee the results are as per epetatios ad satisfyig. The proposed approah has the followig features: ) This approah is easy to uderstad ad to apply for fidig the optimal solutio of fuzzy assigmet problem ourrig i real life situatio. 2) It is easy ad simple to use for the deisio maker ad a be easily implemeted to solve other type of problems like, trasportatio problems, projet shedules ad etwork flow problems. 3) This approah solves all types of assigmet problems, the miimum assigmet problem ad the maimum assigmet problem. 4) This approah proposes a effetive ad effiiet way for hadlig the fuzzy assigmet problem. 5) This approah solves the fuzzy assigmet problem without usig ay rakig futio. Referees Belma, R., & Zadeh, L. A. (970). Deisio makig i a fuzzy eviromet. Maagemet Siee, 7, he, M. S. (985). O a fuzzy assigmet problem. Tamkag J, 22, Liebma, J., Lasdo, L., Shrage, L., & Ware, A. (986). Modelig ad Optimizatio with GINO. The Sietifi Press, Palo Alto, A. Li,. J., & We, U. P. (2004). A labelig algorithm for the fuzzy assigmet problem. Fuzzy Sets ad Systems, 42, Liu, L., & Gao, X. (2009). Fuzzy weighted equilibrium multi-job assigmet problem ad geeti algorithm. Applied Mathematial Modellig, 33, Majumdar, J., & Bhuia, A. K. (2007). Elitist geeti algorithm for assigmet problem with impreise goal. Europea Joural of operatioal Researh, 77, Sakawa, M., Nishizaki, I., & Uemura, Y. (200). Iterative fuzzy programmig for two- level liear ad liear fratioal produtio ad assigmet problems: a ase study. Europea Joural of operatioal Researh, 35, Shrage, L. (984). Liear, iteger ad quadrati programmig with LINDO. The Sietifi Press. Palo Alto, A. Taha, H. A. (992). Operatios Researh, A Itrodutio, 5th ed. (Mamilla, New York). Wag, X. (987). Fuzzy optimal assigmet problem. Fuzzy math, 3, Yaakob, S. B., & Watada, J. (2009). Fuzzy approah for assigmet problem. IEEE, Yag, L., & Liu, B. (2005). A multi-objetive fuzzy assigmet problem: New model ad algorithm. IEEE Iteratioal oferee o Fuzzy Systems, Ye, X., & Xu, J. (2008). A fuzzy vehile routig assigmet model with oetio etwork based o priority-based geeti algorithm. World Joural of Modelig ad Simulatio, 4, Zimmerma, H. J. (99). Fuzzy Set Theory ad Its Appliatios, 2 d ed., Kluwer Aademi Publishers, Bosto/Dorgreht/Lodo. 8 ISSN E-ISSN

8 Moder Applied Siee Vol. 6, No. 3; Marh 202 Table. Fuzzy osts (I Dollars) Jobs A B Persos Table 2. The solutio for above multi-objetive liear programmig problem for various weights Sr o. w w 2 w 3 (, 2, 3, 2, 22, 23, 3, 32, 33) 0 (0,0,,0,,0,,0,0) (0,0,,0,,0,,0,0) 3 0 (0,0,,0,,0,,0,0) (0,0,,0,,0,,0,0) (0,0,,0,,0,,0,0) 6 0 (0,0,,0,,0,,0,0) (0,0,,0,,0,,0,0) (0,0,,0,,0,,0,0) (0,0,,0,,0,,0,0) 0..3 (0,0,,0,,0,,0,0) (0,0,,0,,0,,0,0) Table 3. The fial result of fuzzy assigmet problem Jobs A B (4.5,5,5.5) (8.,9,9.9) (2.7,3,3.3) Persos 2 (7.2,8,8.8) (6.3,7,7.7) (7.2,8,8.8) 3 (5.4,6,6.6) (9,0,) (0.8,2,3.2) Published by aadia eter of Siee ad Eduatio 9

9 Moder Applied Siee Vol. 6, No. 3; Marh 202 Table 4. Result of ovetioal assigmet problem A Jobs B Persos ISSN E-ISSN