An Application of Assignment Problem in Laptop Selection Problem Using MATLAB

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1 Applied themtics d Scieces: A Itertiol Jourl (thsj ), Vol., No., rch 05 A Applictio of Assigmet Problem i ptop Selectio Problem Usig ATAB ABSTRAT Ghdle Kirtiwt P, uley Yogesh The ssigmet selectio problem used to fid oe-to- oe mtch of give Users to ptops, the mi objective is to miimize the cost s per user requiremet. This pper presets stisfctory solutio for rel ssigmet ptop selectio problem usig ATAB codig. KEYWORD Assigmet problem, lier iteger progrmmig, Revised Oes Assigmet ethod (ROA), ATAB progrmmig 00 themtics Subject lssifictio: 90-0, 90-08, 90B06, 90B0, 9005, 9008, INTRODUTION Now dy s lptop is very essetil equipmet for college, uiversity studets s well s fculties. It is lso commo to observe tht t ech home hvig either lptop or P. osider tht idividul wt to purchse lptop, but i mrket vrious types of product & compies vilble, where yoe c be cofused. If we wt to purchse lptop seller gives us ltest high cofigured system, which cost pproximtely betwee Rs. 45,000 to Rs 90,000. So problem rises here, if ower is geerl user d his purpose d requiremet re bsic the whether he should purchse give mout lptop? To overcome this problem we defied four ctegories of users, d lso recommed which lptop he should purchse s per his ctegory d bsic requiremet. To solve this lptop selectio problem Assigmet odel used. Assigmet Problem (AP) is kow s degeerte form of trsporttio problem, i AP totl umber of ssig elemet is. It ppers i some decisio-mkig situtios. The typicl problems re to ssig ctivities to resources, workers to jobs etc. ptop selectio problem is i ublced form, where four types of users hve to choose four lptops from described lptops (see tble 3). To solve this problem ROA method is used which gives some oes i Assigmet mtrix d usig Ghdle d uley rule will fid out optiml solutio. To verify the result ATAB Progrm is used which will sve clcultig time of problem d give ccurte optiml solutio withi sec. ATAB is specilly used for mtrix, usig ATAB softwre vrious mtrix opertios c be performed for e.g. mtrix dditio, subtrctio, multiplictio, elemet to elemet multiplictio, iverse etc d vrious fuctios vilble for solvig mtrix. I ublced lptop selectio problem ROA lgorithm is used s progrm. The time complexity for this lgorithm is less th O ( log ). ATAB progrmmig is stroger d sophisticted th 4

2 Applied themtics d Scieces: A Itertiol Jourl (thsj ), Vol., No., rch 05 covetiol computer lguges (e.g., FORTRAN) for solvig techicl problems. ATAB is iterctive system whose bsic dt elemet is rry tht does ot require dimesioig. ATHEATIA ODES FOR ASSIGNENT PROBE To miimize the overll cost or time, let cosider Assigmet Problem of resources to ctivities i such wy tht oe d oly oe job should ssig to ech resources. Where is cost mtrix give s, A R R Resource R Required Activity A O A Avilble I Assigmet odel, the vilbility t ech of the resources d the requiremet t ech of the destitios is uity. et x Assig resources to ctivity such tht, x = ; if ssigmet of i resources to j ctivity. 0 ;otherwise The the mthemticl model for ssigmet problem is, = () Subject to, = = =0 () For ll i=,,, d j=,,,. Prelimiries: Blce Assigmet Problem: I blced model ctivities equl resources. Ublce Assigmet Problem: I ublced model ctivity does ot equl resource. 4

3 Applied themtics d Scieces: A Itertiol Jourl (thsj ), Vol., No., rch 05 3 Revised Oes Assigmet ethod (ROA) for Assigmet Problem. [, 3] I ROA method, by dividig miimum elemet from ech row d colum to correspodig rows d colum we get t lest oe vlue i ech rows d colums. The fid complete ssigmet i terms of oes. Now, i ssigmet mtrix is the cost or effectiveess of ssigig mchie. The ew lgorithm is s follows. b b O et (-) be ssigmet problem i which the objective fuctio c be miimized or mximized. Step I miimiztio cse, fid the miimum elemet of ech row (sy i ) d write it o the right hd side of the mtrix. O After dividig ech elemet of i th row of the mtrix by i. We get t lest oe oes i ech rows. I term of oe for ech row d oe for ech colum, do ssigmet. Otherwise go to step. Step Fid the miimum elemet of ech colum (sy b j ), d write it below j th colum, by dividig ech elemet of j th colum of the mtrix by b j. We get t lest oe oes i ech colums. This gives ssigmet mtrix i terms of oes. If o fesible ssigmet c be chieved from step () d () the go to step 3. O b 43

4 Applied themtics d Scieces: A Itertiol Jourl (thsj ), Vol., No., rch 05 Note: I mximiztio cse, the of step we hve fuzzy mtrix. Which ll elemets re belog to [0, ], d the gretest is oe []. Step 3 To fid optimlity test drw the miimum umber of lies which cover ll the oes i the Assigmet mtrix. If the umbers of drw lies re less th, the optimlity coditio fils i.e. complete ssigmet is ot possible. While optimlity coditio stisfied, if the umber of lies is exctly equl to, the the complete ssigmet is possible. Step 4 If optimlity coditio fils i step 3, the select the smllest elemet (sy d ij ) out of those which do ot lie o y of the lies i the bove mtrix. By dividig smllest elemet d ij to ech elemet of the ucovered rows or colums. This gives some ew oes to this row or colum. If still optimlity coditio fils i this ew mtrix, the use step 4 d 3 itertively. By repetig the sme procedure the optiml ssigmet will be obtied. (To ssig oe Ghdle d uley Rule is used, which is metioed below.) Step 5 (Ghdle d uley Rule to ssig oe i Assigmet trix) i) For miimiztio problem select mx vlue from clculted mtrix d write it o right hd side s well s bottom side. To ssig oe, strt from mi vlue of colums (metioed t bottom side) d select oes. If there re more th oe oes i y colum the igore temporrily, d give lst priority to tht colum. If still there re ideticl oes i colum the give the priority to mx vlue of rows (metioed t right hd side). Or Vice Vers. Remrks: I Ublced Assigmet Problem, if there re more th oe oes i y row the give priority to first elemet. Priority rule To solve Ublced Assigmet Problem (o squre mtrix), we covert it ito Blced Assigmet trix form by ddig rtificil row or colum, which hvig ll elemets oe. Hece we c solve the problem usig ROA method. After performig the steps reduces to mtrix which hs oes i ech rows d colums. So, the optiml ssigmet hs bee reched. 44

5 Applied themtics d Scieces: A Itertiol Jourl (thsj ), Vol., No., rch 05 4 APPIATIONS: Now dy vrious techiques hve bee used to ttrct cdidtes i colleges d uiversities for dmissio. et cosider tht college mgemet decide to give the lptops to dmitted studets s per their requiremets d courses. To reduce lptop purchse cost we hve select lptop s per courses d there requiremet, for lptop cofigurtio we divide users ito four ctegories, like Geerl users, Professiols, Progrmmers, Egieers. Requiremets of Users: As per our observtios we coclude followig. Geerl Users: ike sciece, commerce, d rts studet, used lptop for preprig otes, presettio d lso usig some softwre like s-office, Tlly, DTP etc. for tht sufficiet cofigurtio is : sus/ dul core processor, or GB RA, 500 GB HDD, 5 GD (for high grphics, like gmes).. Professiol Users: ike BA (ll courses), IWA, A, used lptop for ccoutig, presettio d lso usig some softwre like s-office, Tlly, SAP d some olie softwre which required high cofigurtio th Geerl users. For tht sufficiet cofigurtio is: dul core processor, or GB RA, 500 GB HDD. 3. Progrmmers: ike A, S,.E / B.E (omputer) used lptop for progrmmig, mkig softwre s. Geerlly they used softwre like - progrmmig, VB, VB.et, Orcle, Jv, SAP etc. which required high cofigurtio th Professiol users. For tht sufficiet cofigurtio is: core i3 processor, or 4 GB RA, 500 GB HDD, GB GD (for high grphics). 4. Egieers: ike echicl, Electricl, ivil etc. used lptop for grphics desig. Geerlly they used softwre like Auto d, UG, PRO E, ATIA which required higher grphics resolutio. For tht sufficiet cofigurtio is: ore i5 or i7 processor, 4 or 8 GB RA, 500 GB HDD, or GB GD (for high grphics). To overcome this situtio there re severl credit poits uder cosidertio, firstly cosider the competecies ecessry to develop the users. Obviously some competecies like processor, RA hvig more importce d some competecies like HDD, Scree size, Grphics rd hvig less importt compre to first oe. oreover, usully the users re ot idepet; therefore reltioship mog them could pper. Filly, ll prticulr with credit poits re show i tble s per their ecessry requiremets. 45

6 Applied themtics d Scieces: A Itertiol Jourl (thsj ), Vol., No., rch 05 Prticulrs redit Poits Essetil 5 Firly High 4 High 3 oderte ow Firly ow 0 Tble Prticulrs\ Users Geerl Professiols Progrmmers Egieers Dul ore ore i ore i Processor ore i Asus Brzo Asus Itel Dul ore Asus ore i Asus ore i GB RA 4 GB GB HDD 500 GB TB 0 0 Scree 5.6" Grphics rd 5 B GB GB Oce the users ivolved i the selectio procedure hve bee determied, the lptops must ext to be cosidered. et it be imgied tht there re lptops (see tble 3) tht might be ble to select for differet Users. ptops ptop ptop.. Nme 46

7 Applied themtics d Scieces: A Itertiol Jourl (thsj ), Vol., No., rch 05 ptop For ech oe it is ecessry to fid out by some pproprite, mes the levels i ech of the prticulr required for the users. Filly s there re liks betwee the users d lptops (usig tble credit poits i tble 3) must be looked t i order to fid out the reltioships tht there would be mog them, s show i tble. Tble Ublced Assigmet Problem is solved by usig ROA method d ATAB Progrmmig (See Appix ). This gives fil optiml solutio s: Users Geerl Professiols Progrmmers Egieers ptop ONUSIONS: To select pproprite lptops s per user s requiremet d bsic eeds. The give problem coverted ito Assigmet odel. The give iformtio i tble 3 is coverted ito umericl by usig credit poits give i tble, to solve this relistic problem ROA method is used d t lst it verified by ATAB progrm which gives optiml solutio withi sec. 6 REFERENES [] Hdi Bsirzdeh. Oes Assigmet ethod for solvig Assigmet Problems, Applied themticl Scieces, Vol 6 (0), No. 47, pp [] Frcisco Herrer, Erique ópez, risti ñ, iguel A. Rodríguez. Solvig ssigmet selectio problem with verbl iformtio d usig geetic lgorithms, Europe Jourl of Opertio Reserch, 9 (999), pp [3] Ghdle Kirtiwt P, uley Yogesh (03), Revised Oes Assigmet ethod for Solvig Assigmet Problem, Jourl of Sttistics d themtics, Volume 4, Issue (03), pp

8 Applied themtics d Scieces: A Itertiol Jourl (thsj ), Vol., No., rch 05 Tble 3: (shows vrious types of lptops with specifictios d cost) Appix : % ATAB PROGRA FOR ASSIGNENT - APTOP SEETION PROBE. % DEVEOPED BY : Dr. Kirtiwt Ghdle, r. Yogesh uley clc tic; x= [ ] % mtrix x is copied i xv d x vrible xv=x; x=x; % clcultig ech row miimum o. mxr= mxr(x) % fid row d colum o [r c]=size(x) % dividig ech elemet of row 48

9 for i=:r for j=:c x(i,j)=x(i,j)/mxr(i); x y=x; for i=:r for j=:c if x(i,j)< x(i,j)=0; x l=; for i=:r for j=:c [xr xc]= fid(x==l,''); k=legth(xr); x; z=xv.*x toc; for m=:k x(xr(m),xc(m))=; if x(xr(m),j)~= x(xr(m),xc(m)) x(xr(m),j)=0; Dr. Kirtiwt P Ghdle Deprtmet of themtics, Dr. Bbsheb Ambedkr rthwd Uiversity, Aurgbd drkp.ghdle@gmil.com r. Yogesh uley Deprtmet of themtics, Dr. Bbsheb Ambedkr rthwd Uiversity, Aurgbd yogesh.m.muley@gmil.com 49