Site Selection Using Optimization Techniques

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1 Iteratioal Joural of Scietific & Egieerig Research, Volue 4, Issue 4, April Site Selectio Usig Optiizatio Techiques Vadaa Bagla, Aaa Gupta Abstract The process of selectio of sites for coercial activities ivolves yriad of qualitative, quatitative ad fiacial factors. I geeral, ere are ultiple diversified factors. Due to e hua tedecy to deped ore o eotios a reasos, ere is every chace of reachig a irratioal coclusio. This paper presets a foral syste to evaluate coparative raks of available sites for coercial activities ad ereby to deterie e log tered profitable decisio usig Leicographic Approach. A versatile solutio is provided to give proble usig Weighted Pealty Meod. Ide Ters Aalytic Hierarchy Process (AHP), Cosistecy, Eige value, Eige vector, Leicographic Approach, Mied Iteger Prograig, Weighted Peality Meod. 1 INTRODUCTION A lost all ivestet decisios ivolve ultiple, diverse ad cople set of social ad fiacial factors which are quite hard to be overcoe by ere ituitio. The site selectio process is a eeplificatio of a ivestet decisio which ivolves e evaluatio of attributes for aiizatio of profit ad iiizatio of cost which are e ost iportat requireets for e successful fuctioig of a particular busiess activity. Qualitative factors ust also be cosidered while selectio of a site for a activity. A uber of allocatio probles have bee solved i e recet past; for ready referece see Carlsso & Fuller[1], Igizio[4], Igizio & Cavalier [5], Serfii[7] ad Azar[8]. I is paper, we evisio a site selectio odel for coercial activities which efficietly eplores a ulti-criteria decisio-akig odel ivolvig ree obectives. Maiizatio of profit ad iiizatio of set-up cost are e aor obectives which are take up as first ad secod obectives alteratively. Last but ot e least obective is to rak various attributes such as capacity, eighborhood, coectivity, trasport availability ad proiity which are cosiderably iportat factors for a flourishig busiess. The solutio procedure cosists of two phases. I e first phase, weights are allocated to various available sites based o various iportat aspects such as eighborig locality, broad or arrow coectig roads, area/capacity of e available sites, proiity factors such as copetitive busiess rivals i Vadaa Bagla is curretly pursuig her Doctoral progra i Operatios Research ad is workig as Assistat Professor i Maharaa Agrese Istitute of techology(mait), Delhi, Idia. E-ail: vadaa_6928e@yahoo.co Aaa Gupta is workig as a Associate Professor i Delhi Techological Uiversity (DTU), Delhi, Idia. E-ail: guptaaaa2003@yahoo.co.i e earby areas, trasport availability such as etro or oer public trasports. To accoplish is, a approach of Aalytic Hierarchy Process (AHP) give by Saaty [10] is used. I e secod phase, e proposed proble seekig to aiize e profit, iiize e set-up cost ad aiize e allotted weights, is odeled as ied iteger prograig proble. Sice e obective is to aiize e profit ad weights. Also at e sae tie to iiize e cost, reversed costs are beig take after oralizatio. The proble is solved usig hierarchical optiizatio eod. The said proble is also be solved usig Weighted Pealty Meod developed by Prakash ad Gupta[9]. The paper is orgaized as follows. Sectio 1 is itroductory. Sectio 2 eplais e Aalytic Hierarchy Process (AHP). Sectio 3 describes e proble aeatically. Sectio 4 eplais e proposed eodology to fid e set of solutios. Sectio 5 illustrates e eod via a eaple. The paper cocludes i Sectio 6 for furer applicatio i real estate ad ay oer iportat fields 2 ANALYTIC HIERARCHY PROCESS (AHP) The aalytical hierarchy process (AHP) is a decisio akig approach desiged to aid i e solutio of cople ultiple criteria probles i uber of applicatio doais. The outcoe of AHP is a prioritized weightig of each decisio alterative. The first step i e aalytical hierarchy process is to odel e proble as a hierarchy. The hierarchy is a structured ea of describig e proble at had. It cosists of a overall goal at e top level, a group of optios or alteratives for reachig e goal ad a group of factors or criteria at relate e alteratives to e goal. I ost cases e criteria are furer broke dow ito sub criteria, sub-sub criteria ad so o i ay levels as per e requireet of e proble. Oce e hierarchy has bee costructed, e participats use e AHP to establish priorities for all its odes. I is, e eleets of a proble are copared i pairs wi respect to

2 Iteratioal Joural of Scietific & Egieerig Research, Volue 4, Issue 4, April eir relative ipact o a property ey share i coo. The pair wise copariso is quatified i a atri for by usig e scale of Relative Iportace give i Saaty [10] as show i Table 1. This scale has bee validated for effectiveess, ot oly i ay applicatios by a uber of people, but also rough eoretical copariso wi a large uber of oer scales. Durig e elicitatio process, a positive reciprocal atri is fored i which (i,) eleet a is filled by e correspodig uber fro e Table 1. TABLE 1 Aalytic Hierarchy Measureet Scale e e derived priority vector w satisfies w i / w = a, i < (2) Ay reciprocal atri satisfyig (1) is called cosistet. However i practice, e priority atri seldo satisfies (1), ereby akig it ore iportat to defie soe rela easurig of cosistecy check, Saaty [10] itroduced e cocept of Cosistecy Ide (CI) of a reciprocal atri as e ratio a 1 where λ a ad, respectively stad for e aiu eige value ad order of e reciprocal atri. The obtaied CI value is copared wi e Rado Ide (RI) give i Table 2. The table 2 had bee calculated as a average of CI s of ay ousads atrices of e sae order whose etries were geerated radoly fro e scale 1 to 9 wi reciprocal force. The siulatio results of RI for atrices of size 1 to 10 had bee developed by Saaty [10] ad are give i Table 2. TABLE 2 RANDOM INDEX (RI) The ratio of CI ad RI for e sae order atri is called e Cosistecy Ratio (CR). The uber is chose accordig to e followig criterio. a, if i doiates 1/a, if doiates i 1, if i ad do ot doiate over oe aoer The atri so fored is called e reciprocal atri. This reciprocal atri is used to calculate e local priority weight of each criterio. The local priority weight (w) is e oralized eige vector of e priority atri correspodig to e aiu eige value of e atri. For detailed reasoig of is accout we refer to Lugig [2], Ball & Sriivasa[3], Bryso & Moboluri[6] ad Saaty[10]. A iterestig property of e priority atri is at if i additio its eleets are such at a a k = a ik, i k (1) 3 PROBLEM DESCRIPTION AND MATHEMATICAL FORMULATION Suppose a corporate body/m.n.c. deals i differet busiess/outlets ad ere are available sites. Proble is to allocate a suitable site out of e available sites to each busiess/outlet. While doig is, e two ai obectives of e copay are to aiize e overall profit ad to iiize e overall cost. At e sae tie, e copay wats to prioritize e sites carryig ore weights. Let p (i=1,,; =1,,) deote e epected profit, whe i busiess is set up o site. Also let c be e overall cost ad w be e weight of site (=1,,) calculated by AHP. Let deote e oralized cost of e site. The (1- )= r deotes e reversed cost of e site. It is to be oted at we have reversed e cost as our obective is to iiize e cost ad o e cotrary, we are dealig wi a aiizatio proble. The e above described odel is forulated as e followig ree obective proble. Maiize Z( )= (P( ), R( ), W( )) Where P( ) = i p 1 1

3 Iteratioal Joural of Scietific & Egieerig Research, Volue 4, Issue 4, April Subect to 1 i1 R( ) = i r 1 1 W( ) = i w 1 1 = 1, i=1,, (1) 1, =1,, (2) {0,1}, i=1 ; =1,, (3) The costrait (1) esures at each kid of outlet is allotted wi a site. It is clear i costrait (2) at sae site is ot allotted for ore a oe outlet. I costrait (3) e value of is oe if is zero. i outlet is allotted wi activity, oerwise 4 SOLUTION PROCEDURE 4.1 Allottig Weights Usig AHP I e site selectio odel, we costruct hierarchy of attributes which are ost iportat i decisio akig usig AHP. To evaluate e hierarchy, various surveys are coducted to rate each attribute to oers at e sae level i a series of pair wise coparisos usig a scale fro 1 to 9 (Table 1).We rak each of e available site i e fial set by evaluatig e site wi respect to upper level attributes separately as a illustrated i Table 4 ad Table 5. The evaluatio process fially geerates e global weights for each available site of iterest, as show i Table 6 of e illustratio. 4.2 Procedure to obtai optial solutio usig Leicographic Approach Cosider e ree- obective liear prograig proble Maiize Z( )= (P( ), R( ), W( )) subect to give costraits. The eod requires at e obective fuctios are to be prioritized i decreasig order of iportace. Let P( ) be e ost prioritized ad W( ) is e least prioritized obective. The e eod cosists of followig procedure. Optiize e sigle obective proble cosistig of P( ) as e obective fuctio subect to give costraits. All oer obectives are igored. Let P( ) = k 1 be e optial solutio obtaied usig iteger prograig i e first iteratio. Fid e optial solutio of recostructed sigle obective proble wi R( ) as e obective fuctio ad a added costrait P( ) k 1,wi e origial costrait equatios. Let R( ) = k 2 be e optial solutio obtaied i e secod iteratio. Fially fid optial solutio to e give proble i fial iteratio by reforig e proble as: Maiize W( ) Subect to P( ) k 1 R( ) k 2 1 i1 =1, i=1,, (1) 1, =1,, (2) {0,1}, i=1 ; =1,, (3) The above stated procedure provides us wi a optial solutio to e give ree obective prograig proble wi prioritized obectives. A siilar eodology ay be adopted while cosiderig cost as e first priority obective. But e above stated odel has its liitatios. It does ot provide alteratives to e aspirat which suits best to his pocket. 4.3 Procedure to obtai a set of efficiet solutios usig Weighted Pealty Meod Based o e feedback provided by e ivestor/corporate body/m.n.c, priorities are assiged to each of e ree obectives. Here we have take aiizatio of overall profit ad iiizatio of cost as first ad secod priority obectives alteratively. Also aiizatio of qualitative raks (weights) is assiged ird priority. This ree-obective proble is reduced to a equivalet sigle-obective iteger prograig proble followig e procedure developed by Prakash ad Gupta [9]. Here p deotes e epected aual profit if i busiess activity is set up at site. Now we partitio e set { p : i=1,, ; =1,,} ito e subsets Lk (k = -1, 1,,q) i e followig way. L -1 cosists of ose p for which i busiess activity ca ot be set up at site. For eaple a petrol pup ca ot be set up i a ultistoried shoppig cople. Cosequetly we block allocatio i at particular ( i, ) cell. Afterwards we follow e leicographic arrageet of p s aog e re-

4 Iteratioal Joural of Scietific & Egieerig Research, Volue 4, Issue 4, April aiig p. Let L 1 cosists of ose p havig e largest uerical value. L 2 cosists of p havig e et largest uerical value. Cotiuig i is way, fially L q cosists of p havig e sallest uerical value. Now to deal wi e cost fuctio siultaeously, we calculate oralized cost, for each potetial site ad cosequetly respective reversed cost r = (1- ) for each available site. Weights w have already bee calculated for each site S 1, S 2, S via AHP, as eplaied i sectio 4.1. Now Sice e profit fuctio P( ) is e first priority factor followed by cost fuctio R( ) ad e weight factor W( ). Assigig positive priorities M 1,,Mq, M c, M w, M -1 to each of e su L 1,...,, r Lq respectively. Here Lk i1 1 is e su of, i1 1 w, L 1 s correspodig to p s belogig to L k. Followig poits should be observed while allottig e priorities. (i) No allocatio ca be ade i set L -1. (ii) Cost factor has bee reversed as our obective is aiizatio of e ree give factors. Now e priority weights assiged are M 1 > > M 2 > > M 3 >> M q > > M C >> M w >> M -1 The sybol a >> b idicates a is arbitrarily large copared to b. Havig doe is, e proble wi aiizatio of P( ) as e first priority obective, aiizatio of R ( ) (iiizatio of C()) as e secod priority obective ad aiizatio of W( ) as e ird priority obective, is reduced to a sigle obective iteger prograig proble. Maiize Z( )= k i1 w 1 Subect to 1 q M k 1 + M -1 L1 1, i=1,, i1 {0,1}, 1, =1,, L K i=1,,; =1,, + M c i1 1 r + M w The optial solutio for e above proble yields e first efficiet solutio. Estiated aual profit of e busiess activities at e selected site is deteried by addig e profits i e allocated cells. Miiu cost is calculated by addig e costs correspodig to allocated sites. Also correspodig total weights ca be foud out i e siilar aer. Now to obtai secod efficiet solutio, associate a cost M -1 (zero) wi each of e variables for which p is aiu ad rest of e proble reais uchaged. This will soehow reduce e profit but at e sae tie reduce e cost bor by e corporate body/ M.N.C. i geeral. Solve e resultat proble by adoptig e sae procedure. The ird ad subsequet efficiet solutios for e proble are obtaied by repeatedly odifyig e obective fuctio ad proceedig i e siilar aer described above. This will provide a variety of solutios to e ivestor which suits best to his pocket. Reark:- The proposed eodology provides a alterative to goal prograig. As evidet i goal prograig, secod ad ird priorities ay be partially fulfilled. 5 ILLUSTRATION We will ow illustrate e proposed eodology via a eaple. Suppose a ulti atioal copay is seekig to ivest i its differet proposals like glossary stores, apparel outlets, gold outlets ad petrol pups. Each proposal requires a suitable site to be established. As a saple survey, Table 3 shows a set of available sites. TABLE 3 BUISINESS PROPOSALS VERSES AVILABLE SITES Here S 2 ad S 7 are sites o upper levels of a ultistoried shoppig cople. Last row shows e costs of e available sites i requisite uits. Itercellular costs show epected aual profit. For eaple e cell (1,1) shows at if B 1 proposal is setup o S 1 site, e aual epected profit is 3.5 uits. The

Iteratioal Joural of Scietific & Egieerig Research, Volue 4, Issue 4, April-2013 1691 cells (4,2) ad (4,7) are left blak because a petrol pup ca ot be set up o e upper levels of a ultistoried buildig.

5 Iteratioal Joural of Scietific & Egieerig Research, Volue 4, Issue 4, April cells (4,2) ad (4,7) are left blak because a petrol pup ca ot be set up o e upper levels of a ultistoried buildig. The copay wats a optial set of solutios which should aiize e overall profit at e iiu cost which suits its pocket. At e sae tie e copay does ot wat to coproise fro quality poit of view for e sake of its reputatio ad log ter profitable busiess. To eet e requireet give by e M.N.C., all available sites were raked as eplaied i sectio 4.1 by takig ito accout all iportat affectig attributes give i Fig. 1. TABLE 4 RANKING OF ATTRIBUTES a = C.I.= C.R.= TABLE 5 COMPARISON MATRIX FOR NEIGHBORHOOD Fig. 1 Hierarchy for Site Selectio For is purpose a survey o irty two people was coducted ad reciprocal atrices were geerated by takig ea of all values of atrices (havig C.R < 0.1 ) for furer calculatios. The iportat attributes were eighborhood, coectig roads, capacity, trasport availability ad Proia. Neighborhood takes ito accout stadard of e surroudig localities. Broad Coectig roads were give prefereces. Capacity/Area of e available site also plays a iportat role. Availability of trasport (Metro/Bus) is a iportat attribute for iddle class people. Proia plays a very iportat role i ruig of a successful busiess activity. Based o various surveys coducted o e available sites, weights were calculated for each attribute at each level. As a eaple Table 4 shows e copariso atri ad calculated weights at level two ad Table 5 shows e copariso atri for e eighborhood. Furer ore e global weights are calculated i Table 6. a = C.I.= C.R.= TABLE 6 CALCULATION OF GLOBAL WEIGHTS FOR SITES Table 7 gives e reversed cost ad calculated oralized weights for each site.

6 Iteratioal Joural of Scietific & Egieerig Research, Volue 4, Issue 4, April TABLE 7 CALCULATION OF REVERSED COST FOR SITES 13=1, 22=1, 37=1, 41=1 yieldig P() = 14.5, R() = ad W()= Table 8 shows e efficiet solutio by takig aiizatio of profit as first priority obective. TABLE 8 PRIORITIZING MAXIMIZATION OF PROFIT First phase of e solutio procedure has bee accoplished usig AHP. 5.1 Solutio Usig Leicographic Approach Now to solve e give proble usig Leicographic Approacheplaied i sectio 4.2, takig aiizatio of profit as e sigle obective fuctio subect to give costraits, e proble reduces to followig iteger prograig proble. Maiize P() = Subect to i1 1, i=1,2,3,4 1, =1,,7 {0,1}; i=1,,4 ; =1,,7 Solvig e above liear prograig proble usig iteger prograig, we get 13=1, 22=1, 37=1, 41=1 yieldig P() = Recostructig e above proble i e secod iteratio as Maiize R() = Subect to P() 14.5 ad all above costraits i iteratio 1. Agai solvig e proble usig iteger prograig, we get 13=1, 22=1, 37=1, 41=1 yieldig P() = 14.5 ad R() = Reforig e above proble i e ird iteratio as show: Maiize P() = Subect to R() ad all above costraits i iteratio 2. Solvig above usig iteger prograig Therefore e required optial solutio usig Leicographic Approachis give by [Optial profit = 14.5 uits, Optial cost = 250 uits ad Optial weight = ] Siilarly e said proble is solved by e eplaied eodology takig iiizatio of cost as e first priority obective, aiizatio of profit as e secod ad aiizatio of weights as e ird priority obectives. The efficiet solutio is give i Table 9. TABLE 9 PRIORITIZING MINIMIZATION OF COST [Optial cost = 160 uits, Optial profit = 10 uits ad Optial weight = ] 5.2 Solutio Usig Weighted Pealty Meod Now to solve e above proble usig Weighted Pealty Meod, e reversed costs ad oralized weights are calculated by adoptig e siilar procedure as eplaied before i sectio 4.1. we ow provide priorities to respective profits viz. M1 to 4 uits, M2 to 3.5 uits, M3 to 3 uits, M4 to 2.5 uits, M5 to 2 uits, M6 to 1.5 uits ad M7 to 1 uit as aiizatio of profit is e first priority obective. Also assigig Mc to respective reversed costs ad Mw to oralized weights such at M1 >> M2 >> >>M7 >>Mc >> Mw ad adoptig e procedure obtaied i sectio 4.3, followig set of values are obtaied for various cells as show i Table 10.

7 Iteratioal Joural of Scietific & Egieerig Research, Volue 4, Issue 4, April TABLE 10 Assigig Priorities Usig Weighted Pealty Meod Siilarly a set of efficiet solutios is obtaied by takig aiizatio of reversed cost (iiizatio of cost) as e first priority obective, aiizatio of profit as e secod ad aiizatio of weights as e ird priority obective. Table 12 shows e set of efficiet solutios by prioritizig iiizatio of cost. TABLE 12 PRIORITIZING MINIMIZATION OF COST The give iteger prograig proble reduces to MaiizeZ()=(M Mc Mw) 11+ (M Mc Mw) (M Mc+.0631Mw) 46 Subect to i1 1, i 1, 2, 3, 4 1, 1,, 7 {0,1}; i=1,,4 ; =1,,7 By solvig e above proble, e set of efficiet solutios prioritizig aiu profit is obtaied as show i Table 11. TABLE 11 PRIORITIZING MAXIMIZATION OF PROFIT 5.3 Copariso of e Two Meodologies Leicographic Approach provides a optial solutio to e give proble accordig to e priorities assiged to obectives. But it does ot provide choices to e aspirat which suit best to his pocket. Whereas usig Weighted Pealty Meod, a set of efficiet solutios is obtaied ad ivestor has a wider choice. As evidet fro Tables 8 ad 11, e optial solutio give by Leicographic Approachby prioritizig aiizatio of profit is sae as e first efficiet solutio obtaied by Weighted Pealty Meod. Sae is e tred show i Tables 9 ad 12, while cosiderig iiizatio of cost as e first priority obective. Also oe ay otice at e si efficiet solutio i Table 11 shows a profit of 5 uits at a set up cost of 160 uits ad qualitative weights While at e sae cost ad weight, first efficiet solutio i Table 12 gais a profit of 10 crores. So e ivestor is provided wi a lucrative optio at e sae set up cost. 6 CONCLUSION While strategically aagig our locatio decisios, we eed solutios at address ultiple logistics ad ecooic factors ivolvig real estate ad our custoers. Maiizatio of profit ad iiizatio of set-up cost are e two aor obectives. Third obective is to rak e sites qualitatively, for a flourishig busiess. The eod also suggested at oce startig up wi a low profit/low cost odel, high futuristic grow ay be epected if qualitative raks are high. For eaplefif efficiet solutio give i Table 11 ad secod efficiet solutio give i Table 12 gaied highest raks i spite of low cost ad low profit, but e proect ay ear higher profits i ear future i spite of low oe tie set up cost due to qualitative aspects. Preset paper ca be solved alterately by chagig e level of priorities. For eaple high raks ay be prioritized to low cost etc. The proposed eodology ay prove to be a powerful tool for efficiet decisio akig. The odel preseted has potetial applicatio i e area of real estate, portfolio aageet, ivestet eory etc.

8 Iteratioal Joural of Scietific & Egieerig Research, Volue 4, Issue 4, April REFERENCES [1] Carlsso, C. ad Fuller, R., Multiple criteria decisio akig: The case for iterdepedece, Coputers & Operatios Research, 22 (3), , [2] F. Lugig, Aalytical Hierarchy i Trasportatio Probles, A applicatio for Istabul, Urba Trasportatio Cogress of Istabul, vol 2, pp , [3] J. N. Ball ad V. C. Sriivasa, Usig e Aalytic Hie- rarchy Process i House Selectio, Joural of real Estate Fiace ad Ecooics Vol. 9, pp , [4] J.P. Igizio, Liear Prograig i Sigle ad Multi-obective Systes, Pretice-Hall, Eglewood Cliffs, New Jersy, pp , , [5] J.P. Igizio ad T.M. Cavalier, Liear Prograig, Pretice- Hall, Eglewood Cliffs, New Jersy, pp , [6] N. Bryso ad A. Moboluri, Aalytic Hierarchy Process for solvig Multiple Criteria Decisio Makig Probles, Europea Joural of OperatioalResearch, vol.76, pp , [7] P. Serafii, Maeatics of Multi Obective Optiizatio, New York : Spriger-Verlag, [8] S. Azar, Reductio Meod wi Syste Aalysis for Multiobective Optiizatio-Based Desig, Structural Optiizatio, vol. 7, pp.47 54, 1994 [9] S. Prakash, A. Gupta, Efficiet solutios for e proble selectig sites for pollig statios ad assigig voter-area to e wi two obectives, Proc. of Natioal coferece o aageet sciece ad practice (MSP-2006), pp , March 31 - April 1, 2006 [10] T. L. Saaty, The Aalytic Hierarchy Process, New York: McGraw Hill, pp.55-57,

AUTOMATIC GENERATION OF FUZZY PAYOFF MATRIX IN GAME THEORY

AUTOMATIC GENERATION OF FUZZY PAYOFF MATRIX IN GAME THEORY Dr. Farha I. D. Al Ai * ad Dr. Muhaed Alfarras ** * College of Egieerig ** College of Coputer Egieerig ad scieces Gulf Uiversity * Dr.farha@gulfuiversity.et;