INCORPORATING WAITING TIME IN COMPETITIVE LOCATION MODELS: FORMULATIONS AND HEURISTICS 1

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1 INCORPORATING WAITING TIME IN COMPETITIVE LOCATION MODELS: FORMULATIONS AND HEURISTICS 1 Francsco Slva a, Danel Serra b a GREL, IET, Unverstat Pompeu Fabra, Ramon Tras Fargas, 25-27, Barcelona, Span. CEEAplA, Unversdade dos Açores, Rua da Mae de Deus, 9502 Ponta Delgada, Portugal emal: francsco.slva@upf.edu b GREL, IET, Unverstat Pompeu Fabra, Ramon Tras Fargas, 25-27, Barcelona Span. emal: danel.serra@upf.edu Abstract In ths paper we propose a metaheurstc to solve a new verson of the Maxmum Capture Problem. In the orgnal MCP, market capture s obtaned by lower travelng dstances or lower travelng tme, n ths new verson not only the travelng tme but also the watng tme wll affect the market share. Ths problem s hard to solve usng standard optmzaton technques. Metaheurstcs are shown to offer accurate results wthn acceptable computng tmes. Keywords: Market capture, queung, ant colony optmzaton. JEL:C61,L80. 1 Ths research has been possble thanks to the grant SFRH/BD/2916/2000 from the Mnstéro da Cênca e da Tecnologa, Fundação para a Cênca e a Tecnologa of the Portuguese government. 1

2 Introducton. ReVelle s Maxmum Capture Problem (1986) ntated a seres of studes on the locaton of retal facltes n dscrete space (see Serra and ReVelle (1995)). The MAXCAP model makes the followng assumptons: (1) the product sold s homogeneous, (2) the consumer s decson on patronzng the store s based on dstance and (3) unt costs are the same n all stores regardless of ownershp. Examples of servces that best ft these three assumptons can be found manly n the fast food sector, n convenence stores and n the bankng sector. However, n all these examples, not only the dstance but also watng tme seems to determne the consumer s decson. The number of persons the consumer fnds n queue, when he or she arrves at the store, can be a measure for the consumer s percepton of watng tme. Furthermore, the watng tme for one vst may affect future decsons as to whch store to patronze the next vst. Ths seems to be qute relevant for some retal stores, fast food restaurants or ATM machnes. Kohlberg (1983), n poneer work n the same lne of research, consders a varant of the classcal Hotellng model for store locatons. The author assumes that when choosng a store, consumers take nto account not only travel tme but also watng tme for the servce at each store, whch n turn depends on the number of consumers patronzng that store. Assumng that each consumer makes the decson that mnmzes travel tme plus watng tme, stores market shares are shown to be contnuous functons of ther locatons. There s also a general consensus that the dstances may be nterpreted n a functonal, proxmty, or smlarty context rather than n a geometrcal one. Our clam s that n some types of servces, watng tme has a strong mpact on the consumer s percepton of proxmty. In chapter 1 we wll revse some lterature on compettve spatal modelng. In chapter 2 we descrbe a model, whch ncorporates explctly watng tme, and n chapter 3 we propose a metaheurstc to solve the model. Some results of our computatonal experments are descrbed n chapters 4 and 5. 2

3 1 Lterature Revew. In ts smplest scenaro the game works as follows: the leader frm locates a number of facltes, antcpatng that the follower wll react to the locaton pattern. The follower, n turn, wll then solve the condtonal locaton problem of locatng hs own facltes gven the leader s chosen locatons. Followng Hakm (1983), we refer to the leader s problem of locatng a fxed number of facltes, knowng that the follower wll subsequently locate hs own facltes, as an (r p) centrod problem. The follower, n turn, wll then face a locaton pattern of the facltes of the leader and, gven that, optmze the locaton of hs own facltes. Ths s known as the (r Xp) medanod problem. A typcal model n the former category s the MaxCap (maxmum capture) model ntroduced by ReVelle (1986). The model formulated by ReVelle fnds the optmal locaton on a network consderng that each demand pont wll patronze the closest faclty. Several authors have expanded ReVelle s formulaton: Eselt and Laporte (1989) generalze ReVelle s fndngs n two drectons: they allow dfferental weghts for the facltes and they leave a parameter of the cost functon varable so as to facltate senstvty analyss, Serra and ReVelle (1993) ntroduce n the model facltes that are herarchcal n nature and where there s competton at each level of the herarchy, the same authors, Serra and ReVelle (1994), account the possble reacton from compettors to the enterng frm n the preemptve locaton problem, n whch the leader wshes to preempt the enterng frm n ts bd to capture market share to the maxmum extent possble. Serra, Ratck and ReVelle (1996) offer a modfcaton of the MaxCap problem n whch they consder uncertanty. The authors consder dfferent future scenaros wth respect to demand and/or the locaton of compettors. Most compettve locaton problems were at frst developed under the hypothess that dfferent frms provde the same ndstngushable product and that all customers have the same preferences,.e., the same determnstc utlty functon. Some lterature refers to the topc of droppng the hypothess of the homogenety of the product. 3

4 In Drezner (1994), customers base faclty choce on a utlty functon that ncorporates a faclty s attrbutes and the dstance to the faclty. Although customers are no longer assumed to patronze the closest faclty, customers at a certan demand pont apply the same utlty functon. Drezner and Drezner (1996) assume the utlty functon to change from one consumer to another for customers located at the same demand pont. Usng ths assumpton the all or nothng property dsappears. Serra, Eselt, Laporte and ReVelle (1999) developed two models allowng dfferent customer choce rules. One model assumes that customers consder the closest faclty of each frm and then patronze the two facltes n proporton to the customer-faclty dstance. The other model assumes that the demand captured by a faclty s affected by the exstence and locaton of all facltes of both frms. Other mprovements over the ntal maxmum capture model refer to mnmum market shares that frms need to capture n order to survve. Carreras and Serra (1998) present a model that locates the maxmum number of servces that can coexst n a gven regon wthout havng losses, takng nto account that they need a mnmum demand level n order to survve. Serra, ReVelle and Rosng (1999) consdered the problem of locatng several facltes such that each faclty attracts a mnmum threshold of customers. Drezner and Eselt (2002) consder a mnmum market share threshold to be captured, below whch the frm cannot survve and propose the obectve of mnmzng the probablty that revenues fall short of the threshold necessary for survval. 2 The model. The MAXCAP problem seeks the locaton of a fxed number of stores belongng to a frm n a spatal market where there are other stores belongng to other frms already competng for clents. The obectve of the enterng frm s to maxmze ts profts. Whenever the prces charged at the dfferent facltes are equal and there are no locaton-specfc cost dfferences, the proft-maxmzng obectve reduces to maxmzaton of sales. 4

5 A customer s an ndvdual or a group wth a unque and dentfable locaton and behavor. Snce a customer has a locaton and ssues demand, the term demand pont s also used. The expresson pont demand as defned by Plastra (2001) refers to dscrete demand concentrated n a fnte set of ponts. We consder a dscrete locaton space n the sense that there s only a fnte lst of canddate stes and the market s characterzed by pont demand. Each customer feels some attracton towards each of the competng facltes, usually referred as patronzng behavor. The attracton functon descrbes how a customer s attracton, also called utlty, towards a faclty s obtaned. When we ncorporate watng tme n the MAXCAP, customers wll patronze a gven frm f the sum of the travelng tme plus the watng tme at one of ts stores s the lowest when compared wth other frms stores. Let us assume an enterng frm (frm A) that wants to locate p new outlets when there are q other outlets from another frm (frm B) already competng at the market place. In order to solve the problem we consder that the enterng frm wants to maxmze ts market share, that s Max Z = I A J a X (1) Where,,I ndex and set of demand ponts,j ndex and set of potental locatons J A set of frm A s (entrant frm) store locatons a demand at node X =1 f demand pont patronzes a store at =0 otherwse 5

6 Consderng an ndependent M/M/1 queue for each server, the average watng tme at s gven by: w = µ λ ( µ λ ) (2) Where, f frequency of persons from demand node that wll buy the product/servce (e.g. persons per hour) µ servce rate As n Maranov and Serra (1998) let us accept the assumpton that request for servce at each demand pont appear accordng to a Posson process wth ntensty f. Each center serves a set of demand ponts, therefore the requests for servce at that center are the unon of the requests for servce of the nodes n the set. Thus they can be descrbed as a stochastc process equal to the sum of several Posson processes. The new stochastc process s also a Posson process, wth an ntensty λ equal to the sum of the ntenstes of the processes at the nodes served by the center. Ths set of nodes wll result from the problem s soluton. Varables X are used n order to rewrte parameter λ : λ = l f X (3) If a partcular varable X s one, meanng that node s allocated to a center at, the correspondng ntensty f wll be ncluded n the computaton of λ. Let us also assume an exponentally dstrbuted servce tme, wth an average rate of µ so that, assumng steady-state each center can be modeled as an M/M/1 queung system. Equaton (2) can then be rewrtten as w f X = µ µ f X (4) 6

7 In order to compute the value of frm A s obectve, we need addtonal nformaton concernng the allocaton of demand nodes to the stores defned through varables X. Assumng that all customers wll patronze the store locaton that mnmzes travelng tme plus watng tme, a good estmate for the allocaton varables value wll result from the mnmzaton of average total tme (average travelng tme from a demand pont to an outlet + average watng tme at a outlet). For each of frm A s potental store locatons, and n order to obtan the value of the X, we solve the followng p-medan type model: Mn Z = λ s. t. J I X wth X f 1 I X = 1 J < C a d X + λ I J 2 J µ µ (5) (6) (7) { 0,1} I, J (8) f X f X 1 λ 1 = and λ2 = a 1 J Where the addtonal notaton s the followng: d dstance from node to node C capacty at store locaton. Constrant (6) lmts the allocaton of one demand pont to only one store and constrant (7) fxes the capacty of each store (n order to obtan a fnte queue capacty we mpose C to be smaller or equal to µ ). Once the allocatons of all the demand ponts to the stores locaton are known t s possble to compute the market share of frm A as gven by equaton (1). 7

8 Karv and Hakm (1979) prove that the p-medan problem s a NP-Hard problem on a general graph. Besdes that, notce that the p-medan obectve s non-lnear and that we need to solve a p-medan model for each of the possble locatons of a frm A store. Ths explans the mportant role played by the metaheurstcs descrbed n the followng secton. 3 Metaheurstcs to solve the model. 3.1 Descrpton of Metaheurstcs. Ant Colony Optmzaton (ACO) ntroduced by Colorn, Dorgo and Manezzo (1991) s a cooperatve search algorthm nspred by the behavor of real ants. In analogy to the bologcal example, ACO s based on the ndrect communcaton of a colony of smple agents, called ants, medated by pheromone trals. The pheromone trals n ACO serve as dstrbuted, numercal nformaton, whch the ants use to probablstcally construct solutons to the problem, and whch the ants adapt durng the executon of the algorthm to reflect ther search experence. For a recent descrpton of these metaheurstcs, ther applcatons and advances refer to Dorgo and Stützle (2003). For the applcaton to the partcular case of an assgnment problem, refer to Manezzo and Colorn (1999) and to Lourenço and Serra (1998). The problem descrbed can easly be cast nto the framework of the ACO metaheurstc. It can be represented by a graph n whch the set of components comprses the set of demand ponts and the set of faclty locatons. Each assgnment wll consst of a couplng (, ) of demand ponts and store locatons and t corresponds to an ant s walk on the graph. Lourenço and Serra (1998) present new metaheurstcs for the Generalzed Assgnment Problem. The best result was found usng a MAX-MIN Ant System (MMAS), based on an algorthm suggested by Stützle (see as an example Stützle (1998)). Also, Stützle and Hoos (1997) refer the MMAS as one of the most effcent algorthms for the Quadratc Assgnment Problem. The MMAS s an mprovement of the more general Ant System metaheurstc, whch ntroduces upper and lower bounds to the values of the pheromone trals, as well as a dfferent ntalzaton of ther values. 8

9 The pseudo code for the metaheurstcs we used to solve the problem n secton 2 s descrbed n Fgure 1: procedure ant 1 Intalze MAX-MIN ant systems upper and lower bounds; 2 for ter=1 to n_ter do 3 allocaton ntal _ soluton( tau ) ; 4 allocaton local _ search( allocaton) ; 5 Update_allocaton(Allocaton,Best_Allocaton); 6 Update_attractveness(tau ); 7 enddo; 8 end ant Fgure 1: Ant s Algorthm Pseudo Code In pont 1 of the algorthm MMAS upper and lower bounds are ntalzed. Wth ths purpose we used the followng procedure: 1. For each demand pont compute τ, the attractveness to a store located at where: τ 1 = 1 + d The closer t s located, the more attractve the store. At ths pont of the algorthm t s not possble to compute the watng tme snce we do not have nformaton about the allocaton of the demand ponts to the stores. 2. Compute the mnmum of τ and the maxmum of τ 3. Compute the lower and upper bounds for the pheromone trals accordng to the followng expressons: τ max = max ( τ ) number of demand ponts ( ) τ mn = 0.1 mn τ These are the same expressons used n Lourenço and Serra (2000) and they gve us ntal values for the lmts n the MMAS. At each of the teratons an ntal soluton s constructed as a functon of attractveness (pont 3) and a local search procedure s mplemented (pont 4). 9

10 The pseudo code for the ntal soluton procedure s llustrated n fgure 2. procedure ntal_soluton (tau ) {allocate every demand pont to a store locaton} 1 for =1 to N do {actualze watng tme at each store} 2 for =1 to NP do 3 W _ W _ ( allocatons) ; 4 enddo; {ncorporate watng tme n the stores attractveness} 5 for =1 to N do 6 for =1 to NP do 7 tau tau + 8 enddo; 9 enddo; {compute probabltes} 10 for =1 to N do 11 for =1 to NP do 12 prob 1 W _ ; tau ; tau 13 enddo; 14 enddo; {allocate demand pont to a potental faclty locaton} 15 alloc_ alloc( prob ) ; 16 enddo; 17 end ntal_soluton; Fgure 2: Intal Soluton s Algorthm Pseudo Code Startng wth the frst demand pont n the demand ponts lst, each demand pont wll be allocated to a store locaton accordng to the followng three steps: a) actualze watng tmes at the stores, b) actualze stores attractveness and c) compute new probabltes. One of the man characterstcs of the algorthm s that we are ncorporatng watng tme at a store locaton n the attractveness of that store for all demand ponts. Attractveness s nversely correlated wth watng tme: τ τ = τ new new + 1 w f w 0 otherwse Whenever there s a new allocaton, watng tme vares and the stores attractveness s updated. Snce probabltes are postvely related to attractveness, also the probabltes wll be updated. 10

11 Each of the demand ponts are allocated to a potental store locaton accordng to the probablty rule: P = τ J τ where, J s the set of both frms store locatons. P s the probablty that one ant wll assgn demand pont to a potental faclty locaton at. At ths pont of the algorthm t ths possble to obtan solutons volatng constrant (5),.e. the resultng arrval rate to a store s bgger than the servce rate. In order to avod ths soluton we opted to penalze the obectve wth a large value M. As suggested n Stützle and Hoos (1997) we decded to add a local search phase to the ACO algorthm, n whch ants are allowed to mprove ther solutons. Ths may mprove the performance of the algorthm wth respect to qualty and convergence speed. The Pseudo Code for the local search phase s llustrated n Fgure 3. The local search phase conssted n the followng procedure: de-allocate each demand pont from potental store locaton, and allocate ths demand pont to each one of the other potental locatons. Keepng new allocaton, de-allocate each of the other demand ponts, one at a tme, and check for all possble alternatve allocatons always computng the respectve obectve. Whenever the obectve mproves accept new obectve and allocatons. procedure local_search (allocaton) 1 for all 1 D do 2 1* alloc_ 1 ; 3 for all 1 S \ { 1 *} do 4 alloc _ 1 1 ; 5 for all D\ { } 2 1 do 6 2* alloc_ 2 ; 7 for all 2 S \ { 2 *} do 8 alloc _ 2 2 ; 9 evaluate obectve; 10 f ob_best>ob do 11 ob_best : = ob; 12 else 13 alloc_ 1 1*; 11

12 14 alloc_ 2 2*; 15 endf 16 enddo 17 enddo 18 enddo 19 enddo 20 end local_search Fgure 3: Local Search Algorthm Pseudo Code In lne 6 of the ant procedure (fgure 1), the pheromone trals (attractveness of each demand pont to a potental store locaton) s updated accordng to the followng expresson: where: τ new = ρτ + Q = τ max, 0, f node s allocated to a faclty at otherwse and, 0.01, Q = 0.05, f the soluton s nfeasble f the soluton s feasble Parameter ρ works out as the persstence of the tral; the same s to say that 1-ρ gves the evaporaton of the pheromone tral. Ths parameter must be fxed to a value smaller than one to avod an unlmted accumulaton of trace. In the MMAS pheromone trals must be restrcted wthn upper and lower bounds,.e.: f new ( τ τ ) τ max new = τ max f new ( τ τ ) τ mn new = τ mn For a more detaled exposton of MAX-MIN ant systems see as an example Stützle and Hoos (1998). 3.2 Analyss of the Metaheurstc performance. In order to obtan a measure of the metaheurstcs precson we randomly generated 100 examples and solved the problem of allocatng 20 demand ponts to 3 stores, whose locatons 12

13 are known, n order to mnmze the sum of average travel tme and average watng tme as descrbed through the model n secton 3. For each example we solved the nteger problem defned through equatons (3)-(6) wth a commercal package (LINGO 6) and compared the results wth the ones obtaned usng the metaheurstc suggested n secton 2. The results are descrbed n table The examples are dvded nto two groups. The examples defned as regular examples conssted of generatng both the coordnates as well as the populatons from a unform dstrbuton. The other group of examples results from the use of the procedure descrbed n Cordeau et al (1997). The latter procedure generates nstances n whch customers tend to be clustered around some fxed centers, as s often the case n real lfe. Table 3.2.1: examples wth 20 demand ponts and 3 facltes Iteratons Regular examples % dentcal obectves 78% 80% 82% Average Devaton (% optmal ob.) 2.23% 2.03% 1.71% % dentcal allocatons 97% 97% 97% Average computng tme LINGO s s s Average computng tme Heurstcs 3.19 s 7.28 s s Cordeau et al (1997) % dentcal obectves 70% 72% 72% Average Devaton (% optmal ob.) 1.77% 1.73% 1.65% % dentcal allocatons 97% 97% 97% Average computng tme LINGO 16.5 s 16.5 s 16.5 s Average computng tme Heurstcs 2.34 s 4.41 s 9.17 s For each one of the examples the metaheursc was mplemented wth 25, 50 and 100 teratons. The results seem to be qute close n terms of dentcal allocatons, whch concdes wth our ntal nterest n the metaheurstc. In respect to computng tmes, the metaheurstc s advantages are clear even for small examples. 13

14 4 Computatonal experments. 4.1 Comparson of the results obtaned wth and wthout watng tme. In the MaxCap model as defned by ReVelle (1986), snce watng tme depends on market share and the obectve of the frms maxmzes market share, there s a tendency for the entrant frm to accumulate large watng tmes. We llustrate ths tendency wth 30 examples n whch frm A wants to locate a new store when there are already two other stores pertanng to frm B operatng n the market. In all examples we randomly generated the coordnates and the populatons of 20 demand ponts from a unform dstrbuton. The coordnates where generated n a 6 6 square and the populatons n the nterval [6000,8000]. The frequency of people lookng for the servce by unt of tme was fxed at 10% of the populaton. Servce rate was fxed at 1000 / unt of tme. In the examples, we consdered that every demand pont s also a potental store locaton. Let us call the orgnal ReVelle (1986) MaxCap model, model 1, and the model descrbed n secton 3, model 2. Results for model 1 were obtaned solvng the respectve nteger program n LINGO 6. Results for model 2 were obtaned usng the metaheurstc defned n secton 3 and solvng the model for all possble locatons for the new frm s store, from whch we choose the best one (maxmzes market capture). Table shows the man results obtaned wth our experments. In ths table, we see how small the percentage s of our 30 examples from whch the use of both models resulted n the same locaton. Table 4.1.1: Results from the computatonal experments. Model 1 Model 2 Average watng tme n one outlet Standard devaton for the watng tme n one outlet

15 Average watng tme n the new outlet % of examples wth the same locaton n both models 10% 4.2 A numercal example. The problem s also llustrated wth Swan s (1974) well-known 55-node network. In ths example we consder an entrant frm (frm A) that wants to locate a new store when there are already two stores of another frm (frm B) operatng n the two demand ponts locaton wth the hgher populatons. Then, we vary the servce rate from 0.5 customers per mnute to 0.6, 0.7 and 0.8 customers per mnute. In Table 4.2.1, we compare the results obtaned wth model 1 and model 2. Once agan results presented as model 1 result from the applcaton of the orgnal formulaton of ReVelle s (1986) MaxCap model and the results presented as model 2 result from the applcaton of the model suggested n secton 3, evaluatng all possble new frm s locaton. In all the examples, the arrval rates orgnatng from each of the demand ponts by unt of tme (mnute) were fxed at 0.02% of the respectve populatons. The Eucldean dstances computed from the orgnal coordnates fulfll the dstance matrx, measured as travelng tme n mnutes. In order to smplfy the problem the potental store locatons were restrcted to the 15 demand ponts wth the hgher populatons. Table 4.2.1: results for Swan s 55-node network. µ=0.5 µ=0.6 µ=0.7 µ=0.8 Model 1 Locaton: 3 Locaton: 3 Locaton: 3 Locaton: 3 Obectve: 1673 Obectve: 1673 Obectve: 1673 Obectve: 1673 W 3 =5.06 W 3 =2.47 W 3 =1.5 W 3 =1.08 W 1 =0.83 W 1 =0.54 W 1 =0.38 W 1 =0.28 W 2 =0.10 W 2 =0.07 W 2 =0.05 W 2 =0.04 Model 2 Locaton: 3 Locaton: 3 Locaton:3 Locaton: 3 Obectve: 1354 Obectve: 1409 Obectve: 1509 Obectve: 1579 W 3 =2.59 W 3 =1.59 W 3 =1.16 W 3 =0.87 W 1 =1.62 W 1 =0.99 W 1 =0.67 W 1 =

16 W 2 =1.82 W 2 =1.02 W 2 =0.59 W 2 =0.46 Average travelng Tme :10.74 Average travelng tme :10.61 Average travelng tme :10.59 Average travelng Tme :10.68 W 1 average watng tme at store 1; W 2 average watng tme at store 2; W 3 average watng tme at store 3 (entrant) We can verfy how the tendency for the watng tmes n the three faclty locatons becomes smlar wth ncreases n the servce rate. For lower levels of servce rate, the devaton from the watng tme n the new store and the watng tme n the other two stores s clearly greater for model 2. The obectves resultng from both models are dfferent n all the examples. Watng tme has no mpact on the obectve of model 1 whle reducng the obectve n model 2. We gve addtonal nformaton on the average travelng tmes resultng from model 2. 5 A Heurstc Concentraton algorthm to solve larger problems. An obvous lmtaton of the methodology proposed n the prevous sectons s the tme requred to solve larger problems. A possble strategy to dmnsh ths problem s the use of a heurstc concentraton algorthm. Heurstc concentraton was developed specfcally to deal wth larger problems. HC s a two stage process. Stage 1 nvolves dong some number (q) of random start runs of an nterchange heurstc. A number of these solutons are then subected to a smple analyss n order to develop the concentraton set. Stage 2 s the constructon of a (heurstcally derved) good soluton or the best soluton (by an exact method) from the concentraton set. For a detaled descrpton of ths methodology, see Rosng and ReVelle (1997) as an example. A general descrpton of the heurstc concentraton algorthm proposed to solve the problem formulated n secton 2 conssts of the followng: Stage 1: 1. Fnd p random ntal locatons for frm A s stores; 2. Allocate each demand node to ts closest store locaton. Fnd the demand served by each frm A outlet as well as total frm A market capture. If the utlzaton factor s bgger than one, set the market capture to zero and go to step 3. 16

17 3. Choose the frst of frm A s outlets from a lst of ts stores and trade ts locaton to an empty node wthn the set of potental locatons. 4. Fnd agan the demand served by each of frm A s outlets. Compute market capture. If the utlzaton factor s bgger than one, set the market capture to zero. If market capture has mproved, store the new locatons. If not, restore the old soluton. 5. Repeat steps 3 and 4 untl all potental empty locatons have been evaluated one at a tme for each outlet. 6. If frm A mproved ts market share to a value greater than n Step 2, go to Step 3 and restart the procedure. 7. When no mprovement s acheved for a complete set of one-at-a-tme trades, store fnal soluton. 8. Go to Step 1 untl a number q of teratons of Stage 1 s met. Stage 2: 9. Use all fnal locatons obtaned from all startng solutons or use the fnal locatons from the best k out of the multple startng solutons n Stage 1 to form the new, reduced set of potental locatons (the concentraton set - CS). 10. Fnd p random ntal locatons n the CS for frm A s stores; 11. Solve the P-Medan model: fnd the demand served by each of frm A s outlets as well as total market capture of frm A usng the ant algorthm descrbed n secton 3. If the utlzaton factor s bgger than one, set the market capture to zero and go to step Choose the frst of frm A s outlets from a lst of ts stores and trade ts locaton to an empty node wthn the set of potental locatons n the CS. 13. Fnd agan the demand served by each of frm A s outlets usng the ant algorthm descrbed n secton3. Compute market capture. If the utlzaton factor s bgger than one, set the market capture to zero. If market capture has mproved, store the new locatons. If not, restore the old soluton. 17

18 14. Repeat steps 3 and 4 untl all potental empty locatons have been evaluated one at a tme for each outlet. 15. If frm A mproved ts market share to a value greater than n Step 11, go to Step 12 and restart the procedure. 16. When no mprovement s acheved for a complete set of one-at-a-tme trades, store fnal soluton. 17. Go to Step 10 untl a number p of teratons of Stage 2 s met. In stage one we hope to elmnate some of the potental store locatons due to ther perphery, ncreased travelng dstances and consequent penalzaton on the P-Medan obectve. We used the heurstc concentraton algorthm n order to locate 2 and 3 stores of an entrant frm when there s another frm operatng wth two stores located n the two demand ponts wth the larger populatons. In our experments, we compare the solutons obtaned usng an algorthm that consders all possble combnatons for the locaton of new stores (algorthm 1) wth the ones obtaned usng the above algorthm. For each dfferent combnaton of number of demand nodes and number of new stores, we randomly generated 10 numercal examples. As n secton 4, the examples were generated usng the procedure descrbed n Cordeau et al (1997). Coordnates where randomly generated from a unform dstrbuton on a 6 6 square, dstances are Eucldean, populatons were generated from a unform dstrbuton between 6000 and 8000 and the arrval rates at each demand pont were fxed at 10% of the respectve populatons. Every demand pont s also a potental store locaton. Table 5.1: Results from concentraton heurstcs. 20 nodes 35 nodes 2 stores 3 stores 2 stores 3 stores Algorthm 1 Average computng tme (seconds) Algorthm 2 Algorthm 3 Number of dfferent obectves Average number of elements n the CS Average computng tme (seconds) Number of dfferent obectves

19 Average number of elements n the CS Average computng tme (seconds) Gven the small sze of the examples (20 and 35 nodes) we only consdered 100 teratons n stage 1. The dfference between algorthms 2 and 3 conssts of the fact that n algorthm 3, we adopted the procedure of ncorporatng a new soluton n the CS whenever the obectve s greater or equal to 90% of the best obectve found at the moment and n the second stage we used complete enumeraton for the potental locatons n the CS. Table 5.1 resumes the results obtaned wth our experments. In general the HC shows nterestng results allowng sgnfcant reductons n the problem. Conclusons. The model proposed n ths paper seems to be qute useful n the locaton decsons of new stores for servces n whch watng queues are common, as s the case of fast food restaurants, supermarkets or commercal banks. When the servce rate s not large enough relatve to the arrval rate whch, n turn, results from the market share, watng tme may have a sgnfcant mpact on the optmal locaton of a new outlet of an entrant frm. The metaheurstcs we propose n ths paper produce results that are close to optmal, offerng mportant savngs n computatonal processng tmes. 19

20 References Carreras, Mquel and Danel Serra On optmal locaton wth threshold requrements. Soco-Economc Plannng Scences. Vol. 33.pp Colorn, A., M. Dorgo and V. Manezzo. (1991). Dstrbuted Optmzaton by Ant Colones. Proceedngs of ECAL91- European Conference on Artfcal Lfe. Elsever Publshng.pp Cordeau, J. F., M. Gendreau and G. Laporte A tabu search algorthm for perodc and mult-depot vehcle routng problems. Networks. Vol Dorgo, Marco and Thomas Stützle The ant colony optmzaton metaheurstcs: algorthms, applcatons and advances n Handbook of Metaheurstcs. Edted by Fred Glover and Gary Kochenberger. Kluwer Academc Publshers. Drezner, Tammy and H.A. Eselt Consumers n compettve locaton models. In Faclty Locaton: Applcatons and Theory. Edted by Zv Drezner and H.W. Hemacher. Sprnger Verlag. Drezner, Tammy Locatng a sngle new faclty among exstng, unequally attractve facltes. Journal of Regonal Scence. Vol. 34. N. 2. pp Eselt, H.A. and Glbert Laporte The maxmum capture problem n a weghted network. Journal of Regonal Scence. Vol. 29. N. 3.pp Hakm, S.L On locatng new facltes n a compettve envronment. European Journal of Operatonal Research. Vol Karv, O and S.L. Hakm An algorthm Approach to Network Locaton Problems, Part II: the P-medan. SIAM ournal of Appled Mathematcs. Vol Kohlberg, Elan Equlbrum store locatons when consumers mnmze travel tme plus watng tme. Economcs Letters. Vol. 11.pp Lourenço, H. and Danel Serra Adaptve approach heurstcs for the generalzed assgnment problem. Economcs Workng Paper 288. Unverstat Pompeu Fabra. Manezzo, Vttoro and Alberto Colorn The ant system appled to the quadratc assgnment problem. IEEE Transactons on knowledge and data engneerng. Vol. 11. N.5. ReVelle, Charles The maxmum capture or sphere of nfluence locaton problem: Hotellng revsed on a network. Journal of Regonal Scence. Vol. 26. N. 2. Rosng, K.E. and Charles ReVelle Heurstc Concentraton: Two Stage Soluton Constructon. European Journal of Operatonal Research. Vol. 97.pp Plastra, Frank Statc compettve faclty locaton: an overvew of optmzaton approaches. European Journal of Operatonal Research..Vol pp Serra, Danel and Charles ReVelle The pq-medan problem: locaton and dstrctng of herarchcal facltes. Locaton Scence. Vol. 1. N. 4.pp Serra, Danel and Charles ReVelle Market capture by two compettors: the preemptve 20

21 locaton problem. Journal of Regonal Scence. Vol. 34. N. 4.pp Serra, Danel and Charles ReVelle Compettve locaton n dscrete space. In Faclty Locaton: Applcatons and Methods. Edted by Zv Drezner. Sprnger Verlag. Serra, Danel, Charles ReVelle and Ken Rosng Survvng n a compettve spatal market: the threshold capture model. Journal of Regonal Scence. Vol. 39. N.4.pp Serra, Danel, H. A. Eselt, Glbert Laporte and Charles ReVelle Market capture models under varous customer choce rules Envronment and Plannng B. Vol. 26. N. 5.pp Serra, Danel, S. Ratck and Charles ReVelle The maxmum capture problem wth uncertanty. Envronment and Plannng B. Vol. 23. N. 4.pp Stützle and Hoos MAX-MIN ant system for Quadratc Assgnment Problems. TH Darmstadt, FB. Informatk Darmstadt. Germany. Stützle and Hoos Improvements on the Ant System: Introducng MAX-MIN Ant System. n Artfcal Neural Networks and Genetc Algorthms. Edted by G. D. Smth and N.C. Steele, R.A., pages Stützle, Thomas An ant approach for the flow shop problem. TH Darmstadt, FB. Informatk Darmstadt. Germany. Swan, Ralph A parametrc decomposton algorthm for the soluton of uncapactated locaton problem Management Scence. Vol