Finding location equilibria for competing firms under delivered pricing

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1 Jounal of the Opeational Reeach Society (2011) 62, Opeational Reeach Society Ltd. All ight eeved /11 Finding location equilibia fo competing fim unde deliveed picing BPelegín-Pelegín 1, P Dota-González 2 and P Fenández-Henández 1 1 Univeity of Mucia, Mucia, Spain; and 2 Univeity of La Palma de Gan Canaia, La Palma de Gan Canaia, Spain We adde the poblem of finding location equilibia of a location-pice game whee fim fit elect thei location and then et deliveed pice in ode to maximize thei pofit. Auming that fim et the equilibium pice in the econd tage, the game i educed to a location game fo which a global minimize of the ocial cot i a location equilibium if demand i completely inelatic and maginal poduction cot i contant. The poblem of ocial cot minimization i tudied fo both a netwok and a dicete location pace. A node optimality popety when the location pace i a netwok i hown and an Intege Linea Pogamming (ILP) fomulation i obtained to minimize the ocial cot. It i alo hown that multiple location equilibia can be found if maginal deliveed cot ae equal fo all competito. Two ILP fomulation ae given to elect one of uch equilibia that take into account the aggegated pofit and an equity citeion, epectively. An illutative example with eal data i olved and ome concluion ae peented. Jounal of the Opeational Reeach Society (2011) 62, doi: /jo Publihed online 14 Apil 2010 Keywod: location; game theoy; intege pogamming; deliveed picing 1. Intoduction Majo deciion fo fim that ell the ame type of poduct and compete fo cutome ae whee to locate thei facilitie and what pice to et. The pofit each fim get i affected not only by the location of it facilitie and the pice that the fim et in the maket, but alo by the facility location and the pice et by it competito. The maximization of pofit fo each competing fim can be een a a location pice game, which ha been tudied ince the wok by Hotelling (1929). Much exiting liteatue deal with a linea maket (ee d Apemont et al, 1979; Obone and Pitchik, 1987; Gabzewicz and Thie, 1992), which i in pat due to the complexity of olving the aociated location poblem in othe location pace a the plane o a netwok (ee the uvey pape Eielt et al, 1993; Platia, 2001; Revelle and Eielt, 2005). Pofit i etimated in mot model in thi context auming that cutome buy at the cheapet facility. A efinement of the Nah equilibium by uing a two-tage poce i taken a olution of the coeponding game. In the fit tage, fim imultaneouly chooe location. Given any outcome of the fit tage, fim then imultaneouly chooe pice in the econd tage. The coeponding two-tage olution i called a ubgame pefect Nah equilibium. It captue the Coepondence: B Pelegín-Pelegín, Depatment of Statitic and Opeation Reeach, Univeity of Mucia, Faculty of Mathematic, Campu de Epinado, Epinado, Mucia, Spain. pelegin@um.e idea that, when fim elect a location, they all anticipate the conequence of thei choice on pice. The diviion into two tage i motivated by the fact that the choice of location i uually pio to the deciion on pice. Futhemoe, the location deciion i elatively pemanent wheea the pice deciion can be eaily changed. The exitence of a pice equilibium in the econd tage of the game depend on the pice policy to be conideed, among othe facto. When each fim et a factoy pice equal fo all the cutome in the maket and the buye take cae of the tanpotation (f.o.b. o mill picing policy) a pice equilibium aely exit (ee Gacía et al, 2004). In thi cae, the aociated location poblem ha been tudied in nonlinea location pace by taking pice a paamete (ee Eielt, 1992; Gacía and Pelegín, 2003; Sea and ReVelle, 1999; Zhang, 2001). On the othe hand, thee fequently exit a pice equilibium when each fim chage a pecific pice in each maket aea, which include the tanpotation cot (deliveed picing policy). The exitence of a pice equilibium wa hown fo the fit time by Hoove (1936), who analyed patial diciminatoy picing fo fim with fixed location and concluded that a fim eving a paticula maket would be contained in it local pice by the delivey cot of the othe fim eving that maket. In ituation whee demand elaticity i not too high, the equilibium pice at a given maket i equal to the delivey cot of the fim with the econd lowet delivey cot. Thi eult wa extended late to patial duopoly (ee Ledee and Hute, 1986; Ledee and Thie, 1990) and to patial oligopoly (ee Gacía et al, 2004;

2 730 Jounal of the Opeational Reeach Society Vol. 62, No. 4 Dota-González et al, 2005) fo diffeent type of location pace. In a duopoly with completely inelatic demand and contant maginal poduction cot, Ledee and Thie (1990) how that a location equilibium exit that i a global minimize of the ocial cot. The ocial cot i defined a the total deliveed cot if each cutome wee eved with the lowet maginal deliveed cot. In oligopoly, the ame eult i obtained by Dota-González et al (2005), who peent a model whee fim take location and delivey pice deciion along a netwok of connected but patially epaated maket. Unde eaonable aumption, they how that a location equilibium can be found at the node. Then, a location equilibium can be detemined by global minimization of the ocial cot, but, to ou knowledge, no pocedue ha been popoed fo finding uch equilibium. If demand i pice enitive o maginal poduction cot ae not contant, the ocially optimal location may not be an equilibium of the location-pice game a ha been hown in Gupta (1994) and Hamilton et al (1989). Thi wok extend the pape by Dota-González et al (2005), to a multifacility cenaio, whee each fim et up multiple facilitie, and it peent a olution pocedue to find a location equilibium. Fo any location pace, it i hown that the location of the fim ae in equilibium if and only if each fim minimize the ocial cot with epect to the competito fixed location. If the location pace i a netwok, it i poved that a location equilibium exit at the node. An intege linea pogamming (ILP) fomulation i popoed to find a location equilibium in dicete location pace. In the cae that all competito have the ame location candidate and equal maginal deliveed cot, it i hown that multiple location equilibia exit. A dicuion on the election of a location equilibium and an illutative example ae alo peented. The et of the pape i oganized a follow. In Section 2, the notation and the model ae intoduced fo an abitay location pace and two competing fim. The equilibium analyi i dicued in Section 3, whee a node optimality popety i hown when the location pace i a netwok. The ILP fomulation fo finding a location equilibium in dicete pace i peented in Section 4. Section 5 i devoted to the exitence of multiple location equilibia. Two pocedue ae peented to elect one of the multiple equilibia. An illutative example with eal data i alo hown. Finally, an extenion to oligopoly and ome concluion ae peented in Section The location pice game We conide eveal fim playing a location-pice game in an abitay location pace (eg a line, a netwok, the plane, etc.) in which thee i a et of patially epaated maket aea. We aume that maket ae aggegated at n demand point (ee Fanci et al, 2002 fo demand aggegation). At each demand point a given homogeneou pice-inelatic poduct will be old by the competing fim. The fim manufactue and delive the poduct to the cutome, which buy fom the fim that offe the lowet pice. Fit the fim have to chooe thei facility location in ome pedetemined location pace, and then, once thei facility location ae et, the fim will et deliveed pice at each demand point. In thi way, each fim ha to make deciion on location and pice in ode to maximize it pofit. Fo implicity, we conide two fim locating eveal facilitie each, but the eult obtained can be extended to any numbe of competing fim, a will be hown in Section 6. The following notation will be ued: k, K ={1,...,n} index and et of maket q k demand in maket k u = 1, 2 index of the fim L u et of poible facility location fo fim u X u et of facility location choen by fim u, X u L u cx u maginal poduction cot of fim u at location x, x L u txk u maginal tanpotation cot of fim u fom location x to maket k pxk u = cu x + t xk u maginal deliveed cot (o minimum deliveed pice) of fim u fom location x to maket k pk u(x u ) = min{pxk u : x X u }, minimum pice that fim u can offe in maket k Note that L u may be a finite et of point (Dicete Location), the point in a netwok, node o point on the edge (Netwok Location), o a egion in the plane (Plana Location) Pice equilibium In thi ubection, we how the exitence of a pice equilibium fo the econd tage of the game. It i aumed that cutome do not have any pefeence concening the upplie and they buy fom the fim that offe the lowet pice. We conide that, each fim cannot offe a pice below it maginal deliveed cot and each facility can upply all demand placed on it. Thu, fo each maket k, once the et of location X u, u=1, 2, ae fixed, fim u, u=1, 2, will et a pice geate than, o equal to, p u k (X u ). If the two fim offe the ame pice in maket k, the one with the minimum maginal deliveed cot can lowe it pice and obtain all the demand in maket k. Then we conide that tie in pice ae boken in favou of the fim with lowe maginal deliveed cot. If the tied fim have the ame maginal deliveed cot in maket k, no tie beaking ule i needed to hae demand becaue they will obtain zeo pofit fom maket k a a eult of pice competition. In the long-tem competition, cutome in maket k will not buy fom fim u if p u k (X u )>min{p u k (X u ) : u = 1, 2}, in uch a cae it competito can offe the lowet pice. In ode to maximize it pofit, the fim with the minimum maginal

3 B Pelegín-Pelegín et al Location equilibia fo competing fim 731 deliveed cot in maket k will et a pice equal to the minimum maginal deliveed cot of it competito, which in tun i, a eult of the pice competition poce, the pice et by it competito in maket k. With uch pice, none of the two fim will eact by changing the pice and theefoe a pice equilibium i obtained. Theefoe, fo any fixed et of location, X 1 and X 2, pice competition lead to the following equilibium pice: { p 2 p k 1 (X 1, X 2 k (X 2 ) if pk 1 ) = (X 1 )<pk 2(X 2 ) pk 1(X 1 ) othewie { p 1 p k 2 (X 1, X 2 k (X 1 ) if pk 2 ) = (X 2 )<pk 1(X 1 ) pk 2(X 2 ) othewie 2.2. Reduction to a location game We aume that, once the et of facility location X u, u=1, 2, ae fixed, the fim will et the equilibium pice in each maket k. The maket in which each fim get a poitive pofit ae detemined a follow: M 1 (X 1, X 2 )={k K : p 1 k (X 1 )<p 2 k (X 2 )} eved by fim 1 M 2 (X 1, X 2 )={k K : p 2 k (X 2 )<p 1 k (X 1 )} eved by fim 2 Othe maket ae eved by both fim, but the pofit coming fom thee maket i zeo. The pofit function ae: Π 1 (X 1, X 2 ) = (pk 2 (X 2 ) pk 1 (X 1 )) q k Π 2 (X 1, X 2 ) = k M 1 (X 1,X 2 ) k M 2 (X 1,X 2 ) (p 1 k (X 1 ) p 2 k (X 2 )) q k Then the location-pice game i educed to a location game whee deciion ae on location and Π u (X 1, X 2 ) i the payoff fo playe u, u = 1,2. Thi game i tudied in the following ection. 3. Exitence of location equilibia The ocial cot i the total cot incued to upply demand to cutome if each cutome would pay fo the poduct the minimum deliveed cot. Fo any fixed et of location, X u, u = 1, 2, the ocial cot i given by: S(X 1, X 2 ) = n min{pk 1 (X 1 ), pk 2 (X 2 )} q k Popety 1 If the fim et the equilibium pice in each maket, then: Π 1 (X 1, X 2 ) = Π 2 (X 1, X 2 ) = n pk 2 (X 2 )q k S(X 1, X 2 ) n pk 1 (X 1 )q k S(X 1, X 2 ) Poof The payoff fo fim 1 can be expeed a follow: Π 1 (X 1, X 2 ) = (pk 2 (X 2 ) pk 1 (X 1 )) q k = = k M 1 (X 1,X 2 ) + k / M 1 (X 1,X 2 ) n pk 2 (X 2 )q k p 2 k (X 2 )q k k / M 1 (X 1,X 2 ) p 2 k (X 2 )q k n min{pk 1 (X 1 ), pk 2 (X 2 )}q k n pk 2 (X 2 )q k S(X 1, X 2 ) Thu, the pofit obtained by fim 1 i the total cot that would be expeienced by it competito eving the entie maket with minimum deliveed cot minu the ocial cot. A imila expeion i obtained fo fim 2. Popety 2 (X 1, X 2 ) i a location equilibium if: S(X 1, X 2 ) S(X 1, X 2 ), X 1 S(X 1, X 2 ) S(X 1, X 2 ), X 2 Poof The pai (X 1, X 2 ) i a location equilibium if and only if: Π 1 (X 1, X 2 ) Π 1 (X 1, X 2 ), X 1 Π 2 (X 1, X 2 ) Π 2 (X 1, X 2 ), X 2 and fom Popety 1, thee inequalitie ae equivalent to the following one: S(X 1, X 2 ) S(X 1, X 2 ), X 1 S(X 1, X 2 ) S(X 1, X 2 ), X 2 Popety 3 Any global minimize of S(X 1, X 2 ) i a location equilibium. Poof It follow fom Popety 2. Fom Popety 3 it follow that we can detemine location equilibia by olving the following poblem: (P) : min {S(X 1, X 2 ) : X 1 L 1, X 2 L 2 } If L u, u = 1, 2, i a egion in the plane, poblem (P) i vey difficult to olve and equie vey complex global optimization technique. Thi cae i outide the cope of thi pape and it will not be conideed in the following Location on a netwok If L u, u = 1, 2, i the et of point in a netwok (node and point on the edge), in Dota-González et al (2005) it i hown that, if each fim locate a ingle facility, thee exit a et of node that i a global minimize of the ocial cot.

4 732 Jounal of the Opeational Reeach Society Vol. 62, No. 4 Thu, the poblem of finding a location equilibium in a netwok i educed to the ame poblem a in dicete pace. In the following ection we deal with the poblem of finding location equilibia in dicete pace. 4. Detemination of location equilibia in dicete pace Figue 1 Example of a netwok. We extend thi eult to the cae whee each fim opeate eveal facilitie. Let N = (V, E, l) denote a netwok, whee V i a finite et of node, E i the et of edge (pai of node), and l i the function that aign to each edge a poitive numbe (length, time, cot,..) aociated to the edge. A diagam of a netwok i hown in Figue 1. Aumption 1 Fo u = 1, 2, the maginal poduction cot, cx u, i a poitive concave function when x vaie along any edge in the netwok, and it i independent of the quantity poduced. Aumption 2 Fo u = 1, 2, the maginal tanpotation cot, txk u, i a poitive, concave and inceaing function with epect to the ditance fom x to each maket k. Popety 4 Unde Aumption 1 and 2, thee exit a et of node which i a global minimize of the ocial cot. Poof Let X 1 and X 2 be abitay et of facility location in the netwok. If x X 1 i not a node, then x i in the inteio of ome edge e = (a, b) E. Aume that all point in X 1 and X 2 ae fixed, but the point x vaie in edge e. Unde Aumption 1 and 2, it eult that the minimum pice to eve maket k, min{pk 1(X 1 ), pk 2(X 2 )}, i a concave function when x vaie in edge e. A the um of weighted concave function, with non-negative weight, i alo concave, it follow that the ocial cot, S(X 1, X 2 ), i concave when x vaie in edge e, and the othe location ae fixed. Theefoe, the ocial cot eache it minimum value on edge e fo x = a o x = b. A imila eult i obtained if x X 2 i not a node. Theefoe, if we eplace each non-node point in X 1 and X 2 by the coeponding minimize node of the ocial cot, we will obtain two et of node V 1 and V 2 fo which S(V 1, V 2 ) S(X 1, X 2 ). Conequently, thee exit a et of node that minimize the ocial cot. In thi ection, we peent an ILP fomulation to find location equilibia when L u, u = 1, 2, i a finite et of point. In a netwok, by Popety 4, we can ue L u = V, u = 1, 2, in ode to find location equilibia. A L u i finite fo u = 1, 2, X u can be epeented by a vecto x u with component xi u, i Lu, with 0 1 value, whee xi u = 1 indicate that facility location i i choen by the fim u, thatii X u,andxi u = 0 mean that i / X u.we aume that the numbe of facilitie to be located by fim 1 and 2 i and, epectively. The numbe of facilitie of the fim ae not deciion vaiable, but they ae detemined in each ituation depending on ome exogenou facto (budget, egulation, abitation, etc). Fo any fixed location et x 1 and x 2, we conide the 0 1 vaiable zik u, i Lu,wheezik u =1 indicate that fim u eve maket k fom a facility at location i. Ifzik u = 0, it mean that maket k i not eved fom a facility of fim u at location i. Then, the ocial cot i given by olving the following poblem in vaiable zik u S(x 1, x 2 ) = min ( n pik 1 z1 ik + i L 1 zik 1 + zik 2 = 1, i L 1 i L 2.t. i L 2 p 2 ik z2 ik ) q k k K zik u x i u, u = 1, 2 k K, i Lu z u ik, x u i {0, 1}, k K, i L u, u {1, 2} Theefoe, the ocial cot minimization poblem become: n n (P) : min q k pik 1 z1 ik + q k pik 2 z2 ik i L 1 i L 2.t. zik 1 + zik 2 = 1, k K (1) i L 1 i L 2 z 1 ik x 1 i, k K, i L1 (2) z 2 ik x 2 i, k K, i L2 (3) i L 1 x 1 i = (4) i L 2 x 2 i = (5) z u ik, x u i {0, 1}, k K, i Lu, u {1, 2} (6)

5 B Pelegín-Pelegín et al Location equilibia fo competing fim 733 Once the above poblem i olved, the pofit of each fim i obtained by taking into account that pofit of a fim plu ocial cot equal total deliveed cot of it ival (ee Popety 1). Poblem (P) can be olved by any tandad ILP-optimize (XPe-Mp, Cplex,...), howeve computational difficultie may occu when the numbe of binay vaiable i lage. To olve poblem (P), the containt zik u {0, 1} in (6) can be elaxed to zik u 0, which make that lage intance of the poblem can be olved in mall un time. The ILP-optimize geneate one optimal olution ( x 1, x 2 ) of poblem (P), which i a location equilibium. In ome cae, the exitence of multiple global minimize of ocial cot may be poible, and theefoe the exitence of multiple location equilibia. In the following, we how the exitence of multiple equilibia in a paticula cae of the location game, and we peent two pocedue to elect one of uch equilibia. 5. Exitence of multiple location equilibia When the location pace i dicete, ocial cot minimization i a combinatoial optimization poblem that may have multiple global optima and then moe than one location equilibium could exit. In paticula, thi occu if fim locate a fixed numbe of facilitie, have a common et of location candidate (L 1 = L 2 ), and the minimum deliveed pice that fim can offe ae equal (pik 1 = p2 ik, fo all i and all k). In uch a cae, the fim will locate at diffeent ite and the ILP fomulation of poblem (P) i educed to the well-known ( + ) MEDIAN poblem, whee i the numbe of facilitie fo fim 1 and i the numbe of facilitie fo fim 2 (ee Rolland et al, 1996; Avella et al, 2007 fo ome efeence on thi poblem). Due to ymmety, it i veified that any patition of the optimal olution et, which contain + location, into two ubet of cadinality and, epectively, i a location equilibium. Theefoe, a lage numbe of location equilibia can be obtained ( (+)! combination).!! When moe than one location equilibium ae found, the competing fim would agee to elect a Paeto optimum equilibium. Othewie both fim would obtain le pofit than the one obtained by chooing a Paeto optimum. In the following, we dicu the cae in which the fim chooe an equilibium that maximize the aggegated pofit. Thi i jutified by the fact that thi equilibium i a Paeto optimum fo the fim, and it i not unuual fo competito to each ageement that benefit both fim. Futhemoe, an equity citeion i alo conideed to elect a location equilibium that guaantee a minimum level of aveage pofit pe facility to each competing fim ILP fomulation fo electing a location equilibium In the above mentioned ituation, if X i an optimal olution of poblem (P) (an ( + ) MEDIAN poblem), any patition of X into two ubet, X 1 and X 2,of and location in X, epectively, i a location equilibium. Thee exit a plethoa of poible citeia to elect one of the many patition of Table 1 Facility location Label Node Name of city Population 1 14 Albatea (Alicante) Bacelona Sta. Coloma de Cevello (Bacelona) Letona (Alava) Ooo (La Couña) Hueto Taja (Ganada) Cato del Rei (Lugo) A Fonagada (Lugo) Alcocón (Madid) Madid Manzanae el Real (Madid) Te Canto (Madid) Oviedo (Atuia) Aahal (Sevilla) La Puebla de Cazalla (Sevilla) Sevilla Aielo de Malfeit (Valencia) Calet (Valencia) Ontinyent (Valencia) Valencia Ugao-Miaballe (Vizcaya) Zaagoza a ocial cot minimize (optimal olution of (P)). We will deal with the election of one patition that maximize the aggegated pofit the competing fim get. The aggegated pofit coeponding to a patition X 1, X 2 i: Π a (X 1, X 2 ) = n (pk 1 (X 1 ) + pk 2 (X 2 ))q k 2S(X 1, X 2 ) whee S(X 1, X 2 ) = S, S being the minimum ocial cot (optimal value of poblem (P)). Let p ik = pik 1 = p2 ik. If we define the 0 1 vaiable d e, l X, whee d l =1 indicate that location l i aigned to fim 1, and d l = 0 mean that l i aigned to fim 2; and the continuou vaiable pk u, u = 1, 2, that epeent the minimum deliveed pice to eve maket k fom ome facility location of fim u. Then, the poblem of finding the patition of X that maximize the aggegated pofit can be fomulated a: (P a (X)) : max n (pk 1 + p2 k )q k 2S.t. p 1 k p lkd l +D(1 d l ), k K, l X (7) pk 2 p lk(1 d l )+Dd l, k K, l X (8) d l = (9) l X d l {0, 1}, pk 1 0, p2 k 0, k K, l X (10) whee D i a fixed numbe geate than the maximum maginal deliveed pice. Containt (7) guaantee that pk 1 i not geate

6 734 Jounal of the Opeational Reeach Society Vol. 62, No. 4 than the minimum maginal deliveed pice of fim 1. Similaly, containt (8) guaantee that pk 2 i not geate than the minimum maginal deliveed pice of fim 2. The numbe D i ued o a containt (7) and (8) hold fo any aignment to vaiable d l. In the optimal olution, both vaiable pk 1 and pk 2 ae equal to the minimum deliveed pice in maket k of fim 1 and fim 2, epectively. Thu, the optimal value of poblem (P a (X)) i the maximum aggegated pofit the fim can obtain. If d l, p 1 k, and p2 k, i an optimal olution to poblem (P a (X)), then the optimal patition i: X 1 ={l X : d l = 1} X 2 ={l X : d l = 0} and the maximum aggegated pofit i: Figue 2 Demand point and facility location. Table 2 Π a (X 1, X 2 ) = Location equilibia with maximum aggegated pofit + S(X) X 1 X 2 Π1 (X 1,X 2 ) n (p 1 k + p2 k )q k 2S Π 2 (X 1,X 2 ) , ,15, ,15 2, ,10,15, ,18 8,10, ,4,10,14, ,14 2,4,10, ,14,19 4,7, ,4,6,10,16, ,16 2,4,7,10, ,7,16 2,4,10, Fo the fim to agee on the choice of the location equilibium peviouly obtained ome equity citeion hould be atified. Fo intance, the aveage pofit pe facility each fim obtain hould be cloe to the global aveage pofit pe facility that i defined a the aggegated pofit divided Π a ,4,6,10,13,16, ,13 2,4,6,10,16, ,13,16 2,4,6,10, ,6,13,16 2,4,10, ,6,10,13,16,19,21, ,13 2,6,10,16,19,21, ,13,21 2,6,10,16,19, ,6,13,16 2,10,19,21, ,2,6,10,13,16,20,21, ,13 1,2,6,10,16,20,21, ,13,21 1,2,6,10,16,20, ,6,13,16 1,2,10,20,21, ,6,10,13,16 1,2,20,21,

7 B Pelegín-Pelegín et al Location equilibia fo competing fim 735 Table 3 Reult fo λ = X 1 X 2 Π 1 Π 2 λ Πa + Π a λ Π a , ,15, ,18 11, ,10,15, ,18 8,10, ,7,10,14, ,14 2,4,10, ,14,19 4,7, ,6,7,10,16, ,16 2,4,7,10, ,7,16 2,4,10, ,5,6,10,13,16, ,16 2,4,5,10,13, ,6,16 2,4,10,13, ,6,13,16 2,4,10, ,6,10,13,16,19,21, ,16 2,5,10,13,19,21, ,6,16 2,10,13,19,21, ,6,13,16 2,10,19,21, ,5,6,10,13,16,20,21, ,16 1,2,5,10,13,20,21, ,13,21 1,2,6,10,16,20, ,6,13,16 1,2,10,20,21, ,6,10,13,16 1,2,20,21, by the total numbe of facilitie (Π a (X 1, X 2 )/( + )). Howeve, it i poible that, fo a patition that maximize the aggegated pofit, one of the fim get much le pofit pe facility than it competito, and theefoe ome kind of compenation would be equied in ode to obtain an ageement. An altenative way of electing a patition i by including equity containt in the above fomulation. The aim of uch containt i to detemine a location equilibium, o that both fim get imila pofit pe facility, if uch equilibium exit. Let Π a denote the maximum aggegated pofit, which can be obtained by olving poblem (P a (X)). Fo any λ, 0 λ 1, we conide the following equity containt: the aveage pofit pe facility each fim obtain i geate than, o equal to, λπ a /( + ). Then, we can obtain a location equilibium veifying that containt, fo which aggegated pofit i maximum, by olving the following ILP poblem: n (Pλ a (X)) : max (pk 1 + p2 k )q k 2S.t. pk 1 p lkd l + D(1 d l ), k K, l X (11) pk 2 p lk(1 d l ) + Dd l, k K, l X (12) d l = (13) l X 1 ( n ) pk 2 q k S λ Πa + (14)

8 736 Jounal of the Opeational Reeach Society Vol. 62, No. 4 Table 4 Reult fo λ = X 1 X 2 Π 1 Π 2 λ Πa + Π a λ Π a , ,18 11, ,18 8,10, ,7,10,14, ,14 2,4,10, ,14,19 4,7, ,6,7,10,16, ,17 4,6,7,10, ,7,16 2,4,10, ,5,6,10,13,16, ,17 4,5,6,10,13, ,5,13 4,6,10,16, ,5,10,13 2,6,16, ,6,10,13,16,19,21, ,19 5,6,10,13,16,21, ,19,22 5,6,10,13,16, ,6,13,16 2,10,19,21, ,5,6,10,13,16,20,21, ,16 1,2,5,10,13,20,21, ,6,16 1,2,10,13,20,21, ,6,13,16 1,2,10,20,21, ,6,10,13,16 1,2,20,21, ( n 1 p 1 k q k S ) λ Πa + (15) d l {0, 1}, pk 1 0, p2 k 0, k K, l X (16) Obeve that (P λ a(x)) educe to (Pa (X)) fo λ = 0. In ode to elect a location equilibium, a equence of poblem (P λ a (X)) can be olved fo fixed inceaing λ value until one not feaible poblem i found. If λ i the geate value of λ fo which (P λ a (X)) i feaible, fim could elect the location equilibium given by the following patition: X 1 ={l X : d l = 1} X 2 ={l X : d l = 0} whee d l ae the optimal value fo vaiable d l in poblem (P ā λ (X)) An illutative example We peent an example with eal data in ode to illutate ocial cot minimization and how to elect a location equilibium when multiple location equilibia exit. Both maket aea and location candidate ae the Spanih citie on the Ibeian Peninula with a population of ove 4000 people (ie 1046 citie). A demand, q k, a popotion of the total population of city k wa taken. Deliveed pice p ik wa taken popotional to the Euclidean ditance between citie i and k. The geogaphical coodinate and population of each city (fom the 2001 cenu) wee obtained fom and epectively. The ocial cot minimization poblem (( + ) MEDIAN poblem) wa olved fo + = 2, 3,..., 10.

9 B Pelegín-Pelegín et al Location equilibia fo competing fim 737 Table 5 Reult fo λ = X 1 X 2 Π 1 Π 2 λ Πa + Π a λ Π a ,18 11, ,18 8,10, ,19 4,7,10, ,7,10 2,14, ,6,7,10,16, ,17 4,6,7,10, ,7,10 2,6,16, ,5,6,10,13,16, ,17 4,5,6,10,13, ,5,13 4,6,10,16, ,5,10,13 2,6,16, ,19,22 5,6,10,13,16, ,19,21,22 5,6,10,13, ,5,6,10,13,16,20,21, ,5,13 1,6,10,16,20,21, ,2,20,22 5,6,10,13,16, ,6,10,13,16 1,2,20,21, Fo the optimal olution X of each ( + ) MEDIAN poblem, we olved both poblem (P a (X)) and poblem (P λ a(x)), λ = 0.5, 0.6, 0.7, 0.8, 0.9, fo all combination of and. Optimal olution to thee poblem ae location equilibia. All thee ILP poblem wee olved by uing the optimize FICO Xpe-Moel (2009). The diffeent optimal location obtained when olving all the above mentioned ( + ) MEDIAN poblem ae hown in Table 1. The fit column i a label to epeent each location. Column 2 and 3 epeent the numbe of the node and the name of the coeponding city. The fouth column i the numbe of inhabitant of each city. The geogaphical ituation of the citie i hown in Figue 2. The eult fo the (P a (X)) poblem (location equilibia maximizing aggegated pofit) ae hown in Table 2, whee the ocial cot (S(X)), the location equilibium (X 1 and X 2 ), the aveage pofit pe facility each fim obtain ( Π1 (X 1,X 2 ) and Π2 (X 1,X 2 ) ), and the maximum aggegated pofit (Π a ),ae given fo each one of the 25 cenaio conideed. It i poible to obeve how ocial cot S(X) educe a the numbe of facilitie to locate ( + ) inceae. The eduction in ocial cot pe each additional facility i moe impotant when the numbe of facilitie to locate i mall. The location equilibium i quite untable to change in the numbe of facilitie to locate. Nevethele, a the numbe of facilitie inceae, thee eem to be a table goup of location coincident in all cenaio. The mot fequently appeaing node ae 2, 10, 16, and 6, in thi ode. Note that diffeence in aveage pofit pe facility between fim ae in ome cae vey impotant. Fo example, when + = 8, aveage pofit pe facility of fim 2 i moe than even time the aveage pofit pe facility of fim 1 when the diffeence in the numbe of facilitie i high ( = 1, = 7), and moe than two time when both fim locate the ame numbe of facilitie ( = = 4). Fo any fixed

10 738 Jounal of the Opeational Reeach Society Vol. 62, No. 4 Table 6 Reult fo λ = X 1 X 2 Π 1 Π 2 λ Πa + Π a λ Π a ,18 11, ,18 8,10, ,19 4,7,10, ,7,10 2,14, ,17 4,6,7,10, ,7,10 2,6,16, ,5,17 4,6,10,13, ,5,10,13 2,6,16, ,19,22 5,6,10,13,16, ,19,21,22 5,6,10,13, ,6,16 1,5,10,13,20,21, ,2,20,22 5,6,10,13,16, ,6,10,13,16 1,2,20,21, value of +, it i obeved that aveage pofit pe facility of a fim inceae when the numbe of it facilitie inceae. On the othe hand, it i poible to ee that the aggegated pofit educe a educe too. Thi i motivated by the effect of the inceae in competition, and theefoe the eduction in individual pofit. Obeve that diffeent pofit ae obtained when the fim have the ame numbe of facilitie. Thi i explained by the fact that cutome buy fom the cloet facility, then the maket captued by the fim ae diffeent (fo intance, ee in Table 2 the optimal facility location obtained fo = = 2), and the fim obtain diffeent pofit. The eult fo the (P λ a (X)) poblem ae hown in Table 3 to 7. Thee table contain location equilibia maximizing aggegated pofit but veifying the equity containt. The optimal value of (P λ a(x)) i denoted by Πa λ and it i hown in column 9. Thee eult ae ummaized in Figue 3, whee fo each λ value the fit ba contain thee level: numbe of cenaio in which the found equilibium i diffeent fom the one obtained fo λ = 0 (top level); numbe of cenaio in which the found equilibium i the ame a the one obtained fo λ = 0 (mid level); and numbe of cenaio in which the equilibium i lot (bottom level). The econd and thid ba ae the maximum pecentage deviation and the aveage pecentage deviation of Π a λ with epect to Π a, epectively. The pecentage deviation i meaued by 100(Π a Π a λ )/Πa. We ee that fo ome cenaio, a λ inceae, it i not poible to find any equilibium that atifie the equity containt. Futhemoe, new location equilibia fo othe cenaio may appea when equity containt ae conideed. Thu, fo λ = 0.5 thee i one cenaio without location equilibium and 12 cenaio with a new location equilibium, and fo λ = 0.9 thee ae ix cenaio whee the location equilibium i lot in which

11 B Pelegín-Pelegín et al Location equilibia fo competing fim 739 Table 7 Reult fo λ = X 1 X 2 Π 1 Π 2 λ Πa + Π a λ Π a ,7,10 2,14, ,17 4,6,7,10, ,5,10,13 2,6,16, ,19,22 5,6,10,13,16, ,19,21,22 5,6,10,13, ,2,20,22 5,6,10,13,16, ,2,20,21,22 5,6,10,13, thee wa a location equilibium fo λ = 0.8. Fo λ = 0.9 thee i a location equilibium in even out of 25 cenaio. Obeve that the width of the bottom level inceae while the width of the mid level deceae when λ inceae. The maximum pecentage deviation of Π a λ with epect to Πa i 25, which i obtained fo λ = 0.8. The aveage pecentage deviation inceae when λ inceae and it vaie fom 5.9 to Fo each λ value, we have alo evaluated the pecentage deviation of the aveage pofit pe facility of each fim with epect to the global aveage pofit pe facility which ae given by 100 (Π1 / Π a /+) and 100 (Π2 / Π a /+), epectively. Thu, the pecentage deviation obtained fom Table 2, Π a /+ Π a /+ coeponding to + = 4ae 55,3 ( = 1) and 19,7 ( = 2) fo fim 1 and 18,4 ( = 3) and 19,8 ( = 2) fo fim 2, then the aveage deviation ae 17,8 fo fim 1 and 0,7 fo fim 2. The aveage deviation coeponding to the Figue 3 Reult fo the (P λ a (X)) poblem. two fim fo each + value, without equity containt (λ = 0) and with equity containt (λ = 0.5,...,0.9), ae hown in Figue 4, whee it i obeved that deviation above

12 740 Jounal of the Opeational Reeach Society Vol. 62, No. 4 Figue 4 the global aveage pofit pe facility ae malle than deviation below the global aveage pofit pe facility. Nomally, one fim deviate above and anothe deviate below, but in ome cae both fim deviate below the global aveage pofit pe facility (fo intance, λ = 0.7 and + = 7, 8,10). Obeve that deviation deceae when λ inceae. The maximum deviation fo λ = 0.9 vaie fom 2% ( + = 6) to 8% ( + = 9). Finally, a a way of evaluating the lo of efficiency (eduction in aggegated pofit) in the popoed olution when intoducing equity containt, we compae Table 2 and 7 (λ = 0 and λ = 0.9, epectively) with epect to the even cenaio with equilibium. It can be obeved how in two cenaio the olution popoed in each one of them i the ame in both table, and theefoe efficiency i not lot ( = = 3and = = 5); in thee cenaio the eduction in aggegated pofit i cloe to 2% ( = 2, = 5; = = 4; = 4, = 6); and in the othe two cenaio thi eduction i cloe to 5% ( = 3, = 6; = 4, = 5). Aveage deviation fo each + value and each λ value. 6. Extenion to oligopoly and concluion The peviou eult fo two fim can diectly be extended to oligopoly. In fact, long-tem pice competition fo a fixed numbe U of fim, which locate eveal facilitie each, lead to a pice equilibium in which each fim et the lowet deliveed cot of it competito a the pice in maket k, if that fim i the only one with minimum deliveed cot in maket k. Othewie, the fim et it minimum deliveed cot a the pice in maket k. In a imila way to the duopoly cae, equilibium pice ae detemined fo the choen facility location. Fo any location pace, it i poible to ee that the location of the fim ae in equilibium if each fim minimize the ocial cot with epect to the competito fixed location. In the cae of the location pace being a netwok, a location equilibium alo exit at the node. A in the duopoly cae, a location equilibium can be obtained by minimizing ocial cot. Thi poblem can be fomulated in dicete pace a an ILP poblem by taking

13 B Pelegín-Pelegín et al Location equilibia fo competing fim 741 vaiable xi u and zik u, u = 1,...,U, which in tun become a ( U u=1 u ) MEDI AN poblem if location candidate and deliveed pice ae the ame fo all competing fim, whee u i the numbe of facilitie to be located fo fim u. Futhemoe, ILP fomulation imila to the one in the duopoly cae, can be ued to elect a location equilibium when equity containt ae conideed. We have peented a geneal famewok of the locationpice Betand game whee minimize of the ocial cot ae location equilibia, and hown that a location equilibium in dicete location pace can be found by olving an ILP poblem. A cae whee multiple location equilibia exit ha been conideed, fo which ILP fomulation, without and with equity containt that depend on a paamete λ, 0 λ 1, have been popoed to elect one location equilibium. The highe the value of λ the moe equity in pofit pe facility fo each competing fim i obtained, howeve a eduction in aggegated pofit may occu. An example with eal data ha been olved in 25 cenaio to point out the exitence of multiple equilibia and how the equity containt affect them. Fo the highet value of λ tudied (λ=0.9), thee exit a location equilibium in moe than 25% of the cenaio conideed. In all the cenaio with equilibium fo λ = 0.9, the lo of efficiency (eduction in aggegated pofit) in le than 5%, and in 5 of thee 7 cenaio the lo of efficiency i le than 2%. In a netwok we have hown that, unde two common aumption, the poblem of detemining a location equilibium i educed to a dicete poblem in which the location candidate ae the node. Then the popoed pocedue can be ued when the location pace i a netwok. Acknowledgement Thi eeach ha been uppoted by the Minity of Science and Technology of Spain unde the eeach poject ECO /ECON and ECO /ECON, in pat financed by the Euopean Regional Development Fund (ERDF). Refeence Avella P, Saano A and Vailev I (2007). Computational tudy of lage-cale p-median poblem. Math Pogam Se. A 109: d Apemont C, Gabzewicz JJ and Thie JF (1979). On Hotelling tability in competition. Econometica 47: Dota-González P, Santo-Peñate DR and Suáez-Vega R (2005). Spatial competition in netwok unde deliveed picing. Pap Reg Sci 84: Eielt HA (1992). Hotelling duopoly on a tee. Ann Opn Re 40: Eielt HA, Lapote G and Thie JF (1993). Competitive location model: A famewok and bibliogaphy. Tanpot Sci 27: FICO Xpe-Moel (2009). Fai Iaac Copoation. Bliwoth: Nothamptonhie, UK. Fanci RL, Lowe TJ and Tami A (2002). Demand point aggegation fo location model. In: Dezne Z and Hamache H (ed). Facility Location: Application and Theoy. Spinge: Belin-Heidelbeg, pp Gabzewicz JJ and Thie JF (1992). Location. In: Aumann R and Hat S(ed).Handbook of Game Theoy with Economic Application. Elevie: Amtedam, pp Gacía MD and Pelegín B (2003). All Stackelbeg location equilibia in the Hotelling duopoly model on a tee with paametic pice. Ann Opn Re 122: Gacía MD, Fenández P and Pelegín B (2004). On pice competition in location-pice model with patially epaated maket. TOP 12: Gupta B (1994). Competitive patial pice dicimination with tictly convex poduction cot. Reg Sci Uban Econ 24: Hamilton JH, Thie JF and Wekamp A (1989). Spatial dicimination: Betand v. Counot in a model of location choice. Reg Sci Uban Econ 19: Hoove EM (1936). Spatial pice dicimination. Rev Econ Stud 4: Hotelling H (1929). Stability in competition. Econ J 39: Ledee PJ and Hute AP (1986). Competition of fim: Diciminatoy picing and location. Econometica 54: Ledee PJ and Thie JF (1990). Competitive location on netwok unde deliveed picing. Opn Re Lett 9: Obone MJ and Pitchik C (1987). Equilibium in Hotelling model of patial competition. Econometica 55: Platia F (2001). Static competitive facility location: An oveview of optimiation appoache. Eu J Opl Re 129: Revelle CS and Eielt HA (2005). Location analyi: A ynthei and auvey.eu J Opl Re 165: Rolland E, Schilling DA and Cuent JC (1996). An efficient tabu each pocedue fo the p-median poblem. Eu J Opl Re 96: Sea D and ReVelle C (1999). Competitive location and picing on netwok. Geog Anal 31: Zhang S (2001). On a pofit maximizing location model. Ann Opn Re 103: Received Octobe 2008; accepted Novembe 2009 afte two eviion