Capactated Locaton-Allocaton Problem n a Compettve Envronment At Bassou Azz, 2 Blal Mohamed, 3 Solh Azz, 4 El Alam Jamla,2 Unversty Mohammed V-Agdal, Laboratory of Systems Analyss, Informaton Processng and Industral Management (LASTIMI), Mohammada School of Engneers, Rabat, Morocco. 3 Prof., Natonal Hgh School of Mnes, Rabat, Morocco. 4 Prof., Unversty Mohammed V-Agdal, LASTIMI, Superor School of Technology Sale, Sale, Morocco. ORCID : 0000-000-5550-687, 2 ORCID: 0000-000-7837-8909 Abstract In ths paper, we propose a game theoretc model to solve the Capactated locaton-allocaton problem n a compettve envronment for a two level dstrbuton Network. The problem s a varant of the TLCLAP problem. We study the locaton allocaton decsons for two frms that wsh to extend ther dstrbuton networks. Through the ntegraton of several economc parameters, we propose an algorthm for fndng Nash equlbrum n two-player games from the proposed mathematcal model. Thus, we assumed the case of mxed strateges to establsh the frms' best response functons. An optmzaton approach s proposed to fnd equlbrum under capacty constrant. To provde a better analyss of the problem, fve scenaros were gven. Computatonal results show the effectveness of the proposed approach. Keywords: Locaton allocaton problem, Nash equlbrum, Game locaton, Dstrbuton Network INTRODUCTION Large-scale dstrbuton s experencng a boomng n recent years despte a slower growth n economes, saturaton of markets, a contnues costs rsng, fragmented markets and an ncreasng compettve envronment [].In ths context, the openng of one or more stores s one of the key decsons that must be taken to deal wth ths stuaton. In addton, the market study for the mplantaton, the number of stores to be opened and the fnancal capacty of a dstrbuton busness must take nto account the projected market share, whch s mpacted by the competton factor. Thus, the present paper ams to solve the problem of locaton-allocaton n a compettve envronment. LITERATURE REVIEW The problems of faclty locaton have been studed thoroughly durng the last decades, due to the large varety of applcatons based on ths knd of problems. The problem of locaton and allocaton was ntroduced by Alfred Weber who focused n ndustral factory ste mplantaton. Later, the noton of spatal competton n a stuaton of duopoly was ntroduced by []. In hs book, he has hghlghted the compettve nteractons between two stores n a dstrbuton network, based on the prncple of homogeneous dstrbuton of clents to determne the optmal locaton of two stores wth smlar characterstcs. In the same research focus, [2] ntroduced the competton wth mult-stores n the orgnal model [] and shows that there s no equlbrum to a pure strategy. The study of the problem of mult-store locaton on a dscrete space was the subject of the work of [3] that have generated a great nterest among researchers and a large-scale dstrbuton companes. Moreover, the assumpton that transport costs are quadratc n dstance, [4] show that companes tend to open one store. of [5] who frst ntroduce the basc elements of locaton models n a compettve envronment followed by the basc model ntroduced by []. Lately, the noton of competton n the problems of locaton/allocaton has become ncreasngly studed. For the locaton of facltes n a logstcs network, the approach adopted s to make a modelng n the form of a game whose soluton s gven by a Nash equlbrum [6]. In addton, t s mportant n any decson of locaton that the compettors revew ther strateges to constantly nnovate n order to dfferentate from the others players (compettors) [7]. The proxmty of the clent, the nteractons wth the market of the large dstrbuton referrng especally to the strategy of localzaton, and response to changes n demand offers an advantage for the company n the decson of localzaton [8]. Game theory and competton In general, the models of competton n economcs are based on the applcaton of the game theory. The objectve s to study the nteractons of the behavors of several ndvduals, FRIEDRICH, Theory of the Locaton of Industres, Unversty of Chcago Press, 929 334
called players, who are aware of these nteractons. The Nash equlbrum", s a fundamental noton n ths theory. In a game theory, a strategy can be reduced to an elementary decson, but can also be a complex acton plan. The notons of Nash equlbrum and the best response functon are fundamental. Indeed, they represent the solutons to the problems studed n game theory. Duopoly and olgopoly In the context of compettve strateges, frms make decsons n response to compettors. Ths reacton creates a stuaton of strategc nteractons that s accentuated by an atomstc demand n the market. In the case of two frms that compete wth one another, we speak of a duopoly. we try to llustrate the nterest of ths theory n these compettve stuatons. In the case of Bertrand's duopoly, frms compete by prces. We denote by p and p 2, the prces that the frms F and F2 must apply. These prces defne the requests addressed to F and F2. We denote by D (p, p 2 ) the demand addressed to frm, =,2, D the aggregate demand on the market and C, C 2 are respectvely the constant unt costs for frms F and F2. Each frm seeks to maxmze ts proft n the market. Therefore, the problem of the frm s formulated as follows: max π (p C )D (p, p 2 ), =,2 We assume that the frms are symmetrc n costs, C = C 2 = C and each frm serves the entre demand addressed to t. In attemptng to analyze how prce polcy can affect the compettve stuaton, we can notce that: - If p > p 2, customers wll turn to F2, whch offers low prces for the same products offered by F. Then the total demand wll be satsfed by F2. We thus have: D (p, p 2 ) = 0 and D 2 (p, p 2 ) = D; - If p < p 2, customers wll go to the frm F whch offers low prces for the same products offered by F2. Then the total demand wll be satsfed by F. We thus have: D (p, p 2 ) = D and D 2 (p, p 2 ) = 0; - If p = p 2, the demand wll be shared equally between F and F2, so we have : D (p, p 2 ) = D 2 (p, p 2 ) = D 2. Thus, the unque equlbrum of Bertrand's duopoly wth symmetrc costs s characterzed by: p = p 2 = C, whch gves: π = (p C)D (p, p 2 ) = 0. Locaton models n a compettve envronment In the models of duopoly presented by Cournot, Bertrand, the decsons of the two competng companes are taken smultaneously. The optmal solutons for these concurrent decsons are called "Nash equlbrum" or sometmes called Cournot-Nash equlbrum.the prelmnary work of locatng facltes smultaneously has been proposed n [9]. Thus, snce sequental decson-makng leads to an asymmetry between decson-makers, we must dfferentate the dentty of decson-makers. In the case of a duopoly, the company that starts wth makng the locaton decson s called the leader and the other s called the follower. The Stackelberg model s based on three major assumptons: Decsons are made once and for all, Decsons are made sequentally, The leader and the follower have a complete knowledge of the localzaton game. An extenson of the aforementoned model s gven by the sequental localzaton model proposed by [0]. It deals wth the case where a frm locates n nstallatons, but the compettor frm locates only one nstallaton. Recently, a cooperatve competton n faclty locaton problems n whch potental nvestors are n competton over acqurng sutable stes and clents s studed n []. An acceptance threshold constrant s appled to faclty allocaton that s based on a combnaton of dstance between a faclty and clents, and nvestors product prces. PROBLEM DESCRIPTION The problem consdered n the paper s a two level supply chan network nvolvng multple demand ponts and two frms that have ther own warehouses. Each frm tres to expand ts dstrbuton network by openng new stores. The objectve for both frms s to have addtonal market shares. Each frm n our paper faces demand from customers. In our problem, t s assumed that one of the two frms already has stores that are open to the market. The problem for every frm s to determne the exact locaton of the stores that wll open. Then, these stores wll have to supply a set of demand ponts. Smlarly, we wll have to determne for each store ts warehouse that wll supply t. Thus, the objectve s to determne a dstrbuton network that wll provde maxmum proft on the market. Modelng problem In ths secton, we wll formulate the Capactated locatonallocaton problem n a compettve envronment for a two level dstrbuton Network. Indeed, t s a more realstc model than the model TLCLAP [2]. The problem s consdered as localzaton game that s played between two players (frms) who want to locate stores among a predefned set of potental stes so as to satsfy demand ponts and maxmze proft. Formulaton Now we defne the notaton used n: Number of potental stes 335
m: Number of frm pre-exstng opened stores N: Number of demand ponts : Index of potental stes : {,2,.., n} j: Index of demand ponts j = {,2,.., N} f: Index of frms f = {,2} w j : Demand quantty expected by ste from demand pont j C f : Transportaton cost pad by frm f to supply stores from warehouses Pr f : The prce fxed by the frm f F f : Fxed cost assocated wth frm f for openng a store at ste Γ f : Capacty avalable of store, estmated by a frm f. A j : Attractveness of demand pont j wth respect to store Decson varables: f X j : Bnary varable ndcatng whether demand pont j s assgned to store of frm f Z f : Bnary varable ndcatng whether store s opened by frm f Prce polcy s not defned by frms. We may consder that we are dealng wth a factory prce polcy [3]snce the prce n each locaton s fxed and the customers nsure ther own transport. The prce s ndependent of the chosen stes and s set by each frm. The fact that customers provde ther own transport justfes the fact that demand wll not ncrease wth dstance, and customers of a demand pont wll tend to frequent the store wth greater attractveness. Smlarly, t s assumed that both frms have the same Transportaton costng product from warehouses to ther stores. For smplcty, ths cost s assumed to be fxed. As a game, we consder certan rules of the game. These rules are defned as follows: - The pre-establshed stores of frm reman open; - Whenever the two frms decde to locate a servce on a gven ste, only frm s able to do so. Thus, co-locaton s forbdden. Ths can be wrtten as follows: Z =(-Z 2 ); - A customer belongng to a demand pont j wll be served by a store wth more attractveness. Ths can be wrtten as follows: f X j = { Z f f A j A kj 0 otherwse Frm decdes to open exactly p stores, ths s expressed by the equaton: n Z = p It can also be seen that the total cost of openng stores can be wrtten as follows: φ(z, Z 2 n ) = F Z To explan the problem, we defne the proft equatons for each frm. Ths proft corresponds to the dfference between the revenues and operatng cost of each frm. Takng the case of frm, we have ts charge whch corresponds to the Transportaton costng the total quantty of products between an open node j and warehouse and the cost of openng node j. Moreover, the frm's total revenue corresponds to the product of the quanttes and prces appled by the frm. The proft formula s gven as follow: π (σ, σ 2 ) = (Pr C ) N j n+m w j X j n F Z Wth σ and σ 2 are the locaton and allocaton strateges respectvely for frm and frm 2. Thus, σ can be expressed as a par (X j, Z ) and σ 2 can be expressed as a par (X 2 j, Z 2 ). For frm 2, ts proft equaton s as follows: π 2 (σ, σ 2 ) = (Pr 2 C 2 ) w j N j n 2 X j n 2 F 2 Z In the context of a game of Nash, σ, σ 2 are actons for frm and frm 2, respectvely, whch we wll call player and player 2. These actons concern store locatons and ther assgnments to a set of demand ponts. We want to make predctons about the outcome of the game. In some cases we wll be able to fnd an acton for each player so that each player's acton s the best response to the other's acton. Ths means that even f one player can guess the other's acton, he wll not be able to change hs chosen acton, and ths s vald for both players. Calculatng Nash Equlbrum Nash's theorem guarantees the exstence of a Nash equlbrum, possbly nvolvng mxed strateges for one or both actors, n each non-cooperatve fnte set such as the one consdered n ths paper. Moreover, t s mportant to note that, gven the strateges, we do not assume that frms choose a random acton, but only that the acton of each frm can be seen as random by the other frm [4]. Based on the approach adopted by [5], a Nash equlbrum s calculated for the game. Thus, an algorthm based on the best 336
responses of each player to the strategy of the other. Let σ t and σ 2 t be the vectors of strategy such that σ t s strategy t taken by the frm and σ 2 t by the frm 2. To smplfy the mplementaton of the algorthm, we assume that we wll start wth the localzaton. Snce we consder that the game s played smultaneously we suppose that the Player always has the prvlege of startng the game. Ths player starts by choosng, randomly, a locaton strategy σ. Then, t wll have to buld ts dstrbuton network by: - Assgn to each open ste a set of demand ponts so as to respect the capacty constrant; - Assgn to each open ste ts supply warehouse The second player looks for the best response for the strategyσ. We note ths strategy σ 2. Player 2 have also to buld hs network dstrbuton wth the same procedure adopted by player. Faced wth ths strategy, player must fnd the best response that wll be noted by σ 2. The game contnues, untl we fnd the Nash equlbrum. To better understand how ths game wll enable us to seek the resoluton of the problem, we present the matrx payoff as follows: Frm Frm2 σ σ 2 σ s sequence where all the demand ponts are satsfed (assgned to a frm). Algorthm : Locaton allocaton game progress Whle (t<s) t=, Strategy ( σ f t ) Best response to Strategy ( σ f t ) t=t+ End whle The strategy of a player conssts, frstly, to locate the stes and then to allocate them to the demand ponts and the warehouses. Suppose t s frm that wll start the game, then t wll choose p canddate stes to open. Ths choce s made randomly. Then, the frm wll try to allocate each selected ste. The algorthm s based on the method of fndng the demand ponts havng a good attractveness wth respect to the chosen ste. In addton, ths allocaton must take nto account the capacty parameter of the nstallaton. However, t can be noted that n the algorthm the assgnment of the stes to the warehouses was omtted. Ths s due to the fact that the transport cost s fxed for each frm whch wll change nothng n the calculaton of the proft. Algorthm 2 gves how a frm's strategy s constructed. Note that we have marked a frm by f, to say that t s possble to start the game ether by frm or frm 2. σ 2 π, π 2 π 2, π 2 π s, π 2 Algorthm 2: Gettng a strategy locaton allocaton.. σ 2 s π s, π 2 s Fgure : Matrx of game payoffs Where, π s = π ( σ s, σ 2 s ) and π 2 s = π 2 ( σ s, σ 2 s ) Moreover, by referrng to the work of (Godnho and Das 200), when one begns wth an acton n whch a gven player does not open any store, he tends to lead the algorthm to reach a balance whch slghtly benefts the other player. In order to avod ths bas, n computatonal experments we wll always apply the algorthm twce: The frst conssts n startng wth a zero acton for frm (e the acton n whch the frm does not open a store) and the second conssts of a null acton for frm 2. As we can see n algorthm, a sequence of actons s played. A player chooses hs strategy then the other player seeks the best response to ths strategy. The game contnues untl reachng the maxmum value S, whch corresponds to the Strategy ( σ f t ) Begn End Let E p be a set of p potental stes, E p {,2,.., n}; Frm f: Choose randomly E p For each k n E p do N Whle (Γ kf j= w kj X kj ) Begn for j= to N do If A kj s maxmal then f X kj = End End whle Return σ f t 337
Aganst the strategy of frm, frm 2 must also take ts strategy whch wll present the correct answer to that taken by frm. Indeed, the dea descrbed n ths paper conssts n maxmzng the proft of frm 2 by choosng stes wth a low ste establshment cost. Ths choce wll concern a set of canddates' stes whose cardnalty s r. Ths set corresponds to the stes to be opened by frm 2. Algorthm to get a best response to a strategy σ f t Algorthm : Best response to Strategy σ f t Begn Choose the potental stes for whch capacty s mnmal F f s mnmal Let E r be a set of r potental stes, E r {{,2,.., n} E p }; For each k n E r do Whle (Γ kf j= w kj X kj ) Begn N For each j n Not Selected demand pont {,.,N} f X kj = If A kj s maxmal then End End whle Return Br(σ f t ) End As ths game s n a strategc form, the payoff matrx, establshed by the strateges gven by the algorthms, provdes all necessary nformaton for fndng the Nash equlbrum. Algorthm : Nash equlbrum procedure Begn For t= to S For t2= to S If ( Br(σ t )= σ 2 t2 ) and ( Br(σ 2 t2 )= σ t ) End Then ( σ t, σ 2 t2 ) s a Nash equlbrum σ t, σ 2 t2 are the solutons for the problem PROBLEM RESOLUTION To allow the model to be used as a tool of analyss, we developed an algorthm whch solves several scenaros. It was then mplemented n a computer va a programmng language. Experments Desgn In order to provde better solutons for a good analyss, an Object-Orented Programmng approach was chosen. Indeed, we have defned several classes of objects that represent all the elements of the game. Whch are as follows : The Node class: It s used to manage node objects. It has all the propertes for a node: node type (stores, warehouse or request pont), node ndex, node setup cost f t s to be opened as a store, and fnally the capacty. In addton to the propertes, ths class also ncludes the method of retrevng the ndex from the node, snce the matrces related to the dstrbuton network requre t. The Network class: It s used to manage the dstrbuton network. It contans all the propertes relatng to a network: the lst of objects of the node class, the number of nodes of the network, the matrx of requests for each node wth respect to another, the attractveness matrx for each node wth respect to another. In addton, two methods are mplemented: the method that allows to know the state of a node durng the course of a game, and the method of ntalzaton of the network. The latter conssts n creatng the network accordng to the number of fxed nodes, categorzng the nodes by type and nsertng logstc data of each node and defnng the nodes already opened by a gven frm. The Frms class: It allows to manage the frms that performs the allocaton locaton set. It ncludes all the propertes relatve to a player: The player's ndex, the prce he apples to the stores, the Transportaton costng the merchandse the network object representng the dstrbuton network of each frm, the lst of the stes to be opened and the assgnment matrx of each node. As for the methods, ths class ncludes two essental functons that are to locate and allocate. The Games class: Ths object class s used to manage the game. Ths class s responsble for generatng the game nstances related to stores to be opened by a frm, the number of nodes, logstcs cost lmts to allow generaton of random values. Consequently, t mplements the algorthm, and makes t possble to recover the results of each game. Smlarly, ths class allows you to calculate the profts for each player n a gven experment of the game. Instantaton of the problem The procedure for generatng each problem nstance runs as follows: A gven number of canddate stes are chooses among the nodes of the network. Ths choce s made accordng to the 338
capacty of each ste. Thus, only those nodes wth a large capacty are retaned. Ths crteron allows to satsfy as much as possble the capacty constrant. A frm defnes the number of stes to open. Through a random functon, we choose arbtrarly, among the canddate stes, those to be opened for each frm. For allocaton, a ste s assgned to a node wth the greatest attractveness. Ths operaton s repeated untl the maxmum of the network s covered. Program sequence The program was mplemented wth "C #" language, on a 2.53 Intel 5 machne wth 4 GB of RAM. A random functon was used to generate the problem nstances. Durng the ntalzaton phase,we defne the global logstc network wth the nodes and the dfferent logstc parameters, we create the players through ther parameters, and then the game starts wth an arbtrary chosen player. He performs two operatons: selectng stes to open and allocatng each ste to demand ponts. The other player performs the same approach untl fndng the best response to the choces made by the frst player. Numercal tests We conducted numercal tests to evaluate the relevance of the chosen approach to determne the equlbrum of the game. We start by the nstantaton of the tests objects. Indeed, a set of nstances based on a dstrbuton network has been generated n a random manner that nclude the number m and n. Those data are related to potental stes, logstcs values, warehouse locatons and demand ponts. Smlarly, an attractveness value has been assgned for each potental ste wth respect the nodes. Furthermore, n order to evaluate the varaton of the results accordng to partcular parameters of the problem, we defned 9 sets of experments. Each set ncludes nstances. On the other hand, because the objectve of the problem presented s to look for the localzaton and allocaton Table contans the results obtaned for the case of the scenaro. We note that n all the strateges consdered for frm, proft s always lower than that of frm 2. Ths s explaned by the number of open nodes. Indeed, ncreasng the number of stes to be opened for frm 2 enhance ts proft advantage over that of frm even f the preferental advantage has been granted to frm. Ths experment has been repeated several tmes and a proft rato of less than s solutons n the case of competton, we wll then consder how to look for possble Nash equlbrum. For ths purpose, we wll consder nstances of small sze so that we can use a smple algorthm wthout facng the constrant of the executon tme. Therefore, we defned a network wth 200 nodes. For each experment, we retans the proft of the frm and the proft of the frm 2 whch corresponds to the best response to the strategy of locaton and allocaton of the frm. We wll also be nterested n the relatve advantage of the Frm on frm 2, whch we measure as the rato between the two profts. To better analyze the establshed model, we have assumed 5 scenaros whch are as follows: Scenaro : Both frms opt for the same prce and the same freght cost. Frm opens p stores and frm 2 opens r stores wth p r 0,25; Scenaro 2: Both frms opt for the same prce and for the same freght cost. Frm opens p stores and frm 2 opens r stores wth 0,75 p r ; Scenaro 3: Frm 2 adopts a double prce compared to frm. Frm opens p stores and frm 2 opens r stores wth 0,75 p r ; Scenaro 4: Frm 2 whch adopted a double prce compared to frm. Frm opens p magaznes and frm 2 opens r magaznes wth 0,75 r p ; Scenaro 5: Frm 2 apples the same prce as frm. Frm opens p magaznes and frm 2 opens r magaznes wth 0,75 p wth advantage to frm 2. r Results and analyss In ths secton, we wll analyze the results obtaned through the varous pre-establshed scenaros n order to seek not only the solutons of the problem but also to examne the mpact of the mportant parameters defnng the problem. Thus, we wll study the mpact of the number of stores to be opened and the prces appled n the frms dstrbuton network. always obtaned and the average of the proft ratos always approxmates to 0.2. In Table 2, a smlar experence to that of scenaro was carred out except that ths tme we assgn 6 stes to be opened by frm and 8 stes by frm 2. It s noted that Always frm 2 has a compettve advantage snce only for strateges 4 and 8 t s surpassed by frm. In addton, there s always an average proft rato of 0.2. 339
Table : Scenaro results Strateges - Number of stes to be opened: (Frm: 2, Frm2: 0) - Transportaton cost: (Frm: 200, Frm2: 200) - Appled Prce: (Frm: 42, Frm2: 42) Proft Frm Proft best response Frm 2 Proft/Proft2 220,85997 6623,90342 0,8436523 2 266,52826 76844,37974 0,6476844 3 3387,70493 57447,30732 0,2330432 4 0038,09483 66249,2853 0,552005 5 6807,92038 6938,49872 0,2424738 6 950,558347 7442,56607 0,3299576 7 5686,2732 7246,2702 0,2647846 8 983,8235 59829,70603 0,64394 9 477,95583 60494,09202 0,24329576 0 609,06846 6783,25244 0,7279706 8598,23475 57007,63865 0,508260 2 828,65645 69520,36775 0,2620624 3 5593,0393 7028,24445 0,22206535 4 2042,67023 85478,5587 0,4088527 5 6597,22273 77033,59373 0,2545435 6 3666,37096 79599,63442 0,768887 7 896,66 56543,55659 0,2038854 8 2049,6269 63474,7666 0,8982603 9 9547,0788 64593,6523 0,3026827 Strateges Table 2: Scenaro 2 results - Number of stes to be opened: (Frm: 6, Frm2: 8) - Transportaton cost: (Frm: 200, Frm2: 200) - Appled Prce: (Frm: 42, Frm2: 42) Proft Frm Proft best response Frm 2 Proft/Proft2 3037,7286 5020,734 0,6039789 2 40504,69928 4693,34804 0,86306277 3 3708,7325 46589,3264 0,68060086 4 40529,4423 37609,864,07762804 5 3449,56309 40976,22004 0,83998873 340
6 32652,48806 4995,6996 0,6536822 7 396,06858 45349,90653 0,8625392 8 27558,62228 4302,38673 0,6407363 9 3637,0332 45306,8938 0,8057853 0 33242,6274 4397,9054 0,80300264 37,2923 43088,78969 0,73595226 2 24843,0977 4234,80748 0,58960984 3 36808,94592 39350,0409 0,9354282 4 3354,87079 4043,89092 0,872957 5 40342,98 3974,3439,02982993 6 478,0764 4738,96406 0,8863545 7 3552,22 42788,8492 0,82994095 8 3735,3532 44509,6785 0,83432082 9 2724,2446 49042,2935 0,55307802 In Experment 3, the objectve s to see the mpact of prce on equlbrum. Values gven n Table 3 ndcate that the prce has a certan mpact on the proft of the frms. As shown n ths table, the frm 2 whch has adopted a double prce compared to frm have a hgh proft for all strateges. But lookng at the prevous experments ths polcy wll be n van n case the frm adjusts ts opened nodes number. It s nterestng to study ths case and analyze the mpact of approprate number of nodes to reach a game equlbrum aganst a hgh prcng polcy of the compettor. To confrm the remark rased n scenaro 3, we conducted an experment n whch we keep the same assumptons as the prevous scenaro, except for the number of magaznes to open ( Table 2).. Indeed, we opted for 80 stores for frm wth a low prce compared to that of frm 2. We note that n strateges 8 and 3, the gan of frm s clearly hgher than that of the frm 2. Smlarly, t has been observed that, on average, the proft rato s close to 0.93. Ths means that aganst an unfavorable prce polcy, t s possble to revew the number of stores to reverse the stuaton to ts beneft and reach equlbrum of close profts. In ths way, the stes to open are qualfed as a best strategy for a frm to deal wth unfar prcng polcy of compettors. Table 3: Scenaro 3 results Strateges - Number of stes to be opened: (Frm: 80, Frm2: 40) - Transportaton cost: (Frm: 200, Frm2: 200) - Appled Prce: (Frm: 400, Frm2: 800) Proft Frm Proft best response Frm 2 Proft/Proft2 4350,8567 928,88 0,38873649 2 45504,92074 9868,977 0,4635367 3 43363,95078 94888,237 0,45700082 4 47380,5983 435,827 0,452696 5 46838,7938 0848,3729 0,43309753 34
6 48487,4365 06570,464 0,45498005 7 44820,6032 02236,5785 0,43840085 8 4254,90509 0354,6757 0,4097463 9 48482,0464 0643,462 0,4769862 0 4796,69936 04430,842 0,459425 42563,86026 99532,34249 0,42763849 2 45896,38975 06584,79 0,430666 3 4853,7029 02239,28 0,40937008 4 53709,80043 0969,008 0,5267267 5 452,53387 0287,546 0,4394967 6 48692,8330 0680,6229 0,4559847 7 42364,493 09835,2352 0,38570948 8 4550,3425 098,543 0,4698975 9 43460,40703 8534,97802 0,509422 In Scenaro 5, t s assumed that both frms decde to open the same number of stores, to use the same prce, and they have the same transport cost. Table 5, shows the results obtaned for ths scenaro. It can be noted that the frm always has the advantage and t has a consderable gan compared to the frm 2. Thus, the frm 2 has no nterest n keepng the same prces as those appled by the frm, or to downsze the number of opened stores compared to the compettor frm. Table 4: Scenaro 4 results Strateges - Number of stes to be opened: (Frm: 80, Frm2: 40) - Transportaton cost: (Frm :200, Frm2 :200) - Appled Prce: (Frm : 400, Frm2 : 800) Proft Frm Proft best response Frm 2 Proft/Proft2 56409,53223 678,0470 0,9398764 2 5938,88675 65395,9657 0,9080466 3 56803,64857 6498,02285 0,874575 4 566,0076 64900,38559 0,86464829 5 5945,3949 6094,52599 0,96809646 6 58200,599 64906,687 0,89668825 7 5929,634 57740,72382,02685985 8 58278,38024 58204,6238,002672 9 59389,95386 7328,67667 0,8326238 0 59292,63053 62858,68045 0,94326878 5650,6576 56867,9922 0,98739964 2 5893,6405 66053,22029 0,8800 3 6966,23709 59043,67786,04949826 4 59690,69943 60943,3255 0,97944605 342
5 59903,6373 66444,4086 0,9056352 6 57973,5002 6302,7643 0,9200279 7 5904,2457 62438,64038 0,9455583 8 58324,74033 64707,9978 0,9035289 9 58432,84624 65057,46209 0,8987285 Strateges Table 5: Scenaro5 results - Number of stes to be opened: (Frm: 35, Frm2: 40) - Transportaton cost: (Frme :200, Frme2 :200) - Appled Prce: (Frme : 400, Frme2 : 400) Proft Frm Proft best response Frm 2 Proft/Proft2 43300,60677 3806,50483,36375825 2 44599,09582 32029,0294,392458549 3 4938,24856 3276,2233,532759232 4 44642,98259 32766,70393,362449598 5 45868,27374 32844,6295,3965234 6 43,39625 378,8603,38272383 7 4643,2462 333,62397,482780839 8 49987,59868 32259,424,549550249 9 46043,35848 33000,94388,3952338 0 44569,70096 322,96873,387546 43088,68356 398,0049,3473227 2 42908,0734 3522,0462,28562339 3 44653,7005 30427,65835,467536542 4 44684,4890 28937,00393,54498879 5 4555,66552 36048,58845,252633389 6 45676,09503 3057,06476,54606789 7 45373,8374 30064,99899,5099383 8 4247,90866 3460,39079,22743285 9 4902,34863 33783,2559,45054593 It should be ponted out that besde the studed scenaros; we analyze through other experments the mpact of transport costs n the frst level of the dstrbuton network. It appears that ths parameter has lttle mpact on the profts of frms n relaton to the prce and the number of stes to be opened. As a matter of fact, n a problem of locaton allocaton n a compettve envronment, the prce to apply remans an mportant parameter that gves a real compettve advantage. Nevertheless, ths parameter must be studed accordng to the number of stores to be opened by the compettor. But n realty, ths number depends closely on the nvestment budget of the frm. Ths budgetary constrant lmts the leeway of frms n settng the number of stores to open. Therefore, emphass should be placed on the prcng polcy to be adopted. 343
CONCLUSION The model we have developed deals wth the problem of localzaton of new stores and proposes a modelng of the dstrbuton network, an enumeraton of the parameters and logstc constrants lnked to the dstrbuton actvty as well as a resoluton approach by the game theory. The purpose s to determne the optmal locaton of the extensons to operate on the dstrbuton network that leads to a mnmzaton of the overall cost. The choce of locaton n an exstng dstrbuton network obvously takes n account a customer demand and servce objectves n the relevant geographcal area and t s concerned also by nternal and external logstcs and techncal parameters related to competton. The resoluton of the model requres an analyss of the locaton and allocaton strategy. Thus, face to a strategy of a gven frm, the compettor must develop the best response that lead to the Nash equlbrum of the game. Ths model, whch can be used to search for nvestment opportuntes n optmal areas, also proposes an analyss of the commercal actvty and the detecton of geographcal locaton possbltes, based on logstc performance crtera derved from the strategy of the company and takng nto account those of the compettors. The game theory used for ths problem takes nto account nteractons wth stores through the attractveness parameter whch has an mportant role n any locaton and allocaton strategy. However, ths work can be mproved n the future by other work ncludng the selecton of other specfc parameters, testng for large problems and comparson of results wth other methods wthn the spatal approach. REFERENCES [] H. Hotellng, Stablty n competton, Econ. J., vol. 39, no. 53, pp. 4 57, 929. [2] M. B. Tetz, Locatonal strateges for compettve systems, J. Reg. Sc., vol. 8, no. 2, pp. 35 48, 968. [3] D. L. Huff, Defnng and estmatng a tradng area, J. Mark., pp. 34 38, 964. [4] X. Martnez-Gralt and D. J. Neven, Can prce competton domnate market segmentaton?, J. Ind. Econ., pp. 43 442, 988. [5] H. A. Eselt and G. Laporte, EQUILmRIUM RESULTS IN COMPETITIVE LOCATION MODELS, 996. [6] N. Sadan, F. Chu, and H. Chen, Compettve faclty locaton and desgn wth reactons of compettors already n the market, Eur. J. Oper. Res., vol. 29, no., pp. 9 7, 202. [7] J. Vogel, Servce Management:Strategc Servce Innovaton Management n Retalng. Sprnger New York, 202, pp. 83 95. [8] Ó. González-Bento, P. A. Muñoz-Gallego, and P. K. Kopalle, Asymmetrc competton n retal store formats: Evaluatng nter- and ntra-format spatal effects, J. Retal., vol. 8, no., pp. 59 73, 2005. [9] H. Von Stackelberg, The theory of the market economy. Oxford Unversty Press, 952. [0] M. B. Tetz and P. Bart, Heurstc methods for estmatng the generalzed vertex medan of a weghted graph, Oper. Res., vol. 6, no. 5, pp. 955 96, 968. [] M. Rohannejad, H. Navd, B. V. Nour, and R. Kamranrad, A new approach to cooperatve competton n faclty locaton problems: Mathematcal formulatons and an approxmaton algorthm, Comput. Oper. Res., vol. 83, pp. 45 53, 207. [2] A. AIT BASSOU, M. Hlyal, A. Soulh, and J. El Alam, New varable neghborhood search method for a two level capactated locaton allocaton problem, J. Theor. Appl. Inf. Technol., vol. 83, no. 3, p. 442, 206. [3] H. A. Eselt, Equlbra n compettve locaton models, n Foundatons of locaton analyss, Sprnger, 20, pp. 39 62. [4] R. Gbbons, A prmer n game theory. Harvester Wheatsheaf, 992. [5] R. Porter, E. Nudelman, and Y. Shoham, Smple search methods for fndng a Nash equlbrum, Games Econ. Behav., vol. 63, no. 2, pp. 642 662, 2008. 344