EMPIRICAL EVALUATION OF A METAHEURISTIC FOR COMMERCIAL TERRITORY DESIGN WITH JOINT ASSIGNMENT CONSTRAINTS

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1 Proceedings of the 12th Annul Interntionl Conference on Industril Engineering Theory, Applictions nd Prctice Cncun, Mexico EMPIRICAL EVALUATION OF A METAHEURISTIC FOR COMMERCIAL TERRITORY DESIGN WITH JOINT ASSIGNMENT CONSTRAINTS Súl I. Cbllero-Hernández 1*, Roger Z. Ríos-Mercdo 1, Fbián López 2, nd Stu Elis Scheffer 1 1 Grdute Progrm in Systems Engineering Universidd Autónom de Nuevo León Sn Nicolás de los Grz, Nuevo León, Mexico Corresponding uthor s e-mil: sul@ylm.fime.unl.mx 2 Grupo ARCA Monterrey, México Abstrct: In industry, territory design is motivted by chnges in the number or properties of customers served by given compny. The gol of territory design is to group the customers into mngeble-sized territories. It is often required to blnce the demnd mong the territories in order to delegte responsibility firly. In this pper, we present metheuristic solution pproch bsed on GRASP (Greedy Rndomized Adptive Serch Procedure) to prticulr commercil territory design problem motivted by rel-world ppliction in beverge distribution compny in the city of Monterrey, Mexico. Our empiricl wor includes n evlution of the overll lgorithmic performnce. In ddition, we study the effect of the weight prmeter of the GRASP greedy function (which is convex combintion of the originl dispersion-bsed obective function nd the reltive violtion of the blnce constrints) on the qulity of the finl solution. The experiments were crried out over set of rndomly generted instnces bsed on rel-world dt from the industril prtner. 1. INTRODUCTION The territory design problem is concerned with grouping smll geogrphic res into lrger geogrphic clusters clled territories in such wy tht the ltter fulfill certin plnning criteri. This problem belongs to the fmily of districting problems tht hve brod rnge of pplictions lie politicl districting nd the design of sles nd services territories. The wor of Klcsics, Nicel, nd Schröder (2005) is n extensive survey on pproches proposed for territory design tht presents n up to dte stte of the rt on the topic. The problem ddressed in this pper is motivted by rel-world ppliction in the city of Monterrey, Mexico. The firm wishes to prtition the city re into disoint territories tht re suitble for their commercil purposes. In prticulr, the firm wishes to design territories tht be blnced (similr in size) with respect to ech of two different ctivity mesures (number of customers nd sles volume). Additionl plnning criteri include: (i) contiguity of ech territory, so tht ech bsic unit (BU) cn rech ech other by trveling within the territory; (ii) territory compctness, so tht customers within territory re reltively close to ech other; (iii) oint ssignment of BUs, so tht specified pirs of BUs must be ssigned to the sme territory in the design, nd (iv) fixed number of territories. Although severl commercil territory design pproches hve ppered in the literture, the specific fetures present in this concrete problem me it very unique, nd it hs not been ddressed before to the best of our nowledge. Vrgs- Suárez, Ríos-Mercdo, nd López (2005) studied similr problem without compctness, contiguity nd oint ssignment constrints nd where the obective ws to minimize the unblnce mong territories. Ríos-Mercdo nd Fernández (2006) studied the problem considering compctness nd contiguity but without oint ssignment constrins. In this wor we propose nd develop solution pproch bsed on GRASP (Feo nd Resende, 1995) to construct high-qulity solutions. Our empiricl wor includes n evlution of the overll lgorithmic performnce of the proposed pproch. Our computtionl experiments show tht this method yields good results. 2. PROBLEM DESCRIPTION Let G = (V, E) be grph where ech BU is represented by node i V nd n rc connecting nodes i nd exists in E if units i nd re locted in dcent blocs. Ech node hs three properties: geogrphicl coordintes (c i x, ci y ), nd two ISBN:

2 Proceedings of the 12th Annul Interntionl Conference on Industril Engineering Theory, Applictions nd Prctice Cncun, Mexico mesurble ctivities. Let ω i be the vlue of ctivity t node i, where the first ctivity, = 1, represents the number of customers within tht BU nd the second ctivity, = 2, represents the totl sles volume of the BU. A territory is subset of nodes V V. The number of territories p is fixed nd given s prmeter. It is required tht ech node is ssigned to only one territory. One of the desired properties in solution is tht the territories be blnced with respect to ech of the ctivity ω ( V ) =, =1, 2 mesures. Let us define the size of territory V with respect to ctivity s: ω i V i. Due to the discrete structure of the problem nd to the unique ssignment constrint, it is prcticlly impossible to hve perfectly blnced territories with respect to ech ctivity mesure. To overcome this difficulty we mesure the degree of blnce by computing the reltive devition of ech territory from its verge vlue of ctivity tht is, = ω ( V )/ p, = 1, 2. We re lso given collection H of pirs of nodes such tht [i, ] H implies tht node i nd node must be ssigned to the sme territory. In ddition, the industril prtner requires tht in ech of the territories, the units must be geogrphiclly locted reltively close to ech other. One wy to chieve this is to ssign one node in ech territory to serve s territory center, denoting the center of territory by c() nd then to define distnce mesure such s f ( V V, K, V ) mx{ ( ) }, p = d, 1 2 c 1,, p = K V (1) over the p territories, d c() represents the Eucliden distnce from node to center of territory. All prmeters of the model re ssumed to be nown with certinty. The problem cn be thus described s finding p-prtition of V tht stisfies the specified plnning criteri of blncing: ( 1 τ ) ω ( V ) (1 + τ ), (2) contiguity, nd oint ssignment, tht minimizes the bove distnce mesure (1). This problem cn be modeled in terms of integer progrmming s p-center problem with dditionl constrints on cpcity, contiguity nd oint ssignment. Given the complexity of grph prtitioning problems nd the nture of the dditionl constrints, such s contiguity, we resort to metheuristic tht hndles some of the difficulty lredy in the construction phse, nd further ssuring in ech step tht the contiguity is not broen. 3. PROPOSED ALGORITHM GRASP (Feo nd Resende, 1995), well-nown metheuristic tht cptures good fetures of both pure greedy lgorithms nd rndom construction procedures, hs been widely used for solving mny combintoril optimiztion problems. GRASP is two-phse itertive process. In GRASP itertion, we first perform construction phse for building fesible solution nd then post-processing phse tht ims to improve the solution. The post-processing typiclly consists of locl serch within suitble neighborhoods. The proposed lgorithm strts with n infesible territory ssignment in which ech BU is ssigned to singleton territory, V o = (V 1,, V n ) with V i = {i}. Before continuing with the construction, we force the oint ssignment of the specified pirs of BUs (v, w) in H: we find pth P in G from v to w nd merge into single territory ll the territories tht contin one or more nodes included in P. In ech subsequent step of the construction phse, we merge two dcent territories until we rech the desired number of territories. By construction, ll the territories will be contiguous. However, this solution might not be fesible with respect to the blnce constrints (2). The post-processing phse ttempts to reduce if not completely overcome the blnce-constrint infesibility. The post-processing phse lso ims to reduce the mximum distnces within the territories. The reltive importnce of these two tss is controlled by weight prmeter. We now describe in detil ech of the components of the lgorithm. 3.1 Pre-Processing Phse To ddress the oint ssignment of bsic units, we need to te into ccount lso the contiguity requirement the finl solution must fulfill both requirements. One wy to do this is to find, for ech (v, w) in H, pth P in G between v nd w 423

3 Proceedings of the 12th Annul Interntionl Conference on Industril Engineering Theory, Applictions nd Prctice Cncun, Mexico tht must be ssigned to the sme territory nd then by merging ll territories tht contin one or more nodes in the pth P. The newly obtined territory then contins both of the desired bsic units nd is necessrily contiguous if ll the merged territories were contiguous. The initil territory ssignment consists of singleton territories nd the merging is done with respect to pths in G, nd hence ll creted territories re contiguous s well. For dditionl diversity of solutions, in the spirit of GRASP, insted of using lwys the shortest pth, we compute shortest pths (Mrtins, Pscol, nd Sntos, 1999) between the BUs nd rndomly select one of these pths. 3.2 Construction Phse After the pre-processing phse, the number of territories is different from p. For the mority of nturl instnces, the number of territories is significntly lrger thn p; our lgorithm does not hndle the rre cses where the number of territories is smller thn p. We decrese the number of territories by itertively merging dcent territories two t time until p territories remin. The two territories to merge re chosen using greedy function tht weighs both distnce-bsed dispersion mesure f V ) = mx { d } nd the reltive violtion of the blnce constrints (2). Let us define ( i, V i ω ( V ) = i ω V i s the size of the territory V with respect to the ctivity. Let V i nd V be two dcent territories (tht is, territories connected by t lest one edge in G), let us define their greedy function s φ( Vi, V ) = λ f ( Vi V ) + (1 λ) G( Vi V ), (3) where V i V is the new territory tht is the union of the two smller territories, λ is prmeter, nd G ( V V ) = g ( V V ), (4) i A i ω + τ with g ( V ) = (1/ )mx{ ( V ) (1 ),0} s the sum of the reltive infesibilities with respect to the upper bound of the blnce constrint of ctivity. In the spirit of GRASP, the list of ll possible cndidte moves re sorted by non-decresing vlue of its greedy function, nd restricted cndidte list (RCL) is constructed by selecting ll those moves which re within α (or 100 α %) of the best move. A move from the RCL is chosen rndomly, the greedy function of the remining moves is updted nd we proceed itertively until the number of territories equls p. 3.3 Post-Processing Phse After the construction phse hs reched the desired number of territories, the post-processing phse consisting of locl serch is performed. The gol of this phse is both to reduce the (possible) infesibility with respect to the blnce constrints nd to improve the vlue of the obective function. The locl serch uses merit function tht weighs both the infesibility of the blnce constrints nd the obective function. For prtition S = {V 1,, V p }, the merit function is given by ψ ( = λf( + (1 λ) I( (5) where F( is the dispersion mesure given by F( = mx mx dc = 1, K, p V ( ) (6) nd I( is the sum of the reltive infesibilities of the blnce constrints, 424

4 Proceedings of the 12th Annul Interntionl Conference on Industril Engineering Theory, Applictions nd Prctice Cncun, Mexico I( = p = 1 A i ( V ), (7) ω τ τ with i ( V ) = (1/ )mx{ ( V ) (1 + ),(1 ) ( V ),0}. We define neighborhood N( of prtition S to consists of ll prtitions S rechble from S by moving bsic unit i from its current territory t(i) to the territory of nother BU such tht the edge (i, ) E nd t() t(i). By only considering territories directly dcent to the BU i, we cn ssure tht lso the prtition S hs only contiguous territories. ω 4. EXPERIMENTAL RESULTS In this section we present the experimentl results obtined with C++ implementtion of the proposed lgorithm, compiled with the Sun C++ compiler worshop 8.0 under the Solris 9 operting system nd executed on SunFire V440 server. For the experiments, we generted rndomly problem instnces bsed on rel-world dt provided by the industril prtner. Ech instnce consists of node coordintes generted uniformly t rndom in [0; 500] x [0; 500] plne nd set of edges plced mong the nodes to me plnr grph. The node ctivities were generted s the sum of d rndom numbers with uniform distribution, where d depends on the number of nodes, contrcting together nodes from the rel-world dt. We experimented with 20 instnces of size n = 500 nd p = 10. The number of oint ssignment constrints considered in the problem instnces is bout 5 % of n, this ws suggested by the industril prtner. For ll instnces, the llowble devition for the blncing constrints ws set t 5 % (τ = 0.05). The locl serch phse is executed until we rech locl optimum; tht is, when we cnnot find n improvement movement in the neighborhood. 4.1 Experiment A: GRASP Itertion Count As GRASP is n itertive procedure, we need to determine for how mny itertions we need to run the lgorithm to rech good solution in resonble time of computtion. In order to determine the itertion count to use for the rest of the experiments, we first executed the lgorithm five times over three instnces for 1,000 GRASP itertions. Figure 1 shows how the qulity of the best solution found improves over the itertions performed for two of these instnces (ech of the five repetitions shown s seprte line); the third instnce behved very similrly. Instnce 1 Instnce 2 ob. fn. ob. fn Itertion Itertion Figure 1. GRASP Convergence Figure 1 shows tht ll mor improvements in the vlue of the obective function tend to occur during the first itertions, for which we find it sfe to conclude tht running the lgorithm up to 500 itertions suffices for finding good solution. 4.2 Experiment B: The effect of the weight prmeter (λ) 425

5 Proceedings of the 12th Annul Interntionl Conference on Industril Engineering Theory, Applictions nd Prctice Cncun, Mexico We then studied the effect of the weight prmeter λ (5) in the qulity of the solutions found in the locl serch phse. We exmined four different vlues of the prmeter: 0.0, 1/3, 2/3, nd 1.0, in order to ssess this effect. Tble 1 shows for ech vlue of the prmeter studied, the reltive men devition from the best solution found, the number of times this vlue of the prmeter led to the best solution, nd the number of infesible solutions found. We exclude from the tble the cse λ = 1 due to not hving found ny fesible solutions with this vlue. Tble 1. Clibrtion of Prmeter λ Vlue of the prmeter λ 0 1/3 2/3 Reltive devition from best solution (verge) Frequency of being the best solution Number of infesible solutions As we cn lern from the tble, lrger vlues of the prmeter hd better effect. This mens tht giving more weight to the obective function rther thn to the violtion of the constrints led to better solutions. 4.3 Experiment C: Evlution of the locl serch In this experiment we studied the behvior of the solution qulity during the locl serch procedure. We computed the verge reltive improvement on the qulity of the solutions obtined by the locl serch t the end of ech GRASP itertion. The results re shown in Figure 2. Ech curve shows the pth followed by the locl serch (in terms of the best solution found until tht point) for five specific itertions (shown in the right). Locl serch behvior 3.5 Obective vlue Movement Iter. 0 Iter. 99 Iter. 199 Iter. 299 Iter. 399 Figure 2. Behvior of the Locl Serch The numericl results of this experiment show tht the locl serch produces n improvement of bout 90 percent in the qulity of the solution. Moreover, s we cn see, there re cses where convergence is reltively slow nd other s where locl optimum is found reltively fst. 5. CONCLUSIONS 426

6 Proceedings of the 12th Annul Interntionl Conference on Industril Engineering Theory, Applictions nd Prctice Cncun, Mexico In this wor we hve presented GRASP lgorithm for this territory design problem. One of the min contributions of this wor is the incorportion of oint-ssignment constrints tht to the best of our nowledge hve not been treted before. Experimentl results show the effectiveness of the method in finding good-qulity solutions for problems of size n = 500 in resonbly short computtion times (bout 170 seconds) using 500 GRASP itertions. In ddition, we observed n excellent performnce of the locl-serch phse. Our experiments indicte tht vlue of the weight prmeter (λ) of 2/3 leds to the best solutions for the instnces of the problem out of the vlues studied. As future wor we pln to execute more extensive experimenttion considering instnces of size n = 1,000 nd up, nd study the behvior of nother locl-serch procedure bsed on swp moves. Acnowledgements: This reserch hs been supported by the Mexicn Ntionl Council for Science nd Technology (grnt SEP-CONACYT Y). In ddition, the first uthor is supported by fellowship for grdute studies from CONACYT. 6. REFERENCES 1. T. A. Feo nd M. G. C. Resende (1995). Greedy rndomized dptive serch procedures. Journl of Globl Optimiztion, 6(2): J. Klcsics, S. Nicel, nd M. Schröder (2005). Towrd unified territoril design pproch: Applictions, lgorithms, nd GIS integrtion. Top, 13(1): E. Q. V. Mrtins, M. M. B. Pscol, nd J. L. E. Sntos (1999). Devition lgorithms for rning shortest pths. Interntionl Journl of Foundtions of Computer Science, 10(3): R. Z. Ríos-Mercdo nd E. Fernández (2006). A rective GRASP for sles territory design problem with multiple blncing requirements. Technicl report PISIS , Grdute Progrm in Systems Engineering, Universidd Autónom de Nuevo León, Sn Nicolás de los Grz, Mexico, September. 5. L. Vrgs-Suárez, R. Z. Ríos-Mercdo, nd F. López (2005). Usndo GRASP pr resolver un problem de definición de territorios de tención comercil. In M. G. Arens, F. Herrer, M. Lozno, J. J. Merelo, G. Romero, nd A. M. Sánchez, editors, Memoris del IV Congreso Espñol sobre Metheurístics, Algoritmos Evolutivos y Bioinspirdos (MAEB), pp , Grnd, Spin, September. In Spnish. 427