A CONTINUOUS DISTRICTING MODEL APPLIED TO LOGISTICS DISTRIBUTION PROBLEMS

Challenges and Maturty of Producton Engneerng: compettveness of enterprses, workng condtons, envronment. São Carlos, SP, Brazl, to 5 October 00. A CONTINUOUS DISTRICTING MODEL APPLIED TO LOGISTICS DISTRIBUTION PROBLEMS Antôno Galvão Nacléro Novaes (UFSC) novaes@deps.ufsc.br Enzo Morosn Frazzon (BIBA) enzo.frazzon@gmal.com Bernd Scholdz-Reter (BIBA) enzo.frazzon@gmal.com Orlando Fontes Lma Jr (Uncamp) oflmaj@fec.uncamp.br The am of dstrctng problems s to get an optmzed partton of a terrtory nto smaller unts, called dstrcts or zones, subject to some sde constrants such as balance, contguty, and compactness. Logstcs dstrctng problems usuually nvolve addtonal optmzaton crtera and constrants. Dstrctng problems are called contnuous when the underlyng space, both for faclty stes and demand ponts, are determned by varables that wll vary contnuously. Vorono dagrams can be successfully used n assocaton wth contnuous approxmaton models to solve locaton-dstrctng problems. We dscuss n the paper the applcaton of non-ordnary Vorono dagrams n logstcs dstrctng problems, partcularly the Power Vorono dagram, assocated wth a contnuous demand approach, whch allows for the ntroducton of physcal barrers nto the vehcle dsplacement representaton, such as rvers, reservors, hlls, etc. Palavras-chaves: Logstcs dstrbuton, Dstrctng, Vorono dagrams, Optmzaton

Challenges and Maturty of Producton Engneerng: compettveness of enterprses, workng condtons, envronment. São Carlos, SP, Brazl, to 5 October 00.. Introducton The vehcle routng problem (VRP) s the one of desgnng a set of routes from a central depot to varous demand ponts, each havng servce requrements, n order to mnmze the total dstance. The total dstance travelled s often substtuted by cycle tme or by a cost functon. When customer demands or some other element of the problem are random varables, we have the stochastc vehcle routng problem (SVRP) (BASTIAN AND RINNOOY KAN, 99). Common examples are stochastc demands and stochastc travel tmes. In addton, sometmes the set of customers to be vsted s not known wth certanty. The prmary objectve of such models s to fnd optmal tours,.e. the best sequence of vsts n order to mnmze the total travelled dstance, cycle tme, or the total transportaton cost, respectng, at the same tme, servce requrements (STEWART AND GOLDEN, 983; BERTSIMAS, 99). The am of dstrctng problems, on the other hand, s to get an optmzed partton of a terrtory nto smaller unts, called dstrcts or zones, subject to some sde constrants (HOJATI, 996; MEHROTRA, JOHNSON AND NEMHAUSER, 998; BOSKAYA, ERKUT AND LAPORTE, 003). The constrants reflect a number of common sense crtera. One of them s to balance demand among dstrcts. Furthermore, the resultng dstrcts must be contguous and geographcally compact (WILLIAMS, 995). Logstcs and transportaton dstrctng problems usually nvolve addtonal optmzaton crtera and constrants. In general, apart from the basc balance, contguty, and compactness prncples, there s not a set of general crtera that are common to all dstrctng problems. Dstrctng problems are called contnuous when the underlyng space, both for faclty stes and demand ponts, are determned by one or more varables that wll vary contnuously. Dstrctng problems are assocated wth a number of practcal applcatons. Poltcal dstrctng, n whch one s nterested n drawng of electoral dstrct boundares, has receved much attenton n the lterature (HOJATI, 996; MEHROTRA, JOHNSON AND NEMHAUSER, 998; BOSKAYA, ERKUT AND LAPORTE, 003, WILLIAMS, 995). School dstrctng (SCHOEPFLE, AND CHURCH, 99) and polce dstrctng (D AMICO et al., 00) are two other areas of research nterest. In addton, the lterature presents artcles on the desgn of sales terrtory (ZOLTNERS AND SINHA, 997), as well as emergency, health-care, and logstcs dstrctng. Among the latter, we menton the balanced allocaton of customers to dstrbuton centers (ZHOU, MIN AND GEN, 00) and the desgn of multvehcle delvery tours (LANGEVIN AND SAINT-MLEUX, 99; NOVAES, DE CURSI AND GRACIOLLI, 000). Contnuous approxmatons to dstrctng problems are based on the spatal densty and dstrbuton of the demand rather than on precse nformaton on every demand unt. It allows for smple, yet robust models that are useful when plannng a new servce or the expanson of an exstng one (LANGEVIN, MBARAGA AND CAMPBELL, 996; NOVAES, DE CURSI AND GRACIOLLI, 000; DASCI AND VERTER, 00). The assocaton of contnuous approxmaton technques wth Vorono dagrams opens the way to solve a number of real-lfe dstrctng problems. In partcular, the use of non-ordnary Vorono dagrams to solve logstcs and transportaton problems has been reported n the lterature (OKABE, BOOTS AND SUGIHARA, 995; SUZUKI AND OKABE, 995; GALVÃO et al., 006; NOVAES et al., 009). Boots and South (997) used a multplcatvely weghted Vorono dagram approach for modelng retal trade areas. Galvão

Challenges and Maturty of Producton Engneerng: compettveness of enterprses, workng condtons, envronment. São Carlos, SP, Brazl, to 5 October 00. et al. (006) defned a multplcatvely weghted Vorono dagram model to solve an urban freght dstrbuton problem. The utlzaton of non-ordnary Vorono dagrams n logstcs dstrctng problems, assocated wth a contnuous demand approach, also allows for the ntroducton of physcal barrers nto the model (NOVAES et al., 009). Ths s an mportant property because t permts to treat problems wth obstacles mposed by thoroughfares, hghways, rvers, reservors, hlls, etc. The purpose of ths paper s to apply a contnuous dstrctng model to a dstrbuton logstcs problem, combnng a power Vorono dagram approach wth an optmzaton algorthm.. Contnuous approxmaton A large part of dscrete dstrctng problems can be converted nto problems nvolvng contnuous functons, wth good practcal results. Wth such formulaton, demand potentally arses at any pont n a plane and feasble locatons of facltes are equally any pont n a plane. Contnuous demand approxmaton models are usually based on the spatal dstrbuton varables rather than on precse nformaton on every servcng pont (LANGEVIN, MBARAGA AND CAMPBELL, 996; DASCI et al., 00). A number of urban dstrctng problems have been solved wth ths approach (NEWELL AND DAGANZO, 986a; NOVAES, DE CURSI AND GRACIOLLI, 000; GALVÃO et al., 006; NOVAES et al., 009). Before choosng an approprate algorthm to solve a dstrctng problem wth ths approach, t s necessary to represent the data n a contnuous format. In the applcaton analyzed n ths paper, the varables to be contnuously represented can be the number of vstng ponts n a sub-regon, the demand ntensty at pont (x, y), or other attrbute of nterest. Methods for the constructon of a regular contnuous approxmaton to a functon may be found n the lterature. Usually a good approxmaton s attaned wth a bquadratc splne (BOOR, 00) combned wth a fnte element dscretzaton of the regon under analyss (BATHE, 98). These technques were adopted n ths research snce the use of fnte elements saves computatonal tme. But the method may be mplemented wth any knd of fnte element mesh by usng the approprate weghts n order to evaluate the functon to be approxmated. For more detals, the reader s referred to Galvão et al. (006)... Gudelnes for dstrct desgn Dstrctng problems are assocated wth one or more actvtes or operatons to be performed wthn the served regon and consderng each dstrct ndvdually. From a plannng pont of vew, dstrctng should be performed at the strategc or tactcal level, whle the detaled optmzaton of actvtes should be performed at the operatonal level. In another words, dstrctng should nvolve a more global vew and s usually related to the manageral and admnstratve levels. Thus, dstrct borders should not change too frequently but, nstead, should be modfed only when major system and demand changes take place (MUYLDERMANS et al., 00). In general, the methods for solvng dstrctng problems follow a sequence of procedures (FLEISCHMANN et al., 988): Defnton of one or more actvty measures, or queres (OKABE, BOOTS AND SUGIHARA, 000); Set up a functon (or functons) to represent the ultmate objectve of the dstrctng process; Set up a number of constrants to be respected n the dstrctng process; Solve the dstrctng problem wth an approprate algorthm. 3

Challenges and Maturty of Producton Engneerng: compettveness of enterprses, workng condtons, envronment. São Carlos, SP, Brazl, to 5 October 00. In some cases, the underlyng query s qute smple. For example, n a smple poltcal dstrctng problem one may wsh to partton the regon nto a number of dstrcts such that each dstrct has almost the same number of voters. The basc query s then compute the total number of voters n each dstrct, and the representng functon s the ntegral of the densty of voters over the dstrct area. If V s the total number of voters n dstrct, the assocated balance constrants would be V V,,..., m, and j, () where m s the number of dstrcts and s a tolerance level. j In a more complex case, the query may be fnd the maxmum dstance from the depot to any clent premse wthn the dstrct, for whch the optmzng functon could be the sum of the squared devatons of such dstances, to be mnmzed,.e. mn S m ( F ) ( F ) ( d d ), () (F ) where d s the maxmum dstance from the depot to any clent premse wthn the dstrct. Some transportaton and logstcs problems nvolve more complex queres because the operatons to be performed n a dstrct depend on a number of endogenous and exogenous varables. Each problem has ts own set of queres, assocated wth functons and constrants, but some general recommendatons should be kept n mnd: As ponted out by Muyldermans et al. (00), most dstrctng problems requre a mult crtera approach, and the queres and related functons must reflect ths requste when necessary. Fleschmann et al. (988), for nstance, n order to measure salesman workload to perform a marketng effort, adopted a compound score that takes nto account the sales revenue and the frequency of vsts to retaler customers n each dstrct; Snce dstrctng should preferably be performed at the strategc or tactcal level, the functon (or functons) to represent the ultmate objectve of the dstrctng process should not nvolve too many varables and detaled modelng. More effort should be appled to reduce computng tme due to the large number of teratons when runnng such models; Balance among dstrcts must be approprately represented by one or more constrants n the model; In order to avod volatng the requrement that the served unts n each dstrct are contguous and each dstrct s geographcally compact, the model must ncorporate approprate heurstcs to satsfy such requstes. The Vorono dagram approach facltates ths task snce, f the Vorono dagram type s well chosen and well employed, contguty and compactness condtons are naturally respected. 3. Vorono dagrams In ths secton t s presented a summary of the basc concepts and propertes of Vorono dagrams that wll further be used to solve a logstcs dstrctng problem. Vorono dagrams comprse a vast subject, and the reader s referred to Aurenhammer (99) and Okabe, Boots and Sughara (000) for more detals. Although Vorono dagram generators can be ponts, lnes, crcles, or areas of dverse shapes, our applcaton deals wth pont generators only. In fact, logstcs problems usually nvolve pont-lke facltes such as depots, clent premses, trucks, etc. Presently, Vorono dagrams are extensvely used n computatonal geometry, computer graphcs, robotcs, pattern recognton, games, etc (AURENHAMMER, 99). The 4

Challenges and Maturty of Producton Engneerng: compettveness of enterprses, workng condtons, envronment. São Carlos, SP, Brazl, to 5 October 00. basc concept of Vorono dagrams s qute smple: gven a fnte set of dstnct and solated ponts n a contnuous space (generator ponts), we assocate all locatons n that space wth the closest member of the pont set (OKABE, BOOTS AND SUGIHARA, 000). In the smplest case of Vorono dagram the dstances from any pont to the generator ponts are represented by the Eucldean norm. The edges of the resultng Vorono polygons n a plane are lne segments. There are stuatons, however, when the Eucldean dstance does not represent well the attractng process. For nstance, suppose that the sx generator ponts exhbted n Fgure are retal stores sellng the same knd of product. Assume further that, n addton to dstance, the attracton of such stores depends on a set of features, leadng to the weghtng coeffcents shown n Fgure. In order to take these elements nto account, several knds of weghted Vorono dagrams have been developed. These dagrams use a famly of weghts w ( w, w,..., w m ) such that the domnance regon ncreases wth the weght w. For nstance, multplcatvely weghted planar Vorono dagrams correspond to ( X, P ) X P, (3) w where w s a famly of strctly postve weghts. In the case wth only two generator ponts, the locus of the ponts X satsfyng (3) s the Apollonus crcle (OKABE, BOOTS AND SUGIHARA, 000), except f w w, when the bsector becomes a straght lne. Fg. shows an example of MW-Vorono dagram for the weghts there ndcated. In general, a MW-Vorono regon s a non-empty set and need not be convex, or connected; and t may have holes (OKABE, BOOTS AND SUGIHARA, 000). Fgure An example of a multplcatvely-weghted Vorono dagram Analogously, the addtvely weghted Vorono dagram s represented by ( X, P ) X P w. (4) Here, the sgn of w s not restrcted. The combnaton between addtve and multplcatve weghts leads to the compoundly weghted Vorono dagram, whch s assocated to ( X, P ) X P w w (5) 5

Challenges and Maturty of Producton Engneerng: compettveness of enterprses, workng condtons, envronment. São Carlos, SP, Brazl, to 5 October 00. We recall that the sgn of w î s not restrcted. In ths case, the boundary of the domnance regon s a fourth-order polynomal functon, and ts shape s farly complex. The power Vorono dagram corresponds to ( X, P ) X P w (6) In ths case, only postve values of w are usually used. The lne segment connectng P and P s a straght lne perpendcular to the lne segment P P. An mportant property of power j Vorono dagrams, useful n applcatons, s that the resultng Vorono polygons are always convex. Power Vorono dagrams are especally useful to solve dstrctng problems wth barrers, as n the case descrbed n Secton 4.. When solvng Vorono dagram problems wth obstacles, the Eucldean norm s not acceptable. If an obstacle les on the lne lnkng an orgn and a destnaton, t s not possble to traverse t straght. Instead, a detour around the obstacle must be taken. Followng Okabe, Boots and Sughara (000), let us consder a generator set P and a set of c closed regons O O,..., Oc ( c ). The set O represents a set of obstacles that are not traversable. These obstacles are assumed not to overlap each other and ponts of P are not allowed to le wthn the obstacles (Fgure a). Furthermore, each obstacle s assumed to be connected and wth no holes. For computatonal convenence t s assumed that O (,..., c) s a polygon, but t s not assumed that O s necessarly convex. Lne segments are also accepted as obstacles. The vsblty-shortest-path dstance between a generc pont X and a generator pont P, expressed as d SP (X, P ), s obtaned consderng all possble contnuous paths connectng X and P that do not traverse obstacles. A vsblty polygon wth respect to P, and denoted by Vs (P ), s the set of ponts that are vsble from P. An example s shown n Fgure a, where the vsblty polygon wth respect to the generator pont P s ndcated by the hatched area. To compute the vsblty-shortest-path dstance between a pont X and a generator pont P one uses the correspondent vsblty graph, whch s formed by all possble paths connectng X and P (Fgure b). On the vsblty graph one solves the classcal shortest-path problem wth the ad of an approprate algorthm such as, for example, the well-known Djstra method. For the example of Fgure b, the shortest-path between P and X s P B F X. j 6

Challenges and Maturty of Producton Engneerng: compettveness of enterprses, workng condtons, envronment. São Carlos, SP, Brazl, to 5 October 00. Fgure The shortest-path vsblty graph from pont P to pont X 4. An urban dstrbuton problem wth a geographcal barrer Most logstcs dstrbuton and collectng problems nvolve spatal varables assocated wth operatonal and economc elements, such as routng, vehcle capacty, vehcle costs, servcng tmes, etc. As ponted out by Muyldermans et al. (00), dstrctng assocated wth transportaton and logstcs problems should be performed at the strategc and tactcal level, whereas routng should be performed at the operatonal level. In other words, dstrctng nvolves a more global vew and s often related to the manageral and admnstratve levels, whle routng processes are more detaled and lnked to day-to-day operatons. In many applcatons, locatons are characterzed by Cartesan coordnates and dstances are estmated by an L (Eucldean) metrc, corrected by a routng factor (GALVÃO et al., 006). Addtonally, n one-to-many dstrbuton and collectng problems wth multple tours (DAGANZO, 996), an dealzed dense rng-radal network pattern s frequently adopted as a theoretcal modelng bass. In such cases, the deal confguraton of the dstrcts should be wedge-shaped and elongated toward the depot (NEWELL AND DAGANZO, 986a). Whle nterestng from a theoretcal pont of vew, ths approach s not readly applcable to real lfe stuatons. For more generc metrcs, n fact, the optmal orentaton of the dstrcts and ts shape are not obvous. The non defnton of the real local network metrc makes the deal shape of the zones unclear. Furthermore, snce the real transportaton nfrastructure usually presents a coarse network of roads wth varyng velocty, the deal orentaton of the dstrcts s also unclear (NEWELL AND DAGANZO, 986b). Some computatonal tests (GALVÃO et al., 006) have ndcated that the adopton of a Vorono dagram parttonng process changes the resultng total dstrbuton cost only margnally when compared wth the 7

Challenges and Maturty of Producton Engneerng: compettveness of enterprses, workng condtons, envronment. São Carlos, SP, Brazl, to 5 October 00. correspondng rng-radal results, and therefore the Vorono formulaton may be used as an adequate dstrctng approxmaton n a varety of practcal applcatons In ths secton we descrbe and dscuss one applcaton of a generalzed Vorono dagram to a logstcs dstrbuton problem wth barrers n order to llustrate the possbltes of the method. 4.. Problem descrpton The utlzaton of generalzed Vorono dagrams n logstcs dstrctng problems has some advantages. The fttng process, for nstance, leads to more equalzed load factors among the dstrcts, meanng the vehcles assgned to the zones wll show more balanced utlzaton levels. Ths happens because the generalzed Vorono dagrams have more degrees of freedom when searchng for the dstrct contours when compared to the wedge-shaped, geometrcal parttonng scheme. Furthermore, as mentoned, the resultng total dstrbuton cost s only margnally affected by such an approxmaton (GALVÃO et al., 006). Addtonally, the utlzaton of an approprate Vorono dagram approach opens the way to solve dstrctng problems wth geographcal barrers mposed by thoroughfares, hghways, rvers, reservors, parks, steep hlls, etc. The problem concerns an urban dstrbuton servce coverng a specfc area. The objectve s to defne the number of dstrcts and ther boundares to be assgned to the delvery vehcles n order to: (a) mnmze total daly delvery costs; (b) balance the dstrbuton effort among the vehcles, and (c) respect capacty constrants. Addtonally, the resultng dstrcts must be contguous and geographcally compact (MEHROTRA, JOHNSON, AND NEMHAUSER, 998). A homogeneous fleet of m delvery vehcles s assumed and each vehcle s allocated to a dstrct, performng a complete cycle per workng day from a central depot, n a one-tomany dstrbuton scheme (DAGANZO, 996). The served urban regon R may be of rregular shape and has an area A. The densty of servcng ponts vares over R but s nearly constant and Posson dstrbuted over dstances comparable wth a dstrct sze (NEWELL AND DAGANZO, 986a). The demand s formed by: (a) the total of number servcng ponts, (b) the average weght of cargo delvered at each clent s locaton (kg), and (c) the mean stoppng tme per vstng pont (mn). Galvão et al. (006) developed a multplcatvely weghted Vorono dagram model to solve ths problem wth no geographcal barrers. That same problem was extended to the stuaton n whch geographcal obstacles restran dstrctng lmts (NOVAES et al., 009). In ths paper the obstacle s represented by a freeway, wth crossng ponts spaced along the road (vaducts and underpasses). Those crossng ponts are congested most of the tme. As a practcal consequence, logstcs operators do not desgn delvery dstrcts coverng areas stuated n both sdes of the freeway. Ths practcal stuaton s handled, n the model, wth the ntroducton of one obstacle represented by the mentoned freeway (Fgure 3). The problem was solved wth a power Vorono dagram formulaton assocated wth the vsblty-shortestpath metrc (Secton 3). 4.. Vehcle cycle modelng Vehcle travel dstances wthn the dstrcts are assumed to be approxmately represented by a Eucldean metrc, wth the real dstances beng estmated wth the ad of a mathematcal functon (DAGANZO, 984), usually a multplyng route factor greater than unt. Followng Sten (978), the expected dstance E[ DZ ] travelled by a vehcle wthn a dstrct of area A, and n vstng ponts, can be approxmated as 8

Challenges and Maturty of Producton Engneerng: compettveness of enterprses, workng condtons, envronment. São Carlos, SP, Brazl, to 5 October 00. / E DZ k A n k n, (7) 0 0 where n A s the densty of ponts over A. Expresson (7) can be appled to most metrcs (Novaes and Gracoll, 999) and presupposes that the ponts are unformly and ndependently scattered over the area, and the dstrct s farly compact and farly convex. The coeffcent k 0 can be expanded nto two multplcatve factors (NOVAES AND GRACIOLLI, 999). The frst one depends solely on the adopted metrc and routng strategy. The second factor s a correctve coeffcent (route factor) reflectng the road network mpedance. The vehcle starts from the depot, goes to the assgned dstrct, does the delvery or pck up, and comes back to the depot when all the vsts are completed, or when the maxmum allowed workng tme per day s reached, whchever occurs frst. Ths complete sequence makes up the vehcle cycle. In some practcal crcumstances more than one tour per day can be assgned to the same truck. Ths mples extra lne-haul costs, but dependng on the cargo characterstcs, vehcle sze restrctons, and other factors, multple daly tours per vehcle mght sometmes be approprate. For the sake of smplcty, we assume that the vehcles perform just one cycle per day. The model can be easly modfed to take nto account multple daly cycles. The total cycle length D s the sum of the lne-haul dstance (ether way) and the local travel dstance gven by (7). The total cycle tme T, on the other hand, s the sum of the lne-haul tme, the local travel and the total handlng tme. The latter s the sum of the tmes spent n delverng the cargo at the customer s locatons. The expected value of T for a generc dstrct s ED [ ] k A n T E T p n E t (8) LH 0 [ ] ( S ) vl vz where ED [ LH ] s the expected lne-haul travel dstance (one way) from the depot to the dstrct, v L s the average lne-haul speed, v Z s the average local speed, p s the probablty that a customer be vsted, and E ( t S ) s the expected stop tme spent n one delvery (Novaes and Gracoll, 999). Assumng statstcal ndependence of the elements whch form the cycle tme, the varance of T s gven by T h Z S var[ T] var( t ) var( t ) p n var( t ), (9) where th s the lne-haul travel tme (one way), and t Z s the local travel tme. Usng the central lmt theorem, T can be represented by the normal dstrbuton T ~ N( T, T ). We assume that the cycle tme cannot exceed a maxmum of H 0 workng hours per day, mposed by labor restrctons and company polces. Let ~ N(0, ) be the unt normal varate. Adoptng a 98 percentle (mono tal dstrbuton), =.06, and thus g T.06 T H0 (0) s a restrcton that must be respected. Let [ ] and u Eu, on the other hand, be respectvely the mean and the standard devaton of the quantty u of product delvered per vstng pont n the generc dstrct. Then, assumng statstcal ndependence of the customer s demands, the 9

Challenges and Maturty of Producton Engneerng: compettveness of enterprses, workng condtons, envronment. São Carlos, SP, Brazl, to 5 October 00. expected value and the varance of the total vehcle load U for one tour n the dstrct s gven by U E[ U] p n E[ u] and Var[ U] U p n u. () Accordng to the central lmt theorem, U can be represented by the normal dstrbuton U ~ N( U, U ). If W s the truck capacty, and adoptng a 98 monotal percentle, another restrcton that must be respected s g U U.06 W. () In place of restrctons (0) and (), two loadng factors assgned to dstrct (,,..., m) are used wth the same objectve, the frst takng nto account tme utlzaton and, the second, vehcle capacty utlzaton () ( T ) g H0 and () ( U ) g, (,,..., m). (3) W The load factor for dstrct s the largest of (T ) and ( U ) ( T) ( U) max{, }, (,,..., m) (4) Thus, the balancng crteron s to equalze load factors among the m dstrcts s j (, j,..., m), (5) where ε > 0 s a small tolerance factor. 4.3. Defnng the number of dstrcts Assumng a ntal value for m and applyng the model, t wll be necessary to ncrease m f the resultng values of (,,..., m) are greater than one. Conversely, f the resultng load factors are too low, the value of m should be reduced. Ths process contnues untl one gets a sutable soluton respectng (5). Snce the computng process takes tme, t s recommended to choose a better estmate of m to be ntally used n the model. Let Q R be the total quantty of cargo carred per day n regon. If W s the cargo capacty of a vehcle, a rough estmate of m s m Q / W. (6) U R Wth regard to cycle-tme restrcton, a rough estmate of m s gven by the followng formula (Novaes et al, 009) 0

Challenges and Maturty of Producton Engneerng: compettveness of enterprses, workng condtons, envronment. São Carlos, SP, Brazl, to 5 October 00. m T k A N v H R D 0 R kd v L pn (L) R, (7) where A R s the area of regon, NR s the total number of delvery ponts n, s the average stoppng tme spent n one clent, and p s the probablty a customer s vsted n the tour. If the regon s approxmately crcular and the depot s farly centralzed, the average (L) dstance D may be assumed to be (Novaes et al, 009) D ( L) 3 A, (8) Where A s the average area of one dstrct. Applyng (6) and (7) one estmates m T and m respectvely. A good ntal approxmaton for the number of dstrcts m s the largest U value of m T and m U. Applyng the model and analyzng the resultng values of (,,..., m) n a recursve way, one may change the value of m, runnng the model agan untl an acceptable soluton s obtaned. 4.4 The teratve process The dstrctng problem wth obstacles s solved wth a power Vorono dagram ( X, P ) [ d ( X, P )] w, (,..., m), (9) SP where d ( X, P ) s the vsblty-shortest-path dstance between pont X and the generator pont weghts SP P (Secton 3). Let k represent the stage of the teratve process. At each stage the ( ) w ( w k 0,,..., m) are modfed accordng to the followng convergence rule (k ) ( k) ( k ) ( k ) w w v (0) where v ( k ) s gven by m ( k ) m v ( k ) ( k ) d ( k ), d 0 () ( k ) where and d s a control parameter. The value of d s changed emprcally n order to control the convergence of the model. Snce d 0, the weght (k ) w of dstrct wll ( ) ncrease f k ( k ) s greater than the mean. Puttng w wth a postve sgn n (9), the domnance regon of P (the dstrct area) wll decrease (OKABE, BOOTS AND ( k ) SUGIHARA, 000), tendng to lead to a more balanced soluton. If s the standard ( ) devaton of the observed k (,..., m), the value of d s chosen as to guarantee a decreasng sequence of (k ) () () ( k)... (5)

Challenges and Maturty of Producton Engneerng: compettveness of enterprses, workng condtons, envronment. São Carlos, SP, Brazl, to 5 October 00. At stage k the teratve process may termnate, n accordance to (5), f ( k) ( k) Max Mn, (53) () where ε > 0 s a small tolerance factor. For k, the weghts w,,..., m are set equal to zero, leadng to an ordnary Vorono dagram confguraton. At each stage of the process the power Vorono dagram s constructed wth an approprate algorthm (NOVAES et al., 009), and the resultng relevant attrbutes are computed, n specal, the center of mass of each dstrct. The center of mass s related here to the concentraton of delvery ponts wthn the dstrct, snce the number of stops s the prevalng varable when dmensonng ths knd of servce. The centers of mass of the dstrcts are then taken as the generator ponts of the Vorono dagram for the next stage of the teratve process. 4.5. Results and conclusons We have presented a method to solve a logstcs dstrctng problem n whch the dsplacement of vehcles s restrcted by geographcal barrers, employng a Power Vorono dagram algorthm. The resultng Vorono tessellaton for ths example resulted nto 57 balanced dstrcts, as shown n Fgure 3. The resultng dstrct contours are smooth and closer to the confguraton contours encountered n practcal stuatons, when compared to the tradtonal wedge-shape formulaton. Fgure 3 Power Vorono dagram logstcs dstrctng wth barrers The resultng partton of the regon led to more balanced tme/capacty utlzaton (load factors) across the dstrcts. Consderng the coarse underlyng road-network approxmaton (Eucldean metrc assocated wth a route factor), and the fact that a small error n the parameters tend to gve only a much smaller ncrease n the cost, we conclude that the Power Vorono dagram fttng process s a vald methodology to solve a number of practcal dstrbuton dstrctng problems. Apart from obtanng smoothed dstrct parttons, the utlzaton of Vorono dagrams opens the possblty for further explorng some of ts propertes n order to get better approxmatons to real-world problems. In specal, Vorono dagrams allow for the ntroducton of physcal obstacles nto the model, as shown n the present work. Ths knd of

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