Single-Allocation Hub Network Design Model with Consolidated Traffic Flows

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1 See dscussons, sas, and auhor profles for hs publcaon a: hps:// Sngle-Allocaon Hub Nework Desgn Model wh Consoldaed Traffc Flows ARTICLE n TRANSPORTATION RESEARCH RECORD JOURNAL OF THE TRANSPORTATION RESEARCH BOARD JANUARY 2007 Impac Facor: 0.54 DOI: / CITATIONS 3 READS 57 3 AUTHORS, INCLUDING: Dong-Kyu Km Seoul Naonal Unversy 33 PUBLICATIONS 39 CITATIONS Tschangho John Km Unversy of Illnos, Urbana-Champagn 50 PUBLICATIONS 622 CITATIONS SEE PROFILE SEE PROFILE Avalable from: Dong-Kyu Km Rereved on: 09 Aprl 2016

2 Sngle Allocaon Hub Nework Desgn Model wh Consoldaed Traffc Flows Dong Kyu Km, Chang Ho Park, and Tschangho John Km Economes of scale acheved by consoldang raffc flows are fundamenal characerscs of hub neworks. In he hub nework desgn problem, hen, s parcularly mporan o quanfy cos savngs semmng from economes of scale, he operang coss of hub facles, and he opporuny coss of delays n hub facles. Because of he NP (nondeermnsc polynomal me)-complee naure of he hub nework desgn problem, mos prevous researchers have focused on he developmen of heursc algorhms for approxmae soluons. The purpose of hs paper s o develop a hub nework desgn model consderng economes of scale semmng from consoldaon of raffc flows. The model s desgned o deermne endogenously he cos-decreasng effec resulng from economes of scale and o nclude several cos componens and capacy consrans relave o he parculary of hub neworks. The model n hs paper s compared wh recenly publshed sudes usng real daa. The resuls of he analyss show ha he proposed model can provde a more relable and realsc represenaon of real hub neworks. Ths sudy can no only form he heorecal bass of an effecve and raonal hub nework desgn bu also conrbue o he assessmen of exsng and planned logscs sysems. In a hub nework, cenrally locaed servce facles serve as hubs. Traffc flows from a se of oulyng nonhub nodes arrve a hubs, and afer regroupng, leave he hub facles bound eher o oher hubs or o her ulmae desnaons. Thus, he flows from he same orgn wh dfferen desnaons are consoldaed on he roue o a hub facly, and flows wh dfferen orgns bu he same desnaon leave ogeher ou of a hub facly. Because raffc flows are consoldaed, economes of scale preval. However, very lle consderaon has so far been gven o heorecal and analycal approaches o economes of scale n hub neworks. The hub nework problem, whch nvolves deermnaon of shppng roues and hub locaons for mnmzng oal cos, s known o be NP (nondeermnsc polynomal me)-complee, whch means ha s mpossble o solve he exac soluon n raonal me on a real-sze nework; hus, mos prevous work has focused on he developmen of approxmaon algorhms. Several properes of he hub nework desgn problem, herefore, have been nroduced n he prevous leraure wh smplfed forms. The dscoun rae resul- D. K. Km and T. J. Km, Deparmen of Urban and Regonal Plannng, Unversy of Illnos a Urbana Champagn, 111 Temple Buell Hall, 611 Lorado Taf Drve, Champagn, IL C. H. Park, Deparmen of Cvl, Urban, and Geosysem Engneerng, Seoul Naonal Unversy, Shllm-dong, Gwanak-gu, Seoul, Souh Korea, Correspondng auhor: D. K. Km, rafkdk@gmal.com. Transporaon Research Record: Journal of he Transporaon Research Board, No. 2008, Transporaon Research Board of he Naonal Academes, Washngon, D.C., 2007, pp DOI: / ng from he consoldaon of flows has been assumed consan or varable wh an arbrarly specfed parameer. The objecve funcon has ncluded only ransporaon coss or consrucon coss (or boh), and he capacy consran has been consdered neglgble n mos cases. The prevous leraure relaed o hs paper s presened n Table 1. The purpose of hs paper s o develop a hub nework desgn model wh consoldaed raffc flows ha creae economes of scale. The model n hs paper s formulaed o reflec he parculary of hub neworks as follows. Frs, he cos-decreasng effec caused by consoldaon of flows n hub neworks s he general characersc, resulng from he ncrease of operaon performance (flows dsance) n shppng sysems. The effec s deermned endogenously by ncludng he opmal frequences no he decson varable of he model. Second, n addon o ransporaon and consrucon coss, nvenory and handlng coss are ncluded n he objecve funcon. Ths enables he model o reflec several condons caused by consoldaon of flows and o analyze he effcency among several nework servces such as drec servce, ermnal servce, and hub nework servce. Thrd, consrans on hub and lnk capaces are nroduced o he formulaon. Hub capacy consrans affec congeson and delay n hub facles, and he lnk consrans have an nfluence on he deermnaon of dscoun raes n he opmal frequency process. The dsncve properes of he model descrbed above are presened n Fgure 1. The remander of hs paper s organzed as follows. The nex secon dscusses assumpons, a decson-makng srucure, cos funcons, and formulaon of he model. The followng secon compares he resuls beween he proposed model and recenly publshed sudes. Nex, emprcal sudes are presened for evaluaon of he proposed model. Las, a summary and dscusson are presened. MODEL DEVELOPMENT Assumpons The hub nework desgn model for opmal hub locaons, nonhub allocaons, and raffc roues has several srucures wh hub nework polces (5). A frs, hub nework polces are dvded no nonsrc hubbng polcy and src hubbng polcy, dependng on wheher drec servce s o be allowed or no. Furher, src hubbng polcy has wo ypes. Sngle allocaon hub locaon problems requre ha each node be assgned o one hub for servce on all nbound and oubound roues; mulple allocaon hub locaon problems allow each node o send and receve va several hubs (see Fgure 2). Hub locaon research has focused prmarly on he sngle allocaon hub locaon problem because conans he fundamenal properes of hub neworks whle possessng a smple srucure. Furhermore, he sngle allocaon hub locaon problem requres he raffc volume 51

3 52 Transporaon Research Record 2008 TABLE 1 Exsng Pernen Leraure Decson Objecve Dscoun Capacy Model Leraure Varables Funcon Raes Consrans Srucure O Kelly (1) L, A, R T C N QIP Klncewcz (2) L, A, R T C LC IP Campbell (3) L, A, R T, H, L N N IP Skorn-Kapov and Skorn-Kapov (4) L, A, R T C N IP Aykn (5) L, A, R T C N IP Campbell (6) L, A, R T, H, L C N IP Erns and Krshnamoorhy (7) L, A, R T C N IP O Kelly e al. (8) L, A, R T, H, L V N IP Skorn-Kapov e al. (9) L, A, R T, H, L C N QIP O Kelly and Bryan (10) L, A, R T V N IP Klncewcz (11) L, A, R T V N IP Arnold e al. (12) L, A, R T N N MILP Racunca and Wyner (13) L, A, R T V N MILP L: hub locaon varable A: nonhub allocaon varable R: raffc roue varable T: ransporaon cos H: hub consrucon cos L: lnk consrucon cos C: consan dscoun rae V: varable dscoun rae N: no consderaon QIP: quadrac neger programmng LC: lnk capacy consran MILP: mxed neger lnear programmng IP: neger programmng Approach Mehod Model Formulaon Objecve Funcon Decson Varables Facly Consrans Tradonal Model Heursc approach Exogenously assumed dscoun raes Transporaon cos Consrucon cos Hub locaon Non-hub allocaon Uncapacaed hubs and lnks Proposed Model Analycal approach Endogenously deermned consoldaon effecs Transporaon cos Consrucon cos Invenory cos Handlng cos Hub locaon Non-hub allocaon Lnk frequency Flow capacy of hubs & lnks Frequency capacy of lnks FIGURE 1 Dsncve properes of model. Hub Nework Polcy Src Hubbng Polcy Sngle Allocaon Mulple Allocaon Non-Src Hubbng Polcy FIGURE 2 Oulne of hub nework polces.

4 Km, Park, and Km 53 from each nonhub node o be moved va only one hub. Ths may no only brng abou a cos-decreasng effec due o consoldaon of flows, bu also cause more congeson n specfc hubs or lnks beween hubs, and herefore generae addonal requred coss. Fnally, capacy consrans have more nfluence on nework desgn and s opmal soluon. Therefore, hs paper wll formulae he hub nework desgn model usng sngle allocaon src hub nework polcy. In hs polcy, flows are assumed o pass hrough one or more hubs. Bu handlng pahs wh more han wo hubs would ncrease subsanally he number of pahs n he model, and s beleved ha mulple pahs would resul n only a fracon of demand appearng n any opmal soluon (13). Therefore, we wll focus on wo or fewer hub pahs. Decson-Makng Srucure The decson-makng srucure of he hub nework desgn model dffers from ha of general logsc models. Frs, he former ncludes raffc roues beween nodes as endogenous decson varables, whereas he laer s composed of sraegc level (facly locaon), accal level (flow roung and shppng plannng) and operaonal level (producon and operaon rae) decsons sequenally, so neher flow roung nor shppng plannng have any nfluence on he decson of facly locaon (14). Ths s caused by oversmplfcaon, such as unlmed facly capacy and here beng no addonal congeson delays or coss. I s known, however, ha roung decsons are affeced by hub locaons, and he choce of hub locaons by roung decsons (5). Hence, hub locaon and roung problems should be consdered ogeher n desgnng hub neworks. Second, one of he unque properes of hub neworks s consoldaon of flows. Consoldaon faclaes economes of scale and hus decreases un cos. The change n un cos has a sgnfcan effec on opmal roung decsons. Consoldaon, however, may also brng abou addonal coss and delays assocaed wh he operaon of hub facles. The consoldaon of flows s closely relaed o opmal frequences and shpmen szes; hus, opmal frequences should be ncluded n he decson varables of he model. Las, lnk and hub capacy consrans affec hub locaon and nonhub allocaon. Especally n he sngle allocaon hub nework problem, flows are shpped va a conneced hub, so hub locaon and nonhub allocaon should be recalculaed f assgned flows a opmal- y are larger han he capacy of lnks or hubs. Therefore, he srucure of he hub nework desgn problem should nclude hub locaon, nonhub allocaon, and opmal frequency decsons o reflec he neracon beween sraegc decsons and accal decsons. In hs paper, he decson varables n he model are (a) hub locaon varables, (b) nonhub allocaon varables, and (c) opmal frequency varables. The unqueness of he precedng hub nework desgn decson s shown n Fgure 3. Cos Funcons The general classfcaon of logscs coss has been developed by many scholars. Operaonal and consrucon coss of hub neworks wll be quanfed on he bass of he classfcaon of logscs coss by Blumenfeld e al., as shown n Fgure 4 (15). Fgure 5 shows he pah and he cos componens of an em from orgn node o j ha s roued va hubs k and m. The cos componens conss of lnk and hub consrucon coss, ransporaon coss, frequency delay coss, ravel me delay coss, dsrbuon ransshpmen coss, and servce delay coss. The cos componens over he enre hub neworks are calculaed as follows [see Campbell, O Kelly and Bryan, Cranc e al., and Daganzo, among many ohers (3, 10, 14, 16)]: L LC p uc C = dz dklzkzl cf dp + + O O C TR TR C = uc df + dkl fkl + dl jflj O C HC FR H p uc = cf dp p v = + c 2 O j P z j zw j j s f + zw jl j s f lj j zz jlwj h f kl dz lj jl () 1 ( 2) () 3 ( 4) Tradonal Model Proposed Model Sraegc Facly Locaon/Scale Resource Developmen Hub Locaon Non-Hub Allocaon Taccal Roue Choce of Flows Shppng Plannng Vehcle/Flee Sze Roue Choce of Flows Opmal Frequency Opmal Shpmen Sze Operaonal Producon & Operaon Rae Economes of Scale Capacy Consrans FIGURE 3 Unqueness of hub nework desgn decson.

5 54 Transporaon Research Record 2008 Handlng Cos Moon Cos Dsrbuon/Transshpmen Cos Transporaon Cos Cos Componens Vehcle Operang Cos Holdng Cos Ren Cos Hub and Lnk Consrucon Cos Invenory Cos FIGURE 4 Classfcaon of logscs coss by Blumenfeld e al. (15). Spen Tme n Vehcle and n Node C DT DT C = z + zjl uc Wj () 6 C TT SD = = O + O / l k v d c vcd lj j zw j s v + zw jl j s v lj v d DT vc p u z + zjl Wj j k l l k O D P P/ p u DT + z zjl / l k W j O The noaons are as follows: O, D, P = se of orgn, desnaon, and hub node, respecvely; n = number of nodes; p = number of hubs deermned exogenously; h, s = lnk ype beween hubs or hub and nonhub, respecvely; v c = me value of a commody c; cf = converson facor of operang days; dp = duraon perod; p = me horzon of neres; d j = dsance of a dreconal lnk (, ); W j = amoun of flow from o j; C LC = oal lnk consrucon coss; C HC = oal hub consrucon coss; c kl j zzw jl j h v kl () 5 ( 7) C TR = oal ransporaon coss; C DT = oal dsrbuon ransshpmen coss; C FR = oal frequency delay coss; C TT = oal ravel me delay coss; C SD = oal servce delay coss; uc L = un cos for lnk consrucon per lengh; uc H = un cos for hub consrucon per spo; uc TR = un ransporaon coss per lengh-frequency on lnk ype ; uc DT = un dsrbuon ransshpmen cos; u DT = un servce delay me; z j = 1 f node j s a hub and 0 oherwse; z = 1 f node s allocaed o a hub k and 0 oherwse; f kl = opmal frequency of a lnk (k, l), of whch he ype s {h, s}; f max kl = maxmum frequency of a lnk (k, l); vkl = velocy of a lnk (k, l), he ype of whch s {h, s}; and K max j = maxmum shpmen sze of a lnk (, j). Equaon 1 llusraes he sum of he deprecaon coss of lnk consrucon; he erms n brackes sgnfy he lnk dsances beween orgn and hub, beween hubs, and beween hub and desnaon. Equaon 2 represens he sum of he deprecaon coss of hub consrucon. Equaons 3 and 4 represen oal ransporaon coss and oal frequency delay coss, respecvely. Boh nclude a lnk frequency varable, so he sum of he ransporaon coss and frequency delay coss of a lnk (r, s) s rewren as follows. TR FR TR vcp Γ Crs + Crs = uc drs frs + 2 f rs TR 2uc d v p Γ () 8 rs c Ren Cos (lnk) Transporaon Cos Invenory Cos (n veh.) Orgn Ren Cos (lnk) Transporaon Cos Invenory Cos (n veh.) HUB 1 HUB 2 Ren Cos (lnk) Transporaon Cos Invenory Cos (n veh.) Desnaon Invenory Cos (n orgn) Ren Cos (hub) Handlng Cos Invenory Cos (n hub) Ren Cos (hub) Handlng Cos Invenory Cos (n hub) FIGURE 5 Pah and cos componens of an em from orgn o desnaon.

6 Km, Park, and Km 55 where Equaon 8 s he arhmec geomerc mean nequaly, so he opmal frequency o mnmze he summaon s calculaed as follows: f rs zw rs rq f r O s P q D Γ= zprzqswpq f r P s P p O q D zw sr ps f r P s D p O = The capacy consran dummy varable and he maxmum frequency of a lnk (r, s) are κ rs and f rs max, respecvely, and he mnmum cos of he sum of he ransporaon coss and frequency delay coss s as follows: { } = ( ) TR FR TR C + C * 1 κ 2uc d v pγ where κ rs equals 0 f f rs = rs rs rs rs c f rs max and 1 oherwse. Equaon 5 llusraes he sum of ravel me delay coss, and he hree erms represen he cos on he lnk beween orgn and hub, beween hubs, and beween hub and desnaon, respecvely. Equaon 6 presens he sum of dsrbuon and ransshpmen coss. Equaon 7 llusraes he sum of servce delays n hubs as has been presened by Cranc e al. (14). Formulaon In hs paper, he auhors formulae he hub nework desgn model reflecng he unqueness of hub nework servce on he bass of he sngle allocaon hub locaon model presened by Campbell (3). In addon, hub capacy consrans and lnk frequency and capacy consrans are added. The proposed model s as follows. Consrans 12, 13, 19, and 20 are he same as hose found n Campbell (3). Consran 12 ensures exacly p, he prescrbed number of hubs ha should be chosen. Consrans 13 and 19 enforce sngle allocaons for each node and preven allocaons o nonhub nodes. Consran 20 represens he nonnegave neger consran. Equaons 14 and 15 make flows on every lnk equal o or smaller han maxmum shpmen sze. Consrans 16 and 17 ensure ha lnk frequency on lnks beween hubs and beween hub and nonhub should no exceed he maxmum frequency, respecvely. Equaon 18 represens hub capacy consrans. LC HC TR DT FR TT SD mn Z = C + C + C + C + C + C + C ( 11) subjec o vcp Γ 2uc TR d vcp Γ 2uc TR d rs rs + κ rs uc d f TR max rs rs () 9 vcp Γ + ( ) max f 10 2 rs zk = p ( 12) k z = 1 ( 13) k O h max f z z f k, l ( 16) kl s max f z f, k ( 17) DT u z + z jl Wj p < z z 1, k ( 19) f, z, z 0 neger, k, l, j, ( 20) max s zw K f, ( 15) j O / l k max h zzw K f kl, ( 14) jl kl k jl j kl kl COMPARISON OF ECONOMIES OF SCALE CAUSED BY CONSOLIDATION OF FLOWS k ( 18) Prevous researchers have appled varous ypes of cos-decreasng effecs o her models o reflec economes of scale semmng from consoldaon of flows. Some have assumed ha he un ransporaon cos per operaon performance (flows dsance) of every lnk s consan, causng economes of scale n hub neworks o be nvaldaed. O Kelly, Klncewcz, Skorn-Kapov and Skorn-Kapov, Aykn, Campbell, Erns and Krshnamoorhy, and Skorn-Kapov e al. have appled consan dscoun raes assumed exogenously on lnks beween hubs (1, 2, 4 7, 9). The un cos s consan, bu s decreased by α%. O Kelly and Bryan have formulaed her model o allow he un cos per operaon performance o be lnearly decreased, causng oal coss o ncrease a a decreasng rae as flows ncrease (10). Racunca and Wyner have refleced he dscoun rae n hub neworks wh he square roo of he amoun of flows regardless of ravel dsance (13). In addon o he lnks beween hubs, hey have also added an exponenal funcon of he flows o hub-o-desnaon lnks, because he lnks from hub o desnaon are assumed o have lower dscoun raes. Thus far, no prevous research has appled endogenous dscoun raes. For he model here, dscoun raes are deermned endogenously by ncludng he opmal frequency decson process. Fgures 6 and 7 compare he dscoun raes of oal coss when he shpmen sze and frequency, respecvely, are resrced. If he opmal shpmen sze, whch equals oal flows on a lnk dvded by s opmal frequency, s larger han he maxmum shpmen sze on he lnk, hen he frequency on he lnk s ncreased. Therefore ransporaon cos under he shpmen sze consrans s larger han s on unconsraned condons (Fgure 6). However, f opmal frequency on a lnk s larger han he maxmum frequency on he lnk, hen ransporaon cos s smaller han n he unconsraned condon. Bu frequency delay cos s more han unconsraned, so oal cos s ncreased (Fgure 7).

7 56 Transporaon Research Record 2008 (Type I) α = 0.8 (Type II) (Type III) α = 0.7 Toal Transporaon Coss α = 0.6 Effecs of he shpmen sze consrans (Type IV) Operaon Performance (flows x dsance) Proposed Model (shpmen sze consraned) Proposed Model (unconsraned) O'Kelly (1) Racunca & Wyner (13) O'Kelly & Bryan (10) FIGURE 6 Comparson of dscoun raes of oal coss (maxmum shpmen sze consraned). The dscoun rae s no appled o he lnks beween nonhub and hub n O Kelly and beween orgn and hub n Racunca and Wyner (Type I) (1, 13). The lnk beween hubs n O Kelly reflecs scale economes represened by he nerhub dscoun rae α, whch ranges from 0 o 1, wh he dscoun ncreasng as α decreases (1). The same dscoun rae s appled o all nerhub lnks regardless of he amoun of flow ha ravels across he lnk (Type II). The model by O Kelly and Bryan allows he oal cos funcon o be concave (10). The oal coss ncrease a a decreasng rae as flows ncrease, bu he oal coss may be decreased f hey exceed a hghes pon. The pon vares wh he un cos decreasng rae, whch s deermned arbrarly, so he objecve and dealed esmaon of he rae s necessary for applyng he model (Type III). (Type III) (Type I) Toal Transporaon Coss α = 0.8 α = 0.7 α = 0.6 (Type II) (Type IV) Effecs of he frequency consrans Operaon Performance (flows x dsance) Proposed Model (shpmen sze consraned) Proposed Model (unconsraned) O'Kelly (1) Racunca & Wyner (13) O'Kelly & Bryan (10) FIGURE 7 Comparson of dscoun raes of oal coss (maxmum frequency consraned).

8 Km, Park, and Km 57 TABLE 2 Daa Ses for Emprcal Sudes [flow (on/day), dsance (km), speed (km/h)] Facor Chcago Frankfur Hong Kong Inchon London Los Angeles New York Pars Sngapore Tokyo Chcago, Illnos F D 6,965 12,539 10,555 6, ,656 15,253 10,107 S Frankfur, F Germany D 6,965 9,171 8, ,324 6, ,256 9,368 S Hong Kong, F Chna D 12,539 9,171 2,045 9,643 11,654 12,970 9,608 2,590 2,938 S Inchon, F Souh Korea D 10,555 8,544 2,045 8,858 9,643 11,100 8,929 4,632 1,263 S London F D 6, ,643 8,858 8,756 5, ,861 9,587 S Los Angeles, F Calforna D 139 9,324 11,654 9,643 8, ,104 14,154 8,765 S New York F D 58 6,191 12,970 11,100 5, ,835 15,260 10,838 S Pars F D 6, ,608 8, ,104 5,835 10,703 9,713 S Sngapore F D 15,253 10,256 2,590 4,632 10,861 14,154 15,260 10,703 5,375 S Tokyo F D 10,107 9,368 2,938 1,263 9,587 8,765 10,838 9,713 5,375 S F = flow; D = dsance; S = speed. Racunca and Wyner produced much smaller coss han ohers because he cos funcon s exponenal of flows regardless of ravel dsance, alhough s well known ha economes of scale are relaed o boh amoun and ravel dsance of flows (13). Furhermore, hey have no consdered ha he dscoun resulng from consoldaon of flows s appled no only o he lnks beween hub and desnaon, bu also o he lnks beween orgn and hub (Type IV). In he proposed model, he dscoun s appled o every lnk. The dscoun rae s relaed o boh flows and dsance and resrced by he maxmum lnk shpmen sze and frequency. Furhermore, he model does no need any esmaon of parameers, so he resul from he proposed model s nerpreed o be objecve and reflecve of realy. EMPIRICAL ANALYSIS Daa Sources To llusrae he dfferences beween he soluons suggesed by he proposed model and hose suggesed by prevous models, Inernaonal Cvl Avaon Organzaon arlne daa from he arlne commody n 2001 were used. Ten nodes were seleced from he se of he 25 larges arpor ces n 2001 (shown n Table 2). Un coss and defaul values are shown n Table 3. Resuls Table 4 compares he resuls from he proposed model wh O Kelly s resuls and Racunca and Wyner s (1, 13). The resul from he pro- posed model s smlar o O Kelly s, bu he laer may vary wh he dscoun rae α, whch s deermned exogenously. Therefore, addonal esmaon s requred. In conras, he resul from Racunca and Wyner s sgnfcanly dfferen from ha obaned n oher research (13). Fgures 8 and 9 show he resuls of he opmal nework desgn. The fndngs of O Kelly wh α =0.8 produced he same resul as TABLE 3 Un Coss and Defaul Values for Emprcal Sudes Iem Un Value Node no. 10 Hub no. 3 Tme horzon Hour 24 Tme value $/on 660 Deprecaon cos of $/km 0 lnk consrucon Deprecaon cos of $/pon 31,960 hub consrucon Dsrbuon $/on 860 ransshpmen cos Servce delay me h/on Transporaon cos $/freq-on $/on-km 206 Shpmen sze Ton/flgh (beween hubs) 107 Ton/flgh (beween hub 60 and nonhub)

9 58 Transporaon Research Record 2008 TABLE 4 Comparson of Resuls from Seleced Research Nework Desgn Resuls Rank of Resuls from Research Hub 1 Hub 2 Hub 3 Proposed Racunca (allocaed nonhubs) (allocaed nonhubs) (allocaed nonhubs) Model O Kelly (1) and Wyner (13) 4 (3, 4, 9, 10) 5 (2, 5, 8) 7 (1, 6, 7) (1, 6, 7) 4 (3, 4, 9, 10) 5 (2, 5, 8) (3, 4, 9, 10) 7 (1, 6, 7) 8 (2, 5, 8) (2, 5, 8) 4 (3, 4, 9, 10) 7 (1, 6, 7) (1, 6, 7) 4 (3, 4, 9, 10) 8 (2, 5, 8) (3, 4, 9, 10) 5 (2, 5, 8) 7 (1, 6, 7) (1, 6, 7) 2 (2, 5, 8) 4 (3, 4, 9, 10) (1, 6, 7) 3 (3, 4, 9, 10) 5 (2, 5, 8) (3, 4, 9, 10) 7 (1, 6, 7) 8 (2, 5, 8) (2, 5, 8) 3 (3, 4, 9, 10) 7 (1, 6, 7) (1, 6) 4 (3, 4, 9, 10) 5 (2, 5, 7, 8) (1, 6, 7) 3 (3, 4, 9, 10) 8 (2, 5, 8) (3, 4, 9, 10) 5 (1, 2, 5, 8) 7 (6, 7) (1, 6, 7) 2 (2, 5, 8) 3 (3, 4, 9, 10) (1, 6) 4 (3, 4, 9, 10) 8 (2, 5, 7, 8) FIGURE 8 Opmal soluon found n model of O Kelly (1) FIGURE 9 Opmal soluon found n model of Racunca and Wyner (13).

10 Km, Park, and Km 59 dd he proposed model (1). In conras, Racunca and Wyner s model seleced Node 3 as a hub, no Node 4, so he flows beween Node 4 and Node 10 wll requre a roundabou (13). Ths may be caused by excessve consoldaon of flows o he hub whou consderaon of he delay caused by he roundabou. The emprcal sudes show ha O Kelly s model requres addonal esmaon of he dscoun rae, whle Racunca and Wyner s model may produce excessve consoldaon of flows (1, 13). The proposed model, however, can deermne he dscoun rae endogenously, ncludng he opmal frequency decson, and smulae he delay accordng o he roundabou as well as economes of scale caused by consoldaon of flows. The proposed model s hus nerpreed o solve he hub nework desgn problem and reflec he unqueness of hub neworks raonally and objecvely. CONCLUSIONS Ths paper develops a hub nework desgn model wh consoldaed raffc flows ha faclae he realzaon of economes of scale. The model s desgned o endogenously deermne cos dscoun raes by ncludng he decson-makng process of he opmal frequences. The model also ncludes several addonal cos componens and hub and lnk capacy consrans ha reflec he unqueness of hub neworks. These addonal coss and consrans have no been consdered n prevous sudes. The resuls of he model have been compared wh hose of some oher recenly proposed models. The proposed model can deermne he dscoun rae endogenously by ncludng decsons on he opmal frequency. The model can herefore realze economes of scale semmng from consoldaon of flows whou any esmaon of parameers. Furhermore, ncludes several cos componens and capacy consrans n relaon o he unqueness of hub neworks, so can smulae and analyze several condons of more relable and realsc hub neworks, such as rade-offs beween economes of scale and he delay caused by a roundabou, congeson from he operaons of hub facles, shppng plannng wh lnk capacy consrans, and so on. The model could also be appled o assess he effcency of exsng and planned logsc sysems of hub operaons, whch s a subjec for fuure sudy. ACKNOWLEDGMENTS The auhors hank he Korea Research Foundaon, whose gran, funded by he governmen of Souh Korea, suppored hs research. The auhors also hank he anonymous referees who revewed an earler verson of hs manuscrp for her consrucve, helpful commens. REFERENCES 1. O Kelly, M. E. A Quadrac Ineger Program for he Locaon of Ineracng Hub Facles. European Journal of Operaonal Research, Vol. 32, 1987, pp Klncewcz, J. G. Heurscs for he P-Hub Locaon Problem. European Journal of Operaonal Research, Vol. 53, 1991, pp Campbell, J. F. Ineger Programmng Formulaons of Dscree Hub Locaon Problems. European Journal of Operaonal Research, Vol. 72, 1994, pp Skorn-Kapov, D., and J. Skorn-Kapov. On Tabu Search for he Locaon of Ineracng Hub Facles. European Journal of Operaonal Research, Vol. 73, 1994, pp Aykn, T. Neworkng Polces for Hub-and-Spoke Sysems wh Applcaon o he Ar Transporaon Sysem. Transporaon Scence, Vol. 29, 1995, pp Campbell, J. F. Hub Locaon and he P-Hub Medan Problem. Operaons Research, Vol. 44, 1996, pp Erns, A. T., and M. Krshnamoorhy. Exac and Heursc Algorhms for he Uncapacaed Sngle Allocaon P-Hub Medan Problem. Locaon Scence, Vol. 4, 1996, pp O Kelly, M. E., D. L. Bryan, D. Skorn-Kapov, and J. Skorn-Kapov. Hub Nework Desgn wh Sngle and Mulple Allocaon: A Compuaonal Sudy. Locaon Scence, Vol. 4, 1996, pp Skorn-Kapov, D., J. Skorn-Kapov, and M. E. O Kelly. Tgh Lnear Programmng Relaxaons of Uncapacaed P-Hub Medan Problems. European Journal of Operaonal Research, Vol. 94, 1996, pp O Kelly, M. E., and D. L. Bryan. Hub Locaon wh Flow Economes of Scale. Transporaon Research Par B: Mehodologcal, Vol. 32, 1998, pp Klncewcz, J. G. Enumeraon and Search Procedures for a Hub Locaon Problem wh Economes of Scale. Annals of Operaons Research, Vol. 110, 2002, pp Arnold, P., D. Peeers, and I. Thomas. Modelng a Ral/Road Inermodal Transporaon Sysem. Transporaon Research Par E: Logscs and Transporaon Revew, Vol. 40, 2004, pp Racunca, I., and L. Wyner. Opmal Locaon of Inermodal Fregh Hubs. Transporaon Research Par B: Mehodologcal, Vol. 39, 2005, pp Cranc, T. G., J.-A. Ferland, and J.-M. Rousseau. A Taccal Plannng Model for Ral Fregh Transporaon. Transporaon Scence, Vol. 18, 1984, pp Blumenfeld, D. E., L. D. Burns, J. D. Dlz, and C. F. Daganzo. Analyzng Trade-Offs Beween Transporaon, Invenory and Producon Coss on Fregh Neworks. Transporaon Research Par B: Mehodologcal, Vol. 19, 1985, pp Daganzo, C. F. Logscs Sysems Analyss. Sprnger-Verlag, New York, The opnons expressed n he paper are hose of he auhors and do no necessarly reflec hose of he fundng agences. The Fregh Transporaon Plannng and Logscs Commee sponsored publcaon of hs paper.