Conference Paper Assignment techniques on Virtual Networks. Performance considerations on large multi-modal networks

Transcription

1 econstor Der Open-Access-Publiktionsserver der ZBW Leibniz-Informtionszentrum Wirtschft The Open Access Publiction Server of the ZBW Leibniz Informtion Centre for Economics Jourquin, Brt; Limbourg, Sbine Conference Pper Assignment techniques on Virtul Networks. Performnce considertions on lrge multi-modl networks 43rd Congress of the Europen Regionl Science Assocition: "Peripheries, Centres, nd Sptil Development in the New Europe", 27th - 30th August 2003, Jyväskylä, Finlnd Provided in Coopertion with: Europen Regionl Science Assocition (ERSA) Suggested Cittion: Jourquin, Brt; Limbourg, Sbine (2003) : Assignment techniques on Virtul Networks. Performnce considertions on lrge multi-modl networks, 43rd Congress of the Europen Regionl Science Assocition: "Peripheries, Centres, nd Sptil Development in the New Europe", 27th - 30th August 2003, Jyväskylä, Finlnd This Version is vilble t: Nutzungsbedingungen: Die ZBW räumt Ihnen ls Nutzerin/Nutzer ds unentgeltliche, räumlich unbeschränkte und zeitlich uf die Duer des Schutzrechts beschränkte einfche Recht ein, ds usgewählte Werk im Rhmen der unter nchzulesenden vollständigen Nutzungsbedingungen zu vervielfältigen, mit denen die Nutzerin/der Nutzer sich durch die erste Nutzung einverstnden erklärt. Terms of use: The ZBW grnts you, the user, the non-exclusive right to use the selected work free of chrge, territorilly unrestricted nd within the time limit of the term of the property rights ccording to the terms specified t By the first use of the selected work the user grees nd declres to comply with these terms of use. zbw Leibniz-Informtionszentrum Wirtschft Leibniz Informtion Centre for Economics

2 Assignment techniques on Virtul Networks Performnce considertions on lrge multi-modl networks Europen Regionl Science Assocition ERSA 2003 Congress University of Jyväskylä - Finlnd Brt JOURQUIN I, II nd Sbine LIMBOURG I II Fcultés Universitires Ctholiques de Mons (FUCM), Group Trnsport & Mobility (GTM) 151 Ch. de Binche, B-7000, Mons, Belgium Tel : , Fx : , gt&m@fucm.c.be II Limburgs Universitir Centrum (LUC) Universitire Cmpus, gebouw D, B-3590 Diepenbeek, Belgium Tel : , Fx : ABSTRACT Multi-Modl freight models re trditionlly built following the well known fours steps model in which genertion, distribution, modl-split nd ssignment re seen s seprted modules. An lterntive pproch, implemented in some softwres, is to represent the multi-modl network by mens of mono-modl one, representing ech prticulr trnsport opertion (loding or unloding opertion, trnshipments,...) by dedicted virtul link. This pproch is proven to give interesting results, but s the drwbck to generte much lrger networks s their geogrphic representtion. These huge networks (often lrger thn severl hundreds of thousnds links) mde it difficult to implement loop-bsed equilibrium ssignment techniques. The incresing computtion power of recent hrdwre mke it now much more esy to test severl lterntive ssignment techniques. This pper presents some results obtined on lrge multi-modl network, using different equilibrium ssignment lgorithms

3 1 Introduction Until few yers go, trnsport models were essentilly focused on pssengers flows. More recently, some freight specific network models hve been developed but, like pssengers models, they re essentilly nlysing networks from "link" point of view rther thn from "node" point of view. Even if some of them dels to some extend with the different opertions performed t the nodes, i.e. loding /unloding, trnshipment or simple trnsit, their output still trgets minly trnsport flows on the networks. As result of trend towrds economic globlistion, just in time deliveries nd rod trnsports expnsion, gret del of ttention is pid nowdys to reorgnistion of networks, intermodl trnsports nd new technicl bundling concepts in order to substitute to direct rod trnsport lterntive trnsport solutions with less negtive externl effects. Thus, there is need for better modelling of the functions ssumed by nodes, i.e. terminls nd trnshipment pltforms. Beyond this brief introduction, this pper will present n overview of the Virtul Network methodology implemented in the Nodus softwre developed t FUCM-GTM. It will then discuss the most used ssignment techniques nd how they cn be pplied to Virtul networks. Finlly, the performnces of these methods will be discussed on smll nd lrge networks. 2 NODUS nd Virtul networks A geogrphicl multimodl trnsport network is mde of links like rods, rilwys or wterwys, on which vehicles moves; t its nodes it is mde lso of connecting infrstructures these links like terminls or logistics pltforms where goods re loded, unloded, trnshipped or processed in different wys. To nlyse trnsport opertions over the network, costs or weights must be ttched to these geogrphicl links over which goods re trnsported nd to the connecting points where goods re hndled. However, most of these infrstructures cn be used in different wys nd with different costs. For exmple, bots of different sizes nd operting costs cn use the sme wterwy; t terminl truck's lod cn be trnshipped on trin, bundled with some others on bot or simply unloded s it reched its destintion. Normlly, the costs of these lterntive opertions should be different

4 2.1 The Building of Virtul Network: n Intuitive Approch A simple geogrphic network does not provide n dequte bsis for detiled nlyses of trnsports opertions where the sme infrstructure is used in different wys. To solve this problem, the bsic ide, initilly proposed by Hrker (1987) nd Crinic et l. (1990), is to crete virtul link with specific cost for prticulr use of n infrstructure. The NODUS softwre proposes methodology nd n lgorithm which cretes in systemtic nd utomtic wy complete "virtul network" with ll the virtul links corresponding to the different opertions which re fesible on every rel link or node of geogrphic network. The softwre nd its underlying methodology re discussed in Jourquin (1995) nd Jourquin & Beuthe (1996). This permits to pply the methodology to extensive multimodl netwoks. The solution cn be presented first in n intuitive wy by using the exmple of simple wterwys network, s shown in Figure 1. This network consists of 4 nodes nd 5 links. The W links represent wterwys nd the R links ril trcks. The numbers fter these letters correspond to the trnsporttion mens tht cn be used on the links. So, W1 represents wterwy tht cn support smll brges nd W2 wterwy tht cn be used by both smll nd lrge brges. To go from node to d, it could be possible tht the route 1+3, using lrge brges nd trins, is less expensive thn the route 5, using exclusively smll brges. It ppers tht computing costs nd routes on this kind of networks is not immedite: different costs cn be ssigned on single link, depending on which trnsporttion mens is used. In this exmple, the use of smll brge on link W1 hs different cost thn the use of lrge brge on the sme link; the sme is true for the nodes: in the given exmple, the simple trnsit of smll brge come from link 1 nd going on link 2 cn be done t no cost, but the trnshipment from lrge brge coming from link 1 onto trin tht will go on link 3 represents n importnt cost. This problem cn be usefully hndled on the corresponding virtul network illustrted in Figure 1, provided tht the relevnt costs re ttched to ech of the virtul links. As cn be seen, the solution involves the cretion of set of virtul nodes nd set of virtul links connecting these nodes. Ech rel link hs been split in s mny virtul links s - 3 -

5 there re possible uses; their end-nodes re connected by new virtul links corresponding to simple trnsit or trnshipment opertions t the rel nodes. In this wy, this network with multiple mens use is represented by unique but more complex network on which ech link corresponds to unique opertion with specific cost. Then, one chepest pth cn be computed by mens of n lgorithm such s the one proposed by Dijkstr (1959) for low density grphs. The resulting solution is n exct solution, tking ll the possible choices into ccount. We cn now demonstrte how the virtul network is built on the bsis of rther simple, though somewht complex nottion, which provides convenient wy to link cost functions to virtul links. Tble 1 enumertes the elements of the rel network. Figure 1: Multimodl network 1 (W2) b 2 (W1) c 5 (W1) 3 (R1) 4 (R1) d In first step, the virtul links corresponding to the rel links, i.e. ril trcks, etc., must be generted. These re defined in Tble 2 by their end-nodes, the nottion of which indictes successively the node, the rel link, the mode nd the mens. Tble 1: Rel network Link Node 1 Node 2 Type of link 1 A B W2 2 B C W1 3 B D R1 4 D C R1 5 A D W1 Tble 2: Trvelling links Rel links End-nodes of virtul links 1 1W1 b1w1 1W2 b1w2 2 b2w1 c2w1 3 b3r1 d3r1 4 d4r1 c4r1 5 5W1 d5w1-4 -

6 Tble 3: Connecting virtul links to node b Rel node End-nodes of virtul links b b1w1 b2w1 b1w2 b2w1 b1w1 b3r1 b1w2 b3r1 b2w1 b3r1 Figure 2: Prtil virtul network 1W1 b1w1 b2w1 c2w1 5W1 c4r1 1W2 b1w2 b3r1 d3r1 d5w1 d4r1 In second step, these virtul links must be connected by trnsit or trnshipment virtul links. To keep things s simple s possible, we just enumerte in Tble 3 the connecting virtul links relted to node b. They cn be viewed in Figure 2, where the virtul network resulting from tht procedure is presented when it is pplied to ll the nodes. In this network, the boldfced links represent the links of the rel network, possibly split up. The dotted links represent the trnsit virtul links, while the trnshipment links re indicted by thin continuous line. This network is not yet complete becuse it does not contin entry nd exit nodes in/from the network. This cn be done by the cretion of dditionl virtul nodes ssocited to loding nd unloding opertions t nodes where they re possible. Those entry or exit points in the virtul network re referenced by dding 000 to the rel node number. They must lso be connected to other nodes by pproprite virtul links. In generl, the weight given on link cn very well vry with the direction it is used; loding nd unloding opertions for instnce don t hve necessrily the sme cost, nd - 5 -

7 the cost of going upstrem on river is normlly higher thn going downstrem. To solve tht problem, ll virtul nodes re doubled t genertion time by dding + or sign to their code; by the sme token, ll links re split into two oriented rrows connecting these new nodes. This is illustrted for rel node b in figure 3. The virtul network requires the development of four types of cost functions, which re ssocited with specific virtul links through their nottion: The use of trvelling cost function is indicted by difference between the virtul link's indices denoting the rel end-nodes; Trnsit costs re pplied when the link's indices denoting the two connected rel links vry, wheres the mode nd mens indices remin the sme; For trnshipment costs, the link's indices of the connected rel links should vry, s well s the mode nd/or mens indices; For loding/unloding costs, one of the two indices of the rel links is "00000". Those doubled virtul nodes lso mke it possible to void unwnted movements, like n unloding followed by loding to circumvent forbidden trnshipment opertion. Figure 3: Detiled virtul network t node - + b000 b1w b2w1 b1w b3r1-6 -

8 Obviously, the codifiction used in the preceding figures is not suitble for rel pplictions; using single letter s node lbel would limit the size of the rel network to 26 nodes! Tht s why the virtul nodes re coded in the following wy: plus or minus sign, 5 digits for the node number, 5 digits for the link number, 2 digits for the mode nd 2 digits for the mens. Ech lbel is thus represented by 14 digits number preceded by sign. 2.2 Cost Functions on Virtul Networks After this overview of the bsic methodology, it is necessry to explin how the cost functions re connected with the virtul network nd which re their chrcteristics. As usul in trnsporttion nlysis (see, for exmple, Kresge nd Roberts (1971), or Wilson nd Bennet (1985)), "generlised cost", which llows to integrte ll fctors relevnt for trnsport decision mking in terms of monetry units, re used. The concept cn be defined in different wys ccording to whether it is the point of view of the shipper which is tken or the one of the crrier, nd ccording to the unit of reference which is used, i.e. tons, tons-km, vehicles, distnces, etc. The specific cost functions which compose the generlised cost, obviously, must be coherent cross modes nd mens, but their functionl forms cn be freely chosen. The virtul network requires the development of four types of cost functions, which re ssocited with specific virtul links through their nottion: The use of trvelling cost function is indicted by difference between the virtul link's indices denoting the rel end-nodes; Trnsit costs re pplied when the link's indices denoting the two connected rel links vry, wheres the mode nd mens indices remin the sme; For trnshipment costs, the link's indices of the connected rel links should vry, s well s the mode nd/or mens indices; For loding/unloding costs, one of the two indices of the rel links is "00000". These four types of function re mde of the following elements: All the costs relted to moving vehicle between trip's origin nd destintion, like lbour, fuel, insurnce, mintennce costs, or triffs; The cost of inventory of the goods during trnsporttion; Hndling nd storge costs or triffs, including pckging, loding nd unloding nd services directly linked to trnsport

9 All residul indirect costs like generl dministrtive services which my be ssigned to trnsports on n verge bsis. 2.3 Computing shortest pths on networks The contemporry trnsport systems re used in n intensive wy nd re often congested t vrious degrees, prticulrly in urbn res. Trnsport models hve to determine how the trffic is distributed over the trnsport network of which the structure nd the cpcity re known. This is the ssignment problem. Unless no cpcity constrints re tken into ccount (All-Or Nothing ssignment), n ssignment consists in the distribution of the trffic on network considering the demnd s well s the flows supported by the network (cpcity). The ssignment methods try to model the trffic s distribution over network tking into ccount set of constrints, such s cpcity nd trnsport costs, in order to obtin n equilibrium. This type of problems cn be solved by mens of optimistion methods. We re interested here in the methods intended to model flows of commodities on lrge multimodl networks, using virtul network, which, s explined erlier, mke it possible to combine modl choice nd ssignment in single step. The results of the ssignment, which depend on the sophistiction of the implemented method, include n estimte of flows, trvel durtion nd/or corresponding costs, for ech link of the network. All the equilibrium ssignment methods re bsed on the ll-or-nothing lgorithm. The principle of this lgorithm is to compute the minimum weight pth between ech pir of origin nd destintion, nd to llocte the totl demnd to be trnsported between these nodes onto this single pth. As shown in Tble 4, the Dijkstr s lgorithm, with binry hep implementtion, seems to be the best lgorithm to be used on virtul networks (which is grph with low density, s the number of links is fr below the squre of the number of nodes). The Dijkstr s lgorithm solves the problem of the shortest pth from n origin to ll the possible destintions. The lgorithm of Johnson finds ll the shortest pths between ll the pirs of nodes of the grph, it using the lgorithm of Bellmn-Ford nd Dijkstr. If we tke the detiled virtul network generted from the Belgin network, we hve pproximtely virtul links (M) nd virtul nodes (N), of which

10 (X) re potentil (un)loding nodes. The ltest chrcteristic involves tht the Dijkstr s lgorithm is the most suitble, becuse only smll mount of nodes re relevnt to compute pths from, becuse they re potentil origins. Tble 4: Complexity of shortest-pth lgorithms Liner (worst cse) Used priority queue Binry hep Fiboncci hep (worst cse) (mortized nlyze 1 ) Dijkstr s lgorithm Executed X times O(XN²) O(XM log 2 N) O(XN log 2 N+XM) Johnson s lgorithm O(N³) O(NM log 2 N) O(N² log 2 N+NM) Given N : number of nodes M : number of links X : number of nodes tht re potentil origin or destintion 1 A mortised nlysis gurntees n verge performnce for ech opertion in the worst cse. Among the vrious implementtions of this lgorithm, the binry hep implementtion gives the best results, becuse our grph is not dense (M is much lower thn N²). The Fiboncci hep implementtion gives smller execution times only in the most unfvourble cses (15% of the cses tested on our networks) nd is, on verge, 50% slower thn the binry heps for our problems. We hve lso introduced stop criterion in the Dijkstr s lgorithm. It stops indeed, s soon s the lgorithm determined the shortest pth to ll the destintions to rech strting from given origin. This improves the computing time by more thn 50%, becuse the flows strting from node re, in most cses, sent to only few destintions tht re reltively close to the origin. 2.4 Cost functions nd congested links The All-Or-Nothing (AON) lgorithm presents however some limits becuse the flow of trnsport between two given nodes should be distributed between vrious lternte routes. The multi-flow lgorithms mke it possible to clculte those pths, but they don t tke into ccount ny constrint. However, certin xes of the network suffer from congestion nd this congestion lso led in the serch of lternte routes, in order to blnce the trffic on the network. The equilibrium lgorithms try to model this phenomenon in trnsport systems. These models, which tke into ccount the vrition of the trnsporttion generlized costs ccording to the ssigned flows, consider tht the distribution of the trffic on the network is the result of n interction between the supply nd the demnd of trnsport

11 They try to pproch the equilibrium conditions stted formlly by Wrdrop in The second principle (or the design principle) sttes tht: Under equilibrium conditions trffic should be rrnged in congested networks in such wy tht the verge (or the totl) trvel cost is minimised. It is importnt to note tht the fct of using these methods on virtul network mkes it possible to observe trnsfers of flow between the vrious trnsporttion modes. Indeed, in opposition to the clssicl four stges models (genertion, distribution, modl-split nd ssignment), nd s specified higher, the virtul network combines the modl split nd the ssignment in single step. Thus, n lterntive route cn very well be used, totlly or prtilly, by nother mode nd/or mens of trnsport. This is probbly the principl contribution of the use of equilibrium ssignment methods on virtul networks nd will be illustrted in the next section. The trffic ssignment models which tke the effects of the congestion into ccount require reltion between the cost nd flow on the network, which cn be described in generl wy like : C =C ({V}). The cost on link should be function of flow V on the network nd not only on the link itself. This is simplified by considering tht C =C (V ) i.e. tht the cost on link is function of the flow V on it. A good cost function must hve the following properties: it must be relistic, non decresing, monotonous, continue, differentible nd should not generte infinite costs if the flow is equl or higher thn the cpcity. Mny functions (Ortúzr nd Willumsen (1990)) were proposed to describe this reltion. The most used re : Smock (1962) : C = C 0 e V / K where C : cost for given flow V C 0 : cost t free flow K : cpcity of the link C= C Overgrd (1967) generlises the previous function : α β ( The Bureu of Public Rods (1964), USA, proposes the stndrd function which β ( ) is certinly the most used: C = C 1 α( V / K ) 0 + We used this ltest function with α= 0.15 nd β= 4, which re the most used vlues for the Europen networks. 0 V / K)

12 2.5 Flow equilibrium lgorithms Vrious ssignment lgorithms which try to obtin distribution of flows on the network pproching the equilibrium conditions stted by Wrdrop cn be found in the literture. The implementtion of these lgorithms on virtul network is not immedite. Indeed, in the relity, congestion is observed on rel links, not on virtul links. However, in virtul network, the sme rel link is represented by vrious virtul links, ccording to the number of trnsporttion mens tht re defined. It is thus necessry to consolidte the flows obtined on the virtul links before obtining flows indeed ssigned on the rel links. A first technique, bsed on the method of the successive verges (MSA) ws implemented in Nodus. During the initilistion step, for ech link, the flow null nd its ssocited cost V is set to C is computed for free flow sitution. The process then enters in loop tht is repeted until stop condition is stisfied. At given itertion n, nd for ech link, the cost n C is computed, tht depends on the flow n 1 V found on the link t itertion n-1. A set of uxiliry flows n F is then obtined by mens of n AON ssignment bsed on the just re-computed costs. The new flow n V is then obtined : V n = n n 1 n n ( 1 λ ) V + λ F, where λ = n n 1 The lgorithm of Frnk-Wolfe (FW) (1956) is very similr to the previous one. It differs from it only by the wy λ is computed : insted of being fixed t 1/n, s it ws the cse in the method of the successive verges, it is clculted to optimise the displcement in the descent direction F n V n-1. After ech itertion, one hve to compute the depth of the descent: n λ min V where 0 λ n 1 0 C ( V ) dv The effects of the congestion cn lso be tken into ccount by n incrementl ssignment (INC). At ech itertion, only restricted proportion of the demnd mtrix is ssigned on the network. The incrementl method chrges the network grdully. The

13 totl quntity to trnsport is split by fctor p i, such s p = 1 nd, during ech itertion, n dditionl increment is loded on the network. We clculted these fctors s follows: If n represents the number of itertions, then, i n i + 1 p i = where i=1,2,,n n * ( n + 1) / 2 The p s i re thus decresing rithmetic progression in which the difference between two successive terms is 1 n * ( n + 1) / 2 The min disdvntge of the incrementl method is tht once tht flow is ssigned on link, it is not more possible to withdrw prt of it in order to ssign it to nother link. This will be illustrted in the next section. The incrementl method cn lso be ssocited the lgorithm of Frnk-Wolfe: to ccelerte the convergence of the lgorithm of Frnk-Wolfe, the incrementl method is used to obtin the initil flows (Inc+FW). When should the itertions be stopped? Except with regrd to the incrementl method, for which the number of itertions must be fixed priori, we use the stop-rule of Le Blnc et l (1975) : Stop if mx ( n) ( n 1) ( V V ) < ε, n V becuse the shpe of the cost-flow curve indictes tht flows re better indictors of the differences between successive itertions thn costs s it is discussed in Roy Thoms book (1991). We used ε =0,05, which is very low vlue which ensures to be very close to the rel equilibrium. 3 Some results on smll nd lrge networks These vrious lgorithms were implemented in the new version of the Nodus softwre, currently under development. This version (5.0) uses file formt tht is comptible with Arcinfo. Nodus uses the BBN Technologies' OpenMpTM pckge, tht is n Open Source JvBensTM bsed progrmmer's toolkit, providing the mens to llow users to see nd mnipulte vrious geosptil informtion. Entirely written in Jv, Nodus will be vilble on wide rnge of computer pltforms. i

14 3.1 A study cse on smll network A first exercise ws performed on mono-modl rod network of 17 nodes nd 36 links. In this network, 7 nodes cn be used s potentil origins nd/or destintions. In first step, smll rndom Origin-Destintion (O/D) mtrix ws ssigned onto the network. For this mtrix, nd fter n All-Or-Nothing ssignment, no link ppers to be congested. For ll the tested ssignment techniques, we obtined the sme set of computed pths nd the sme totl flows on the links. The All-Or-Nothing ssignment is obviously the fstest ssignment method. The difference on the totl costs on the network (see Tble 5) is explined by the fct tht, in the itertive methods, one uses costs clculted on the flows obtined during the preceding itertion. Indeed, these methods use stop criteri which compres two successive itertions. A minimum of two itertions is thus necessry, nd ll these dditionl itertions ssign flows on, t lest prtilly, loded network. The sme results re obtined for the MSA, FW nd Inc + FW methods becuse in the MSA method with two itertions, λ is fixed, by definition to 0,5, nd tht the sme vlue is obtined for λ in the two FW bsed methods. Tble 5: Results on non congested smple network Itertions Comp. Time (s) Totl cost AON , ,2 MSA ,4 FW Inc Inc+FW , , , , , , , ,4 In second exercise, ll the quntities of the sme O-D mtrix were multiplied by four, so tht the most importnt flow fter the first AON ssignment represents 70% of the link cpcity. In this test cse, the results obtined by the different ssignment techniques re now different from the results of simple AON. This is illustrted by Figure 4, in which the 100% cost level corresponds to n AON ssignment

15 A four step incrementl ssignment ssocited with Frnk-Wolfs lgorithm gives the fstest results s the equilibrium is obtined twice s fst s for simple FW. The vrition of cost between the solution obtined by n AON ssignment nd the optiml solution is bout 8%. Figure 4: ssignment results on lightly congested smll network 108,50% 108,00% Totl cost (100% = AON) 107,50% 107,00% 106,50% 106,00% 105,50% MSA FW Inc+FW Inc 105,00% Computing time (seconds) Figure 5: ssignment results on congested smll network 124,00% 122,00% Totl cost (100% = AON) 120,00% 118,00% 116,00% MSA FW Inc+FW Inc 114,00% Computing time (seconds) For quntities ten times the one ssigned during the first exercise, we hve, fter n AON ssignment, cpcity overshooting on three links. Here lso, the incrementl

16 Frnk Wolfe ssignment rrives more quickly t the equilibrium solution thn ny other method. The vrition of cost between the solution obtined by n AON ssignment nd the optiml solution is bout 19.5%. 3.2 An ppliction on rel network The Virtul Newtork methodology ws designed to develop multi-modl models, i.e. models in which different trnsporttion modes cn be used or combined between the origins nd destintions of the O-D mtrixes. Moreover, the methodology is intended to be used on lrge geogrphic networks, covering severl regions or even countries. The following exercise, bsed on the network represented in Figure 6, implements the different ssignment techniques discussed erlier on rel cse, in which rod, ril nd wterborne trnsport cn be used. Figure 6: Rel network The implementtion of the cpcity restrined methods is not immedite, becuse the rods re shred by privte crs nd lorries. To give resonble nswer to this problem, the cpcity of the rods ws fixed to theoreticl residul cpcity tht is left over when the privte crs re lredy on the network. The residul cpcity for the trucks ws estimted in the following wy : Averge lod of the lorries (including empty return trips) : 12,3 tons

17 58% of the trffic is done by smll trucks (equivlent to 1,5 privte vehicles) 42% of the trffic is performed by lrge trucks (equivlent to 2 privte vehicles) Thus, n verge truck represents 1,71 privte vehicles nd trnsports 12,3 tons. Knowing tht n verge 80% of the number of vehicles observed on the network re light vehicles, the residul cpcity for trucks, expressed in tons, cn be estimted s 0,2 x (Cpcity of the rod section/ 1,17) x 12,3. The origin destintion mtrixes tht were used nd the originl model re thoroughly discussed in Geerts nd Jourquin (2001). These mtrixes contin nnul mounts of goods tht moves between the origins nd destintions, nd only contins inter-urbn nd inter-regionl dt. This is quite different to urbn O-D mtrixes for pek hours for different resons, but minly becuse no informtion is vilble on the deprture nd/or rrivl time. Moreover, s distnces re long, single trip often occurs during different periods of the dy, including pek hours. No cpcity restrictions were introduced for ril or wterborne trnsport, i.e., their respective cost functions re not dependent on the lredy ssigned flows. In this exercise, we simulted trffic during n verge pek period. To do tht, the sturtion of the rod network ws estimted on the bsis of the well-known criterion of the 30th pek hour. (Trffic observed t the 30 th rnk, ccording to the clssifiction of the observed pek hours during one yer, ordered by decresing vlue). This method consists in compring the flow of the 30th pek hour with the theoreticl cpcity of the considered rod section. This method, simple in its design, hs the following prticulr dvntge: the flow of the 30 th pek hour represents qusi constnt shre whtever the plce or time of the verge dily flow. For the Belgin network, this vlue is proven to be 13,6% of the dily flow. Knowing tht smll truck is in opertion 241 dys/yer nd lrge trucks work 264 dys/yer, we cn estimte the verge worklod: (58 x x 264)/100 = 250,66 dys. The quntity to be ssigned on the network during n verge pek hour cn thus be estimted to (Annul quntity/250,66) x 0,136. Now tht both the residul cpcity on the network is expressed in tons nd tht we hve O-D mtrixes for n verge pek hour, we cn test the different ssignment methods on the rel network. The results re shown in Figure 7. As could be expected, the

18 equilibrium solution is only slightly more expensive (+1%) tht the AON initil solution. Tble 6: impct on the estimted modl split Modl split Method Rod Ril Wter MSA -2,56% + 0,80% + 5,75% FW -3,23% + 1,15% + 7,09% Inc+FW -3,34% + 1,16% + 7,16% Inc (n=4) -1,62% + 0,35% + 3,64% The ssignment methods lso hve n impct on the mrket-shre of the different trnsporttion modes (see Tble 6). Indeed, cheper pth on virtul network cn very well be found using nother trnsporttion mode. Figure 7: ssignment results on rel cse 103,50% 103,00% Totl cost (100% = AON) 102,50% 102,00% 101,50% 101,00% MSA FW INC+FW INC 100,50% 100,00% Computing time (seconds) The vrious methods give different modl splits thn the one obtined by n AON. Ech time, there is reduction of the tons.km trnsported by rod due to the effects of congestion. For the incrementl method, one notes significnt difference of the modl splits compred to the other methods. Indeed, once tht flow is ffected on link, it is not ny more possible to withdrw prt of it nd to ssign it on nother link

19 The model ws further more stressed, doubling the ssigned quntities, to see how the different lgorithms rect on significntly more loded network. The results of this exercise re presented in Figure 8 nd Tble 7. This time, the equilibrium solution is bout 4,5% more expensive thn the AON totl cost. Tble 7: new modl split Modl split Method Rod Ril Wter MSA - 4,21% + 3,77% + 8,60% FW - 4,89% + 4,27% + 9,79% INC+FW - 4,79% + 4,01% + 9,19% INC (n=4) - 2,30% + 1,38% + 4,70% Figure 8: ssignment results on stressed rel cse 140,00% 135,00% Totl cost (100% = AON) 130,00% 125,00% 120,00% 115,00% 110,00% MSA FW Inc+FW Inc 105,00% 100,00% Computing time (seconds) 4 Conclusions Severl ssignment techniques nd lgorithms were implemented on virtul networks to model multi-modl freight trnsport networks. The virtul network pproch is different from the clssicl four steps models becuse it combines modl choice nd ssignment in single step. The essentil drwbck of this methodology is tht it genertes huge networks, mking computing times much more longer thn those observed on clssicl rel networks

20 Modern computers now give the possibility to compute such lrge models, becuse both the power of their processor(s) nd the mount of memory tht is vilble re not ny more to be compred with the hrdwre tht ws vilble some yers go. All the equilibrium ssignment methods tht were implemented nd tested in this pper re bsed on the All-Or-Nothing lgorithm. Cpcity constrints re tken into ccount by the incrementl method, the Method of Successive Averges (MSA) nd the Frnk- Wolfe lgorithm in order to pproch the equilibrium condition stted formlly by Wrdrop. A vrition bsed on the Frnk-Wolfe lgorithm, but using the result of n incrementl solution s strting point ws lso tested. One of the most interesting results obtined in this reserch is impct of the equilibrium models on the modl split. Indeed, s virtul network cn be seen s mono-modl representtion of multi-modl network, n lterntive pth cn very well involve the use of nother (combintion of) trnsporttion mode. However, the incrementl loding technique will not converge to the correct solution if one of the initil itertions ssigns too much flow on given link. This method cn thus very well led to incorrect modl split solution nd is therefore not recommended. It is lso interesting to note tht, on rel cse, the equilibrium solution is very close to one obtined fter single All-Or-Nothing ssignment. In the cse commented in this pper, the totl cost on the system is just 1% higher thn the initil AON cost. This is linked to the nture of the problem tht is solved : one hs to ssign nnul mounts of goods on long distnces. Moreover, the trns-europen network tht is used doesn t contin the detils of the network t the very locl level, mking it difficult to find close lterntive routes. But isn t it relistic option, s truck driver coming from Pris nd going to Antwerp doesn t know the detils of the locl networks round ech city? 5 Bibliogrphy 1. P.T. Hrker. Predicting intercity freight flows. VNU Science press, T.G. Crinic, M. Florin, J. Guélt, nd H. Spiess. Strtegic plnning of freight trnsporttion: Stn, n interctive grphic system. Trnsporttion reserch record, 1283, B. Jourquin. Un outil d'nlyse économique des trnsports de mrchndises sur des réseux multi-modux et multi-produits: Le réseu virtuel, concepts, métho

21 des et pplictions. PhD thesis, Fcultés Universitires Ctholiques de Mons, B. Jourquin nd M. Beuthe. Trnsporttion policy nlysis with geogrphic informtion system: the virtul network of freight trnsporttion in Europe. Trnsporttion Reserch C, 4(6): , D.B. Johnson. A note on Dijkstr's shortest pth lgorithm. Journl A.C.M., 20: , E.W. Dijkstr. A note on two problems in connection with grphs. Numerische Mthemtik, 1: , D.T. Kresge nd P.O. Roberts. Techniques of Trnsporttion Plnning: Systems Anlysisnd Simultion Models. Brooking Institution, Wshington DC, A.G. Wilson nd R.J. Bennet. Mthemticl Methods in Humn Geogrphy nd Plnning. John Wiley & Sons, N-Y, Wrdrop,J.G. Some Theoreticl Aspects of Rod Trffic Reserch Proc. Inst. Civ. Eng.,Prt II, 1, p J. de D. Ortúzr nd L.G. Willumsen. Modelling Trnsport. John Wiley & Sons, N-Y, p M. Frnk nd Wolfe. An lgorithm for qudrtic progrmming, Nvl Reserch Logistics Qurterly, 3 (1956), p Le Blnc, L.J., Morlock, E.K. nd Pierskll, W.P. An Efficient Approch to Solving the Rod Network Equilibrium Assignment Problem Trnspn. Res., 9, p Smock R.J. An itertive ssignment pproch to cpcity restrint on rteril networks. Highwy Reserch Bord Bulletin 156, 1962, Overgrd K.R. Urbn trnsporttion plnning : trffic estimtion. Trffic Qurterly, XXVI(2), Bureu of Public Rods Trffic Assignment Mnul. Urbn Plnning Division, US Deprtment of Commerce, Wshington DC, Roy Thoms. Trffic Assignment Techniques. Avebury Technicl, p Geerts J.F. nd Jourquin B., Freight Trnsporttion Plnning on the Europen Multimodl Network: the cse of the Wlloon Region, Europen Journl for Trnsport Infrstructure Reserch, Vol 1, No. 1,