Optimal Redesign of the Dutch Road Network

Transcription

1 Optmal Redesgn of the Dutch Road Network M. Snelder a,b,, A.P.M. Wagelmans c, J.M. Schrjver a, H.J. van Zuylen b, L.H. Immers a,d a TNO, Van Mourk Broekmanweg 6, P.O. Box 49, 2600 AA Delft, The Netherlands b Delft Unversty of Technology, Faculty of Cvl Engneerng and Geoscences, Department of Transport and Plannng P.O. Box 5048, 2600 GA Delft, The Netherlands c Econometrc Insttute, Erasmus Unversty Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands d Catholc Unversty of Leuven, Department of Transportaton Plannng and Hghway Engneerng, Kasteelpark Arenberg, 40, B-3001 Heverlee, Belgum Econometrc Insttute Report EI Abstract The Dutch natonal road network has been developed over several decades. In the past, roads were constructed accordng to the then current spatal and transportaton plannng phlosophes. Because the exstng road network s a result of a long process of successve developments, the queston can be asked whether ths network s the most approprate from the current pont of vew, especally takng n consderaton the current soco-economc structure of the Netherlands. To answer ths queston an optmzaton algorthm for desgnng road networks has been developed. Wth ths algorthm the Dutch road network has been redesgned based on mnmzaton of the travel and nfrastructure costs and by takng nto account the soco-economc structure of the Netherlands. A comparson between the exstng network and the new desgn shows that the redesgned Dutch natonal road network has sgnfcantly lower total costs than the exstng road network. It s found that the constructon of less roads wth more lanes on dfferent locatons leads to a reducton of the total travel tme and the total vehcles klometers traveled. Keywords: Network Desgn Problem; road network; Netherlands; redesgn; scratch; motorway 1 Introducton The Dutch natonal road network has been developed over several decades. In the past, roads were constructed accordng to the then current spatal and transportaton plannng phlosophes. Because the exstng road network s a result of a long process of successve developments, the queston can be asked whether ths network s the most approprate from the current pont of vew, especally takng nto account the current soco-economc structure of The Netherlands. To answer ths queston we redesgned the road network from the begnnng, usng the present soco-economc and spatal structure of the Netherlands as the startng pont. The redesgn shows the deal road network gven certan economc objectves. Fgure 1 llustrates the sgnfcance of congeston on the Dutch natonal road network. Ths congeston leads both to annoyance and to drect costs, quantfed n the order of 0.6 bllon euros n 2002, and estmated as hgh as 1.7 bllon euros n 2020 (AVV, 2004). The occurrence of a certan degree of congeston s optmal from an economc pont of vew, but most lkely, wth the same total lane length, the amount of congeston could be smaller f the structure of the network were dfferent. Correspondng author: Address: TNO, Van Mourk Broekmanweg 6, P.O. Box 49, 2600 AA Delft, The Netherlands. Tel.: ; Fax: E-mal addresses: maake.snelder@tno.nl, wagelmans@few.eur.nl, jeroen.schrjver@tno.nl, h.j.vanzuylen@ctg.tudelft.nl, ben.mmers@tno.nl 1

2 IC-rato peak >1.2 Fgure 1: Intensty/Capacty rato (IC-rato) n the peak perod on the motorways 2001 There s nteracton between land use and nfrastructure. On the one hand, the transport nfrastructure s partly determned by the spatal structure, and, on the other hand, the (economc) spatal structure s partly determned by the transport nfrastructure. The background between such nteractons s that both persons and companes usually have a preference for settlng at well accessble places, and that extra nfrastructure s usually bult at places were congeston s worst. Because the current soco-economc fgures are taken as a startng pont, t s lkely that the deal road network resembles the current road network n many ways. However, there wll also be dfferences. A comparson between ths deal network and the current road network gves an dea whch locatons n the current network should be mproved n order to better deal wth the demand for traffc. Besdes, t could also show whch network characterstcs are optmal. For example, t could be used to answer questons lke whether t s better to enlarge the capacty of exstng roads or to construct entrely new roads. Or n other words: Is t better to buld two parallel roads wth both two lanes or s t better to buld one road wth four lanes? It s very complex to desgn a road network startng from the begnnng. Therefore, some smplfcatons are requred. The man choce was to base the desgn method on the user optmum, whch means that the resultng optmal redesgn of the Dutch road network s only optmal from the pont of vew of the costs for the road users. Only the costs of drvng are taken nto account. These costs consst of travel tme costs (computed wth an average value of tme for passenger and 2

3 freght transport), varable car/truck costs (for nstance, fuel, tres, ol and mantenance), and nfrastructure costs. External and envronmental costs are only ndrectly ncluded. Land use s ndrectly ncluded n ths study because the locatons of both lvng areas and ndustral and commercal areas determne travel demand. On the other hand, no extra costs are ncluded for crossng rvers, lakes, and areas of ecologcal mportance. Furthermore, the road network s only desgned at a natonal and regonal level. Local roads are not taken nto account, although, obvously, n realty roads at all dfferent levels should be complementary to each other. Fnally, relablty of travel tmes s becomng an ncreasngly mportant ssue. If the road network s desgned n a robust way, ths mproves the relablty of travel tmes. However, the robustness of the network s not ncluded n the appled optmzaton crtera. Stll, we dd compute the effects of an ncrease of the demand on the qualty ndcators (the ndcators are descrbed n secton 4.1). Ths gves an mpresson of the robustness of the network wth respect to varatons n demand. The Network Desgn Problem (NDP) and methods to solve ths problem are presented n secton 2. Secton 3 descrbes the algorthm and ts nput, as used to make a redesgn of the Dutch road network. In Secton 4 the results are presented together wth a comparson between the exstng road network of the Netherlands (2001) and ts redesgn. Fnally, secton 5 contans the conclusons. 2 Network Desgn Problem The problem of redesgnng complete networks can be seen as a specal case of the Network Desgn Problem (NDP). The NDP s a well known problem n the lterature. It has been recognzed as one of the most dffcult and challengng problems n transport (Yang and Bell, 1998). The problem nvolves the optmal decsons on the expanson of a road network. Usually there s a certan budget restrcton. The NDP can be formulated n a dscrete and contnuous way. The dscrete verson of the problem s a NP-complete problem (Johnson et al., 1978), whch means that t s unlkely that the problem can be solved wthn polynomal computaton tme. In dfferent artcles and books overvews are presented of methods that solve the NDP: (Magnant and Wong, 1984), (Yang and Bell, 1998), (Steenbrnk, 1974b). These overvews clearly show that there are a many possble methods. Although the methods dffer n many ways they also have a number of smlartes. For example, the NDP s usually formulated as a b-level problem. The top level addresses the queston where new lnks should be constructed or where the capacty of exstng roads should be extended gven the transport flows. The objectve functon on ths level consders the total costs of road nvestments and the travel costs. The lower level problem s the assgnment problem. In the assgnment problem the traffc s assgned to the network, where the objectve functon consders the ndvdual travel tme. Ths b-level problem can be seen as a Stackelberg game, n whch the network desgner (on the top level) s the leader and the travelers (at the lower level) are the followers. The b-level problem can be solved n dfferent ways. The exstng approaches for solvng the upper level problem can be dvded nto three groups. The frst group s the group of the dscrete and exact approaches. In LeBlanc (1975) a Branch and Bound algorthm s used and n Poorzahedy and Turnqust (1982) a Branch and Backtrack algorthm s used for solvng an approxmaton of the NDP. The second group s the group of the dscrete and heurstc approaches. In Xong and Schneder (1992) a cumulatve genetc algorthm s used. The thrd group contans the contnuous and heurstc approaches. These approaches are used most frequently. For example, n Pearman (1979) a very smple approxmaton method s used. Steenbrnk (1974a and 1974b) uses a decomposton method. In Fresz et al. (1992) smulated annealng s used and n Abdulaal and LeBlanc (1979) the methods of Hooke and Jeeves and Powell are used. The method of Fbonacc, Golden secton and Bolzano search are appled n Suwansrkul et al. (1987). They show that ths algorthm s faster than the algorthm of Hooke and Jeeves n case of convex nvestment functons. The lower level problem can be solved by dfferent types of assgnments. Choces have to be made between statc and dynamc assgnments, between determnstc or stochastc assgnments, between 3

4 sngle user or mult user class assgnments and a decson has to be made as to whether or not an equlbrum assgnment s used. In Ortúzar and Wllumsen (2001) the dfferent assgnment methods are descrbed n detal. In prevous research all knds of assgnment methods are used for solvng the lower level problem of the NDP. For example, n Fresz et al. (1992) and LeBlanc (1975) the Frank- Wolfe algorthm s used to carry out a determnstc user equlbrum assgnment. In Xong and Schneder (1992) a neural network approach s used to carry out ths same assgnment. The stochastc user equlbrum assgnment s used n Lo and Tung (2001). In Davs (1994) and Chen and Alfa (1991) ths knd of assgnment s used n combnaton wth respectvely the contnuous and dscrete verson of the NDP. 3 Complete redesgn of road networks In ths study a method was developed and mplemented for the complete redesgn of road networks, that s, takng the magnary stuaton as a startng pont where land use exsts, however, wthout any road network present. Ths method was tested by desgnng a road network for The Netherlands whch mnmzes the costs of the drvers gven the current-soco economc stuaton. For that case, n ths chapter an adjusted formulaton of the NDP s presented. An teratve optmzaton-assgnment (IOA) algorthm was used that teratvely solves the upper and lower level problems. It was chosen to use a contnuous formulaton of the NDP. In the contnuous NDP formulaton capactes can take all values larger than or equal to 0 pcu/hour (passenger car unts per hour). In contrast, n practce the capacty can only take a lmted number of values, dependent on the number of lanes, road type and the maxmum speed. For ths, a dscrete formulaton of the NDP appears the more approprate. However, the dscrete decson verson of the problem s a NP-complete problem (Johnson et al., 1978). Ths mples that the optmzaton verson of the dscrete NDP s NP-hard. Thus, a contnuous formulaton of the NDP (also called mxed contnuous formulaton (Yang and Bell, 1998)) s requred, certanly for the desgn of large networks. The method of Golden secton search (Press et al., 1999) was used to compute the optmal lnk capactes (upper level) gven the flows on each lnk. An all or nothng (AON) assgnment was used n the lower level problem. In ths assgnment method all travelers choose the route wth the lowest generalzed travel cost, and congeston s not taken nto account. It s clear that ths type of assgnment does not descrbe the route choce of travelers n realty too well. However, for our network desgn problem ths assgnment algorthm s the most sutable: t s very fast and, more mportantly, assgns the travelers to the routes that are preferred n the non-congested stuaton and avods that cut-through routes are unnecessarly created. The am s to redesgn the Dutch road network n such a way that the preferences of travelers are taken nto account n a satsfactory way. Ths mples that the network s to be adjusted to the preferences of travelers. Because road users prefer travelng va the shortest/fastest/cheapest route, an all or nothng assgnment s the approprate assgnment method n the desgn phase (f the network s desgned from the begnnng). It could be argued that ths assgnment results n a stuaton wth congeston and thus, that some travelers could be better off by usng other routes. However, snce n the next teraton the capactes are adjusted to the flows, the congeston s reduced to an economcally optmal level. In the evaluaton phase (after the desgn phase has been completed) a determnstc user equlbrum assgnment s used to assess the qualty of the network. An accurate assessment of congeston effects requres a dynamc traffc assgnment wth dynamc demand profles. Ths study, however, dd not have the ambton to go nto so much detal. The objectve was to determne the optmal road structure wthn a well defned, but stll lmted framework. In ths secton the desgn method s descrbed n detal. Frst, the appled notaton s presented, thereafter the nput s dscussed, and fnally a mathematcal formulaton of the upper and lower level problem s presented. 3.1 Notaton The followng notaton wll be used n the remanng sectons: 4

5 I = {1..n}: set of lnks K = {1..nk}: set of nodes k S(m) - = {1..n(m) - }: set of lnks endng n node m (m K Z) S(m) + = {1..n(m) + }: set of lnks startng n node m (m K Z) P = {1..np}: set of perods p S = {1..ns}: set of ntervals s on the speed flow curve SF = {1..nsf}: set of ponts sf on the speed flow curve Z = {1..nz}: set of zones z n = number of lnks nk = number of nodes n(m) - = number of lnks endng n node m (m K Z) n(m) + = number of lnks startng n node m (m K Z) np = number of perods ns = number of ntervals on the speed flow curve nsf = number of ponts on the speed flow curve nz = number of zones Y : capacty of lnk (pcu/hour) X : flow on lnk n perod p (pcu/hour) δ = 1 f flow from orgn o to destnaton d uses lnk n perod p od 0 otherwse θ = 1 f lnk s constructed 0 otherwse 3.2 Network A road network conssts of access and egress nodes, lnks and junctons. Access and egress nodes are places where travelers or freght can enter or leave the network. Junctons are locatons where two or more lnks come together. Lnks are connectons between two access or egress nodes, between two junctons or between access or egress nodes and junctons. The desgn of networks rases the followng questons: 1 How many access or egress nodes should the network contan and where should these nodes be located? 2 how many junctons should the network contan and where should these junctons be located? 3 what must be the road type of the lnks? 4 whch capacty s requred for each of the lnks whch connect the nodes and access or egress nodes? There are many possble answers to these questons. We have chosen to use a grd as a startng pont. A part of ths grd s shown n fgure 2. It contans almost 30,000 lnks and about 5,000 nodes. Fgure 2: Grd 5

6 The grd s bult up from equlateral trangles wth a horzontal orentaton and wth sdes of 3 klometers. Of course, the grd can be bult up wth other shapes lke squares and the length of the lnks can also vary. However, trangles seem to be the best opton snce the number of lnks that come together n one node (6) s relatvely large compared to other shapes. Furthermore, Bolt (1982) shows that the trangle structure has the hghest captal costs (buldng costs) and the lowest varable costs (travel dstances and travel tme) compared to other shapes. For rch countres wth a hghly developed nfrastructure network, lke the Netherlands, ths structure s probably the most sutable. Fgure 3 shows the captal costs and varable cost for dfferent network structures on a relatve scale Low captal cost Hgh captal cost Low varable cost Hgh varable cost Fgure 3: Captal cost and use cost for dfferent network structures (Bolt, 1982) The optmal dstance for nodes of a network on a regonal level s crca three klometers (van Nes, 2002). Because ths research focuses on the natonal level the dstance of 3 klometers s an approprate lower bound for the dstance between nodes. All the lnks n the grd can be used. Intally, all lnks are motorways wth speeds accordng to the speed-flow curve of fgure 5. The curve has been specfcally ft to measured speeds on the Dutch motorways. The defntve choce of the nfrastructure types for the lnks (regonal road, trunk road or motorway wth 2, 3, 4 or 5 lanes) depends on the fnal capacty of the lnks. In table 1 all the lnk types wth ther capactes are shown. The capactes are optmzed durng the desgnng process. If the capacty s larger than 0 passenger car unts (pcu) per hour the lnk s ncluded n the network. Table 1: Capacty of roads per lnk type Lnk Type Number of lanes Capacty (y) (pcu/hour) Regonal road 1 0 < Y 1000 Trunk road < Y 1575 Motorway < Y 4650 Motorway < Y 7250 Motorway < Y 9700 Motorway 5 or more Y > 9700 The SMART 99 zonng system (consstng of 500 zones) was used n ths study. SMART (Schrjver, 2004) s a four stage transport model made and owned by TNO. It was appled n ths research to create an OD matrx. Its zonng system as well as ts assgnment model (Smartass) were used. In fgure 4 ths zonng system s shown. The fgure shows the number of nhabtants (12 years and older) per square klometer. All the zones are connected to the nearest node n the grd by feeders. The locatons where the feeders meet the network are the access or egress nodes. 6

7 Inhabtants (12+) per km or more Fgure 4: SMART 99 zonng system wth populaton densty 3.3 Demand The optmal network depends on the demand for transport and the demand depends, n ts turn, on the qualty of the network. Ths mples that the optmal network structure and the demand should deally be computed teratvely. However, the choce was made to use two fxed OD-matrces: OD and OD-deal. OD-2001 contans the demand for transport n the exstng road network of the Netherlands of 2001 and OD-deal represents the demand for transport n the deal stuaton. Both matrces were constructed wth SMART. OD-2001 s a result of a full model run for the year OD-deal was constructed wth almost the same nput as s used for the constructon of OD-2001 (same cost data, and soco-economc fgures). The only dfference s that the trangular grd network (wth very large lnk capactes on all lnks) was used nstead of the network of The shape of the grd network and the large lnk capactes ensure that the demand n OD-deal s not lmted by shortcomngs of the network. OD-deal contans all the trs that people would lke to make and, therefore, t can be argued that the deal network should adjust to ths demand and not the other way around. Ths approach compensates for the possble shortcomngs of the use of a fxed OD matrx nstead of elastc demand. However, nduced demand, that s, demand created by changes n the soco-economc and spatal structure by another road nfrastructure, was left out n ths study. OD-2001 and OD-deal contan the same demand matrx for freght transport. Ths s n fact the exstng demand for In the deal case the freght volumes transported by road are probably hgher. A generaton model for freght transport s, however, not yet ncluded n SMART and the 7

8 ncreased freght transport was, consequently, not taken nto account. Furthermore, OD-deal was constructed wth the exstng network for publc transport of Because the qualty of the road network used to construct the matrx was very hgh, road transport had a more compettve poston compared to publc transport. The demand for publc transport (and other modes) was, therefore, underestmated, and the demand for transport by car overestmated. 3.4 Upper level problem: determnng the optmal capacty of the lnks The objectve of the man problem s shown n equaton 1. In the upper level problem the optmal capactes are determned by mnmzng the total lnk costs (tk) of the lnks gven the flows of the prevous assgnment. Y, θ I MIN tk ( X, Y, θ ) (1) The costs of a lnk (tk ) are computed as the product of the lnk length (d ) wth the total costs per klometer (travel tme costs, varable vehcle costs, and nfrastructure costs). The total varable vehcle costs and the total travel tme cost are computed by multlyng the cost of travelng over the lnk for one user wth the total number of users per week. In the formula below p s the ndex for the dfferent perods (off-peak, mornng peak and evenng peak), wf p s a parameter for the weght of a perod n a week, ca are the varable vehcle costs for a user, and g s a functon for the nfrastructure costs. tk( X, Y, θ) = d { wf px ( t( X, Y) + ca) } + g( Y, θ) (2) p The travel tme costs for one user (t ) for drvng over a road of 1 km s computed by dvdng the value of tme of that user (ct) by the realzed speed on the lnk. The travel tme costs vary per perod and per lnk. s ( ) ( X, Y ) t X Y = ct v X Y (3) (, ) / (, ) The speed v s (X,Y ) on lnk s a functon of the capacty Y and the flow X. Fgure 5 shows the speed flow curve that s used (SMART). Because all roads were assumed motorways durng the desgn process only one curve was used. From ths fgure t can be seen that the curve conssts of several parts. Wthn these parts the curves are lnear. The total functon s a contnuous nondfferentable functon. If the part n whch an IC-rato les s known, the accompanyng speed can be computed wth equaton 4. The part n whch the IC-rato les can be determned wth equaton 8. The parameters a, b and c can be computed wth equaton 5, equaton 6 and equaton 7. In these formulas sp sf s the speed on pont sf of the curve and c sf s the IC-rato on pont sf of the curve. 8

9 Fgure 5: Speed flow curve s ( X, Y ) (, ) = 12 f s X Y v X Y 10 (, ) = 0 2 mn ( a + b ( X / Y (, ) (, ) (, )),10 ) (, ) 1.. s X Y s X Y c f s s X Y ns X Y The values of and 10-2 were chosen n such a way that the speeds are also well defned for hgh and negatve IC-ratos. a = sp s(x,y ) s(x,y ) ( ) / + 1 ( + 1 ) b = sp sp c c s (X,Y ) s (X,Y ) s (X,Y ) s (X,Y ) s (X,Y ) c = c s(x,y ) s(x,y ) 0 f / 0 X Y < s (X,Y ) = sf f c(sf) ( X / Y ) < c(sf + 1) sf SF (4) (5) (6) (7) (8) The costs of nfrastructure g (Y, θ ) are also part of the total lnk costs (tk ). The costs of nfrastructure consst of fxed and varable costs (equaton 9). These costs only have to be pad f a lnk s constructed. That s, f θ s equal to 1. Cc are the constant nfrastructure costs and cv are the varable nfrastructure costs. Intally, these costs were set to euro/km per week and 1.31 euro/km per pcu per week (Meeuwssen, 2003). In a senstvty analyss, the nfluence of these values s consdered. g ( Y,θ ) = θ cc + cv Y (9) Constrants The flow on a lnk s the optmal flow of the prevous assgnment (equaton 10). X = flow assgnment (10) 9

10 A lnk s ether constructed (Ө =1) or not constructed (Ө =0) (equaton 11). { 0, } I θ 1 (11) The capacty of lnk can only be larger than 0 pcu/hour f the lnk s constructed. MaxY was gven a very large value. Y θ maxy I (12) 3.5 Lower level problem: assgnment The objectve of the assgnment problem s shown n equaton 13. The optmal lnk flows (X) are determned by mnmzng the costs per user (equaton 14) gven the capactes of the lnks. Ths s done wth an AON-assgnment. The formulaton of the objectve functon and the restrctons presented n ths secton are only correct for ths type of assgnment. X MIN lc ( X, Y ) X I p P (13) The costs for one user for travelng over a lnk lc( X, Y, θ ) are specfed n equaton 14. The route choce of travelers depends on the costs of each lnk. These costs consst of travel tme costs (equaton 3), varable vehcle costs (ca = 0.11 euro/km), nfrastructure costs, extra costs (equaton 15) and a penalty f a lnk exsts n only one drecton (equaton 16). lc ( X, Y ) = d ( t ( X, Y ) + ca + nfra ( X, Y ) + extra ( Y ) + penalty ( Y )) (14) In the current stuaton (also n the stuaton of 2001) the users of the roads have to pay a fxed car ownersh tax every year and there s an excse on fuel consumpton. These taxes are not drectly related to nfrastructure costs. Nevertheless, t was decded that n the redesgn process the user has to pay for the nfrastructure costs. In ths way roads are only constructed f they are used by a suffcent number of travelers. Further, a large porton of the capacty s only used by a small group of users. Ths was taken nto account by passng the costs of that extra capacty only onto those travelers who actually use the capacty. That s, travelers n the off-peak perod do not have to pay for the capacty that s only used n the peak perod. Ths concept s presented n (Meeuwssen, 2003). A mathematcal formulaton for the computaton of the nfrastructure costs for one user, that s, nfra ( X, Y ), as well as a more detaled descrton can be found n hs paper. As stated above some extra costs were ncluded n the users cost functon. These costs decrease f the lnk capacty ncreases. The functon for the extra costs s shown n equaton 15. The extra costs were ncluded to make sure that some concentraton of lnks takes places. If these costs were not ncluded many small roads would be constructed nstead of a smaller number of more sgnfcant lnks. Concentraton of traffc on lnks leads to a loss of drectness n the connecton of zones, but the advantages of concentraton easly compensate for ths. Concentraton results n economes of scale. Wth the same total capacty concentraton leads to better usage of the nfrastructure, more effcent technologes can be used, and t results n less envronmental damage (Bovy and Van Nes, 2002). The user does not really have to pay for these extra costs and, therefore, they were not ncluded n the optmzaton of the lnk capactes. The parameters w1 and w2 were estmated based on the lower speeds, the hgher rsks of accdents and the hgher varable car costs on roads of a lower level than a 10

11 motorway. Ths results n values of and -1.3 for w1 and w2 respectvely. These values are further consdered n a senstvty analyss. extra ( Y ) = w1y (15) w2 The OD-matrces do not have to be symmetrcal, because t s, for example, possble to make a tr n one drecton n the off-peak perod and to make a return tr n the peak perod. Ths can result n lnks that are only constructed n one drecton. Ths s not desrable and, therefore, a penalty s ncluded f a lnk exsts n just one drecton. The penalty s computed wth equaton 16. penalty( Y) = 6 10 / Y f Y 0 and the lnk n opposte drecton does not exst. 0 otherwse (16) Constrants The capactes of the lnks are the optmal capactes computed n the optmzaton of the upper level problem (equaton 17). Ths means that Y s fxed n the lower level problem. Y = optmal capactes (17) All the flows that enter a node also have to leave that node (equaton 18). δ trs = δ od od od S ( k ) + S ( k) trs od k K, o Z, d Z, p P (18) The total flow that goes to centrod z and comes from orgn o must be equal to the number of trs from orgn o to centrod z (equaton 19). oz δ S ( z) trs oz = trs oz z Z, o Z /( z), p P (19) The total flow that comes from centrod z and goes to destnaton d must be equal to the number of trs from centrod z to destnaton d (equaton 20). zd δ + S ( z) trs zd = trs zd z Z, o Z /( z), p P (20) The total flow on lnk n perod p must be equal to the sum of the trs on OD-relatons that use that lnk n perod p. X = o Z d Z od δ trsod I, p P (21) All trs from orgn o to destnaton d ether use (1) or do not use (0) lnk n perod p. od { 0,1} I, p P, o Z d Z δ, (22) 11

12 4 Results Ths secton presents the optmal redesgn of the Dutch road network. Because an teratve approach was used n combnaton wth a numercal method (Golden secton) to solve the man problem, the resultng network most lkely represents not a global but a local optmum. The algorthm converges to ths optmum n about 20 teratons. The queston can be asked how close the resultng local optmum s to the global optmum. Usually ths queston s answered based on lower bounds. Ths approach s not followed n ths paper because t appeared very dffcult to fnd a lower bound of good qualty. Another approach to get an mpresson of the qualty of the desgn method s to compare the resultng network wth networks desgned wth other methods. Besdes the qualty of the desgn method, the qualty of the resultng networks s also mportant. In secton 4.1 we descrbe the qualty ndcators that were used. In secton 4.2 and 4.3 the current network and the optmal redesgn of ths network are descrbed together wth ther qualty ndcators. Fnally secton 4.4 gves a comparson between both. 4.1 Network qualty ndcators Besdes based on the total costs, the qualty of a network can be measured wth the followng ndcators: a measure for detours, n tme and dstance (Q) Q ( trs dcf )/ ( trs t ) = (23) p odp od odp odp o Z d Z o Z d Z In ths formula t odp s the realzed travel tme from orgn o to destnaton d n perod p and dcf s the drect dstance (that s, n a straght lne). Q can take a maxmum value of 110 km/h (based on the speed flow curve of fgure 5). Lower values (speeds) are caused by detours made n dstance and delays (congeston). Q must be computed separately for every tme perod p. vehcle klometers traveled Freght transport s ncluded n the vehcle klometers traveled, but the vehcle klometers traveled by ntra zonal traffc are not ncluded n ths ndcator. One vehcle klometer traveled by a truck counts for two. Ths means that the vehcle klometers traveled are actually pcu klometers traveled. total travel tme speeds frequency table The speed can be seen as an ndcator for congeston. The lower the speed, the heaver the congeston. The level of the speeds and the number of lnks on whch these speed levels occur can be read from a frequency table. The frequences are only reported for speeds lower than 20, 50, 70, 80 and 100 km/h. total capacty and the total road length number of lnks. Because all lnks have a length of 3 km (wth the excepton of the Afslutdjk and the Markerwaarddjk (fgure 6)), ths ndcator s almost equal to the total road length. For clarty reasons both are computed. Many of these ndcators requre that the ntenstes on the lnks are known. These ntenstes are computed wth a determnstc user equlbrum assgnment. 4.2 Exstng road network In ths secton we present the exstng road network of the Netherlands n 2001 (as modeled n SMART) and ts qualty ndcators. Ths s done to obtan an mpresson of the qualty of the desgn method and to enable a comparson between the exstng road network and ts redesgn. The ndcators are computed for both the OD-matrx OD-2001 and OD-deal. 12

13 To ensure a far comparson between the exstng road network and ts redesgn, the ndcators for the exstng road network were computed under the assumpton that all lnks n the exstng network are motorways and thus have the same speed flow curve as used n ths study. Table 2 shows all qualty ndcators for the total road length and capacty, whle the same s shown graphcally n Fgure 6. Ths fgure also contans nformaton about the numbers and names of the Dutch roads. These wll be used n the remander of ths paper. Table 2: Qualty ndcators for road length and capacty for the exstng road network of 2001 Descrton Number of lnks Total capacty pcu/h 29 mllon Total road length secondary roads (km) Total road length man roads (km) Total road length motorways 1 lane (km) 347 Total road length motorways 2 lanes (km) Total road length motorways 3 lanes (km) 642 Total road length motorways 4 lanes (km) 92 Total road length motorways 5 lanes (km) 2 Number of lanes A7 Afslutdjk Heerenveen Emmeloord Markerwaarddjk A7 A32 The Hague A13 A29 A9 A44 A4 Rotterdam A17 A16 Amsterdam A10 A12 A1 A2 Utrecht A15 A59 Den Bosch Breda Tlburg A58 A6 A28 Endhoven A50 Apeldoorn Arnhem Njmegen A73 A67 Maastrcht Fgure 6: Number of lanes of the Dutch road network 13

14 Table 3 contans all qualty ndcators wth respect to the traveled klometers. Ths table shows that all qualty ndcators are worse for OD-deal than for OD The total travel tme ncreases by a factor of 1.8 n the peak perod and 1.5 n the off-peak perod. The total vehcle klometers traveled ncreases by a factor of 1.5 n the peak perod and 1.3 n the off-peak perod and for Q the ncrease s a factor of 1.3 n the peak perod and 1.1 n the off-peak perod. Ths ndcates that the demand for travel n the current stuaton n the Netherlands has adjusted tself to the exstng nfrastructure. Table 4 shows the number of lnks wth speeds lower than 20, 50, 70, 80 and 100 km/h. It appears that for the OD-2001 matrx even n the off-peak perod low speeds are found for many lnks. These low speeds can n most cases be explaned by the fact that many lnks have speed-lmts lower than 110 km/h. Nevertheless, fgure 1 shows hgh IC-ratos and, therewth, congeston on many motorways n the peak perod. In the case of OD-deal the speeds are even lower. Table 3: Qualty ndcators for traveled klometers for the exstng road network of 2001 OD-2001 OD-deal Offpeak hour Mornng peak hour Evenng peak hour Offpeak hour Mornng peak hour Evenng peak hour Q (km/h) Total travel tme (hours x1 000) Total vehcle klometers traveled (mllon km) Table 4: Frequency table speed for the exstng road network of 2001 OD-2001 OD-deal Offpeak Mornng peak Evenng peak Offpeak Mornng peak Evenng peak Speed < 20 km/h Speed < 50 km/h Speed < 70 km/h Speed < 80 km/h Speed < 100 km/h The total costs (travel tme costs, nfrastructure costs and varable vehcle costs) of ths network n the case of OD-2001 and OD-deal are respectvely 798 mllon euro/week and mllon euro/week. These total costs dffer approxmately 37%. 4.3 Redesgn of the road network of the Netherlands To redesgn the Dutch road network, the demand pattern OD-deal was used, together wth the followng values for the parameters: fxed nfrastructure costs (cc): 4076 euro/km per week (deprecaton) (50%, 110%, 200%, 300%) varable nfrastructure costs (cv ): 1.31 euro/(pcu km) per week (50%, 110%, 200%, 300%) varable vehcle costs (ca): 0.11 (euro/(pcu km)) (110%) value of tme (ct): (euro/(pcu hour)) (110%) parameter w1 of extra costs functon for concentraton of lnks: 1000 (euro/(pcu km)) (2000, 3000) parameter w2 of extra costs functon for concentraton of lnks: -1.3 (-1.1, -1.5) These values are estmatons. It s possble that dfferent values hold n realty than appled n ths model approach. Ths s especally true for the parameters w1 and w2. A senstvty analyss was carred out to further study ths. The numbers between brackets show the percentage changes n costs, respectvely the parameter values that were used n the senstvty analyss. Besdes the costs and parameters above we also ncreased the demand for transport wth 10% on all relatons wthout 14

15 redesgnng the network. Ths gves a frst mpresson of the robustness of the network. In the evaluaton of the dfferent desgns, the total costs of each desgn were computed wth the orgnal settngs of the costs. The total costs are the costs as defned n the objectve functon of the upper level problem. The conclusons from the senstvty analyss are presented below. The senstvty analyss shows that changes of the parameters of the extra cost functon for concentraton have the largest effects on the network structure and the total costs. The maxmum dfference n the total cost compared to the network wth the lowest costs s 2.3%. Furthermore, there s an enormous dfference n the number of lnks and, therewth, the geographcal densty of the roads n the desgns wth dfferent parameters for the extra cost functon. For example, a change of w2 from -1.3 to -1.1 leads to a decrease n the number of lnks of 22.9% and a change of w2 from to -1.5 leads to an ncrease of the total number of lnks of 33.0%. The total capacty shows much less varaton (maxmum of 13.7%). From the senstvty analyss t becomes clear that overall the ndcators for the network desgned wth w1 and w2 respectvely equal to 1000 en -1.3 gve the lowest total cost. Consequently, the network presented as the optmal redesgn n ths artcle s the network desgned wth parameters 1000 en Varaton of the fxed and varable nfrastructure costs showed a maxmum devaton of 2.2% n the total costs and a maxmum devaton of 2.3% n the ndcators for the traveled klometers. From ths we conclude that the desgn algorthm s nsenstve to changes n the fxed and varable costs. A smlar concluson can be drawn for the value of tme and the varable vehcle costs. An ncrease n the demand wth 10% (wthout changng the desgn of the network) results n an ncrease of the total travel tme of 10% n the off-peak perod and 13% n the peak perod. The total vehcle klometers traveled ncrease wth 9.8% n the off-peak perod and wth about 10.2% n the peak perods. There are two possble explanatons for the small ncrease n travel tme. The frst explanaton s that the model does not take congeston related effects lke a capacty drop and blockng back effects nto account. These effects could result n larger travel tme ncreases. The robustness of the network (or the redundancy n the network) s the second explanaton. The small ncrease n travel tme mples that the desgned network s capable of dealng wth an ncrease n the demand and, therefore, s qute robust. However, the number of lnks wth speeds lower than 70, 80 and 100 km/h almost doubled and Q decreased wth 0.2% n the off-peak perod and 2.6% n the peak perod. Ths means that a further ncrease n the demand probably causes more mportant performance decreases. Fnally, the senstvty analyses showed that the optmal network found by the appled desgn algorthm had the lowest total cost (computed wth the dentcal cost parameters) of all networks desgned n the senstvty analyss. If the optmum found by the algorthm s a local optmum nstead of a global optmum then t s lkely that a change n one of the costs would result n a network wth lower total costs. Ths does not prove that the algorthm actually found the global optmum, but t does mply that the soluton found s lkely to be a good one. In fgure 7 the optmal redesgn s shown. Ths network s desgned based on optmzaton of the costs mentoned before. The three most remarkable characterstcs of ths network are the large number of lanes of dfferent roads compared to the present stuaton, the many route alternatves and the absence of the Afslutdjk. The qualty ndcators for ths network are shown n the tables 5, 6 and 7. In the followng secton a comparson between the exstng road network of 2001 and ts redesgn s presented. 15

16 Regonal road Trunk road Motorway wth 2 lanes Motorway wth 3 lanes Motorway wth 4 lanes Motorway wth 5 lanes Fgure 7: Redesgned Road network for the Netherlands Table 5: Qualty ndcators for road length and capacty for the redesgned road network Descrton Number of lnks Total capacty pcu/h 62.5 mllon Total road length regonal roads (km) Total road length trunk roads (km) Total road length motorways 2 lanes (km) Total road length motorways 3 lanes (km) Total road length motorways 4 lanes (km) Total road length motorways 5 lanes (km) 973 Table 6: Qualty ndcators for traveled klometers for the redesgned road network OD-deal Off-peak hour Mornng peak hour Evenng peak hour Q (km/h) Total travel tme (hours x1000/week) Total vehcle klometers traveled (mllon km/h)

17 Table 7: Frequency table speed for the redesgned road network OD-deal Off-peak Mornng peak Evenng peak Speed < 20 km/h Speed < 50 km/h Speed < 70 km/h Speed < 80 km/h Speed < 100 km/h Comparson wth the current network In ths secton a comparson s made between the exstng road network of 2001 of the Netherlands and ts redesgn. The comparson s based on the qualty ndcators and on the structure of the networks. The ten most remarkable smlartes and dfferences n structure are presented. A comparson between the qualty of two networks can only be made f dentcal nput s used. We decded to compare the networks desgned wth OD-deal. It seems unfar to use OD-deal for computng the ndcators of the exstng network because n realty a much smaller number of trs was made n However, n 2001 latent demand exsted. A far larger number of trs would have been made by car f the qualty of the exstng network had been better. In contrast to OD-2001, ODdeal does contan these trs and ths matrx s therefore used n the comparson. The total travel costs of the exstng network are mllon euro/week and the travel costs of the redesgned network are 972 mllon euro/week. Ths means that the costs of the exstng network could be mproved by at least 11%. The total road length of the exstng network as modeled n SMART s 27 thousand km wth a total capacty of 29 mllon pcu/hour (table 2). The redesgn has a total road length of 16 thousand km wth a total capacty of 62.5 mllon pcu/hour. Of course, the road length depends heavenly upon the desgn level, so a comparson of the road length only s not very meanngful. However, n combnaton wth the total capacty t ndcates that the roads n the redesgn have much more lanes/capacty than the roads n the exstng network. Ths can also be concluded from the total road length per road type. For example, the exstng road network contans 736 km of roads wth 3 or more lanes and the redesgn contans about km of roads wth 3 or more lanes. Furthermore, n the redesgn the total vehcle klometers traveled and the total travel tme are respectvely 8% and 17% less than the total vehcle klometers traveled and the total travel tme n the exstng network. Overall, the concluson s that the ndcators of the redesgned network are sgnfcantly better than the ndcators of the exstng network. Ths mples that the redesgned network s better suted to the demand for travel by car. The structure of the redesgned network does not dffer that strongly from the exstng road network. Ths was to be expected because of the nterdependency between the nfrastructure and the spatal structure (secton 1). However, a closer look at both networks does show some dfferences. Below, the ten most remarkable smlartes and dfferences (n random order) between the exstng network of 2001 and ts redesgn are presented. 1) Rng roads around ctes are not found n the redesgn. The number of regons (zones) used and the lengths of the lnks n the grd should contan enough detal to make t possble that rng roads result from the desgn process. In the exstng network the ctes of Amsterdam (fgure 8), Rotterdam and Endhoven have complete or partal rng roads. Rng roads are often bult for reasons of lvablty. Local authortes and ctzens have a strong averson of heavy traffc streams through ther ctes. In ths study, the lvablty aspect s not explctly ncluded n the cost functons and, therefore, rng roads do not occur n the redesgn, 2) the Afslutdjk s not ncluded n the redesgn. The Afslutdjk was constructed for hydraulc reasons and not for means of accessblty. Nevertheless, about travelers make use of 17

18 ths road every day. In the redesgn, these travelers have to use the Markerwaarddjk or drve completely around the lake (IJsslemeer) (fgure 8), 3) the A6 (fgure 8), the road from Amsterdam to the north-east provnces, appears both n the exstng network and ts redesgn. The frst dfference s that n the redesgn ths road splts at Emmeloord nto two separate smaller roads, whle n the exstng network ths road leads all the way to Heerenveen (about 45 km further to the North). The second dfference s that the road conssts of 4 to 5 lanes n the redesgn whle t only conssts of 2 lanes n the orgnal network. Ths s, partly, a consequence of the fact that the Afslutdjk s not used n the redesgn, Afslutdjk Heerenveen A7 Markerwaarddjk Emmeloord 9 Amsterdam A10 A5 Fgure 8: The rng road of Amsterdam, the Afslutdjk, the Markerwaarddjk and the A6 4) n the exstng network the two economcally most mportant ctes of the Netherlands (Amsterdam and Rotterdam) are not drectly connected. To travel from Rotterdam to Amsterdam or the other way around, the avalable route s from the A13 from Rotterdam to The Hague and then the A4 from The Hague to Amsterdam (fgure 9). In the 1960 s the government started the constructon of the A3, whch was supposed to connect Amsterdam and Rotterdam drectly. Soon thereafter ths project was stopped because n the end the government decded to preserve the exstng landscape. In the redesgn the road from Rotterdam to Amsterdam does exst, 5) n the exstng network Amsterdam s connected wth The Hague by two motorways (fgure 9): the A4, and to the west the A44. The A44 begns a lttle to the north of The Hague and jons the A4 about 15 to 20 km from Amsterdam. Ths stuaton makes t dffcult to reach coast places (such as, Katwjk and Noordwjk) when they are approached from the southern/eastern orgns. Ths s caused by the fact that the A4 s only lnked to the A44 by a regonal road through Leden and another regonal road n between The Hague and Leden. In the redesgn the man motorway connectng The Hague and Amsterdam s located close to the A4. The dfference s that there are many parallel roads and that these roads are ntertwned at many places. Ths mples that there are many alternatve routes to travel between Amsterdam and The Hague and even Rotterdam. Ths llustrates one of the most mportant dfferences between the exstng road network and ts redesgn: between the bg ctes there s a second coherent road network whch s complementary to the man motorway network, 6) the A13 s n the exstng road network the only motorway that connects Rotterdam and The Hague. Ths motorway s one of the most congested motorways n the Netherlands, especally n the drecton from The Hague to Rotterdam. The constructon of a second motorway (A4) south-west of the A13 has been planned for many years, but t s not clear when (f ever) the constructon wll start. In the redesgned network the A13 can be recognzed wth another motorway runnng parallel (about 5 klometers to the south-west): ths s the mssng part of the A4. It runs from Delft to Schedam\Vlaardngen (smaller A6 18

19 suburbs of Rotterdam on the west sde). Surprsngly, the part of the A4 that exsts n the current stuaton (from The Hague to Delft) s not part of the redesgn (fgure 9), A9 Amsterdam A10 Noordwjk KatwjkA44 Leden A4 A2 A1 The Hague Delft A13 Rotterdam A12 Utrecht A15 A29 A16 Fgure 9: Connecton between Amsterdam, Rotterdam, The Hague and Utrecht 7) the largest ctes of the Netherlands (Amsterdam, The Hague, Rotterdam, Utrecht, Den Bosch, Endhoven, Tlburg, Breda, Njmegen and Arnhem) are connected by motorways wth 4 or 5 lanes per drecton. In the exstng road network ths s not the case: less lanes are avalable (fgure 6). Motorways wth 2x5 lanes hardly exst and motorways wth 2x4 lanes are already qute exceptonal. Even 6 lane motorways (3 per drecton) are not common. In the exstng road network roads of such sgnfcant dmensons are avalable only between Amsterdam, The Hague, Rotterdam and Utrecht, 8) n the exstng road network there are two motorways that connect the northern part of the country (Fresland, Gronngen and Drenthe) wth the western part (A6/A7 and A28). The roads n the redesgn do not match the exstng network exactly, but the structure s more or less the same, 9) n the exstng road network there are fve man west to east connectors. The most northern connecton runs from Amsterdam to Enschede (A1), the mddle northern road (A12) from The Hague to Utrecht and Arnhem, the mddle from the port of Rotterdam to Njmegen (A15), the mddle southern from Breda to Den Bosch (A59), and the most southern runs near the border between the Netherlands and Belgum (A58). In the redesgn the A1, the A12 and the A59 can be recognzed clearly. The A15 can also be dstngushed, but t runs a slghtly more southward. In fact, ths motorway can be seen as a combnaton of the A15 and the A59 and A50 (the A59 and A50 are located parallel at the south of the A15), 10) the north-western part of the country s connected wth the south-east by one long motorway n the exstng network (A2). It starts n Amsterdam and goes all the way to the most southern part of Lmburg. In the redesgn ths road s replaced by two parallel motorways, whch consst of 4 and 5 lanes per drecton from Amsterdam to Utrecht and surroundngs. From there on, one of these motorways jons a motorway n the redesgn road that s very smlar to the A1. The other motorway contnues n the south-eastern drecton and s smlar to the A2. 5 Dscusson, concluson and recommendatons Redesgnng a complete network for an entre country s a complex task. Desgnng networks from the very begnnng can be seen as a specal case of the so-called Network Desgn Problem. Ths problem s complex even for small networks, let alone a network of about 30 thousand lnks, 5 thousand nodes and 500 zones. Furthermore, there are many actors nvolved, such as, travelers, local resdents, and the government, all wth dfferent demands. Land use and the spatal structure are other key elements. On the one hand, they form a restrcton for the locatons avalable for the constructon of nfrastructure, and, on the other hand, there s a strong nteracton between the 19

20 locatons where people lve and enterprses operate, and the nfrastructure. Fnally, many factors should be ncluded n the optmzaton process. There are costs drectly related to travelng by car, but there are also external costs. It s very dffcult to take nto account all the factors mentoned above. Therefore, some smplfcatons are requred. When the optmal redesgn presented n ths paper s consdered, these smplfcatons have to be kept n mnd. Nevertheless, t appears possble to meanngfully compare the exstng road network wth ts redesgn. The redesgn shows that the current road network can be mproved. The redesgn of the Dutch natonal road network has sgnfcantly lower total costs than the exstng road network (11% lower). The total vehcle klometers traveled and the total travel tme are respectvely 8% and 17% lower than n the exstng network. Ths s realzed through the constructon of roads wth more lanes and located somewhat dfferently, and through the constructon of a second coherent road network whch s complementary to the man motorway network between the major ctes. A senstvty analyss showed that especally the parameters of the functon that determnes the concentraton of roads have mportant mpacts on the structure of the desgned network. The total costs showed a varaton of 2.3% n dfferent runs and a varaton of about 30% n the number of lnks constructed. The algorthm appears to be nsenstve to changes n the fxed and varable nfrastructure costs and to changes n the varable vehcle costs and the value of tme. Furthermore, the optmal redesgn of the Dutch road network s robust n the sense that t s capable of dealng wth an ncrease n demand of 10%. Ths ncrease does not sgnfcantly nfluence the travel tme of ndvdual travelers and ther vehcle klometers traveled. On the bass of the nfrastructure network redesgn effort, the followng recommendatons follow: Envronmental ssues, land use and safety play an mportant role n the decsons on new roads or extra lanes, and, therefore, these aspects should be taken nto account more explctly. Ths could for example be done by extendng the objectve functons or by makng the nfrastructure costs dependent upon the locaton, the exstng road network s not used as a factor n the desgn method. Although the redesgn of the complete network gves an mpresson of the man opportuntes for mprovement of the current network, t s not suffcently specfc for decson makers. A possblty s to adjust the desgn algorthm n such a way that t can take the exstng road network nto account and compute the most valuable mprovements of the exstng network, the demand matrx OD-deal s sutable for the desgn of the deal road network when a stuaton wthout exstng nfrastructure s taken as the bass, but t s advsable to make the desgn algorthm capable of dealng wth elastc demand (demand that depends upon the qualty of the network). Ths s necessary to model small changes to exstng networks, the OD-matrces that are used to redesgn the Dutch road network do not contan nternatonal traffc. Off course, the capacty of roads n the Netherlands of nternatonal mportance depends also on nternatonal traffc. Therefore, nternatonal traffc should be ncluded n the ODmatrces, fnally, the relablty of travel tme needs to be taken nto account n future research. The relablty of travel tme s a complcated term because t s nfluenced by many factors and many ndcators can be used to measure relablty. The robustness of the network s one of the factors of nfluence. Future research could show f t s possble to take robustness and/or relablty nto account n the desgn phase. The mplementaton of ths mprovement and the mprovements mentoned above s not easy, because t requres more sophstcated route choce algorthms. 20