Transportation Research Part A 41 (2007) 644–654

www.elsevier.com/locate/tra

Tolling traffic links under stochastic assignment: Modellingthe relationship between the number and price level

of tolled links and optimal traffic flows

Kathryn Stewart *

Transport Research Institute, Napier University, Edinburgh, EH10 5DT, United Kingdom

Received 16 February 2006; received in revised form 19 September 2006; accepted 22 September 2006

Abstract

The classical road-tolling problem is to toll network links such that under the principles of Wardropian User Equilib-rium Assignment a System Optimising (SO) flow pattern is obtained. Stochastic assignment methods are accepted to bemore realistic than deterministic and it is of interest to examine the potential for optimal tolling in the case of StochasticUser Equilibrium (SUE). In examining the case of Stochastic User Equilibrium the ‘desired flow pattern’ to be createdmust first be determined. The classical economics solution of replacing unit-cost flow functions with marginal-cost flowfunctions which under deterministic assignment produces the System Optimal solution (where Total Network Travel Cost(TNTC) is minimised) does not generally result in TNTC being minimised in the Stochastic Case. Instead such tolls pro-duce a ‘Stochastic System Optimal’ (SSO) solution where the Total Perceived Network Travel Cost (TPNTC) is minimised.

This paper examines and compares link-based tolling solutions to achieve both the SSO (TPNTC minimised) and trueSO (TNTC minimised) under SUE and illustrates the concept with numerical examples. Such link-based tolling schemesproduce network benefit by re-routing rather than traffic suppression as opposed to the cordon-based charging schemeswhich have been implemented in practice. Equity issues relating to charging schemes are discussed and the desirabilityof zero-toll routes is highlighted associated with greater potential political acceptability of charging schemes that donot impose excessive charges upon users (such as minimal or low revenue tolls). A heuristic is developed to toll networklinks in such a way as to balance the number of links tolled against the revenue required to produce a desired reduction inTNTC such that optimal network flow patterns are approached.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Congestion Charging; Traffic assignment; Stochastic user equilibrium; Optimal tolls; Marginal social costs

0965-8564/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.tra.2006.09.015

* Tel.: +44 131 455 2618; fax: +44 131 455 2239.E-mail address: k.stewart@napier.ac.uk

K. Stewart / Transportation Research Part A 41 (2007) 644–654 645

1. Introduction

1.1. General background to road user charging

Road tolling as a general theme is particularly topical currently in the UK and whilst there are not manyactual schemes in existence, there have been numerous trials and much public debate. Despite having been onthe road transport agenda in a variety of forms for the past 40 years the only established schemes which arecurrently operational are those in Singapore, the Norwegian cities of Bergen, Oslo and Trondheim (May andMilne, 2000) and more recently UK schemes in Durham (2002) and London (2003). Whilst the Norwegianschemes have been designed mainly to raise revenue, the Singapore scheme was aimed more at reducing con-gestion, as were the recent UK schemes.

The Singapore scheme is particularly interesting not only as it is aimed to reduce congestion, but alsobecause it successfully utilises an electronic road pricing system at a time when UK studies still seem scepticalabout the practical implementation of electronic systems (Road Charging Options for London, 2000). Singa-pore initially used an Area Licensing Scheme (ALS) introduced in 1975, but replaced this with Electronic RoadPricing (ERP) in 1998 (Seik, 2000). Singapore’s ERP system is the first system of its kind to be used permanentlyand extensively. The fitting of In Vehicle Units (IVUs) for all motor vehicles was undertaken in 1997–1998, andwas free of charge and voluntary. However by July 1998, 96% of 680,000 vehicle owners had fitted their vehicleswith the IVUs (Seik, 2000). This is noteworthy as much opposition to schemes which rely on electronic pricing,such as link based tolling schemes, rests on the claim that the fitting of vehicles with the necessary devices wouldbe impracticable and that such devices would have to be fitted during the vehicles’ manufacture creating anunacceptable time lag (Ison, 1998). The success of the fitting of the devices for the Singapore scheme contradictssuch claims and suggests that electronic schemes are currently feasible and not just some mythical possibility forthe future. Whilst this scheme is still cordon based it offers the technological potential for a link-based scheme,as detection beacons could potentially be situated on links where a toll was to be levied. Thus the implemen-tation of a minimal revenue link-based tolling scheme may currently be feasible.

Despite the success of the Singapore ERP the London study concluded that it was unlikely that a fullyelectronic system which would comply with national standards could be made operational for the plannedimplementation date, which was constrained by political considerations. It was recognised however thatdespite a high initial set up cost such a scheme would have lower running and enforcement costs, and wouldhave the potential for much greater flexibility of the charging scheme, such as higher charges at peak times.There is no reason however why the London scheme could not adapt its payment procedures at a later date.

Whilst the existing schemes are fixed price and cordon based, there has been research and many trialsconsidering a variety of pricing measures (May and Milne, 2000; PROGRESS project, 2000). The UKGovernment’s Commission for Integrated Transport recently published a report ‘Paying for Road Use’(CfIT 2002), which suggests the introduction of nationwide road user charging, although this would notbe implemented within the current 10 year plan (i.e. before 2010). The report is of interest in that it suggeststhe use of marginal social cost pricing on all roads (i.e. motorways, A Roads, minor roads, city centres etc.),but this would be balanced by a reduction (or abolition) of Vehicle Excise Duty combined with a reductionin fuel duty, so that the result would be fiscal neutrality. Thus they are suggesting a theoretical economics-based approach which would rely on charging for travel along a link rather than passing across a cordonand as such are suggesting an ERP/ITS solution such as would be required if minimal revenue tolls were tobe implemented in practise.

1.2. Acceptability and equity issues related to road user charging

Political acceptability and how to achieve it is a major consideration in the planning and successful imple-mentation of road user charging schemes. Whilst to date most urban schemes have utilised toll rings, there isno reason to assume that link-based schemes may not be preferred in the future particularly if national roaduser charging schemes (be they revenue neutral or otherwise) are introduced. Issues of assessing acceptabilitymay be problematic. In Edinburgh, despite a successful public enquiry, and whilst there was no legal need, thedecision to assess public support by means of a dedicated public referendum was made. This referendum

646 K. Stewart / Transportation Research Part A 41 (2007) 644–654

showed a high level of opposition to the scheme (about 3:1 with turnout 61.8%) and plans were consequentlydropped. It is not clear whether the negative referendum result was wholly due to the unpopularity of conges-tion charging in general and this scheme in particular, or if it was purely a negative consequence of holding areferendum for this sort of political decision. It is possible though that many of the perceived negative featuresof this cordon scheme would have been less severe had a link-based tolling scheme been proposed. Opinionpolls in other cities have shown that public opinion can be quite supportive of charging schemes when theyare initially proposed, but that this support gradually diminishes up to the point where the scheme is actuallyintroduced. Post implementation support and acceptability tends to rise again whilst people get ‘used to’ thechanges. This has been the trend for most of the Norwegian toll rings (Tretvik, 2003). The acceptability issuehas been claimed by some to be analogous to the issues surrounding acceptability of parking charges, suchcharges are unpopular when first introduced but tend to be accepted in time, particularly when such measuresare so widely used that they become commonplace (Jones, 2003).

There are two types of equity impacts which are commonly considered; Spatial (or horizontal) equity andSocial (or vertical) equity. It is perceived to be important that Charging schemes are seen to be ‘fair’ in someway (Jones, 2003), and whilst economic MSCP-tolls might be viewed as fair from one perspective (in that theycharge all road users for their externalities), the fact that such costs might be much harder to bear (i.e. agreater proportion of income), for lower income groups might make them ‘socially’ unfair, also socially mar-ginal journeys are not equal to economically marginal journeys. The concept of link-based rather than cordon-based charging schemes might make charging more acceptable from a social equity perspective; the potentialexistence of untolled links and untolled routes imply that there will be a trade-of in routing based upon a per-son’s own value of time, those of higher income groups may choose to take the quicker tolled links in theirroute where as the lower income groups have the option of taking untolled links and possibly untolled routes(depending on their journey) albeit with a greater time cost. Issues of spatial equity arise from the differences inwhere people live and travel with respect to the charging points. This is a particularly problematic issue withrespect to cordon or area schemes as it is obviously possible for a person to live just outside a boundary andneed to travel to just within it. This is particularly inequitable if the cordon charge is set up in such a way thata similar person who lived just within the boundary with a similar trip distance did not have to pay. Againlink-based schemes would considerably reduce the impact of spatial equity, as it is primarily a cordon schemeproblem. A system whereby everyone who used a particular link (at a particular time) being charged the sameamount would certainly appear fairer.

1.3. Modelling the effect of road tolls

The tolling schemes which are the main subject of this paper are ‘optimal’ schemes (or ones that closelyapproach optimality). That is the toll sets determined will all force a User Equilibrium (UE) assignment toproduce the System Optimal (SO) flow pattern or a Stochastic User Equilibrium (SUE) assignment to producea Stochastic Social Optimum (SSO) or SO flow pattern. This is commonly referred to as first best tolling whilesecond best tolling refers to sub-optimality where not all network links may be tolled (such as a cordon).

First-best schemes have not been implemented in practise as link or path-based tolling schemes wouldrequire substantial investment in infrastructure in order to charge vehicles for travel along particular linksor paths. Thus operational tolling schemes to date have been cordon based. Such schemes do not seek to opti-mise traffic flow within a network but rather to reduce congestion by discouraging travel and thus reducingdemand. They also generally seek to raise revenue. In general various fixed scenarios for cordon placementand charging levels are compared and a decision on which to implement made by comparing projected reduc-tion in traffic with projected raising of revenue and a politically acceptable balance is determined. It is howeverpossible to mathematically optimise for second-best tolls under certain network restrictions namely where notall links in a congested network may be tolled (e.g. Verhoef, 2002; Zhang and Yang, 2004; Hearn and Law-phongpanich, 2003). If the links tolled are required to form a cordon the location of such a cordon will have asignificant impact on the benefits that may be derived (May et al., 2002).

Most extant schemes have been the result of judgemental cordons based on the actual location of conve-nient ‘ring’ roads, but genetic algorithms have been developed to design and generate optimal cordons whererevenue is to be maximised under assumptions of elastic demand (Sumalee, 2004a; Sumalee, 2004b). Such

K. Stewart / Transportation Research Part A 41 (2007) 644–654 647

optimal cordons have then been compared with judgmental cordons (Sumalee et al., 2005) where it is shownthat these optimal cordons can potentially significantly increase the benefits which could be achieved by judg-emental cordons (to the order of two or three times the benefit). Thus it may be observed that second bestoptimality as opposed to judgmental pricing may be highly beneficial in terms of raising revenue. Such cordonschemes however do not create the beneficial effects that might be achieved by re-routing within the cordonarea. This work concentrates on optimising for efficient network flow rather than optimising the potential rev-enue and is not based on cordon assumptions.

Traffic assignment models seek to replicate the traffic pattern that is created when drivers choose theirroutes across a network from their origin to their destination. In the case of deterministic assignment it isassumed that drivers, with perfect network knowledge, will act selfishly to minimise their personal travel costresulting in the Wardropian User Equilibrium (UE) flow pattern. Tolls may then be imposed to ‘force’ a UEassignment to result in an alternative desired flow pattern. The System Optimal (SO) is one such desired flowpattern where the Total Network Travel Cost (TNTC) is minimised and occurs when all used routes haveequal marginal cost. Tolls that result in such a flow pattern being created are non-unique; the classical eco-nomics solution of Marginal Social Cost Price (MSCP) tolls whereby a toll equal to the difference betweenthe marginal social cost (that is imposed upon the network by the driver) and marginal private cost (thatthe driver experiences) is levied on each link, is one such toll set that produces the desired flow pattern. Otherconstraints may however be imposed; such as toll revenue being minimised, resulting in Minimal-Revenue(Min-Rev) tolls, or fiscal neutrality being obtained resulting in Robin Hood tolls (Hearn and Ramana,1998) where the non-negativity condition normally imposed upon toll sets is relaxed.

The calculation of minimal revenue tolls is of interest as it is often claimed (Frey, 2003; Mayeres, 2003) thatthe imposition of MSCP tolls will be politically unacceptable owing (amongst a range of reasons) to their‘high’ cost. In finding Min-Rev tolls that create the same desired network flow effects as MSCP-tolls a possiblebaseline position may be reached. If Min-Rev tolls are still deemed too high for political acceptability then itwould be assumed that all such toll sets would be too expensive and System Optimality could not be achievedby politically acceptable tolling. If however Min-Rev tolls were not deemed to be too high, then it would besimple to add a constant link-toll vector to the Min-Rev link-toll vector and still achieve system optimality,but allowing the policy maker to extract the desired amount of revenue from the users.

The Minimal Revenue toll problem has in the case of deterministic assignment been solved (such that theSystem Optimal solution is obtained) by various methods: Linear Programming (Bergendorff et al., 1997),reduction to multi-commodity max-flow problem (Dial, 2000), and although an efficient link based methodthat does not rely on Mathematical Programming solutions has yet to be developed in the multiple origin case,a single origin algorithm exists (Dial, 1999).

Deterministic assignment models assume that all drivers have a perfect knowledge of travel costs on everylink and model route spreading by the use of unit cost-flow relations that simulate congestion. Stochasticassignment methods however assume that instead of drivers having a ‘perfect’ knowledge of the varyingOD costs of a network that they have a variable perception of these costs. Stochastic user equilibrium(SUE) assignment is based on the premise that each driver will act to minimise their perceived route cost(which follows a distribution such as those given in the logit or probit models) and Stochastic User Equilib-rium (SUE) methods have been developed (Maher and Hughes, 1997). It would seem logical that drivers doperceive costs differently from each other, either because of different levels of network knowledge or differentpriorities (e.g. avoidance of right turns or roundabouts, minimising distance or time) and so the use of a sto-chastic method would seem to be more realistic. Thus it is useful to extend the concept of optimal tolling to thestochastic case.

In the Stochastic case it is not obvious what the desired flow pattern to be obtained by tolling should be.This paper considers two options:

1. That the desired flow pattern should be that obtained by marginal social cost pricing. Tolls that are anal-ogous to MSCP tolls in the deterministic case do not however give the system optimal solution in the sto-chastic case (Yang, 1999) and instead of the actual Total Network Travel Cost being minimised the ‘TotalPerceived Network Travel Cost’ is minimised (Maher et al., 2005). This flow pattern is termed the Stochas-tic System Optimum (SSO). Min-Rev tolls associated with the SSO can then be obtained.

648 K. Stewart / Transportation Research Part A 41 (2007) 644–654

2. That the desired flow pattern should be the true System Optimal flow pattern where the actual Total Net-work Travel Cost is minimised. This paper determines tolls that will result in the SO flow pattern beingobtained for small networks under stochastic assignment.

2. Optimal tolling under SUE

2.1. Desired flow pattern under SUE

In the Stochastic environment the ‘desired’ flow pattern is not as readily obvious as in the deterministiccase. The directly analogous position is to seek the SSO (based on MSCP tolls), the potential then existingto create Min-Rev tolls to result in an SSO flow pattern. The SSO though is the flow pattern where the per-ceived network travel costs are minimised, and whilst this might make most obvious sense from the perspectiveof the users, the SO, where actual network costs are minimised could still be argued to give the most ‘optimal’solution from the perspective of the planner, i.e. that which would result in least congestion for a certain fixeddemand of traffic flowing through the network). It is not immediately obvious whether tolls to force an SUEassignment to result in the deterministic SO flow pattern do in fact exist; a discussion of this and a heuristic tofind an approximation to such tolls is presented in Stewart and Maher, 2006). The two tolling possibilitiesunder Stochastic Assignment are presented in Fig. 2.1.

2.2. Min-Rev SSO tolls under SUE

As mentioned in Section 1.3 Marginal Social Cost Price (MSCP) tolls are a toll set such that (in the deter-ministic case) both economic benefit is maximised and Total Network Travel Cost (TNTC) is minimised. Ifonly the condition for TNTC to be minimised is to be considered, then the optimal flow pattern (SO) producedunder MSCP-tolls may also be produced by other toll sets, such as the Minimal-Revenue toll set where TNTCis minimised and Total revenue to be collected from the users is also minimised.

MSCP tolls may be derived in the stochastic case in an analogous manner to the deterministic case. A nine-node network frequently used in the literature (Bergendorff et al., 1997; Dial, 2000), is used throughout thispaper for illustrative purposes, and is shown in Fig. 2.2. In all examples logit-based SUE has been used, withvariability parameter h = 0.1, although this analysis is not limited to logit-based SUE and is equally applicableto all stochastic assignments.

In the case of deterministic assignment, it is well known that that the Total Network Travel Cost is mini-mised and the System Optimal flow pattern is obtained, when cost-flow functions are replaced by marginalcost-flow functions. Recent work (Maher et al., 2005) has shown that the analogous case is true under stochas-tic assignment. Thus MSCP tolls may be easily found using existing link-based assignment methods.

The minimal revenue toll problem is thus similar to that in the deterministic case, and may be solved bylinear programming. For the nine-node network given this may be simplified by looking explicitly at path-tolls. MSCP path tolls may be calculated and a zero toll tree for each OD pair defined (shown by dashed linesin Fig. 2.3) where the path cost is set to zero and the remaining path-tolls adjusted accordingly. Path-tolls arethen translated back into link tolls (this solution may not be unique and in the example below the solution

Deterministicenvironment

Stochasticenvironment

UE

SO

SUE

SSOminimises

total travel costs

drivers’ ownroute choice

+ optimal tolls + optimal tolls

Fig. 2.1. Desired flow pattern in deterministic vs stochastic environment.

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(5,12) (2,11) (3,25)

(6,18)

(3,35)

(8,26) (4,26)

(7,32) (8,30) (6,24)

(8,39)

(9,35) (6,33) (6,43)

OD Pair: [1,3][ 1,4] [2,3] [2,4]Demand: 10 20 30 40ca(xa) = ca

(0)(1 + 0.15(xa/Ya)4)

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OD Pair: [1,3][ 1,4] [2,3] [2,4]Demand: 10 20 30 40ca(xa) = ca

(0)(1 + 0.15(xa a)4)

Fig. 2.2. Nine-node network diagram, showing OD demand and link cost-flow relations.

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4.1

2.2

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14.1 1.3

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4.4 0.2

4.7 2.2 1.6

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2.9

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00.7

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0

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0

TNTCSUE: 2441 TNTCSSO: 2332

MSCP Rev: 1186

TNTCSUE: 2441 TNTCSSO: 2332

Min-Rev: 402

Fig. 2.3. Nine-node network diagram, MSCP and Min-Rev tolls to achieve SSO.

K. Stewart / Transportation Research Part A 41 (2007) 644–654 649

with the maximum number of zero-tolled paths has been chosen). MSCP tolls and Min-Rev tolls for the nine-node network are illustrated in Fig. 2.3.

It may be seen that Min-Rev tolls may achieve the SSO flow pattern and network travel cost, whilst costingthe user only 34% of the MSCP-toll cost. Both toll-sets reduce the total network cost by 4.92%. In addition tothe reduced cost of Min-Rev tolls there exists the advantageous effect of zero-toll paths and zero-toll links,which are beneficial from equity considerations as there is no obligation to be charged an additional monetarytoll for travel through a network (although a time penalty of having to choose a slower route will exist), whereas for a cordon or area-based charging scheme, payment will be taken irrespective of routing choices.

2.3. SO tolls under SUE

In attempting to determine a set of tolls that will result in the deterministic System Optimal flow patternbeing produced under an SUE assignment either path-based or link-based methodologies may be attempted.In Stewart and Maher (2006) a path-based methodology is presented that shows it may be possible to derive aset of tolls to replicate the deterministic SO precisely under certain network conditions. These conditionsinclude the fact that zero-flow paths should not exist at the SO solution. Whilst this condition may be over-come by attempting to replicate not the deterministic SO but rather a stochastic SSO (with variability param-eter set so that the solution is tending towards the deterministic solution), this will result in path-tolls whichtend to infinity. Clearly it is possible to set a desired flow pattern to be such that the tolls produced are allfinite (with any desired upper bound), but in comparative terms these tolls tend to still be disproportionatelyhigh and have therefore limited potential political acceptability (the effect is such that even if certain linkswere considered closed or pedestrianised that the remaining tolls are still unjustifiably high). Further itwas found that such tolls do not generally produce a viable set of associated link-tolls and would therefore

650 K. Stewart / Transportation Research Part A 41 (2007) 644–654

result in a path-based tolling scheme being required. This would create some logistical problems for imple-mentation that are further discussed in Section 4. Such issues resulted in a link-based methodology beingfavoured.

The link-based methodology derived is based on a heuristic which seeks to match the flow on each link withthe desired link flow at the SO solution. This heuristic does not guarantee a Minimal Revenue solution in atheoretical sense, but applies additional tolls to links in small increments so that at each stage the additionalrevenue required to produce a reduction in TNTC may be monitored and assessed. This will enable a trafficplanner to potentially ‘trade-off’ a reduction in TNTC (approaching the SO solution) against an increase inrevenue to be extracted from the users. After a certain point, the relative benefit from imposing more tollsis outweighed by the relative disadvantage of imposing higher tolls for less additional effect. Such tolls maybe thought of as ‘low-revenue’ tolls rather than ‘minimal-revenue’ tolls.

The heuristic procedure is given below and Table 1 shows the stepwise construction of a toll-set for thenine-node network used previously.

Step 1: Find the SO solution and let FSO, CSO and TNTCSO denote the corresponding flow pattern, link cost,and total network travel cost.

Step 2: Link toll vector set to zero: T0 = 0.Step 3: Set n = 0.Step 4: Perform SUE assignment with current link tolls Tn: Cn and Fn obtained.

Step 5: Calculate: Pj = ðF ðnÞj – F ðSOÞj Þ(jCðnÞj – CðSOÞ

j j) " j.

Step 6: Determine link j where P(j) is greatest.Step 7: *Perform iteration to calculate tj s.t F ðnÞj ¼ F ðSOÞ

j to required degree of accuracy.

7a: Set tj0¼ Cj0

� Cð9mÞj

���

��� where Cj0

is the current cost on link j (as per step 4)

7b: Set m = 17c: Perform SUE assignment, calculate Cjm

� CðSOÞj

���

���

7d: Set tjm¼ tjm�1

þ Cjm� CðSOÞ

j

���

���

7e: Calculate P jm: Stop if sufficiently close to zero and let tj ¼ tjm

, or set m = m + 1 and repeat fromstep 7c.

Step 8: Tn + 1 = Tn + t; where ti = tj when i = j and ti = 0 otherwiseStep 9: Calculate TNTC: Stop if TNTC sufficiently close to TNTCSO or set n = n + 1 and repeat from Step 4.

Table 1Iterative building of ‘Optimising’ toll set

Iteration 0 1 2 3 4 5 6 7 8 9 10 11 12

t1(1-5) – – – – – – – – – – 0.9 0.9 0.9t2(5-7) – 7.2 7.2 7.2 7.2 7.2 7.2 8 8 8.9 8.9 8.9 8.9t3(7-3) – – – – – – – – – – – – –t4(1-6) – – – – – – – – – – – – –t5(2-5) – – – – – – – – – – – – –t6(5-9) – – – – – – – – 0.6 0.6 0.6 0.6 0.6t7(9-7) – – – – – – 1.4 1.4 1.4 1.4 1.4 1.4 1.4t8(6-9) – – – – – – – – – – – – –t9(9-8) – – – 13 13 13 13 13 13 13 13 13.8 13.8t10(7-4) – – 7.9 7.9 12.9 12.9 12.9 12.9 12.9 12.9 12.9 12.9 12.9t11(8-3) – – – – – – – – – – – – –t12(2-6) – – – – – 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.8

t13(6-8) – – – – – – – – – – – – –t14(8-4) – – – – – – – – – – – – –

TNTC 2441 2385 2337 2285 2268 2262 2259 2258 2257 2256 2255 2255 2254REV 0 154 307 449 568 705 746 759 777 797 813 819 822

TNTC and Revenue: θ = 0.1

0

1000

2000

3000

0 5 10 15Iteration

TNTC

Rev

Fig. 2.4. Total toll revenue required for reduction in TNTC.

K. Stewart / Transportation Research Part A 41 (2007) 644–654 651

It may be seen from Fig. 2.4 that the first few iterations are by far the most significant with little additionalbenefit being obtained from approaching the SO to a high level of agreement of link-flows/costs. It is also anobjective from equity considerations to leave as many links toll-free as possible, thus there is a benefit to notadding small tolls onto additional links to get only a small degree closer to the SO solution (where TNTCwould be minimised).

The link toll set produced from the 12 iterations shown in Table 1 is given in Fig. 2.5 (link width is pro-portional to the link toll). The TNTC achieved after these 12 iterations is only 0.02% higher than the minimumTNTC possible (at the SO solution), but if the process were terminated after only 4 iterations, only 4 links intotal would need to be tolled and the TNTC obtained would still only be 0.6% higher than that at the true SOsolution. It would seem reasonable therefore, to stop the process when an acceptable balance is obtainedbetween reduction in TNTC, revenue required, and number of links which would require a toll. This is furtherillustrated in the more general numerical example in Section 3.

In comparison with the SSO min-rev toll set shown in Fig. 2.3, it may be seen that the low-rev toll solutionafter 12 iterations of the heuristic results in a reduction in TNTC (compared to the SUE cost of 2441) of 7.64%(the true SO would result in a reduction of 7.66%), where as the SSO Min-Rev tolls produced a reduction of4.47%. The SSO Min-Rev tolls however required less revenue to be extracted from the user, 402 as opposed to822. For comparative purposes reductions in TNTC, revenue required, and an Index I, (where I = percentagereduction in TNTC per 1000 units of revenue), are summarised in Table 2 for SSO with MSCP-tolls and Min-Rev tolls, and for SO with both 4 and 12 Iterations of Low-Rev tolls.

TNTC SO : 2253.9 TNTC: 2254.4

Low-Rev: 882

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8.90.9

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Fig. 2.5. logit toll-set for Bergendorff’s network (h = 0.1) – 12 iterations.

Table 2Comparison of SSO and SO toll sets

SSO: MSCP- tolls SSO: Min-Rev tolls SO: Low-Rev tolls4-Iterations

SO: Low-Rev tolls12-Iterations

% Reduction TNTC 4.47 4.47 7.09 7.64Revenue 1186 402 568 822Index: I 3.77 11.11 12.48 9.29Links Tolled 14 6 3 7

652 K. Stewart / Transportation Research Part A 41 (2007) 644–654

It can be readily seen that the extra reduction in TNTC from using a 12-Iteration toll set, does not seemjustifiable in terms of revenue required. The value of I is not much greater for the SSO: Min-Rev tolls thanthe SO: Low-Rev tolls with 4-Iterations, but the required number of links to be tolled is clearly least underthe SO low-Rev tolls with 4 iterations, resulting in that tolling methodology seemingly resulting in the mostdesirable results. This comparison is not however necessarily generalisable, but the benefit of such a heuristicto derive tolls lies in the ability to halt the process when the desired number of links to be tolled is reached, italso tolls the ‘worst’ links first, i.e. those where the product of cost and flow is most different to that at the SOsolution. In larger networks, linear programming solutions to explicitly derive SSO Min-Rev tolls may be rel-atively costly in terms of CPU requirements, making efficient heuristics more attractive. It should be notedhowever that the heuristic presented is not specific to the SO solution and may in fact be used to seek tollsto approach any desired flow pattern, which could if desired include the SSO, and may result in toll sets similarto the Min-Rev tolls derived by linear programming.

3. Numerical example

The 9-node network in Section 2 whilst useful for comparative purposes with results in the literature is acy-clic and contains only one-way links. The heuristic presented is however applicable to more general networksas illustrated in the numerical example in this section which gives an indication of how link-based schemescould be presented in reality. The network used is a 13-node network, based around a notion of an outerand inner ring city ring; the cost flow functions used are synthetic. Fig. 3.1 shows the MSCP-toll set whichwould create the SSO flow pattern under SUE and Fig. 3.2 shows a Low-Rev toll set with 2-iterations usingthe SO heuristic.

The heuristic concentrates the tolls on the links which are deemed to be ‘worst’ i.e. those with a product oflink-cost and flow most greatly in excess of the desired SO value. The MSCP tolls reduce the TNTC by 3.1%,whereas the maximum achievable reduction at the theoretical SO is 10.5%. However tolling using the ‘SO heu-ristic’ produces a reduction in TNTC of 6.1% after only 2 full iterations at a cost to the user of only 1.8% of theMSCP-toll value (MSCP-toll revenue = 757,300; Low-Rev-toll revenue = 13,500). Thus the low-Rev tolls canpotentially produce a significant improvement in network performance (given a fixed level of demand) whilstimposing less cost upon the user and tolling only a few links. Such a scheme with obvious equity benefits mightbe more politically acceptable than a cordon-based scheme.

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9.2

Fig. 3.1. SSO: MSCP-tolls for numerical example.

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Fig. 3.2. SO: Low-Rev tolls for numerical example.

K. Stewart / Transportation Research Part A 41 (2007) 644–654 653

4. Conclusions

This paper has discussed attempting to achieve a true SO flow pattern under the principles of stochastic userequilibrium assignment. Whilst path-based methods would be required to precisely replicate such a flow pat-tern (to any desired level of accuracy), such a tolling scheme would result in excessively large tolls that wouldbe politically difficult to implement. Consequently work has concentrated on developing link-based heuristicswhich produce toll sets that will balance the reduction in total network travel cost (whist aiming for the SOsolution) against the revenue required to create this routing change. Limiting the number of links that needto be tolled is also a major consideration. Such toll sets have been compared with both MSCP-tolls andMin-Rev tolls to create the SSO flow pattern, whilst MSCP-tolls appear too high to be politically acceptable,MIN-Rev SSO tolls appear to produce good network results (although not quite as beneficial as Low-rev SOtolls). The desired flow pattern in the stochastic case would though have to be decided by the traffic plannerwhen they considered what nature of ‘optimality’ they might wish to produce, i.e. minimising real or perceivedcosts.

In the case of road user charging, it must be considered whether a path based tolling system would be eithersensible or implementable. A link-based system would be more obviously possible by use of a smart cardwithin the vehicle and roadside beacons on the tolled links. A path-based system however would requirethe vehicle to be tracked throughout the network and the appropriate toll for the complete route chosen tobe charged at the point of exit; this may be feasible with a GPS on-vehicle navigational system, but such asystem would need to be installed into all vehicles so this would not be a viable solution in the immediatefuture. In addition it should be considered whether a path based system would be seen to be ’fair’ in that itwould essentially charge different prices for travel along a certain link depending on which overall path werechosen. From both practical and technical perspectives it appears that link-based schemes would be generallypreferable.

The work presented here has assumed a fixed level of demand. The validity of such an assumption is obvi-ously debateable as it is generally accepted that demand is elastic and that an increase in costs to the users willresult in a decrease in demand. Thus the ‘real’ effects of a link-based charge would be two-fold, a re-routingeffect as described in this paper and a traffic suppression effect to account for elastic demand. Most cordonschemes are based on reducing congestion by the suppression of traffic within the cordon boundary as thereis no further incentive for re-routing to occur once the decision to enter the boundary has been taken(although re-routing will obviously occur outwith the boundary). The two-fold effects of link-based tollingassuming elastic demand are presented in Stewart and Maher, 2005.

654 K. Stewart / Transportation Research Part A 41 (2007) 644–654

References

Bergendorff, P., Hearn, D.W., Ramana, M.V., 1997. Congestion toll pricing of traffic networks, network optimization. In: Pardalos, P.,Hearn, D.W., Hager, W.W., (Eds.), Lecture Notes in Economics and Mathematical.

Commission for Integrated Transport 2002. Paying for Road Use, <www.cfit.gov.uk>.Dial, R.B., 1999. Minimal-revenue congestion pricing part I: A fast algorithm for the single origin case. Transportation Research B 33 (3),

189–202.Dial, R.B., 2000. Minimal-revenue congestion pricing part II: An efficient algorithm for the general case. Transportation Research B 34

(8), 645–665.Frey, B.S., 2003. Why are efficient transport policy instruments so seldom used? In: Schade, J., Schlag, B. (Eds.), Acceptability of

Transport Pricing Strategies. Elsevier, pp. 27–62.Hearn, D.W., Lawphongpanich, S., 2003. Solving second best toll pricing problems, In: Proc. the Theory and Practice of Congestion

Charging: An International Symposium, Imperial College London.Hearn D.W, Ramana M.V., 1998. Solving Congestion Toll Pricing Models. In: Marcotte, P., Nguyen. (Eds.), Equilibrium and Advanced

Transportation Modeling, Kluwer Academic Publishers, pp. 109–124.Ison, S., 1998. A concept in the right place at the wrong time: congestion metering in the city of Cambridge. Transport Policy 5 (3), 139–

146.Jones, P., 2003. Acceptability of road user charging: meeting the challenge. In: Schade, J., Schlag, B. (Eds.), Acceptability of Transport

Pricing Strategies. Elsevier, pp. 27–62.Maher, M.J., Hughes, P.C., 1997. A probit-based stochastic user equilibrium assignment model. Transportation Research B 31 (4), 341–

355.Maher, M.J., Stewart, K., Rosa, A., 2005. Stochastic social optimum traffic assignment. Transportation Research B 39 (8), 753–767.May, A.D., Liu, R., Shepherd, S.P., Sumalee, A., 2002. The impact of cordon design on the performance of road pricing schemes.

Transport Policy 9, 209–220.May, A.D., Milne, D.S., 2000. Effects of alternative road pricing systems on network performance. Transportation Research A 34 (6),

407–436.Mayeres, I., 2003. Reforming transport pricing: an economic perspective on equity efficiency and acceptability. In: Schade, J., Schlag, B.

(Eds.), Acceptability of Transport Pricing Strategies. Elsevier, pp. 27–62.PROGRESS project., 2000. <www.progress-project.org>.Road Charging Options for London: A Technical Assessment, <http://www.open.gov.uk/glondon/transport/rocol.htm>.Seik, F.T., 2000. An advanced demand management instrument in urban transport Electronic road pricing in Singapore. Cities 17 (1), 33–

45.Stewart, K.J., Maher, M. J., 2006. Minimal Revenue Network Tolling: System Optimisation under Stochastic Assignment, In: Hearn,

D.W., Lawphongpanich, S., Smith, M. (Eds.), Mathematical and Computational Models for Congestion Charging, AppliedOptimization. Springer, New York, pp. 201–218.

Stewart, K., Maher, M.J., 2005. Minimal Revenue Network Tolling: System Optimisation under Stochastic Assignment with ElasticDemand. In: 4th IMA International Conference on Mathematics in Transport, 7–9 September, University College London.

Sumalee, A., 2004a. Optimal road user charging cordon design: a heuristic optimisation approach. Computer-Aided Civil andInfrastructure Engineering 19, 377–392.

Sumalee, A., 2004b. An innovative approach to option generation for road user charging scheme design: Constrained and multi-criteriadesign, In: Proc. 10th World Conference on transport research, Istanbul.

Sumalee, A., May, A., Shepherd, S., 2005. Comparison of judgemental and optimal road pricing cordons. Transport Policy 12 (5), 384–390.

Tretvik, T., 2003. Urban road pricing in norway: public acceptability and travel behaviour. In: Schade, J., Schlag, B. (Eds.), Acceptabilityof Transport Pricing Strategies. Elsevier, pp. 77–92.

Verhoef, E., 2002. Second-best congestion pricing in general static transportation networks with elastic demands. Regional Science andUrban Economics 32, 281–310.

Yang, H., 1999. System optimum, stochastic user equilibrium, and optimal link tolls. Transportation Science 33 (4), 354–360.Zhang, X., Yang, H., 2004. The optimal cordon-based network congestion pricing problem. Transportation Research 38B, 517–537.