Analysis of regulation and policy of private toll roads in a build-operate-transfer scheme under demand uncertainty

Transportation Research Part A 41 (2007) 537–558

www.elsevier.com/locate/tra

Analysis of regulation and policy of private toll roads in abuild-operate-transfer scheme under demand uncertainty

Anthony Chen a,*, Kitti Subprasom b

a Department of Civil and Environmental Engineering, Utah State University, Logan, UT 84322-4110, USAb Planning Division, Department of Highways, Bangkok 10400, Thailand

Received 30 April 2005; received in revised form 11 July 2006; accepted 21 November 2006

Abstract

The build-operate-transfer (BOT) approach is one of the privatization mechanisms for promoting transportation infra-structure developments by using private funds to construct new infrastructure facilities. In a BOT scheme, it often involvesthree parties: the government, whose objective is to maximize the benefit defined in terms of social welfare added to thesociety; the private investors, whose objective is to maximize the profit generated from the investment; and the road users,whose objective is to minimize the inequality of benefit distribution among the road users traveling from different origin–destination pairs. Each of these parties has different objectives that often conflict with each other. In this paper, we developvarious optimal road pricing models under demand uncertainty for analyzing the tradeoffs among the three objectives. Inaddition, a project evaluation framework is developed for assessing the effects of government policy and regulation on theBOT project. Seven cases of the BOT road pricing problem are analyzed: (1) BOT without regulation, (2) BOT with pricecontrol regulation, (3) BOT with equity regulation, (4) BOT with construction cost subsidy, (5) BOT with concession per-iod extension, (6) BOT with construction cost subsidy and concession period extension, and (7) BOT with multiple objec-tives. Numerical results using a real case study of the Ban Pong–Kanchananburi Motorway (BMK) in Thailand areprovided to examine the above seven cases.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Build-operate-transfer scheme; Multi-objective optimization; Network design; Pricing schemes

1. Introduction

Rapid economic expansion and increasing in travel demands require the massive development of transpor-tation infrastructures. In recent years, the governments in many countries have begun privatizing transporta-tion infrastructures to the private investors. Some of the forces driving this movement include: a scarcity ofpublic resources, an increase in the demand for better service, and political trend toward the deregulationof infrastructures from public monopoly. Confronted with these forces, the government, therefore, is in favor

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

doi:10.1016/j.tra.2006.11.009

* Corresponding author. Tel.: +1 435 797 7109; fax: +1 435 797 1185.E-mail address: [email protected] (A. Chen).

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538 A. Chen, K. Subprasom / Transportation Research Part A 41 (2007) 537–558

of having the private investors play an increased role in the investment and operation of transportation infra-structure projects. One of the privatization alternatives is the build-operate-transfer (BOT) approach. In aBOT scheme, the government grants the private investors the rights to finance, develop, and operate a revenueproducing infrastructure for a defined period called the concession period, after which the infrastructure istransferred back to the government (Walker and Smith, 1995). The BOT scheme is gaining popularity andacceptance as an innovative way to finance the construction of infrastructures in both developed and devel-oping countries (see Subprasom, 2004 for a list of BOT projects).

The BOT concept originated during the industrial revolution that took place in Europe in the eighteenthcentury. It brought dramatic social changes such as urbanization and its associated infrastructure needs. Atthat time, European governments had only a basic taxation system to support monarchs and to fight wars.Infrastructures such as canals, turnpikes, and railroads were financed and built by the private investors. They,in turn, would collect tolls or user charges from the users to recoup their investment and earn a profit. This isthe earliest predecessor of the BOT concept (Chang, 1996). In 1847, a group of French, British, and Austrianinvestors investigated the feasibility of constructing the Suez canal which would link the Mediterranean andRed Seas; this became the world’s first international enterprise using the BOT approach in the 19th century(Sidney, 1996). After studying European roads that were financed by the private investors in the 1960s and1970s, governments in Italy and Spain also allowed the private investors to operate toll roads in a bid toattract additional capital for road construction (Yates, 1992). The concept of toll roads in the United Statesdates back to the eighteenth century. Recent examples of the BOT concept include the Dulles Greenway, a14-mile toll road extension of the Dulles airport access road to Leesburg, Virginia (Euritt et al., 1994), andthe 10-mile express lanes built in the median strip of the existing California State Route 91 (Sidney, 1996).In the developing countries, most BOT project developments were funded by the World Bank, the AsianDevelopment Bank, or by the developed countries such as Japan and the United States. Some examples ofBOT projects in the developing countries are as follows: five major toll automobile tunnels in Hong Kong(Downer and Porter, 1992); the superhighway project connecting the booming industrial cities of the PearlRiver Delta region in China (Yang and Meng, 2000); the Don Muang Tollway, an elevated toll expresswayin metropolitan Bangkok, Thailand (Ogunlana, 1997); the Kepong toll road in Malaysia (Walker and Smith,1995); and the Mexico City-Guadalajara project, a toll road in Mexico (Huang, 1995; Swan, 1993).

A number of studies have been undertaken on the modeling of BOT as a network design problem. Yangand Meng (2000, 2002) provided bi-level mathematical programming formulations to determine the optimaltoll charge and roadway capacity of the private toll roads in a transportation network under various marketconditions. Subprasom and Chen (2007) examined the effects of regulation on highway pricing and capacitychoice of a BOT scheme. Profit and welfare gain from both the private and public sectors were evaluated withrespect to different toll-capacity combinations, but demand uncertainty was not considered in both studies.Travel demand was assumed to be known exactly in the future. However, there is no guarantee that the traveldemand forecast would precisely materialize under uncertainty. This is because travel demand forecast isaffected by many factors such as economic growth, land-use pattern, socioeconomic characteristics, etc. Allthese factors cannot be measured accurately, but can only be roughly estimated. To account the uncertainty,Chen et al. (2001) incorporated demand uncertainty into the BOT network design problem, but only oneobjective was considered in the stochastic bi-level mathematical programming formulation, which is to max-imize the expected profit. Using the classical mean-variance model in portfolio analysis (Markowitz, 1927),Chen et al. (2003) extended the BOT network design problem to consider two objectives: maximizing expectedprofit and minimizing variance of profit. The variance associated with profit is considered as a risk. In theirstudy, however, only the perspective of the private investors is considered.

In this paper, the BOT network design models are extended to study the key characteristics of different pric-ing schemes (i.e., toll design) from three party perspectives: the government, whose objective is to maximizesocial welfare; the private investors, whose objective is profit maximization; and the road users, whose objec-tive is to minimize the spatial equity. In addition, policy and regulation are typically imposed by the govern-ment to ensure that the BOT project satisfies certain requirements. A project evaluation framework isdeveloped to assess the effects of government policy and regulation on the optimal toll level, project outcomes,and financial viability of the BOT project. A case study of the Ban Pong–Kanchanaburi Motorway (BKM)project in Thailand is used to examine seven cases of the BOT road pricing problem: (1) BOT based on

A. Chen, K. Subprasom / Transportation Research Part A 41 (2007) 537–558 539

individual party perspective without regulation, (2) BOT with price control regulation, (3) BOT with equityregulation, (4) BOT with construction cost subsidy, (5) BOT with concession period extension, (6) BOT withconstruction cost subsidy and concession period extension, and (7) BOT with multiple objectives.

The remainder of this paper is organized as follows. Section 2 presents a project evaluation framework forassessing project performances, financial feasibility indices, and effects of government policy and regulation onthe BOT project. Section 3 provides an overview of the BKM project in Thailand. Numerical results of theseven cases are provided in Section 4. Section 5 provides some concluding remarks. An Appendix is also pro-vided for readers interested in the mathematical formulations for BOT road pricing.

2. Project evaluation framework

The BOT project represents a major capital investment for which project performances and financial anal-ysis must be carried out during the planning stages. In this paper, the project performances refer to the objec-tives from different party perspectives that include profit, social welfare, and equality of benefits to road users.The goal of the financial analysis is to evaluate the financial viability of the BOT project. Regardless of thepublic’s needs, a BOT project will not be financed unless the financial position of the project is sufficient toattract investment from the private investors. For this reason, the project performances and financial evalu-ation are the cornerstone of the entire feasibility analysis of the BOT project.

The methodology used for the project performances and financial feasibility evaluation is shown in Fig. 1and can be summarized as follows.

Step 1. A BOT road pricing problem with one or more objectives under demand uncertainty, formulated as astochastic bi-level programming problem, is solved using the simulation-based genetic algorithm pro-cedures developed by Chen et al. (2003, 2006). This step determines the optimal toll based on differentroad pricing models, while considering the travel choice of road users (see Appendix for the details ofvarious BOT road pricing models and mathematical formulations).

Pricing and annual trafficcharacteristics

Investment costs and annualoperating expenses

Project performances

Cash flow

Financial indicesand project performance

Stop

Consider the policiesImpose the regulations

not feasible with

present configuration

feasible but project

performances are not

Financial indices and projectperformances are satisfied

Stochastic bi-level programming

Simulation-optimization

Financial indices are

satisfied

Financial indices are

Step 1 Step 2

Step 3

Step 4 Step 5

Fig. 1. Project performances and financial feasibility evaluation framework.


Step 2. From the annual traffic characteristics and cost of the BOT project, the project performances and thecash flow are determined.

Step 3. Project performances (i.e., profit, social welfare, and spatial equity) and three criteria for financialevaluation (i.e., the net present value (NPV), the internal rate of return (IRR), and the breakeven year)are all considered in the decision process.

Step 4. If the project is financially feasible but other project performances (e.g., equity) are not satisfied, theregulation or constraint on these outcomes may be imposed so that the toll structure is modified tosatisfy certain requirements.

Step 5. If the project is not financially feasible with the present configuration, certain policies from the gov-ernment (e.g., construction cost subsidy, concession period extension, etc.) may be considered to mod-ify the project cash flow in order to make the project financially viable. The process of projectevaluation is repeated until the financial indices and project performances are satisfied.

2.1. Project performances

Project performances refer to the objectives (goals) of the decision makers from different party viewpoints.In this paper, three different perspectives are considered that include the private investors, the government,and the road users. Three pricing strategies from different perspectives are summarized in Table 1.

2.1.1. Profit

For the private investors, the main concern is profit when determining the viability of a BOT project. Theannual profit for a given realization of demand uncertainty (e) is defined as the difference between revenue andcost.

TableComp

Pricing

ProfitSocialInequi

pnðx; zðx; eÞÞ ¼ wn � Cn; ð1Þ
where wn is annual revenue in year n; and Cn is annual cost in year n. The annual revenue of a BOT project isthe number of road users patronizing the BOT roads ðvn
aðx; eÞÞ multiplied by the toll charge ðxnaÞ:

wnðx; zðx; eÞÞ ¼Xa2A

nvnaðx; eÞxn

a; ð2Þ

where n is parameter that transforms the hourly link flow into annual link flow; A is set of BOT links. The costof a BOT project consists of the construction cost and maintenance-operating cost (see Section 3.2 for a sum-mary of the BKM project costs).

2.1.2. Social welfare

For the government, the main concern of a BOT project is the benefit defined in terms of social welfareadded to the society. The social welfare for a given realization of demand uncertainty (e) is defined as the dif-ference between consumer surplus and cost of the BOT project (Yang and Meng, 2000):

Snðx; zðx; eÞÞ ¼ #n � Cn: ð3Þ
The annual consumer surplus in monetary terms is calculated as:
#nðx; zðx; eÞÞ ¼ nb

Xw2W

Z dnw

0

D�1w ðxÞdx�

Xa2A

vnaðx; eÞtn

aðvnaðx; eÞÞ

( ); ð4Þ

1arison of three pricing strategies

strategies Perspectives Objectives

maximization Private investors Maximize expected profitwelfare maximization Government Maximize expected social welfarety minimization Road users Minimize expected Gini coefficient


where b is parameter that transforms toll into equivalent time value; A is set of links; W is set of origin–des-tination (O–D) pairs; D�1

w ð:Þ is the inverse demand function of O–D pair w; dnw is demand between O–D pair w

in year n; and tna is travel time on link a in year n. The consumer surplus measures the difference between what

consumers (road users) are willing to pay for travel and what they actually pay. The first term in Eq. (4) is thetotal travel cost that the road users of demand ðdn

wÞ are willing to pay whereas the second term is the totaltravel cost that they actually pay. A higher social welfare implies a better system performance for the society.In this study, we are interested in the social welfare improvement so the net social welfare is measured by com-paring the case of implementing the BOT project with the do-nothing case. It should be noted that the objec-tive function of the social welfare maximization is purely related to economic efficiency and does not includethe other benefits (e.g., environment improvement).

2.1.3. Inequality

For the road users, the main concern is the spatial equity of benefit distribution of the BOT project. Thespatial equity impact can be described as the distribution of the benefits and costs of the pricing scheme acrossthe road users traveling from different O–D pairs in the network. If the pricing scheme benefits only a smallgroup of road users from some areas, while the rest of the population experiences a decline in benefits, thepricing scheme is considered inequitable.

In this paper, the spatial distribution of the benefits and costs expressed in terms of consumer surplusimprovement of the BOT project is considered. The Gini coefficient that is commonly used as an incomeinequality measure is applied to evaluate the spatial equity impact of the BOT project development (Lorenz,1905). The annual Gini coefficient is calculated as:

/nðx; zðx; eÞÞ ¼PW

i¼1

PWj¼1dn

i ðx; eÞdnj ðx; eÞj#̂n

i ðx; eÞ � #̂nj ðx; eÞj

2ðdnðx; eÞÞ2 �#nðx; eÞ; ð5Þ

where W is the total number of O–D pairs; dni is travel demand of O–D pair i without the BOT project for year

n, dn isP

idni is total travel demand in the whole network for year n; #̂n

i is consumer surplus improvement inyear n for O–D pair i; and �#n is average consumer surplus improvement in year n. The value of the Gini coef-ficient is between 0 and 1; the lower the Gini coefficient, the more equitable the BOT scheme.

2.2. Financial feasibility indices

Financial feasibility for the BOT project development is based on the discounted cash flow (DCF) model,which is one of the widely used techniques for financial evaluation. The DCF model brings together all thecash flow profiles of a project over the planning horizon (adjusted for time value of money), and combinesthem into a measure of NPV, IRR, and breakeven year.

2.2.1. Net present value

When an investment is made, the decision makers look forward to gaining benefits over the planning hori-zon against what might be gained if the money was invested elsewhere. The investor’s required rate of return(RRR) for a capital investment is selected to reflect this opportunity cost of capital, and it is used to discountthe estimated future cash flow to the present time. The net present value is the discounted value of the netreturn at the end of the planning horizon above what might have been gained by investing elsewhere at theRRR. In other words, it is the difference between the present values of the benefits (revenues) minus the pres-ent value of the costs of a project. In general, if the BOT project has a NPV greater than or equal to zero, it isconsidered feasible. If the NPV is greater than zero, the proposed project will earn a return on the investmentgreater than the RRR used as a discount rate.

2.2.2. Internal rate of return

The IRR has been proposed as an index of the desirability of project (Hendrickson and Au, 1989). In gen-eral, the higher the rate, the better the project. By definition, it is the discount rate at which the net presentvalue of benefits equals the net present value of costs. This method is usually applied by comparing the


RRR to the IRR values. The IRR rule is to accept a project if its IRR > RRR and to reject a project if itsIRR 6 RRR.

2.2.3. Breakeven year

The breakeven year represents the amount of time that it takes for the project revenue to recover its initialcost. The use of the breakeven year as a decision rule specifies that the BOT project with a breakeven year lessthan a specified number of years should be accepted. When comparing among the pricing strategies of theBOT project, the scheme with the quickest payback is preferred.

3. The Ban Pong–Kanchanaburi Motorway (BKM) case study

3.1. General background and traffic characteristics

In 1996, the Thai government reviewed the results of the Motorway Study conducted by the Japan Inter-national Cooperation Agency (JICA) for the preparation of a master plan for the construction of intercitymotorways in Thailand. At present, some of the intercity motorways have been constructed and openedfor service such as the Bangkok–Chonburi motorway and the Eastern Outer Bangkok Ring Road. Due tothe present constraint of the national budget, the Thai government has implemented an important policyon investment in the transportation infrastructure. The policy states that the private investors are invitedto participate in the transportation infrastructure investment under the promotion and support of the govern-ment, so that the government’s investment burden can be relieved. Consequently, part of the budget can beused for development in other parts of the country.

The BKM project, developed under the BOT scheme, has a total length of approximately 45 km, startingfrom the City of Ban Pong, Ratchaburi Province and extending to Kanchanaburi Province, which already hasa detailed design as shown on the project location map in Fig. 2. A four-lane highway (two on each direction)with a capacity of 1900 PCU/h/lane will be built to connect the City of Ban Pong and the City of Tha Maka,and from the City of Tha Maka to the Kanchanaburi Province. The BKM project is one of the high prioritymotorways in the master plan. Once its construction is completed, it will provide another option for trafficdistribution from Bangkok and other regions to the west and the south of Thailand and the neighboring coun-

Fig. 2. The BKM project location map.


tries. This will also distribute the development to the region including time and cost savings for overall travelof road users.

The design years for the analysis of the BKM project consist of 2009, 2019, 2029, and 2034. As shown inFig. 3, the highway network is comprised of 17 nodes, 48 links, and 156 O–D pairs. The BKM project will bebuilt to connect node 1 (Ban Pong) and node 3 (Tha Maka), and from node 3 (Tha Maka) to node 6 (Kan-chanaburi). The entry–exit points are located at nodes 1, 3, and 6. For the traffic demand pattern, the baseyear of 2000 is used for developing the O–D trip matrices based on the existing travel pattern (from O–D sur-veys). The traffic characteristics of the base year network have been calibrated to reflect travel characteristicsof the study area. The trip matrices in the design years of 2009, 2019, 2029, and 2034, which are based on thebase year trip matrix, have been projected according to the projection of socioeconomic planning parameters.The link characteristics and the trip matrices of the design years can be referred to Department of Highways(2001) and Subprasom (2004).

3.2. Costs of BKM project and value of time of road users

The main part of the investment costs of the project is the construction cost. The construction cost of theBKM project is estimated from the quantities and the unit prices of work items using the information of theproject area. Other cost items of the BKM project include maintenance cost, land acquisition cost, operatingcost, and environment monitoring cost. The private investors are responsible for all project costs except landacquisition and compensation costs, which are the responsibility of the government. The BKM project costsare summarized in Table 2 (Department of Highways, 2001).

The data used to estimate the value of time (VOT) of road users was obtained from the National StatisticOffice of Thailand and includes per capita income, hourly income per employed person, and average hourlyincome in the study area. It is assumed that the VOT for purposes other than business is 40% of the passen-ger’s income, while the VOT is assumed to be equal to the passenger’s income for business trips. The resultingVOT in the year 2000’s value are as follows: 64.4 Baht per PCU-hour for 2000, 95.4 Baht per PCU-hour for2009, 147.6 Baht per PCU-hour for 2019, 213.9 Baht per PCU-hour for 2029, and 302.3 Baht per PCU-hourfor 2039 (Department of Highways, 2001).

Ratchaburi

8

7

1

2

3

4

5

6

9

131211

10 17 16 15

14

1

2

3

4

5

6

7

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33

34 35 36

37

38

39

40

41

42

43

44

4546

47

48 KamphaengSa en

Nakorn Pathom

PhanomThuan

Kanchanaburi

Ban Pong

Tha Maka

Tha Muang

BKM Project

BKM Project

Fig. 3. Network topology of the BKM project.

Table 2Summary of the BKM project costs

Project cost Million Baht Million $US

Construction cost 5541.92 138.55Detailed design cost 96.98 2.42Construction supervision cost 96.98 2.42Land compensation cost 569.82 14.25Property compensation cost 49.71 1.24Tree compensation cost 2.24 0.06Setting out right of way cost 14.70 0.37Routine maintenance cost 24.70 0.62Periodic maintenance cost (7th year and 22nd year) 153.38 3.83Periodic maintenance cost (15th year) 190.30 4.76Project operating cost 45.20 1.13Environmental mitigation and monitoring cost 38.33 0.96

Note: 40 Baht equals 1 $US.


3.3. Problem setting

The pricing models for the BKM project are conducted for 30 years (concession period). The basic assump-tion for the analysis period is a 30-year term, starting in the beginning of 2005, with a design and constructionperiod of 4 years, bringing it to the end of 2008. The service to road users is available in the beginning of 2009through the end of 2034. A 12% discount rate that the Department of Highways uses in their project evalu-ation is adopted for this case study.

A negative exponential demand function is used for the annual O–D travel demand of the BKM projectcase study:

dnwðeÞ ¼ �dn

wðeÞ expð�ccnwÞ; 8 w 2 W ; n 2 N ; ð6Þ

where dnwðeÞ is random travel demand between O–D pair w in year n; �dn

wðeÞ is random potential demand be-tween O–D pair w in year n; cn

w is average travel time, which includes the equivalent time of toll, for all trav-elers between O–D pair w in year n; and c is a scaling parameter reflective of the sensitivity of demand to thefull trip price. The potential O–D demands of design years are given in Department of Highways (2001) andSubprasom (2004). The value of c is set to be 0.95 for all cases.

To handle travel demand uncertainty in the BOT network design problem, a stochastic simulation is used tosimulate the uncertainty of O–D demands based on a probability distribution function (pdf) with pre-definedmean and variance. In this study, the Latin Hypercube Sampling technique is used to generate random O–Ddemand variates according to a predefined pdf (e.g., normal distribution). The potential demand is chosen asthe only key exogenous input variable to reflect the uncertainty of travel demand. Random samples of thepotential demand for each O–D pair w in year n can be generated according to the following equation:

�dnwðeÞ ¼Meanð�dn

wÞ � Zrnw; ð7Þ

where Meanð�dnwÞ is mean value of potential demand between O–D pair w in year n; Z is random variable gen-

erated from N (0, 1); and rnw is standard deviation of potential demand between O–D pair w in year n.

The link travel time function used in the traffic assignment problem is the standard Bureau of Public Road(BPR) function given below.

tnaðvn

aÞ ¼ t0a 1:0þ 0:15

vna

cna

� �4( )

; 8 a 2 A; n 2 N ; ð8Þ

where t0a and ca are the free-flow travel time and capacity of link a.

In the BKM case study, the following parameters are used:

a. Population size is 200.b. The maximum number of generations is 50.


c. The maximum number of samples is 500.d. Probability of crossover is 0.50.e. A uniform crossover scheme is adopted. The population is divided into two groups, arranged in descend-

ing order of fitness values. The top half will be used as parents for mating.f. Probability of mutation is 0.15.g. The lower bound and upper bounds for the toll charge are [0 Baht, 200 Baht].h. n is 2440.8 h/year (Department of Highways, 2001).i. RRR and discount rate is 12% (Department of Highways, 2001).

4. Numerical results

4.1. Optimal road pricing schemes

This section presents the numerical results of the optimal road pricing schemes for different party perspec-tives. It is expected that different road pricing schemes will yield different project performances. Since thecapacity of the BKM project is already determined (i.e., 3800 PCU/h in each direction), cost of the BOT pro-ject is fixed. Therefore, the term ‘‘performances’’ refer to the revenue, consumer surplus, and user’s benefitinequality (Gini coefficient). Table 3 presents the optimal toll charge corresponding to the expected perfor-mances for each design year. The numbers in brackets indicate the standard deviation of project performances.It can be seen that the three objectives are conflict with each other and the tradeoffs among the different objec-tive can be observed.

The key characteristics of each pricing scheme are different. It is found that the pricing structure of profitmaximization greatly depends on the levels of travel demand of each design year. From Table 3, a relationshipexists between the toll level and the level of revenue, in which a higher toll charge generates higher net revenue.The profit maximization scheme is found to impose the highest toll charge when compared with the other twoschemes. Obviously, this characteristic of the pricing structure will not be suitable for the social welfare max-imization scheme since it will overprice the trips. This is verified by the fact that the profit maximizationscheme generated a lower consumer surplus. For the social welfare maximization scheme, the level of demandis an important factor. The optimal toll charge of this scheme must balance the induced and suppresseddemands. If the toll charge is set too low, it may cause an increase in travel demand; therefore, traffic condi-tions on the BOT links may reach a highly congested level that result in a decrease in consumer surplus.

Table 3Results of optimal pricing schemes for different party perspectives

Pricing strategy Toll Revenue Consumer surplus Gini(Baht) (million Baht) (million Baht)

Year 2009

Profit maximization 7.95 338.08 [3.34] 614.83 [5.94] 0.642 [0.095]Welfare maximization 6.75 259.24 [2.05] 630.75 [6.00] 0.638 [0.088]Inequality minimization 1.50 58.06 [0.92] 493.92 [4.87] 0.606 [0.080]

Year 2019

Profit maximization 51.66 895.58 [151.74] 1,270.96 [112.37] 0.738 [0.124]Welfare maximization 12.92 454.16 [5.69] 1,671.39 [179.54] 0.431 [0.077]Inequality minimization 12.49 439.23 [5.46] 1,665.81 [175.40] 0.424 [0.071]

Year 2029

Profit maximization 111.60 2,154.77 [271.04] 2,641.31 [250.56] 0.689 [0.099]Welfare maximization 84.50 1,723.16 [219.18] 2,840.81 [279.09] 0.485 [0.078]Inequality minimization 19.17 358.03 [2.83] 1,978.66 [130.50] 0.397 [0.066]

Year 2034

Profit maximization 142.69 3,104.92 [314.77] 3,156.07 [294.07] 0.705 [0.118]Welfare maximization 105.71 2,372.89 [230.51] 3,595.03 [326.03] 0.474 [0.072]Inequality minimization 23.06 436.76 [3.27] 2,006.51 [202.05] 0.403 [0.070]


Likewise, if too many trips are suppressed, it causes a reduction in consumer surplus. The last objective is thespatial equity measured by the Gini coefficient. The equity minimization scheme tries to distribute the users’benefit of the pricing scheme equally to all road users in the network. The toll charge under equity minimiza-tion is found to be very low, even with a high travel demand level. Toll charges of the social welfare maximi-zation and profit maximization schemes may be too high for the inequality minimization scheme. These resultsreveal the typical characteristics of the social welfare maximization and equity minimization schemes. Thesocial welfare maximization scheme focuses on generating benefits for some particular O–D pairs (those thatcontribute most to the social welfare improvement). This will naturally create an equity problem due to itsunequal treatment. On the other hand, the equity minimization pricing scheme does not generate a substan-tially higher benefit for any particular O–D pair; therefore, it reduces the unequal treatment. Similar relation-ships between profit and equity are also observed.

As mentioned previously, regardless of the project performances, the BOT project will not be financedunless the financial position of the project is sufficient to attract the private investors to invest in the BOT pro-ject. Fig. 4 plots the stochastic cumulative profiles of the project’s cash flow against time over the concessionperiod for the profit maximization scheme. These plots comprise of the mean value of cumulative cash flowand the lower- and upper-bound values that give the 95 percent confident interval. The results also includethe mean and standard deviation of NPV, IRR, and breakeven year. There are three phases of interest in thisfigure. The first phase is the construction phase, which is characterized by the negative cumulative cash flowthat results from the project disbursements on construction. At the end of this phase, the expected cumulativecash flow reaches its lowest point. The second phase encompasses the operation period when benefits increaseuntil the expected cash flow line reaches the point where the expected cumulative cash flow is zero. This pointis known as the expected breakeven year, which is 14.4 years. The last phase starts at the expected breakevenyear and continues until the end of the concession period. The project cash flow in this phase goes upward dueto the increasing revenue from toll collection of the BOT project.

4.2. Price control regulation

This section analyzes the effects of price control regulation on the performances and financial feasibility ofthe BOT project. For the BKM project, the Thai government had set regulation on the pricing structure. Thebase toll rate for the opening year 2009, based on the Ministry of Transport and Communication (MOTC)

-1.00E+03

-7.50E+02

-5.00E+02

-2.50E+02

0.00E+00

2.50E+02

5.00E+02

7.50E+02

1.00E+03

1.25E+03

1.50E+03

1.75E+03

2.00E+03

2005

2007

2009

2011

2013

2015

2017

2019

2021

2023

2025

2027

2029

2031

2033

Investment Year

Cum

ulat

ive

Cas

h F

low

(m

illio

n B

aht)

Mean

Upper bound

Lower bound

95%

Mean NPV = 1058.205 million Baht

STDEV of NPV = 301.552 million Baht

Mean IRR = 11.68%

STDEV of IRR = 1.51%

Mean Breakeven Year = 14.4 years

STDEV of Breakeven Year = 1.84 years

Discount Rate (RRR)= 12.0%

beginning of operation period to breakeven year

breakeven year to the end of concession period

construction phase

Fig. 4. Stochastic cumulative cash flow profile for profit maximization.


announcement No. 19 (Department of Highways, 2001), is 30 Baht. Once the initial toll rate has been estab-lished, toll rates are supposed to increase every 3 years at an annual rate of 3%. Table 4 presents the effects ofprice control regulation on the corresponding expected project performances in each design year. In addition,the percent differences in project performances compared with the optimal pricing scheme without regulationare also provided.

Under the profit maximization strategy, toll charges in the design years 2019, 2029, and 2034 with pricecontrol regulation are lower than the optimal pricing scheme without regulation. In the opening year 2009,however, the toll charge of regulated pricing is higher than non-regulated pricing. Since the demand of year2009 is low, it is not necessary to impose a high toll. A toll charge of 30 Baht is found to be too high for theopening year and could cause some road users to choose the free roads instead of the tolled roads. From thegovernment perspective, the optimal toll charge of this scheme must balance the induced and suppresseddemands. If the toll charge is too low, it may cause an increase in travel demand; and traffic conditions onthe BOT roads may reach a highly congested level that result in a decrease in consumer surplus. Likewise,if too many trips are suppressed, it also causes a reduction in consumer surplus. From Table 4, it is found thatthe pricing structure in the years 2009 and 2019 are above the optimal pricing rate. On the other hand, tollcharges are below the optimal pricing for the years 2029 and 2034. For the inequality minimization scheme,the concept of minimizing spatial equity is to encourage a pricing scheme with well distributed benefits to theroad users of different areas. It can be observed that toll charges of regulated pricings of all design years arehigher than the optimal pricing scheme. The price control regulation results in an increase of the expected Ginicoefficient that brings more inequality of benefits to the road users traveling from different O–D pairs.

Fig. 5 shows the effects of price control regulation on the financial feasibility of the private investors. Notethat the mean and standard deviation of the financial indicators in the figure are for the case with price controlregulation. As observed, the cumulative cash flow under price control regulation is inferior compared to thenon-regulated case. With price control regulation, it yields an expected IRR of 6.40% with a standard devia-tion of 1.92%. As noted earlier, the RRR of the BKM project is 12%. Based on the investment rule, price con-trol regulation causes the BKM project to be infeasible in terms of the expected IRR. The private investors arenot satisfied with this condition; therefore, the Thai government may need to propose other policies in orderfor the financial status of private investors to become feasible.

4.3. Equity regulation

Apart from the optimal pricing scheme of each perspective, several outcome constraints may be imposed inthe BOT project development. Here outcome constraints refer to the constraints on the project performancesof the BOT scheme (e.g., consumer surplus, revenue, or equity). To illustrate the effects of the outcome

Table 4Effects of price control regulation on project performances

Design year 2009 2019 2029 2034

Toll charge of pricing regulation (Baht) 30 40.32 54.18 62.81

Profit maximization

Percent difference in toll charge +277.36% �21.95% �51.45% �55.98%Expected annual revenue (million Baht) 154.63 858.08 1027.18 1302.20Percent difference in revenue �54.26% �4.20% �52.33% �58.06%

Social welfare maximization

Percent difference in toll charge +344.44% +212.07% �35.88% �40.58%Expected annual consumer surplus (million Baht) 352.22 1,272.53 1,997.25 2,600.46Percent difference in consumer surplus �44.16% �23.86% �29.69% �27.67%

Inequality minimization

Percent difference in toll charge +900% +222.82% +182.63% +172.38%Expected Gini 0.735 0.642 0.595 0.63Percent difference in Gini +21.29% +47.93% +49.87% +56.33%

-1.00E+03

-5.00E+02

0.00E+00

5.00E+02

1.00E+03

1.50E+03

2.00E+03

2005

2007

2009

2011

2013

2015

2017

2019

2021

2023

2025

2027

2029

2031

2033

Investment Year

Cum

ulat

ive

Cas

hFlo

w (

mill

ion

Bah

t)

95%

95%

Without pricing regulation

With pricing regulation

Mean NPV = 412.819 million BahtSTDEV of NPV = 83.592 million BahtMean IRR = 6.40%STDEV of IRR = 1.92%Mean Breakeven Year = 17.0 yearsSTDEV of Breakeven Year = 1.68 yearsDiscount Rate (RRR) = 12.0%

Fig. 5. Effects of price control regulation on stochastic cumulative cash flow profile for profit maximization.


constraints, the optimal pricing of the profit maximization scheme with equity constraints is specifically con-sidered. It is assumed that the upper bound of the expected Gini coefficient is set at 0.500 for all design years,except year 2009, which is set at 0.606 (the best value obtained from the inequality minimization scheme fromTable 3).

Fig. 6 shows the effects of equity regulation on the profit maximization scheme. The solutions are plottedaccording to the expected value of the project outcomes. The arrows correspond to the direction in which eachof the solutions moves to when the equity constraints are imposed in the profit maximization scheme. Thenumbers in parentheses indicate the toll charges for each design year. It is found that toll charge decreases

Fig. 6. Effects of equity regulation on profit maximization.


from 13% to 82%, expected revenue decreases from 11% to 83%, expected consumer surplus decreases from4% to 25%, and expected Gini coefficient decreases (or improves) from 5% to 39% when the equity constraintsare imposed in the profit maximization scheme.

The goal of equity regulation is to encourage a pricing structure with well distributed benefits for roadusers. Fig. 7 shows the users’ benefit in terms of consumer surplus improvements for each O–D pair underthe profit maximization scheme for year 2019. As shown in the figure, the profit maximization scheme withoutequity regulation (Gini = 0.738) generates non-uniform benefits to road users traveling from different O–Dpairs (i.e., some gain a lot of benefits while some are worse off as a result of the BOT project), whereas thebenefits with equity regulation (Gini = 0.448) are more uniform (i.e., less O–D pairs are worse off). Clearly,equity regulation on the profit maximization scheme causes the toll charge to be lower and tends to better dis-tribute the costs and benefits to road users traveling from different O–D pairs.

The effects of equity regulation on the financial status of the private investors are shown in Fig. 8. It is evi-dent that the cumulative cash flow profile with equity regulation is inferior compared to that of the profit max-imization scheme without equity regulation. Under the equity regulation, the financial status of privateinvestors is not feasible since the cumulative cash flow profile falls below the zero-profit region. The expectedNPV is 36.601 million Baht with a standard deviation of 119.16 million Baht (a decrease of 96.54%); theexpected IRR is 0.210% with a standard deviation of 0.608% (a decrease of 98.20%); and the expected break-even year is 29.2 years with a standard deviation of 1.94 years (an increase of 100.02%).

4.4. Construction cost subsidy

Since imposing regulations on pricing structure or equity makes the project financially infeasible, the gov-ernment may propose policies to support the private investors in order for them to obtain a higher profit andfor the financial performance to become feasible. One of the government support policies is construction costsubsidy. In this section, it is assumed that the government continues imposing the price control regulation.Since the financial performance of the private investors in terms of the expected IRR is not feasible underthe price control regulation, the government needs to quantify the amount of construction cost subsidy neededto make the expected IRR reach the minimum requirement (i.e., the RRR of 12.0%). It is necessary to pointout that government support on the construction cost does not affect the pricing structure because the tollcharge is fixed by the price control regulation in each design year. Likewise, it only affects the financial per-formances of the private investors.

-3.00E+05

-2.00E+05

-1.00E+05

0.00E+00

1.00E+05

2.00E+05

3.00E+05

4.00E+05

5.00E+05

6.00E+05

0 20 40 60 80 100 120 140 160

O-D Number

Con

sum

er S

urpl

us (

Bah

t)

Without equity regulation

With equity regulation

Fig. 7. Effects of equity regulation on O–D benefits for profit maximization in year 2019.

-1.00E+03

-5.00E+02

0.00E+00

5.00E+02

1.00E+03

1.50E+03

2.00E+03

2005

2007

2009

2011

2013

2015

2017

2019

2021

2023

2025

2027

2029

2031

2033

Investment Year

Cum

ulat

ive

Cas

h F

low

(m

illio

n B

aht)

Without equity constraint

With equity constraint

95%

95%


Fig. 8. Effects of equity regulation on stochastic cumulative cash flow profile for profit maximization.


It is found that the government must subsidize at least 55% of the construction cost to make the expectedIRR reach 12%. Fig. 9 shows the stochastic cumulative cash flow profile for profit maximization with andwithout government subsidy. With this amount of construction cost subsidy, the expected IRR increases to12.18%. If the government subsidy is lower than 55% of the construction cost, the expected IRR is not finan-cially viable. However, if the subsidy is higher than 55%, the expected IRR will exceed the minimum require-ment and the private investors will benefit from the government subsidy.

4.5. Concession period extension

An essential element of a BOT project is the duration of the concession period. The concession period isnormally set to allow the private investors sufficient time to build the project and to operate it to make a

-1.00E+03

-5.00E+02

0.00E+00

5.00E+02

1.00E+03

1.50E+03

2.00E+03

2005

2007

2009

2011

2013

2015

2017

2019

2021

2023

2025

2027

2029

2031

2033

Investment Year

Cum

ulat

ive

Cas

hFlo

w (

mill

ion

Bah

t)


With government subsidy:55% grant of construction cost

Without government subsidy

95%

Fig. 9. Effects of construction cost subsidy on stochastic cumulative cash flow profile for profit maximization.


reasonable profit before returning the infrastructure to the government. If the concession period is set toolong, the potential profits beyond that period must be so heavily discounted to the present value that theyare become negligible. A short concession period, however, may result in a high toll charge for the users(i.e., to recover the cost of the project and earn some profit), which could be unacceptable. In this section,we assume that the Thai government regulates the toll charge of the BKM project to protect the road usersfrom the monopoly power of private investors. Since the expected IRR of the private investors is not feasibleunder the price control regulation, the government may propose a policy to allow the duration of the conces-sion period to be extended until the expected IRR of the BKM project reaches 12%. Because the analysis ofthis policy approach is beyond 30 years, certain assumptions are made. It is assumed that the O–D demandafter year 2029 will increase 2.2% per year. The remaining costs of the BKM project after year 2034 include theroutine maintenance cost and project operating cost. The value of time of the road users after year 2034increases 3.4% annually. The toll charge under the price control regulation after year 2034 increases every3 years at 3% annually.

From the results, the duration of the concession period must end at year 2067. Fig. 10 shows the relationshipbetween the investment year and the expected IRR. The expected IRR increases significantly from the years 2034to 2054. After that, the expected IRR remains fairly steady due to the effects of the discount factor used to dis-count the estimated future cash flow to the present time. If the investment period is too long, profits of the BKMproject are so heavily discounted to the present value that they are generally small and disregarded. At year 2067,however, the expected IRR of the BKM project is exactly 12% with a standard deviation of 1.01%. Therefore, thegovernment must extend the concession period from 30 years to 63 years to fulfill the IRR obligation.

In summary, the concession period extension policy can ensure that the private investors obtain therequired rate of return (i.e., 12%). The advantage of this approach is that it is not necessary for the govern-ment to provide a direct subsidy to the private investors in which the concession period extension policy issuitable, especially when the government has a limited budget. On the contrary, from the private investors’viewpoint, the BOT project is not an attractive investment if the required investment period is too long toaccomplish the economic goal.

4.6. Combination of construction cost subsidy and concession period extension

It is evident that the price control regulation causes the financial infeasibility for the private investors (i.e.,expected IRR < 12%). The possible government-support policies on construction cost subsidy and concession

0%

2%

4%

6%

8%

10%

12%

14%

16%

2034 2039 2044 2049 2054 2059 2064 2069

IRR

(%

)

95%

2067

Total concession period = 63 years

Extension Period

Fig. 10. Effects of concession period extension to the project IRR.


period extension could make the private investors to obtain a higher profit as well as the financial performanceof their cash flow to become feasible. However, these policies are difficult to implement; the 55% constructioncost subsidy is considered an impossible budget for the government to accommodate. For the concession per-iod extension policy, the private investors feel that the BOT project is not an attractive investment, because theperiod required to accomplish the return of investment is too lengthy.

The government, therefore, can do better by combining construction cost subsidy and concession periodextension to make the policy feasible. It is assumed that the government allows the concession period to beextended by 10 years, in which the concession period will end at year 2044 with a total concession periodof 40 years. With this condition, the government needs to quantify the amount of construction cost subsidyrequired to make the expected IRR reaches 12%. It is found that the government only needs to subsidize about13% of the construction cost to make the expected IRR become 12%.

The extension period of 10 years (for a total concession period of 40 years) is reasonable for the privateinvestors and 13% of construction cost subsidy is an attainable budget for the government. This is one ofthe feasible policies; however, it is not the best policy. In reality, there are several alternative policies and pol-icy combinations that the government can choose to implement in the BOT scheme.

4.7. Three-objective road pricing model

In the previous sections, results of various single-objective road pricing models with and without con-straints (i.e., price control regulation and equity regulation) have been presented. The results revealed thatthe optimal toll charge obtained from each road pricing scheme is different, resulting in conflict among thethree objectives. A toll charge that is optimal for a certain party may not be appropriate for the other parties.In this section, we present results that simultaneously optimize all three objectives: profit maximization for theprivate investors, social welfare maximization for the government, and inequality minimization for the roadusers. In principle, multi-objective optimization problems are very different from single-objective optimizationproblems. This is because solving multi-objective optimization problems often requires a set of Pareto (or non-dominated) solutions, not just a single best solution as in the single-objective problems. Here we used the sim-

Fig. 11. Pareto solutions of three-objective road pricing model for year 2019.


ulation-based multi-objective genetic algorithm (SMOGA) procedure developed by Chen et al. (2006) toexplicitly solve the three-objective road pricing model described in Appendix A.4. Fig. 11 presents the setof Pareto solutions for year 2019. There are 31 distinct Pareto solutions. The revenue ranges from 102.258million Baht to 779.008 million Baht; the consumer surplus ranges from 472.940 million Baht to 1,422.239million Baht; and the Gini coefficient ranges from 0.436 to 0.624. These 31 Pareto solutions can be groupedinto two different clusters based on their objective values to facilitate the decision makers in choosing the bestcompromised pricing scheme. Group A contains the solutions with a higher preference on the inequality min-imization objective. Most of the solutions in this group have a lower value in revenue and consumer surplusbut with a better Gini coefficient value (more equitable). On the other hand, solutions in group B are mostlyclustered in the area that has a higher value in revenue and consumer surplus but with a poorer Gini coefficientvalue (less equitable).

Fig. 12 further shows the pair-wise comparison scatter plots of the 31 Pareto solutions. These plots allowone to visualize the tradeoffs between each pair of objectives and to understand the results of a multi-objectiveoptimization better. As can be seen from the figure, the number of solutions on the Pareto front for each pairof objectives is different. There are 8 solutions on the social welfare-profit frontier, 12 solutions on the socialwelfare-equity frontier, and 4 solutions on the profit-equity frontier. These plots clearly reveal that the threeobjectives are conflict with each other. A gain in one objective must be inevitably traded off by a loss of at leastone of the other objectives. There exists no single best solution that can accomplish all three objectives simul-taneously. Using the Pareto solutions obtained from a multi-objective optimization model, the decision mak-ers can assess the tradeoffs among different pairs of objectives, analyze the solutions in terms of the minimumrequirement for project performances, and/or introduce additional criteria to select the most preferred solu-tion for implementation. In addition, the decision makers can use different visualization methods to commu-nicate with the involved parties easily on the issue of the conflict between the requirements from differentgroups.

Fig. 12. Pair-wise comparison of Pareto solutions.


5. Concluding remarks

In this study, we presented various road pricing models for determining the optimal toll charge and a pro-ject evaluation framework for assessing the related project outcomes and financial indices of a BOT schemeunder demand uncertainty from different parties involved. Specifically, three different party perspectives areconsidered: (1) the government is concerned about the social welfare improvement (benefit to the society) gen-erated from the BOT project development; (2) the private investors are more concerned about the profit gen-erated from the BOT project investment; and (3) the road users may prefer the BOT project that equallydistributes the benefits and costs of the BOT project across the road users from different areas in a network.Seven cases were analyzed: (1) BOT based on individual party perspective without regulation, (2) BOT withprice control regulation, (3) BOT with equity regulation, (4) BOT with construction cost subsidy, (5) BOTwith concession period extension, (6) BOT with construction cost subsidy and concession period extension,and (7) BOT with multiple objectives from all party perspectives.

The results showed that different road pricing schemes based on different objectives yielded different tolldesigns, project performances, and financial outcomes. In case (1), it was found that the profit maximizationtolls are the highest, the equity minimization tolls are the lowest, and the social welfare maximization tolls arein the middle. When government regulation is imposed as in cases (2) and (3) in the profit maximizationscheme, profit is decreased while social welfare and equity are increased from the non-regulated case. Thisresult generally agrees with the purpose of imposing regulation, which is to ensure the benefit of the society.However, it also causes the financial status of the project to be infeasible. In order to make the project finan-cially viable, cases (4), (5), and (6) examined various policies to support the private investors. It was found thatthe combination of construction cost subsidy and concession period extension is a viable policy. Case (7)examined the tradeoffs among the three objectives via a multi-objective optimization method. The resultsindeed revealed that the three objectives are conflict with each other. In summary, the analysis revealed someinteresting results on the effects of regulation and policy of private toll roads in a BOT scheme that explicitlyaccounts for demand uncertainty when considering the tradeoffs among the objectives of the government, theprivate investors, and the road users.

Acknowledgements

This research is partially supported by the National Science Foundation (CMS-0134161). The secondauthor also would like to acknowledge the financial support from the Royal Thai Government Scholarship.

Appendix A

This appendix describes the BOT road pricing models and formulations for determining the optimal tollcharge based on three party perspectives under demand uncertainty. Notation is provided first, followed bythe general stochastic bi-level programming formulation, the lower-level program, which is the traffic equilib-rium problem with elastic demand, and various upper-level programs that represent the objectives of differentparties.

A.1. Notation

For convenience, the sets, variables, and parameters used throughout the paper are defined as follows:

Sets

A set of linksA set of BOT linksW set of origin–destination (O–D) pairsRw set of routes between O–D pair w


R set of all routes in the network, andN set of design years

Variables

x vector of tolls for all BOT linksz vector of decision variables of the lower-level programxn

a toll charge on BOT link a in year n

vna hourly flow on link a in year n

tna travel time on link a in year n

dnw travel demand between O–D pair w in year n

D�1w ð�Þ inverse of the demand function associate with O–D pair w

f wr flow on route r between O–D pair w

dwar 1 if route r between O–D pair w uses link a, and 0 otherwise

cnw average travel time, which includes equivalent time of toll for all travelers between O–D pair w in year

n

Cn costs of BOT project in year n

E[pn(x,z(x,e))] expected profit in year n

pn(x,z(x,e)) profit of realization e in year n

wn(x,z(x,e)) revenue of realization e in year n

E[Sn(x,z(x,e))] expected social welfare in year n

Sn(x,z(x,e)) social welfare of realization e in year n

#n(x,z(x,e)) consumer surplus of realization e in year n

E[/n(x,z(x,e))] expected Gini coefficient in year n, and/n(x,z(x,e)) Gini coefficient of realization e in year n

Parametersc scaling parameter that reflects the sensitivity of demand to the full trip priceb parameter that transforms toll into equivalent time valuen parameter that transforms the hourly link flow into annual link flowt0a free-flow travel time of link a

e vector of random variables�dn

w potential demand between O–D pair w in year n

rnw standard deviation of potential demand between O–D pair w in year n, and

Z random variable generated from N (0,1)

A.2. Stochastic bi-level mathematical programming formulation

Network design problems in transportation planning and management can be posed as a Stackelberg game(Fisk, 1984). In this game, the traffic manager (or leader) is assumed to have knowledge on how the road users(or follower) would respond to a given strategy (i.e., design variables determined by the leader). However, it isimportant to recognize that the strategy set by the leader can only influence (not control) the route choice deci-sions of the road users. In other words, the leader’s strategy and the follower’s route choices are inter-depen-dent and this interaction can be represented by a bi-level mathematical program. In addition, the leadersometimes has to make the decision under uncertainty where certain inputs are not known exactly. The generalstochastic bi-level mathematical program is formulated as follows:

minx

F ðx; zðx; eÞÞ ðA1Þ

subject to : Gðx; zðx; eÞÞ 6 0; ðA2Þ

where z(x, e) is implicitly defined for each realization e by


minz

f ðx; zðx; eÞÞ ðA3Þ

subject to : gðx; zðx; eÞÞ 6 0; ðA4Þ

where F is objective function of the upper-level program; x is decision variables of the upper-level program(i.e., toll charge); G is constraint set of the upper-level program; f is objective function of the lower-level pro-gram; z is decision variables of the lower-level program; g is constraint set of the lower-level program; and e israndom variables in the lower-level program.

In this paper, the upper-level program determines the optimal toll charge of a BOT project by optimizingone or more objectives under demand uncertainty; while the lower-level program determines the travel choiceof network users for a given toll charge with uncertain demand. In the BOT network design problem, thelower-level program is modeled as a standard user equilibrium with elastic demand (Sheffi, 1985) and canbe described as follows.

For a given toll charge strategy (x) and each realization of the random variable vector (e), the lower-levelprogram solves a traffic assignment problem with elastic demand.

mind;v2zðx;eÞ

Xa2A�A

Z va

0

taðxÞdxþXa2A

Z va

0

ðtaðxÞ þ bxaÞdx�Xw2W

Z dw

0

D�1w ðxÞdx ðA5Þ

subject to:

gðzðx; eÞÞ ¼

Pr2Rw

f wr ¼ dwðeÞ; 8 w 2 W

va ¼Pr2R

f wr dw

ra; 8 a 2 A

f wr P 0; 8 r 2 Rw;w 2 W

dw P 0; 8 w 2 W

8>>>>><>>>>>:

ðA6Þ

Eq. (A5) is the objective function, which consists of the sum of the integrals of the link performance func-tions for both free links and BOT links (first two terms) and the sum of the integrals of the inverse demandfunctions (third term). The demand function (Dw(cw)) is assumed to be dependent on the generalized travelcost (cw) between that O–D pair alone. Constraint set in Eq. (A6) represents the O–D demand conservations,the incidence relationship between link flows in terms of path flows, and non-negativity conditions of pathflows and O–D flows. The solution to the above minimization problem is (z(x, e)), which consists of a setof O–D demands (dw(x, e)) and a set of link flows (va(x, e)). Both solutions are a function of x (i.e., a vectorof toll charge) in the upper-level program, and e (i.e., random potential demands) in the lower-level program.

A.3. Single-objective road pricing models

This study analyzes the BOT network design problem from three perspectives: the private investors, thegovernment, and the road users. The objectives of different parties are expressed as follows.

A.3.1. Profit maximizationThe objective in the upper-level program for the private investors is to maximize the sum of the expected

annual profit subject to the non-negative constraint on the toll charge for each year n.

maxXn2N

E½pnðx; zðx; eÞÞ� ðA7Þ

subject to : xna P 0; 8 a 2 A; n 2 N : ðA8Þ

A.3.2. Social welfare maximization

The objective in the upper-level program for the government is to maximize the sum of the expected annualsocial welfare subject to the non-negative constraint on the toll charge for each year n.


maxXn2N

E½Snðx; zðx; eÞÞ� ðA9Þ


A.3.3. Inequality minimization

The objective function in the upper-level program for the road users is to minimize the sum of the expectedGini coefficient subject to the non-negative constraint on the toll charge for each year n.

minXn2N

E½/nðx; zðx; eÞÞ� ðA11Þ


A.4. Multi-objective road pricing model

In practice, there is no guarantee that the optimal solutions obtained from each individual objective areimplementable. A solution that is suitable for a certain party may not be appropriate for the others. Therefore,it is meaningful to simultaneously optimize profit, social welfare, and equity of the three parties involved. Theobjective function in the upper-level program for the multi-objective road pricing model is to optimize all threeobjectives subject to the non-negative constraint on the toll charge for each year n.

minx

F ðx; zðx; eÞÞ ¼

maxPn2N

E½wnðx; zðx; eÞÞ�

maxPn2N

E½#nðx; zðx; eÞÞ�

minPn2N

E½/nðx; zðx; eÞÞ�

8>>>>><>>>>>:

ðA13Þ


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Documents

Analysis of regulation and policy of private toll roads in a build-operate-transfer scheme under demand uncertainty