Mean-Variance Model for the Build-Operate-Transfer Scheme Under ...

Transportation Research Record 1857 ■ Paper No. 03-3401

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Mean-Variance Model for the Build-Operate-Transfer Scheme Under Demand Uncertainty Anthony Chen, Kitti Subprasom, and Zhaowang Ji A mean-variance model was developed for determining the optimal toll and capacity in a build-operate-transfer (BOT) roadway project subject to traffic demand uncertainty. This mean-variance model involves two objectives: maximizing mean profit and minimizing the variance (or standard deviation) of profit. The variance associated with profit is considered as a risk. Because maximizing expected profit and minimizing risk are often conflicting, there may not be a single best solution that can simultaneously optimize both objectives. Hence, it is necessary to explicitly consider this as a multiobjective problem so that a set of nondominated solutions can be generated. In this study, the optimal toll and capacity selection for the BOT problem under demand uncertainty is formulated as a special case of the stochastic network design problem. A simulation-based multiobjective genetic algorithm was developed to solve this stochastic bilevel mathematical programming formulation. Numerical results are also presented as a case study.

In developing countries where governments are under severe financial constraints to improve the transportation infrastructure system to keep up with fast-growing economies, the build-operate-transfer (BOT) approach is an attractive avenue for building new transportation infrastructures. In recent years, BOT projects have become fashionable in Southeast Asia. Some examples of BOT projects include the following: Taiwan’s high-speed rail project linking Taipei to Kaoshiung, its largest harbor and the second biggest city in Taiwan (1); five major toll automobile tunnels in Hong Kong (2); the Superhighway project connecting the booming industrial cities of the Pearl River delta in China (3); the Bangkok second-stage expressway in Thailand; the Kepong toll road in Malaysia; and toll road Highway 1 in Vietnam (4). The BOT approach is one of the public–private partnership models for transportation infrastructure development by using private funds to undertake new infrastructure facility. It involves assembling private investors (concessionaires) to finance, design, build, and operate the infrastructure; in return, the private investors receive revenue from toll charges for a certain number of years called a concession period. After the concession period is expired, the facility is returned to the government. Because of numerous financial, institutional, and political constraints to meet the growing needs of providing better transportation systems, the BOT scheme is gaining popularity and acceptance as an innovative way to finance the construction of new infrastructures even in developed countries. Recent examples in the United States include the Dulles Greenway, a 14-mi (22.4-km) toll road extension of the Dulles Airport access road to Department of Civil and Environmental Engineering, Utah State University, Logan, UT 84322-4110.

Leesburg, Virginia (5), and the 10-mi (16-km) express lanes built in the median of existing California State Route 91 (1). Although the BOT concept appears simple, there are many factors affecting the success of a BOT project. One of these factors is selection of the roadway capacity and toll charge of the BOT links (roads). From the viewpoint of private investors, the main concerns are cost and revenue (i.e., profit). Cost depends on the roadway capacity of the BOT links, while revenue is a function of the toll charge and the traffic volume that will patronize the BOT links. In a general transportation network, users may have a choice between choosing BOT roads with a toll charge or the free access route. This choice behavior introduces a complex interdependent relationship between the private investors, who decide the optimal capacity and toll charge of the BOT links to maximize profit, and the road users, who choose the routes that minimize their generalized costs. In addition, profitability of a BOT project is highly dependent on accurate market demand forecasting, which is usually highly uncertain. Hence, demand uncertainty is a great concern to the private investors. The selection of roadway capacity and toll charge of the BOT links combined with uncertain traffic forecasts makes the investment highly uncertain and risky. As can be observed, the optimal selection of capacity and toll in a BOT problem under demand uncertainty is quite intricate and it may not be adequate to consider it by either risk analysis or network optimization alone. Some earlier risk-based studies of the BOT problem include those of Lam and Tam (6), Malini and Raghavendra (7), and Seneviratne and Ranasinghe (8); all used Monte Carlo simulation to assess the effects of risks and uncertainties of various parameters such as construction cost, operation cost, traffic volume, and toll. The basic premise is to examine the effects of profit, internal rate of return, net present value, and so forth, by changing one parameter at a time while keeping all other parameters constant. However, risk analysis by itself does not optimize the selection of toll charge and roadway capacity of a BOT problem. For the network optimization approach, Yang and Meng (3) formulated a bilevel program to determine the optimal combination of toll capacity for a BOT network design problem. However, demand uncertainty is not considered in their bilevel programming formulation. Recently, Chen et al. (9) combined both risk analysis and network optimization into a simulation-optimization framework for optimal selection of capacity and toll in a BOT problem under demand uncertainty. Only one objective was considered in the stochastic bilevel programming formulation, which is to maximize the expected profit. A simulation-based genetic algorithm (GA) procedure was also developed to solve the stochastic version of the bilevel programming formulation (10). In this paper, a mean-variance model is developed for determining the optimal toll and capacity in a BOT roadway project subject to traffic demand uncertainty. This mean-variance model involves two

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objectives: maximizing mean profit and minimizing variance (or standard deviation) of profit. The variance associated with profit is considered as a risk. Because maximizing expected profit and minimizing risk are often conflicting, there may not be a single best solution that can simultaneously optimize both objectives. Hence, it is necessary to explicitly solve this as a bi-objective problem to enumerate a set of nondominated solutions. For this task, a distance-based method is developed and incorporated into the simulation-based GA procedure. A case study is also conducted to demonstrate the feasibility of the mean-variance model. Also the results are compared with those obtained by solving the classic weighted-sum method.

Transportation Research Record 1857

E{π[u, v(u, )]} = expected profit, and V{π[u, v(u, )]} = variance of profit.

Parameters α = parameter that transforms the capital cost of the project into unit period cost, β = parameter that transforms toll into equivalent time value, ϕ = ratio of maintenance-operating costs to the capital cost, t 0a = free-flow travel time of link a, and k = proportionality parameter to convert free-flow travel time into length.

FORMULATION This section describes the mean-variance model for optimal selection of the toll and capacity BOT problem under demand uncertainty. Notation is provided first for convenience, followed by the general stochastic bilevel programming formulation, and the mean-variance BOT model.

Notation For convenience, the sets, variables, and parameters are defined as follows:

Sets A – A W Rw R

= = = = =

set of links, set of BOT links, set of origin-destination (O-D) pairs, set of routes between O-D pair w ∈ W, and set of all routes in the network.

Stochastic Bilevel Programming Formulation Many decision-making problems for transportation planning and management can be posed as a Stackelberg game (11). In this game, the traffic manager (or leader) is assumed to have knowledge of how the users (or follower) would respond to a given strategy. However, it is important to recognize that the strategy set by the traffic manager can only influence (not control) the route choice decisions of the users. In other words, the leader’s strategy and the follower’s route choices are interdependent, and this interaction can be represented by a bilevel programming formulation. In addition to the leader–follower structure of the transportation decision-making problem, the decision sometimes has to be made under uncertainty where certain inputs are not known exactly. The general stochastic bilevel programming formulation can be stated as follows: min F [u, v(u, )] u

subject to G[u, v(u, )] ≤ 0

Variables xa ya x y  va ca ta dw σw

= = = = = = = = = =

Z Dw f wr δ war

= = = =

Ia(ya) = Cmo = Cc = π[u, v(u, )] =

(1)

(2)

v(u, ) is implicitly defined by toll charged on BOT link a, capacity on BOT link a, vector of tolls for all BOT links, vector of capacities for all BOT links, vector of random variables, flow on link a, capacity on link a, travel time on link a, expected demand between O-D pair w ∈ W, standard deviation of demand between O-D pair w ∈ W, random variable generated from N (0,1), random demand between O-D pair w ∈ W, flow on route r ∈ Rw, w ∈ W, 1 if route r between O-D pair w ∈ W uses link a and 0 otherwise, construction cost function with respect to capacity of BOT links, maintenance-operating costs for a unit period after conversion, capital cost of a BOT project for a unit period after conversion, profit (revenue-cost) of realization ,

min f [u, v()] v

(3)

subject to g[u, v()] ≤ 0

( 4)

where F u G f v g 

= = = = = = =

objective function of the upper level (i.e., decision maker), decision variables of the upper level, constraint set of the upper level, objective function of the lower level (i.e., travelers or users), decision variables of the lower level, constraint set of the lower level, and random variables in the lower level.

Mean-Variance BOT Model The mean-variance model is one of the oldest finance areas, dating back to the work of Markowitz (12). The basic assumption is that risk is measured by variance, and the decision criteria (or objectives)

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are to maximize expected return and to minimize variance. In many cases, there does not necessarily exist a best solution with respect to both objectives because of a conflict between the two objectives. A solution may be best in one objective and worst in the other objective. Therefore, there usually exists a set of solutions, called nondominated solutions or Pareto optimal solutions, which cannot be directly compared with each other. The mean-variance model in the upper level is simply to maximize the expected profit and to minimize the variance of profit subject to the nonnegative constraints on the toll-capacity combination on the BOT links. max E{π[u, v(u, )]} min F[u, v(u, )] =  u min V {π[u, v(u, )]}

(5)

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links. For each realization of the random variable vector , v(u, ) is said to be feasible if and only if w ∀r ∈ Rw , w ∈ W fr ≥ 0   ∑ f rw = dw () ∀w ∈ W v(u, ) = r ∈Rw  va = ∑ f rw δ war ∀a ∈ A  r ∈R

(12)

For each realization of the random variable vector , the lower level minimizes the following objective: min

v ∈v ( u ,  )

∑ ∫

a ∈A − A

va

0

ta (ω )dω + ∑ ∫ [ta (ω, ya ) + βxa ]dω va

a ∈A

0

(13)

subject to xa ≥ 0

ya ≥ 0

a ∈A

(6)

where u = {x, y} is a vector of toll-capacity combination in the upper level; and v(u, ) is a vector of link flows obtained by solving the lower level. Profit in the BOT project is the difference between revenue and cost. As mentioned before, revenue is a function of x (toll charge) and v(u, ) (the number of users patronizing the BOT links), while cost depends on y (capacity of BOT links). The cost of a BOT link consists of the construction cost (Cc) and maintenance-operating cost (Cmo): cost = Cc + Cmo

( 7)

The construction cost is a function of the number of lanes (or roadway capacity). Following the study by Yang and Meng (3), the construction cost function is assumed to be linear: Ia ( ya ) = kta0 ya

(8)

Other appropriate construction cost functions can also be applied. Further, the maintenance-operating cost is assumed to be proportional to the construction cost according to the parameter ϕ. Hence, the final cost function can be expressed as a function of road capacity as follows: cost = (1 + ϕ )[αIa ( ya )]

( 9)

Revenue is the number of users multiplying the toll charge: revenue = va (u, ) xa

(10)

Hence, profit of the BOT project is π[u, v(u, )] =

∑ v (u, ) x −∑ (1 + ϕ)[αI ( y )] a

a ∈A

The solution to the preceding minimization problem is a set of link flows, va(u, ), which is a link-flow vector as a function of the random variable vector  and the toll-capacity combination determined by the upper level. The lower level is a standard network equilibrium problem that can be readily solved by many efficient algorithms (13). Here the path-based gradient projection algorithm is used, which has been found to outperform the state-of-the practice algorithms, such as the Frank–Wolfe and PARTAN algorithms, and is at least as good as or superior to the state-of-the-art simplicial decomposition algorithms like disaggregate simplicial decomposition and restricted simplicial decomposition algorithms (14, 15).

a

a

a

(11)

a ∈A

Note that the preceding profit equation is a function of not only the decision variables (i.e., toll and capacity) in the upper level but also the decision variables (i.e., flow on the BOT links) of the lower level, which are random because the link flows are a function of the uncertain demand. Thus, one can take expectation and variance of the profit function to optimize the mean-variance model. The lower level is a standard network equilibrium problem that determines the equilibrium link flows for both BOT links and free

SOLUTION PROCEDURE Two issues need to be dealt with when solving the stochastic bilevel programming formulation: (a) how to handle the demand uncertainty, and (b) how to solve the bilevel program. Stochastic simulation is used to simulate the uncertainty of traffic demands based on probability distribution with predefined mean and variance. Bilevel programming problems generally are difficult to solve because evaluation of the upper-level objective function requires solving the lower-level problem. For network design problems, the lower-level problem can be considered as nonlinear constraints. This often makes the bilevel programs nonconvex. To tackle the nonconvexity issue, GA is used because it can work with continuous and discrete parameters, differentiable and nondifferentiable functions, and unimodel and multimodal functions as well as convex and nonconvex feasible regions (16 ). Previously, Yin (17 ) showed that GA can be used to solve bilevel programming formulations. Chen et al. (10) extend it to a simulationbased GA procedure for solving the BOT network design problem with demand uncertainty. However, only one objective was considered. In this study, the simulation-based GA procedure is extended to consider multiple objectives. The multiobjective optimization problem is solved with the distance-based method (18). A flow diagram describing the simulation-based multiobjective GA procedure is presented in Figure 1, and it can be summarized as follows: • Step 1. Define GA’s parameters: mutation probability, crossover probability, population size (P), maximum number of generations (Nm), and maximum number of sample sizes (Snsp). Initialize N (counter for the generation number) and a set of solutions of size P [i.e., design variables (toll and capacity of BOT links)] to be optimized. Initialize p (counter for the number of solutions).

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• • • •

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Generate initial solution Define population size, P Define maximum number of generations, Nm Define maximum number of sample sizes, Snsp

N=1

Update solution by GA operators • Reproduction • Crossover • Mutation

N=N+1

p=1 p=p+1 S=1

S=S+1

Generate random traffic demand

Calculate relative distance to all existing Pareto solutions, dl (x)

Evaluate solution Yes Collect statistical inferences (mean and variance of profit)

Compare new solution with all existing Pareto solutions, and calculate fitness value, F

S < Snsp No No

First Solution?

Update Pareto solution set

Yes Assign potential value, d1 Fitness value, F = d1

p P (population size). • Step 4. Improve all solutions via GA operators: reproduction, crossover, and mutation. Increment N = N + 1. Repeat Step 2 and Step 3 until N > Nm. • Step 5. Report the nondominated solution set. As mentioned previously, the distance-based method (18) is used to solve the multiobjective optimization problem by explicitly gen-

erating the nondominated solutions in each generation. The basic idea is to assign fitness values to each solution according to a distance measure with reference to the nondominated solutions obtained in the previous generation. The general solution procedure of the distance-based method is adapted from Osyczka and Kundu (18) and is provided as follows. • Step 3.1. The first generated solution is taken as a Pareto solution with a potential value d1, which is an arbitrarily chosen value called the starting potential value. The first generated solution has a fitness value of F, which is set to d1. • Step 3.2. For a new solution u, calculate the relative distances to all existing Pareto solutions:

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 fkl − fk (u)  ∑   fkl  k =1  q

d1 (u) =

2

l = 1, 2, . . . , m

7

where q m fk(u) fkl

Guangzhou

1

(14)

1 8

= = = =

the number of objectives, the number of Pareto solutions obtained by genetic search, the value of the kth objective of the new solution u, and the value of the kth objective for the lth Pareto solution.

l = 1, 2, . . . , m

6 2 10

(16)

where pmax is the maximum potential value. Then set pmax = F. Update the set of Pareto solutions. Set the potential value of the new solution to be F. – If the solution is a new Pareto solution, calculate its fitness value: F = pl * + dl * (u)

(17)

add it to the Pareto solution set with a potential value of F. If F > pmax, set pmax = F. – If the solution is not a new Pareto solution, calculate its fitness value: F = pl * − dl * (u)

(18)

If F < 0, set F = 0 to avoid negative fitness values.

3 4

FIGURE 2

Hong Kong

Pearl River delta regional network.

30 years. To facilitate the analysis, a 30-year concession period is converted into a unit period. This annual worth conversion is similar to the equivalent uniform annual cost that is used in engineering economic analysis (19). Hence, the BOT feasibility analysis in terms of profit (revenue − cost) is converted into a unit period value. To simulate demand uncertainty, a Latin hypercube sampling (LHS) technique is used. LHS is a stratified sampling method that has been found to outperform the Monte Carlo method (20). LHS partitions the input distribution into intervals of equal probability. Only one random variate is sampled within each interval. This sampling technique significantly reduces the number of samples and still achieves a reasonably level of accuracy. In this study, LHS is used to generate random traffic demand variates according to a predefined normal distribution. The random O-D demands are presented in Table 2. Standard deviation (σw) of the O-D demand is set to half the expected O-D demand (σw − dw /2). Hence, random samples for each O-D pair can be generated according to the following standard normal distribution: dw () = Dw = dw ± Zσ w

(20)

whenever a negative value is generated, the O-D demand is resampled. In this case study, the following parameters are used:

NUMERICAL EXPERIMENTS Problem Setting and Data Set Characteristics The simulation-based multiobjective GA procedure proposed in this study is demonstrated with the case study of an intercity expressway in the Pearl River delta region of South China given by Yang and Meng (3). The network is presented in Figure 2. It consists of 4 nodes, 10 links, and 12 O-D pairs. The case study involves construction of an expressway between Node 3 and Node 4, leading to two new links, Link 9 and Link 10. Because the two new links connect the same nodes in opposite directions, the same capacity and toll charge are assumed for both. Thus, there are only two decision variables in the problem: toll (x) and capacity (y) for the new BOT roads. The link travel time function used in the traffic assignment problem is the standard Bureau of Public Roads function: v 4  ta (va ) = ta0 1.0 + 0.15 a    ca   

4

9

(15)

• Step 3.3. Compare the new solution u with all existing Pareto solutions: – If the solution is a new Pareto solution and it dominates at least one of the existing Pareto solutions, calculate its fitness value:

Shenzhen

5

where l* indicates the nearest existing Pareto solution to the new solution u.

F = pmax + dl * (u)

2

3

Zhuhai

Then find the minimum distance: dl * = min{dl (u)}

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(19)

The basic inputs of the link travel time function are presented in Table 1. The concession period of this project is assumed to be

• Population size is 25 chromosomes. • Maximum number of generations is 50. TABLE 1 Input Data for Link Travel Time Function

Link No.

Free-Flow Travel Time (h)

Link Capacity (veh/h)

1 2 3 4 5 6 7 8 9 10

1.5 1.5 0.4 0.4 0.6 0.6 1.2 1.2 0.5 0.5

9,000 9,000 7,200 7,200 3,400 3,400 9,000 9,000 To be determined To be determined

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TABLE 2

Origin

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Two sets of results are provided here. One set is generated according to the weighted-sum approach with the simulation-based GA procedure for a single-objective optimization problem developed by the authors (10). Another set is generated based on the distancebased approach combined with the simulation-based GA procedure for the multiobjective problems proposed in this paper.

Distributions of O-D Demands

Destination

Random O-D Demand (veh/h)

1

1

1

2

0 N(4500, σ w2 )

1

3

N(4500, σ w2 )

1

4

N(3000, σ w2 )

2 2

1 2

2

N(4500, σ w2 )

Results from Distance-Based Method

3

0 N(1500, σ w2 )

2

4

N(3000, σ w2 )

3

1

N(4500, σ w2 )

3 3

2 3

3

4

0 N(3000, σ w2 )

4

1

N(3000, σ w2 )

4

2

N(3000, σ w2 )

4 4

3 4

In the distance-based method, both objectives are explicitly considered simultaneously when generating the nondominated solutions. First, convergence characteristics of the simulation-based multiobjective GA procedure using the distance-based method are provided. Figure 3 indicates the best values of the two objectives as a function of the number of generations. As can be observed, the expected profit increases significantly and converges at the 18th generation, while the standard deviation of profit (or risk) decreases in the first few generations and converges at the 20th generation. Figure 4 presents the evolution of the nondominated solutions resulting from the 1st, 3rd, and 50th generations. It indicates how the nondominated solutions migrate to the Pareto frontier. As indicated in Figure 4, the nondominated solutions are well spread, covering both objectives uniformly. Together with the convergence curve presented in Figure 4, it appears that the nondominated solutions generated by the distance-based method have achieved the two distinct goals in multiobjective optimization problems (21):

N(1500, σ w2 )

N(3000, σ w2 ) 0

• Maximum number of samples is 1,000. • A uniform crossover scheme is adopted. The population is divided into two groups, arranged in descending order of the fitness values. The top half will be used as parents for mating. • Probability of mutation is 0.15. • α = 3.4 × 10−5 (1/h): Parameter that transforms the capital cost of the project into a unit period cost. • ϕ = 0: For simplicity, the ratio of maintenance-operating costs to the capital cost is set to zero. • k = 1.0 × 106 [Hong Kong dollars (HK$)/h (in 1998 dollars here and throughout this paper), vehicles (veh)/h]: Proportionality parameter. • β = 1/120 (h/HK$): Value of time. • Lower bound and upper bound for toll are [5 HK$, 100 HK$]. • Lower bound and upper bound for capacity are [1,000 veh/h, 9,000 veh/h].

1. Discover solutions as close to the Pareto optimal solutions as possible, and 2. Find solutions as diverse as possible in the Pareto frontier. Figure 5 presents a two-dimensional conditional space diagram to view the Pareto optimal solutions. In this figure, each solution contains four pieces of information: toll, capacity, expected profit, and standard deviation of profit. The diagram is divided into four panels in terms of the ranges of standard deviation. Capacity and toll are represented by the x and y axes, respectively. The number next to each solution indicates expected profit and the size of the circle represents the magnitude of standard deviation of profit (i.e., larger circle has higher standard deviation). From the figure, it appears that higher expected

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FIGURE 3 Convergence curve of the distance-based method. (STDEV  standard deviation.)

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FIGURE 4 Evolution of nondominated solutions by the distance-based method. (STDEV  standard deviation.)

profits are often accompanied by larger standard deviation of profits (see the highest-risk region in the upper right corner). Likewise, the lower-risk regions also provide lower expected profits.

Results from Weighted-Sum Method In principle, multiobjective optimization problems are very different from single-objective optimization problems. This is because solving multiobjective optimization problems often requires a set of non-

FIGURE 5

dominated solutions, not just a single best solution as in single optimization problems. This requirement leads to overwhelming complexity when the number of objectives increases. One way to reduce the complexity is to reformulate the multiobjective optimization problem by scaling the objectives into a single objective by multiplying each objective with a predefined weight. This method is known as the weighted-sum approach. In the weighted-sum approach, the two objectives (i.e., maximizing expected profit and minimizing risk) are transformed into a composite objective function with predefined weights. Optimizing this

Two-dimension conditional diagram with mean-variance information. (STDEV  standard deviation.)

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ber of nondominated solutions, it is necessary to repeatedly solve the single-objective optimization problem with different combinations of weight values. Because the true weights are unknown, there is no guarantee that the weighted-sum method can achieve the Pareto set of nondominated solutions.

CONCLUSIONS AND FUTURE RESEARCH In this study, a mean-variance model was presented to determine the optimal toll and capacity of a BOT project under demand uncertainty. This model considers that private investors are interested in not only maximizing expected profit but also minimizing risk.

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composite objective is equivalent to a single-objective optimization problem. To generate multiple nondominated solutions, different combinations of weight values are uniformly assigned to both objectives (e.g., 0.0:1.0, 0.1:0.9, . . . , 1.0:0.0). The nondominated solutions for selected combinations of weights are presented in Figure 6. It is found that the pattern of nondominated solutions is highly dependent on the selection of weight combination. The number of nondominated solutions generated is different from each weight combination. For the 11 combinations of weights examined here, none can fully satisfy the two criteria of Pareto optimal solutions. There may exist a weight combination that can generate a good set of nondominated solutions to cover the entire Pareto optimal frontier. However, it is difficult to choose a priori such a weight combination. To have a sufficient num-

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FIGURE 6 Nondominated solutions with different weights: (a) 0.0:1.0, (b) 0.2:0.8, (c) 0.4:0.6, (d) 0.6:0.4, (e) 0.8:0.2, (f ) 1.0:0.0. (STDEV  standard deviation.)

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Because BOT projects are often subject to great uncertainties, minimizing the risk of not making a particular profit level or even loss is of great interest to private investors. Hence, the mean-variance model provides a more realistic assessment of BOT projects. A simulationbased multiobjective GA procedure was also presented for solving this stochastic multiobjective BOT network design problem. Based on the numerical experiments, it was found that the distance-based method performs better than the weighted-sum method. The nondominated solutions obtained from the distance-based method cover the entire Pareto optimal region, satisfying the two distinct goals of multiobjective optimization problems. The mean-variance model proposed in this paper for the BOT network design problem can be extended to consider multiple BOT projects operating by multiple private firms to examine competition and equilibrium effects of private toll roads in a traffic network (22). It can also be extended to consider user heterogeneity in the value of time (i.e., multiclass network equilibrium model with distinct group-specific incomes) (23). Furthermore, it is worthwhile to consider not only profit maximization of the private firm but also welfare maximization of the society. Finally, the simulation-based multiobjective GA procedure should be enhanced by incorporating local search, better evolutionary strategies, and improved sampling techniques to speed up the process of finding Pareto optimal solutions.

ACKNOWLEDGMENT This work was partly supported by a National Science Foundation grant.

REFERENCES 1. Sidney, M. L. Build, Operate, Transfer: Paving the Way for Tomorrow’s Infrastructure. John Wiley and Sons, New York, 1996. 2. Downer, J. W., and J. Porter. Tate’s Cairn Tunnel, Hong Kong: South East Asia’s Longest Road Tunnel. Proc., 16th Australian Road Research Board Conference, Part 7. Australian Road Research Board, Perth, 1992, pp. 153–165. 3. Yang, H., and Q. Meng. Highway Pricing and Capacity Choice in a Road Network Under a Build-Operate-Transfer Scheme. Transportation Research A, Vol. 34, 2000, pp. 207–222. 4. Walker, C., and A. J. Smith. Privatized Infrastructure: The Build Operate Transfer Approach. Thomas Telford, London, 1995. 5. Euritt, M. A., M. H. Robert, and E. J. James. An Overview of Highway Privatization. Center for Transportation Research, University of Texas at Austin, 1994. 6. Lam, W. H. K., and M. L. Tam. Risk Analysis of Traffic and Revenue Forecasts for Road Investment Projects. Journal of Infrastructure Engineering, Vol. 4, No. 1, 1998, pp. 19–27.

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