A simulation-based multi-objective genetic algorithm (SMOGA) procedure for BOT network design problem

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DOI 10.1007/s11081-006-9970-y

A simulation-based multi-objective genetic algorithm (SMOGA) procedure for BOT network design problem

Anthony Chen·Kitti Subprasom·Zhaowang Ji

Received: 23 June 2004 / Revised: 12 April 2005

CSpringer Science+Business Media, LLC 2006

Abstract Solving optimization problems with multiple objectives under uncertainty is generally a very difficult task. Evolutionary algorithms, particularly genetic algorithms, have shown to be effective in solving this type of complex problems. In this paper, we develop a simulation-based multi-objective genetic algorithm (SMOGA) procedure to solve the build-operate-transfer (BOT) network design problem with multiple objectives under demand uncertainty. The SMOGA procedure integrates stochastic simulation, a traffic assignment algorithm, a distance-based method, and a genetic algorithm (GA) to solve a multi-objective BOT network design problem formulated as a stochastic bi-level mathematical program. To demonstrate the feasibility of SMOGA procedure, we solve two mean-variance models for determining the optimal toll and capacity in a BOT roadway project subject to demand uncertainty.

Using the inter-city expressway in the Pearl River Delta Region of South China as a case study, numerical results show that the SMOGA procedure is robust in generating

‘good’ non-dominated solutions with respect to a number of parameters used in the GA, and performs better than the weighted-sum method in terms of the quality of non-dominated solutions.

Keywords Network design problem . Multiple objectives . Demand uncertainty . Simulation . Genetic algorithm

A. Chen ()·Z. Ji

Department of Civil and Environmental Engineering, Utah State University, Logan Utah 84322-4110, USA

e-mail: [email protected] K. Subprasom

Planning Division, Department of Highways, Bangkok 10400, Thailand

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1 Introduction

The Build-Operate-Transfer (BOT) approach is one of the privatization mechanisms for promoting transportation infrastructure development by using private funds to con- struct new infrastructure facility. The fundamental principal of BOT is a concession granted by the government to the private investors as a concessionaire to build an infrastructure facility; when completed, the private investors are allowed to operate the infrastructure by collecting toll charges for a certain number of years called a concession period. After the concession period expires, the facility is transferred back 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 in both the developing and developed countries (Sidney, 1996).

Although the BOT concept appears simple, there are many factors affecting the success of a BOT project. Two of these factors are the selection of toll charge and roadway capacity of a BOT project and the evaluation of relevant benefits to the private investors and to the government. From the private investors’ viewpoint, the main concern is profit while under the government’s viewpoint, social welfare gain to the society is of interest. Recently, Yang and Meng (2000, 2002) formulated the BOT problem as a bi-level mathematical program, which is a special transportation network design problem. This BOT network design problem is concerned with the optimal selection of toll charge and capacity of the private toll roads in a transportation network to optimize certain objectives while accounting for route choice behavior of network users. Profit and welfare gain from both the private investors and the government were evaluated with respect to different toll-capacity combinations, but demand uncertainty was not considered in their study. Travel demand is assumed to be known exactly in the future. However, there is no guarantee that the travel demand forecast would precisely materialize under uncertainty. This is because travel demand forecast is affected by many factors such as economic growth, land-use pattern, socioeconomic characteristics, etc. All these factors cannot be measured accurately, but can only be roughly estimated. Chen et al. (2001) incorporated demand uncertainty into the BOT network design problem, but only one objective was considered in the stochastic bi-level mathematical programming formulation, which is to maximize the expected profit. Using the classical mean-variance model in portfolio analysis (Markowitz, 1952), Chen et al. (2003) extended the BOT network design problem to consider two objectives: maximizing expected profit and minimizing variance 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. A solution may be best in one objective but worst in other objective, which cannot be directly compared with each other. Therefore, it may not be appropriate to combine the multiple objectives into a single composite objective using pre-defined weights for each objective. It is necessary to explicitly solve this as a multi-objective problem by generating a set of non-dominated solutions (also known as “Pareto” optimal solutions). In this paper, we develop a simulation-based multi-objective genetic algorithm (SMOGA) procedure to solve two mean-variance models for determining the optimal selection of toll charge and roadway capacity of a BOT project under demand uncertainty. One model is

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the profit maximization problem for the private investors, and another is the social welfare maximization problem for the government. Due to the complexity of the BOT network design problem (i.e., multiple objectives, bi-level optimization structure, and demand uncertainty), traditional optimization methods are not appropriate. In this paper, we integrate stochastic simulation, a traffic assignment algorithm, a distance- based method, and a genetic algorithm into the SMOGA procedure to solve multi- objective BOT network design problem under demand uncertainty.

The remainder of the paper is organized as follows. In Section 2, the BOT network design problem and formulation are introduced. Specifically, two mean-variance BOT models are formulated for the private investors and the government, respectively.

Section 3 presents the SMOGA procedure for solving the mean-variance BOT models.

Numerical results are provided in Section 4 to illustrate the feasibility of the proposed SMOGA procedure. Finally, Section 5 provides some conclusions and future research.

2 BOT network design problem

This section briefly describes the BOT network design problem as a stochastic bi- level mathematical program formulation and the two mean-variance BOT models that will be used to demonstrate the feasibility of the SMOGA procedure. For details, the readers can refer to Chen et al. (2003).

2.1 Stochastic bi-level mathematical program

Many 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 users (or follower) would respond to a given strategy (i.e., design variables determined by the leader). However, it is important to recognize that the strategy set by the leader can only influence (not control) the travel choice decisions of the users. In other words, the leader’s strategy and the follower’s travel choices are inter-dependent, and this interaction can be represented by a bi-level mathematical program. In addition, the leader sometimes has to make the decision under uncertainty where certain inputs are not known exactly. The general stochastic bi-level mathematical program can be formulated as follows:

minu F (u,z (u, ε)) (1)

subject to: G (u,z (u, ε))≤0, (2) where z(u,ε) is implicitly defined for each realizationεby solving

minz f (u,z (u, ε)) (3)

subject to: g (u,z (u, ε))≤0, (4) where F=objective function of the upper-level program; u=decision variables of the upper-level program; G=constraint set of the upper-level program; f =objective

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function of the lower-level program; z=decision variables of the lower-level program;

g=constraint set of the lower-level program; andε=random variables in the lower- level program.

In the BOT network design problem, the upper-level program determines the optimal toll charge and roadway capacity (u) of a BOT project by optimizing one or more objectives under demand uncertainty; while the lower-level program determines the travel choice behavior of network users for a given toll-capacity combination under demand uncertainty z(u,ε). In the two mean-variance models to be described later, the lower-level program is modeled as a standard user equilibrium traffic assignment problem with elastic demand (Sheffi, 1985) and can be described as follows.

For a given toll-capacity combination determined by the upper-level program (u) and for each realization of the random demand vector (ε), the lower-level program solves a traffic assignment problem with elastic demand.

d,v∈minz(u,ε)

a∈A−A¯

_v_a

0

ta(ω)dω+

a∈A¯

_v_a

0

(ta(ω,ya)+βxa)dω

−

w∈W

_d_w

0

D_w⁻¹(ω)dω (5)

subject to:

g (z(u, ε))=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

r∈Rw

f_r^w=d_w(ε), ∀w∈W va =

r∈R

f_r^wδra^w, ∀a∈ A

f_r^w≥0, ∀r∈ R_w, w∈W

d_w≥0, ∀w∈W

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where A = set of links; ¯A = set of BOT links; W = set of origin-destination (O-D) pairs; R_w=set of routes between O-D pair w; R=set of all routes in the network; u = (x, y) is a vector of toll-capacity combination in the upper-level program; xa =toll charge on BOT link a; ya = capacity on BOT link a; ta(.) = travel time on link a;β =parameter that transforms toll into equivalent time value;

D_w⁻¹(.) = inverse of the demand function associated with O-D pair w; d_w(ε) = random demand between O-D pair w;va =flow on link a; f_r^w =flow on route r between O-D pairw; andδ_ra^w =1 if route r between O-D pairwuses link a, and 0 otherwise.

Equation (5) is the objective function, which consists of the sum of the integrals of the link performance functions for both free links and BOT links (first two terms), and the sum of the integrals of the inverse demand functions (third term). The demand function D_w(c_w) is assumed to be dependent on the generalized travel cost c_w between that O-D pair alone. The constraint set in Equation (6) represents the demand conservations, the incidence relationship between link flows in terms of path flows, and non-negativity conditions. The solution to the above minimization problem is z(u, ε), which consists of a set of O-D demands d_w(u,ε) and a set of link flowsva(u,ε).

Both solutions are a function of u (i.e., a vector of toll-capacity combination) in the

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upper-level program, and ofε(i.e., a random demand vector) in the lower-level program. As mentioned before, the upper-level program represents the decision maker.

Here we consider two mean-variance BOT models: the profit maximization model for the private investors and the social welfare maximization model for the government.

2.2 A mean-variance BOT model for profit maximization

The mean-variance model is one of the oldest models in the finance area, dating back to the work of Markowitz (1952). The basic assumption is that risk is measured by variance, and the decision criteria (or objectives) are to maximize the expected return and to minimize the variance. The mean-variance model for the private investors is to maximize the expected profit and to minimize the variance of profit subject to non-negative constraints on the toll-capacity combination of the BOT links.

minu F (u,z (u, ε))=

max E [π(u,z (u, ε))]

min V [π(u,z (u, ε))] (7)

subject to : xa ≥0, ya≥0, ∀a∈ A,¯ (8) whereπ(u, z(u,ε))=profit of realizationε; E[π(u, z(u,ε))]=expected profit; and V[π(u, z(u,ε))]=variance of profit. Profit of the BOT roads for a realizationεis defined as the difference between revenue and cost:

π(u,z (u, ε))=

a∈A¯

va(u, ε)xa−

a∈A¯

(1+ϕ) [αIa(ya)] (9)

The first term is the revenue, which is a function of xa (toll charge) andva(u,ε) (the number of users patronizing the toll links), and the second term is the cost, which depends on ya(capacity of toll links) used in the construction cost function Ia(ya).α is a parameter that transforms the capital cost of the project into a unit period cost, andϕis a ratio of maintenance-operating costs to the capital construction cost.

2.3 A mean-variance BOT model for social welfare maximization

For the government, the main concern of a BOT project is the benefit defined in terms of social welfare added to the society. The mean-variance model for the government is to maximize the expected social welfare and to minimize its variance:

minu F(u,z(u, ε))=

max E [S (u,z (u, ε))]

min V [S (u,z (u, ε))] (10)

subject to: xa ≥0, ya ≥0, ∀a∈ A,¯ (11)

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where S(u, z(u,ε))=social welfare of realizationε; E[S(u, z(u,ε))]=expected social welfare; and V[S(u, z(u,ε))]=variance of social welfare. The social welfare for a given realizationεis defined as the difference between consumer surplus and cost of the BOT project:

S (u,z (u, ε))= 1 β

w∈W

_d_w

0

D_w⁻¹(ω)dω−

a∈A−A¯

va(u, ε)ta(va(u, ε))

−

a∈A¯

va(u, ε)ta(va(u, ε),ya) −

a∈A¯

αIa(ya) (12)

The consumer surplus measures the difference between what consumers (users) are willing to pay for travel and what they actually pay. The first term in Equa- tion (12) is the total travel cost that the users of demand (d_w) are willing to pay whereas the second and third terms are the total travel cost that they actually pay.

The fourth term is the construction cost function of the private investors. 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 comparing implementing the BOT project with the do-nothing case.

It should be noted that the social-welfare objective function is purely related to the economic efficiency and does not include the other benefits (e.g., environment improvement).

3 A simulation-based multi-objective genetic algorithm (SMOGA) procedure

Stochastic bi-level mathematical programs with multiple objectives are generally difficult to solve by traditional methods. There are three issues that need to be considered when solving this complex problem: (1) how to handle the demand uncertainty, (2) how to solve the bi-level mathematical program, and (3) how to simultaneously optimize all objectives by solving for a set of non-dominated solutions. In this section, we present the SMOGA procedure that integrates stochastic simulation, a traffic assignment algorithm, a distance-based method, and a genetic algorithm to tackle the different issues involved in solving the mean-variance BOT network design models under demand uncertainty. Stochastic simulation is used to simulate the uncertainty of traffic demands based on a probability distribution with pre-defined mean and variance. Bi-level mathematical programs are generally difficult to solve because evaluation of the upper-level objective function requires solving the lower-level program. We use a standard traffic assignment algorithm (known as the Frank-Wolfe algorithm) for solving the lower- level program (Sheffi, 1985). For network design problems, the lower-level program can be considered as nonlinear constraints. This often makes the bi-level mathematical programs non-convex and difficult to solve by the standard optimization methods (Yang and Bell, 1998). To tackle the non-convexity issue in the network design problems, Friesz et al. (1992) and Meng and Yang (2002) used simulated annealing (SA), while Cree et al. (1998) and Yin (2000) employed genetic algorithm (GA). Both

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meta-heuristics are stochastic search methods that have the potential of obtaining the global optimal solution by providing a means to escape local optima (i.e., accept moves that worsen the objective value). The main mechanism driving the optimization process is a simple search operator inspired by different natural based phenomena (i.e., physical annealing process with solids for SA and natural selection based on the principle of evolution-survival of the fittest for GA). Other meta-heuristics (e.g., tabu search, ant colony optimization, memetic algorithms, etc.) are also possible candidates for tackling the non-convexity issue in bi-level mathematical programs (see Glover and Kochenberger (2003) for the details about these meta-heuristics); however, GA is adopted in this study because it can work with continuous and discrete parameters, differentiable and non-differentiable functions, and uni-modal and multi-modal functions as well as convex and non-convex feasible regions (Goldberg, 1989). In addition, GA has been widely applied in many fields because of its globality, parallelism, and robustness features. Examples of GA applications in Civil Engineering include design of road construction excavation plans (Haidar and Naoum, 1997), road maintenance planning (Chan et al., 1994; Fwa et al., 1994), structural optimization (Yang and Soh, 1997), water system design (Lippai et al., 1999), multi-objective urban planning (Balling et al., 1999), transportation network design (Cree et al., 1998; Yin, 2000), and many others.

For the multi-objective optimization problem, we use the distance-based method (Osyczka and Kundu, 1995) to generate an approximate Pareto solution set. The overall SMOGA procedure is presented in Figure 1 and can be summarized as follows.

Step 1: Define GA parameters: mutation probability (Pm), crossover probability (Pc), population size (P), maximum number of generations (Nm), and maximum number of sample sizes (Snsp). Initialize N (counter for the number of generations) and a set of initial solutions of size P. Initialize p (counter for the number of solutions).

Step 2: Evaluate the objective function of solution p with the maximum number of samples. Collect statistics (e.g., mean and standard deviation of objective value).

Step 3: Use the distance-based method to solve the bi-objective optimization problem and update the non-dominated (or Pareto) solution set. Increment p= p+1.

Repeat Step 2 until 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 non-dominated solution set.

3.1 Simulation module

To handle travel demand uncertainty in a multi-objective BOT network design problem, stochastic simulation is used to simulate the uncertainty of O-D demands based on a probability distribution with pre-defined mean and variance. Latin Hypercube Sampling (LHS) technique, one of the sampling techniques, is employed in this paper. LHS is a stratified sampling method that has shown to outperform the Monte Carlo (MC) method (McKay, 1988). LHS partitions the input distribution into intervals of equal probability. Only one random variate is sampled within each interval. This sampling technique significantly

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·Generate initial solution

·Define population size, P

·Define number of generations, N_m

·Define maximum number of sample sizes, S_nsp

N = 1

p = 1 S = 1

Simulation module

S > S_nsp Distance-

based module P > p

N > N_m GA module ·Reproduction ·Crossover ·Mutation

Report the nondominated solutions

N = N+1

p = p+1 S = S+1

No

No No Yes

Yes

Yes Traffic

assignment module Step 1

Step 2

Step 3

Step 4

Step 5

Fig. 1 Simulation-based multi-objective genetic algorithm (SMOGA) procedure

reduces the number of samples while still achieves a reasonably level of accuracy.

In this study, we use LHS to generate random traffic demand variates according to a predefined Normal distribution. It should be noted that other distributions could also be used. The potential O-D demand of the demand function is chosen as the only key exogenous input variables to reflect the uncertainty of travel demand. Random samples of potential demand ˆd_wfor each O-D pair w can be generated according to the following equation:

dˆ_w=d¯_w±Zσw, ∀w∈W, (13)

where ¯d_w=mean potential O-D demand between O-D pair w; Z=random variable generated from N(0, 1); andσ_w =standard deviation of potential demand between O-D pairw.

3.2 Traffic assignment module

The lower-level program is a standard traffic assignment problem with elastic demand. For each realization of potential O-D demands generated in the simulation module, we use the Frank-Wolfe algorithm (also known as the convex combinations

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method) to solve the lower-level program. The algorithmic steps are summarized below.

Step 0: Initialization.

Set iteration counter n=1.

Find an initial feasible flow pattern{va⁽¹⁾},{d_w⁽¹⁾}. Step 1: Link travel time update.

Calculate t_a⁽ⁿ⁾=ta(v⁽ⁿ⁾a ), ∀a∈ A.

Compute D_w⁻¹(d_w⁽ⁿ⁾), ∀w∈W . Step 2: Direction finding.

Compute the shortest path s between each O-D pairwbased on t_a⁽ⁿ⁾. Obtain the minimum travel time c⁽ⁿ⁾_w at iteration n between O-D pairw.

Execute the following assignment rules:

If c⁽ⁿ⁾_w <D⁻¹_w (d_w⁽ⁿ⁾), set g^w(n)_s =dˆ_wand g_r^w(n)=0,∀r∈ R_w,r =s.

If c⁽ⁿ⁾_w >D⁻_w¹(d_w⁽ⁿ⁾),set g_r^w⁽ⁿ⁾=0,∀r ∈R_w. If c⁽ⁿ⁾_w =D⁻_w¹(d_w⁽ⁿ⁾),set either of the above.

where ˆd_w=random potential demand between O-D pairw.

Compute the auxiliary flows:

u⁽ⁿ⁾_a =

w∈W

r∈R_w

g_r^w(n)δ_ra^w, ∀a∈ A; q_w⁽ⁿ⁾=

r∈R_w

g_r^w(n), ∀w∈W (14)

where g^w_r,u_a,q_ware the auxiliary variables of f_r^w, va,d_w. Step 3: Move size.

Findα⁽ⁿ⁾which minimizes the objective function along its descent direction.

minα⁽ⁿ⁾

a∈A−A¯

_v⁽ⁿ⁾_a _+α⁽ⁿ⁾_(u⁽ⁿ⁾_a _−v⁽ⁿ⁾_a ₎

0

ta(ω)dω+

a∈A¯

_v⁽ⁿ⁾_a _+α⁽ⁿ⁾_(u⁽ⁿ⁾_a _−v⁽ⁿ⁾_a ₎

0

ta(ω,ya)dω

+

a∈A¯

β

v⁽ⁿ⁾_a +α⁽ⁿ⁾

u⁽ⁿ⁾_a −v_a⁽ⁿ⁾

xa−

w∈W

_d_w⁽ⁿ⁾_+α⁽ⁿ⁾(^qw⁽ⁿ⁾−d_w⁽ⁿ⁾)

0

D_w⁻¹(ω)dω (15) Step 4: Flow update.

Compute the following:

v⁽ⁿ⁺¹⁾_a =v_a⁽ⁿ⁾+α⁽ⁿ⁾

u⁽ⁿ⁾_a −v⁽ⁿ⁾_a

, ∀a ∈ A (16)

d_w⁽ⁿ⁺¹⁾=d_w⁽ⁿ⁾+α⁽ⁿ⁾

q_w⁽ⁿ⁾−d_w⁽ⁿ⁾

, ∀w∈W (17)

Step 5: Convergence test.

The equilibrium conditions can be checked by calculating the closeness of the O-D travel time to those implied by the inverse demand function, in addition to the

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closeness of the consecutive O-D travel times. The following criterion is used for convergence:

w∈W

D_w⁻¹ d_w⁽ⁿ⁾

−c⁽ⁿ⁾_w

c⁽ⁿ⁾_w +

w∈W

c_w⁽ⁿ⁾−c⁽ⁿ_w⁻¹⁾

c_w⁽ⁿ⁾ ≤η, (18)

whereηis the tolerance (e.g., 0.01). If the test is satisfied, terminate; otherwise, set n =n+1 and go to step 1.

3.3 Distance-based module

The distance-based method (Osyczka and Kundu, 1995) is used to solve the multi- objective optimization problem by explicitly generating the non-dominated solutions in each generation. The basic idea is to assign a fitness value to each solution according to the distance measure with reference to the existing non-dominated solutions obtained in the previous generation of the SMOGA procedure. A higher fitness value is assigned to a solution if it is farther away from the existing non-dominated solution set. The distance-based procedure is given below.

Step 1: The first generated solution is taken as a non-dominated solution with a potential value d1, which is an arbitrarily chosen value called the starting potential value. The first generated solution has the fitness value of F, which is set to d₁.

Step 2: For a new solution u, calculate the relative distances to all existing non- dominated solutions:

dl(u)= ^q

k=1

fkl− fk(u) fkl

2

, for l=1,2, . . . ,m, (19)

where q is the number of objectives, m is the number of non-dominated solutions obtained by genetic search, fk(u) denotes the value of the kth objective of the new solution u, and f_kldenotes the value of the kth objective for the lth non-dominated solution. Then find the minimum distance:

dl^∗=min{dl(u)}, for l=1,2, . . . ,m, (20) where l^∗indicates the nearest existing non-dominated solution to the new solution u.

Step 3: Compare the new solution u with all existing non-dominated solutions:

(a) If the solution is a new non-dominated solution and it dominates at least one of the existing non-dominated solutions, calculate its fitness value:

F= pmax+dl^∗(u), (21)

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where p_maxis the maximum potential value. Then set p_max=F. Update the set of non-dominated solutions. Set the potential value of the new solution to be F.

(b) If the solution is a new non-dominated solution, calculate its fitness value:

F= pl^∗+dl^∗(u), (22)

where p_l∗ is the fitness value of l^∗; add it to the non-dominated solution set with a potential value of F. If F > pmax, set p_max=F.

(c) If the solution is not a new non-dominated solution, calculate its fitness value:

F= pl^∗−dl^∗(u), (23)

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

3.4 GA module

The GA is an intelligent stochastic search method for optimization problems based on the mechanics of natural selection and natural genetics (Goldberg, 1989). The GA was pioneered by Holland (1975) and has been further developed by many researchers to solve a variety of problems. The GA is different from traditional optimization techniques in three important aspects. First, the GA operates by manipulating a pool of solutions instead of one single solution each time. Working with a pool of solutions enables the GA to identify and explore properties simultaneously in different search directions. Second, the GA employs probabilistic transition rules to generate new solutions from the existing pool of solutions. This introduces perturbations to move out of local optima. Third, the GA selects better solutions in each step by comparing directly the objective function values (or fitness values) of generated solutions. The search process is not gradient-based and does not require any information on differentiability, convexity, or other auxiliary properties.

The problem-solving process of the GA begins with the identification of problem solutions and the genetic representation of these solutions. The search process for the GA solutions that best satisfy the objective function involves generating an initial random pool of solutions to form a parent solution pool, followed by obtaining new solutions (so called offspring) and forming new parent pools through an iterative process. This iterative process consists of reproduction, crossover, and mutation operators. This process continues until, ideally, a set of optimal solutions is found. In this section, we provide a brief description of the GA implementation. Readers can refer to Goldberg (1989), Deb (2001), and Coello Coello et al. (2002) for more details.

Chromosome representation: In general, there are two chromosome representations:

binary and real. Since the decision variables in the upper-level program are real, we adopt the real representation to represent the toll charge and capacity of BOT roads.

Reproduction: The tournament mechanism and the half replacement strategy are used for reproduction. After the fitness value for each chromosome is calcu- lated, they are ranked based on the fitness value. Chromosomes in each generation will be equally divided into two parts. Chromosomes in the upper

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part are eligible to be selected as parent chromosomes to generate offspring chromosomes by using the tournament competition scheme. Chromosomes in the lower part are replaced by the offspring chromosomes generated from the crossover and mutation operations. For the tournament selection, two chromosomes are randomly paired for mating. The fitter chromosome has a better chance to enter the mating pool. This process is repeated until the mating pool is filled.

Crossover: A uniform crossover operator is used in this study. In the uniform crossover, a crossover mask, which consists of 0s and 1s and has the same length as the chromosome structure, determines the positions between the parent chromosomes to exchange genetic materials. The value of each bit in the crossover mask indicates the parent supplying genetic units to the offspring. Figure 2 gives a graphical illustration of the uniform crossover operator.

Mutation: The purpose of the mutation operator is to prevent GA trapped in a local minimum. Mutation is the means of alternating the value of genetic units, hopefully, to introduce new genetic structures to the new offsprings. All new offspring are subjected to the mutation operator with a predefined mutation rate (Pm). Genes are randomly selected to change their values as shown in Figure 3. Mutation allows the GA to explore new region of the solution space and to prevent premature convergence to a sub-optimal solution.

4 Numerical experiment

4.1 Problem and parameters setting

The SMOGA procedure proposed in this study is demonstrated using the case study of an inter-city expressway in the Pearl River Delta Region of South China given in Yang and Meng (2000). Here we want to determine the optimal toll charge and capacity for

A₁ A₂ A₃ A₄ A₅ A₆

B₁ B₂ B₃ B₄ B₅ B₆

Parent chromosome 1

Parent chromosome 2

1 0 1 0 0 1

Crossover mask

A₁ A₃ B₄ B₅ A₆

B₁ B₂

B₃ A₄ A₅

A₂ B₆

Offspring 1

Offspring 2 Crossover

A₁ A₂ A₃ A₄ A₅ A₆

B₁ B₂ B₃ B₄ B₅ B₆

Parent chromosome 1

Parent chromosome 2

1 0 1 0 0 1

Crossover mask

A₁ A₃ B₄ B₅ A₆

B₁ B₂

B₃ A₄ A₅

A₂ B₆

Offspring 1

Offspring 2 Crossover

Fig. 2 Uniform crossover operator

Mutation

A₁ B₂ A₃ B₄ B₅ A₆

Offspring 1

Selected gene

A₁ B₂ A'₃ B₄ B₅ A₆

Offspring 1

Fig. 3 Mutation operator Springer

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both profit maximization and social welfare maximization models described in Section 2. The network is depicted in Figure 4. It consists of 4 nodes, 10 links, and 12 O-D pairs. The case study involves construction of a toll road 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 links.

The link travel time function used in the lower level problem is the standard Bureau of Public Road (BPR) function.

ta(va)=t_a⁰

1.0+0.15 va

ca

4

, ∀a∈ A, (24)

where ca=capacity of link a. The O-D demand function is:

d_w(ε)=dˆ_w exp(−γc_w), ∀w∈W, (25) where d_w(ε)=the random O-D demand; ˆd_w=the random potential O-D demand generated according to equation (13);γ=scaling parameter which reflects the sensitivity of demand to full trip price; and c_w=O-D travel time (inclusive of equivalent time of toll). The basic inputs of the link travel time function and parameters of the demand function can be found in (Yang and Meng, 2000). In this case study, the following parameters are used:

rPopulation size is 100 chromosomes.

rThe maximum number of generations is 50.

rThe maximum number of samples is 500.

rProbability of mutation (Pm) is 0.15.

rProbability of crossover (P_c) is 0.50.

Hong Kong 1

4

2 3

7 8

1 2 6 5

9

10 4

3 Guangzhou

n e h z n e h S i

a h u h Z

BOT links

Fig. 4 Pearl River Delta Region network

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r_ϕ₌0: For simplicity, the ratio of maintenance-operating costs to the capital cost is set to zero.

r_α₌₃_.₄_×₁₀⁻⁵ _{(1/h), k}₌₁₀_×₁₀⁶ (HK$/h.veh/h), β =1/120 (h/HK$), andγ =1.

rStandard deviation of potential demand is set to be one third of mean value.

rThe lower bound and upper bound for toll are [5 HK$ , 100 HK$].

rThe lower bound and upper bound for capacity are [1000 veh/h, 10000 veh/h].

4.2 Convergence and evolution characteristics

In the distance-based method, both objectives (mean and standard deviation) are considered simultaneously when generating the non-dominated solutions. First, convergence curves of the SMOGA procedure are provided. The convergence curve shows the best values of the two objectives as a function of the number of generations. Figure 5 depicts the convergence curve for the profit maximization problem. As can be seen, expected profit increases significantly in the early generations and converges at the 15th generation, while the standard deviation of profit decreases in the early generations and converges around the 19th generation. Figure 6 shows the convergence curve for the social welfare maximization problem. Similar convergence characteristics are also observed in the social welfare maximization case.

Figures 7 and 8 display the evolution of the non-dominated solutions result- ing from the first, third, and fiftieth generations. These two figures show how the non-dominated solutions migrate to the Pareto frontier. Initially only a few non-dominated solutions are generated. As the search process proceeds, the number of non-dominated solutions increases with greater coverage. The number of non-dominated solutions for the first, third, and fiftieth generation are 5, 25, and

90000 100000 110000 120000 130000 140000 150000

0 5 10 15 20 25 30 35 40 45 50

Number of Generations

Expected Profit (HK$)

0 1000 2000 3000 4000 5000 6000 7000 8000

STDEV of Profit (HK$)

Expected profit

STDEV of profit

Fig. 5 Convergence curve for profit maximization Springer

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210000 220000 230000 240000 250000 260000 270000 280000 290000

0 5 10 15 20 25 30 35 40 45 50

Number of Generations

Expected Welfare (HK$)

16000 17000 18000 19000 20000 21000 22000 23000 24000 25000

STDEV of Welfare (HK$)

Expected welfare

STDEV of welfare

Fig. 6 Convergence curve for social welfare maximization

49 for profit maximization and 7, 15, and 36 for social welfare maximization.

It appears that the non-dominated solutions obtained at the fiftieth generation are well-converged and well-distributed in the objective space, and the solutions from the fiftieth generation are considered to be a good approximation to the Pareto frontier.

0 20000 40000 60000 80000 100000 120000 140000 160000

0 5000 10000 15000 20000

Expected Profit (HK$)

First generation Third generation Fiftieth generation

Fig. 7 Evolution of non-dominated solutions of profit maximization

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0 50000 100000 150000 200000 250000 300000

15000 20000 25000 30000 35000 40000

Expected Welfare (HK$)

First generation Third generation Fiftieth generation

Fig. 8 Evolution of non-dominated solutions of social welfare maximization

4.3 Effects of GA parameters

In this section, we perform different tests to examine the effects of GA parameters used in the SMOGA procedure.

4.3.1 Effects of population sizes

Figures 9 and 10 show the effect of population sizes 30, 50, 100, and 200 on the quality of non-dominated solutions for profit maximization and social welfare maximization, respectively. The numbers of non-dominated solutions obtained using population sizes of 30, 50, 100, and 200 are 21, 29, 49, and 50 for profit maximization and 19, 19, 36, and 36 for social welfare maximization. It can be observed that the non-dominated solutions obtained from population sizes of 30 and 50 are significantly inferior compared to those obtained from population sizes of 100 and 200. However, there is no significant difference between population sizes of 100 and 200 for both profit maximization and social welfare maximization. Therefore, we use a population size of 100 for the rest of the numerical results.

4.3.2 Effects of random seeds

We test the robustness of the SMOGA procedure by performing multiple runs with different random seeds. The range of non-dominated solutions for 10 different random seeds with a population of 100 under profit maximization and social welfare maxi-

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0 20000 40000 60000 80000 100000 120000 140000 160000

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 STDEV of Profit (HK$)

Expected Profit (HK$)

Population = 30 Population = 50 Population = 100 Population = 200

Fig. 9 Non-dominated solutions of profit maximization with different population sizes

50000 75000 100000 125000 150000 175000 200000 225000 250000 275000 300000

15000 20000 25000 30000 35000 40000

Expected Welfare (HK$)

Population = 30 Population = 50 Population = 100 Population = 200

Fig. 10 Non-dominated solutions of social welfare maximization with different population sizes

mization are presented in Figures 11 and 12. The results show that the non-dominated solutions have little variation. The small variation of the non-dominated solution sets shows that the SMOGA procedure is robust with respect to different random seeds.

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0 20000 40000 60000 80000 100000 120000 140000 160000

0 5000 10000 15000 20000 25000

Expected Profit (HK$)

Range of non-dominated solutions found with SMOGA for 10 different random seeds and a population of 100

Fig. 11 Collective non-dominated solutions for profit maximization with 10 different random seeds

50000 100000 150000 200000 250000 300000

15000 20000 25000 30000 35000 40000

Expected Welfare (HK$)

Range of non-dominated solutions found with SMOGA for 10 different random seeds and a population of 100

Fig. 12 Collective non-dominated solutions for social welfare maximization with 10 different random seeds

4.3.3 Effects of mutation and crossover rates

To examine the effects of mutation and crossover rates on the quality of non-dominated solutions, we perform 10 combinations of mutation and crossover rates. Because a single best solution in the multi-objective optimization problems may not exist,

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assessing and comparing the quality of non-dominated solution sets require different measures. Here we use three measures to quantify how ‘good’ the non-dominated solution set represents the Pareto frontier. These three measures are: (1) number of non-dominated solutions, (2) diversity, and (3) convergence to the Pareto frontier.

However, note that other appropriate measures may also be used (see Coello Coello et al., 2002 for other measures). The first measure can be obtained by simply counting the number of non-dominated solutions in the set. For the second measure, we select the spacing metric (S) (Schott, 1995):

S = 1

m−1 m

l=1

(¯s−sl)², (26)

where m is the number of non-dominated solutions, and slis a spacing index defined as:

sl=min

j (|f1l(u)− f1 j(u)| + |f2l(u)− f2 j(u)|), for l=1, . . . ,m, (27) where ¯s is the mean of all sl, f1l(u) is the value of objective 1 (expected outcome) of solution u, and f2l(u) is the value of objective 2 (variance of outcome) of solution u. A value of zero for this spacing metric indicates that all members in the non-dominated solution set are equidistantly spaced. For the third measure, we use the two set coverage (CS) (Coello Coello et al., 2002):

CS (X1,X2)=|{u2 ∈X2;∃u1∈ X1: u₁u2}|

|X2| (28)

where X₁is the non-dominated solutions of set 1; X₂, is the non-dominated solutions of set 2; u₁ is a member in the non-dominated solution set 1; u₂is a member in the non-dominated solution set 2. u₁ u2means u₁dominates u₂if f (u₁)> f (u2), and u1u2means u₁weakly dominates u₂if f (u₁)≥ f (u2). If all points in X₁dominate or are equal to all points in X₂, then by definition CS=1; otherwise, CS=0 implies the opposite.

Tables 1 and 2 present the results of the three measures mentioned above with different combinations of mutation and crossover rates for profit maximization and social welfare maximization, respectively. For comparison purpose, we use 0.15 and 0.5 for mutation and crossover rates as the base case. Note that the percentage appeared in the spacing metric column indicates a percentage change of spacing metric compared with the base case. The two set coverage is computed by comparing the non-dominated solution set of each mutation- crossover rate combination with the base case and vice versa (i.e., CS (X1,X2) and CS (X2,X1)). Here X1 denotes the non-dominated solution set of the base case and X₂ denotes the non-dominated solution set of each mutation-crossover rate combination.

It is found that the number of non-dominated solutions for each combination is quite similar for the 10 combinations of mutation and crossover rates. The percent-

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Table 1 Effects of mutation and crossover rates on the quality of non-dominated solutions for profit maximization

Two set coverage GA parameters Number of non- Spacing

[Pm: Pc] dominated solutions metric CS (X1, X2) CS (X2, X1)

Base case [0.15 : 0.5] 49 0.1795 – –

[0.05: 0.4] 47 0.1810 (+0.84%) 1.0000 0.9592

[0.05: 0.6] 45 0.1845 (+2.79%) 0.9111 0.9184

[0.05: 0.8] 45 0.1798 (+0.17%) 0.9778 0.9184

[0.10: 0.4] 46 0.1863 (+3.79%) 0.9348 0.9388

[0.10: 0.6] 46 0.1879 (+4.68%) 0.9130 0.9388

[0.10: 0.8] 47 0.1874 (+4.40%) 1.0000 0.9592

[0.20: 0.4] 44 0.1889 (+5.24%) 0.9545 0.8980

[0.20: 0.6} 45 0.1835 (+2.23%) 0.9556 0.9184

[0.20: 0.8] 48 0.1858 (+3.51%) 1.0000 0.9796

Table 2 Effects of mutation and crossover rates on the quality of non-dominated solutions for social welfare maximization

Two set coverage GA parameters Number of non- Spacing

[Pm: Pc] dominated solutions metric CS (X1, X2) CS (X2, X1)

Base case [0.15 : 0.5] 36 0.2155 – –

[0.05 : 0.4] 34 0.2252 (+4.51%) 1.0000 0.9444

[0.05 : 0.6] 35 0.2097 (−2.70%) 0.9429 0.9722

[0.05 : 0.8] 37 0.2267 (+5.18%) 0.9730 1.0000

[0.10 : 0.4] 33 0.2181 (+1.21%) 0.9091 1.0000

[0.10 : 0.6] 34 0.2255 (+4.63%) 1.0000 0.9444

[0.10 : 0.8] 35 0.2220 (+3.02%) 0.9714 0.9722

[0.20 : 0.4] 35 0.2220 (+3.02%) 1.0000 0.9722

[0.20 : 0.6} 37 0.2158 (+0.15%) 0.9730 1.0000

[0.20 : 0.8] 33 0.2058 (−4.48%) 1.0000 0.9167

age change of the spacing metric relative to the base case varies around±5%. The values of CS (X1,X2) and CS (X2,X1)are close 1.0, implying that the non-dominated solutions in the two sets are of similar quality. From these three measures, it seems that the variation of mutation and crossover rates does not affect the quality of the non-dominated solutions. In other words, the proposed SMOGA procedure is reliable with respect to the variation of mutation and crossover rates.

4.4 Comparison with the weighted-sum method

In this section, we compare the weighted-sum method with the distance-based method used in the SMOGA procedure. In the weighted-sum approach, the two objectives (i.e., maximizing expected outcome and minimizing risk) are transformed into a composite objective function with predefined weights. Optimizing this composite objective is equivalent to solving a single objective optimization problem. To generate multiple

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non-dominated solutions, different combinations of weight values are assigned to both objectives (e.g., 0.0:1.0, 0.2:0.8, 0.4:0.6, 0.6:0.2, 0.8:0.2, and 1.0:0.0). Tables 3 and 4 present the three measures (i.e., number of non-dominated solutions, spacing metric, and two set coverage) for quantifying the quality of the non-dominated solution sets for profit maximization and social welfare maximization, respectively. To compute the two set coverage, we use the base case from Tables 1 and 2 as the reference set. Based on the three measures, it is found that the quality of non-dominated solutions is highly dependent on the selection of weight combination. The numbers of non-dominated solutions generated for each weight combination are quite different and less than that of the distance-based method. The spacing metric for all weight combinations is much higher than that of the base case using the distance-based method. This implies that the non-dominated solutions generated by the weighted-sum method are not as diverse as those generated by the distance-based method. Furthermore, the two set coverage measure consistently shows that the non-dominated solutions of the base case are considerably better than those generated by the weighted-sum method as indicated by the values of CS (X₁,X2) and CS (X₂,X1). Since the values of CS (X₁,X2)are closer to 1.0, it indicates that the solutions in the base case dominate or are equal to the solutions generated by the weighted-sum method. For the six combinations of weights examined here, none can produce a ‘good’ set of non-dominated solutions as indicated by the

Table 3 Results of weighted-sum method for profit maximization

Two set coverage Number of non- Spacing

dominated solutions metric CS (X1, X2) CS (X2, X1)