[ieee fourth international symposium on uncertainty modeling and analysis, 2003. isuma 2003. -...

A Simulation-based Multi-Objective Genetic Algorithm (SMOGA) for

Transportation Network Design Problem

Anthony Chen, Kitti Subprasom, and Eric Z. Ji Assistant Professor, PhD Candidate, PhD Candidate

Department of Civil and Environmental Engineering, Utah State University

[email protected]

Abstract

In the conventional transportation network design

problem, travel demand is assumed to be known exactly

in the future. However, there is no guarantee that the

travel demand forecast would be precisely materialized

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. Another issue in many

existing transportation network design problems

considers only one objective or a composite objective

with a priori weights. It may be more realistic to

explicitly consider multiple objectives in the

transportation network design problem. In this paper, we

incorporate both travel demand uncertainty and multiple

objectives into the transportation network design

problem. It is formulated as a stochastic bi-level

programming problem (SBLPP) where the upper level

represents the traffic manager and the lower level

represents the road users. To solve this SBLPP, a

simulation-based multi-objective genetic algorithm

(SMOGA) is developed. Numerical results are provided to

demonstrate the feasibility of SMOGA.

1. Introduction

The transportation network design problem is

concerned with the configuration of the network to

maximize a certain objective while accounting for the

route choice behavior of transportation network users.

[1]. Most of the network design problems found in the

literature consider only one objective or a composite

objective with a prior weights [2-11]. This weighted-sum

approach to multi-objective problems is, in principle,

different from multi-objective optimization problems that

explicitly consider multiple objectives. This is because

solving multi-objective optimization problems often

requires a set of non-dominated solutions, not just a

single best solution as in the single optimization

problems. Nowadays, multi-objective optimization is

gaining interests in transportation network design

problems. Advantageous information from multiple

objective optimization would facilitate and enhance the

decision making process.

Another important issue is that most network design

problems do not consider travel demand uncertainty. It

implicitly assumes travel demand is known exactly in the

future. However, there is no guarantee that such travel

demand forecast would be precisely materialized under

uncertainty. This is because travel demand forecast is

affected by many factors such as economic growth, land

use pattern, socioeconomic characteristics, etc. In this

paper, we incorporate travel demand uncertainty and

multi-objective optimization in transportation network

design analysis. It is formulated as a stochastic bi-level

programming problem (SBLPP) where the upper level

represents the traffic manager and the lower level

represents the road users. To solve this SBLPP, a

simulation-based multi-objective genetic algorithm

(SMOGA) is developed. A case study is conducted to

demonstrate the application of the SMOGA framework.

2. Stochastic bi-level programming

formulation

Most existing network design problems are formulated

as a bi-level program that has a leader-follower structure.

In this structure, the upper level program is the leader

(decision makers) and the lower level program is the

follower (users). The leader is assumed to have

knowledge on how the follower would respond to a given

strategy (design variables determined by the upper level

program). However, it is important to recognize that the

strategy set by the leader can only influence (not control)

the follower’s strategy (travel choice of the users). In

addition, the leader sometimes has to make decision

under uncertainty where certain inputs are not known

exactly. The general stochastic bi-level programming

problem (SBLPP) can be formulated as follows:

)),(,(min uzuFu

(1)

Proceedings of the Fourth International Symposium on Uncertainty Modeling and Analysis (ISUMA’03) 0-7695-1997-0/03 $17.00 © 2003 IEEE

subject to: 0)),(,( uzuG , (2)

where ),(uz is implicitly defined for each realization

by

))(,(min zufz

(3)

subject to: 0))(,( zug , (4)

where F = objective function of the upper level (i.e.,

decision maker); u = decision variables of the upper level;

G = constraint set of the upper level; f = objective

function of the lower level (i.e., travelers or users); z =

decision variables of the lower level; g = constraint set of

the lower level; and = random variables in lower level.

In this paper, we consider a special network design

problem called the Build-Operate-Transfer (BOT)

problem [6]. The BOT NDP holds special characteristics

that differ from conventional NDPs investigated in the

literature: the road toll and road capacity have to be

optimized jointly under specific market condition.

For transportation planning and management

problems, the lower level program represents the user’s

route choice behavior responding to the leader’s strategy.

Basically, the route choice problem can be formulated as

a standard user equilibrium with elastic demand. For

each realization of the random variable vector , the

lower level solves the following network equilibrium

problem with elastic demand.

Aa

v

aaa

AAa

v

auzvd

aa

dxytdt00,,

}),({)(min

dDWw

w

d w 1

0

(5)

subject to:

Rr

w

ra

w

ra

Rr

w

w

r

w

w

r

Aafv

Wwdf

WwRrf

uzw

,

,

,,0

),(, (6)

where A = set of links; A = set of toll links; W = set of O-

D pairs; Rw= set of routes between O-D pair w W ; R =

set of all routes in the network; },{ yxu is a vector of

toll-capacity combination in the upper level; xa= toll

charged on toll link a; ya= capacity on toll link a;

at =

travel time on link a; = parameter that transforms toll

into equivalent time value; )(1

ww dD = the inverse of the

demand function;wd = travel demand between O-D

pair w W ; va= flow on link a; fr

w = flow on route

r R w Ww , ; and ar

w = 1 if route r between OD pair

w W uses link a, and 0 otherwise.

The solution to the above minimization problem is

),(uz which consists of a set of OD demands

),(ud wand a set of link flows ),(uva

. Both solutions

are a function of u (i.e., a vector of toll-capacity

combination) in the upper level, and the random variable

vector in the lower level.

As mentioned before, the upper level program

represents the decision-makers. Here we consider two

mean-variance BOT models: the profit maximization

problem and the welfare maximization problem.

2.1. A mean-variance BOT model for profit

maximization

The mean-variance model is one of the oldest finance

areas, dating back to work of Markowitz [12]. The basic

assumption is that risk is measured by variance, and the

decision criteria (or objectives) are to maximize expect

return and to minimize variance. In many cases, there

does not necessary exist a best solution with respect to

both objectives because of conflict between the two

objectives. A solution may be best in one objective but

worst in other objective. Therefore, there usually exists a

set of solutions, called non-dominated solutions or Pareto

optimal solutions, which cannot be directly compared

with each other.

Profit in the private toll road project is the difference

between revenue and cost. Revenue is a function of x

(toll charge) and ),(uv (the number of users patronizing

the toll links), while cost depends on y (capacity of toll

links). The cost of a toll link consists of the construction

cost (cC ) and maintenance-operating cost (

moC ):

moc CCCost (7)

The construction cost is a function of the number of lanes

(or roadway capacity). Following the study by Yang and

Meng [6], the construction cost function is assumed to be

linear:

aaaa ytkyI 0 (8)

whereaa yI = construction cost function with respect to

capacity of toll links; k = proportionality parameter to

convert free-flow travel time into length; and 0

at = free-

flow travel time of toll link a. 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 , which is a ratio of maintenance-operating

costs to the capital cost. Hence, the final cost function

can be expressed as a function of road capacity as

follows:

)]()[1( aa yICost (9)

where = parameter that transforms the capital cost of

the project into unit period cost.


Revenue is the number of users multiplying the toll

charge:

Revenueaa xuv , (10)

Hence, profit of the private toll road for a realization

is

aaAa

aaAa

yIxuvuvu 1,,, (11)

Hence, the mean-variance model for profit maximization is to maximize the expected profit and to minimize the variance of profit subject to the non-negative constraints on the toll-capacity combination on the toll links.

,,min

,,max)),(,(min

uvuV

uvuEuvuF

u

(12)

subject to: x y a Aa a0 0, , (13)

where ,, uvu = profit (revenue-cost) of realization ;

))],(,([ uvuE = expected profit; and ))],(,([ uvuV =

variance of profit.

2.2. A mean-variance BOT model for social

welfare maximization

The social welfare in dollars per unit period generated from the BOT project is defined to be the sum of consumer and producer surplus as given below:

Ww

d

AAa

aaa

w

uvtuvdDuzuS0

1 ),(),()({1

),,(

Aa

aa

Aa

aaa yIuvtuv )(),,(),( (14)

Hence, the mean-variance model for social welfare maximization is to maximize the expected social welfare and to minimize its variance

,,min

,,max)),(,(min

uzuSV

uzuSEuzuF

u

(15)

subject to: x y a Aa a0 0, , (16)

where ,, uzuS = social welfare of realization ;

))],(,([ uzuSE = expected social welfare; and

))],(,([ uzuSV = variance of social welfare.

3. Solution procedure

We use stochastic simulation to simulate the uncertainty of travel demand function based on probability distribution with pre-defined mean and variance. A Latin Hypercube Sampling (LHS) technique is adopted to generate random traffic demand variates according to a Normal distribution. LHS is a stratified sampling method that has shown to outperform the Monte Carlo (MC) method [13]. It partitions the input distribution into intervals of equal probability. Only one random variate is sampled within each interval. This

sampling technique can significantly reduce the number of samples while still achieve a reasonably level of accuracy.

Bi-level programming problems are generally difficult to solve because evaluation of the upper-level objective function requires solution of the lower-level problem. For network design problems, the lower-level problem can be considered as nonlinear constraints. This often makes the bi-level programs non-convex. To tackle the non-convexity issue, we use genetic algorithm (GA) [14]. For multi-objective optimization problem, it is solved using the distance-based method [16]. The simulation-based multi-objective genetic algorithm (SMOGA) procedure is summarized below:

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. Initialize p

(counter for the number of solutions). Step 2. Evaluate the objective function of solution p

with the maximum number of samples. Mean and standard deviation of objective value is collected.

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 >

mN .

Step 5. Report the non-dominated solution set. As mentioned above, the distance-based method [15]

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 fitness values to each solution according to a distance measure with reference to the non-dominated solutions obtained in the previous generation. The general solution procedure of the distance-based method is adapted from Osyczka and Kundu [15] and is provided below.

Step 3.1. The first generated solution is taken as a Pareto solution with a potential value

1d , 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

1d .

Step 3.2. For a new solution u, calculate the relative distances to all existing Pareto solutions:

mlf

uffud

q

k kl

kkl

l ,...,2,1,)(

)(

2

1

(17)

where m is the number of Pareto solutions obtained by genetic search, uf k

denotes the value of the kth

objective of the new solution u, and klf denotes the value


of the kth objective for the lth Pareto solution. Then find the minimum distance:

mludd ll,...,2,1,)(min*

(18)

where *l indicates the nearest existing Pareto solution to

the new solution u.Step 3.3. Compare the new solution u with all

existing Pareto solutions: (a). If the solution is a new Pareto solution and it

dominates at least one of the existing Pareto solutions, calculate its fitness value:

)(*max udpFl

(19)

wheremaxp is the maximum potential value. Then set

maxp = F. Update the set of Pareto solutions. Set the

potential value of the new solution to be F.(b). If the solution is a new Pareto solution, calculate

its fitness value:

)(** udpFll

(20)

add it to the Pareto solution set with a potential value of F. If F >

maxp , set maxp = F.

(c). If the solution is not a new Pareto solution, calculate its fitness value:

)(** udpFll

(21)

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

4. Numerical experiment

4.1. Problem setting

The simulation-based multi-objective GA (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 [6]. Here we want to determine the optimal toll and capacity to satisfy both objectives under profit maximization and social welfare maximization schemes. The network is depicted in Figure 1. 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.

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

4

0 15.00.1)(a

a

aaac

vtvt

(22)

whereac = capacity of link a. The O-D demand function

is:

)exp( www cDd (23)

wherewD = the potential demand; = scaling parameter

which reflects the sensitivity of demand to full trip price;

andwc = 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 [6].

Hong Kong

1

4

23

7

8

1

2

6

5

9

10 4

3

Guangzhou

Zhuhai Shenzhen

Figure 1. Pearl River Delta regional network

In this case study, the following parameters are used:

Population size is 25 chromosomes.

The maximum number of generations is 50.

The maximum number of samples is 1000.

Probability of mutation is 0.15.

0 : For simplicity, the ratio of maintenance-

operating costs to the capital cost is set to zero. 5104.3 (1/h), 61010k (HK$/h.veh/h),

120/1 (h/HK$), and = 1

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

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

The lower bound and upper bound for capacity are [1000 veh/h, 9000 veh/h].

4.2. Numerical results

In the distance-based method, both objectives (mean and variance) are explicitly 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 2 depicts the convergence curve for profit maximization. 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 18th generation. Figure 3 shows the convergence curve for the social welfare maximization


problem. Similar convergence characteristics are also observed in the social welfare maximization case.

95000

96000

97000

98000

99000

100000

101000

102000

103000

104000

105000

0 5 10 15 20 25 30 35 40 45 50

Number of generations

Ex

pe

cte

d p

rofi

t (H

K$

)

10000

10500

11000

11500

12000

12500

13000

13500

14000

ST

DE

V o

f p

rofi

t (H

K$

)

Expected profit STDEV of profit

Figure 2. Convergence curve for profit

maximization

210000

215000

220000

225000

230000

235000

240000

0 5 10 15 20 25 30 35 40 45 50

Number of generations

Ex

pe

cte

d w

elf

are

(H

K$

)

47000

49000

51000

53000

55000

57000

ST

DE

V o

f w

elf

are

(H

K$

)

Expected welfare STDEV of welfare

Figure 3. Convergence curve for social welfare

maximization

Figures 4 and 5 display the evolution of the non-dominated solutions resulting from the first, third, and fiftieth generations. These two figures show how the non-dominated solutions migrate to the Pareto frontier. It appears that the non-dominated solutions obtained at the fiftieth generation are well-converged and well-distributed in the objective space.

Due to space limitation, only a limited set of approximate Pareto optimal solutions are provided in Tables 1 and 2. Table 1 provides the design variables (toll-capacity combination) corresponding to some approximate Pareto optimal solutions under the profit maximization scheme depicted in Figure 4. The first solution is the best solution obtained from a single objective (i.e., maximizing the expected profit without considering the variance of profit). This solution gives the maximum expected profit and serves as a benchmark solution for comparing the non-dominated solutions

obtained from solving both objectives simultaneously. From Table 1, we observe that the tradeoff between the two conflicting objectives is dependent on the selection of the toll-capacity combination for the BOT project. There are many such instances from the set of non-dominated solutions that the decision makers can choose based on their risk preferences and criteria. Table 2 provides the design variables (toll-capacity combination) corresponding to some approximate Pareto optimal solutions under the social welfare maximization scheme depicted in Figure 5. Again, the first solution is the best solution obtained from a single objective (i.e., maximizing the expected social welfare without considering the variance of social welfare). Similar tradeoff between the two objectives is also observed in the social welfare maximization problem.

0

20000

40000

60000

80000

100000

120000

140000

160000

0 5000 10000 15000 20000 25000

STDEV of profit (HK$)

Ex

pe

cte

d p

rofi

t (H

K$

)

First generation Third generation Fiftith generation

Figure 4. Evolution of non-dominated solutions under

profit maximization

0

50000

100000

150000

200000

250000

300000

350000

45000 50000 55000 60000 65000 70000

STDEV of welfare (HK$)

Exp

ecte

d w

elf

are

(H

K$)

First generation Third generation Fiftith generation

Figure 5. Evolution of non-dominated solutions under

for social welfare maximization


Table 1. Non-dominated solutions for profit

maximization

Toll Capacity Expected STDEV

(HK$) (veh/h) profit (HK$) profit (HK$)

55.65 2,600 161,657 -

66.10 4,500 143,820 15,752

40.76 2,400 120,222 8,011

47.77 3,200 111,469 6,883

50.34 7,100 132,848 10,324

51.38 7,700 106,481 5,791

57.70 3,800 136,544 11,575

64.93 2,500 128,004 9,149

80.15 1,100 146,093 21,671

93.47 1,600 54,446 1,896

Table 2. Non-dominated solutions for social

welfare maximization

Toll Capacity Expected STDEV

(HK$) (veh/h) welfare (HK$) welfare (HK$)

19.40 4,400 310,457 -

6.70 3,600 301,223 64,504

20.15 5,500 172,934 50,496

25.78 2,300 299,230 64,203

26.02 6,400 262,514 56,319

27.34 3,100 234,406 53,753

39.22 7,400 269,147 58,122

40.12 6,200 238,144 55,219

45.54 1,500 292,544 62,900

14.49 4,900 138,186 48,941

5. Conclusions and future research

In this study, we considered multiple objectives and demand uncertainty in transportation network design problem. The SMOGA framework was developed to solve the stochastic multi-objective network design problem. Using a special network design problem called the Build-Operate-Transfer (BOT) problem, we demonstrated the feasibility of using the SMOGA framework on two mean-variance BOT models: the profit maximization problem and the welfare maximization problem. It was found that the proposed framework could search simultaneously the Pareto optimal solutions under travel demand uncertainty. For future research, we plan to extend the study to consider more than two objectives.

6. References

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