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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
achen@cc.usu.edu
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.
Proceedings of the Fourth International Symposium on Uncertainty Modeling and Analysis (ISUMA’03) 0-7695-1997-0/03 $17.00 © 2003 IEEE
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
Proceedings of the Fourth International Symposium on Uncertainty Modeling and Analysis (ISUMA’03) 0-7695-1997-0/03 $17.00 © 2003 IEEE
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
Proceedings of the Fourth International Symposium on Uncertainty Modeling and Analysis (ISUMA’03) 0-7695-1997-0/03 $17.00 © 2003 IEEE
problem. Similar convergence characteristics are also observed in the social welfare maximization case.
95000
96000
97000
98000
99000
100000
101000
102000
103000
104000
105000
0 5 10 15 20 25 30 35 40 45 50
Number of generations
Ex
pe
cte
d p
rofi
t (H
K$
)
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
Proceedings of the Fourth International Symposium on Uncertainty Modeling and Analysis (ISUMA’03) 0-7695-1997-0/03 $17.00 © 2003 IEEE
Table 1. Non-dominated solutions for profit
maximization
Toll Capacity Expected STDEV
(HK$) (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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Proceedings of the Fourth International Symposium on Uncertainty Modeling and Analysis (ISUMA’03) 0-7695-1997-0/03 $17.00 © 2003 IEEE
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