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A reliability‐based network design problem
Article · June 2005
DOI: 10.1002/atr.5670390303
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Journal ofAdvanced Transportation, Vol. 39, No. 3, pp. 247-270 www. advanced-transport. corn
A Reliability-Based Network Design Problem
Piya Chootinan S. C. Wong Anthony Chen
This paper presents a reliability-based network design problem. A network reliability concept is embedded into the continuous network design problem in which travelers’ route choice behavior follows the stochastic user equilibrium assumption. A new capacity-reliability index is introduced to measure the probability that all of the network links are operated below their capacities when serving different traffic patterns deviating from the average condition. The reliability- based network design problem is formulated as a bi-level program in which the lower level sub-program is the probit- based stochastic user equilibrium problem and the upper level sub-program is the maximization of the new capacity reliability index. The lower level sub-program is solved by a variant of the method of successive averages using the exponential average to represent the learning process of network users on a daily basis that results in the daily variation of traffic-flow pattern, and Monte Car10 stochastic loading.
The upper level sub-program is tackled by means of genetic algorithms. A numerical example is used to demonstrate the concept of the proposed framework.
Keywords: Reliability analysis, capacity reliability, continuous network design problem, bi-level program, stochastic user equilibrium.
1. Introduction
The network design problem (NDP) is one of optimizing the improvement of a transportation network with respect to a system-wide objective while considering the route choice behavior of network users.
P. Chootinan and A. Chen, Department of Civil & Environmental Engineering, Utah State Universiv, Logan, UT, USA
S.C. Wong, Department of Civil Engineering, The Universiw of Hong Kong, Hong Kong, China
Received: August 2004; Accepted: March 2005
248 P. Chootinan, S.C. Wong, and A . Chen
It involves making the optimal decisions at the strategic, tactical, and operational levels as to how to choose improvements for the network in such a way as to make efficient use of limited resources to achieve the stated objective (e.g., the minimization of total travel time or the maximization of social welfare). Based on the design variables, NDPs can be categorized into three types: discrete NDPs, continuous NDPs, and mixed NDPs. Discrete NDPs involve binary design variables such as the addition of new roads (LeBlanc, 1975; Poorzahedy and Turnquist, 1982; Chen and Alfa, 1991), and the selection of one-way and two-way network designs (Drezner and Wesolowsky, 1997, 2002). Continuous NDPs involve continuous design variables. Examples of continuous NDPs include capacity enhancement (Abdulaal and LeBlanc, 1979;
Suwansirikul and Friesz, 1987; Davis, 1994; Meng and Yang, 2002), road pricing (Ferrari, 1995, 2002; Yang and Lam, 1996; Yang and Bell, 1997; Szeto and Lo, 2005), signal timing (Allsop, 1974; Allsop and Charlesworth, 1977; Cantarella et al., 1991; Wong and Yang, 1997, 1999; Chiou, 1999; Gao and Song, 2002), and ramp metering (Yang et al., 1994; Yang and Yagar, 1994). Mixed NDPs involve both discrete and continuous design variables. Applications in this category include combined landuse-network design problems (Lin and Feng, 2003), simultaneous location-network design problems (Melkote and Daskin, 2001), and cordon-based network congestion pricing (Zhang and Yang, 2004).
The NDP has long been recognized to be one of the most difficult and challenging problems in transportation literature. It has received much attention as indicated by the growing body of literature mentioned previously. For a comprehensive survey of the modeling and algorithm development related to network design problems over the past two decades, see Boyce (1984), Magnanti and Wong (1984), Friesz (1985), and Yang and Bell (1998). Despite the extensive development of the NDP, the emerging technique of reliability analysis, however, has received relatively little attention in designing reliable networks.
Clark and Watling (2005) have comprehensively classified reliability analyses into the categories of connectivity reliability (Bell and Iida, 1997), travel time reliability (Du and Nicholson, 1997; Bell et al., 1999;
Clark and Watling, 2005), capacity reliability (Chen et al., 1999, 2000, 2002), behavioral reliability (Mirchandani and Soroush, 1987; Yin and Ieda, 2001; Lo and Tung, 2003), and potential reliability (Berdica, 2002;
Bell, 2000; Bell and Cassir, 2002). For connectivity reliability NDPs, Bell and Iida (1997) investigated the discrete NDP of maximizing the
A Reliability-Based Network ... 249
reliability of a network so that it will function after a major natural disaster, such as an earthquake, that would affect the connectivity of the road network. For behavioral reliability NDPs, Lo and Tung (2003) studied also the problem of allocating design capacities to maximize the multiplier on the origin-destination (0-D) demand matrix (network reserve capacity), subject to the probabilistic user-equilibrium constraints on network links and the budgetary constraint.
Conventional capacity reliability analysis utilizes the concept of reserve capacity that was originally applied to the isolated junctions (Webster and Cobbe, 1966; Allsop, 1972; Wong, 1996) and was then extended to the signal-controlled networks (Wong and Yang, 1997).
Chen et al. (1999,2000, and 2002) provided an assessment methodology, which combines the reliability and uncertainty analysis, network equilibrium models, sensitivity analysis of the equilibrium network-flow and the expected performance measure, as well as Monte Carlo methods, to assess the capacity reliability of a degradable road network. Yang and Wang (2002) compared and contrasted performances of the network that were designed based on the capacity maximization and travel time minimization. However, they did not deal with reliability measures in their paper. In contrast, Lo and Tung (2003) maximized the 0-D multiplier (network reserve capacity) in the context of probabilistic user equilibrium. However, the capacity reliability measure that is based on the reserve capacity concept is rather restrictive because it requires that travel demands of all 0-D pairs increase or decrease by the same proportion. Gao and Song (2002) criticized this restriction and generalized the reserve capacity definition by allowing different multipliers for individual 0-D pairs. Although the use of a common multiplier has the advantages of problem tractability and allows the creation of buffers to accommodate the possible increase in traffic in different parts of the network, the network reserve capacity is not determined by explicitly taking into account the variation of traffic-flow pattern (e.g., links may be more or less vulnerable), thus its relationship to the network reliability is yet to be established.
In this paper, we introduce a new capacity reliability index, which measures the probability that all of the traffic links are operating below their respective capacities. The reliability-based NDP is formulated as a bi-level program of which the lower level sub-program is the probit- based stochastic user equilibrium problem and the upper level sub- program is the maximization of the new capacity reliability index. The lower level sub-program is solved by a variant of the method of
250 P. Chootinan, S. C. Wong, and A . Chen
successive averages using the exponential average to represent the day- to-day learning process of network users, and Monte Carlo stochastic loading. The upper level sub-program is tackled by means of genetic algorithms. A numerical example is used to demonstrate the concept of the proposed methodology.
2. Mathematical Formulation
In the NDP, there are two sets of decision makers: the network users and the network planner. The network users are free to choose their routes such that their individual (perceived) travel costs are minimized, whereas the planner aims to make the best use of limited resources to optimize network performances (e.g., reducing congestion, minimizing environmental impact, maximizing throughputs), taking into account users’ route choice behavior. However, it is important to recognize that the decisions made by the planner can only influence, not control, the decisions of the network users. This problem has a leader-follower structure, and is analogous to the Stackelberg game (Fisk, 1984). In a Stackelberg game, the planner is the leader, and the network users are grouped together as the follower. Typically, the NDP can be formulated as the following bi-level program.
Minimize F(x,u) G(x,u) I 0 (UP) u
subject to
where x = x(u) is implicitly defined by:
where F and u are respectively the objective function and the decision vector of the upper level sub-program (UP), G is the constraint set of UP, fand x are respectively the objective function and the decision vector of the lower level sub-program (LP), and g is the constraint set of LP. The upper level sub-program describes the leader or planner problem, and the lower level sub-program represents the follower or user’s behavioral
A Reliability-Based Network.. . 251
problem. In this paper, we consider the continuous NDP, where link capacity enhancements are treated as continuous variables.
2. I Upper level sub-program
Consider a transportation network G(N,A), where N is the set of nodes and A is the set of links. Let qrs be the travel demand between zones r E R c
N
and s E S c N, where R is the set of origins and S the set of destinations. On the assumption of a stochastic user equilibrium, the flow assigned to any path or link is a random variable. Let v, and C, be respectively the flow and capacity of link a. The probability of all network links operating below their capacities, given below, defines the network reliability index.L = Pr(v, 5 C,,VU E A )
(3)
The problem studied here is that of determining the optimal capacity enhancements by maximizing the reliability of the network (or minimizing the probability of overloading the network links) under a budget constraint, while taking into account the behavior of network users. The problem can now be formulated as a bi-level program.
Maximize L = Pr(v,(u) 5 C,(u),Vu E A ) subject to:
g(u> 2 B
U,in I u 5 u,,
where V,(U'E) is a random variable of traffic flows on link a. The probability can be evaluated by repeatedly solving the stochastic traffic assignment problem (lower level sub-program) for a given design vector u. Constraint (4b) ensures that the expenditure associated with the improvement strategy is less than the available budget (23). Constraint (4c) defines the possible range of design variables.
2.2 Lower level sub-program
252 P. Chootinan, S. C. Wong, and A. Chen
The lower level sub-program is a stochastic user equilibrium problem that can be formulated as follows.
Minimize Z(v) =-cqr3 . E rnin{T;"} T"(v) + ~ v a t a ( y a , u ) - ~ b a ( w , u ) d o
"0
( 5 )
rs [ k
I- 1
a a owhere qrs is the demand between origin r and destination s,
*;
is theperceived travel cost on path k connecting origin r and destination s,
Trs
is the set of measured travel costs of all paths between origin r and destination s, v is the set of link flows, and ta is the cost of link a, which is a function of link flows and design variables.
A variant of the method of successive average (MSA) combined with a Monte Car10 stochastic loading is adopted to solve the lower level sub- program (Sheffi, 1985). The link flow pattern obtained during each MSA iteration is interpreted a sample of link-flow pattern in a day-to-day context and forms the distributions of link flows (see Figure 1). In the MSA step, the exponential average is adopted instead of the commonly used arithmetic average. This is because the arithmetic average assumes that travelers can memorize every piece of information (e.g., travel time experienced) fiom the beginning and average them out equally, thus it may not be a good representation of travelers' behavior on a day-to-day basis. The idea behind the exponential average is to average out all pieces of information but put more emphasis on the recent information.
The last data point (travel time, t,J has a weight of 1 while the data point before it (tn-2) has a weight of w (the value between 0 and 1). The data point before tn-2 has a weight of w 2 and the weights for the remainders can be determined in the same manner. The exponential average of n-1 data point (e.g., travel time experienced up to the beginning of day n) is given by:
2 3
?,-I
+
W?,-2+
w ?n-3+
w ?"-4+...+
wn-lt0z, =
l + W + W 2 + W 3 +
...+
wn-l (6)From the above expression, the exponential average is quite suitable in the context of day-to-day behavior. First, as already stated, it gives higher weight to the recent data point, which implies that travelers make decision based very much on the recent experience because it is difficult to assume that they can memorize every single piece of past experience (Arentze and Timmermans, 2005). Second, the length of memory (i.e.,
A Reliability-Based Network. .. 253
- - - -
I I I I I I I I I I I I I I *
I
-
I
the number of days of which the information can be processed) can be controlled by the value of w. Basically, w should be set in such a way that the data point beyond the limitation (length of memory) would not significantly influence the computation of the average. Here, w can be interpreted as a learning rate (Walting and Hazelton, 2003). It is known from the geometric series that the denominator of Equation (6) is well approximated by l/l-w. The recursive formulae of the exponential average can be written as:
~
Each outer Initialization of travel time iteration
I llnk flow
- - -
-- - - -
t -- - - - - - - - -
I generates aA sample link flow Stochastic network loading
(Probit model with Monte Carlo Simulation)
Perfom MSA to update the
experience
- - - ---- - - - -
1 flows+
I I~ l l sample link flow
average travel time perceived by
i
~ ~ ~ m f ~ ~ ~ e * e Probab'l'ryI I probability distributions of link
k
Flowthe users from their past
r, = (1 - w) rn-l + wt,_, + w2t,-, + ...+ w"-lt, = (1 - w)t,_, + wr,-, , V n = 1 ,..., N where T~ = to.
( 1
Figure 1. Development of link-flow distribution
(7)
3. Reliability-based Genetic Algorithm Procedure
Many heuristic algorithms have been developed to solve the bi-level network design problem. A summary survey is provided by Yang and Bell (1998). Bi-level problems are generally difficult to solve, because the evaluation of the upper level objective functiodconstraints requires the solution of the traffic assignment at the lower level sub-program. For an NDP, the lower level sub-program can be considered as a system of nonlinear (equilibrium) constraints. Although the upper and the lower level sub-programs are individually convex mathematical programs, their combination is usually non-convex, which is difficult to solve using standard optimization methods. The genetic algorithm (GA) based
254 P. Chootinan, S. C. Wong, and A . Chen
approach has been proposed as a powerful technique for solving difficult problems that do not possess nice mathematical properties (e.g., continuity, differentiability, uni-modal, convexity). Detailed descriptions of the GA can be found in Goldberg (1989) and Gen and Cheng (2000).
Yin (2000) demonstrated that a GA-based approach could be used to solve a continuous NDP posed as a bi-level program. Chen et al. (2003) developed a simulation-based GA procedure for the solution of the Build-Operate-Transfer (BOT) network design problem with demand uncertainty. In this paper, the GA is used to solve the proposed reliability-based (capacity reliability) NDP.
3. I GA implementation
Typical GA implementation involves the definition of the solution (or chromosome) representation, fitness evaluation, and GA operators (i.e., reproduction, crossover, and mutation). The GA implementation is described in detail in the followings.
3. I . I Chromosome representation
In the GA, decision variables can be coded as binary, integer, or real representation (Rothlauf, 2000). For the reliability-based NDP proposed here, the solutions (or design variables u,) are represented by a string of real numbers with a length equal to the number of network links
IAl.
Thevalue of each gene represents the link capacity expansion, which is limited by the upper and lower limits of Constraint (4c).
3. I .2 Fitness evaluation
The fitness value determines the probability of each individual solution reproducing a new solution. For unconstrained problems, the fitness value is strictly proportional to the objective value (L in this case).
A solution with a higher objective value will have a higher chance of being selected. However, the feasibility of the solution must be taken into account for constrained problems. In this study, the penalty method is used to handle the budget constraint in Equation (4b). Basically, the fitness of the solution is penalized whenever the solution (improvement plan) violates the budget constraint. However, no special constraint handling is required for the boundary constraints of the design variables in Equation (4c), because they are explicitly used to limit the search
A Reliability-Based Network.. . 255
space being explored by the GA. Equation (8), given below, is the fitness function used in this study.
where L is the original objective value as defined earlier, and p is the penalty value for penalizing the infeasible solutions (p > 0). To evaluate the fitness of each solution (e.g., one improvement plan) requires the solution of the probit-based traffic assignment problem, which will be described later.
3.1.3 Reproduction
Reproduction is the process of selecting solutions from a pool of populations for mating purposes. It directs the genetic search toward the promising area of the search space. The solutions are selected on the basis of their fitness. The higher the fitness value, the higher the chance of being selected. After the fitness of solutions ( L ) are computed, the chance of an individual solution being selected (or the probability of survival) is determined by:
Y
V n (9)
where n is the index of chromosomes in the population (in each generation). Here, we use the roulette wheel selection to randomly select the candidate solutions. The solution with the higher chance will occupy a larger portion on the roulette wheel. The selection process is based on a random number drawn between 0 and 1, and the solution associated with the intercepted portion of the wheel will enter the mating pool.
3.1.4 Crossover and mutation
Crossover and mutation provide the means to stochastically manipulate the existing solutions to generate new offspring. The crossover plays a major role in the exchange of genetic materials (or the characteristics of solutions) between a selected pair of solutions. In this
256 P. Chootinan, S.C. Wong, and A. Chen
way, two new offspring, each of which inherits some of the characteristics of its parents, will be created. As real-code representation is used to represent the chromosomes, the arithmetic crossover is used.
This method is similar to the linear combination of two solution vectors with a random fraction. The number of solutions exchanging their genetic materials is determined by a so-called probability of crossover
After the crossover is performed, the mutation is applied. The major role of mutation is to introduce new genetic materials to the pool of solutions. In turn, the mutation gives the genetic search the ability to jump out of the local optima whenever it is getting trapped. There is also a probability associated with this operator, the probability of mutation
(Pd,
which is generally set at a very small value. The values of selected genes are alternated within the possible ranges defined by the lower and upper limits (e.g. Constraint 4c).3.2 GA solution procedure 3.2.1 Upper level sub-program
The upper level sub-program is solved by the genetic algorithm procedure. A flowchart is provided in Figure 2, and can be summarized as follows.
Step UO.
Step U1.
Step U2.
Step U3.
step u4.
Starting with an empty network (i.e. free-flow travel costs), perform the lower level sub-program by a variant of the method of successive averages to obtain the perceived average travel costs in the original network,
Define the GA parameters, such as crossover rate, mutation rate, population size, and maximum number of generations.
Generate a set of initial solutions (i.e., design variables), and initialize the generation counter.
Evaluate the solutions by solving the lower level sub- program by a variant of the method of successive averages.
If the stopping criterion is met (e.g., the maximum number of generations), then terminate the procedure and report the solution set; otherwise, go to Step U4.
Using the current solutions as parents, perform the GA operators (i.e., reproduction, crossover, and mutation) to
zy’, V a
A Reliability-Based Network ... 25 7
generate offspring (new solutions). Increase the generation counter and return to Step U2.
Defme problem and GA parameters
.
Generate initial population1.
New set of solutionsEvaluate solutions - Probit trafllc assignment
.
- Penalty methodI
Mutation
I I
t
I I I I I
Cross over
1
,
+
Report solutionFigure 2. Reliability-based GA flowchart.
3.2.2 Lower level sub-program
In Step U2 of the GA procedure, the evaluation of the fitness of each solution requires the solution of the probit-based stochastic traffic assignment procedure (Sheffi, 1985). The exponential average combined with a Monte Carlo stochastic loading is adopted to solve the lower level sub-program. The steps are provided as follows.
Step LO. Initialization. Use "la'? (obtained from Step UO) as the initial average travel costs. Set n: =l.
258
Step L 1 .
Step L2.
Step L3.
P. Chootinan, S. C. Wong, and A. Chen
Determine the link $’ow pattern. Perform a stochastic network loading (Probit model with Monte Carlo Simulation) based on the current set of link travel costs
.
This yields the link flow pattern, v‘”), which will rr-‘), V abe collected to form the probability distributions of link flows (for the determination of capacity reliability index).
Update the link costs. Update the average travel cost pattern according to a learning rate w:
5:) = (1 - w)
.
t , ( v y ,”)+
w. rr-1)
, v uConvergence testing. Set n: = n + l . If n is greater than the number of MSA iterations, go to Step L1; otherwise, terminate the algorithm.
3.2.3 Interpretation of results
The probit-based traffic assignment adopted in the lower level sub- program resembles the day-to-day variation in the route choice of road users in the context of a stochastic assignment framework, in which all travelers are assumed to be rational and to have the same information acquisition (or learning) mechanism. Travelers do not know in advance the actual costs they will experience during a trip. They make their route choice decisions based on their personal experience, which results from their past experiences and learning processes (Walting and Hazelton, 2003).
The distribution of the link flow pattern from the lower level sub- program can be interpreted as a day-to-day variation. In each iteration, we compare the link flows and the corresponding link capacities in the entire network. If all of the links are operated below their capacities, then we call it a network “success”; otherwise, it is a network “failure”. The number of network successes divided by the total number of iterations is a good estimator for the network reliability index in Equation (3).
4. Numerical Example
4. I Problem description
The network depicted in Figure 3 is used to illustrate the proposed framework. It consists of 5 nodes, 7 links, and 2 origin-destination (0-D)
A Reliability-Based Network.. . 259
pairs (1-4 and 1-5). The travel demands of the 0 - D pairs 1-4 is 21.2, and of pairs 1-5 is 26.5. The characteristics of the network links are provided in Table 1. The link cost function, fa(.), used in this study is the standard Bureau of Public Roads (BPR) function (given below) where a is 0.15 and
p
is 4.0.(1 1) I , (v, ) = I:
.
(1+
a. (v, /c,y)
0
where is the travel cost at free flow condition. In the stochastic network loading (Probit model with Monte Carlo Simulation) procedure, the standard deviation is assumed to be one-third of link travel time. The variance-covariance matrix is a diagonal matrix, and the number of inner iterations is 1,000 (simulation iteration). In this study, the convergence criterion of the MSA for the lower level sub-program is 1,000 repetitions (outer iterations).
Figure 3. Topology of the test network.
260 P. Chootinan, S.C. Wong, and A . Chen
Table 1. Characteristics of Network links
Node Free-flow
travel cost Capacity Link no.
From To
1 1 2 4.0 25.0
2 1 3 5.2 25.0
3 2 3 1
.o
15.04 2 4 5.0 15.0
5 2 3 5.0 15.0
6 3 4 4.0 15.0
7 3 5 4.0 15.0
To improve the performance of the current network, it is assumed that there are 10 units of budget (B = 10) available. In addition, the capacity of the network links is allowed to expand by up to 20 percent of the existing capacity (5 units for Links 1 and 2, and 3 units for the rest).
The cost of the capacity expansion is given by:
g , ( u , ) = 0.30-u,2, 'da
where ua is the capacity expansion of link a.
In this s ~ d y , 200 generations of the GA were performed with 8 chromosomes in each generation (population size). Real code is used for the chromosome representation (a chromosome of length 7 for the capacity expansion on 7 links). The probabilities (rates) of arithmetic crossover and random mutation are 0.50 and 0.20, respectively. The penalty value, p, is set at 100000, which is much larger than the maximal possible value of L (network reliability: 1 .OOO).
4.2 EHect of learning rate (w)
This section investigates the effect of learning rate (w) on the distributions of link flows (i.e., link-flow pattern). In the probit-based stochastic traffic assignment, link costs used for making route choice decision for the current iteration (e.g., today) are based on the exponential average of link costs experienced up to the last iteration (e.g., yesterday). Using the test network depicted in Figure 3, Tables 2 and 3 present the characteristics of link-flow distributions (ie., mean and
A Reliability-Based Network.. . 261
standard deviation) with different w values of learning rates. Note that the maximum number of inner iterations is 1,000. It is clear from the results that as length of the memory becomes shorter (lower w value), the variation of traffic flow is higher (i.e., travelers make their decisions based very much on the most recent information - yesterday, thus the length of memory is irrelevant).
Table 2. Mean link flows at different learning rates w Value (Learning Rate)
0.1 0.3 0.5 0.7 0.9
Link
1 25.6418 27.8942 27.9225 27.9304 27.9306 2 22.0581 19.8057 19.7775 19.7696 19.7694
3 5.3216 8.8915 8.8973 8.9225 8.9 195
4 8.2654 8.0764 8.0829 8.0706 8.0815
5 12.0548 10.9263 10.9423 10.9372 10.9296 6 12.9346 13.1236 13.1 171 13.1294 13.1 185 7 14.4452 15.5737 15.5577 15.5628 15.5704
Table 3. Standard deviations of link flows at different learnine rates w Value (Learning Rate)
0.1 0.3 0.5 0.7 0.9
Link
1 20.5093 1.501 1 0.91 16 0.7578 0.7208 2 20.5093 1.501 1 0.91 16 0.7578 0.7208
3 2.5296 0.6059 0.5267 0.5194 0.5441
4 6.6930 0.4794 0.3602 0.3458 0.3363
5 1 1.3463 0.8417 0.5154 0.4568 0.4307
6 6.6930 0.4794 0.3602 0.3458 0.3363
7 1 1.3463 0.8417 0.5 154 0.4568 0.4307
262 P. Chootinan, S.C. Wong, and A. Chen
4.3 Numerical results
Based on the current network condition and demand level, using w =
0.90, the link flows, which are the results of lower level sub-program, are obtained as shown in Table 4. The mean link flows on Links 1 and 7 clearly exceed the link capacities, whereas those on the other links do not. Further consideration of the variation in link flows (e.g., day-to-day variation), reveals that the capacity of Link 1 is not sufficient to handle the current demand level in any circumstances. However, there is about a 9 percent chance that the capacity of Link 7 will be sufficient. As Link 1 always fails to handle the current demand level, the likelihood that all network links will operate below their capacity is zero.
The reliability-based network design problem is solved by the bi- level programming approach described in Section 3. Figure 3 depicts the convergence curve of the GA. Network links can be categorized into two groups: critical links (Links 1 and 7) and non-critical links. As the generation increases, the reliability of the critical links (Link 1) is improved. At the final generation (20Oth), the solution given below is obtained:
u = [5.0000 1.1 139 0.0000 0.0000 0.4745 0.8692 2.14391
Table 4. Link flows obtained from a probit-based stochastic traffic assignment
Link Canacitv _ _
interval Link flow
( V a l Ca)
1 ,
M e a n LB UB
99% confidence Pr
.a
25.00 25.00 15.00 15.00 15.00 15.00 15.00
27.9306 19.7694 8.9 195 8.08 15 10.9296 13.1185 15.5704
0.7208 0.7208 0.5441 0.3363 0.4307 0.3363 0.4307
26.0739 29.7873 17.9127 21.6261 7.5180 10.3210 7.2152 8.9478 9.8202 12.0390 12.2522 13.9848 14.46 10 16.6798
0.0000 1
.oooo
1
.oooo
1
.oooo
1
.oooo
1
.oooo
0.0930
*
The(1-a) x 100% confidence interval is [p-z,-,,, . ~ , c ( + z , - , / ~ .o],where p is the mean and o the standard deviation.A ReliabiliQ-Based Network ... 263
It should be noted that the globality of the solution obtained here cannot be guaranteed. This is due to the non-convex nature of bi-level transportation problems, and the fact that the GA is a heuristic approach.
The expenditure that corresponds to this near-optimal strategy found by the GA is 9.55 units. Capacity expansions are introduced for Links 1 and 7 of which the existing capacities are insufficient. Under the new network configuration, the likelihood that all network links will operate below their capacities is 88.20 percent; Pr(vu-(C,) is 88.52 percent for Link I, and 100.0 percent for the rest. It can be observed that the small capacity expansions are also recommended for links with sufficient capacity (Links 2, 5, and 6). This is because the travel cost fhction adopted in Equation (9) depends on the link capacity. Hence, any increase in link capacity will reduce the congestion sensitivity of the travel cost with respect to the link flow. The small capacity expansions on the non-critical links help to draw a certain amount of traMic from the critical links and, in turn, enhance their reliability. In this example, the improved capacities on Links 2 and 6 help to divert the traffic from critical Links 1 and 7 to non-critical links on path 1+3+4. This results in an overall improvement in network reliability.
To verify the postulation above, the capacity allocation strategy obtained from the GA is modified. As only Links 1 and 7 are likely to be overloaded, we still increase the capacity of Link 1 to 5 units (upper limit), and spend the remaining budget to expand the capacity of Link 7 (equivalent to 2.8867 units). The capacities of all of the non-critical links remain unchanged. Stochastic traffic assignment is performed based on this modified strategy, and the resultant network reliability is only 84.80 percent. It is clear that, without the complements from the small capacity expansions, the modified strategy could not drive the system to the highest level of reliability, even though the budget is hlly utilized (10 units compared to 9.55 units recommended by the GA). This shows the necessity of such small capacity expansions. Table 5 compares the performances of the capacity allocation strategy obtained from the GA and the modified strategy.
264 P. Chootinan, S. C. Wong, and A . Chen
Table 5. Performances of the GA and modified capacity allocation strategies
Modified strategy GA strategy
Mean S.D
Link Link flow Pr Link flow Pr
Capacity (vai Capacity ( V a l
C*) c a )
Mean S.D
1 30.0000 29.3526 0.6731 0.8480 30.0000 29.2600 0.6689 0.8820 2 25.0000 18.3474 0.6731 1.0000 26.1139 18.4400 0.6689 1.0000 3 15.0000 10.4114 0.5123 1.0000 15.0000 10.2848 0.5223 1.0000 4 15.0000 8.4557 0.3382 1.0000 15.0000 8.2964 0.3207 1.0000 5 15.0000 10.4855 0.4368 1.0000 15.4745 10.6787 0.4207 1.0000 6 15.0000 12.7443 0.3382 1.0000 15.8692 12.9036 0.3207 1.0000 7 17.8867 16.0145 0.4368 1.0000 17.1439 15.8213 0.4207 1.0000
Figures 5 , 6, and 7 show the distributions of the flows (before and after improvement) on Links 1, 2 and 7, respectively. It should be noted that the flow patterns of these links are also changed following an improvement in their capacities. However, the capacity enhancement already accounts for the behavioral changes of network users in the lower level sub-program.
2. 3.4, 5, d 6 1.00
D
o.80vl
CI
2. 0.60
.-
- 3 8
0.40-Link 1 -Link2 -Link3
-- Link4 --Links -Link6 +Link7
0 40 80 120 160 200
Generation Number
Figure 4. Convergence curve of the GA
A Reliability-Based Network ... 265
1.200
1.ooO
g
0.8001
cn 3 c
& 0.600
3
& 0.400
0.200
0.ooO 0.8620
Exatilgcapciy
24.00 25.00 26.00 27.00 28.00 29.00 30.00 31.00 32.00 33.00 Link Flow
Figure 5. Distributions of flows on Link 1 (before and after improvement)
1.200
1.ooO
g
0.800cn E c
& 0.800
3
B P
g
0.4000.200
O.Oo0
ExbmgCapcty -
15.00 17.00 19.00 21.00 23.00 25.00 27.00
Link Flow
Figure 6. Distributions of flows on Link 2 (before and after improvement)
266
1.200
1.OW
1
0.8003
P
*
0.6002
v) CI
0.400
0.200
0.OW
._ ________.__..._...--
I
”-0.0830
P. Chootinan, S.C. Wong, and A . Chen
17.1439
13.00 14.00 15.00 16.00 17.00 18.00 19.00
Link Flow
Figure 7. Distributions of flows on Link 7 (before and after improvement)
5. Conclusion
This paper presents the reliability-based network design problem (NDP), which is based on the concept of capacity reliability. The new capacity reliability index is introduced to measure the probability that all of the network links are operating below their capacities (i.e., a network success). The reliability-based NDP studied here is formulated as a bi- level program in which the lower level sub-program is the probit-based stochastic user equilibrium problem and the upper level sub-program is the maximization of new reliability index. The lower level sub-program not only represents the stochastic behavior of network users, but also accounts for their day-to-day learning process. The exponential average is used to replace the arithmetic mean commonly used in the MSA step of the probit-based stochastic traffic assignment. This bi-level program is solved by means of genetic algorithms. The numerical example demonstrates the concept of the proposed framework. So far, although only the day-to-day route choice variability is considered, the proposed framework can easily be extended to incorporate the demand variability and the degradation of network capacity.
A Reliability-Based Network ... 267
Acknowledgements
The work described in this paper was supported by a CAREER grant from the National Science Foundation of the United States (CMS- 0134161) and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7 134/03E).
References
Abdulaal, M. and LeBlanc, L.J. (1979) Continuous equilibrium network design models. Transportation Research, 13B, 19-32.
Allsop, R.E. (1972) Estimating the traffic capacity of a signalized road junction. Transportation Research, 6,245-255.
Allosp, R.E. (1974) Some possibility for using traffic control to influence trip distribution and route choice. Proceedings of the Sixteenth International Symposium on Transportation and Traflc Theory.
Allsop, R.E. and Charlesworth, J.A. (1977) Traffic in a signal-controlled road network: an example of different signal timings inducing different routes. Trafic Engineering and Control, 18,262-264.
Arentze, T. and Timmermans, H. (2005) Modelling learning and adaptation in transportation contexts. Transportmetrica, 1, 13-22.
Bell, M.G.H. (2000) A game theory approach to measuring the performance reliability of transport networks. Transportation Research, 34B, 533-546.
Bell, M.G.H. and Cassir, C. (2002) Risk-averse user equilibrium traffic assignment: an application of game theory. Transportation Research, Bell, M.G.H., Cassir, C., Iida, Y. and Lam, W.H.K. (1 999) A sensitivity- based approach to network reliability assessment. Proceedings of the 14th International Symposium on Transportation and Traflc Theory, Jerusalem, pp. 283-300.
Bell, M.G.H. and Iida, Y. (1 997) Transportation Network Analysis. John Wiley and Sons, Chichester, U.K.
Berdica, K. (2002) An introduction to road vulnerability: what has been done, is done and should be done. Transport Policy, 9, 117-127.
Boyce, D.E. (1 984) Urban transportation network-equilibrium and design models: recent achievements and future prospects. Environment and Planning A , 16, 1445-1474.
36B, 67 1-682.
268 P. Chootinan, S. C. Wong, and A . Chen
Cantarella, G.E., Improta, G., and Sforza, A. (1991) Iterative procedure for equilibrium network traffic signal setting. Transportation Research, 24A, 241-249.
Chen, A., Ji, Z., and Recker, W. (2002) Travel time reliability with risk- sensitive travelers. Transportation Research Record, 1783,27-33.
Chen, A., Subprasom, K. and Ji, Z. (2003) A mean-variance model for the build-operate-transfer scheme under demand uncertainty.
Transportation Research Record, 1857,93- 101.
Chen, A., Tatneni, M., Lee, D.H. and Yang H. (2000) Effect of route choice models on estimating network capacity reliability.
Transportation Research Record, 1733,63-70.
Chen, A., Yang, H., Lo, H.K. and Tang, W.H. (1999) A capacity related reliability for transportation networks. Journal of Advanced
Transportation, 33, 183-200.
Chen, A., Yang, H., Lo, H.K. and Tang, W.H. (2002) Capacity reliability of a road network: an assessment methodology and numerical results. Transportation Research, 36B, 225-252.
Chen, M. and Alfa, A.S. (1991) A network design algorithm using a stochastic incremental traffic assignment approach. Transportation Science, 25,215-224.
Chiou, S.W. (1999) Optimization of area traffic control for equilibrium network flows. Transportation Science, 33,279-289.
Clark S. and Watling D. (2005) Modeling network travel time reliability under stochastic demand. Transportation Research, 39B, 1 19- 140.
Davis, G.A. (1994) Exact local solution of the continuous network design problem via stochastic user equilibrium assignment.
Transportation Research, 28B, 6 1-75.
Drener, Z. and Wesolowsky, G.O. (1997) Selecting an optimum configuration of one-way and two-way routes. Transportation Science, 31,386-394.
Drener, Z. and Wesolowsky, G.O. (2002) Network design: selecting and design of links and facility location. Transportation Research, 37A, Du, Z.P. and Nicholson, A.J. (1997) Degradable transportation systems:
sensitivity and reliability analysis. Transportation Research, 31B, Ferrari, P. (1 995) Road pricing and network equilibrium. Transportation Ferrari, P. (2002) Road network toll pricing and social welfare.
241-256.
225-237.
Research, 29B, 357-372.
Transportation Research, 36B, 47 1-483.
A Reliability-Based Network.. . 269
Fisk, C.S. (1984) Game theory and transportation systems modeling.
Transportation Research, 18B, 30 1-3 13.
Friesz, T.L. (1985) Transportation network equilibrium, design and aggregation: key developments and research opportunities.
Transportation Research, 19A, 41 3-427.
Gao, Z.Y. and Song, Y.F. (2002) A reserve capacity model of optimal signal control with user-equilibrium route choice. Transportation Research, 36B, 313-323.
Gen, M. and Cheng, R. (2000) Genetic Algorithms and Engineering Optimization. John Wiley and Sons, Inc., New York.
Goldberg, D. (1 989) Genetic Algorithms in Search, Optimization, and Machine Learning. Addison- Wesley, Reading, MA.
LeBlanc L.J. (1975) An algorithm for the discrete network design problem. Transportation Science, 9, 183- 199.
Lin, J.J. and Feng, C.M. (2003) A bi-level programming for the land use network design problem. Annals of Regional Science, 37,93-105.
Lo, H.K. and Tung Y.K. (2003) Network with degradable links: capacity analysis and design. Transportation Research, 37B, 345-363.
Magnanti, T.L. and Wong, R.T. (1984) Network design and transportation planning: models and algorithms. Transportation Science, 18, 1-55.
Melkote, S. and Daskin, M. (2001) An integrated model of facility location and transportation network design. Transportation Research, 35A, 515-538.
Meng, Q. and Yang, H. (2002) Benefit distribution and equity in road network design. Transportation Research, 36B, 19-35.
Mirchandani, P. and Soroush, H. (1987) Generalized traf?ic equilibrium with probabilistic travel times and perceptions. Transportation Science, 21, 133-152.
Poorzahedy, H. and Turnquist, M.A. (1 982) Approximate algorithms for the discrete network design problem. Transportation Research, 16B, 45-55.
Rothlauf, F. (2000) Representations for Genetic and Evolutionary Algorithms. Studies in Fuzziness and Soft Computing, Vol. 104, Physica-Verlag, New York.
Sheffi, Y. (1 985) Urban transportation networks: equilibrium analysis with mathematical programming methods. Prentice Hall, England Cliffs, NJ.
Suwansirikul, C. and Friesz, T.L. (1987) Equilibrium decomposed
2 70 P. Chootinan, S.C. Wong, and A . Chen
optimization: a heuristic for the continuous equilibrium network design problem. Transportation Science, 21,254-263.
Szeto, W.Y. and Lo, H.K. (2005) Strategies for road network design over time: robustness under uncertainty. Transportmetrica, 1,47-63.
Watling, D. and Hazelton, M.L. (2003) The dynamics and equilibria of day-to-day assignment models. Networks and Spatial Economics, 3, Webster, F.V. and Cobbe, B.M. (1966) Traflc Signals. Road Research
Technical Paper, No. 56, HMSO, London.
Wong, S.C. (1996) On the reserve capacity of priority junctions and roundabouts. Transportation Research, 30B, 44 1-453.
Wong, S.C. and Yang, C. (1999) An iterative group-based signal optimization scheme for traffic equilibrium networks. Journal of Advanced Transportation, 33,20 1-2 17.
Wong, S.C. and Yang, H. (1997) Reserve capacity of a signal-controlled road network, Transportation Research, 3 lB, 397-402.
Yang, H. and Bell, M.G.H. (1997) Traffic restraint, road pricing and network equilibrium. Transportation Research, 31B, 303-3 14.
Yang, H. and Bell, M.G.H. (1998) Models and algorithms for road network design: A review and some new developments. Transport Reviews, 18,257-278.
Yang, H. and Lam, W.H.K. (1996) Optimal road tolls under conditions of queuing and congestion. Transportation Research, 30A, 3 19-332.
Yang, H. and Wang, J.Y.T. (2002) Travel time minimization versus reserve capacity maximization in the network design problem.
Transportation Research Record, 1783, 1 7-26.
Yang, H. and Yagar, S. (1994) Traffic assignment and traffic control in general freeway-arterial corridor systems. Transportation Research, Yang, H., Yagar, S. Iida, Y. and Asakura, Y. (1994) An algorithm for inflow control problems on urban freeway networks with user- optimal flows. Transportation Research, 28B, 123-1 39.
Yin, Y. (2000) Genetic Algorithms based approach for bilevel programming models. Journal of Transportation Engineering, 126, Yin, Y. and Ieda, H. (2001) Assessing performance reliability of road networks under nonrecurrent congestion. Transportation Research Record, 1771, 148-155.
congestion pricing problem. Transportation Research, 38B, 5 17-537.
349-370.
28B, 463-486.
115-120.
Zhang, X. and Yang, H. (2004) The optimal cordon-based network
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