Comparative Analysis of Transportation Network Design Problem under Stochastic Capacity

JOURNAL OF TRANSPORTATION SYSTEMS ENGINEERING AND INFORMATION TECHNOLOGY Volume 14, Issue 3, June 2014 Online English edition of the Chinese language journal Cite this article as: J Transpn Sys Eng & IT, 2014, 14(3), 85-90,116.

RESEARCH PAPER

Comparative Analysis of Transportation Network Design Problem under Stochastic Capacity JIANG Yang 1, SUN Hui-jun 1,*, WU Jian-jun2 1 Beijing Jiaotong University, MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology; 2 Beijing Jiaotong University, State Key Laboratory of Rail Traffic Control and Safety

Abstract: The method to accurately simulate users travel behavior in the network design problem (NDP) is one of the most crucial problems. Most researches in the network equilibrium based approach to model NDP ignore the unreliability aspect of travel time. The uncertain events result in the spatial and temporal variability of network travel times, which directly contributes to the crucial decision of NDP. Specifically, the mean travel time (MTT), the travel time budget (TTB), and the

- reliable mean-excess travel

time (METT) are employed in the transportation network design problem under uncertain environment due to stochastic link capacity. Numerical results are presented to examine how these models affects decisions under the condition of travel time variability. The comparative analyses show that the performance of DRUE and METTUE is better than DUE which is employed in network design problem under variation degrees because of considering travel time variability. Key words: urban traffic; transportation network design; link capacity variation; travel time reliability; particle swarm optimization

1 Introduction Transportation network design problem is to make a crucial decision of the network construction. The decisions are allocation of a limited financial budget to enhance the existing links and/or to the addition of new candidate links. The aim is to cope with the rapidly growing travel demand and the congestion problem[1]. NDP is firstly proposed by Morlok in 1973 has been continuously studied during the last five decades. The number of related publications has grown over time. In 1975, Leblanc[2] firstly studied the urban traffic network design problems and formulated the problem with a mixed integer optimization model. In recent years, the study of urban traffic network design problem has made great progress. Recent empirical studies[3 4] revealed that travelers actually consider the travel time variability as a risk in their route choice decisions. They are interested in not only travel time saving but also the risk reduction. However, the traditional user equilibrium (DUE) model[5] neglects travel time variability in the route choice decision process. It uses only the expected travel time (Mean Travel Time-MTT) as the criterion for

making route choices, which implicitly assumes all travelers to be risk-neutral. Uncertainty is unavoidable in real life. Various uncertainty factors can contribute to the travel time variability, such as nature disaster, traffic accident, and recurrent congestion. These uncertain events result in the variations of traffic flow, which directly contributes to the spatial and temporal variability of network travel times. Such travel time variability introduces uncertainty for travelers such that they do not know exactly when they will arrive at the destination. Lo et al.[6] proposed a key concept adopted in models is the travel time budget (TTB), which is defined as the average travel time plus an extra time as an acceptable travel time, such that the probability of completing the trip within the TTB is no less than a predefined reliability threshold (or a confidence level  ). The concept of TTB is analogous to the Value-at-Risk (VaR), which is by far the most widely applied risk measure in the finance area. Furthermore, in order to describe travelers’ route choice decision process under travel time variability, it is not adequate to describe travelers’ risk preferences only considering the reliability aspect. On the one hand, the FHWA report[7]

Received date: Sep 6, 2013; Revised date: Nov 6, 2013; Accepted date: Nov 12, 2013 *Corresponding author. E-mail: [email protected] Copyright © 2013, China Association for Science and Technology. Electronic version published by Elsevier Limited. All rights reserved. DOI: 10.1016/S1570-6672(13)60138-5

JIANG Yang et al. / J Transpn Sys Eng & IT, 2014, 14(3), 85-90,116

documented that travelers, especially commuters, do add a ‘buffer time’ to their expected travel time to ensure more frequent on-time arrivals when planning a trip. It represented the reliability aspect in the travelers’ route choice decision process. On the other hand, the impacts of late arrival and its explicit link to the travelers’ preferred arrival time were also examined in the literature [8]. It represented the travelers’ concern of the unreliability aspect of travel time variability in their route choice decision process[9]. Recently, Zhou and Chen[10] proposed a new model called the  -reliable mean-excess travel time (METT) user equilibrium model or the mean-excess travel time user equilibrium (METTUE) model for short that explicitly considers both reliability and unreliability aspects of travel time variability in the route choice decision process. Recent empirical studies on the value of time and reliability reveal that travel time variability plays an important role on travelers’ route choice decision process. It can be considered as a risk to travelers making a trip. Therefore, travelers are not only interested in saving their travel time but also in reducing their risk. The uncertainty events from supply side sources (stochastic link capacity variations) typically lead to uncertainty of network travel times. It could influence the travelers’ trip decision, including their departure time, destination, mode, and route choice, which consequently affect the flow pattern. MTT, TTB, and METT are employed in the transportation network design problem under an uncertain environment due to stochastic link capacity. Numerical are presented for the comparative analysis to examine how these models affects decisions under the condition of travel time variability.

2 Route choice formulation under stochastic link capacity Consider a strongly connected network G  ( N , A) , where N and A denote the sets of nodes and links, respectively. Let R and S denote a subset of N for which travel demand qrs is generated from origin r  R to destination s  S , and P rs denote the set of paths from origin r to destination s . Since link capacity is random, and assumes it follows to uniform distribution ua ~ B(a (Ca  ya ),(Ca  ya )) . Therefore, the travel time on path p Tp is also a random variable. The travel time on each link is depicted 0 n by ta ( xa )  ta  1  ( xa / ua )  . Similar to Lo et al.[6], all travelers are assumed to have knowledge of the variability of path travel time acquired from past experiences and incorporate this information along with their risk-preferences into their route choice decisions. Therefore, to study the user equilibrium problem under an uncertain environment, a key factor is to understand the travelers’ route choice decision process under travel time variability. Definition 1: (MTT) In the traditional DUE model, travelers

are assumed to be risk-neutral since they make their route choice decisions based on the expected travel time[6]. Expected travel time variability is induced by stochastic link capacity noted in Eq.(1).    1  1 n     (1) E (Tp )    a , p ta0   ta0 ( xa )n n  a  Ca (1   )(1  n)     where, Tp is the random travel time on path p ; E[Tp ] is the expected travel time; xa is the flow on link a ;  is confidence level;  a is the degree of degraded capacity for link a ; Ca is the design capacity of link a ;  a , p is link-path incidence parameter, 1 if link a is on path p , zero otherwise; ta0 is free-flow travel time on link a ;  and n are the parameter in BPR function, respectively. Definition 2: (TTB) Link capacity degradations cause link and route travel time variability. Travelers, therefore, do not know their exact travel times. Most travelers would depart earlier to allow for additional time, or add a travel time margin to the expected trip time, to avoid late arrivals[6]. In other words, travelers allow for a longer travel time budget to hedge against travel time variability. We define the travel time budget as Eq.(2). (2) Bp  E(Tp )   Tp Where, Bp is the travel time budget;  Tp is the extra time added to the mean travel time as a ‘buffer time’ to ensure more frequent on-time arrivals at the destination under the travel time reliability requirement;  is degree of risk aversion;  Tp is noted in Eq.(3).  Tp 

2     1  a1 2n 1  a1 n 2 0 2 2n      ( t ) ( x )   (3) a a, p a a  2n n  Ca (1  a )(1  2n)  Ca (1  a )(1  n)  

Definition 3: (METT) By considering the travel time reliability requirement, travelers are searching for a path such that the corresponding travel time budget allows for on-time arrival with a predefined confidence level  . Meanwhile, they are also considering the impacts of excessively late arrival (i.e., the unreliable aspect of travel time variability) and its explicit link to the travelers’ preferred arrival time in the route choice decision process. Therefore, it is reasonable for travelers to choose a route such that the travel time reliability is ensured most of the time and the expected unreliability impact is minimized. This trade-off between the reliable and unreliable aspects in travelers’ route choice decision process can be represented by the mean-excess travel time (METT)[10]. It is defined as Eq.(4). (4)  p ( )  E[Tp | Tp  Bp ( )] where, Bp ( ) is the minimum travel time budget on path p with a predefined confidence level  defined in Eq.(5). (5) Bp  min{B | Pr(Tp  B)  } The mean-excess travel time  p ( ) for a path p with a predefined confidence level  is equal to the conditional

JIANG Yang et al. / J Transpn Sys Eng & IT, 2014, 14(3), 85-90,116

expectation of the travel time exceeding the corresponding path travel time budget B p ( ) noted in Eq.(6). (6)  p ( )  Bp ( )  E[Tp  Bp ( ) | Tp  Bp ( )] According to the definition above, it is easy to see that the METT can be represented in Eq.(7).  Tp  ( 1 ( ))2  (7)  p ( )  E (Tp )  exp    2 2 (1   )   where, symbols definition are described above.

3

The problem is usually formulated as a bi-level problem. The upper level problem is the leader’s problem which plans or manages the transportation network. The upper-level problem includes the measurable goal of reducing total travel time, budget restrictions and the design decisions to be made. The lower-level results are flow patterns under the conditions of minimizing the Eq. (1), Eq. (2) and Eq. (7). The bi-level structure allows the decision-maker to consider the reaction of the travelers and improve the network to influence the travel choice of travelers but has no direct control on their choice. Mathematically, the problem can be represented as follows: (8) min Z   E (Ta ) xa

wmax  wmin t , Iter _ max t c1  (c1 f  c1i )   c1i , Iter _ max t c2  (c2 f  c2i )   c2i . Iter _ max

c1 f  0.5 , c1i  2.5 , c2 f  2.5 c2i  0.5 , wmax  1.4 ,

wmin  0,35 ,  and  are random numbers in (0,1). Step 5 Termination. After the particle moves into a new position, heuristic assignment is performed to generate a new feasible solution, the performance of each particle is evaluated, and so on. The same procedures from Step 2 to 5 are repeated until the maximum number of iterations Iter _ max  500 is reached.

4

Numerical Results

a

G( y )  B,

a

(8a)

ya  ya  ya ,

a

(8b)

a

yait 1  yait  Vait 1 where, wt  wmax 

Mathematical model and solution method

ya

Step 4 Update position value. PSO searches for optima by updating its position using velocity function. The historical information is used to calculate the velocity and the particle changes position according to the following equation: Vait 1  wt  Vait  c1    ( pbestait  yait )  c2   ( gbest t  yait )

a

where, B is the total budget; G(ya) is cost function with respect to ya; y and y are the low bound and up bound of ya. a a According to (8), it is the objective function of the upper problem to minimize the total cost. Condition (8a) is the investment budget constraint and (8b) constitutes the non-negativity condition. Let xa be the optimal solution value of minimizing Eq. (1), Eq. (2), Eq. (7) given the stochastic capacity. The network design problem has been proved to be NP-hard. We thus have to seek a heuristic method to solve the model. Step1 Initialization. Randomly determine an initial PSO population consisting of m particles satisfying constraint conditions (8a) and (8b) and denote the population by Yait  { ya , a  A, i  m} . The max velocities and iterations are Vmax and Iter _ max . Let the number of iterations t = 1. Step 2 Fitness value evaluation. The fitness value of each particle is calculated with the Eq. (8). Step 3 Previous best particle and global best particle. In each iteration, the position giving the best fitness value is recorded as the previous personal best solution and represented as pbest t . The best value obtained so far by the whole swarm is tracked and memorized as global best gbest t .

To illustrate the differences among the three user equilibrium models (i.e., DUE, DRUE and METTUE models) employed in NDP, a simple network with 7 nodes, 10 links, and single OD pairs is adopted (Fig. 1) for demonstration purpose. 1

Origin

1

2 3

2 4

3 5

4

6

5

8

7

9

6

10

7 Destination

Fig. 1 The test network

The parameters of the network are listed in Table 1. The link travel time function adopted in this study is the BPR function. Table 1 Network parameters OD

Routes

（1,7）

1-4-9 1-3-6-9 1-3-8-10 2-5-6-9 2-5-8-10 2-7-10

Demand

n  4,   0.15,a  0.6 G( ya )  2*( ya )2 ,  =0.95 B  10000

100

Link

ta0

1 2 3 4 5 6 7 8 9 10

4 6 2 2 2 2 2 2 7 5

Design capacity 60 60 60 60 60 60 60 60 60 60

JIANG Yang et al. / J Transpn Sys Eng & IT, 2014, 14(3), 85-90,116

To further illustrate the efficiency of the solution algorithm, a convergence test of absolute deviations between iterations is given in the numerical example as shown in Fig. 2. The results indicate that the algorithm can converge to a steady state within a short step.

degradation level

In the following, the effect of demand variations on the equilibrium results for NDP is explored, where B = 10000, θ = 0.5. The performance measure used for comparison is the total travel time budget, as normalized by that of METT. It is clear that the impact of MTT, TTB, and METT on total travel time budget increase as the demand level increases and MTT ≥ TTB ≥ METT. Furthermore, the differences among them get larger when the demand level increases. Due to the congestion effect, all of the measures have a higher rate of increase at higher demand levels than that at lower demand levels. This implies that the consideration of both reliability and unreliability aspects of travel time variability may has a more significant effect on travelers’ route choice decision under heavier congestion levels, and it is similar to the total travel time budget. ( Fig.4)

Fig. 2 The convergence of the PSO algorithm

We study the sensitivity of travel time budget with stochastic link capacity. For illustration purposes, we set the link degradation parameter θa to be uniform across the whole network, as denoted by one single θ. We vary θ and plot total travel time budget associated with θ. Meanwhile, B = 10000 and demand is 150. As shown in Fig.3, the performance of MTT and TTB is decreased when θ varies from 0.5 to 0.9. The x-axis refers to θ, and when θ = 0.9, the network is perfect without too much degradation. On the other hand, θ = 0.5 represents a highly degradable network. The y-axis represents the total travel time budget. The performance measure used for comparison is the total travel time budget, as normalized by that of METT. It can be seen that the discrepancies of the simplifications strongly depend on the degradation level. When degradation is light (θ = 0.8 and θ = 0.9), TTB produces smaller travel time budgets and MTT is the highest, implying that one does not lose too much accuracy by ignoring the unreliable aspect of unacceptable risk. However, as the degradation level increases (θ = 0.5 and θ = 0.6), their differences become pronounced. That is, ignoring the effect of reliability aspect or unreliable aspect can result in a sizeable underestimation of the actual travel time budget.

Fig. 4 The network total travel time budget with traffic volume

Now, we fix the OD demand to be 150 and θ = 0.5, and examine the effect of different budget levels. The performance measure used for comparison is the total travel time budget, as normalized by that of METT. We can see that when the budget level is high, the results of three user equilibrium are similar. When the budget level is low, the MTT has much erroneous judgement of travel time index. This means travelers need to set aside a larger amount of extra time to ensure both reliability and unreliability aspects of the larger travel time variability induced by the lower budget level. (Fig.5)

Fig. 5

Fig. 3 The network total travel time budget with capacity

The network total travel time budget with total budget

The models developed can be used to identify the optimal

JIANG Yang et al. / J Transpn Sys Eng & IT, 2014, 14(3), 85-90,116

locations for link improvements while considering the stochastic nature of the problem. For illustration purposes, we conduct the numerical study with the network demand is 100, B = 10000 and θ = 0.6.

Fig. 6 Sensitivity of different rout-select patterns with respect to link capacity enhancement

In Fig. 6, the x-axis represents the link where improvement is introduced; whereas the y-axis is the corresponding enhancement on link capacity. We introduce the link capacity enhancement under MTT, TTB and METT. The results for MTT are shown by the blank bars. Based on these results, one can identify that Link 10 is the most productive location for improvement and the total travel time budget of the network is minimum; whereas the decision of the candidate is Link 1. The models that ignored the reliability aspect or unreliable aspect would come up with a very different set of results, sometimes misidentifying the most beneficial improvement locations. Say, in the case of MTT, the model suggests that the most productive link for improvement is Link 10; however, in the most realistic representation of METT, Link 1 would bring about a much lower reduction in total travel time of the network. This result again illustrates the importance of capturing the reliability aspect or unreliable aspect of unacceptable risk. Ignoring them could lead to erroneous prediction and investing resources in a sub-optimal way.

5

be satisfied. However, the DRUE model does not assess the magnitude of the unacceptable travel times exceeding the TTB. Hence, it is possible that the travelers in the TTB will encounter unacceptable travel times in some bad days. It is suitable for NDP at higher degradation level than that at lower degradation level. (3) In addition to the travel time reliability requirement defined by TTB, the METT further considers the possible risk associated with the travel times beyond the TTB. The limited numerical results showed that the three models are indeed different and highlighted the essential ideas of the METT, which is to consider both reliability and unreliability aspects of handling travel time variability in the route choice decision process. Due to the complex situations in the real world, no UE model is sufficient in capturing all aspects of the travelers’ risk-taking behavior in the route choice decision process, and whether the MTT, TTB and METT is the most realistic is an open question.

Acknowledgements This work is supported by A Foundation for the Author of National Excellent Doctoral Dissertation of PR China (201170), National Basic Research Program of China (2012CB725406) and the Fundamental Research Funds for the Central Universities (2013YJS049).

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Conclusions

In this study, three different user equilibrium models under stochastic link capacity were employed in NDP. The differential effects are demonstrated by comparative analysis, and some valuable conclusions are drawn. (1) The MTT ignores the travel time variation in the travelers’ route choice decision process. Thus, it may not be able to reflect travelers’ risk preference under an uncertain environment. That is, ignoring the effect of reliability aspect or unreliable aspect can result in a sizeable underestimation of the actual travel time budget. (2) By considering the reliability aspect of stochastic travel time, the TTB assumes that all travelers minimize their TTB, such that a predefined confidence level of on-time arrival can

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