Optimal transit routing with partial online information

Transportation Research Part B 72 (2015) 40–58

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Transportation Research Part B journal homepage: www.elsevier.com/locate/trb

Optimal transit routing with partial online information Peng (Will) Chen, Yu (Marco) Nie ⇑ Department of Civil and Environmental Engineering, Northwestern University, United States

a r t i c l e

i n f o

Article history: Received 19 December 2013 Received in revised form 11 November 2014 Accepted 12 November 2014

Keywords: Routing Hyperpath Partial information Attractive set Cycles

a b s t r a c t This paper studies the routing strategy in a transit network with partial online information at stops. By partial online information, we mean that the arrival time of the incoming transit vehicles is available for a subset of the lines serving a stop. To cope with the partial information assumption, a new routing strategy is proposed and closed form formulae for computing expected waiting times and line boarding probabilities are derived. The proposed strategy unifies existing hyperpath-based transit route choice models that assume either no information or full information. Like many existing models, it ensures optimality when all information is available or the headway is exponentially distributed. The problem of determining the attractive set is discussed for each of the three information cases. In particular, a new heuristic algorithm is developed to generate the attractive set in the partial information case, which will always yield a solution no worse than that obtained without any information. The paper also reveals that, when information is available, an optimal hyperpath may contain cycles. Accordingly, the cause of such cycles is analyzed, and a sufficient condition that excludes cycles from optimal hyperpaths is proposed. Finally, numerical experiments are conducted to illustrate the impact of information availability on expected travel times and transit line load distributions. Among other findings, the results suggest that it is more useful to have information on faster lines than on slower lines. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction In a transit network where multiple lines share several stops and segments of routes, passengers often face the choice of transit lines when they arrive at a stop. Chriqui and Robillard (1975) first studied this ‘‘common bus lines’’ problem using a simple corridor example. In this seminal work, Chriqui and Robillard (1975) show that passengers may select a set of attractive lines to minimize the expected total travel time. The corresponding route choice strategy is boarding the first arriving line in that attractive set. Spiess and Florian (1989) extended this notion of strategy to a general transit network, namely, the choice of an attractive set of lines is considered at each node where boarding occurs. Nguyen and Pallottino (1988) interpreted a strategy as a hyperpath, which is defined as an acyclic directed graph connecting an origin to a destination. Methods for finding optimal hyperpaths have been proposed and incorporated into various frequency-based transit assignment models (see e.g. Spiess and Florian, 1989; Nguyen and Pallottino, 1988; de Cea and Fernández, 1993; Wu et al., 1994; Marcotte et al., 1998; Cominetti and Correa, 2001; Kurauchi et al., 2003; Hamdouch et al., 2004; Cepeda et al., 2006; Schmöcker et al., 2008; Trozzi et al., 2013). The notion of hyperpath has also been used in building reliable routing models for highway networks (see e.g., Bell, 2009; Bell et al., 2012; Schmöcker et al., 2009; Ma et al., 2013).

⇑ Corresponding author. Tel.: +1 847 467 0502. E-mail address: [email protected] (Y. (Marco) Nie). http://dx.doi.org/10.1016/j.trb.2014.11.007 0191-2615/Ó 2014 Elsevier Ltd. All rights reserved.

P. (Will) Chen, Y. (Marco) Nie / Transportation Research Part B 72 (2015) 40–58

41

The classical hyperpath model relies on several assumptions. First, transit line headway is typically assumed to be exponentially distributed. Recent studies (Li et al., 2015; Ruan and Lin, 2009), however, suggest that empirical headway data fit other types of distributions (such as Gamma and Erlang) much better than the exponential distribution. Second, passengers are assumed to arrive at stops randomly, which approximates the reality well when headway does not exceed certain threshold (around 10–13 min) (O’FLAHERTY and Mancan, 1970; Okrent, 1974; Seddon and Day, 1974; Bowman and Turnquist, 1981; Fan and Machemehl, 2009). Third, passengers are also assumed to have no access to on-line information (i.e., the knowledge of the arrival times of incoming transit vehicles). Yet, such information is readily available nowadays via display boards at stops or transit apps on mobile devices (see e.g., BusTracker offered by Chicago Transit Authority1). As the availability of such online information may affect the boarding decision significantly, it is important to incorporate it into transit route choice models. The role of real-time information in the common bus lines problem was studied in Hickman and Wilson (1995), which assumed that waiting time distributions vary as passengers gather more information after arriving at a stop. Accordingly, passengers would only board the transit vehicle if the expected travel time upon boarding is smaller than the minimum expected travel time if the vehicle is skipped. Because each boarding decision invokes an evaluation of the expected travel time, computing boarding probabilities is not analytically tractable. Instead, simulations are used to evaluate the value of information in a rather small case study. Gentile et al. (2005) assumed that the arrival time of next transit vehicle can be accurately projected once passengers arrive at a stop. Thus, passengers will always board the transit vehicle that has the minimum total travel time. This study is focused on the impact of information availability on transit route choices. Specifically, we consider a situation in which passengers can only access online waiting time information for a subset of the available lines at a stop. In contrast, Gentile et al. (2005) assumed that such information is available for all lines. The ability to cope with partial information is important because limitations in the current vehicle location technology and random incidents could render incomplete information in the transit system. The partial information case may also be regarded as a generalization of two special cases: no information and full information. As it turns out, the availability of partial information leads to very different route choice strategies that pose greater analytical and numerical challenges for computing expected waiting time and boarding probabilities. Tackling these challenges is a focus of this paper. The rest of the paper is organized as follows. The existing stop model of route choice in transit networks is reviewed in Section 2. Section 3 proposes the route choice strategy in the partial information case. Closed-form formulae for computing boarding probabilities and expected waiting times are presented for the exponential headway distribution in Section 4. Section 5 discusses the attractive set problem in all three information cases, and Section 6 addresses the cycle issue in optimal hyperpaths when information is available. Results of numerical experiments are reported in Section 7. Section 8 concludes the paper.

2. Stop model Consider a stop in a transit network that is shared by multiple transit lines L ¼ f1; 2; . . . ; ng. For each line i 2 L, let si denote the line travel time and hi denote the headway of the line i. The basic assumptions of the stop model, widely adopted in the literature to simplify the analysis (e.g. Spiess and Florian, 1989; Gentile et al., 2005), are as follows:

Assumption 1. Transit line headways are statistically independent with given continuous distributions. Assumption 2. Passengers arrive randomly at the stop, i.e., they do not coordinate their arrival times at the stop with the scheduled arrival times of transit vehicles. Assumption 3. Transit line travel time is deterministic. Assumption 4. Passengers can reliably estimate the remaining line travel time, i.e., the expected travel time from the stop to the destination, once boarded a vehicle of the line. Assumption 5. Passengers are always able to board the next coming transit vehicles, i.e., congestion effects are not considered.

1

http://www.ctabustracker.com/bustime/home.jsp

42

P. (Will) Chen, Y. (Marco) Nie / Transportation Research Part B 72 (2015) 40–58

The above assumptions will be adopted throughout the paper regardless of information availability. With Assumptions 1 and 2, the line waiting time probability density function can be related to the headway distribution through the formula (Chapter 2.13 Larson and Odoni, 1981):

R þ1

g i ðhÞdh ¼ ki ð1  Gi ðwÞÞ; E½hi 

w

f i ðwÞ ¼

ð1Þ

where f i ðwÞ is the probability density function of the line waiting time, g i ðhÞ and Gi ðhÞ are the probability density function and the cumulative distribution function of the line headway, respectively, and 1=E½hi  ¼ ki the inverse of the mean headway, generally referred as the mean frequency of the line. Let F i ðwÞ denote the cumulative distribution function of the line waiting time and F i ðwÞ its complement, then we have

Z

w

f i ðv Þ dv ; Z F i ðwÞ ¼ 1  F i ðwÞ ¼ F i ðwÞ ¼

ð2Þ

0

þ1

f i ðv Þ dv :

ð3Þ

w

With the knowledge of the line waiting time distribution, the line boarding probabilities as well as the expected waiting time at the stop can be derived. Depending on what assumptions we make regarding the information availability, the following three cases may arise:  No information case Passengers do not have any online information about the line waiting time.  Full information case Passengers have access to the waiting times of all available lines at the stop.  Partial information case Passengers have access to the waiting times of a subset of all available lines at the stop. The first two cases have been studied in the literature. A brief review of the existing results is given below. In the no information case, passengers are typically assumed to board the first arriving line that belongs to a fixed attractive set (see e.g. Spiess and Florian, 1989; Gentile et al., 2005).2 For narrative convenience, we assume for now that passengers will consider all lines in L attractive. In reality, of course, only a subset of L may be attractive to passengers. The problem of determining the attractive set will be discussed later. With the above strategy, the probability of boarding line i is equal to the probability that line i has the minimum waiting time among the lines in L. By Assumption 1, the line waiting times are independent from each other. Thus, this probability can be computed as

pi ¼

Z

f i ðwÞ

0

where

Q

Y

þ1

Prðwj P wÞdw ¼

Z

þ1 0

j2Lni

f i ðwÞ

Y j2Lni

F j ðwÞdw ¼

Z

0

þ1

ci ðwÞdw;

ð4Þ

Prðwj P wÞ is the joint probability that line i has the minimum waiting time among the lines in L, when the

j2Lni

realized waiting time of line i equals w. Note that

ci ðwÞ ¼ f i ðwÞ

Y

F j ðwÞ

ð5Þ

j2Lni

may be interpreted as the probability density function of the waiting time at the stop conditional on boarding line i. With full information, passengers will always board the line with the minimum total travel time (Gentile et al., 2005). Consequently, the probability of boarding line i equals the probability that line i has the minimum total travel time, which is

pi ¼

Z

f i ðwÞ

0

¼

Z Z 0

Prðwj P w þ si  sj Þdw

j2Lni þ1

f i ðwÞ

0

¼

Y

þ1

Y

F j ðw þ si  sj Þdw

j2Lni þ1

ci ðwÞdw;

where

ci ðwÞ ¼ f i ðwÞ

Y

F j ðw þ si  sj Þ:

ð6Þ

ð7Þ

j2Lni

2 It is worth noting that this strategy ensures optimality only if the headway follows exponential distribution, which has the memorylessness property. See remarks at the end of this section for more details.

P. (Will) Chen, Y. (Marco) Nie / Transportation Research Part B 72 (2015) 40–58

43

With ci ðwÞ properly defined, the expected waiting time, the expected remaining line travel time and the expected total travel time associated with a set of lines L at a stop can be computed as

xL ¼

XZ i2L

lL ¼

XZ i2L

þ1

0

0

wci ðwÞdw;

ð8Þ

si ci ðwÞdw;

ð9Þ

þ1

and

uL ¼ xL þ lL ¼

XZ i2L

þ1 0

ðw þ si Þci ðwÞdw:

ð10Þ

Remark 1. Because travelers always board the first arriving line in a fixed attractive set, ci and pi (hence xL ) can be computed with closed form formulae, which is appealing for large-scale applications. This strategy, however, may not be optimal when the headway is not exponentially distributed. For general headway distributions, ensuring optimality may require the attractive set be changed with the elapsed waiting time, leading to a so-called ‘‘dynamic strategy’’ (Hickman and Wilson, 1995; Billi et al., 2004; Nökel and Wekeck, 2009). Yet, finding ci and pi under such a strategy seems analytically intractable, as it involves solving multiple nonlinear equations using an iterative algorithm, and each iteration requires numerical integrations (Billi et al., 2004). Moreover, information can alleviate the problem of sub-optimal solutions arising from the ‘‘taking-the-first-arriving-line’’ strategy. To see this, note that the best possible outcome is achieved when passengers know exactly when each line will arrive (i.e. with full information), which is true regardless of the type of headway distribution. This implies that a dynamic strategy cannot offer a better solution than that obtained in the full information case. In light of the above observations, this paper will not pursue the dynamic strategy approach. In the next section, a new routing strategy will be proposed as a computationally viable improvement to the classical strategy by taking advantage of information availability. We emphasize that, unless all information is available or all headway distributions are exponential, the strategy may not ensure optimal solutions for the reason explained above. 3. Route choice with partial information Suppose that passengers can only obtain the line waiting times for a subset of all available lines. Let Ly and Ln denote the subsets of lines with and without waiting time information, respectively, and L ¼ Ly [ Ln . For simplicity we shall continue to assume that passengers will consider all lines in L attractive. This assumption will be relaxed in Section 5. The question is what is a good strategy for passengers to choose among the available lines. Based on the aforementioned well-known strategies for the no information case and full information case, we propose the following strategy. When a passenger arrives at the stop, he/she can immediately retrieve the waiting times of those lines in the subset Ly and thus can   identify the best line (ly ) with the minimum total travel time in Ly . Then the passenger will wait for the line ly to come. While  the passenger is waiting for ly , lines in the subset Ln may arrive. If the realized waiting time of the coming line yields a smaller  total travel time than that of line ly , the passenger will board this incoming line from Ln instead; otherwise, the passenger will   board ly when it arrives. In a nutshell, the target for a passenger is boarding ly , although a better option may arise in the process of waiting for it.3 We now analyze how to calculate the probability of boarding each line i under the above route choice strategy. Depending on which subset the line i belongs to, the analysis is conducted in two separate cases. 3.1. Lines with waiting time information For line i 2 Ly , i.e., the waiting time information is available for line i, the passengers will board the line if the following two conditions are satisfied:  line i has the minimum total travel time (wi þ si ) among all j 2 Ly ; and  the realization of waiting time wj for each line j 2 Ln is such that wj þ sj > wi þ si or wj > wi , i.e., wj > minðwi þ si  sj ; wi Þ. The first condition directly draws from the case with full information, which must be satisfied if the line i with waiting time information is the chosen line. The second condition states that the realization of waiting time for line j 2 Ln must ensure that either the realized total travel time is larger than that of line i or the realized waiting time is larger than wi . It indicates that the passengers should not wait any longer than wi where i ¼ arg minðwi þ si Þ, because if they skip the line i2Ly



3 Since the strategy assumes that the first arriving line in Ln will be taken if it outperforms ly , it may still not guarantee optimality. However, the problem is expected to be less severe as more information becomes available.

44

P. (Will) Chen, Y. (Marco) Nie / Transportation Research Part B 72 (2015) 40–58

i, they may end up with a worse choice without knowing the arrival information for the lines in Ln . The probability of boarding line i 2 Ly can be computed as

pi ¼

Z

f i ðwÞ

0

¼

Z Z

Prðwj > w þ si  sj Þ

j2Ly ni

Y

þ1

f i ðwÞ

0

¼

Y

þ1

Y

Prðwj > minðw þ si  sj ; wÞÞdw

j2Ln

F j ðw þ si  sj Þ

j2Ly ni

Y

F j ðminðw þ si  sj ; wÞÞdw

j2Ln

þ1

ci ðwÞdw;

0

ð11Þ

where

ci ðwÞ ¼ f i ðwÞ

Y

F j ðw þ si  sj Þ

j2Ly ni

Y F j ðminðw þ si  sj ; wÞÞ;

ð12Þ

j2Ln

for i 2 Ly and can be interpreted as the probability density function of the waiting time at the stop conditional on boarding line i 2 Ly . 3.2. Lines without waiting time information For line i 2 Ln , i.e., no waiting time information is available for line i, the passengers will board the line if the following two conditions are satisfied:  wi 6 wj and wi þ si 6 wj þ sj , i.e., wi 6 minðwj þ sj  si ; wj Þ, where j ¼ arg min ðwj þ sj Þ; j2Ly

 wk > wi or wk þ sk > wj þ sj , i.e., wk > minðwj þ sj  sk ; wi Þ, for all k 2 Ln where j ¼ arg min ðwj þ sj Þ. j2Ly

The first condition ensures that the realized waiting time of line i 2 Ln is less than the waiting time of the best line from Ly and also yields a smaller total travel time. The second condition guarantees that line i is the first coming line from the subset Ln to satisfy the first condition. The probability of boarding line i 2 Ln can thus be written as

pi ¼

Z

þ1

f i ðwi Þ

0

¼

Z

XZ

þ1

f i ðwi Þ

0

¼

Z

f j ðwj Þ

max

j2Ly

XZ

Prðwk > wj þ sj  sk Þ

k2Ly nj

Y

ðwi ;wi þsi sj Þþ1

f j ðwj Þ

max

j2Ly

Y

ðwi ;wi þsi sj Þþ1

Y

Prðwk > minðwj þ sj  sk ; wi ÞÞdwj dwi

k2Ln ni

Y

F k ðwj þ sj  sk Þ

k2Ly nj

F k ðminðwj þ sj  sk ; wi ÞÞdwj dwi

k2Ln ni

þ1

0

ci ðwÞdw;

ð13Þ

where

XZ

ci ðwÞ ¼ f i ðwÞ

ðw;wþsi sj Þþ1

f j ðwj Þ

max

j2Ly

Y

F k ðwj þ sj  sk Þ

k2Ly nj

Y

F k ðminðwj þ sj  sk ; wÞÞdwj ;

ð14Þ

k2Ln ni

for i 2 Ln and can be interpreted as the probability density function of the waiting time at the stop conditional on boarding line i 2 Ln . 3.3. Discussions The expected waiting time, the expected remaining line travel time and the expected total travel time in the partial information case can be computed similarly using Eqs. (8)–(10). The main difference lies in the definition of ci ðwÞ. It is worth noting that the partial information case is reduced to no information case when all the lines in Ly have infinite waiting times. To see this, note that if we set wj ¼ þ1 for j 2 Ly , then ci ðwÞ in Eq. (14) becomes

XZ

ci ðwÞ ¼ f i ðwÞ

j2Ly

¼ f i ðwÞ

Y Y

f j ðwj Þ

max

XZ F k ðwÞ

k2Ln ni

¼ f i ðwÞ

ðw;wþsi sj Þþ1

j2Ly

F k ðwÞ;

k2Ln ni

Y

F k ðwj þ sj  sk Þ

k2Ly nj

F k ðwÞdwj

k2Ln ni

ðw;wþsi sj Þþ1

max

Y

f j ðwj Þ

Y

F k ðwj þ sj  sk Þdwj

k2Ly nj

ð15Þ

45

P. (Will) Chen, Y. (Marco) Nie / Transportation Research Part B 72 (2015) 40–58

which equals Eq. (5). Similarly, the partial information case can be reduced to the full information case by setting the wailing times for any line j 2 Ln to be infinite. The reader can verify that when wj ¼ þ1 for j 2 Ln in Eq. (12), it collapses to Eq. (7). Thus, no information and full information cases may be regarded as two special cases of the partial information case. 4. Exponential headway distribution For the exponentially distributed headway, we have for each line i an exponential waiting time distribution with the following probability density function

f i ðwÞ ¼



ki eki w ; if w P 0; 0;

ð16Þ

otherwise;

where ki is the mean frequency of line i. The complement of the cumulative distribution function is given by

 F i ðwÞ ¼

eki w ; if w P 0; 1;

ð17Þ

otherwise:

Thanks to the special structure of the waiting time distribution resulting from the exponentially distributed headway, closed form results may be obtained for line boarding probabilities (pi ) and expected waiting times (x). The derivation for the full information case and partial information case is presented below. 4.1. Full information case Suppose that we rearrange the lines in the set L such that s1 6 s2 6 . . . 6 sn . Define dij ¼ si  sj (i; j ¼ 1; 2; . . . ; n). It is obvious that dji ¼ dij and dij  dkj ¼ dik . According to our indexing, clearly we have

8 > < < 0; if i < j; dij ¼ 0; if i ¼ j; > : > 0; if i > j:

ð18Þ

In addition, define dnþ1;i ¼ þ1 (i ¼ 1; 2; . . . ; n). With full information at stops, the probability of taking each line i may be expressed as

pi ¼

Z

f i ðwÞ

0

Z

¼

þ1

Z

Prðwj > w þ dij Þdw

j2Lnfig

f i ðwÞ

0

¼

Y

þ1

Y

F j ðw þ dij Þdw

j2Lnfig þ1

f i ðwÞ

0

Y

F j ðw þ dij Þ

j¼1;...;i1

Y

F j ðw þ dij Þdw

ð19Þ

j¼iþ1;...;n

According to the definition of dij , the interval ½0; þ1 can be divided into several sub-intervals, i.e.,

½0; þ1 ¼ ½dii ; diþ1;i  [ ½diþ1;i ; diþ2;i  [ . . . [ ½dni ; dnþ1;i :

ð20Þ

For the sub-interval ½dki ; dkþ1;i  (k ¼ i; i þ 1; . . . ; n), it is easy to see that

w þ dij



P 0; j 6 k; 6 0;

ð21Þ

j > k;

where w 2 ½dki ; dkþ1;i . Recall the definitions of f i ðwÞ and F i ðwÞ for exponential headway distribution in Eqs. (16) and (17), for w 2 ½dki ; dkþ1;i , we have

f i ðwÞ

Y

j¼1;...;i1

F j ðw þ dij Þ

Y j¼iþ1;...;n



F j ðw þ dij Þ ¼ ki eki w e

i1 X

kj ðwþdij Þ 

j¼1

e

k X

kj ðwþdij Þ

j¼iþ1



¼ ki e

k X

kj ðwþdij Þ

j¼1

:

ð22Þ

46

P. (Will) Chen, Y. (Marco) Nie / Transportation Research Part B 72 (2015) 40–58

With Eqs. (20) and (22), the integration for calculating

pi ¼

n Z X

k X



dkþ1;i

pi can be simplified as

kj ðwþdij Þ

j¼1

ki e

dw 1 0 X k k X  k d  k d  k d j ij j j ki kþ1;i n C B X ki C B e j¼1 Be j¼1  e j¼1 ¼ C Pk A @ kj k¼i dki

k¼i

k X

j¼1

¼

n X k¼i

1 0 X k k X  kj ðdki dji Þ  kj ðdkþ1;i dji Þ C B ki B j¼1 C  e j¼1 C Be Pk A kj @ j¼1

1 0 X k k X  k d  k d j kj j kþ1;j C n X ki B C B j¼1 ¼  e j¼1 C Be Pk A @ k j k¼i

ð23Þ

j¼1

where the third equality follows from that dji ¼ dij and the last equality follows from that dij  dkj ¼ dik . Let k X



Xlk  e

ðkj dlj Þ

j¼1

! k k X X ¼ exp  ðkj dlj Þ ; 8l P k; and; Kk ¼ kj j¼1

ð24Þ

j¼1

then we have

pi ¼

 n  X ki ðXkk  Xkþ1 k Þ Kk k¼i

Similarly, we can derive the formula to calculate the expected waiting time. According to Eq. (8), the expected waiting time can be expressed as

EW ¼

n Z X i¼1

þ1

wfi ðwÞ

0

Y

Prðwj > w þ dij Þdw

ð25Þ

j2Lnfig

Following the previous derivation, the above expression can be further simplified as

EW ¼

n X n Z X



dkþ1;i

ki we

k X

kj ðwþdij Þ

j¼1

dki

i¼1 k¼i

dw ¼

n X n X





ki e

Pk

kd j¼1 j ij

Kk

i¼1 k¼i

Z

dkþ1;i

wdeKk w :

ð26Þ

dki

Integrating by parts, we have

Z

dkþ1;i

1 Kk dkþ1;i ðe  eKk dki Þ Kk 1 Kk dkþ1;i 1 Kk dki e ðKk dkþ1;i þ 1Þ  e ðKk dki þ 1Þ ¼ Kk Kk k k X X

wdeKk w ¼ dkþ1;i eKk dkþ1;i  dki eKk dki þ

dki



¼

1 e Kk



kj dkþ1;i

j¼1

ðKk dkþ1;i þ 1Þ 

1 e Kk

kj dki

j¼1

ðKk dki þ 1Þ

ð27Þ

Plugging the above expression into Eq. (26) yields

EW ¼

n X n X i¼1 k¼i

3 2 X k k X  kj ðdki dji Þ  kj ðdkþ1;i dji Þ 7 6 ki 6 j¼1 7 e ðKk dki þ 1Þ  e j¼1 ðKk dkþ1;i þ 1Þ7 26 5 Kk 4

3 2 X k k X  k d  k d j kj j kþ1;j n X n 7 X ki 6 7 6 j¼1 ¼ e ðKk dki þ 1Þ  e j¼1 ðKk dkþ1;i þ 1Þ7 26 5 4 K i¼1 k¼i k ¼

n X n X ki h i¼1 k¼i

2 k

K

i

Xkk ðKk dki þ 1Þ  Xkþ1 k ðKk dkþ1;i þ 1Þ :

ð28Þ

P. (Will) Chen, Y. (Marco) Nie / Transportation Research Part B 72 (2015) 40–58

47

4.2. Partial information case As before we rearrange the lines in the set L such that s1 6 s2 6 . . . 6 sn . dij (i ¼ 1; 2; . . . :; n þ 1; j ¼ 1; 2; . . . ; n) are defined as in the previous section. Ly and Ln are the set of lines with and without online information, respectively. Clearly, we have L ¼ Ly [ Ln . According to Eq. (11), for the exponential case, the probability of taking each line i (i 2 Ly ) can be expressed as

pi ¼

Z

f i ðwÞ

0

¼

Y

þ1

Z

j2Ly ni

f i ðwÞ

Z

Y

Y

f i ðwÞ

0

Y

Prðwj > w þ dij Þ

j
Prðwj > minðw þ dij ; wÞÞdw

j2Ln

Y

þ1

0

¼

Prðwj > w þ dij Þ

Prðwj > w þ dij Þ

j>i;j2Ly

Y

Prðwj > w þ dij Þ

j
Y

Prðwj > wÞ

j
Prðwj > wÞ

j
Y

Y

Prðwj > w þ dij Þdw

j>i;j2Ln

Prðwj > w þ dij Þdw

ð29Þ

j>i;j2L

As in the full information case, we also divide the interval ½0; þ1 into sub-intervals ½dki ; dkþ1;i  (k ¼ i; i þ 1; . . . ; n). Similarly, the above expression can be simplified as

pi ¼

n Z X k¼i

¼



ki eki w e

n X k¼i

X

kj ðwþdij Þ 

j
e

X

kj w 

j
e

k X

kj ðwþdij Þ

j¼iþ1

dw

dki

n Z dkþ1;i X k¼i

¼

dkþ1;i



X

k X

kj dij 

j
ki e

kj dij 

j¼i

e

k X

kj w

j¼1

dw

dki

X



ki Pk

j¼1 kj

e

kj dij 

j
k X

kj dij

j¼i

1 0 X k k X  kj dkþ1;i C B  kj dki C B j¼1  e j¼1 C Be A @

ð30Þ

Notice that



X

kj dij 

j
k X

kj dij ¼

X

kj dij 

j
j¼i

i1 k k X X X X kj dij  kj dij ¼ kj dij  kj dij ; j¼1

j¼i

j
ð31Þ

j¼1

Eq. (30) can be written as

X

pi ¼

n X k¼i

ki Pk

j¼1 kj

kj dij 

ej
X ¼

n X k¼i

¼

ki Pk

j¼1 kj

 n  X ki k¼i

Kk

kj dij

j¼1

kj dij

ej
exp

k X

1 0 X k k X  kj dki  kj dkþ1;i C B C B j¼1  e j¼1 C Be A @

0 X 1 k k X  kj dkþ1;j B  kj dkj C B j¼1 C  e j¼1 Be C @ A

X

! kj dij





Xkk  Xkþ1 : k

ð32Þ

j
Because finding pi for the lines in Ln in the partial information case involves two-dimension integration, it is difficult to obtain a closed form solution. For readers’ convenience, Table 1 summarizes the closed form solutions under three information scenarios for the exponential headway distribution. Of the analytical results presented in Table 1, those for the no information case is well known. The other results, to the best of our knowledge, are new. 5. Choice of the attractive set Up to now, it is implicitly assumed that all available lines at a stop line are used. However, the formulae presented in the previous sections are applicable for any given subset of the available lines. Let uR denote the expected total travel time from

48

P. (Will) Chen, Y. (Marco) Nie / Transportation Research Part B 72 (2015) 40–58

the given stop to the destination corresponding on the subset R # L. The attractive set L # L is defined as the subset that yields the minimum expected total travel time, that is, 

uL 6 uR ; 8R # L:

ð33Þ

We next discuss the problem of finding the attractive set under the three information cases. 5.1. No information case In the no information case, it is well known that when headway is exponentially distributed, the attractive set can be generated using the greedy method, which ranks all the available lines in the increasing order of the line travel time and successively adds the line into the attractive set until the addition of the next line increases the current expected total travel time Nguyen (1989). A description of the greedy method is given in Algorithm 1 for the convenience of the reader. It is worth noting that this method may not produce the correct attractive set for other headway distributions (see e.g., Gentile et al., 2005). In this case, enumerating and evaluating all subsets may be necessary to avoid sub-optimal solutions. Algorithm 1. Greedy algorithm for finding L 1: initialize: 2: Rank L ¼ f1; 2; . . . ; ng such that s1 6 s2 ; . . . ; 6 sn .  3: Set L ¼ f1g. Compute uL . Set R ¼ L . 4: Set l ¼ 2 and R ¼ R [ l. Compute uR .  5: while uR < uL do   6: Set L ¼ R and uL ¼ uR . 7: l ¼ l þ 1. 8: if l > n then 9: Break. 10: else 11: Set R ¼ R [ l and compute uR . 12: end if 13: End While 14: Return L .

5.2. Full information case With full information, the choice of the attractive set is more straightforward. If the headway distribution is a general continuous distribution with a range from 0 to þ1, each line in L could realize the minimum total travel time, since the line waiting time does not have a finite upper bound. Hence, the attractive set should include all lines in L. However, this might not be the case if the line waiting times are bounded. To see this point, consider the following extreme example. Assume that the headway follows deterministic distribution and there are two lines at the stop, i.e., L ¼ fa; bg. Let ha ¼ 10; sa ¼ 30, and hb ¼ 5 and sb ¼ 45. In this case, passengers will never board line b, because it is impossible that it will have a smaller total travel time due to its long line travel time. Although the attractive set may not include all the available lines, we note that determining the attractive set beforehand is unnecessary in the full information case. One can simply apply the formulae derived before to the entire line set L. If a line’s characteristics are such that it will never be selected, its boarding probability computed from the formulae would be zero. The lines with zero boarding probability may then be excluded in a postprocess. Table 1 Closed form solutions for

pi and x with exponential headway distribution. Line boarding probability (pi ) ki

No information Full information

Kn Pn  ki 

Partial information

Pn  ki 

k¼i Kk

i 2 Ly i 2 Ln

k¼i Kk

ðXkk  Xkþ1 k Þ P  k kþ1 exp j
N/A

Expected waiting time at the stop (x) No information

1 Kn

Full information

Pn Pn

Partial information

N/A

i¼1

ki k¼i K2 k

h

Xkk ðKk dki þ 1Þ  Xkþ1 ðKk dkþ1;i þ 1Þ k

i

P. (Will) Chen, Y. (Marco) Nie / Transportation Research Part B 72 (2015) 40–58

49

5.3. Partial information case In the partial information case, the enumeration of subsets seems necessary to guarantee optimality for the attractive set choice, regardless of the type of headway distribution. To avoid excessive computation time, the greedy algorithm presented in Algorithm 1 may be used as a heuristic. Note that the greedy algorithm requires repeatedly computing the expected waiting and total travel times for each subset examined. This computation, however, is far more expensive for the partial information case than for the no information case, since the former involves a two-dimension integration. To improve the computational efficiency, a new greedy-based heuristic algorithm (Algorithm 2) is proposed. The method initializes the attractive set with the one obtained assuming Ly ¼ £, and then successively evaluate the remaining lines in the ascending order of their line travel times, according to the formulae presented in Section 3. Algorithm 2. Heuristic algorithm for finding L in the partial information case. 1: Find the attractive set Ln assuming no information available. 2: Set L ¼ Ln . Set R ¼ Ln . Set Lua ¼ L n Ln . 3: if Lua is not empty then 4: Sort the lines in Lua ¼ f1ua ; 2ua ; . . . ; mua g such that s1ua 6 s2ua ; . . . ; 6 smua .  5: Compute uL using Eq. (10). 6: Set l ¼ 1ua and R ¼ R [ l. Compute uR using Eqs. (14), (12) and (10).  7: while uR < uL do   8: Set L ¼ R and uL ¼ uR . 9: Set l ¼ l þ 1. 10: if l > mua then 11: Break. 12: else 13: Set R ¼ R [ l and compute uR using Eqs. (14), (12) and (10). 14: end if 15: end while 16: end if 17: Return L . Although Algorithm 2 cannot guarantee finding the optimal attractive set in the partial information case, it will generate an attractive set that never yields a larger expected total travel time than the minimum expected total travel time in the no information case. In other words, the heuristic ensures that passengers will never be worse off with partial information, compared to the case without any information. To prove the result, we introduce the following lemma first. S S Lemma 1. If L ¼ Ly Ln , where Ly and Ln are sets of lines with and without information, respectively, and b L ¼ L^y Ln where L^y is b 4 the same set of lines as Ly , but is deprived of information ; then uL 6 u L . Proof. Assume that we have N (N ! þ1) realizations of waiting times for each line, and line li and l^i (i ¼ 1; 2; . . . ; N) denote L, respectively. Let TT li and TT l^i be the corresponding total travel time which the chosen line of each realization for sets L and b includes the waiting time and the line travel time. Then we have

uL ¼ lim

N!þ1

PN

i¼1 TT li

N

b ; u L ¼ lim

N!þ1

PN

i¼1 TT l^i

N

ð34Þ

:

For the no information case, we know that l^i will be the first arriving line in each realization. For the partial information case, according to the strategy described in Section 3, passengers may skip the first arriving line. Note that when this happens, the chosen line is ensured to have a smaller total travel time, which implies that the following condition must be satisfied:

TT li 6 TT l^i :

ð35Þ

b The above inequality together with Eq. (34) leads to u 6 u L . L

h

The main result is now formally stated below, followed by the proof. Proposition 1. The attractive set generated by Algorithm 2 will never yield a larger expected total travel time than the minimum expected total travel time in the no information case. 4

In other words, the symbol ^: is an operator that removes information from a set of lines.

50

P. (Will) Chen, Y. (Marco) Nie / Transportation Research Part B 72 (2015) 40–58

Proof. Let Ln denote the attractive set in the no information case and L the attractive set generated by Algorithm 2. Note  ^ ^ that Ln  L as per the algorithm. We need to show that uL 6 uLn , where uLn is the expected total travel time associated with   Ln when all lines with information in Ln are assumed to not have information. First, it follows from Lemma 1 that 

^

uLn 6 uLn :

ð36Þ 

Ln .

Note that Algorithm 2 initially sets L ¼ Then, a line that is not initially in expected total travel time. This implies that in the end 

Ln

would be added only if it decreases the



uL 6 uLn :

ð37Þ 

^

Combining the above two inequalities leads to uL 6 uLn .

h

6. Optimal hyperpath in a transit network We now consider the problem of finding the optimal hyperpath in a transit network under different information cases. Following Li et al. (2015), two dummy links and two dummy nodes are created for each line at a transfer node: an alighting link and its tail node, and a boarding link and its head node. Correspondingly, three sets of nodes may be defined: the set of S S transfer nodes N t , the set of dummy alighting nodes (N a ) and the set of dummy boarding nodes (N b ). Let N ¼ N t N b N a . Similarly, two dummy link sets may be defined: the alighting link set Aa and the boarding link set Ab . Note that the network may contain three other types of links: those connecting a boarding node to an alighting node at the next stop (called transit link and denoted as At ), those connecting the alighting to boarding nodes on the same line at the same stop (called dwell links and denoted as Ad ), and those connecting two adjacent transfer nodes through walking (called walking links and S S S S denoted as Aw ). The entire set of links in the network A ¼ At Aa Ab Ad Aw . Table 2 specifies the line travel time and the mean frequency for each link based on their type. 6.1. Optimality conditions 

Let Li be the set of all feasible boarding nodes associated with node i. It is clear that Li ¼ £ for all i R N t . Denote uLi as the minimum expected total travel time from node i to the destination S 2 N t (where Li is the attractive set at i); and IðiÞ and OðiÞ as the sets of incoming and outgoing nodes associated with node i, respectively. Bellman’s principle of optimality dictates that

( XZ



þ1

uLi ¼ min R # Li

j2R

0

)



ðw þ sij þ uLj Þcij ðwÞdw ;

i 2 Nt ;

ð38Þ





uLi ¼ minfuLj þ sij g i R Nt ;

ð39Þ

j2OðiÞ

LS

u ¼ 0:

ð40Þ

This optimality condition naturally gives rise to a labeling-type algorithm (Nguyen, 1989). The reader is referred to Appendix A for a description of a label-correcting algorithm. Note that a label-setting algorithm may be constructed similarly, but is omitted for simplicity. Would an optimal hyperpath contain cycles? The following result gives a sufficient condition to exclude cycles. 



Proposition 2. If for any j 2 Li ; sij þ uLj 6 uLi , the optimal hyperpath must not contain any cycles. Proof. Suppose that the optimal hyperpath contains a cycle that involves a node i. Then one can trace the cycle from one   node i all the way back to itself. Suppose sij þ uLj 6 uLi holds along the cycle, then the cycle may only exist if all sij are non-positive. This is a contradiction because a cycle must involve at least one link in either At or Aw . h Corollary 1. An optimal hyperpath does not contain any cycles if (1) the headway distribution is exponential and (2) no information about the line waiting times is available. Proof. When the two conditions are satisfied, combining the analytical results in Table 1 and Eq. (10) leads to 

uLi ¼ P

1

j2Li kij

þ

X kij ðuLj þ sij Þ P j kij j2L i







Again, we prove it through contradiction. Suppose sip þ uLp > uLi , we need to show that by removing p from Li uLi can be reduced. In other words, Li is not the optimal attractive set at i. When p is removed from Li , we have

51

P. (Will) Chen, Y. (Marco) Nie / Transportation Research Part B 72 (2015) 40–58 Table 2 Travel time and line frequency on different links in the transit network.

kij sij

Aa

Ab

Ad

Aw

At

1 Transfer loss (assumed as 0.1 min in this study)

Line frequency associated with j 0

1 0

1 Walking time

1 Line travel time from i to j

Li =p

u

1þ

¼

P

j2Li =p kij ðu

P

Lj

þ sij Þ

j2Li =p kij

Li


1þ

P

Lj j2Li =p kij ðu

þ sij Þ þ kip ðuLp þ sip Þ

P

þ kip

j2Li =p kij









The above inequality is equivalent to sip þ uLp > uLi =p , which surely holds as per the assumption sip þ uLp > uLi . Therefore, for   any j 2 Li ; sij þ uLj 6 uLi . It then follows from Proposition 2 that the optimal hyperpath must not contain any cycles. h 6.2. Cause and avoidance of cycles 



When the sufficient condition given in Proposition 2 is not satisfied (i.e. 9j 2 Li such that sij þ uLj > uLi . In words, it means that the expected total travel time of the attractive set may be smaller than the line travel time of one of the lines in that set), cycles may arise in an optimal hyperpath. To explain how this seemingly counter-intuitive phenomenon may occur, consider a small network illustrated in Fig. 1. The network has three nodes, each representing a stop. Line headway is assumed to be exponentially distributed for simplicity, but the cases with and without information will be both considered. When the label-correcting algorithm given in Appendix B is applied to find the hyperpath to node C, the solution in each    iteration for both cases are presented in Table 3. In the initialization, we set LA ¼ LB ¼ LC ¼ £; uLA ¼ uLB ¼ 1; uLC ¼ 0, and the scan list Q ¼ fCg. In the first iteration, nodes A and B are updated the same way for both information cases. Note that only C is included in the attractive set for both A and B because the initial line travel time is set to infinity. At the end of the first iteration, both A and B are added into Q. In the second iteration, node B is first examined. It is in this iteration that the two information cases start to differ from each other: 





 In the no information case, node B is not included in LA because sAB þ uLB ¼ 35 > uLA ¼ 30. Hence, after the update uLA remains to be 30 and Q ¼ fAg.  In the full information case, node B is added into LA because it does reduce its expected travel time from 30 to 28.88. Q is still fAg since A remains to be examined anyway. The calculation using the formula presented in Section 2 for the full information case is shown below. 

uLA ¼

Z

þ1

0

¼

Z

0



ðw þ sAC þ uLC Þf AC ðwÞF AB ðw þ sAC  sAB Þdw þ þ1

ðw þ 20 þ 0Þf AC ðwÞF AB ðw þ 20  10Þdw þ

Z 0

Z

þ1

0 þ1



ðw þ sAB þ uLB Þf AB ðwÞF AC ðw þ sAB  sAC Þdw ðw þ 10 þ 25Þf AB ðwÞF AC ðw þ 10  20Þdw  28:88:

ð41Þ

In the third iteration in the no information case, Q becomes empty after A is removed, and the algorithm terminates accord ingly. In the full information case, examining A results A to be added into LB , which in turn reduces uLB from 25 to 24.54. Since LB u is changed in this iteration, B is added back to Q at the end of the iteration. Clearly, from there on, the algorithm enters a   cycle that iteratively reduces uLB and uLA until these values converges to an ‘‘equilibrium’’ state as shown in Table 3. The above example demonstrates that full information may lead to cycles in an optimal hyperpath when the headway is exponentially distributed. Yet, this phenomenon may rise when only partial information is available or the headway distribution is not exponential. Therefore, cycles may not be ruled out in those cases as well. Having cycles in a hyperpath is inconvenient and to some extent counter-intuitive since hyperpaths are often ‘‘defined’’ as acyclic (See page 178 in Nguyen and Pallottino, 1988). If deemed necessary, the following simple heuristic may be introduced to avoid cycles. The idea is to impose the sufficient condition given in Proposition 2 regardless of the information availability and/or the type of headway distributions. Specifically, the strategy will prohibit the passengers from including   any link ij in their attractive set of node i as long as sij þ uLj is no shorter than the current expected travel time at i; uLi . Since the objective here is to minimize the expected travel time, it seems reasonable to assume that a traveler will exclude a line that has an expected travel time higher than what is available to her/him at the current stop (without including that line). Although the excluded line has a higher expected travel time, it may have a smaller realized travel time. Thus, in some circumstances the traveler may be better off by choosing the excluded line based on the given full information. Theoretically assessing how much efficiency might be lost due to the introduction of the above cycle-avoidance strategy is difficult in theory. It appears that the efficiency loss would depend on many parameters, such as the distribution type and network topology. Yet, our numerical experience has been that in real networks cycling is rare and that in most cases it is more of a computational nuisance (since by definition a hyperpath is acyclic) than a major concern for sub-optimal solutions.

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P. (Will) Chen, Y. (Marco) Nie / Transportation Research Part B 72 (2015) 40–58

Fig. 1. A three-node network.

Table 3 Iterations of label-correcting algorithm for solving the three-node problem with and without information 



Iteration

uLA

LA

uLB

LB

Scan list Q at the end of the iteration

No information 0 1 2 3

1 30 30 30

£ fCg fCg {C}

1 25 25 25

£ fCg {C} {C}

{C} {A, B} {A} {£}

Full information 0 1 2 3 4 5 6 ...

1 30 28.88 28.88 28.83 28.83 28.83 ...

£ fCg {B, C} {B, C} {B, C} {B, C} {B, C} ...

1 25 25 24.54 24.54 24.54 24.54 ...

£ fCg {C} {A, C} {A, C} {A, C} {A, C} ...

{C} {A, B} {A} {B} {A} {B} {A} ...

Therefore, the proposed strategy could be potentially useful in large-scale applications where near-optimal hyper-paths are needed. It will be adopted in numerical experiments presented in the following.

7. Numerical experiments 7.1. Impact of information availability To illustrate the impact of information availability on the distribution of line loads and total travel times, a small example with two lines is first constructed. The line attributes are given in Table 4. For simplicity, we assume that both lines have exponential headway distributions. The line boarding probabilities and expected travel times for both no information and full information cases are reported in the first two rows of Table 5. Thanks to the information, passengers always make the right decisions, i.e., boarding the line with the minimum total travel time. Hence, the expected total travel time is lower in the full information case. Also worth noting is that the faster line 1 with lower frequency gains share (up from 33.3% to 59.6%) due to information availability. This is because passengers can skip the slower line 2 and wait for the faster line 1 if the posted waiting time suggests a smaller total travel time. Now we turn to the cases where only one of the two lines has waiting time information, reported in the last two rows of Table 5. At first glance, those results are surprising, because the case with information being available only on line 1 has the same results as full information case and the case with information being available only on line 2 has the same results as the no information case. Yet, those results are indeed expected according to the proposed strategy. In the case only line 1 has information, passengers will know the total travel time of taking line 1 when they arrive at the stop. If line 1 turns out to arrive first, then line 1 must be the best line since line 2 is slower, thus passengers will surely board line 1. If line 2 arrives first, then passengers can compare it with line 1 and then choose to board line 2 or wait for line 1. In both circumstances, passengers will board the line with the minimum total travel time, which is actually the same as the full information case. In the case where line 2 has information, passengers will know the total travel time of taking line 2 when they arrive at the stop. If line 2 arrives first, then passengers will board line 2 since line 1 has no waiting time information. If line 1 arrives

53

P. (Will) Chen, Y. (Marco) Nie / Transportation Research Part B 72 (2015) 40–58 Table 4 Line attributes in the two-line example Line i

Headway mean E[hi ] (min)

Travel time si (min)

1 2

20 10

30 40

Table 5 Numerical results for different information cases in the two-line example Info case

p1

p2

x

l

u

No info Full info Line 1 with info Line 2 with info

0.333 0.596 0.596 0.333

0.667 0.404 0.404 0.667

6.67 7.87 7.87 6.67

36.66 34.04 34.04 36.66

43.33 41.91 41.91 43.33

first, then passengers will surely board line 1 because of its shorter travel time. Hence in this case, passengers will always board the first arriving line, which in turn is the same as the no information case. While simple, this example does offer interesting insights about the impact of the information availability. First, full information helps passengers make better decisions and redistribute the line loads. Second, partial information may achieve the same results as the full information if provided appropriately. Importantly, the waiting time information for the faster lines seems more useful because it can help passengers decide whether or not to skip the slower lines without concerning the extra waiting time for the faster lines. 7.2. Properties of attractive set To explore the properties of the attractive set in the partial information case, a third line is added into the previous example, as shown in Table 6. Again, all three lines are assumed to have exponential headway distributions. The results of the three-line example for both no information and full information cases are reported in Table 7. In this example, the newly-added line 3 is not in the attractive set for the no information case while it is included in the attractive set for the full information case. With full information, passengers will board the slower line 3 when it happens to be the best line. Hence, full information helps an otherwise unattractive line (line 3) gain share. There are in total six possible partial information cases in this three-line example, as enumerated in Table 8. The attractive set is determined by the brute-force enumeration instead of the heuristic algorithm. In all six cases, only case 4 includes all three lines in the attractive set. Interestingly, this case is equivalent to the case with full information, even though the information on line 3 is not available. Again, this result suggests that providing information for the slowest line is unnecessary. For cases 3, 5 and 6 in Table 8, line 3 is excluded from the attractive set even though its waiting time information is available to passengers. Another observation from the table is that, if no information is available on the fastest line 1 (cases 2, 3, 6), information on either line 2, line 3 or both will yield the same results. This indicates that it is not very helpful to provide information for the slower line if information on the faster lines is missing. Therefore, when the resource is limited, the priority of information provision should be given to the lines with shorter travel times. In this example, the attractive set is generated by the enumeration method. However, when the number of lines is large or when the attractive set has to be computed at many stops, the enumeration method is not computationally efficient. In the

Table 6 Line attributes in the three-line example. Line i

Headway mean E[hi ] (min)

Travel time si (min)

1 2 3

20 10 5

30 40 50

Table 7 Numerical results for no information and full information cases in the three-line example Info case

p1

p2

p3

x

l

u

No info Full info

0.333 0.570

0.667 0.353

0 0.077

6.67 6.32

36.66 35.08

43.33 41.40

54

P. (Will) Chen, Y. (Marco) Nie / Transportation Research Part B 72 (2015) 40–58

Table 8 Results for the partial information case in the three-line example Case

Line with info

Attractive set

p1

p2

p3

x

l

u

1 2 3 4 5 6

1 2 3 1,2 1,3 2,3

1,2 1,2 1,2 1,2,3 1,2 1,2

0.596 0.333 0.333 0.570 0.596 0.333

0.404 0.667 0.667 0.353 0.404 0.667

0 0 0 0.077 0 0

7.87 0.67 0.67 6.32 7.87 0.67

34.04 42.66 42.66 35.08 34.04 42.66

41.91 43.33 43.33 41.40 41.91 43.33

practical applications that involve large networks, therefore, heuristic algorithms, such as those presented in Section 5, are necessary. These heuristic algorithms may not always find the correct attractive set. However, we note that, for this threeline example, Algorithm 2 does generate the correct attractive sets for all six partial information cases.

7.3. Hyperpath finding in a real transit network In this section, we test the hyperpath algorithm in the three information cases using a large-scale real world transit network obtained from Chicago Transit Authority (CTA). The bus network, shown in Fig. 2, has 125 routes and 11,179 stops. The headway and link travel time data are extracted from real operation data in morning peak hours in the July of 2011; see Li et al. (2015). for a detailed description of the data. The headway distribution used in the test is Erlang distribution fitted from real headway observations. For the partial information case, we randomly picked about half of all transit links on which waiting time information is assumed to be available. The algorithm was coded in C++ and the tests were conducted on a laptop with Windows 7 Home Premium, Inter(R) Core(TM) i7–2630QM [email protected] GHz and 8.00 GB memory. 7.3.1. Optimal hyperpaths for a given O–D pair We first select one origin–destination (O–D) pair, namely Ashland & Irving Park to Michigan & Grand, to compare optimal hyperpaths obtained in three information cases. Fig. 3a–c show the topology of the optimal hyperpaths in three information

Fig. 2. Topology of Chicago Transit Authority bus network.

P. (Will) Chen, Y. (Marco) Nie / Transportation Research Part B 72 (2015) 40–58

55

Fig. 3. Optimal hyperpath under different information cases from Ashland & Irving Park to Michigan & Grand in the CTA bus network.

cases. The expected total travel times for the no information, partial information and full information cases are 36.96, 36.31 and 35.61 min, respectively. Thus, the information does help improve the expected total travel time. However, the improvement seems not very significant in this example (about 3–4% reduction). Fig. 3a–c reveal that different levels of information do lead to different optimal hyperpaths. In this example, only one line is considered at each stop in the optimal hyperpath when no information is available. With partial information, multiple lines are included in the attractive set at one transfer stop. With full information, multiple lines are included at two transfer stops. To summarize, this real world example confirms the findings from the previous examples, that the information helps reduce the total travel time by including the otherwise unattractive lines.

7.3.2. Average performance in three information cases In this section, the average performance of the three information cases are compared for multiple O–D pairs in the CTA bus network. To assess the average performance, four representative regions are selected in the Chicago metropolitan area covered by the bus network, namely Downtown Chicago, North Suburbs, South Suburbs and West Suburbs. In each region, ten random stops are selected as the origins or destinations. Four routing scenarios, namely, North Suburbs to Downtown Chicago, South Suburbs to Downtown Chicago, West Suburbs to Downtown Chicago and Downtown Chicago to Downtown Chicago, are tested for all three information cases. In each scenario, transit routing between all O–D pairs are performed and the average trip statistics are reported in Table 9. Note that, in total there are 90 O–D pairs for the Downtown to Downtown scenario, and 100 O–D pairs for the other three scenarios. From Table 9 we can see that the total trip time does decrease as more information becomes available, as expected. However, the improvements are generally insignificant. For the trips between West Suburbs to Downtown, for example, the benefit of information is practically negligible. This seemingly disappointing finding about the utility of passenger information has been noted in previous studies (e.g. Hickman and Wilson, 1995; Gentile et al., 2005). Note that the best possible solution is achieved with full information. In other words, even with the dynamic strategy mentioned before, a traveler cannot possibly be better off compared to the case with full information. Therefore, such small differences between the full and partial information results are also good news because it attests that the proposed routing strategy, while not optimal in theory, does produce near-optimal solutions. We close this section by commenting on the relative computational efficiency associated with the three information cases. To solve an all-to-one optimal hyperpath problem, the average CPU time for no information, partial information and full information cases are about 9 s, 400 s, and 7 s, respectively. The reason why the full information case is the least expensive is that the determination of the attractive set is unnecessary with full information. The partial information case is considerably more time consuming than the other two cases because it requires a two-dimension integration to evaluate the expected waiting time even for the Erlang distribution adopted in this study.

56

P. (Will) Chen, Y. (Marco) Nie / Transportation Research Part B 72 (2015) 40–58 Table 9 Average trip statistics for four O–D scenarios in the CTA bus network Trip statistics

Expected Expected Expected Expected Expected Expected Expected

trip time (min) EnRoute time (min) waiting time (min) walking time (min) transfer loss (min) trip distance (mile) average speed (mph)

Trip statistics

Expected Expected Expected Expected Expected Expected Expected

trip time (min) EnRoute time (min) waiting time (min) walking time (min) transfer loss (min) trip distance (mile) average speed (mph)

Trip statistics

Expected Expected Expected Expected Expected Expected Expected

trip time (min) EnRoute time (min) waiting time (min) walking time (min) transfer loss (min) trip distance (mile) average speed (mph)

Trip statistics

Expected Expected Expected Expected Expected Expected Expected

trip time (min) EnRoute time (min) waiting time (min) walking time (min) transfer loss (min) trip distance (mile) average speed (mph)

North to downtown No info

Partial info

Full info

60.86 40.09 10.89 9.43 0.45 9.19 8.98

60.54 39.88 10.72 9.42 0.52 9.23 9.07

60.20 39.63 10.43 9.63 0.51 9.23 9.13

South to downtown No info

Partial info

Full info

36.42 22.58 6.33 7.23 0.28 4.81 7.93

36.33 22.76 6.32 6.96 0.29 4.81 7.95

36.22 22.83 6.21 6.88 0.29 4.81 7.98

West to downtown No info

Partial info

Full info

40.20 26.39 6.76 6.76 0.29 4.30 6.38

40.19 26.44 6.72 6.74 0.29 4.30 6.38

40.18 26.49 6.71 6.69 0.29 4.30 6.38

Downtown to downtown No info

Partial info

Full info

16.51 4.76 1.59 10.03 0.13 1.05 3.72

16.36 4.95 1.68 9.58 0.15 1.05 3.76

16.29 5.01 1.65 9.49 0.14 1.05 3.78

8. Conclusions This paper studies the optimal routing strategy in a transit network with three different levels of online information, namely no information, full information, and partial information. We propose a general analysis framework that unify all three cases. The problem of determining the attractive set is also discussed for each of the three cases. In particular, a new heuristic algorithm is proposed for determining the attractive set with partial information. We show that the proposed heuristic algorithm not only is more efficient, but also guarantees a solution no worse than those obtained in the case of no information. Our analysis reveals that cycles may occur in optimal hyperpaths when information is available. Accordingly, a sufficient condition that would exclude such cycles is proposed and adopted as a heuristic in our case studies to avoid cycles. Our numerical experiments based on both hypothetical and real case studies lead to the following findings:  Information helps reduce the expected total travel time in a transit trip. In general, it also attracts more users to faster lines with lower service frequency, and causes the lines that are otherwise unattractive to be included in the attractive set.  Information on faster lines is more effective than that on slower lines, in terms of minimizing the total travel time. With limited resources, therefore, the priority of information provision should be given to the faster lines.  The benefits of information in reducing the total travel time are relatively insignificant in the CTA bus network. A few possible reasons are offered here to explain why the information was not as helpful as one would have hoped in the CTA network. First of all, in the real world, the alternative lines at any given stop may have rather similar headways and expected total trip times to the destination. Therefore, taking the first arriving bus (as in the no information case) may actually be a near-optimal solution. Second, the cycle avoidance strategy introduced in Section 6 and adopted in our implemen-

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tation eliminates the options that would help further reduce the expected total travel time. Further research is needed to determine which of the two factors plays a more important role. We have noted that, when headway distributions are not exponential, passengers should adopt a dynamic strategy, i.e., they may change their attractive set according to elapsed waiting times (Billi et al., 2004). Due to the computational challenges, the dynamic strategy is not considered here. For future research, efficient numerical schemes can be developed to implement the dynamic strategy and compare it with the strategy proposed in this paper. Another relevant issue concerns the fact that the real-time information itself may be unreliable due to prediction errors. When reported arrival times are random variables following a known distribution, the proposed strategy is no longer applicable  because ly cannot be determined. A more general strategy is needed to integrate both headway and arrival time distributions into passengers’ decision-making, which constitutes another challenging and interesting direction for further research. Finally, the rapidly evolving wireless communication technology has enabled most transit passengers to access passenger information on their smart phones and other mobile devices whenever and wherever they so desire. With such ubiquitous information access, passengers may no longer arrive at stops randomly, as currently assumed in our routing model. Rather, they may well synchronize their departure times with the projected bus arrival time. Moreover, they may make their transfer decisions when they are on board a transit vehicle. It seems that a routing model endogenizing passengers’ departure time and en-route transfer decisions would better capture the true benefits of information. We leave these interesting extensions also to the future studies. Acknowledgements This research was supported by Center for Commercialization of Innovative Transportation Technology at Northwestern University. The authors wish to thank Chicago Transit Authority for providing real-time bus running data used in this research. The generous assistance of Mr. Michael Haynes at CTA is greatly appreciated. We would also like to thank two anonymous reviewers for their constructive comments on an earlier version of the paper. In particular, the literature on the dynamic strategy was brought to our attention by one of the reviewers. The remaining shortcomings are those of the authors’ alone. Appendix A. A label-correcting algorithm Algorithm 3. Label-correcting hyperpath (LCH) algorithm 1: initialize: 2: 8i 2 N, set Li ¼ £; ui ¼ inf 3: Set uS ¼ 0; Q ¼ fSg 4: while Q – £ do 5: Take node j from the front end of Q. Remove j from Q. 6: for all links a 2 IðjÞ do 7: Set i as the tail node of a. 8: if i 2 N t and a 2 Ab then 9: Construct a common-lines problem as follows. 10: Set L ¼ flj8l 2 OðiÞ \ Ab g 11: For each l 2 L, the line travel time sl is set to smk þ uk where l 2 OðmÞ and l 2 IðkÞ.  12: Call Algorithm 1 or 2 to get L and uL with L as the input.  13: if uL < ui then  14: Set ui ¼ uL ; Li ¼ L . 15: if i is not in Q then 16: Insert i to the end of Q. 17: end if 18: end if 19: else 20: Set u ¼ sa þ uj . 21: if u < ui then 22: Set ui ¼ u; Li ¼ fag. 23: if i is not in Q then 24: Insert i to the end of Q. 25: end if 26: end if 27: end if 28: end for 29: end while

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