Transportation Research Part E 47 (2011) 641–647

Contents lists available at ScienceDirect

Transportation Research Part E

journal homepage: www.elsevier .com/locate / t re

Reaching a destination earlier by starting later: Revisited

Chen Qian, Ching-Yuen Chan ⇑, Kai-Leung YungDepartment of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong

a r t i c l e i n f o

Article history:Received 12 April 2010Received in revised form 29 September2010Accepted 23 November 2010

Keywords:TransportationDynamic traffic networkShortest path

1366-5545/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.tre.2011.01.002

⇑ Corresponding author. Tel.: +852 2766 4980; faE-mail address: mfcychan@inet.polyu.edu.hk (C.-

a b s t r a c t

This paper simulates a dynamic traffic network by embedding an extra nonlinear delayfunction to represent traffic lights (or a similar regular delay) in each arc (or link), and itwas shown that a late start driver may catch up with one who started earlier, subject onlyto the condition that they pick the same path (or the same shortest path), and overtakingwill never be made. Moreover, a theoretical lower bound value to guarantee the sustain-ability of the FIFO principle in a dynamic traffic network was also derived.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Arriving earlier at a destination is one of the major concerns of vehicle drivers. Smeed (1967) stated that a vehicle couldreach a destination earlier by starting later. However, Ben-Akiva and De Palma (1986) spotted a problem in Smeed’s argu-ment, and concluded that it was not possible to reach a destination earlier than a driver who set off first. In 1993, Friesz et al.did some research on route choice and departure time decisions of transportation network users, and found that for any lin-ear arc delay function, the resulting arc exit time function was strictly increasing, and hence the time gradient function ex-isted. Friesz et al. (1993) viewed this as a generalization of ‘‘it is impossible to arrive earlier by departing later’’, with theextension of including all continuous flow rate patterns and this also ensured that the FIFO principle would be upheld.

In real life, it is a normal situation that someone who starts later can reach the destination earlier in a dynamic trafficnetwork. This situation seems to conflict with the conclusion being drawn. In this paper, the loophole of previous studiesis investigated. A simple case is provided in Section 4 to illustrate the situation when the changing of the start time will re-sult in different time durations spent in the network, however, overtaking the earlier one is still not possible unless vehicleshad different speeds.

2. Loophole of previous studies

The previous studies did not address the dynamic network issue regarding whether or not the FIFO principle ismaintained. Ben-Akiva and De Palma’s (1986) analysis postulated that all vehicles followed the same arcs from an Originto a Destination (OD), and they used a constant value to represent the path distance in their formulations. In the mathemat-ical proofs from Friesz et al. (1993), only one arc exit time function represented the time cost. All these mean that the arcsconnecting the OD in order to have the shortest path were unchanged (i.e., the same pathway for the same OD all the time).However, it is quite common that more than one set of arcs links the same OD in a traffic network. Furthermore, the

. All rights reserved.

x: +852 2362 5267.Y. Chan).

642 C. Qian et al. / Transportation Research Part E 47 (2011) 641–647

investigation conducted by Friesz et al. (1993) made the precondition that all arcs were represented by linear delay functionsto simulate the time costs. However, another very important feature affecting the traffic is that there are some possible de-lays associated with clock time, like traffic lights. Hence, the arc delay is not linear anymore which also causes the selectionof arcs to form the shortest path to change from time to time.

More and more researches on routing problems are taking into account time constraints in order to move the mathemat-ical models closer to real-world problems, especially in the Shortest Path Problem (SPP) research. The time window appearsto be a common form of time constraint that defines a specified time interval, and a node in a network subjects to a timerelated cause (Desrochers et al., 1992; Dumas and Desrosiers, 1995; Kohl and Madsen, 1997; Chen and Yang, 2000, 2004;Bräysy and Gendreau, 2005a,b; Yang and Chen, 2006; Avarenga et al., 2007; Albiach et al., 2008; Nie and Wu, 2009; Qureshiet al., 2009; Soler et al., 2009). The recent research focuses on more convenient method or algorithms to find the shortestpath in different networks or situations; however, the topics of whether FIFO principle is still valid in these networks orwhether the vehicle drivers should start earlier or later are barely investigated. This paper is to fill this research gap andwe will treat the shortest path analogous to the shortest time duration spent within an OD, rather than the actual shortestphysical distance, to compensate for the nonlinear delays.

3. Examination of FIFO principle in dynamic traffic networks

In this section, the formulation of new time window by taking into consideration nonlinear delays will be illuminated.Then, under the active arc set condition, which results in different arc sets with respect to time for the same OD for providingthe shortest path, the FIFO principle will be examined.

3.1. Time window establishment

While the time windows exist in an arc, the time delay function of this arc can be nonlinear. Fig. 1 shows a time windowon an arbitrary arc k, which lies between nodes x and y with a time duration Tw,k from the inlet x.

Tw,k - the undisturbed time from inlet to the time window in the arc k.

To,k - the undisturbed time from inlet to outlet in the arc k. (Undisturbed means no time delay on the element will be imposed,

by whatever means, including the time window.)

yx

Time Window location

Tw,k

To,k

Fig. 1. Time window allocation in arc k.

Fig. 2. Time window model for arc k.

Fig. 3. Time window delay function for arc k.

C. Qian et al. / Transportation Research Part E 47 (2011) 641–647 643

Also the time window configuration for the arc k is shown in Fig. 2, in which the effect of the time window is representedby a pulse chain with a cyclic time interval (Tp,k) that comprises two zones: time delay (Tpc,k) and the remaining time gap tothe next cycle (Tpo,k); to synchronize the consequences of a time window with the clock time, the initiation clock time (tp,k)has to be employed, and every pulse in the pulse chain is tagged with a number in an ascending order (0, 1, . . . , n, . . .) to sig-nify its position at the clock time axis.

Based on the time window constructed, the time delay function of arc k is illustrated as in Fig. 3.

DTp;k ¼ ðt � tp;kÞmodðTp;kÞ ð1ÞwkðtÞ ¼max½ðTpc;k � DTp;kÞ; 0� ð2Þ

Eqs. (1) and (2) govern the characteristics of this time delay function, where DTp,k stands for the current cycle durationthat has just passed. wk(t) is the time delay caused by the time window. If it falls into the No-Go zone (see Fig. 2), a waitingtime of (Tpc,k � DTp,k) is implied; otherwise, there will be no delay. Now, by taking into consideration the new time window,the exit time function of arc k at time t is:

skðtÞ ¼ t þwkðt þ Tw;kÞ þ To;k ð3Þ

3.2. Single path with time window

For a path formed by linking N arcs (N + 1 nodes) with time sequences: [0, T1], [T1, T2], . . . , [TN�1, TN]. Tw,k is the location ofthe time window, which is a nonlinear delay in arc k (i.e., Tk�1 < Tw,k < Tk). Hence, the exit time function of arc k is sk(t) and isincreasing linearly in the two periods [Tk�1, Tw,k), (Tw,k, Tk] according to Friesz et al. (1993). Therefore, for driver a enteringarc k at time ta, driver b comes at a later time tb (i.e., tb > ta), and there is a time gap DTg = tb � ta. Consequently, at timet = ta,k, the delay time (Wa,k(t)) of driver a at the time window k is:

Wa;kðtÞ ¼max½ðTpc;k � DTp;kÞ;0� ð4ÞDTp;k ¼ ðta;k � tp;kÞmodðTp;kÞ ð5Þ

Now, if the delay on driver a is less than the time gap between these two drivers wa,k(t) < DTg, driver a keeps leading; elsewa,k(t) P DTg, and driver b will catch up with driver a. In the case of wa,k(t) > DTg, both drivers have to wait for the next ‘‘Go’’zone to come before they can proceed together. Consequently, the arc k exit time functions of these two drivers aresk(ta) 6 sk(tb) in the period [Tk�1, Tk] where Tk�1 < Tw,k < Tk. Thus, for f ðtÞ ¼

PNk¼1skðtÞ, f(ta) 6 f(tb) is possible, and thus a driver

who departs later may still catch up with an earlier start driver and so they can reach the destination simultaneously.

3.3. Dynamic traffic network investigation

There are m paths linking an OD, and the path exit time function of path i is fiðtÞ ¼PN

k¼1sk;iðtÞ; ði ¼ 1; . . . ;mÞ. In the case ofthe vehicle arrival time at the origin being equal to time tj (j = 1, 2, 3 . . .) in increasing clock time, the time exit function of theshortest path to reach the destination is the arc set giving the lowest time exit function value at that instant, and it is rep-resented by the equation Fj(tj) = min(f1(tj), f2(tj), . . . , fm�1(tj), fm(tj)) at time tj. In other words, Fj(tj) is the shortest path at timetj in the traffic network. Now, at tj�c < tj (j > c or in other words, tj�c means a time before tj) and with the same path, it is pos-sible for a later start driver to catch up in the single path case, so Fj(tj�c) 6 Fj(tj) is valid (see Section 3.2 for f(ta) 6 f(tb)). Alter-natively, if they are not on the same path and there is only one shortest path existing at one time, then Fj�c(tj�c) < Fj(tj�c) is

No-Go

No-Go

No-Go

Go

Go

Go

TcycTw,max

Window 1

Window 2

Window N

…………..

Fig. 4. Determination of max time delay (Tw,max).

644 C. Qian et al. / Transportation Research Part E 47 (2011) 641–647

resulted. By combining the two results here, one can obtain Fj�c(tj�c) < Fj(tj�c) 6 Fj(tj) and this implies that Fj�c(tj�c) < Fj(tj). Itmeans the later start driver would not be able to catch up once the arc set to formulate the path is different.

3.4. Lower bound value to guarantee FIFO in a dynamic traffic network

According to the analysis in Section 3.2, it is possible for a later start driver to catch up with an earlier start driver in asituation of having the same path, even though they have selected the shortest path at the start of the journey. However,if the time gap between these two drivers is large enough, the FIFO principle can still be upheld. In fact, the later start drivercannot catch up, which means the path exit time functions of these two drivers (a and b) are at the condition that guaranteesfi(ta) < fi(tb) by knowing that they are using the same path i. To examine this, the time exit function of a path has been mod-ified as fi(t) = t + D(t), where t is the time when the journey began and D(t) is the total delay due to the time window effect.

Subsequently, for the FIFO to sustain tb + D(tb) > ta + D(ta) is a requirement, and we can get tb � ta > D(ta) � D(tb). tb � ta isthe time gap, and hence tb � ta >

Pwk(ta) �

Pwk(tb). This means that while the time gap is larger than the difference of the

total delay times between the two drivers, the later start driver can never catch up. Obviously, the minimum delay for driverb is zero, and the maximum time delay

Pwk(ta) for driver a is equal to Tw,max, in Fig. 4. Here, we need to determine the cycle

time of this combined delay function, and fortunately, this can be obtained by Tcyc = LCM(Tp,k), in which k is the arc set in-volved in forming the path. So, to guarantee FIFO, one needs to search for the largest time gap Tw,max in a period of Tcyc; moreoperating details will be given in Section 4.

4. An illustrative experiment

Fig. 5 shows the schematic of a simple traffic network for illustrating the findings in this paper, and Table 1 shows theconfiguration of associated parameters related to this network. An experiment has been conducted by having a driver startthe journey at each 1 unit time interval for a period of 120 time units. The ‘‘Path’’ columns in Table 2 indicate the shortestpath selected at different start times, and the arrival times are shown under the ‘‘Arrival time’’ columns.

It is observed that as time is passing, the shortest path is changing as expected. With increment of the start time, arrivaltimes are increasing nonlinear, and some of them reach the destination at the same time. (E.g., when start times are equal to2–5, the arrival times are 57 for all, and the shortest path is ACE; when start times are 20–22, arrival times are 73, and theshortest path is ABE, etc.) Moreover, it is also found that overtaking can never occur, and each catch up case only happened

A

B

C

D

E

Fig. 5. Experimental network.

Table 1Experimental network configuration.

Arc (k) Tw,k Tp,k tp,k To,k

Tpo,k Tpc,k

A–B 10 5 7 0 24A–C 13 7 5 0 25A–D 10 6 6 0 23B–E 12 6 4 0 27C–E 15 5 5 0 27D–E 15 5 5 0 30

Table 2Shortest paths at different start times.

Start time Path Time delay Arrival time Start time Path Time delay Arrival time Start time Path Time delay Arrival time

0 ABE 51 51 40 ACE 57 97 80 ABE 52 1321 ABE 51 52 41 ACE 56 97 81 ABE 51 1322 ACE 55 57 42 ACE 55 97 82 ABE 51 1333 ACE 54 57 43 ACE 54 97 83 ABE 51 1344 ACE 53 57 44 ACE 53 97 84 ABE 55 1395 ACE 52 57 45 ACE 52 97 85 ABE 54 1396 ACE 52 58 46 ACE 52 98 86 ACE 54 1407 ACE 52 59 47 ABE 52 99 87 ACE 53 1408 ACE 52 60 48 ABE 51 99 88 ACE 52 1409 ABE 51 60 49 ABE 51 100 89 ACE 52 141

10 ABE 51 61 50 ACE 57 107 90 ABE 54 14411 ABE 51 62 51 ACE 56 107 91 ABE 53 14412 ABE 51 63 52 ACE 55 107 92 ABE 52 14413 ABE 51 64 53 ACE 54 107 93 ABE 51 14414 ACE 54 68 54 ACE 53 107 94 ACE 53 14715 ACE 53 68 55 ACE 52 107 95 ABE 54 14916 ACE 52 68 56 ACE 52 108 96 ABE 53 14917 ACE 52 69 57 ACE 52 109 97 ABE 52 14918 ACE 52 70 58 ABE 51 109 98 ACE 59 15719 ACE 52 71 59 ABE 51 110 99 ACE 58 15720 ABE 52 72 60 ABE 51 111 100 ACE 57 15721 ABE 51 72 61 ABE 51 112 101 ACE 56 15722 ABE 51 73 62 ACE 55 117 102 ACE 55 15723 ABE 51 74 63 ACE 54 117 103 ACE 54 15724 ABE 55 79 64 ACE 53 117 104 ACE 53 15725 ABE 54 79 65 ACE 52 117 105 ACE 52 15726 ACE 54 80 66 ACE 52 118 106 ACE 52 15827 ACE 53 80 67 ACE 52 119 107 ABE 52 15928 ACE 52 80 68 ACE 52 120 108 ABE 51 15929 ACE 52 81 69 ABE 51 120 109 ABE 51 16030 ABE 54 84 70 ABE 51 121 110 ACE 57 16731 ABE 53 84 71 ABE 51 122 111 ACE 56 16732 ABE 52 84 72 ABE 51 123 112 ACE 55 16733 ABE 51 84 73 ABE 51 124 113 ACE 54 16734 ACE 53 87 74 ACE 54 128 114 ACE 53 16735 ABE 54 89 75 ACE 53 128 115 ACE 52 16736 ABE 53 89 76 ACE 52 128 116 ACE 52 16837 ABE 52 89 77 ACE 52 129 117 ACE 52 16938 ACE 59 97 78 ACE 52 130 118 ABE 51 16939 ACE 58 97 79 ACE 52 131 119 ABE 51 170

C. Qian et al. / Transportation Research Part E 47 (2011) 641–647 645

when two drivers have the same path. This means that the dynamic traffic network does not keep to the FIFO principlestrictly. Although it is not possible to reach the destination earlier by starting later even though the shortest path has beenchosen at the start time, it is possible to save travel time cost by starting later. For example, the driver who started at clocktime 14 took a total time cost of 54 units, while the driver who started later at time 17 only took 52 units of total time cost.

In this example, there are only two valid Go/No-Go pulse chains, one is from arc AC and the other is from arc CE. The cycletime of the path delay function as mentioned is Tcyc = LCM(Tp,k) = LCM(12, 12, 12, 10, 10, 10) = 60 time units by consideringall arcs, and the pulse chain of each involved arc in the sequence of path order for a period of 60 time units was drawn. Then,the largest square wave overlap was identified by starting with the first pulse chain as in Fig. 6. In the figure, there are fivesquare wave overlap zones labeled as (1), (2), (3), (4), (5), and the largest one is (3), and therefore Tw,max = 11. Subsequently,

Fig. 6. Max time delay (Tw,max) of illustration example.

646 C. Qian et al. / Transportation Research Part E 47 (2011) 641–647

one can observe in Table 2 that for any two drivers with a start time gap larger than 11 time units, the arrival time of the laterstart driver is always larger than the arrival time of an earlier start driver.

5. Conclusion

The contributions of the paper to the literature are that we extend the study of a single path to a dynamic traffic networkwith nonlinear time windows in arcs, and have proven that FIFO principle may not hold in a network situation. We haveshown that it is possible for two drivers starting at different times to reach the destination simultaneously in a dynamic traf-fic network; however, this can happen if, and only if, the paths (or shortest paths) of these two drivers are the same. Other-wise, the FIFO principle is always upheld. In addition, overtaking an early start driver is something not possible! Anotherimportant finding is that a later start driver may be able to catch up when the time gap between two drivers does not exceeda certain limit, and the determination of this time gap has also been established in this research. Lastly, the experimentalresults also showed that it is possible to save travelling time by picking a suitable journey start time, as the total time costin association with the shortest path is changing from time to time.

For the calculation of the time gap, we only eliminated the overlaps in delays and ignored the delay for the catching upvehicle here. Consequently, this value can be overestimated and some further studies will be required to have a better under-standing of this point. In addition, there is simply one time window in an arc, and it is thought that the multiple time win-dows case can be simulated by separating the case into a series of single time window arcs in sequence but would there be abetter way to reduce the computation complexity? Besides, the applications may be more extensive if the regular pulse chainin a time window could be modified to have some forms of variation, but this would also be quite challenging work!

Acknowledgement

The work described in this paper was fully supported by a Grant from the Department of Industrial and Systems Engi-neering of The Hong Kong Polytechnic University (No. RPH4).

References

Albiach, J., Sanchis, J.M., Soler, D., 2008. An asymmetric TSP with time windows and with time-dependent travel times and costs: an exact solution through agraph transformation. European Journal of Operational Research 189, 789–802.

Avarenga, G.B., Mateus, G.R., de Tomi, G., 2007. A genetic and set partitioning two-phase approach for the vehicle routing problem with time windows.Computers & Operations Research 34, 1561–1584.

Ben-Akiva, M., De Palma, A., 1986. Some circumstances in which vehicles will reach their destinations earlier by starting later: revisited. TransportationScience 20, 52–55. Retrieved from Business Source Complete database.

Bräysy, O., Gendreau, M., 2005a. Vehicle routing problem with time windows. Part I: Route construction and local search algorithms. Transportation Science39, 104–118.

Bräysy, O., Gendreau, M., 2005b. Vehicle routing problem with time windows. Part II: Metaheuristics. Transportation Science 39, 119–139.Chen, Y.L., Yang, H.H., 2000. Shortest paths in traffic-light networks. Transportation Research Part B 34, 241–253.Chen, Y.L., Yang, H.H., 2004. Finding the first K shortest paths in a time-window network. Computers & Operations Research 31, 499–513.Desrochers, M., Desrosiers, J., Solomon, M.M., 1992. A new optimization algorithm for the vehicle routing problem with time windows. Operations Research

40, 342–354. Retrieved from Military & Government Collection database.Dumas, Y., Desrosiers, J., 1995. An optimal algorithm for the traveling salesman problem with time windows. Operations Research 43, 367–371. Retrieved

from Military & Government Collection database.Friesz, T., Bernstein, D., Smith, T., Tobin, R., Wie, B., 1993. A variational inequality formulation of the dynamic network user equilibrium problem. Operations

Research 41, 179–191. Retrieved from Military & Government Collection database.Kohl, N., Madsen, O., 1997. An optimization algorithm for the vehicle routing problem with time windows based on lagrangian relaxation. Operation

Research 45, 395–406. Retrieved from Military & Government Collection database.Nie, Y., Wu, X., 2009. Shortest path problem considering on-time arrival probability. Transportation Research Part B 43, 597–613.

C. Qian et al. / Transportation Research Part E 47 (2011) 641–647 647

Qureshi, A.G., Taniguchi, E., Yamada, T., 2009. An exact solution approach for vehicle routing and scheduling problems with soft time windows.Transportation Research Part E 45 (6), 960–977.

Smeed, R.J., 1967. Some circumstances in which vehicles will reach their destinations earlier by starting later. Transportation Science 1, 308–317.Soler, D., Albiach, J., Martínez, E., 2009. A way to optimally solve a time-dependent vehicle routing problem with time windows. Operations Research Letters

37, 37–42.Yang, H.H., Chen, Y.L., 2006. Finding K shortest looping paths with waiting time in a time–window network. Applied Mathematical Modelling 30, 458–465.