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Speed limits, speed selection and network equilibrium
Transportation Research Part C 51 (2015) 260–273
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Transportation Research Part C journal homepage: www.elsevier.com/locate/trc
Speed limits, speed selection and network equilibrium Hai Yang 1, Hongbo Ye ⇑, Xinwei Li, Bingqing Zhao Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
a r t i c l e
i n f o
Article history: Received 29 April 2014 Received in revised form 6 December 2014 Accepted 6 December 2014 Available online 12 January 2015 Keywords: Speed limit Speed selection Safety Traffic equilibrium Traffic network
a b s t r a c t This paper investigates the local and global impact of speed limits by considering road users’ non-obedient behavior in speed selection. Given a link-specific speed limit scheme, road users will take into account the subjective travel time cost, the perceived crash risk and the perceived ticket risk as determinant factors for their actual speed choice on each link. Homogeneous travelers’ perceived crash risk is positively related to their driving speed. When travelers are heterogeneous, the perceived crash risk is class-specific: different user classes interact with each other and choose their own optimal speed, resulting in a Nash equilibrium speed pattern. With the speed choices on particular roads, travelers make route choices, resulting in user equilibrium in a general network. An algorithm is proposed to solve the user equilibrium problem with heterogeneous users under link-specific speed limits. The models and algorithms are illustrated with numerical examples. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Speed limit schemes have been ubiquitous around the world owing to effectiveness in enhancing safety, reducing emissions and/or saving energy consumption in a straightforward manner. However, imposing speed limits will undoubtedly affect travel time and mobility on the roads. Previous studies have overwhelmingly focused on the local impacts of speed limits on performance such as safety and vehicle emissions of the road where the speed limit is imposed; but unfortunately the system-wide impacts of speed limits on a general network have received only very limited attention in both the theoretical and empirical literature (Yang et al., 2012). Although McKnight and Klein (1990), Lave and Elias (1994, 1997) and Grabowski and Morrisey (2007) realized the traffic reallocation effect of speed limits, their studies focus on only local impacts without systematic investigation. Taylor (2000), Woolley et al. (2002) and Madireddy et al. (2011) applied microscopic traffic simulation tools to examine the system-wide impacts of speed limits, and it was reported that traffic reallocation was observed and the travel time disproportionately increased with reduced speed limits. It was not until recently that Yang et al. (2012) made the first attempt to theoretically investigate the traffic reallocation effect of link-specific speed limit schemes on a general network from the viewpoint of macroscopic user equilibrium. The uniqueness conditions of the UE link travel times and link flows were investigated, and the feasibility of using the speed limits as a flow management toll was discussed. In the same spirit, Yang et al. (2013) and Wang (2013) undertook further analysis of the network performance with speed limits. Yang et al. (2013) established the tri-objective, bi-level optimization problem, aiming to minimize the travel time, accident occurrence and emission simultaneously. Wang (2013) also considered the efficiency and equity issues in the implementation of speed limits.
⇑ Corresponding author. Tel.: +852 2358 7175; fax: +852 2358 1534. 1
E-mail address: [email protected] (H. Ye). Part of the research was carried out during the first author’s stay at Beihang University as a Changjiang visiting chair professor.
http://dx.doi.org/10.1016/j.trc.2014.12.002 0968-090X/Ó 2014 Elsevier Ltd. All rights reserved.
H. Yang et al. / Transportation Research Part C 51 (2015) 260–273
261
Nomenclature A a M m
v vm va v ma s sa ^s ^sm ^sm a sm sm a sav e saav e la
set of directed links link a 2 A set of user classes user class m 2 M aggregate flow on a representative link flow of user class m 2 M on a representative link aggregate flow on link a 2 A flow of user class m 2 M on link a 2 A speed limit on a representative link speed limit on link a 2 A optimal speed on a representative link with homogeneous travelers optimal speed of user class m 2 M on a representative link optimal speed of user class m 2 M on link a 2 A speed adopted by user class m 2 M on a representative link speed adopted by user class m 2 M on link a 2 A average speed on a representative link average speed on link a 2 A length of link a 2 A
A key assumption in Yang et al. (2012) and subsequent studies (Yang et al., 2013; Wang, 2013) is road users’ perfect compliance with the speed limits, which is restrictive in practical situations, as reported in many studies (Kanellaidis et al., 1995; Tarko, 2009; Yannis et al., 2013). For an individual traveler, a higher driving speed may weaken control of car and require a longer stopping distance, naturally raising the possibility of crash involvement. The relationship is described by either power functions (Maycock et al., 1998; Quimby et al., 1999) or exponential functions, with the crash risk increasing faster at higher speeds (Fildes et al., 1991; Kloeden et al., 1997, 2001). Besides absolute speed, speed variance among vehicles on a road is also a main factor for crash involvement. Earlier and recent studies found an increased crash risk for vehicles driving faster than the surrounding vehicles. However, the findings diverge for vehicles driving slower than average. An increased risk was reported in earlier studies (Solomon, 1964; Cirillo, 1968; RTI, 1970) but it was not reconfirmed in more recent studies (Kloeden et al., 1997, 2001). It was reported later that the crash frequency at the road section level increases with average speed (Finch et al., 1994; Nilsson, 1982, 2004) and speed variance (Garber and Gadiraju, 1989). Comprehensive reviews of previous studies on the relationship between crash rate and speed can be found in Aarts and van Schagen (2006) and McCarty (1998). This paper is intended to make a useful and substantial extension of the traffic equilibrium model with speed limits, as proposed by Yang et al. (2012), in order to capture road users’ speed selection behavior. The rest of the paper is organized as follows. In Section 2, the traffic equilibrium model with obedient users in Yang et al. (2012) is reviewed. Section 3 analyzes the speed selection of non-obedient, homogeneous users. Section 4 extends this speed selection model to the case with heterogeneous users by including the crash risk increment caused by the speed variance among different user classes. The network-wide user equilibrium problem with speed selection is proposed in Section 5, together with an iterative solution algorithm and some numerical examples for demonstration. General conclusions are provided in Section 6. 2. Traffic equilibrium under speed limits with obedient users Consider a general network G ¼ ðN; AÞ with a set N of nodes and a set A of directed links. Denote W as the set of origindestination (OD) pairs. Each OD pair w 2 W is connected by a set Rw of simple routes serving a given and fixed travel demand dw . Let f r;w denote the flow on path r 2 Rw between OD pair w 2 W, and v A ¼ ðv a ; a 2 AÞT denote the vector of link flows with v a representing the traffic flow on link a 2 A. For a specific link, the speed-flow relationship and the travel time-flow relationship are depicted in Fig. 1. Like most static traffic assignment models, only the normal flow regime is considered in this study, so for any link a 2 A, the travel time function ~ta ðv a Þ is assumed to be separable, continuous, differentiable and strictly increasing with link flow v a . Let sa denote the link-specific speed limit on link a 2 A. If no speed limit is imposed on link a, we can simply set sa to be the free-flow speed on link a. Yang et al. (2012) investigated the properties of traffic equilibrium under a speed limit law by assuming that all travelers strictly adhere to the speed limit on each link. In this case, the speed-flow relationship on link a 2 A under speed limit sa is altered, as shown in Fig. 2(a), where C is the road capacity and sc is the vehicle speed when the link flow rate reaches capacity. Correspondingly, the travel time function t a ðv a Þ on link a 2 A under speed limit sa takes the following form:
t a ðv a Þ ¼
0 6 v a 6 v a ta ; ~ta ðv a Þ; v a < v a 6 C a
ð1Þ
H. Yang et al. / Transportation Research Part C 51 (2015) 260–273
Travel time t
262
0
Traffic volume v
(a)
(b)
Fig. 1. Speed-flow relationship and travel time-flow relationship without speed limits.
Fig. 2. Speed-flow relationship and travel time-flow relationship with a speed limit (Yang et al., 2012).
a is the traffic flow such that ~ta ðv a Þ ¼ ta and C a is the capacity of link where t a is the travel time on link a 2 A at speed sa , v a 2 A. As shown in Fig. 2(b), the travel time function under a speed limit is no longer differentiable or strictly increasing. The user equilibrium (UE) satisfies the following condition:
X ta ðv a Þda;r ¼ lw ;
if f r;w > 0; r 2 Rw ; w 2 W
ð2Þ
a2A
X ta ðv a Þda;r P lw ;
if f r;w ¼ 0; r 2 Rw ; w 2 W
ð3Þ
a2A
where da;r equals 1 if path r uses link a and zero otherwise, and lw is the minimal path travel time between OD pair w 2 W. The UE conditions (2)–(3) are equivalent to the following optimization problem:
minv A 2X
X Z va a2A
t a ðxÞdx
ð4Þ
0
where X is defined by
(
X¼
v A v a ¼
XX w2W r2Rw
f r;w da;r ; a 2 A; f r;w P 0;
X
) f r;w ¼ dw ; r 2 Rw ; w 2 W
ð5Þ
r2Rw
The optimization problem (4)–(5) takes exactly the same form as that of a standard traffic equilibrium model without speed limit (Sheffi, 1985). The only difference is the altered shape of the travel time-flow relationship, which is no longer differentiable and strictly increasing. As proven in Yang et al. (2012), with some common assumptions, the UE travel times on all links are still unique. The UE flows on links with non-binding speed limits are unique, while those on the links with binding speed limits may not be unique, but can be explicitly depicted by a polyhedron or a linear system of equalities and inequalities.
H. Yang et al. / Transportation Research Part C 51 (2015) 260–273
263
Fig. 3. Speed deterrents and enticement in speed selection: (a) ^s 6 s; (b) ^s > s.
3. Speed limits and speed selection with non-obedient homogeneous users In practice travelers do not necessarily abide by the speed limits in a strict manner unless extremely stringent enforcement is in place. The level of travelers’ obedience to a speed limit law depends on the enforcement level (such as the speeding ticket associated with various levels of speeding) as well as travelers’ perception of risk and time and their socioeconomic characteristics. We now relax the assumption of perfect speed limit compliance and extend the model proposed by Yang et al. (2012). Travelers’ noncompliance behavior is taken into account and their speed choice is modeled as a trade-off behavior using a utility-maximizing framework (Tarko, 2009). In the spirit of Tarko (2009), a trip disutility function shown in Fig. 3(a) is adopted to combine the three factors affecting speed choices—safety, time and enforcement. Without necessarily obeying the speed limits, travelers would trade off a portion of their safety, and the risk of receiving a ticket, for a time gain. As shown in Fig. 3, the perceived crash risk and speed enforcement are considered as speed deterrents while the perceived time gain is considered as a speed enticement. It is assumed that the perceived risk of a crash grows with speed, and grows faster at a higher speed, so the crash risk curve is a rising convex curve (it was referred to as U-Shaped in early studies), and the perceived risk of getting a ticket appears at speeds that exceed the speed limit and grows with excessive speed over the posted speed limit. Thus, we make a minor revision of the disutility function given in Tarko (2009).2 Travelers’ disutility function for one unit distance is given by
b uðsÞ ¼ asn þ þ cdðs sÞðs sÞg s
ð6Þ
where s and s are the actual vehicle speed and speed limit, respectively; a and c are positive coefficients accounting for road and weather characteristics and travelers’ risk attitudes; n and g are parameters taking values greater than 1; b represents travelers’ value of time; dðs sÞ is a function which equals 1 if s s > 0 and 0 otherwise. The unit disutility function (6) consists of the perceived crash risk (the first term), the perceived travel time cost (the second term) and the perceived enforcement or risk of receiving a ticket for speeding (the third term). Since g > 1, the third term is differentiable at s ¼ s and thus is continuous, differentiable and convex with respect to s > 0, so the function uðsÞ in (6) is continuous, differentiable and strictly convex with respect to s > 0. Hence, we can always find an optimal speed ^s that minimizes the disutility uðsÞ, i.e., ^s ¼ arg min uðsÞ. The relationship between the optimal speed ^s and speed limit s could be either ^s < s or ^s P s, as shown in Fig. 3. Meanwhile, as shown in Fig. 1(a), the average speed s and traffic volume v follow a physical relationship s ¼ sðv Þ, so travelers can choose their preferred speed ^s only if the traffic volume v is not too high such that sðv Þ P ^s; otherwise, if sðv Þ < ^s, the disutility will decrease with speed, so travelers’ best strategy is to drive at the highest possible speed under the prevailing traffic flow condition. Therefore, travelers’ speed choice can be given by
s¼
^s; sðv Þ;
v 6 v^ v > v^
ð7Þ
^ is the critical traffic volume determined by sðv ^ Þ ¼ ^s. Therefore the new link travel time function is where v
(
tðv Þ ¼
^t; ~tðv Þ;
v 6 v^ v > v^
ð8Þ
^ Þ is constant. where ^t ¼ ~tðv 2 The formulation in Tarko (2009) was uðsÞ ¼ asn þ b=s þ cdðs sÞsg with n ¼ g. We modified the third term for ticket risk so that the disutility function uðsÞ is continuous at s ¼ s, while the new formulation is still reasonable.
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Substituting the speed choice (7) into the disutility function (6), we have the disutility function with respect to the traffic flow:
( uðsðv ÞÞ ¼
a^sn þ b^s þ cdð^s sÞð^s sÞg ; v 6 v^ aðsðv ÞÞn þ sðbv Þ þ cdðsðv Þ sÞðsðv Þ sÞg ; v > v^
ð9Þ
^ , uðsðv ÞÞ is constant, the minimal disutility is achieved within the flow With a predetermined optimal speed ^s, when v 6 v ^ , uðsðv ÞÞ strictly increases with v since sðv Þ < ^s and sðv Þ strictly range where the optimal speed ^s can be reached; when v > v ^ . Consequently, uðsðv ÞÞ is contindecreases with v . On the other hand, uðsðv ÞÞ is still continuous and differentiable at v ¼ v ^. uous, differentiable, increasing and convex with v , but not strictly increasing and not strictly convex when v < v 4. Speed limits and speed selection with non-obedient heterogeneous users We now consider the case with a set of user classes, denoted by M, with different socioeconomic characteristics and risk attitudes. Thus, with overtaking possible, different user classes may choose different optimal speeds on the same road. Furthermore, in comparison with the homogeneous case, the speed variance among user classes sharing the same roadway further increases the risk of crash involvement. Here, for simplicity, we assume that the perceived crash risk r m for user class m 2 M is a function of both the speed selected by the specific user class and the average speed of all users classes, namely, rm ¼ r m ðsm ; sav e Þ, where sm is the speed adopted by user class m 2 M and sav e is the average speed defined by P P sav e ¼ m2M sm v m =v and v ¼ m2M v m with v m the flow of user class m 2 M. Specifically, the following perceived crash risk function is assumed:
(
r m ðsm ; sav e Þ ¼
1
ðsav e sm Þfm þ ðsm Þnm ; sm < sav e
ð10Þ
2
ðsm sav e Þfm þ ðsm Þnm ; sm P sav e
where both f1m and f2m are greater than 1. The crash risk function assumes that the perceived crash risk grows not only with selected speed of the user class but also with its deviation (either positive or negative) from the mean speed, and parameters f1m and f2m reflect different perceptions of risk when driving higher or lower than the average speed. The disutility function for user class m 2 M traveling one unit length of road takes the following form:
um ðsm ; sav e Þ ¼ am rm ðsm ; sav e Þ þ
bm g þ cm dðsm sÞðsm sÞ m sm
ð11Þ
where am and cm account for users’ risk attitude and road conditions, while bm is the value of time of user class m. Since travelers’ disutility functions are inter-dependent among all user classes, their selection of optimal speeds on a specific road segment can be modeled as a Nash game. Given the prevailing traffic volume v m of each class m 2 M on the road, users of class m select their speed according to all other classes’ speed choices. By
@ @ ðsav e sm Þ ¼ m @sm @s
P
i–m s
i
vi
v
sm
P
i–m
v
vi
P
¼
i–m
vi
v
taking derivative of the perceived crash risk function (10) with respect to sm yields
@ 2 rm @ðsm Þ2
¼
8 P i 2 1 v > 1 1 > i–m > ðsav e sm Þfm 2 þ nm ðnm 1Þðsm Þnm 2 ; sm < sav e < fm ðfm 1Þ v
P i 2 > 2 > v > i–m : f2m ðf2m 1Þ ðsm sav e Þfm 2 þ nm ðnm 1Þðsm Þnm 2 ; sm P sav e v
ð12Þ
Since f1m , f2m and nm are all greater than 1, then @ 2 rm =@ðsm Þ2 > 0, which implies r m ðsm ; sav e Þ is continuous, differentiable and strictly convex with respect to sm , as is the whole disutility function (11). Therefore on an individual link, the optimal speeds ^sm of all classes m 2 M exist and are unique. We thus have the following class-specific link travel time function (counterpart of Eq. (8)):
( t ðv ; v m
m
m
Þ¼
^tm ; ~tðv Þ;
v 6 v^ m v > v^ m
ð13Þ
^ m is the critical traffic volume determined by sðv ^ m Þ ¼ ^sm , ^t m ¼ ~tðv ^ m Þ, v m represents the speed selections of all other where v P i user classes except class m, and v ¼ i2M v . The link disutility function (counterpart of Eq. (9)) with respect to the link flows can be written as
( um ðv m ; v m Þ ¼ where ^sav e ¼ ð^sm v m þ
am rm ð^sm ; ^sav e Þ þ b^smm þ cm dð^sm sÞð^sm sÞgm ; v 6 v^ m am rm ðsðv Þ; sav e Þ þ sðbvmÞ þ cm dðsðv Þ sÞðsðv Þ sÞgm ; v > v^ m
P
i–m s
i
v i Þ=v .
ð14Þ
265
H. Yang et al. / Transportation Research Part C 51 (2015) 260–273
Example 1. We use this example to discuss travelers’ speed choices on an individual road. The travel time function is the BPR (Bureau of Public Roads, 1964) function, tðv Þ ¼ 0:05½1 þ 0:15ðv =100Þ4 . The total OD demand is 60, consisting of two user classes. The disutility function of each user class has the form of (10) and (11) with the class-specific characteristics given in Table 1. The parameters are chosen such that they satisfy the assumptions in the paragraphs under Eqs. (6) and (10). We set the speed limit to be 60, 65, 75 and 80, respectively, which gives us four scenarios, as displayed in Fig. 4. In each scenario, we alter the proportions of the two user classes, and the proportion of road users in class 1 is denoted by q. The consequences as to whether each user class obeys the speed limit or not are summarized in Table 2. The single-class speed choices of classes 1 and 2 are highlighted by the red squares and blue circles in Fig. 4, respectively. The following findings are suitable for both homogeneous and heterogeneous cases. Firstly, as indicated in Fig. 4, class 2 travelers always prefer to drive faster than class 1 travelers. This is because, from Table 1, users in class 2 put relatively more 5 4 weight on travel time than users in class 1 40 , which makes the class 2 users prone to reducing the travel > 410 ¼ 210 10 2105 1104 time by speeding and sacrificing safety. Secondly, given a speed limit, the speeds of both classes decrease with the proportion of user class 1 in the whole traffic stream. As class 2 travelers tend to drive faster while class 1 travelers prefer a lower speed, a higher proportion of class 1 lowers the average speed, which will increase the crash risks of class 2 travelers due to the larger deviation from the mean speed. As a result, class 2 travelers will tend to drive at a lower speed to reduce the crash risk. Conversely, for class 1, a higher proportion of class 2 will induce class 1 travelers to drive faster in order to reduce high crash risks. Thirdly, for the same percentage of users in class 1 (q equals 0 and 1 for the single-class cases), the driving speeds
Table 1 User characteristics. User class (m)
am ð105 Þ
bm
cm ð104 Þ
f1m
f2m
nm
gm
1 2
2 4
10 40
1 2
2.5 2.5
2.5 2.5
2 2
2 2
(a) Speed limit of 60
(b) Speed limitof 65
(c) Speed limit of 75
(d) Speed limit of 80
Fig. 4. Speed choices with regard to user composition and speed limit.
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H. Yang et al. / Transportation Research Part C 51 (2015) 260–273
Table 2 Speed selections versus speed limit. Speed limit
60
65
75
80
Class 1 Class 2
Disobey Disobey
Obey if q P 0:47 Disobey
Obey Obey if q P 0:57
Obey Obey
of both classes increase with the speed limit. This is intuitive because increasing the speed limit will reduce the penalty for speeding, which will stimulate drivers to drive faster to save travel time. The increase in crash risk due to increasing speed is moderate because increasing speeds by both classes do not essentially increase speed variance and thereby pertaining crash risks. 5. Network traffic equilibrium with speed selection When incorporating the speed selection model to a network equilibrium framework, the optimal speed selection, link flow composition and route choices are intertwined. In this case, modeling route choice and network equilibrium with speed selection cannot be simply carried out in a conventional way. It depends on whether users are homogeneous or heterogeneous. The equilibrium model with homogeneous but non-obedient users is consistent with the model (4)–(5), by using the link disutility function (9) instead of (1). The same properties of user equilibrium hold as those in Yang et al. (2012): the uniqueness of link disutility at user equilibrium remains valid, and the equilibrium flows on links with non-optimal speeds are still unique. Yet, the solution can be easily obtained using the standard traffic assignment algorithms. The situation with heterogeneous user classes becomes much more complicated. The average speed and aggregate flow on a specific link influence each user class’ speed selection and route choice; the speed selections and route choices in turn m determine the average speeds and link flows on the network. Denote v m a and sa the flow and speed of class m 2 M on link P . Similar to the aforementioned homogeneous traffic assignment problem, a 2 A; la the length of link a 2 A and v a ¼ m2M v m a the multi-class traffic assignment will be conducted based on the link disutility function given by Eq. (14),
8 bm > ^av e ^m gm ; v a 6 v^ m < la am r m ð^sm þ cm dð^sm a ; sa Þ þ ^sm a sa Þðsa sa Þ a a m m um a ðv a ; v a Þ ¼ b > m a v e m : la am r ðsa ðv a Þ; sa Þ þ s ðv Þ þ cm dðsa ðv a Þ sa Þðsa ðv a Þ sa Þgm ; v a > v^ m a a a
ð15Þ
m ^m ^av e ^ m av e where um are defined the same as those in the previous sections, with the subscript a refera ð; Þ, v a , sa , sa , v a , sa ðÞ and sa ring to link a 2 A. From a practical point of view, assume that each link carries positive flow (otherwise this link is unused and thus could be removed from the network). Without loss of realism and generality, we assume that the aggregate flow on each link is posim m tive and bounded away from zero, i.e., v a P ea for all a 2 A, where ea is a small positive number. Denote f r;w and dw as the T flow on path r 2 Rw and demand of user class m 2 M between OD pair w 2 W; v M ¼ ðv m a ; a 2 A; m 2 MÞ the class-specific link flow pattern. The feasible set of the class-specific link flows is
(
XM ¼
v M v ma ¼
XX w2W r2Rw
m
m
f r;w da;r ; f r;w P 0;
X
m
m
f r;w ¼ dw ;
r2Rw
X
)
v ma P ea ; r 2 Rw ; w 2 W; m 2 M
m2M
Apparently, the set XM is convex and bounded. The heterogeneous user equilibrium condition can be described as
X m m m m um a ðv a ; v a Þda;r ¼ lw ; if f r;w > 0; r 2 Rw ; w 2 W; m 2 M a2A
X m m m m um a ðv a ; v a Þda;r P lw ; if f r;w ¼ 0; r 2 Rw ; w 2 W; m 2 M
ð16Þ
a2A
where lm w is the minimum travel disutility of class m 2 M between OD pair w 2 W. The equilibrium state satisfying condition (16) can be obtained by solving the following variational inequality formulation
XX
m m ~ m m ~ M 2 XM um a ðv a ; v a Þðv a v a Þ P 0; 8v
ð17Þ
a2A m2M M m m Since the functions um a ðv a ; v a Þ are continuous on X for all a 2 A, m 2 M, the variational inequality problem (17) has at least one solution, indicating the original heterogeneous user equilibrium flow patterns in (16) do exist (Facchinei and Pang, 2003). It is extremely difficult, if not impossible, to establish the uniqueness of the network traffic equilibrium in terms of link flows and the resulting speed selections, due to their intertwined implicit or explicit nonlinear relationships. Here, we propose an iterative (Gauss–Seidel) heuristic algorithm to calculate the equilibrium speed selections and link flows for the multi-class traffic assignment problem.
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H. Yang et al. / Transportation Research Part C 51 (2015) 260–273
5.1. Solution algorithm The algorithm is summarized as follows. m ð0Þ 1. Initialization. Initialize the flow distribution and speed selection of each class m 2 M on each link a 2 A, ðv m a ; sa Þ , and av e ð0Þ calculate the average speeds ðsa Þ . Denote the set of user classes as f1; 2; ; jMjg. Set k ¼ 0. m ðkÞ 2. (Sub-problem) Multi-class traffic assignment with fixed average speeds. Given ðv m and ðsaav e ÞðkÞ , calculate the new flow a ; sa Þ m m ðkþ1Þ distribution and speed selection ðv a ; sa Þ . 3. Verifying the stopping criteria. Calculate the average speeds ðsaav e Þðkþ1Þ by
ðsaav e Þ
ðkþ1Þ
P ¼
ðkþ1Þ m ðkþ1Þ ð m a Þ m2M ðsa Þ P ðkþ1Þ m m2M ð a Þ
v
v
;
a2A
If the average speeds converge, the user equilibrium solution is obtained; otherwise, set k ¼ k þ 1 and go to step 2. The sub-problem in step 2 is similar to classical multi-class traffic assignment except for the different disutility functions in (15). We adopt the Gauss–Seidel method (Patriksson, 1993; Marcotte and Wynter, 2004) to solve this sub-problem. Compared with the diagonalization method, the Gauss–Seidel method utilizes the most updated network information for each single-class assignment and speeds up the convergence. The algorithm is described as follows.
2 1 3 1
3
2
4
4
Fig. 5. Network structure (Yang et al., 2012).
Table 3 Link characteristics. Link no. (a)
Length (la )
Free flow time (t0a )
Capacity (C a )
Speed limit (sa )
1 2 3 4
55 20 20 32
0.67 0.25 0.20 0.33
200 350 400 300
60 60 70/60/50 60
Table 4 User characteristics. User class (m)
am (105 )
bm
cm (104 )
f1m
f2m
nm
gm
1 2
2 4
10 30
1 2
2.5 2.5
2.5 2.5
2 2
2 2
Table 5 OD demands. User class (m)
Origin
Destination
Demand
1 1 2 2
1 1 1 1
2 4 2 4
200 200 200 200
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H. Yang et al. / Transportation Research Part C 51 (2015) 260–273
Table 6 User equilibrium solutions with different speed limits on link 3. s1a
s2a
sa(va)
^s1a
^s2a
saav e
u1a
u2a
(a) Speed limit of 70 1 0.00 138.95 2 400.00 261.05 3 200.00 61.05 4 200.00 200.00 Total disutility: 21052.50
61.92 27.50 63.57 61.44
64.86 27.50 68.87 64.58
79.71 27.50 97.35 65.13
61.92 44.09 63.57 61.44
64.86 47.92 68.87 64.58
64.86 27.50 64.81 63.01
13.14 7.57 4.76 7.63
34.95 22.42 12.53 20.34
(b) Speed limit of 60 1 0.00 141.24 2 400.00 258.76 3 200.00 58.76 4 200.00 200.00 Total disutility: 20992.23
61.92 27.75 61.23 61.44
64.86 27.75 64.33 64.58
79.53 27.75 97.44 65.13
61.92 44.23 61.23 61.44
64.86 48.05 64.33 64.58
64.86 27.75 61.94 63.01
13.14 7.51 4.77 7.63
34.95 22.23 12.72 20.34
(c) Speed limit of 50 1 0.00 148.43 2 400.00 251.57 3 200.00 51.57 4 200.00 200.00 Total disutility: 20830.07
61.92 28.55 55.38 61.44
64.86 28.55 58.80 64.58
78.91 28.55 97.71 65.13
61.92 44.65 55.38 61.44
64.86 48.47 58.80 64.58
64.86 28.55 56.08 63.01
13.14 7.33 4.90 7.63
34.95 21.66 13.29 20.34
a
v 1a
v 2a
Fig. 6. Sioux Falls network.
Sub-problem: 1. Initialization. The average speeds saav e are given and fixed. Calculate the class-specific optimal speeds ^sm a based on (11). m ð0Þ ðv m are given, set l ¼ 0. a ; sa Þ
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H. Yang et al. / Transportation Research Part C 51 (2015) 260–273 Table 7 Link characteristics. a
Ca
la
t 0a
a
Ca
la
t0a
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
25900.2 23403.47 25900.2 4958.181 23403.47 17110.52 23403.47 17110.52 17782.79 4908.827 17782.79 4947.995 10000 4958.181 4947.995 4898.588 7841.811 23403.47 4898.588 7841.811 5050.193 5045.823 10000 5050.193 13915.79 13915.79 10000 13512 4854.918 4993.511 4908.827 10000 4908.827 4876.508 23403.47 4908.827 25900.2 25900.2
6 4 6 5 4 4 4 4 2 6 2 4 5 5 4 2 3 2 2 3 10 5 5 10 3 3 5 6 4 8 6 5 6 4 4 6 3 3
6 4 6 5 4 4 4 4 2 6 2 4 5 5 4 2 3 2 2 3 10 5 5 10 3 3 5 6 4 8 6 5 6 4 4 6 3 3
39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76
5091.256 4876.508 5127.526 4924.791 13512 5127.526 14564.75 9599.181 5045.823 4854.918 5229.91 19679.9 4993.511 5229.91 4823.951 23403.47 19679.9 23403.47 14564.75 4823.951 5002.608 23403.47 5002.608 5059.912 5075.697 5059.912 5229.91 4885.358 9599.181 5075.697 5229.91 5000 4924.791 5000 5078.508 5091.256 4885.358 5078.508
4 4 5 4 6 5 3 3 5 4 2 3 8 2 2 2 3 4 3 2 4 4 4 6 5 6 2 3 3 5 2 4 4 4 2 4 3 2
4 4 5 4 6 5 3 3 5 4 2 3 8 2 2 2 3 4 3 2 4 4 4 6 5 6 2 3 3 5 2 4 4 4 2 4 3 2
a: link number; C a : link capacity; la : link length; t0a : free flow time. Free flow speeds are uniformly 1; speed limits are uniformly 0.6.
2. Solving a sequence of single-class traffic assignment problems. For each user class i from 1 to jMj, find the traffic equilibrium ðlþ1Þ flow and speed pattern ðv ia ; sia Þ with disutility function (15), while the flows and speeds of all other user classes are ðlþ1Þ given: the flows and speeds of user classes j ¼ 1; 2; ; i 1 are given as ðv ja ; sja Þ , and the flows and speeds of user j j ðlÞ classes j ¼ i þ 1; i þ 2; ; jMj are given as ðv a ; sa Þ . T 3. Checking convergence. If the class-specific link flow pattern v M ¼ ðv m a ; a 2 A; m 2 MÞ converges, stop and get the multiclass user equilibrium flow and speed pattern; otherwise set l ¼ l þ 1 and go to step 2. The following remark is worthy of mention. For the homogeneous case, we are simply dealing with a single-class traffic assignment problem with separable, increasing and convex disutility functions. Many existing well-established, convergent algorithms can be readily used here. For the heterogeneous case, it is complicated yet challenging to find the solution and one has to rely on heuristics. The proposed heuristic algorithm is limited in the sense that it may be not convergent or not converge to a good-quality solution even it is convergent. 5.2. A toy network This example illustrates the equilibrium condition with speed limit and speed choice on a simple toy network. The network structure in Fig. 5 is adopted from Yang et al. (2012) but with different link characteristics (Table 3). The speed limits on links 1, 2 and 4 are fixed at 60 while the speed limit on link 3 varies from 70 to 50. The link travel time functions follow the BPR form,
" t a ðv a Þ ¼
t 0a
1 þ 0:15
va Ca
4 # ð18Þ
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H. Yang et al. / Transportation Research Part C 51 (2015) 260–273
Table 8 Class-specific OD demands (both class 1 and class 2). O
D
#
O
D
#
O
D
#
O
D
#
O
D
#
O
D
#
O
D
#
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 19 20 22 1 2 4 5 6 7 8 9 10 11 12 13 14 15 16 17 22 23 1 2 3 5 6 7 8 9
5 5 25 10 15 25 40 25 65 25 10 25 15 25 25 20 5 15 15 5 20 15 5 5 5 10 5 20 10 20 10 30 10 5 15 5 5 20 10 5 5 5 5 5 10 5 15 5 10 5 15 15 10 5 5 5 10 5 5 5 25 10 10 25 20 20 35 35
4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
18 19 20 21 22 23 24 1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 19 20 21 22 23 1 2 3 4 5 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 8 9 10 11 12 13 14 15 16 17 18
5 10 15 10 20 25 10 10 5 5 25 10 10 25 40 50 25 10 10 5 10 25 10 5 5 5 10 5 15 20 15 20 10 20 40 20 40 20 10 10 5 10 45 25 5 10 15 5 10 5 5 25 10 5 20 10 20 50 30 95 25 35 20 10 25 70 50 10
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 11
3 4 5 6 7 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1
10 35 25 40 50 40 80 40 30 30 20 30 110 70 15 35 45 20 25 15 10 25 10 5 35 40 20 30 40 140 70 30 30 30 45 70 45 10 20 30 15 35 25 10 65 30 15 60 50 40 95 80 140 200 100 95 105 200 220 195 35 90 125 60 130 90 40 25
11 11 11 11 11 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 14 14 14 14 14 14 14 14
10 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8
195 70 50 80 70 70 50 5 20 30 20 55 65 30 10 5 10 30 10 10 35 30 30 100 70 65 35 35 35 30 10 15 20 15 35 35 25 25 15 5 30 10 10 20 30 30 95 50 65 30 35 30 25 5 15 30 30 65 40 40 15 5 5 25 5 5 10 20
14 14 14 14 14 14 14 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17
18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
5 15 25 20 60 55 20 25 5 5 25 10 10 25 30 50 200 70 35 35 65 60 75 10 40 55 40 130 50 20 25 20 10 40 25 45 70 110 70 220 70 35 30 35 60 140 25 65 80 30 60 25 15 20 10 5 25 10 25 50 70 45 195 50 30 25 35 75
18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 21 21 21 21 21
1 4 6 7 8 9 10 11 12 13 14 15 16 17 19 20 21 22 23 1 2 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 23 24 1 2 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 22 23 24 1 4 5 6 7
5 5 5 10 15 10 35 10 10 5 5 10 25 30 15 20 5 15 5 15 5 10 5 10 20 35 20 90 20 15 15 15 40 65 85 15 60 20 60 15 5 15 5 15 5 15 25 45 30 125 30 25 30 25 55 80 85 20 60 60 120 35 20 5 10 5 5 10
21 21 21 21 21 21 21 21 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24
16 17 18 19 20 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 23 24 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 1 4 6 7 8 9 10 11 12 13 14 15 16 17 19
30 30 5 20 60 90 35 25 20 5 5 20 10 10 25 25 35 130 55 35 65 60 130 60 85 15 60 120 90 105 55 15 5 25 5 5 10 15 25 90 65 35 40 55 50 25 30 5 15 35 35 105 35 5 10 5 5 10 10 40 30 25 35 20 20 15 15 5
271
H. Yang et al. / Transportation Research Part C 51 (2015) 260–273 Table 8 (continued) O
D
#
O
D
#
O
4 4 4 4 4 4 4 4
10 11 12 13 14 15 16 17
60 70 30 30 25 25 40 25
7 7 7 7 7 7 8 8
19 20 21 22 23 24 1 2
20 25 10 25 10 5 40 20
11 11 11 11 11 11 11 11
D
#
2 3 4 5 6 7 8 9
10 15 75 25 20 25 40 70
O
D
#
O
D
#
O
D
14 14 14 14 14 14 14 14
9 10 11 12 13 15 16 17
30 105 80 35 30 65 35 35
17 17 17 17 17 17 17 17
16 18 19 20 21 22 23 24
140 30 85 85 30 85 30 15
21 21 21 21 21 21 21 21
8 9 10 11 12 13 14 15
# 20 15 60 20 15 30 20 40
O
D
24 24 24 24
20 21 22 23
# 20 25 55 35
O: origin node; D: destination node; #: demand (10).
Table 9 User characteristics. User class (m)
am
bm
cm
f1m
f2m
nm
gm
1 2
1 2
1 3
2 4
2.5 2.5
2.5 2.5
2 2
2 2
Table 10 User equilibrium solution. a
v 1a
v 2a
s1a
s2a
sa ð v a Þ
^s1a
^s2a
saav e
u1a
u2a
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
1550.0 3350.0 1550.0 2950.0 3350.0 5986.0 3958.4 6008.4 8671.1 2600.0 8693.4 5429.5 6691.6 2950.0 5450.0 6600.0 5466.9 7121.6 6620.5 5171.6 2300.0 4278.4 6693.4 2300.0 10991.6 11043.4 8850.0 10391.6 5550.0 4050.0 2650.0 8750.0 4900.0 6850.0 3936.0 4900.0 6008.4 6036.0 5508.4 6850.0 4433.4
1800.0 3600.0 1800.0 2700.0 3600.0 6581.5 4314.6 6614.6 8581.5 2600.0 8614.6 4201.0 7830.5 2700.0 4185.1 5701.0 5945.9 7404.1 5685.1 6141.5 3505.7 3491.5 7879.5 3381.9 11535.9 11511.2 9509.4 13541.3 5621.9 4050.0 2650.0 9480.7 4685.4 3303.8 4281.5 4718.5 6150.0 6200.0 5650.0 3342.0 4929.4
0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.317 0.600 0.722 0.317 0.144 0.598 0.722 0.143 0.606 0.723 0.542 0.596 0.723 0.493 0.491 0.370 0.404 0.192 0.491 0.722 0.376 0.314 0.262 0.722 0.311 0.722 0.722 0.224 0.259 0.375
0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.317 0.600 0.794 0.317 0.144 0.598 0.794 0.143 0.606 0.792 0.542 0.596 0.794 0.493 0.491 0.370 0.404 0.192 0.491 0.794 0.376 0.314 0.262 0.794 0.311 0.794 0.794 0.224 0.259 0.375
1.000 0.999 1.000 0.798 0.999 0.958 0.998 0.957 0.883 0.841 0.881 0.317 0.600 0.798 0.317 0.144 0.598 0.978 0.143 0.606 0.792 0.542 0.596 0.806 0.493 0.491 0.370 0.404 0.192 0.491 0.831 0.376 0.314 0.262 0.998 0.311 0.993 0.993 0.224 0.259 0.375
0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.676 0.713 0.722 0.676 0.648 0.712 0.722 0.648 0.713 0.723 0.706 0.712 0.723 0.700 0.700 0.684 0.688 0.656 0.700 0.722 0.684 0.675 0.667 0.722 0.675 0.722 0.722 0.661 0.667 0.684
0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.741 0.780 0.794 0.741 0.713 0.779 0.794 0.713 0.780 0.794 0.773 0.779 0.794 0.766 0.766 0.749 0.754 0.721 0.766 0.794 0.750 0.740 0.732 0.794 0.740 0.794 0.794 0.726 0.732 0.749
0.761 0.759 0.761 0.756 0.759 0.760 0.760 0.760 0.758 0.758 0.758 0.317 0.600 0.756 0.317 0.144 0.598 0.759 0.143 0.606 0.765 0.542 0.596 0.765 0.493 0.491 0.370 0.404 0.192 0.491 0.758 0.376 0.314 0.262 0.760 0.311 0.759 0.759 0.224 0.259 0.375
11.62 7.75 11.62 9.68 7.75 7.75 7.75 7.75 3.87 11.62 3.87 13.01 10.13 9.68 13.03 13.97 6.09 3.87 13.99 6.05 19.36 10.69 10.16 19.36 6.82 6.83 14.20 15.84 20.97 18.23 11.62 13.99 19.68 15.55 7.75 19.85 5.81 5.81 18.04 15.72 14.04
31.14 20.76 31.14 25.95 20.76 20.76 20.76 20.76 10.38 31.14 10.38 38.64 28.60 25.95 38.68 41.87 17.20 10.38 41.92 17.05 51.90 30.59 28.71 51.90 19.73 19.76 41.93 46.53 62.77 52.77 31.14 41.27 58.44 46.38 20.76 58.96 15.57 15.57 53.93 46.88 41.42
(continued on next page)
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Table 10 (continued) a
v 1a
v 2a
s1a
s2a
sa ðv a Þ
^s1a
^s2a
saav e
u1a
u2a
42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76
3266.6 10443.4 3800.0 10450.0 10625.0 4003.7 5600.0 5652.9 6354.3 4050.0 5658.0 5874.2 7416.9 6124.5 8125.8 10450.0 5879.2 4842.0 8141.3 4847.1 3000.0 3500.0 2970.5 5893.4 6375.6 9993.4 3500.0 5860.7 5231.0 3900.0 4566.6 2881.0 5486.0 6378.9 2850.0
5342.0 13539.2 5599.1 10245.2 8358.0 3794.9 5578.0 5700.0 6363.3 4050.0 5700.0 5150.0 7208.5 6622.9 9104.8 10231.2 5150.0 4595.2 9118.8 4581.2 3050.0 3500.0 3000.0 3430.9 4182.5 8989.6 3500.0 3432.5 4794.4 4710.4 5427.5 5117.5 5650.0 4130.9 5119.1
0.417 0.402 0.371 0.621 0.303 0.539 0.192 0.231 0.722 0.491 0.231 0.196 0.722 0.722 0.722 0.621 0.196 0.345 0.722 0.346 0.722 0.648 0.722 0.398 0.233 0.304 0.648 0.399 0.293 0.416 0.295 0.522 0.226 0.237 0.524
0.417 0.402 0.371 0.621 0.304 0.539 0.192 0.231 0.794 0.491 0.231 0.196 0.794 0.794 0.794 0.621 0.196 0.345 0.794 0.346 0.765 0.648 0.775 0.398 0.234 0.304 0.648 0.401 0.292 0.416 0.295 0.520 0.226 0.237 0.524
0.417 0.402 0.371 0.621 0.304 0.539 0.192 0.231 0.975 0.491 0.231 0.196 0.978 0.974 0.958 0.621 0.196 0.345 0.958 0.346 0.765 0.648 0.775 0.398 0.234 0.304 0.648 0.401 0.292 0.416 0.295 0.520 0.226 0.237 0.524
0.690 0.688 0.684 0.714 0.674 0.706 0.656 0.662 0.722 0.700 0.662 0.657 0.722 0.722 0.722 0.715 0.657 0.680 0.722 0.680 0.722 0.717 0.722 0.688 0.663 0.674 0.717 0.688 0.672 0.690 0.672 0.704 0.661 0.663 0.704
0.756 0.753 0.749 0.782 0.739 0.772 0.721 0.727 0.794 0.766 0.727 0.722 0.794 0.794 0.794 0.782 0.722 0.745 0.794 0.745 0.793 0.785 0.793 0.753 0.728 0.739 0.785 0.753 0.737 0.756 0.737 0.770 0.726 0.728 0.770
0.417 0.402 0.371 0.621 0.304 0.539 0.192 0.231 0.758 0.491 0.231 0.196 0.758 0.760 0.760 0.621 0.196 0.345 0.760 0.346 0.744 0.648 0.748 0.398 0.233 0.304 0.648 0.400 0.293 0.416 0.295 0.521 0.226 0.237 0.524
10.30 15.90 14.16 5.99 10.16 10.73 21.01 8.77 5.81 18.23 8.78 10.26 3.87 5.81 7.75 5.99 10.27 12.07 7.75 12.05 11.62 9.84 11.62 5.35 12.98 10.16 9.84 5.31 14.04 10.30 13.92 4.39 17.94 12.81 4.37
30.19 46.73 41.78 16.82 30.20 30.74 62.88 26.20 15.57 52.77 26.23 30.70 10.38 15.57 20.76 16.81 30.75 35.75 20.76 35.67 31.20 27.39 31.17 15.73 38.78 30.20 27.39 15.62 41.78 30.21 41.43 12.62 53.61 38.25 12.55
where t0a and C a are the free flow time and link capacity of link a 2 A, respectively. The characteristics of all the user classes are listed in Table 4. Compared with class 1, class 2 has a relatively higher value of time and is less risk-taking. The classrelated OD demands are given in Table 5. The user equilibrium solutions under different speed limit schemes are summarized in Table 6, including the speed selections, link flows and travel disutility of both user classes. Consequently, there are some interesting findings. First of all, both user classes on link 2 cannot drive at their own optimal speeds as restrained by the physical condition of the road; while on the other three links, both user classes can drive at their optimal speeds. Secondly, different classes have different route choices from origin 1 to destination 2: class 1 uses route 1 ! 3 ! 2 only but never uses link 1 while class 2 always uses both routes. Thirdly, decreasing speed limit on link 3 will decrease the driving speeds on link 3 but increase the speeds on link 2, while the speed selections on links 1 and 4 are unchanged. Last but not least, the adjustment of speed limit on a certain road segment will lead to a broad and complex change of speed selection and flow distribution on the whole network, which will consequently influence the individual and social disutility. As in this example, decreasing the speed limit on link 3 actually brings a lower total disutility. How the speed limit affects the system performance and how to design a better speed limit are worthy of further investigation. 5.3. Sioux Falls network In this subsection, we solve the multi-class user equilibrium problem with speed selection under speed limit on the Sioux Falls network in Fig. 6, consisting of 24 nodes and 76 links. The travel time functions are of the BPR form (18) with parameters given in Table 7. The free flow speeds are uniformly 1 and the speed limits are uniformly 0.6 on all links. The class-specific OD demands in Table 8 fall into two user classes, both of which have same demands between all OD pairs. The characteristics of each user class are given in Table 9. The user equilibrium solution is given in Table 10. This example shows that our algorithm is also applicable to a larger network than the toy network in Section 5.2. 6. Conclusions This study investigated both single-class and multi-class user equilibrium problems by considering road users’ speed selection under speed limit conditions. Road users evaluate their travel cost in terms of travel time, crash risk and speeding ticket risk: the crash risk with homogeneous users depends on the absolute speed only, while the crash risk with
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heterogeneous users depends on both the absolute speed and the speed variance among the user classes. Road users are assumed to choose their optimal speed on each particular road to minimize their generalized disutility under the restraint of the flow-speed relationship. An iterative Gauss–Seidel-type algorithm was proposed to solve the traffic assignment problem. The inter-dependence of link flows and speed choices among user classes and the network-wide impacts of speed limits were examined with numerical examples. Further research is expected to apply the model to the optimal design of speed limit schemes for multi-objective network optimization, such as enhancing road safety and reducing network-wide travel time. Acknowledgements The authors wish to express their thanks to three anonymous reviewers for their useful comments on the early versions of the paper. 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Transportation Research Part C journal homepage: www.elsevier.com/locate/trc
Speed limits, speed selection and network equilibrium Hai Yang 1, Hongbo Ye ⇑, Xinwei Li, Bingqing Zhao Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
a r t i c l e
i n f o
Article history: Received 29 April 2014 Received in revised form 6 December 2014 Accepted 6 December 2014 Available online 12 January 2015 Keywords: Speed limit Speed selection Safety Traffic equilibrium Traffic network
a b s t r a c t This paper investigates the local and global impact of speed limits by considering road users’ non-obedient behavior in speed selection. Given a link-specific speed limit scheme, road users will take into account the subjective travel time cost, the perceived crash risk and the perceived ticket risk as determinant factors for their actual speed choice on each link. Homogeneous travelers’ perceived crash risk is positively related to their driving speed. When travelers are heterogeneous, the perceived crash risk is class-specific: different user classes interact with each other and choose their own optimal speed, resulting in a Nash equilibrium speed pattern. With the speed choices on particular roads, travelers make route choices, resulting in user equilibrium in a general network. An algorithm is proposed to solve the user equilibrium problem with heterogeneous users under link-specific speed limits. The models and algorithms are illustrated with numerical examples. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Speed limit schemes have been ubiquitous around the world owing to effectiveness in enhancing safety, reducing emissions and/or saving energy consumption in a straightforward manner. However, imposing speed limits will undoubtedly affect travel time and mobility on the roads. Previous studies have overwhelmingly focused on the local impacts of speed limits on performance such as safety and vehicle emissions of the road where the speed limit is imposed; but unfortunately the system-wide impacts of speed limits on a general network have received only very limited attention in both the theoretical and empirical literature (Yang et al., 2012). Although McKnight and Klein (1990), Lave and Elias (1994, 1997) and Grabowski and Morrisey (2007) realized the traffic reallocation effect of speed limits, their studies focus on only local impacts without systematic investigation. Taylor (2000), Woolley et al. (2002) and Madireddy et al. (2011) applied microscopic traffic simulation tools to examine the system-wide impacts of speed limits, and it was reported that traffic reallocation was observed and the travel time disproportionately increased with reduced speed limits. It was not until recently that Yang et al. (2012) made the first attempt to theoretically investigate the traffic reallocation effect of link-specific speed limit schemes on a general network from the viewpoint of macroscopic user equilibrium. The uniqueness conditions of the UE link travel times and link flows were investigated, and the feasibility of using the speed limits as a flow management toll was discussed. In the same spirit, Yang et al. (2013) and Wang (2013) undertook further analysis of the network performance with speed limits. Yang et al. (2013) established the tri-objective, bi-level optimization problem, aiming to minimize the travel time, accident occurrence and emission simultaneously. Wang (2013) also considered the efficiency and equity issues in the implementation of speed limits.
⇑ Corresponding author. Tel.: +852 2358 7175; fax: +852 2358 1534. 1
E-mail address: [email protected] (H. Ye). Part of the research was carried out during the first author’s stay at Beihang University as a Changjiang visiting chair professor.
http://dx.doi.org/10.1016/j.trc.2014.12.002 0968-090X/Ó 2014 Elsevier Ltd. All rights reserved.
H. Yang et al. / Transportation Research Part C 51 (2015) 260–273
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Nomenclature A a M m
v vm va v ma s sa ^s ^sm ^sm a sm sm a sav e saav e la
set of directed links link a 2 A set of user classes user class m 2 M aggregate flow on a representative link flow of user class m 2 M on a representative link aggregate flow on link a 2 A flow of user class m 2 M on link a 2 A speed limit on a representative link speed limit on link a 2 A optimal speed on a representative link with homogeneous travelers optimal speed of user class m 2 M on a representative link optimal speed of user class m 2 M on link a 2 A speed adopted by user class m 2 M on a representative link speed adopted by user class m 2 M on link a 2 A average speed on a representative link average speed on link a 2 A length of link a 2 A
A key assumption in Yang et al. (2012) and subsequent studies (Yang et al., 2013; Wang, 2013) is road users’ perfect compliance with the speed limits, which is restrictive in practical situations, as reported in many studies (Kanellaidis et al., 1995; Tarko, 2009; Yannis et al., 2013). For an individual traveler, a higher driving speed may weaken control of car and require a longer stopping distance, naturally raising the possibility of crash involvement. The relationship is described by either power functions (Maycock et al., 1998; Quimby et al., 1999) or exponential functions, with the crash risk increasing faster at higher speeds (Fildes et al., 1991; Kloeden et al., 1997, 2001). Besides absolute speed, speed variance among vehicles on a road is also a main factor for crash involvement. Earlier and recent studies found an increased crash risk for vehicles driving faster than the surrounding vehicles. However, the findings diverge for vehicles driving slower than average. An increased risk was reported in earlier studies (Solomon, 1964; Cirillo, 1968; RTI, 1970) but it was not reconfirmed in more recent studies (Kloeden et al., 1997, 2001). It was reported later that the crash frequency at the road section level increases with average speed (Finch et al., 1994; Nilsson, 1982, 2004) and speed variance (Garber and Gadiraju, 1989). Comprehensive reviews of previous studies on the relationship between crash rate and speed can be found in Aarts and van Schagen (2006) and McCarty (1998). This paper is intended to make a useful and substantial extension of the traffic equilibrium model with speed limits, as proposed by Yang et al. (2012), in order to capture road users’ speed selection behavior. The rest of the paper is organized as follows. In Section 2, the traffic equilibrium model with obedient users in Yang et al. (2012) is reviewed. Section 3 analyzes the speed selection of non-obedient, homogeneous users. Section 4 extends this speed selection model to the case with heterogeneous users by including the crash risk increment caused by the speed variance among different user classes. The network-wide user equilibrium problem with speed selection is proposed in Section 5, together with an iterative solution algorithm and some numerical examples for demonstration. General conclusions are provided in Section 6. 2. Traffic equilibrium under speed limits with obedient users Consider a general network G ¼ ðN; AÞ with a set N of nodes and a set A of directed links. Denote W as the set of origindestination (OD) pairs. Each OD pair w 2 W is connected by a set Rw of simple routes serving a given and fixed travel demand dw . Let f r;w denote the flow on path r 2 Rw between OD pair w 2 W, and v A ¼ ðv a ; a 2 AÞT denote the vector of link flows with v a representing the traffic flow on link a 2 A. For a specific link, the speed-flow relationship and the travel time-flow relationship are depicted in Fig. 1. Like most static traffic assignment models, only the normal flow regime is considered in this study, so for any link a 2 A, the travel time function ~ta ðv a Þ is assumed to be separable, continuous, differentiable and strictly increasing with link flow v a . Let sa denote the link-specific speed limit on link a 2 A. If no speed limit is imposed on link a, we can simply set sa to be the free-flow speed on link a. Yang et al. (2012) investigated the properties of traffic equilibrium under a speed limit law by assuming that all travelers strictly adhere to the speed limit on each link. In this case, the speed-flow relationship on link a 2 A under speed limit sa is altered, as shown in Fig. 2(a), where C is the road capacity and sc is the vehicle speed when the link flow rate reaches capacity. Correspondingly, the travel time function t a ðv a Þ on link a 2 A under speed limit sa takes the following form:
t a ðv a Þ ¼
0 6 v a 6 v a ta ; ~ta ðv a Þ; v a < v a 6 C a
ð1Þ
H. Yang et al. / Transportation Research Part C 51 (2015) 260–273
Travel time t
262
0
Traffic volume v
(a)
(b)
Fig. 1. Speed-flow relationship and travel time-flow relationship without speed limits.
Fig. 2. Speed-flow relationship and travel time-flow relationship with a speed limit (Yang et al., 2012).
a is the traffic flow such that ~ta ðv a Þ ¼ ta and C a is the capacity of link where t a is the travel time on link a 2 A at speed sa , v a 2 A. As shown in Fig. 2(b), the travel time function under a speed limit is no longer differentiable or strictly increasing. The user equilibrium (UE) satisfies the following condition:
X ta ðv a Þda;r ¼ lw ;
if f r;w > 0; r 2 Rw ; w 2 W
ð2Þ
a2A
X ta ðv a Þda;r P lw ;
if f r;w ¼ 0; r 2 Rw ; w 2 W
ð3Þ
a2A
where da;r equals 1 if path r uses link a and zero otherwise, and lw is the minimal path travel time between OD pair w 2 W. The UE conditions (2)–(3) are equivalent to the following optimization problem:
minv A 2X
X Z va a2A
t a ðxÞdx
ð4Þ
0
where X is defined by
(
X¼
v A v a ¼
XX w2W r2Rw
f r;w da;r ; a 2 A; f r;w P 0;
X
) f r;w ¼ dw ; r 2 Rw ; w 2 W
ð5Þ
r2Rw
The optimization problem (4)–(5) takes exactly the same form as that of a standard traffic equilibrium model without speed limit (Sheffi, 1985). The only difference is the altered shape of the travel time-flow relationship, which is no longer differentiable and strictly increasing. As proven in Yang et al. (2012), with some common assumptions, the UE travel times on all links are still unique. The UE flows on links with non-binding speed limits are unique, while those on the links with binding speed limits may not be unique, but can be explicitly depicted by a polyhedron or a linear system of equalities and inequalities.
H. Yang et al. / Transportation Research Part C 51 (2015) 260–273
263
Fig. 3. Speed deterrents and enticement in speed selection: (a) ^s 6 s; (b) ^s > s.
3. Speed limits and speed selection with non-obedient homogeneous users In practice travelers do not necessarily abide by the speed limits in a strict manner unless extremely stringent enforcement is in place. The level of travelers’ obedience to a speed limit law depends on the enforcement level (such as the speeding ticket associated with various levels of speeding) as well as travelers’ perception of risk and time and their socioeconomic characteristics. We now relax the assumption of perfect speed limit compliance and extend the model proposed by Yang et al. (2012). Travelers’ noncompliance behavior is taken into account and their speed choice is modeled as a trade-off behavior using a utility-maximizing framework (Tarko, 2009). In the spirit of Tarko (2009), a trip disutility function shown in Fig. 3(a) is adopted to combine the three factors affecting speed choices—safety, time and enforcement. Without necessarily obeying the speed limits, travelers would trade off a portion of their safety, and the risk of receiving a ticket, for a time gain. As shown in Fig. 3, the perceived crash risk and speed enforcement are considered as speed deterrents while the perceived time gain is considered as a speed enticement. It is assumed that the perceived risk of a crash grows with speed, and grows faster at a higher speed, so the crash risk curve is a rising convex curve (it was referred to as U-Shaped in early studies), and the perceived risk of getting a ticket appears at speeds that exceed the speed limit and grows with excessive speed over the posted speed limit. Thus, we make a minor revision of the disutility function given in Tarko (2009).2 Travelers’ disutility function for one unit distance is given by
b uðsÞ ¼ asn þ þ cdðs sÞðs sÞg s
ð6Þ
where s and s are the actual vehicle speed and speed limit, respectively; a and c are positive coefficients accounting for road and weather characteristics and travelers’ risk attitudes; n and g are parameters taking values greater than 1; b represents travelers’ value of time; dðs sÞ is a function which equals 1 if s s > 0 and 0 otherwise. The unit disutility function (6) consists of the perceived crash risk (the first term), the perceived travel time cost (the second term) and the perceived enforcement or risk of receiving a ticket for speeding (the third term). Since g > 1, the third term is differentiable at s ¼ s and thus is continuous, differentiable and convex with respect to s > 0, so the function uðsÞ in (6) is continuous, differentiable and strictly convex with respect to s > 0. Hence, we can always find an optimal speed ^s that minimizes the disutility uðsÞ, i.e., ^s ¼ arg min uðsÞ. The relationship between the optimal speed ^s and speed limit s could be either ^s < s or ^s P s, as shown in Fig. 3. Meanwhile, as shown in Fig. 1(a), the average speed s and traffic volume v follow a physical relationship s ¼ sðv Þ, so travelers can choose their preferred speed ^s only if the traffic volume v is not too high such that sðv Þ P ^s; otherwise, if sðv Þ < ^s, the disutility will decrease with speed, so travelers’ best strategy is to drive at the highest possible speed under the prevailing traffic flow condition. Therefore, travelers’ speed choice can be given by
s¼
^s; sðv Þ;
v 6 v^ v > v^
ð7Þ
^ is the critical traffic volume determined by sðv ^ Þ ¼ ^s. Therefore the new link travel time function is where v
(
tðv Þ ¼
^t; ~tðv Þ;
v 6 v^ v > v^
ð8Þ
^ Þ is constant. where ^t ¼ ~tðv 2 The formulation in Tarko (2009) was uðsÞ ¼ asn þ b=s þ cdðs sÞsg with n ¼ g. We modified the third term for ticket risk so that the disutility function uðsÞ is continuous at s ¼ s, while the new formulation is still reasonable.
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Substituting the speed choice (7) into the disutility function (6), we have the disutility function with respect to the traffic flow:
( uðsðv ÞÞ ¼
a^sn þ b^s þ cdð^s sÞð^s sÞg ; v 6 v^ aðsðv ÞÞn þ sðbv Þ þ cdðsðv Þ sÞðsðv Þ sÞg ; v > v^
ð9Þ
^ , uðsðv ÞÞ is constant, the minimal disutility is achieved within the flow With a predetermined optimal speed ^s, when v 6 v ^ , uðsðv ÞÞ strictly increases with v since sðv Þ < ^s and sðv Þ strictly range where the optimal speed ^s can be reached; when v > v ^ . Consequently, uðsðv ÞÞ is contindecreases with v . On the other hand, uðsðv ÞÞ is still continuous and differentiable at v ¼ v ^. uous, differentiable, increasing and convex with v , but not strictly increasing and not strictly convex when v < v 4. Speed limits and speed selection with non-obedient heterogeneous users We now consider the case with a set of user classes, denoted by M, with different socioeconomic characteristics and risk attitudes. Thus, with overtaking possible, different user classes may choose different optimal speeds on the same road. Furthermore, in comparison with the homogeneous case, the speed variance among user classes sharing the same roadway further increases the risk of crash involvement. Here, for simplicity, we assume that the perceived crash risk r m for user class m 2 M is a function of both the speed selected by the specific user class and the average speed of all users classes, namely, rm ¼ r m ðsm ; sav e Þ, where sm is the speed adopted by user class m 2 M and sav e is the average speed defined by P P sav e ¼ m2M sm v m =v and v ¼ m2M v m with v m the flow of user class m 2 M. Specifically, the following perceived crash risk function is assumed:
(
r m ðsm ; sav e Þ ¼
1
ðsav e sm Þfm þ ðsm Þnm ; sm < sav e
ð10Þ
2
ðsm sav e Þfm þ ðsm Þnm ; sm P sav e
where both f1m and f2m are greater than 1. The crash risk function assumes that the perceived crash risk grows not only with selected speed of the user class but also with its deviation (either positive or negative) from the mean speed, and parameters f1m and f2m reflect different perceptions of risk when driving higher or lower than the average speed. The disutility function for user class m 2 M traveling one unit length of road takes the following form:
um ðsm ; sav e Þ ¼ am rm ðsm ; sav e Þ þ
bm g þ cm dðsm sÞðsm sÞ m sm
ð11Þ
where am and cm account for users’ risk attitude and road conditions, while bm is the value of time of user class m. Since travelers’ disutility functions are inter-dependent among all user classes, their selection of optimal speeds on a specific road segment can be modeled as a Nash game. Given the prevailing traffic volume v m of each class m 2 M on the road, users of class m select their speed according to all other classes’ speed choices. By
@ @ ðsav e sm Þ ¼ m @sm @s
P
i–m s
i
vi
v
sm
P
i–m
v
vi
P
¼
i–m
vi
v
taking derivative of the perceived crash risk function (10) with respect to sm yields
@ 2 rm @ðsm Þ2
¼
8 P i 2 1 v > 1 1 > i–m > ðsav e sm Þfm 2 þ nm ðnm 1Þðsm Þnm 2 ; sm < sav e < fm ðfm 1Þ v
P i 2 > 2 > v > i–m : f2m ðf2m 1Þ ðsm sav e Þfm 2 þ nm ðnm 1Þðsm Þnm 2 ; sm P sav e v
ð12Þ
Since f1m , f2m and nm are all greater than 1, then @ 2 rm =@ðsm Þ2 > 0, which implies r m ðsm ; sav e Þ is continuous, differentiable and strictly convex with respect to sm , as is the whole disutility function (11). Therefore on an individual link, the optimal speeds ^sm of all classes m 2 M exist and are unique. We thus have the following class-specific link travel time function (counterpart of Eq. (8)):
( t ðv ; v m
m
m
Þ¼
^tm ; ~tðv Þ;
v 6 v^ m v > v^ m
ð13Þ
^ m is the critical traffic volume determined by sðv ^ m Þ ¼ ^sm , ^t m ¼ ~tðv ^ m Þ, v m represents the speed selections of all other where v P i user classes except class m, and v ¼ i2M v . The link disutility function (counterpart of Eq. (9)) with respect to the link flows can be written as
( um ðv m ; v m Þ ¼ where ^sav e ¼ ð^sm v m þ
am rm ð^sm ; ^sav e Þ þ b^smm þ cm dð^sm sÞð^sm sÞgm ; v 6 v^ m am rm ðsðv Þ; sav e Þ þ sðbvmÞ þ cm dðsðv Þ sÞðsðv Þ sÞgm ; v > v^ m
P
i–m s
i
v i Þ=v .
ð14Þ
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Example 1. We use this example to discuss travelers’ speed choices on an individual road. The travel time function is the BPR (Bureau of Public Roads, 1964) function, tðv Þ ¼ 0:05½1 þ 0:15ðv =100Þ4 . The total OD demand is 60, consisting of two user classes. The disutility function of each user class has the form of (10) and (11) with the class-specific characteristics given in Table 1. The parameters are chosen such that they satisfy the assumptions in the paragraphs under Eqs. (6) and (10). We set the speed limit to be 60, 65, 75 and 80, respectively, which gives us four scenarios, as displayed in Fig. 4. In each scenario, we alter the proportions of the two user classes, and the proportion of road users in class 1 is denoted by q. The consequences as to whether each user class obeys the speed limit or not are summarized in Table 2. The single-class speed choices of classes 1 and 2 are highlighted by the red squares and blue circles in Fig. 4, respectively. The following findings are suitable for both homogeneous and heterogeneous cases. Firstly, as indicated in Fig. 4, class 2 travelers always prefer to drive faster than class 1 travelers. This is because, from Table 1, users in class 2 put relatively more 5 4 weight on travel time than users in class 1 40 , which makes the class 2 users prone to reducing the travel > 410 ¼ 210 10 2105 1104 time by speeding and sacrificing safety. Secondly, given a speed limit, the speeds of both classes decrease with the proportion of user class 1 in the whole traffic stream. As class 2 travelers tend to drive faster while class 1 travelers prefer a lower speed, a higher proportion of class 1 lowers the average speed, which will increase the crash risks of class 2 travelers due to the larger deviation from the mean speed. As a result, class 2 travelers will tend to drive at a lower speed to reduce the crash risk. Conversely, for class 1, a higher proportion of class 2 will induce class 1 travelers to drive faster in order to reduce high crash risks. Thirdly, for the same percentage of users in class 1 (q equals 0 and 1 for the single-class cases), the driving speeds
Table 1 User characteristics. User class (m)
am ð105 Þ
bm
cm ð104 Þ
f1m
f2m
nm
gm
1 2
2 4
10 40
1 2
2.5 2.5
2.5 2.5
2 2
2 2
(a) Speed limit of 60
(b) Speed limitof 65
(c) Speed limit of 75
(d) Speed limit of 80
Fig. 4. Speed choices with regard to user composition and speed limit.
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H. Yang et al. / Transportation Research Part C 51 (2015) 260–273
Table 2 Speed selections versus speed limit. Speed limit
60
65
75
80
Class 1 Class 2
Disobey Disobey
Obey if q P 0:47 Disobey
Obey Obey if q P 0:57
Obey Obey
of both classes increase with the speed limit. This is intuitive because increasing the speed limit will reduce the penalty for speeding, which will stimulate drivers to drive faster to save travel time. The increase in crash risk due to increasing speed is moderate because increasing speeds by both classes do not essentially increase speed variance and thereby pertaining crash risks. 5. Network traffic equilibrium with speed selection When incorporating the speed selection model to a network equilibrium framework, the optimal speed selection, link flow composition and route choices are intertwined. In this case, modeling route choice and network equilibrium with speed selection cannot be simply carried out in a conventional way. It depends on whether users are homogeneous or heterogeneous. The equilibrium model with homogeneous but non-obedient users is consistent with the model (4)–(5), by using the link disutility function (9) instead of (1). The same properties of user equilibrium hold as those in Yang et al. (2012): the uniqueness of link disutility at user equilibrium remains valid, and the equilibrium flows on links with non-optimal speeds are still unique. Yet, the solution can be easily obtained using the standard traffic assignment algorithms. The situation with heterogeneous user classes becomes much more complicated. The average speed and aggregate flow on a specific link influence each user class’ speed selection and route choice; the speed selections and route choices in turn m determine the average speeds and link flows on the network. Denote v m a and sa the flow and speed of class m 2 M on link P . Similar to the aforementioned homogeneous traffic assignment problem, a 2 A; la the length of link a 2 A and v a ¼ m2M v m a the multi-class traffic assignment will be conducted based on the link disutility function given by Eq. (14),
8 bm > ^av e ^m gm ; v a 6 v^ m < la am r m ð^sm þ cm dð^sm a ; sa Þ þ ^sm a sa Þðsa sa Þ a a m m um a ðv a ; v a Þ ¼ b > m a v e m : la am r ðsa ðv a Þ; sa Þ þ s ðv Þ þ cm dðsa ðv a Þ sa Þðsa ðv a Þ sa Þgm ; v a > v^ m a a a
ð15Þ
m ^m ^av e ^ m av e where um are defined the same as those in the previous sections, with the subscript a refera ð; Þ, v a , sa , sa , v a , sa ðÞ and sa ring to link a 2 A. From a practical point of view, assume that each link carries positive flow (otherwise this link is unused and thus could be removed from the network). Without loss of realism and generality, we assume that the aggregate flow on each link is posim m tive and bounded away from zero, i.e., v a P ea for all a 2 A, where ea is a small positive number. Denote f r;w and dw as the T flow on path r 2 Rw and demand of user class m 2 M between OD pair w 2 W; v M ¼ ðv m a ; a 2 A; m 2 MÞ the class-specific link flow pattern. The feasible set of the class-specific link flows is
(
XM ¼
v M v ma ¼
XX w2W r2Rw
m
m
f r;w da;r ; f r;w P 0;
X
m
m
f r;w ¼ dw ;
r2Rw
X
)
v ma P ea ; r 2 Rw ; w 2 W; m 2 M
m2M
Apparently, the set XM is convex and bounded. The heterogeneous user equilibrium condition can be described as
X m m m m um a ðv a ; v a Þda;r ¼ lw ; if f r;w > 0; r 2 Rw ; w 2 W; m 2 M a2A
X m m m m um a ðv a ; v a Þda;r P lw ; if f r;w ¼ 0; r 2 Rw ; w 2 W; m 2 M
ð16Þ
a2A
where lm w is the minimum travel disutility of class m 2 M between OD pair w 2 W. The equilibrium state satisfying condition (16) can be obtained by solving the following variational inequality formulation
XX
m m ~ m m ~ M 2 XM um a ðv a ; v a Þðv a v a Þ P 0; 8v
ð17Þ
a2A m2M M m m Since the functions um a ðv a ; v a Þ are continuous on X for all a 2 A, m 2 M, the variational inequality problem (17) has at least one solution, indicating the original heterogeneous user equilibrium flow patterns in (16) do exist (Facchinei and Pang, 2003). It is extremely difficult, if not impossible, to establish the uniqueness of the network traffic equilibrium in terms of link flows and the resulting speed selections, due to their intertwined implicit or explicit nonlinear relationships. Here, we propose an iterative (Gauss–Seidel) heuristic algorithm to calculate the equilibrium speed selections and link flows for the multi-class traffic assignment problem.
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5.1. Solution algorithm The algorithm is summarized as follows. m ð0Þ 1. Initialization. Initialize the flow distribution and speed selection of each class m 2 M on each link a 2 A, ðv m a ; sa Þ , and av e ð0Þ calculate the average speeds ðsa Þ . Denote the set of user classes as f1; 2; ; jMjg. Set k ¼ 0. m ðkÞ 2. (Sub-problem) Multi-class traffic assignment with fixed average speeds. Given ðv m and ðsaav e ÞðkÞ , calculate the new flow a ; sa Þ m m ðkþ1Þ distribution and speed selection ðv a ; sa Þ . 3. Verifying the stopping criteria. Calculate the average speeds ðsaav e Þðkþ1Þ by
ðsaav e Þ
ðkþ1Þ
P ¼
ðkþ1Þ m ðkþ1Þ ð m a Þ m2M ðsa Þ P ðkþ1Þ m m2M ð a Þ
v
v
;
a2A
If the average speeds converge, the user equilibrium solution is obtained; otherwise, set k ¼ k þ 1 and go to step 2. The sub-problem in step 2 is similar to classical multi-class traffic assignment except for the different disutility functions in (15). We adopt the Gauss–Seidel method (Patriksson, 1993; Marcotte and Wynter, 2004) to solve this sub-problem. Compared with the diagonalization method, the Gauss–Seidel method utilizes the most updated network information for each single-class assignment and speeds up the convergence. The algorithm is described as follows.
2 1 3 1
3
2
4
4
Fig. 5. Network structure (Yang et al., 2012).
Table 3 Link characteristics. Link no. (a)
Length (la )
Free flow time (t0a )
Capacity (C a )
Speed limit (sa )
1 2 3 4
55 20 20 32
0.67 0.25 0.20 0.33
200 350 400 300
60 60 70/60/50 60
Table 4 User characteristics. User class (m)
am (105 )
bm
cm (104 )
f1m
f2m
nm
gm
1 2
2 4
10 30
1 2
2.5 2.5
2.5 2.5
2 2
2 2
Table 5 OD demands. User class (m)
Origin
Destination
Demand
1 1 2 2
1 1 1 1
2 4 2 4
200 200 200 200
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H. Yang et al. / Transportation Research Part C 51 (2015) 260–273
Table 6 User equilibrium solutions with different speed limits on link 3. s1a
s2a
sa(va)
^s1a
^s2a
saav e
u1a
u2a
(a) Speed limit of 70 1 0.00 138.95 2 400.00 261.05 3 200.00 61.05 4 200.00 200.00 Total disutility: 21052.50
61.92 27.50 63.57 61.44
64.86 27.50 68.87 64.58
79.71 27.50 97.35 65.13
61.92 44.09 63.57 61.44
64.86 47.92 68.87 64.58
64.86 27.50 64.81 63.01
13.14 7.57 4.76 7.63
34.95 22.42 12.53 20.34
(b) Speed limit of 60 1 0.00 141.24 2 400.00 258.76 3 200.00 58.76 4 200.00 200.00 Total disutility: 20992.23
61.92 27.75 61.23 61.44
64.86 27.75 64.33 64.58
79.53 27.75 97.44 65.13
61.92 44.23 61.23 61.44
64.86 48.05 64.33 64.58
64.86 27.75 61.94 63.01
13.14 7.51 4.77 7.63
34.95 22.23 12.72 20.34
(c) Speed limit of 50 1 0.00 148.43 2 400.00 251.57 3 200.00 51.57 4 200.00 200.00 Total disutility: 20830.07
61.92 28.55 55.38 61.44
64.86 28.55 58.80 64.58
78.91 28.55 97.71 65.13
61.92 44.65 55.38 61.44
64.86 48.47 58.80 64.58
64.86 28.55 56.08 63.01
13.14 7.33 4.90 7.63
34.95 21.66 13.29 20.34
a
v 1a
v 2a
Fig. 6. Sioux Falls network.
Sub-problem: 1. Initialization. The average speeds saav e are given and fixed. Calculate the class-specific optimal speeds ^sm a based on (11). m ð0Þ ðv m are given, set l ¼ 0. a ; sa Þ
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H. Yang et al. / Transportation Research Part C 51 (2015) 260–273 Table 7 Link characteristics. a
Ca
la
t 0a
a
Ca
la
t0a
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
25900.2 23403.47 25900.2 4958.181 23403.47 17110.52 23403.47 17110.52 17782.79 4908.827 17782.79 4947.995 10000 4958.181 4947.995 4898.588 7841.811 23403.47 4898.588 7841.811 5050.193 5045.823 10000 5050.193 13915.79 13915.79 10000 13512 4854.918 4993.511 4908.827 10000 4908.827 4876.508 23403.47 4908.827 25900.2 25900.2
6 4 6 5 4 4 4 4 2 6 2 4 5 5 4 2 3 2 2 3 10 5 5 10 3 3 5 6 4 8 6 5 6 4 4 6 3 3
6 4 6 5 4 4 4 4 2 6 2 4 5 5 4 2 3 2 2 3 10 5 5 10 3 3 5 6 4 8 6 5 6 4 4 6 3 3
39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76
5091.256 4876.508 5127.526 4924.791 13512 5127.526 14564.75 9599.181 5045.823 4854.918 5229.91 19679.9 4993.511 5229.91 4823.951 23403.47 19679.9 23403.47 14564.75 4823.951 5002.608 23403.47 5002.608 5059.912 5075.697 5059.912 5229.91 4885.358 9599.181 5075.697 5229.91 5000 4924.791 5000 5078.508 5091.256 4885.358 5078.508
4 4 5 4 6 5 3 3 5 4 2 3 8 2 2 2 3 4 3 2 4 4 4 6 5 6 2 3 3 5 2 4 4 4 2 4 3 2
4 4 5 4 6 5 3 3 5 4 2 3 8 2 2 2 3 4 3 2 4 4 4 6 5 6 2 3 3 5 2 4 4 4 2 4 3 2
a: link number; C a : link capacity; la : link length; t0a : free flow time. Free flow speeds are uniformly 1; speed limits are uniformly 0.6.
2. Solving a sequence of single-class traffic assignment problems. For each user class i from 1 to jMj, find the traffic equilibrium ðlþ1Þ flow and speed pattern ðv ia ; sia Þ with disutility function (15), while the flows and speeds of all other user classes are ðlþ1Þ given: the flows and speeds of user classes j ¼ 1; 2; ; i 1 are given as ðv ja ; sja Þ , and the flows and speeds of user j j ðlÞ classes j ¼ i þ 1; i þ 2; ; jMj are given as ðv a ; sa Þ . T 3. Checking convergence. If the class-specific link flow pattern v M ¼ ðv m a ; a 2 A; m 2 MÞ converges, stop and get the multiclass user equilibrium flow and speed pattern; otherwise set l ¼ l þ 1 and go to step 2. The following remark is worthy of mention. For the homogeneous case, we are simply dealing with a single-class traffic assignment problem with separable, increasing and convex disutility functions. Many existing well-established, convergent algorithms can be readily used here. For the heterogeneous case, it is complicated yet challenging to find the solution and one has to rely on heuristics. The proposed heuristic algorithm is limited in the sense that it may be not convergent or not converge to a good-quality solution even it is convergent. 5.2. A toy network This example illustrates the equilibrium condition with speed limit and speed choice on a simple toy network. The network structure in Fig. 5 is adopted from Yang et al. (2012) but with different link characteristics (Table 3). The speed limits on links 1, 2 and 4 are fixed at 60 while the speed limit on link 3 varies from 70 to 50. The link travel time functions follow the BPR form,
" t a ðv a Þ ¼
t 0a
1 þ 0:15
va Ca
4 # ð18Þ
270
H. Yang et al. / Transportation Research Part C 51 (2015) 260–273
Table 8 Class-specific OD demands (both class 1 and class 2). O
D
#
O
D
#
O
D
#
O
D
#
O
D
#
O
D
#
O
D
#
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 19 20 22 1 2 4 5 6 7 8 9 10 11 12 13 14 15 16 17 22 23 1 2 3 5 6 7 8 9
5 5 25 10 15 25 40 25 65 25 10 25 15 25 25 20 5 15 15 5 20 15 5 5 5 10 5 20 10 20 10 30 10 5 15 5 5 20 10 5 5 5 5 5 10 5 15 5 10 5 15 15 10 5 5 5 10 5 5 5 25 10 10 25 20 20 35 35
4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
18 19 20 21 22 23 24 1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 19 20 21 22 23 1 2 3 4 5 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 8 9 10 11 12 13 14 15 16 17 18
5 10 15 10 20 25 10 10 5 5 25 10 10 25 40 50 25 10 10 5 10 25 10 5 5 5 10 5 15 20 15 20 10 20 40 20 40 20 10 10 5 10 45 25 5 10 15 5 10 5 5 25 10 5 20 10 20 50 30 95 25 35 20 10 25 70 50 10
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 11
3 4 5 6 7 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1
10 35 25 40 50 40 80 40 30 30 20 30 110 70 15 35 45 20 25 15 10 25 10 5 35 40 20 30 40 140 70 30 30 30 45 70 45 10 20 30 15 35 25 10 65 30 15 60 50 40 95 80 140 200 100 95 105 200 220 195 35 90 125 60 130 90 40 25
11 11 11 11 11 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 14 14 14 14 14 14 14 14
10 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8
195 70 50 80 70 70 50 5 20 30 20 55 65 30 10 5 10 30 10 10 35 30 30 100 70 65 35 35 35 30 10 15 20 15 35 35 25 25 15 5 30 10 10 20 30 30 95 50 65 30 35 30 25 5 15 30 30 65 40 40 15 5 5 25 5 5 10 20
14 14 14 14 14 14 14 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17
18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
5 15 25 20 60 55 20 25 5 5 25 10 10 25 30 50 200 70 35 35 65 60 75 10 40 55 40 130 50 20 25 20 10 40 25 45 70 110 70 220 70 35 30 35 60 140 25 65 80 30 60 25 15 20 10 5 25 10 25 50 70 45 195 50 30 25 35 75
18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 21 21 21 21 21
1 4 6 7 8 9 10 11 12 13 14 15 16 17 19 20 21 22 23 1 2 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 23 24 1 2 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 22 23 24 1 4 5 6 7
5 5 5 10 15 10 35 10 10 5 5 10 25 30 15 20 5 15 5 15 5 10 5 10 20 35 20 90 20 15 15 15 40 65 85 15 60 20 60 15 5 15 5 15 5 15 25 45 30 125 30 25 30 25 55 80 85 20 60 60 120 35 20 5 10 5 5 10
21 21 21 21 21 21 21 21 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24
16 17 18 19 20 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 23 24 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 1 4 6 7 8 9 10 11 12 13 14 15 16 17 19
30 30 5 20 60 90 35 25 20 5 5 20 10 10 25 25 35 130 55 35 65 60 130 60 85 15 60 120 90 105 55 15 5 25 5 5 10 15 25 90 65 35 40 55 50 25 30 5 15 35 35 105 35 5 10 5 5 10 10 40 30 25 35 20 20 15 15 5
271
H. Yang et al. / Transportation Research Part C 51 (2015) 260–273 Table 8 (continued) O
D
#
O
D
#
O
4 4 4 4 4 4 4 4
10 11 12 13 14 15 16 17
60 70 30 30 25 25 40 25
7 7 7 7 7 7 8 8
19 20 21 22 23 24 1 2
20 25 10 25 10 5 40 20
11 11 11 11 11 11 11 11
D
#
2 3 4 5 6 7 8 9
10 15 75 25 20 25 40 70
O
D
#
O
D
#
O
D
14 14 14 14 14 14 14 14
9 10 11 12 13 15 16 17
30 105 80 35 30 65 35 35
17 17 17 17 17 17 17 17
16 18 19 20 21 22 23 24
140 30 85 85 30 85 30 15
21 21 21 21 21 21 21 21
8 9 10 11 12 13 14 15
# 20 15 60 20 15 30 20 40
O
D
24 24 24 24
20 21 22 23
# 20 25 55 35
O: origin node; D: destination node; #: demand (10).
Table 9 User characteristics. User class (m)
am
bm
cm
f1m
f2m
nm
gm
1 2
1 2
1 3
2 4
2.5 2.5
2.5 2.5
2 2
2 2
Table 10 User equilibrium solution. a
v 1a
v 2a
s1a
s2a
sa ð v a Þ
^s1a
^s2a
saav e
u1a
u2a
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
1550.0 3350.0 1550.0 2950.0 3350.0 5986.0 3958.4 6008.4 8671.1 2600.0 8693.4 5429.5 6691.6 2950.0 5450.0 6600.0 5466.9 7121.6 6620.5 5171.6 2300.0 4278.4 6693.4 2300.0 10991.6 11043.4 8850.0 10391.6 5550.0 4050.0 2650.0 8750.0 4900.0 6850.0 3936.0 4900.0 6008.4 6036.0 5508.4 6850.0 4433.4
1800.0 3600.0 1800.0 2700.0 3600.0 6581.5 4314.6 6614.6 8581.5 2600.0 8614.6 4201.0 7830.5 2700.0 4185.1 5701.0 5945.9 7404.1 5685.1 6141.5 3505.7 3491.5 7879.5 3381.9 11535.9 11511.2 9509.4 13541.3 5621.9 4050.0 2650.0 9480.7 4685.4 3303.8 4281.5 4718.5 6150.0 6200.0 5650.0 3342.0 4929.4
0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.317 0.600 0.722 0.317 0.144 0.598 0.722 0.143 0.606 0.723 0.542 0.596 0.723 0.493 0.491 0.370 0.404 0.192 0.491 0.722 0.376 0.314 0.262 0.722 0.311 0.722 0.722 0.224 0.259 0.375
0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.317 0.600 0.794 0.317 0.144 0.598 0.794 0.143 0.606 0.792 0.542 0.596 0.794 0.493 0.491 0.370 0.404 0.192 0.491 0.794 0.376 0.314 0.262 0.794 0.311 0.794 0.794 0.224 0.259 0.375
1.000 0.999 1.000 0.798 0.999 0.958 0.998 0.957 0.883 0.841 0.881 0.317 0.600 0.798 0.317 0.144 0.598 0.978 0.143 0.606 0.792 0.542 0.596 0.806 0.493 0.491 0.370 0.404 0.192 0.491 0.831 0.376 0.314 0.262 0.998 0.311 0.993 0.993 0.224 0.259 0.375
0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.722 0.676 0.713 0.722 0.676 0.648 0.712 0.722 0.648 0.713 0.723 0.706 0.712 0.723 0.700 0.700 0.684 0.688 0.656 0.700 0.722 0.684 0.675 0.667 0.722 0.675 0.722 0.722 0.661 0.667 0.684
0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.794 0.741 0.780 0.794 0.741 0.713 0.779 0.794 0.713 0.780 0.794 0.773 0.779 0.794 0.766 0.766 0.749 0.754 0.721 0.766 0.794 0.750 0.740 0.732 0.794 0.740 0.794 0.794 0.726 0.732 0.749
0.761 0.759 0.761 0.756 0.759 0.760 0.760 0.760 0.758 0.758 0.758 0.317 0.600 0.756 0.317 0.144 0.598 0.759 0.143 0.606 0.765 0.542 0.596 0.765 0.493 0.491 0.370 0.404 0.192 0.491 0.758 0.376 0.314 0.262 0.760 0.311 0.759 0.759 0.224 0.259 0.375
11.62 7.75 11.62 9.68 7.75 7.75 7.75 7.75 3.87 11.62 3.87 13.01 10.13 9.68 13.03 13.97 6.09 3.87 13.99 6.05 19.36 10.69 10.16 19.36 6.82 6.83 14.20 15.84 20.97 18.23 11.62 13.99 19.68 15.55 7.75 19.85 5.81 5.81 18.04 15.72 14.04
31.14 20.76 31.14 25.95 20.76 20.76 20.76 20.76 10.38 31.14 10.38 38.64 28.60 25.95 38.68 41.87 17.20 10.38 41.92 17.05 51.90 30.59 28.71 51.90 19.73 19.76 41.93 46.53 62.77 52.77 31.14 41.27 58.44 46.38 20.76 58.96 15.57 15.57 53.93 46.88 41.42
(continued on next page)
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Table 10 (continued) a
v 1a
v 2a
s1a
s2a
sa ðv a Þ
^s1a
^s2a
saav e
u1a
u2a
42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76
3266.6 10443.4 3800.0 10450.0 10625.0 4003.7 5600.0 5652.9 6354.3 4050.0 5658.0 5874.2 7416.9 6124.5 8125.8 10450.0 5879.2 4842.0 8141.3 4847.1 3000.0 3500.0 2970.5 5893.4 6375.6 9993.4 3500.0 5860.7 5231.0 3900.0 4566.6 2881.0 5486.0 6378.9 2850.0
5342.0 13539.2 5599.1 10245.2 8358.0 3794.9 5578.0 5700.0 6363.3 4050.0 5700.0 5150.0 7208.5 6622.9 9104.8 10231.2 5150.0 4595.2 9118.8 4581.2 3050.0 3500.0 3000.0 3430.9 4182.5 8989.6 3500.0 3432.5 4794.4 4710.4 5427.5 5117.5 5650.0 4130.9 5119.1
0.417 0.402 0.371 0.621 0.303 0.539 0.192 0.231 0.722 0.491 0.231 0.196 0.722 0.722 0.722 0.621 0.196 0.345 0.722 0.346 0.722 0.648 0.722 0.398 0.233 0.304 0.648 0.399 0.293 0.416 0.295 0.522 0.226 0.237 0.524
0.417 0.402 0.371 0.621 0.304 0.539 0.192 0.231 0.794 0.491 0.231 0.196 0.794 0.794 0.794 0.621 0.196 0.345 0.794 0.346 0.765 0.648 0.775 0.398 0.234 0.304 0.648 0.401 0.292 0.416 0.295 0.520 0.226 0.237 0.524
0.417 0.402 0.371 0.621 0.304 0.539 0.192 0.231 0.975 0.491 0.231 0.196 0.978 0.974 0.958 0.621 0.196 0.345 0.958 0.346 0.765 0.648 0.775 0.398 0.234 0.304 0.648 0.401 0.292 0.416 0.295 0.520 0.226 0.237 0.524
0.690 0.688 0.684 0.714 0.674 0.706 0.656 0.662 0.722 0.700 0.662 0.657 0.722 0.722 0.722 0.715 0.657 0.680 0.722 0.680 0.722 0.717 0.722 0.688 0.663 0.674 0.717 0.688 0.672 0.690 0.672 0.704 0.661 0.663 0.704
0.756 0.753 0.749 0.782 0.739 0.772 0.721 0.727 0.794 0.766 0.727 0.722 0.794 0.794 0.794 0.782 0.722 0.745 0.794 0.745 0.793 0.785 0.793 0.753 0.728 0.739 0.785 0.753 0.737 0.756 0.737 0.770 0.726 0.728 0.770
0.417 0.402 0.371 0.621 0.304 0.539 0.192 0.231 0.758 0.491 0.231 0.196 0.758 0.760 0.760 0.621 0.196 0.345 0.760 0.346 0.744 0.648 0.748 0.398 0.233 0.304 0.648 0.400 0.293 0.416 0.295 0.521 0.226 0.237 0.524
10.30 15.90 14.16 5.99 10.16 10.73 21.01 8.77 5.81 18.23 8.78 10.26 3.87 5.81 7.75 5.99 10.27 12.07 7.75 12.05 11.62 9.84 11.62 5.35 12.98 10.16 9.84 5.31 14.04 10.30 13.92 4.39 17.94 12.81 4.37
30.19 46.73 41.78 16.82 30.20 30.74 62.88 26.20 15.57 52.77 26.23 30.70 10.38 15.57 20.76 16.81 30.75 35.75 20.76 35.67 31.20 27.39 31.17 15.73 38.78 30.20 27.39 15.62 41.78 30.21 41.43 12.62 53.61 38.25 12.55
where t0a and C a are the free flow time and link capacity of link a 2 A, respectively. The characteristics of all the user classes are listed in Table 4. Compared with class 1, class 2 has a relatively higher value of time and is less risk-taking. The classrelated OD demands are given in Table 5. The user equilibrium solutions under different speed limit schemes are summarized in Table 6, including the speed selections, link flows and travel disutility of both user classes. Consequently, there are some interesting findings. First of all, both user classes on link 2 cannot drive at their own optimal speeds as restrained by the physical condition of the road; while on the other three links, both user classes can drive at their optimal speeds. Secondly, different classes have different route choices from origin 1 to destination 2: class 1 uses route 1 ! 3 ! 2 only but never uses link 1 while class 2 always uses both routes. Thirdly, decreasing speed limit on link 3 will decrease the driving speeds on link 3 but increase the speeds on link 2, while the speed selections on links 1 and 4 are unchanged. Last but not least, the adjustment of speed limit on a certain road segment will lead to a broad and complex change of speed selection and flow distribution on the whole network, which will consequently influence the individual and social disutility. As in this example, decreasing the speed limit on link 3 actually brings a lower total disutility. How the speed limit affects the system performance and how to design a better speed limit are worthy of further investigation. 5.3. Sioux Falls network In this subsection, we solve the multi-class user equilibrium problem with speed selection under speed limit on the Sioux Falls network in Fig. 6, consisting of 24 nodes and 76 links. The travel time functions are of the BPR form (18) with parameters given in Table 7. The free flow speeds are uniformly 1 and the speed limits are uniformly 0.6 on all links. The class-specific OD demands in Table 8 fall into two user classes, both of which have same demands between all OD pairs. The characteristics of each user class are given in Table 9. The user equilibrium solution is given in Table 10. This example shows that our algorithm is also applicable to a larger network than the toy network in Section 5.2. 6. Conclusions This study investigated both single-class and multi-class user equilibrium problems by considering road users’ speed selection under speed limit conditions. Road users evaluate their travel cost in terms of travel time, crash risk and speeding ticket risk: the crash risk with homogeneous users depends on the absolute speed only, while the crash risk with
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heterogeneous users depends on both the absolute speed and the speed variance among the user classes. Road users are assumed to choose their optimal speed on each particular road to minimize their generalized disutility under the restraint of the flow-speed relationship. An iterative Gauss–Seidel-type algorithm was proposed to solve the traffic assignment problem. The inter-dependence of link flows and speed choices among user classes and the network-wide impacts of speed limits were examined with numerical examples. Further research is expected to apply the model to the optimal design of speed limit schemes for multi-objective network optimization, such as enhancing road safety and reducing network-wide travel time. Acknowledgements The authors wish to express their thanks to three anonymous reviewers for their useful comments on the early versions of the paper. 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