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Multi-criteria user equilibrium model considering travel time, travel time reliability and distance
Transportation Research Part D xxx (2017) xxx–xxx
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Transportation Research Part D journal homepage: www.elsevier.com/locate/trd
Multi-criteria user equilibrium model considering travel time, travel time reliability and distance Chao Sun a, Lin Cheng a,⇑, Senlai Zhu b, Fei Han c, Zhaoming Chu d a
School of Transportation, Southeast University, Nanjing 210096, China School of Transportation, Nantong University, Nantong 226019, China c School of Automobile, Chang’an University, Xi’an 710064, China d Road Traffic Safety Research Center of the Ministry of Public Security, Beijing 100062, China b
a r t i c l e
i n f o
Article history: Received 16 August 2016 Accepted 5 March 2017 Available online xxxx Keywords: Multi-criteria user equilibrium Travel time Travel time reliability Travel distance Maximum entropy Partial linearization descent method
a b s t r a c t This paper proposes a multi-criteria user equilibrium model considering travel time, travel time reliability and distance (MUE-TRD). This new model hypothesizes that for each user class and each origin-destination (O-D) pair no traveler can reduce either his or her reliable travel time or travel distance or both without worsening the other objective by unilaterally changing routes in their route choice decision process. Travel time budget which consists of travel time and travel time reliability is used to describe the reliable travel time. A maximum entropy multi-criteria user equilibrium (ME-MUE) model is presented to address the non-uniqueness of the solution in MUE-TRD model. Furthermore, a route-based solution algorithm based on the partial linearization descent method (R-PLD) is developed to solve the ME-MUE model. Numerical examples are also provided to illustrate the essential ideas of the proposed model and the applicability of the developed solution algorithm. The results show that compared to traditional user equilibrium and travel time budget models, ME-MUE model is more consistent with the real trip process that the reliable travel time is increasing with the decreasing of travel distance in used routes; and the road traffic is smoother when using ME-MUE model to design the road network, thus ME-MUE model can reduce road traffic noise and air pollution in the urban road network. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction In transportation systems, travelers may be faced with several criteria to select the routes. Some travelers select routes to minimize their reliable travel time which consists of travel time and travel time reliability according to their past experience, such as the commuters, who travel regularly between the same O-D pair on a daily basis; and some travelers would like to select the routes with minimum travel time or travel distance using the vehicle navigation system (Lima et al., 2016; Wang et al., 2016). Statistical analysis was performed by Abdel-Aty et al. (1995) to examine which route attributes that lead to the choice of a route are considered important by travelers. Three most important factors derived from the analysis are: (1) shorter travel time (40% of respondents); (2) travel time reliability (32% of respondents); and (3) shorter distance (31% of respondents). Due to its theoretical and practical importance, modeling traffic assignment considering these factors is becoming an emerging research subject.
⇑ Corresponding author. E-mail address: [email protected] (L. Cheng). http://dx.doi.org/10.1016/j.trd.2017.03.002 1361-9209/Ó 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Sun, C., et al. Multi-criteria user equilibrium model considering travel time, travel time reliability and distance. Transport. Res. Part D (2017), http://dx.doi.org/10.1016/j.trd.2017.03.002
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C. Sun et al. / Transportation Research Part D xxx (2017) xxx–xxx
For single-objective traffic assignment model, the conventional Wardrop user equilibrium (UE) model (Wardrop, 1952), and the stochastic user equilibrium (SUE) model (Daganzo and Sheffi, 1977), only consider factor 1 (i.e., travel time). The UE model assumes that users try to minimize their travel time, while in the SUE model users are assumed to minimize their perceived travel time, which has a random error component. The traffic equilibrium models under uncertainty consider factors 1 and 2 (i.e., travel time and travel time reliability). To build the traffic equilibrium models under uncertainty, different theories were used, such as travel time budget (TTB) approach (e.g., Lo et al., 2006; Wu, 2015), game theory approach (e.g., Bell and Cassir, 2002), utility-based approach (e.g., Mirchandani and Soroush, 1987), expected residual minimization approach (e.g., Zhang et al., 2011), robust optimization approach (e.g., Ordóñez and Stier-Moses, 2010), prospect theorybased approach (e.g., Xu et al., 2011), schedule delay approach (e.g., Watling, 2006), and mean-excess travel time approach (e.g., Chen and Zhou, 2010). For bi-objective traffic assignment model, Dial (1996, 1997), Nagurney and Dong (2002) and Larsson et al. (2002) propose the bi-objective user equilibrium assignment model based on factors 1 and 3 (i.e., travel time and travel distance which is directly related to vehicle operating cost for the trip). Recently, Wang et al. (2014) and Ehrgott et al. (2015) formulated a general travel time reliability bi-objective user equilibrium model which took factors 1 and 2 (i.e., travel time and travel time reliability) into consideration. This model assumes that users select routes to minimize their travel time and maximize their travel time reliability. The solution of this model is non-unique; and the solution algorithm is not given in their works. Compared to single-objective traffic assignment model, bi-objective traffic assignment model is more realistic. However, it is difficult to find the set of route flow solutions in bi-objective traffic assignment model. Generalized route cost function was used by Dial (1996, 1997) and Larsson et al. (2002) to deal with the bi-objective traffic assignment model. Nagurney and Dong (2002) proposed a weight method in which travelers perceive their generalized cost on a route by travel time and travel distance with different weight. Other bi-objective traffic assignment algorithms (Chen et al., 2010; Chen and Nie, 2013) also combined the two objective functions into a nonlinear generalized cost. All these methods convert the bi-objective problem to single-objective problem using linear combination of two different factors. In fact, combining the two factors into one implicitly assumes the existence of a linear (dis)utility function, and therefore presupposes a certain preference structure. As a result of this, there is the possibility that some reasonable solutions are never considered (Wang et al., 2014). Considering both factors 1, 2 and 3 (i.e., travel time, travel time reliability and travel distance) in the route choice decision process, this paper presents a multi-criteria user equilibrium model considering travel time, travel time reliability and distance (MUE-TRD). This new model hypothesizes that for each user class and each O-D pair no traveler can reduce either his or her reliable travel time or travel distance or both without worsening the other objective by unilaterally changing routes. For computation of the reliable travel time which consists of travel time and travel time reliability, travel time budget (TTB), which is defined as a travel time reliability chance (or on-time arrival) constraint; such that the probability that travel time exceeds the budget is less than a predefined confidence level a to represent the travel time reliability, is used in this paper. For addressing the non-uniqueness of the solution in MUE-TRD model, a maximum entropy multicriteria user equilibrium (ME-MUE) model whose feasible set is the solution set of MUE-TRD model is proposed. This is the first attempt to use the principles of entropy maximizing to deal with the multi-criteria user equilibrium model. A route-based solution algorithm based on the partial linearization descent method (R-PLD) is developed to solve the ME-MUE model. The rest of the paper is organized as follows. In Section 2, MUE-TRD model is presented. In Section 3, ME-MUE model whose feasible set is the solution set of MUE-TRD model is proposed. In Section 4, R-PLD is developed to determine the equilibrium flow pattern. In Section 5, numerical examples are presented to illustrate the essential ideas of the proposed models and the applicability of the solution algorithm. Finally, conclusions are provided. 2. Multi-criteria user equilibrium model This section presents MUE-TRD, the MUE-TRD condition, the existence of solution set and an instance of MUE-TRD model. 2.1. Formulation of MUE-TRD model It is well known from empirical studies that the three most important factors influencing route choice behavior are travel time, travel time reliability and travel distance. Then travelers always would want: (1) to minimize the reliable travel time which consists of travel time and travel time reliability; and (2) to minimize the travel distance. To describe the reliable travel time, travel time budget (TTB) (Lo et al., 2006) which is defined as the average travel time plus a safety margin as an acceptable travel time is used in this paper. Mathematically, the two criteria are: x x minfbk Pr½T x k 6 bk P ag x
min dk 8k 2 K x ; x 2 W
ð1Þ
And the feasible set defined as follows: Please cite this article in press as: Sun, C., et al. Multi-criteria user equilibrium model considering travel time, travel time reliability and distance. Transport. Res. Part D (2017), http://dx.doi.org/10.1016/j.trd.2017.03.002
C. Sun et al. / Transportation Research Part D xxx (2017) xxx–xxx
X
x
f k2K x k;i
¼ qx i 8x 2 W i 2 I
x qx i ¼ li q 8x 2 W i 2 I XX X x x xa ¼ da;k f k;i 8a 2 A
ð2Þ
x k2K x
i
3
x
x
f k;i P 0 8k 2 K x 2 W; i 2 I XX x x da;k da x 2 W dk ¼ x k2K x x
where bk is the travel time budget on route k between O-D pair x at a predefined confidence level a; T x k is the reliable travel x time on route k between O-D pair x; dk is the travel distance on route k between O-D pair x; K x is the set of routes between x O-D pair x; A; W and I are the sets of links, O-Ds and user classes, respectively; f k;i is the route flow of user class i on route k x x between O-D pair x; qi and q are the traffic demand of user class i and total demand between O-D pair x respectively; li is the proportion of class i; xa is the flow on link a; da is the distance on link a; and dx a;k is the route-link incidence parameter whose value is 1 if link a is on route k; 0 otherwise. The travel time budget is the value of cumulative probability under the confidence level of a which represents different classes. As derived in Lo et al. (2006), the route travel time is normally distributed with mean and standard deviation that can be written as follows: 1 x bk ¼ lx k þ U ðai Þrk io Xn x h ð1h1n Þ a lxk ¼ da;k t0a þ bt0a xna cn ð1h a Þð1nÞ
x
a
a
rxk
ð3Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi h i2 P ð1h12n Þ ð1h1n x 2 0 2 2n a a Þ ¼ cn ð1ha Þð1nÞ a da;k b ðt a Þ xa c2n ð1ha Þð12nÞ a
a
1 x where lx k and rk are mean and standard deviation of the route flow; U ðÞ is the inverse of the standard normal cumulative 0 distribution function; ta and C a are the free-flow travel time and capacity on link a, respectively; b and n are the deterministic parameters of bureau of public roads (BPR) (Manual, 2000) link performance function; and cna is the design capacity, while the lower bound capacity is a fraction ha of cna .
2.2. MUE-TRD condition Based on the two criteria functions in Eq. (1), the condition of MUE-TRD can be formulated as follows: Definition 1. The MUE-TRD is a network state such that for each user class and each O-D pair no traveler can reduce either his or her reliable travel time or travel distance or both without worsening the other objective by unilaterally changing routes. The weak condition of MUE-TRD can also be defined as follows: Definition 2. The weak MUE-TRD is a network state such that for each user class and each O-D pair no traveler can reduce both his or her reliable travel time and travel distance by unilaterally changing routes. It can be proved that every solution of MUE-TRD model is also a solution of weak MUE-TRD model, and every solution of the TTB equilibrium model of Lo et al. (2006) is also a solution of MUE-TRD model. Theorem 1. Let F be a route solution to MUE-TRD model. Then F also satisfies the weak MUE-TRD condition. Proof 1. Assume that F does not satisfy the weak MUE-TRD condition. Then, for at least one user class i there must exist two 0 x x x x used route k and k between some O-D pair x such that bk0 ;i < bk;i and dk0 ;i < dk;i . These inequalities contradict the assumption that F satisfies the MUE-TRD condition. h x
x
x
x
The route solution bk0 ;i < bk;i and dk0 ;i ¼ dk;i satisfies the weak MUE-TRD condition while not satisfies the MUE-TRD condition. Because someone will reduce his or her reliable travel time while the travel distance be the same by changing his or her 0 x x x x route from k to k . However, the route solution bk0 ;i ¼ bk;i and dk0 ;i < dk;i is a MUE-TRD model solution. Although someone can 0
reduce his or her travel distance by changing his or her route from k to k , his or her reliable travel time will increase. This is because travel distance is constant, while the reliable travel time (i.e., TTB) is a smooth and convex function (Lo et al., 2006). Theorem 2. Let G be a route solution to TTB model. Then G also satisfies the MUE-TRD condition.
Please cite this article in press as: Sun, C., et al. Multi-criteria user equilibrium model considering travel time, travel time reliability and distance. Transport. Res. Part D (2017), http://dx.doi.org/10.1016/j.trd.2017.03.002
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C. Sun et al. / Transportation Research Part D xxx (2017) xxx–xxx
Proof 2. Assume that G does not satisfy the MUE-TRD condition. Then as described above, for at least one user class i there 0 x x x x must exist two used route k and k between some O-D pair x such that bk0 ;i < bk;i and dk0 ;i 6 dk;i . The first of these inequalities contradicts the assumption that G satisfies the TTB equilibrium condition. h Using this theorem, that G is also a route solution to user equilibrium (UE) model when the confidence level a is equal to 0.5 can be deduced. Under the assumptions that the TTB on each route between each O-D pair for each user class is positive and continuous function of flow, MUE-TRD flow exist. And the qualitative property of solution set is present. Theorem 3. The MUE-TRD flow exists under the assumptions that the TTB on each route between each O-D pair for each user class is positive and continuous with respect to its route flow. And the solution set XBUE-TD of MUE-TRD model is compact.
x
x
Proof 3. Because of the assumption that bk;i is positive and continuous functions of flow, and dk;i is constant, it is known that x
x
x
Gk;i ðfÞ :¼ bk;i þ kdk;i for positive k is positive and continuous. Hence an equilibrium flow f with respect to Gx k;i exists. It can be
0
shown that this equilibrium flow f is a MUE-TRD flow. Assume to the contrary that there exist two used route k and k x x x x x between some O-D pair x such that bk0 ;i < bk;i and dk0 ;i 6 dk;i . Then Gx k0 ;i ðf Þ < Gk;i ðf Þ contradicting the fact that f is an equilibrium flow with respect to Gx k;i . Since the feasible set (i.e., Eq. (2)) is bounded. Thus the solution set XBUE-TD of MUE-TRD model is bounded. According to Ehrgott and Wiecek (2005), the solution set XBUE-TD is closed. Following the Heine-Borel theorem, it has: the solution set XBUE-TD of MUE-TRD model is compact. h
2.3. Instance of MUE-TRD model The network topology and characteristics of the test network are depicted in Fig. 1. The network consists of 2 nodes, 3 links and 1 O-D pair. The route flows are assumed to follow the normal distribution, and the numerical characteristics are depicted in Fig. 1. And the distances of each route are 5, 8 and 9. The equilibrium traffic flow set of MUE-TRD model with the confidence level of 80% and the equilibrium traffic flow of TTB model with the confidence level ranging from 50% to 95% are provided in Fig. 2. It shows that (1) as expected, the TTB model solution with a ¼ 80% satisfy the MUE-TRD conditions. And the TTB solution with a ¼ 80% is on the boundary of MUE-TRD solution set. This is because at this traffic state, the used routes have the same reliable travel time; and it is a limiting case of the solution set. (2) the solution set of MUE-TRD model is bounded and closed; and (3) with the increase of confidence level, the flow on route 3 is always increasing, and the equilibrium traffic time on used routes increase from 9.31 to 11.17. This means that more risk-averse travelers will spend a larger reliable travel time to ensure more on-time arrivals. 3. Maximum entropy multi-criteria user equilibrium model This section proposes the maximum entropy multi-criteria user equilibrium (ME-MUE) model, the existence of solution set and an instance of ME-MUE model. 3.1. Formulation of ME-MUE model It is known that the route solutions to MUE-TRD model proposed in Section 2 are not unique; therefore, an additional behavioral assumption is required to determine the most likely traffic flows among these solutions. The principles of entropy maximizing were used by Rossi et al. (1989), Bar-Gera (2006) and Kumar and Peeta (2015) to generating a single route flow solution under the UE assumption. And it seems the most likely and meaningful route flow solution in reality. In this section, a ME-MUE model whose feasible set is the solution set of MUE-TRD model was proposed.
∼ ∼ ∼ Fig. 1. Network topology and characteristics.
Please cite this article in press as: Sun, C., et al. Multi-criteria user equilibrium model considering travel time, travel time reliability and distance. Transport. Res. Part D (2017), http://dx.doi.org/10.1016/j.trd.2017.03.002
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C. Sun et al. / Transportation Research Part D xxx (2017) xxx–xxx
TTB Solutions of Diffient Efficient Solutions
Flow of Route 3
2.5 2
=0.95
f2
1.5 1 0.5
=0.8, f1 1.69 6.028, f3 2.282
=0.5
0 10
10
5
Flow of Route 2
5 0
0
Flow of Route 1
Fig. 2. Solution set of MUE-TRD model and solutions of TTB model.
Given the route solution set XMUE-TRD of MUE-TRD model, the ME-MUE route flow solution on general road networks is obtained by solving the following nonlinear program:
max EðfÞ ¼ Px
qx ! Pk f x k !
ð4Þ
f 2 XMUE-TRD For minifying the value of objective function, a constant qx can be input to the objective function Px
qx !=qx
x Pk f x k !=q
without
altering the solution of Eq. (4). And it is convenient to maximize the natural logarithmic form of the objective function rather P h qx ! P f xk !i . Since qx is fixed, the first term can be x ln qx k ln qx
than the original one. Then, the objective function becomes
x
dropped from the objective function. And assume that O-D flows qx and route flows f k are usually large, Stirling’s approx i P hP f xk : fx x k imation of ln x!¼x ln x x is used to simplify the objective function x k qx ln qx 1 . Constant q could be dropped without altering the solution. Then, the original issue of ME-MUE route flow solution can be transformed to:
XX x f x max EðfÞ ¼ f k ln qkx x
k
ð5Þ
f 2 XMUETRD The existence of the solution to ME-MUE model is shown as below: Theorem 4. The ME-MUE flow of Eq. (5) exists.
Proof 4. The feasible set XMUE-TRD is non-empty, closed. Moreover, the objective function is positive and continuous with respect to f. Then, the ME-MUE flow of Eq. (5) exists. h It is necessary to point out that the uniqueness of the ME-MUE equilibrium route flow solution cannot be guaranteed. In order to prove the uniqueness, convexity of feasible set XMUE-TRD is required. However, it is difficult to guarantee the convexity of feasible set in general. 3.2. Instance of ME-MUE model The network topology and characteristics of the test network are depicted in Fig. 3. The network consists of 2 O-D pairs and 4 routes. The MUE-TRD model is assumed to have 3 route solutions depicted in Fig. 3. Since each route flow solution is the aggregate outcome of many individual route choice decisions, Eq. (4) is used to find all possible route choice decisions shown in Table 1. It shows that, there are 34 route choice decisions those 4 individual travelers from A-B and 3 from C-D. For the 1th route flow solution, there can be 4 route choice decisions that one individual traveler chooses route 1 and the other 3 individual travelers choose route 2 for O-D pair A; B. And there can be 1 route choice decision that all 3 individual travelers choose route 1 for O-D pair C; D. It is assumed that the occurrence of each route choice decision is equal in the entropy formulation; thus, in this small example, each route choice decision is responsible for a 1/34 probability of occurring. The most likely MUE-TRD route flow solution is the one associated with the largest number of route choice decisions (i.e., maximum entropy). Among all 3 possible MUE-TRD route flow solutions, the 2nd solution, which accounts for 18 out of 34 route choice decisions or a 52.94% probability of occurring, is clearly the most likely one. Please cite this article in press as: Sun, C., et al. Multi-criteria user equilibrium model considering travel time, travel time reliability and distance. Transport. Res. Part D (2017), http://dx.doi.org/10.1016/j.trd.2017.03.002
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f1A, B
A
B
C
D
f f f
f1A, B
f 2A, B
f1C , D
f 2C , D
Solution 1
1
3
3
0
Solution 2
2
2
1
2
Solution 3
3
1
2
1
BUE TD
A, B 2 C ,D 1
C ,D 2
Fig. 3. Network topology and route solutions.
Table 1 All possible route choice decisions associated with the flow solution shown in Fig. 3. Route choice made by individual travelers
From origin A to destination B Traveler 1
Traveler 2
Traveler 3
Traveler 4
Traveler 1
Traveler 2
Traveler 3
RA;B 2 RA;B 2 RA;B 1 RA;B 2
RA;B 2 RA;B 2 RA;B 2 RA;B 1
RC;D 1 RC;D 1 RC;D 1 RC;D 1
RC;D 1 RC;D 1 RC;D 1 RC;D 1
RC;D 1 RC;D 1 RC;D 1 RC;D 1
RA;B 2
RC;D .. 1 . RC;D 2
RC;D 2
RC;D 2
RC;D 2
RC;D 1
RC;D .. 1 . RC;D 2
RC;D 1
RC;D 2
RC;D 1
RC;D 1
1 2 3 4
The 1th route flow solution RA;B RA;B 1 2 RA;B RA;B 2 1 A;B R2 RA;B 2 A;B RA;B R 2 2
5 .. . 22
The 2nd route flow solution RA;B RA;B RA;B 1 2 .. 1 . A;B A;B R2 RA;B R2 1
23 .. . 34
The 3rd route flow solution RA;B RA;B 1 .. 1 . A;B R RA;B 2 1 RA;B 1
From origin C to destination D
RA;B 1
RA;B 1
RA;B 2
RA;B 1
RA;B 1
RA;B 2
Note: = this traveler chooses route 1 between O-D pair A; B; = this traveler chooses route 2 between O-D pair A; B; 1 between O-D pair C; D; RC;D 2 = this traveler chooses route 2 between O-D pair C; D.
RC;D 1
= this traveler chooses route
4. Solution algorithm The reliable travel time (i.e., TTB) in the proposed model is non-additive because it is not possible to decompose the route TTB into the sum of link-based generalized costs. And the route solution set XBUE-TD of MUE-TRD model cannot be expressed using mathematical formulations, because of the complexity of multi-criteria programming. Therefore, a route-based solution algorithm based on the partial linearization descent method (R-PLD) is developed to solve the ME-MUE model as follows. Step 1. (Initialization) Set initial iteration number n ¼ 0, tolerance error e for check convergence, tolerance error e0 for search step size, the working route set K, and initial traffic route flow x x x x qx if d ¼ d where d ¼ min fd g ð1Þ ð0Þ xð0Þ k k k i f ¼ f : f k;i ¼ . 0 otherwise Step 2. (Search Direction and Step Size) For each O-D pair and each user class, choose any two routes ka and kb (i.e., it P P xðnÞ . x ðnÞ needs to choose C 2x;k ¼ kðk 1Þ=2 times); the search direction is set as: tk;i;a;b ¼ 2 ð1; 1ÞT ; and i f k;i;a ðnÞ i f k;i;b ðnÞ
using bisection method to compute the step size ak;i;a;b under the tolerance error e0 . ! ! x xðnþ1Þ f k;i;a ðnÞ f k;i;a ðnÞ ðnÞ Step 3. (Update Traffic Flow) Update the traffic flow ¼ þ ak;i;a;b tk;i;a;b . xðnÞ xðnþ1Þ f k;i;b f k;i;b Step 4. (Check Convergence) If ðkf f
ðnþ2Þ
¼f
ðnþ1Þ
ðnþ1Þ
f
ðnÞ
kÞ=kf
ðnÞ
k 6 e, then stop, f
ðnþ1Þ
is the optimal solutions. Otherwise, set
; n :¼ n þ 1, go to Step 2.
For any two routes ka and kb in an O-D pair, it can be proved that the entropy value reach the maximum when the two route flows are the same using Lagrangian multiplier method. Therefore, the descent direction is the half of difference between the high route flow and the low route flow. For the line search, the classical bisection method is considered to find an approximate step size. The convergence and applicability of R-PLD algorithm are presented using practice instances next section. 5. Numerical examples In this section, a small network is presented to illustrate the essential ideas of the proposed model, and a medium network is presented to illustrate the applicability of the proposed solution algorithm. Please cite this article in press as: Sun, C., et al. Multi-criteria user equilibrium model considering travel time, travel time reliability and distance. Transport. Res. Part D (2017), http://dx.doi.org/10.1016/j.trd.2017.03.002
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C. Sun et al. / Transportation Research Part D xxx (2017) xxx–xxx
5.1. Small network The network topology and characteristics of the test network are depicted in Fig. 4. The network consists of 13 nodes, 19 links, and 4 O-D pairs. The commonly used BPR performance function (b ¼ 0:15, n ¼ 4, Manual, 2000) is adopted. The value of ha in TTB model is equal to 0.3. The free-flow travel time is used as the travel distance for convenience. Link and traffic demand characteristics are provided in Fig. 4. Travelers are partitioned into two classes with the confidence level 50% and 80% whose proportions of demand are 40% and 60%, respectively. The algorithm is coded in Matlab R2015b and tested on a personal computer with Intel(R) Core(TM) i7-5600U CPU 2.60 GHz, 8 GB memory. First, the iterative processes of entropy value of the R-PLD algorithm are shown in Fig. 5. It can be seen that (1) the entropy value can rapidly converge to 2147 with 0.489 s CPU time; (2) the entropy value fluctuates with the characteristics of jump. This is because the jump of transferred traffic flow using bisection method to compute the step size; and (3) the entropy value fluctuates sharply in the middle iterations. This means that a large number of traffic flows transferred at this time. Then, the validity of the solution obtained from the R-PLD algorithm is examined. The equilibrium results of the proposed model are provided in Table 2. As expected, the equilibrium results satisfy the MUE-TRD conditions. Without loss of generality, the equilibrium results in O-D pair (4-2) are shown. For the used routes (i.e., route 15, 16 and 18), the shortest route 15 has the most reliable travel time for all user classes; the longest route 18 has the least reliable travel time for all user classes; and the distance and reliable travel time of route 16 are in the middle. For unused routes (i.e., route 17 and 19), they will not be considered, as they are dominated by route 18, which has no worse distance and reliable travel time. On the other hand, the equilibrium results satisfy conservation constraint. The total demands of class 1 and class 2 in O-D pair (4-2) are 240 and 360, respectively. Table 2 shows that (1) some used routes has the same reliable travel time in the same O-D pair. Because the route flow is on the boundary of MUE-TRD solution set at the process of searching the maximum entropy value; (2) route reliable travel time is increasing with the confidence level. This is because conservative travelers will over estimate their reliable travel time; and (3) compared to traditional UE model and TTB model, ME-MUE model has more used routes, and the reliable travel time is increasing with the decreasing of travel distance in used routes, this result is more consistent with the real trip process.
1 200
1
8/700
7/300
7/900
800
9/700
4
14/900 O 1 4
D
2 400 600
5
5/800
9/600
3 800 200
9
12 15/700
6
13/500 10/700
5/900
7
5/300
9/400
10
8/700
11
8
10/700 9/700
2
1 000
8/700
9/600
Free-flow travel time / Design Capacity 13
11/700
3
1 000
Fig. 4. Network topology, O-Ds and link characteristics.
2500
Entropy Value
2000
time 0.489 entropy 2147
1500 1000 500 0
0
0.2
0.4
0.6
0.8
1
CPU Time (sec) Fig. 5. Iterative processes of entropy value.
Please cite this article in press as: Sun, C., et al. Multi-criteria user equilibrium model considering travel time, travel time reliability and distance. Transport. Res. Part D (2017), http://dx.doi.org/10.1016/j.trd.2017.03.002
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C. Sun et al. / Transportation Research Part D xxx (2017) xxx–xxx
Table 2 Equilibrium results of ME-MUE, UE and TTB models. O-D
Route No.
Sequence of nodes
Distance
Flow of class 1
Flow of class 2
Time of class 1
Time of class 2
UE
TTB
1-2
1 2 3 4 5 6 7 8
1-12-8-2 1-5-6-7-8-2 1-5-6-7-11-2 1-5-6-10-11-2 1-5-9-10-11-2 1-12-6-7-8-2 1-12-6-7-11-2 1-12-6-10-11-2
33 32 35 42 43 35 38 45
121.90 7.64 – 8.57 9.46 – – 12.43
213.87 8.66 1.40 4.80 6.05 – 2.22 3.00
62.10 81.39 65.70 62.10 61.31 87.78 62.10 58.49
89.95 100.26 85.42 75.53 73.14 125.44 79.30 67.85
400 – – – – – – –
400 – – – – – – –
1-3
9 10 11 12 13 14
1-5-6-7-11-3 1-5-6-10-11-3 1-5-9-10-11-3 1-12-6-7-11-3 1-12-6-10-11-3 1-5-9-13-3
34 41 42 37 44 36
45.49 43.66 29.11 63.09 28.82 109.83
159.57 24.03 40.15 6.62 47.03 202.60
69.00 65.39 64.61 65.39 61.78 65.39
89.79 80.36 78.15 83.81 73.23 84.53
429 – – – – 371
339 10 – – 124 327
4-2
15 16 17 18 19
4-5-6-7-8-2 4-5-6-7-11-2 4-5-6-10-11-2 4-9-10-11-2 4-5-9-10-11-2
34 37 44 41 45
109.20 36.00 – 94.80 –
209.80 25.00 – 125.20 –
68.77 63.08 59.47 49.94 58.69
96.84 81.18 70.39 56.76 67.56
408 25 – 167 –
193 – 55 352 –
4-3
20 21 22 23 24 25
4-5-6-7-11-3 4-5-6-10-11-3 4-5-9-10-11-3 4-9-13-3 4-5-9-13-3 4-9-10-11-3
36 43 44 34 38 40
– – – 70.00 – 10.00
– – – 116.00 – 4.00
66.38 62.77 61.99 53.23 62.77 53.23
85.64 75.52 73.03 70.54 80.23 62.71
– – – 200 – –
– – – 175 – 25
5.2. Sioux Falls network The well-known Sioux Falls (Leblanc, 1973) network is used to illustrate the applicability of the proposed algorithm. The network consists of 24 nodes, 76 links, and 550 O-D pairs. A behaviorally generated working route set which has 3441 routes from Bekhor et al. (2008) is used. The values of b, n in BPR performance function, travel distance, ha and classification of users are similar to the small network. Initially, the convergence curve and entropy value of R-PLD algorithm is shown in Fig. 6. The green line is the iterative error curve; it can be seen that after 13 iterations, the error reach to 0 within 93.3 s CPU time. Because of the entropy value is increasing until reach the maximum value (i.e., the search direction is always descending), the error monotonically approaches to zero. The magenta line is the entropy curve; it also illustrates the R-PLD algorithm can rapidly converge to maximum entropy. To further demonstrate the convergence characteristics of the R-PLD algorithm, the evolution processes of reliable travel time and route flow of different user classes are examined. Without loss of generality, Fig. 7 only shows the results on 3 routes (and the distances are 21, 14 and17, respectively) in O-D pair (16-18) where the demand is 200. As expected, the reliable travel time and route flow can rapidly converge to equilibrium results. The equilibrium results satisfy the MUE-TRD conditions: the reliable travel time will increase while some users change his or her route from 2159 to others; the travel
1
3E4
Error Value
1E-4
time 93.30 error 0
Entropy Value
time 93.30 entropy 27788
1E-2
1E-6
Iterative Error Curve Entropy Curve
1E-7 0
0
20
40
60
80
100
2E4
1E4
0
CPU Time (sec) Fig. 6. Convergence curve and entropy value of R-PLD algorithm.
Please cite this article in press as: Sun, C., et al. Multi-criteria user equilibrium model considering travel time, travel time reliability and distance. Transport. Res. Part D (2017), http://dx.doi.org/10.1016/j.trd.2017.03.002
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250 200 150 100 50 0
120
Route 2159 Route 2160 Route 2161
40
100
Route Flow
Route Travel Time
300
20
80
0
60
6.85
6.86
6.87
40 20
0
5
10
15
CPU Time (sec)
20
0
0
5
10
15
20
CPU Time (sec)
Fig. 7. Evolution processes of reliable travel time and route flow. (Note: solid lines, dotted lines represent class 1 and class 2, respectively; the convergence values from top to bottom are class 1 on route 2159, route 2160, route 2161, class 2 on route 2159, route 2160, route 2161).
distance will increase while some users change his or her route from 2160 to others; and either the reliable travel time or distance will increase while some users change his or her route from 2161 to others. The right figure shows that at one iteration, route 2159 and 2160 make a flow transferred for each class, as well as route 2159 and 2161, 2160 and 2161. Overall, at every iteration process of each O-D pair each user class, any two routes have made a flow transferred using R-PLD algorithm. 6. Conclusions This paper proposed MUE-TRD, a new model that assumes travelers minimize their reliable travel time (including travel time and travel time reliability) and their travel distance. The proposed model was then formulated as MUE-TRD conditions, by which, for each user class and each O-D pair no traveler can reduce either his or her reliable travel time or travel distance or both without worsening the other objective by unilaterally changing routes. For obtaining the most likely traffic flows among all the solutions of UE-TRD model, a ME-MUE model whose feasible set is the solution set of MUE-TRD model was presented. Among all MUE-TRD route flow solutions, the entropy-maximizing route flow vector was considered to be the most likely one in reality. Furthermore, a route-based solution algorithm based on the R-PLD was developed to solve the ME-MUE model. In the algorithm, the descent direction was determined according to the difference between the high route flow and the low route flow for any two routes in an O-D pair, and the step size was obtained by a classical bisection method. The essential ideas of the proposed model and the applicability of the R-PLD algorithm were tested in both a small network and the Sioux Falls network. The analysis results indicated that (1) considering both travel time, travel time reliability and travel distance has a significant effect on route choice process for the travelers with different preferences; (2) compared to traditional UE model and TTB model, ME-MUE model has more used routes, and the reliable travel time is increasing with the decreasing of travel distance in used routes; (3) the R-PLD algorithm is an effective method for solving the proposed MEMUE model; and (4) the road traffic is smoother which can reduce road traffic noise and air pollution when using ME-MUE model to design the urban road network. In future research, the stochastic perception error ignored by this paper should be incorporated in the ME-MUE model. For the solution algorithm, embedding a column generation scheme within the R-PLD for testing large networks is needed for real applications of the proposed model. For designing a more reliable network, it is worthwhile to extend the proposed model to the network design problem. Acknowledgements The authors are grateful to the referees for their constructive comments and suggestions to improve the quality and clarity of the paper in 7th International Conference on Green Intelligent Transportation System and Safety. This research was supported by the National Natural Science Foundation of China (No. 51578150, 51378119, and 51608115), the Scientific Research Foundation of Graduate School of Southeast University (No. YBJJ1679), the Natural Science Foundation of Jiangsu Province (No. BK20150613), and the China Scholarship Council (CSC). References Abdel-Aty, M.A., Kitamura, R., Jovanis, P.P., 1995. Investigating effect of travel time variability on route choice using repeated-measurement stated preference data. Transport. Res. Rec. 1493, 39–45. Bar-Gera, H., 2006. Primal method for determining the most likely route flows in large road networks. Transport. Sci. 40 (3), 269–286.
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Bekhor, S., Toledo, T., Prashker, J.N., 2008. Effects of choice set size and route choice models on path-based traffic assignment. Transportmetrica 4 (2), 117– 133. Bell, M.G., Cassir, C., 2002. Risk-averse user equilibrium traffic assignment: an application of game theory. Transport. Res. B: Meth. 36 (8), 671–681. Chen, A., Zhou, Z., 2010. The a-reliable mean-excess traffic equilibrium model with stochastic travel times. Transport. Res. B: Meth. 44 (4), 493–513. Chen, A., Oh, J.S., Park, D., Recker, W., 2010. Solving the bicriteria traffic equilibrium problem with variable demand and nonlinear path costs. Appl. Math. Comput. 217 (7), 3020–3031. Chen, P., Nie, Y.M., 2013. Bi-criterion shortest path problem with a general non-additive cost. Transport. Res. B: Meth. 57, 419–435. Daganzo, C.F., Sheffi, Y., 1977. On stochastic models of traffic assignment. Transport. Sci. 11 (3), 253–274. Dial, R.B., 1996. Bicriterion traffic assignment: basic theory and elementary algorithms. Transport. Sci. 30 (2), 93–111. Dial, R.B., 1997. Bicriterion traffic assignment: efficient algorithms plus examples. Transport. Res. B: Meth. 31 (5), 357–379. Ehrgott, M., Wiecek, M.M., 2005. Multi-objective programming. In: Multiple Criteria Decision Analysis: State of the Art Surveys. Springer, New York, pp. 667–708. Ehrgott, M., Wang, J.Y., Watling, D.P., 2015. On multi-objective stochastic user equilibrium. Transport. Res. B: Meth. 81, 704–717. Kumar, A., Peeta, S., 2015. Entropy weighted average method for the determination of a single representative path flow solution for the static user equilibrium traffic assignment problem. Transport. Res. B: Meth. 71, 213–229. Larsson, T., Lindberg, P.O., Patriksson, M., Rydergren, C., 2002. On traffic equilibrium models with a nonlinear time/money relation. In: Transportation Planning. Springer, US, pp. 19–31. Leblanc, L.J., 1973. Mathematical programming algorithms for large scale network equilibrium and network design problems. Lima, A., Stanojevic, R., Papagiannaki, D., Rodriguez, P., González, M.C., 2016. Understanding individual routing behaviour. J. Roy. Soc. Interface. 13 (116), 20160021. Lo, H.K., Luo, X.W., Siu, B.W., 2006. Degradable transport network: travel time budget of travelers with heterogeneous risk aversion. Transport. Res. B: Meth. 40 (9), 792–806. Mirchandani, P., Soroush, H., 1987. Generalized traffic equilibrium with probabilistic travel times and perceptions. Transport. Sci. 21 (3), 133–152. Manual, H.C., 2000. Transportation Research Board. National Research Council, Washington, DC, p. 113. Nagurney, A., Dong, J., 2002. A multiclass, multicriteria traffic network equilibrium model with elastic demand. Transport. Res. B: Meth. 36 (5), 445–469. Ordóñez, F., Stier-Moses, N.E., 2010. Wardrop equilibria with risk-averse users. Transport. Sci. 44 (1), 63–86. Rossi, T.F., McNeil, S., Hendrickson, C., 1989. Entropy model for consistent impact-fee assessment. J. Urban. Plan. D 115 (2), 51–63. Wang, J.Y., Ehrgott, M., Chen, A., 2014. A bi-objective user equilibrium model of travel time reliability in a road network. Transport. Res. B: Meth. 66, 4–15. Wang, J., He, X., Peeta, S., 2016. Sensitivity analysis based approximation models for day-to-day link flow evolution process. Transport. Res. B: Meth. 92, 35– 53. Wardrop, J.G., 1952. Road paper. Some theoretical aspects of road traffic research. P. I. Civil. Eng. 1 (3), 325–362. Watling, D., 2006. User equilibrium traffic network assignment with stochastic travel times and late arrival penalty. Eur. J. Oper. Res. 175 (3), 1539–1556. Wu, X., 2015. Study on mean-standard deviation shortest path problem in stochastic and time-dependent networks: a stochastic dominance based approach. Transport. Res. B: Meth. 80, 275–290. Xu, H., Lou, Y., Yin, Y., Zhou, J., 2011. A prospect-based user equilibrium model with endogenous reference points and its application in congestion pricing. Transport. Res. B: Meth. 45 (2), 311–328. Zhang, C., Chen, X., Sumalee, A., 2011. Robust Wardrop’s user equilibrium assignment under stochastic demand and supply: expected residual minimization approach. Transport. Res. B: Meth. 45 (3), 534–552.
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Contents lists available at ScienceDirect
Transportation Research Part D journal homepage: www.elsevier.com/locate/trd
Multi-criteria user equilibrium model considering travel time, travel time reliability and distance Chao Sun a, Lin Cheng a,⇑, Senlai Zhu b, Fei Han c, Zhaoming Chu d a
School of Transportation, Southeast University, Nanjing 210096, China School of Transportation, Nantong University, Nantong 226019, China c School of Automobile, Chang’an University, Xi’an 710064, China d Road Traffic Safety Research Center of the Ministry of Public Security, Beijing 100062, China b
a r t i c l e
i n f o
Article history: Received 16 August 2016 Accepted 5 March 2017 Available online xxxx Keywords: Multi-criteria user equilibrium Travel time Travel time reliability Travel distance Maximum entropy Partial linearization descent method
a b s t r a c t This paper proposes a multi-criteria user equilibrium model considering travel time, travel time reliability and distance (MUE-TRD). This new model hypothesizes that for each user class and each origin-destination (O-D) pair no traveler can reduce either his or her reliable travel time or travel distance or both without worsening the other objective by unilaterally changing routes in their route choice decision process. Travel time budget which consists of travel time and travel time reliability is used to describe the reliable travel time. A maximum entropy multi-criteria user equilibrium (ME-MUE) model is presented to address the non-uniqueness of the solution in MUE-TRD model. Furthermore, a route-based solution algorithm based on the partial linearization descent method (R-PLD) is developed to solve the ME-MUE model. Numerical examples are also provided to illustrate the essential ideas of the proposed model and the applicability of the developed solution algorithm. The results show that compared to traditional user equilibrium and travel time budget models, ME-MUE model is more consistent with the real trip process that the reliable travel time is increasing with the decreasing of travel distance in used routes; and the road traffic is smoother when using ME-MUE model to design the road network, thus ME-MUE model can reduce road traffic noise and air pollution in the urban road network. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction In transportation systems, travelers may be faced with several criteria to select the routes. Some travelers select routes to minimize their reliable travel time which consists of travel time and travel time reliability according to their past experience, such as the commuters, who travel regularly between the same O-D pair on a daily basis; and some travelers would like to select the routes with minimum travel time or travel distance using the vehicle navigation system (Lima et al., 2016; Wang et al., 2016). Statistical analysis was performed by Abdel-Aty et al. (1995) to examine which route attributes that lead to the choice of a route are considered important by travelers. Three most important factors derived from the analysis are: (1) shorter travel time (40% of respondents); (2) travel time reliability (32% of respondents); and (3) shorter distance (31% of respondents). Due to its theoretical and practical importance, modeling traffic assignment considering these factors is becoming an emerging research subject.
⇑ Corresponding author. E-mail address: [email protected] (L. Cheng). http://dx.doi.org/10.1016/j.trd.2017.03.002 1361-9209/Ó 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Sun, C., et al. Multi-criteria user equilibrium model considering travel time, travel time reliability and distance. Transport. Res. Part D (2017), http://dx.doi.org/10.1016/j.trd.2017.03.002
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C. Sun et al. / Transportation Research Part D xxx (2017) xxx–xxx
For single-objective traffic assignment model, the conventional Wardrop user equilibrium (UE) model (Wardrop, 1952), and the stochastic user equilibrium (SUE) model (Daganzo and Sheffi, 1977), only consider factor 1 (i.e., travel time). The UE model assumes that users try to minimize their travel time, while in the SUE model users are assumed to minimize their perceived travel time, which has a random error component. The traffic equilibrium models under uncertainty consider factors 1 and 2 (i.e., travel time and travel time reliability). To build the traffic equilibrium models under uncertainty, different theories were used, such as travel time budget (TTB) approach (e.g., Lo et al., 2006; Wu, 2015), game theory approach (e.g., Bell and Cassir, 2002), utility-based approach (e.g., Mirchandani and Soroush, 1987), expected residual minimization approach (e.g., Zhang et al., 2011), robust optimization approach (e.g., Ordóñez and Stier-Moses, 2010), prospect theorybased approach (e.g., Xu et al., 2011), schedule delay approach (e.g., Watling, 2006), and mean-excess travel time approach (e.g., Chen and Zhou, 2010). For bi-objective traffic assignment model, Dial (1996, 1997), Nagurney and Dong (2002) and Larsson et al. (2002) propose the bi-objective user equilibrium assignment model based on factors 1 and 3 (i.e., travel time and travel distance which is directly related to vehicle operating cost for the trip). Recently, Wang et al. (2014) and Ehrgott et al. (2015) formulated a general travel time reliability bi-objective user equilibrium model which took factors 1 and 2 (i.e., travel time and travel time reliability) into consideration. This model assumes that users select routes to minimize their travel time and maximize their travel time reliability. The solution of this model is non-unique; and the solution algorithm is not given in their works. Compared to single-objective traffic assignment model, bi-objective traffic assignment model is more realistic. However, it is difficult to find the set of route flow solutions in bi-objective traffic assignment model. Generalized route cost function was used by Dial (1996, 1997) and Larsson et al. (2002) to deal with the bi-objective traffic assignment model. Nagurney and Dong (2002) proposed a weight method in which travelers perceive their generalized cost on a route by travel time and travel distance with different weight. Other bi-objective traffic assignment algorithms (Chen et al., 2010; Chen and Nie, 2013) also combined the two objective functions into a nonlinear generalized cost. All these methods convert the bi-objective problem to single-objective problem using linear combination of two different factors. In fact, combining the two factors into one implicitly assumes the existence of a linear (dis)utility function, and therefore presupposes a certain preference structure. As a result of this, there is the possibility that some reasonable solutions are never considered (Wang et al., 2014). Considering both factors 1, 2 and 3 (i.e., travel time, travel time reliability and travel distance) in the route choice decision process, this paper presents a multi-criteria user equilibrium model considering travel time, travel time reliability and distance (MUE-TRD). This new model hypothesizes that for each user class and each O-D pair no traveler can reduce either his or her reliable travel time or travel distance or both without worsening the other objective by unilaterally changing routes. For computation of the reliable travel time which consists of travel time and travel time reliability, travel time budget (TTB), which is defined as a travel time reliability chance (or on-time arrival) constraint; such that the probability that travel time exceeds the budget is less than a predefined confidence level a to represent the travel time reliability, is used in this paper. For addressing the non-uniqueness of the solution in MUE-TRD model, a maximum entropy multicriteria user equilibrium (ME-MUE) model whose feasible set is the solution set of MUE-TRD model is proposed. This is the first attempt to use the principles of entropy maximizing to deal with the multi-criteria user equilibrium model. A route-based solution algorithm based on the partial linearization descent method (R-PLD) is developed to solve the ME-MUE model. The rest of the paper is organized as follows. In Section 2, MUE-TRD model is presented. In Section 3, ME-MUE model whose feasible set is the solution set of MUE-TRD model is proposed. In Section 4, R-PLD is developed to determine the equilibrium flow pattern. In Section 5, numerical examples are presented to illustrate the essential ideas of the proposed models and the applicability of the solution algorithm. Finally, conclusions are provided. 2. Multi-criteria user equilibrium model This section presents MUE-TRD, the MUE-TRD condition, the existence of solution set and an instance of MUE-TRD model. 2.1. Formulation of MUE-TRD model It is well known from empirical studies that the three most important factors influencing route choice behavior are travel time, travel time reliability and travel distance. Then travelers always would want: (1) to minimize the reliable travel time which consists of travel time and travel time reliability; and (2) to minimize the travel distance. To describe the reliable travel time, travel time budget (TTB) (Lo et al., 2006) which is defined as the average travel time plus a safety margin as an acceptable travel time is used in this paper. Mathematically, the two criteria are: x x minfbk Pr½T x k 6 bk P ag x
min dk 8k 2 K x ; x 2 W
ð1Þ
And the feasible set defined as follows: Please cite this article in press as: Sun, C., et al. Multi-criteria user equilibrium model considering travel time, travel time reliability and distance. Transport. Res. Part D (2017), http://dx.doi.org/10.1016/j.trd.2017.03.002
C. Sun et al. / Transportation Research Part D xxx (2017) xxx–xxx
X
x
f k2K x k;i
¼ qx i 8x 2 W i 2 I
x qx i ¼ li q 8x 2 W i 2 I XX X x x xa ¼ da;k f k;i 8a 2 A
ð2Þ
x k2K x
i
3
x
x
f k;i P 0 8k 2 K x 2 W; i 2 I XX x x da;k da x 2 W dk ¼ x k2K x x
where bk is the travel time budget on route k between O-D pair x at a predefined confidence level a; T x k is the reliable travel x time on route k between O-D pair x; dk is the travel distance on route k between O-D pair x; K x is the set of routes between x O-D pair x; A; W and I are the sets of links, O-Ds and user classes, respectively; f k;i is the route flow of user class i on route k x x between O-D pair x; qi and q are the traffic demand of user class i and total demand between O-D pair x respectively; li is the proportion of class i; xa is the flow on link a; da is the distance on link a; and dx a;k is the route-link incidence parameter whose value is 1 if link a is on route k; 0 otherwise. The travel time budget is the value of cumulative probability under the confidence level of a which represents different classes. As derived in Lo et al. (2006), the route travel time is normally distributed with mean and standard deviation that can be written as follows: 1 x bk ¼ lx k þ U ðai Þrk io Xn x h ð1h1n Þ a lxk ¼ da;k t0a þ bt0a xna cn ð1h a Þð1nÞ
x
a
a
rxk
ð3Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi h i2 P ð1h12n Þ ð1h1n x 2 0 2 2n a a Þ ¼ cn ð1ha Þð1nÞ a da;k b ðt a Þ xa c2n ð1ha Þð12nÞ a
a
1 x where lx k and rk are mean and standard deviation of the route flow; U ðÞ is the inverse of the standard normal cumulative 0 distribution function; ta and C a are the free-flow travel time and capacity on link a, respectively; b and n are the deterministic parameters of bureau of public roads (BPR) (Manual, 2000) link performance function; and cna is the design capacity, while the lower bound capacity is a fraction ha of cna .
2.2. MUE-TRD condition Based on the two criteria functions in Eq. (1), the condition of MUE-TRD can be formulated as follows: Definition 1. The MUE-TRD is a network state such that for each user class and each O-D pair no traveler can reduce either his or her reliable travel time or travel distance or both without worsening the other objective by unilaterally changing routes. The weak condition of MUE-TRD can also be defined as follows: Definition 2. The weak MUE-TRD is a network state such that for each user class and each O-D pair no traveler can reduce both his or her reliable travel time and travel distance by unilaterally changing routes. It can be proved that every solution of MUE-TRD model is also a solution of weak MUE-TRD model, and every solution of the TTB equilibrium model of Lo et al. (2006) is also a solution of MUE-TRD model. Theorem 1. Let F be a route solution to MUE-TRD model. Then F also satisfies the weak MUE-TRD condition. Proof 1. Assume that F does not satisfy the weak MUE-TRD condition. Then, for at least one user class i there must exist two 0 x x x x used route k and k between some O-D pair x such that bk0 ;i < bk;i and dk0 ;i < dk;i . These inequalities contradict the assumption that F satisfies the MUE-TRD condition. h x
x
x
x
The route solution bk0 ;i < bk;i and dk0 ;i ¼ dk;i satisfies the weak MUE-TRD condition while not satisfies the MUE-TRD condition. Because someone will reduce his or her reliable travel time while the travel distance be the same by changing his or her 0 x x x x route from k to k . However, the route solution bk0 ;i ¼ bk;i and dk0 ;i < dk;i is a MUE-TRD model solution. Although someone can 0
reduce his or her travel distance by changing his or her route from k to k , his or her reliable travel time will increase. This is because travel distance is constant, while the reliable travel time (i.e., TTB) is a smooth and convex function (Lo et al., 2006). Theorem 2. Let G be a route solution to TTB model. Then G also satisfies the MUE-TRD condition.
Please cite this article in press as: Sun, C., et al. Multi-criteria user equilibrium model considering travel time, travel time reliability and distance. Transport. Res. Part D (2017), http://dx.doi.org/10.1016/j.trd.2017.03.002
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Proof 2. Assume that G does not satisfy the MUE-TRD condition. Then as described above, for at least one user class i there 0 x x x x must exist two used route k and k between some O-D pair x such that bk0 ;i < bk;i and dk0 ;i 6 dk;i . The first of these inequalities contradicts the assumption that G satisfies the TTB equilibrium condition. h Using this theorem, that G is also a route solution to user equilibrium (UE) model when the confidence level a is equal to 0.5 can be deduced. Under the assumptions that the TTB on each route between each O-D pair for each user class is positive and continuous function of flow, MUE-TRD flow exist. And the qualitative property of solution set is present. Theorem 3. The MUE-TRD flow exists under the assumptions that the TTB on each route between each O-D pair for each user class is positive and continuous with respect to its route flow. And the solution set XBUE-TD of MUE-TRD model is compact.
x
x
Proof 3. Because of the assumption that bk;i is positive and continuous functions of flow, and dk;i is constant, it is known that x
x
x
Gk;i ðfÞ :¼ bk;i þ kdk;i for positive k is positive and continuous. Hence an equilibrium flow f with respect to Gx k;i exists. It can be
0
shown that this equilibrium flow f is a MUE-TRD flow. Assume to the contrary that there exist two used route k and k x x x x x between some O-D pair x such that bk0 ;i < bk;i and dk0 ;i 6 dk;i . Then Gx k0 ;i ðf Þ < Gk;i ðf Þ contradicting the fact that f is an equilibrium flow with respect to Gx k;i . Since the feasible set (i.e., Eq. (2)) is bounded. Thus the solution set XBUE-TD of MUE-TRD model is bounded. According to Ehrgott and Wiecek (2005), the solution set XBUE-TD is closed. Following the Heine-Borel theorem, it has: the solution set XBUE-TD of MUE-TRD model is compact. h
2.3. Instance of MUE-TRD model The network topology and characteristics of the test network are depicted in Fig. 1. The network consists of 2 nodes, 3 links and 1 O-D pair. The route flows are assumed to follow the normal distribution, and the numerical characteristics are depicted in Fig. 1. And the distances of each route are 5, 8 and 9. The equilibrium traffic flow set of MUE-TRD model with the confidence level of 80% and the equilibrium traffic flow of TTB model with the confidence level ranging from 50% to 95% are provided in Fig. 2. It shows that (1) as expected, the TTB model solution with a ¼ 80% satisfy the MUE-TRD conditions. And the TTB solution with a ¼ 80% is on the boundary of MUE-TRD solution set. This is because at this traffic state, the used routes have the same reliable travel time; and it is a limiting case of the solution set. (2) the solution set of MUE-TRD model is bounded and closed; and (3) with the increase of confidence level, the flow on route 3 is always increasing, and the equilibrium traffic time on used routes increase from 9.31 to 11.17. This means that more risk-averse travelers will spend a larger reliable travel time to ensure more on-time arrivals. 3. Maximum entropy multi-criteria user equilibrium model This section proposes the maximum entropy multi-criteria user equilibrium (ME-MUE) model, the existence of solution set and an instance of ME-MUE model. 3.1. Formulation of ME-MUE model It is known that the route solutions to MUE-TRD model proposed in Section 2 are not unique; therefore, an additional behavioral assumption is required to determine the most likely traffic flows among these solutions. The principles of entropy maximizing were used by Rossi et al. (1989), Bar-Gera (2006) and Kumar and Peeta (2015) to generating a single route flow solution under the UE assumption. And it seems the most likely and meaningful route flow solution in reality. In this section, a ME-MUE model whose feasible set is the solution set of MUE-TRD model was proposed.
∼ ∼ ∼ Fig. 1. Network topology and characteristics.
Please cite this article in press as: Sun, C., et al. Multi-criteria user equilibrium model considering travel time, travel time reliability and distance. Transport. Res. Part D (2017), http://dx.doi.org/10.1016/j.trd.2017.03.002
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TTB Solutions of Diffient Efficient Solutions
Flow of Route 3
2.5 2
=0.95
f2
1.5 1 0.5
=0.8, f1 1.69 6.028, f3 2.282
=0.5
0 10
10
5
Flow of Route 2
5 0
0
Flow of Route 1
Fig. 2. Solution set of MUE-TRD model and solutions of TTB model.
Given the route solution set XMUE-TRD of MUE-TRD model, the ME-MUE route flow solution on general road networks is obtained by solving the following nonlinear program:
max EðfÞ ¼ Px
qx ! Pk f x k !
ð4Þ
f 2 XMUE-TRD For minifying the value of objective function, a constant qx can be input to the objective function Px
qx !=qx
x Pk f x k !=q
without
altering the solution of Eq. (4). And it is convenient to maximize the natural logarithmic form of the objective function rather P h qx ! P f xk !i . Since qx is fixed, the first term can be x ln qx k ln qx
than the original one. Then, the objective function becomes
x
dropped from the objective function. And assume that O-D flows qx and route flows f k are usually large, Stirling’s approx i P hP f xk : fx x k imation of ln x!¼x ln x x is used to simplify the objective function x k qx ln qx 1 . Constant q could be dropped without altering the solution. Then, the original issue of ME-MUE route flow solution can be transformed to:
XX x f x max EðfÞ ¼ f k ln qkx x
k
ð5Þ
f 2 XMUETRD The existence of the solution to ME-MUE model is shown as below: Theorem 4. The ME-MUE flow of Eq. (5) exists.
Proof 4. The feasible set XMUE-TRD is non-empty, closed. Moreover, the objective function is positive and continuous with respect to f. Then, the ME-MUE flow of Eq. (5) exists. h It is necessary to point out that the uniqueness of the ME-MUE equilibrium route flow solution cannot be guaranteed. In order to prove the uniqueness, convexity of feasible set XMUE-TRD is required. However, it is difficult to guarantee the convexity of feasible set in general. 3.2. Instance of ME-MUE model The network topology and characteristics of the test network are depicted in Fig. 3. The network consists of 2 O-D pairs and 4 routes. The MUE-TRD model is assumed to have 3 route solutions depicted in Fig. 3. Since each route flow solution is the aggregate outcome of many individual route choice decisions, Eq. (4) is used to find all possible route choice decisions shown in Table 1. It shows that, there are 34 route choice decisions those 4 individual travelers from A-B and 3 from C-D. For the 1th route flow solution, there can be 4 route choice decisions that one individual traveler chooses route 1 and the other 3 individual travelers choose route 2 for O-D pair A; B. And there can be 1 route choice decision that all 3 individual travelers choose route 1 for O-D pair C; D. It is assumed that the occurrence of each route choice decision is equal in the entropy formulation; thus, in this small example, each route choice decision is responsible for a 1/34 probability of occurring. The most likely MUE-TRD route flow solution is the one associated with the largest number of route choice decisions (i.e., maximum entropy). Among all 3 possible MUE-TRD route flow solutions, the 2nd solution, which accounts for 18 out of 34 route choice decisions or a 52.94% probability of occurring, is clearly the most likely one. Please cite this article in press as: Sun, C., et al. Multi-criteria user equilibrium model considering travel time, travel time reliability and distance. Transport. Res. Part D (2017), http://dx.doi.org/10.1016/j.trd.2017.03.002
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f1A, B
A
B
C
D
f f f
f1A, B
f 2A, B
f1C , D
f 2C , D
Solution 1
1
3
3
0
Solution 2
2
2
1
2
Solution 3
3
1
2
1
BUE TD
A, B 2 C ,D 1
C ,D 2
Fig. 3. Network topology and route solutions.
Table 1 All possible route choice decisions associated with the flow solution shown in Fig. 3. Route choice made by individual travelers
From origin A to destination B Traveler 1
Traveler 2
Traveler 3
Traveler 4
Traveler 1
Traveler 2
Traveler 3
RA;B 2 RA;B 2 RA;B 1 RA;B 2
RA;B 2 RA;B 2 RA;B 2 RA;B 1
RC;D 1 RC;D 1 RC;D 1 RC;D 1
RC;D 1 RC;D 1 RC;D 1 RC;D 1
RC;D 1 RC;D 1 RC;D 1 RC;D 1
RA;B 2
RC;D .. 1 . RC;D 2
RC;D 2
RC;D 2
RC;D 2
RC;D 1
RC;D .. 1 . RC;D 2
RC;D 1
RC;D 2
RC;D 1
RC;D 1
1 2 3 4
The 1th route flow solution RA;B RA;B 1 2 RA;B RA;B 2 1 A;B R2 RA;B 2 A;B RA;B R 2 2
5 .. . 22
The 2nd route flow solution RA;B RA;B RA;B 1 2 .. 1 . A;B A;B R2 RA;B R2 1
23 .. . 34
The 3rd route flow solution RA;B RA;B 1 .. 1 . A;B R RA;B 2 1 RA;B 1
From origin C to destination D
RA;B 1
RA;B 1
RA;B 2
RA;B 1
RA;B 1
RA;B 2
Note: = this traveler chooses route 1 between O-D pair A; B; = this traveler chooses route 2 between O-D pair A; B; 1 between O-D pair C; D; RC;D 2 = this traveler chooses route 2 between O-D pair C; D.
RC;D 1
= this traveler chooses route
4. Solution algorithm The reliable travel time (i.e., TTB) in the proposed model is non-additive because it is not possible to decompose the route TTB into the sum of link-based generalized costs. And the route solution set XBUE-TD of MUE-TRD model cannot be expressed using mathematical formulations, because of the complexity of multi-criteria programming. Therefore, a route-based solution algorithm based on the partial linearization descent method (R-PLD) is developed to solve the ME-MUE model as follows. Step 1. (Initialization) Set initial iteration number n ¼ 0, tolerance error e for check convergence, tolerance error e0 for search step size, the working route set K, and initial traffic route flow x x x x qx if d ¼ d where d ¼ min fd g ð1Þ ð0Þ xð0Þ k k k i f ¼ f : f k;i ¼ . 0 otherwise Step 2. (Search Direction and Step Size) For each O-D pair and each user class, choose any two routes ka and kb (i.e., it P P xðnÞ . x ðnÞ needs to choose C 2x;k ¼ kðk 1Þ=2 times); the search direction is set as: tk;i;a;b ¼ 2 ð1; 1ÞT ; and i f k;i;a ðnÞ i f k;i;b ðnÞ
using bisection method to compute the step size ak;i;a;b under the tolerance error e0 . ! ! x xðnþ1Þ f k;i;a ðnÞ f k;i;a ðnÞ ðnÞ Step 3. (Update Traffic Flow) Update the traffic flow ¼ þ ak;i;a;b tk;i;a;b . xðnÞ xðnþ1Þ f k;i;b f k;i;b Step 4. (Check Convergence) If ðkf f
ðnþ2Þ
¼f
ðnþ1Þ
ðnþ1Þ
f
ðnÞ
kÞ=kf
ðnÞ
k 6 e, then stop, f
ðnþ1Þ
is the optimal solutions. Otherwise, set
; n :¼ n þ 1, go to Step 2.
For any two routes ka and kb in an O-D pair, it can be proved that the entropy value reach the maximum when the two route flows are the same using Lagrangian multiplier method. Therefore, the descent direction is the half of difference between the high route flow and the low route flow. For the line search, the classical bisection method is considered to find an approximate step size. The convergence and applicability of R-PLD algorithm are presented using practice instances next section. 5. Numerical examples In this section, a small network is presented to illustrate the essential ideas of the proposed model, and a medium network is presented to illustrate the applicability of the proposed solution algorithm. Please cite this article in press as: Sun, C., et al. Multi-criteria user equilibrium model considering travel time, travel time reliability and distance. Transport. Res. Part D (2017), http://dx.doi.org/10.1016/j.trd.2017.03.002
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C. Sun et al. / Transportation Research Part D xxx (2017) xxx–xxx
5.1. Small network The network topology and characteristics of the test network are depicted in Fig. 4. The network consists of 13 nodes, 19 links, and 4 O-D pairs. The commonly used BPR performance function (b ¼ 0:15, n ¼ 4, Manual, 2000) is adopted. The value of ha in TTB model is equal to 0.3. The free-flow travel time is used as the travel distance for convenience. Link and traffic demand characteristics are provided in Fig. 4. Travelers are partitioned into two classes with the confidence level 50% and 80% whose proportions of demand are 40% and 60%, respectively. The algorithm is coded in Matlab R2015b and tested on a personal computer with Intel(R) Core(TM) i7-5600U CPU 2.60 GHz, 8 GB memory. First, the iterative processes of entropy value of the R-PLD algorithm are shown in Fig. 5. It can be seen that (1) the entropy value can rapidly converge to 2147 with 0.489 s CPU time; (2) the entropy value fluctuates with the characteristics of jump. This is because the jump of transferred traffic flow using bisection method to compute the step size; and (3) the entropy value fluctuates sharply in the middle iterations. This means that a large number of traffic flows transferred at this time. Then, the validity of the solution obtained from the R-PLD algorithm is examined. The equilibrium results of the proposed model are provided in Table 2. As expected, the equilibrium results satisfy the MUE-TRD conditions. Without loss of generality, the equilibrium results in O-D pair (4-2) are shown. For the used routes (i.e., route 15, 16 and 18), the shortest route 15 has the most reliable travel time for all user classes; the longest route 18 has the least reliable travel time for all user classes; and the distance and reliable travel time of route 16 are in the middle. For unused routes (i.e., route 17 and 19), they will not be considered, as they are dominated by route 18, which has no worse distance and reliable travel time. On the other hand, the equilibrium results satisfy conservation constraint. The total demands of class 1 and class 2 in O-D pair (4-2) are 240 and 360, respectively. Table 2 shows that (1) some used routes has the same reliable travel time in the same O-D pair. Because the route flow is on the boundary of MUE-TRD solution set at the process of searching the maximum entropy value; (2) route reliable travel time is increasing with the confidence level. This is because conservative travelers will over estimate their reliable travel time; and (3) compared to traditional UE model and TTB model, ME-MUE model has more used routes, and the reliable travel time is increasing with the decreasing of travel distance in used routes, this result is more consistent with the real trip process.
1 200
1
8/700
7/300
7/900
800
9/700
4
14/900 O 1 4
D
2 400 600
5
5/800
9/600
3 800 200
9
12 15/700
6
13/500 10/700
5/900
7
5/300
9/400
10
8/700
11
8
10/700 9/700
2
1 000
8/700
9/600
Free-flow travel time / Design Capacity 13
11/700
3
1 000
Fig. 4. Network topology, O-Ds and link characteristics.
2500
Entropy Value
2000
time 0.489 entropy 2147
1500 1000 500 0
0
0.2
0.4
0.6
0.8
1
CPU Time (sec) Fig. 5. Iterative processes of entropy value.
Please cite this article in press as: Sun, C., et al. Multi-criteria user equilibrium model considering travel time, travel time reliability and distance. Transport. Res. Part D (2017), http://dx.doi.org/10.1016/j.trd.2017.03.002
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Table 2 Equilibrium results of ME-MUE, UE and TTB models. O-D
Route No.
Sequence of nodes
Distance
Flow of class 1
Flow of class 2
Time of class 1
Time of class 2
UE
TTB
1-2
1 2 3 4 5 6 7 8
1-12-8-2 1-5-6-7-8-2 1-5-6-7-11-2 1-5-6-10-11-2 1-5-9-10-11-2 1-12-6-7-8-2 1-12-6-7-11-2 1-12-6-10-11-2
33 32 35 42 43 35 38 45
121.90 7.64 – 8.57 9.46 – – 12.43
213.87 8.66 1.40 4.80 6.05 – 2.22 3.00
62.10 81.39 65.70 62.10 61.31 87.78 62.10 58.49
89.95 100.26 85.42 75.53 73.14 125.44 79.30 67.85
400 – – – – – – –
400 – – – – – – –
1-3
9 10 11 12 13 14
1-5-6-7-11-3 1-5-6-10-11-3 1-5-9-10-11-3 1-12-6-7-11-3 1-12-6-10-11-3 1-5-9-13-3
34 41 42 37 44 36
45.49 43.66 29.11 63.09 28.82 109.83
159.57 24.03 40.15 6.62 47.03 202.60
69.00 65.39 64.61 65.39 61.78 65.39
89.79 80.36 78.15 83.81 73.23 84.53
429 – – – – 371
339 10 – – 124 327
4-2
15 16 17 18 19
4-5-6-7-8-2 4-5-6-7-11-2 4-5-6-10-11-2 4-9-10-11-2 4-5-9-10-11-2
34 37 44 41 45
109.20 36.00 – 94.80 –
209.80 25.00 – 125.20 –
68.77 63.08 59.47 49.94 58.69
96.84 81.18 70.39 56.76 67.56
408 25 – 167 –
193 – 55 352 –
4-3
20 21 22 23 24 25
4-5-6-7-11-3 4-5-6-10-11-3 4-5-9-10-11-3 4-9-13-3 4-5-9-13-3 4-9-10-11-3
36 43 44 34 38 40
– – – 70.00 – 10.00
– – – 116.00 – 4.00
66.38 62.77 61.99 53.23 62.77 53.23
85.64 75.52 73.03 70.54 80.23 62.71
– – – 200 – –
– – – 175 – 25
5.2. Sioux Falls network The well-known Sioux Falls (Leblanc, 1973) network is used to illustrate the applicability of the proposed algorithm. The network consists of 24 nodes, 76 links, and 550 O-D pairs. A behaviorally generated working route set which has 3441 routes from Bekhor et al. (2008) is used. The values of b, n in BPR performance function, travel distance, ha and classification of users are similar to the small network. Initially, the convergence curve and entropy value of R-PLD algorithm is shown in Fig. 6. The green line is the iterative error curve; it can be seen that after 13 iterations, the error reach to 0 within 93.3 s CPU time. Because of the entropy value is increasing until reach the maximum value (i.e., the search direction is always descending), the error monotonically approaches to zero. The magenta line is the entropy curve; it also illustrates the R-PLD algorithm can rapidly converge to maximum entropy. To further demonstrate the convergence characteristics of the R-PLD algorithm, the evolution processes of reliable travel time and route flow of different user classes are examined. Without loss of generality, Fig. 7 only shows the results on 3 routes (and the distances are 21, 14 and17, respectively) in O-D pair (16-18) where the demand is 200. As expected, the reliable travel time and route flow can rapidly converge to equilibrium results. The equilibrium results satisfy the MUE-TRD conditions: the reliable travel time will increase while some users change his or her route from 2159 to others; the travel
1
3E4
Error Value
1E-4
time 93.30 error 0
Entropy Value
time 93.30 entropy 27788
1E-2
1E-6
Iterative Error Curve Entropy Curve
1E-7 0
0
20
40
60
80
100
2E4
1E4
0
CPU Time (sec) Fig. 6. Convergence curve and entropy value of R-PLD algorithm.
Please cite this article in press as: Sun, C., et al. Multi-criteria user equilibrium model considering travel time, travel time reliability and distance. Transport. Res. Part D (2017), http://dx.doi.org/10.1016/j.trd.2017.03.002
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C. Sun et al. / Transportation Research Part D xxx (2017) xxx–xxx
250 200 150 100 50 0
120
Route 2159 Route 2160 Route 2161
40
100
Route Flow
Route Travel Time
300
20
80
0
60
6.85
6.86
6.87
40 20
0
5
10
15
CPU Time (sec)
20
0
0
5
10
15
20
CPU Time (sec)
Fig. 7. Evolution processes of reliable travel time and route flow. (Note: solid lines, dotted lines represent class 1 and class 2, respectively; the convergence values from top to bottom are class 1 on route 2159, route 2160, route 2161, class 2 on route 2159, route 2160, route 2161).
distance will increase while some users change his or her route from 2160 to others; and either the reliable travel time or distance will increase while some users change his or her route from 2161 to others. The right figure shows that at one iteration, route 2159 and 2160 make a flow transferred for each class, as well as route 2159 and 2161, 2160 and 2161. Overall, at every iteration process of each O-D pair each user class, any two routes have made a flow transferred using R-PLD algorithm. 6. Conclusions This paper proposed MUE-TRD, a new model that assumes travelers minimize their reliable travel time (including travel time and travel time reliability) and their travel distance. The proposed model was then formulated as MUE-TRD conditions, by which, for each user class and each O-D pair no traveler can reduce either his or her reliable travel time or travel distance or both without worsening the other objective by unilaterally changing routes. For obtaining the most likely traffic flows among all the solutions of UE-TRD model, a ME-MUE model whose feasible set is the solution set of MUE-TRD model was presented. Among all MUE-TRD route flow solutions, the entropy-maximizing route flow vector was considered to be the most likely one in reality. Furthermore, a route-based solution algorithm based on the R-PLD was developed to solve the ME-MUE model. In the algorithm, the descent direction was determined according to the difference between the high route flow and the low route flow for any two routes in an O-D pair, and the step size was obtained by a classical bisection method. The essential ideas of the proposed model and the applicability of the R-PLD algorithm were tested in both a small network and the Sioux Falls network. The analysis results indicated that (1) considering both travel time, travel time reliability and travel distance has a significant effect on route choice process for the travelers with different preferences; (2) compared to traditional UE model and TTB model, ME-MUE model has more used routes, and the reliable travel time is increasing with the decreasing of travel distance in used routes; (3) the R-PLD algorithm is an effective method for solving the proposed MEMUE model; and (4) the road traffic is smoother which can reduce road traffic noise and air pollution when using ME-MUE model to design the urban road network. In future research, the stochastic perception error ignored by this paper should be incorporated in the ME-MUE model. For the solution algorithm, embedding a column generation scheme within the R-PLD for testing large networks is needed for real applications of the proposed model. For designing a more reliable network, it is worthwhile to extend the proposed model to the network design problem. Acknowledgements The authors are grateful to the referees for their constructive comments and suggestions to improve the quality and clarity of the paper in 7th International Conference on Green Intelligent Transportation System and Safety. This research was supported by the National Natural Science Foundation of China (No. 51578150, 51378119, and 51608115), the Scientific Research Foundation of Graduate School of Southeast University (No. YBJJ1679), the Natural Science Foundation of Jiangsu Province (No. BK20150613), and the China Scholarship Council (CSC). References Abdel-Aty, M.A., Kitamura, R., Jovanis, P.P., 1995. Investigating effect of travel time variability on route choice using repeated-measurement stated preference data. Transport. Res. Rec. 1493, 39–45. Bar-Gera, H., 2006. Primal method for determining the most likely route flows in large road networks. Transport. Sci. 40 (3), 269–286.
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Please cite this article in press as: Sun, C., et al. Multi-criteria user equilibrium model considering travel time, travel time reliability and distance. Transport. Res. Part D (2017), http://dx.doi.org/10.1016/j.trd.2017.03.002