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Worst-case traffic assignment model for mixed traffic flow of human-driven vehicles and
connected and autonomous vehicles by factoring in the uncertain link capacity
Jian Wang a, Wei Wang a,
1
, Gang Ren a, Min Yang a
aJiangsu Key Laboratory of Urban ITS, Jiangsu Province Collaborative Innovation Center of Modern Urban Traffic
Technologies, School of Transportation, Southeast University, Nanjing, 211189, China
Abstract: Connected and autonomous vehicles (CAVs) can form platoons to reduce the time headway
and improve the link capacity. However, in a mixed traffic flow environment where both human-driven
vehicles (HDVs) and CAVs exist, the platoon intensity is significantly impacted by the stochastic order
of the HDVs and CAVs (i.e., the fleet sequence). Therefore, the link capacity involves a large uncertainty
even under the same HDV and CAV flow. This uncertain link capacity can cause a large variation in
network flow. In the literature, traffic assignment models for mixed traffic flows of HDVs and CAVs are
developed based on expected link capacity models, in which the computed link capacity is deterministic
for given HDV and CAV flows. These models ignore the impacts of uncertain link capacity on the
network performance and network flow distribution, which can dramatically reduce the effectiveness of
the corresponding planning strategies. To address this problem, this study proposes a worst-case mixed
traffic assignment model. It aims to compute the worst network performance and corresponding
equilibrium flow that may occur due to uncertain link capacity. The worst-case mixed traffic assignment
is formulated as a bilevel programming problem, where the low-level problem is a variational inequality
problem presented to compute the equilibrium results based on a fixed link capacity while the upper-
level problem is to find the optimal input for all link capacities within their ranges to minimize the
network performance. The partition-based norm relaxed method of the feasible direction solution
algorithm is proposed to solve the bilevel worst-case mixed traffic assignment problem. A numerical
application shows that the uncertain link capacity has drastic effects on the network flows and network
performance, and the proposed algorithm can effectively and efficiently solve the bilevel worst-case
mixed traffic assignment problem to compute the worst-case network equilibrium flows and network
performance. These results can help traffic managers design robust planning strategies to ensure a
minimum level of network performance under the impacts of uncertain link capacity.
Keywords: Connected and autonomous vehicles; Mixed traffic assignment model; Bilevel programming
problem; Sensitivity analysis
1. Introduction
Connected and autonomous vehicles (CAVs) are a transformative technology that can dramatically
change travel in the future. They have great potential to improve the mobility of people and goods due
to their characteristics. First, a CAV can detect the surrounding environment using onboard sensors and
can process driving-related information much faster than human beings because it does not have a human
reaction time. It can follow a predecessor vehicle with a smaller time headway than human-driven
vehicles (HDVs). Furthermore, the time headway between CAVs can be reduced if they form a platoon
to drive cooperatively with each other (Wang et al., 2019b). Therefore, the presence of CAVs in mixed
traffic can significantly improve the link capacity. Second, CAVs can access real-time information to
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Corresponding author
Email address: jianw@seu.edu.cn (J. Wang); wangwei@seu.edu.cn (W. Wang); rengang@seu.edu.cn; yangmin@seu.edu.cn;
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help travelers make informed route decisions.
These new characteristics of CAVs will dramatically reshape mixed traffic flow that contains both
HDVs and CAVs. The traveler’s travel behavior and the properties of traffic flow, such as link capacity
and link travel times, are also subject to significant changes. Therefore, the travel demand and network
flow pattern in a mixed traffic flow environment can be very different from those in a pure HDV traffic
flow environment. The existing network equilibrium traffic assignment methods developed on the basis
of pure HDV flow are not able to reproduce mixed network flows due to the new characteristics of the
mixed traffic flow. To design effective planning strategies to improve network performance in a mixed
traffic environment, traffic assignment models for mixed traffic flow have received extensive attention
from researchers in the past few years.
Chen et al. (2016) proposed a mixed traffic assignment model in which the user equilibrium (UE)
principle is used to characterize the route choice behavior of both HDVs and CAVs. They assume that
the link capacity in the context of pure CAV flow can be dramatically increased but remains unchanged
in the context of mixed traffic flow. Levin and Boyles address this gap by proposing a UE-UE mixed
traffic assignment model that factors in the impacts of the market penetration rate of CAVs on link
capacity and link travel time. Liu and Song (2019) proposed an improved UE-UE model considering the
impacts of both the market penetration rate of CAVs and the heterogeneous time headway of different
vehicles on link capacity and network flow distribution. By assuming that the CAV route choice can be
fully controlled, Li et al. (2018) proposed a new mixed traffic assignment model that characterizes the
route choice behavior of HDVs and CAVs using the logit-based stochastic user equilibrium (SUE) model
and system optimal model, respectively. To increase the realism of behavior in mixed traffic flow
modeling, Wang et al. (2019b) proposed a variational inequality-based traffic assignment problem. It
characterizes the route choice behavior of HDVs and CAVs with a cross-nested logit model and UE
model, respectively. Wang et al. (2021) further extended this model by considering demand elasticity.
These studies all used the Bureau of Public Roads (BPR) function to model the link travel time. The
new characteristics of mixed traffic flow are captured through the incorporated link capacity models.
Most of these link capacity models can effectively characterize the impacts of multiple factors on link
capacities, such as the market penetration rate of CAVs and the different time headways of HDVs and
CAVs (Liu and Song, 2019; Wang et al., 2019b; Wang et al., 2021). However, they are deterministic in
the sense that the computed link capacity is fixed for given HDV and CAV link flows. These models
neglect the important fact that rather than being fixed, the link capacity in the context of mixed traffic
flow changes with respect to the CAV platoon intensity, which is determined by the stochastic order of
HDVs and CAVs in the fleet (Ghiasi et al., 2017). Specifically, the mixed traffic flow contains four pairs
of vehicles: an HDV following a CAV, an HDV following an HDV, a CAV following an HDV, and a
CAV following a CAV. Note that a CAV can follow an HDV with a smaller time headway than an HDV
can, and the time headway can be much smaller if it follows a CAV due to platooning. Therefore, the
order of HDVs and CAVs in the flow significantly impacts the platoon intensity, the mean time headway,
and the corresponding link capacity. As the order of HDVs and CAVs in the traffic flow is stochastic,
the link capacity involves a large uncertainty. Ghiasi et al. (2017) showed analytically that depending on
the time headway between CAVs, the difference between the lower bound and upper bound of the link
capacity can be as large as 40% for given traffic conditions.
Due to the uncertain link capacity, the link travel time also varies dramatically even if the flow of
HDVs and CAVs on the link is the same, which impacts the travelers’ route choices and corresponding
network flow. Therefore, the uncertain link capacity can incur a large variation in equilibrium network
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flows. However, mixed traffic assignment methods predict only the equilibrium network flow
corresponding to the expected link capacity (ELC). Due to the uncertain link capacity, large errors may
exist between the equilibrium network flows predicted by these methods and the actual network flows.
Therefore, the traffic planning strategies (e.g., the tolling strategy and autonomous vehicle dedicated lane
(AVDL) deployment strategy) designed based on these mixed traffic assignment methods with ELC may
be too optimistic so that they cannot actually improve the network performance.
To address this problem, this study proposes a worst-case mixed traffic assignment model. It aims to
find the worst network performance and corresponding equilibrium flows due to uncertain link capacity.
The worst-case mixed traffic assignment model can be leveraged by traffic managers to design robust
traffic planning strategies to ensure a minimum level of network performance under the impacts of
uncertain link capacity.
To design the worst-case mixed traffic assignment model, this study quantitatively analyzes the
impacts of the orders of HDVs and CAVs on link capacity and analytically formulates the upper and
lower bounds of the link capacity for given HDV and CAV flows. The worst-case mixed traffic
assignment model is then designed as a bilevel programming problem. The low-level problem is a traffic
assignment model presented to compute the equilibrium flow and network performance for given inputs
of all link capacities under a mixed traffic flow environment. The upper-level problem is to find the
optimal (worst) inputs for all link capacities with their ranges to minimize the network performance. The
output of the worst-case mixed traffic assignment model includes the worst-case inputs of all link
capacities, the worst network performance, and the worst-case equilibrium flows.
Note that depending on the relationship between the HDV and CAV link flows, the lower bound of
the capacity of each link contains two functional forms. The proposed bilevel worst-case mixed traffic
assignment problem cannot be solved using traditional solution algorithms. To address this problem, this
study proposes a partition-based norm relaxed method of feasible direction (NRMFD) solution algorithm.
In each iteration, this algorithm partitions the feasible set of all link capacities into several subsets based
on the HDV and CAV link flows. It can find a descent direction and a corresponding step size to minimize
the objective function at each iteration. We analytically show that this algorithm is globally convergent.
To enable the application of the partition-based NRMFD algorithm, the analytical sensitivity analysis
method for the low-level mixed traffic assignment problem with fixed link capacity is also discussed. A
numerical application indicates that the partition-based NRMFD algorithm can effectively and efficiently
solve the bilevel worst-case mixed traffic assignment problem.
The structures of this study are as follows: Following the introduction section, we discuss link
capacity uncertainty in mixed traffic flow environments. Section 3 formulates the bilevel worst-case
mixed traffic assignment problem after a discussion of its motivation. Section 4 presents the partition-
based NRMFD solution algorithm to solve the proposed bilevel worst-case mixed traffic assignment
problem. Section 5 numerically applies the proposed methods to two networks to compute the worst-
case inputs of all link capacities, the worst network performance, and the corresponding network
equilibrium flows. Section 5 concludes this study and indicates future research directions.
2. Link capacity uncertainty analysis in mixed traffic flow environments
As mentioned previously, a mixed traffic flow contains four types of vehicle pairs: a CAV following
a CAV, a CAV following an HDV, an HDV following a CAV and an HDV following an HDV. Suppose
the time headway between each pair of vehicles is homogeneous. Let and be the time
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headways for the four pairs of vehicles, respectively. According to Liu and Song (2020), it is safe to
assume that an HDV follows an HDV with the same time headway as when it follows a CAV. Note that
compared to an HDV, a CAV can follow an HDV with a smaller time headway, and the time headway
can be much smaller if a CAV follows a CAV due to platooning. Therefore, .
For simplicity, let .
The order of the HDVs and CAVs in a mixed traffic flow dramatically impacts the CAV platoon
intensity and the corresponding link capacity. As the order of HDVs and CAVs is stochastic, the link
capacity and the corresponding link travel time involve great uncertainty. To show this clearly, suppose
link contains only one lane. Figures 1 and 2 demonstrate two scenarios in which the link capacity is
the largest and the smallest, respectively.
In Figure 1, the CAVs and HDVs on link are separated. In this scenario, all CAVs form a long
platoon and drive cooperatively on the road. This scenario minimizes the mean time headway between
vehicles for given traffic flows. Therefore, the capacity under this scenario is maximal. Let and
be the HDV flow and CAV flow, respectively, on this link. Then, the mean time headway (denoted as
) is
(1)
According to Eq. (1), the upper bound of the link capacity (denoted as ) for a given HDV flow
(i.e., ) and CAV link flow (i.e., ) can be characterized as
(2)
Figure 1. The scenario in which the link capacity is largest
Figure 2. The scenario in which the link capacity is minimal
Figure 2 shows another scenario, in which each pair of adjacent CAVs is separated by an HDV; i.e.,
the HDVs are positioned to prevent the formation of a CAV platoon. Clearly, this scenario maximizes
the mean time headway between vehicles. The link capacity under this scenario is minimal. The mean
time headway (denoted as ) in this scenario is
(3a)
CAV HDV
CAV HDV
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(3b)
It should be noted that is continuous at the transition point . Correspondingly, the lower
bound of the link capacity is
(4a)
(4b)
According to Eq. (4), is continuous at the point .
The two scenarios shown in Figures 1 and 2 indicate that the order of HDVs and CAVs in the flow
dramatically impacts the link capacity. The link capacity can change dynamically between and
. This characteristic will result in high uncertainty in the link travel time.
It should be noted that the link capacity varies in both mixed traffic flow and pure HDV flow
environments. Nie (2011) pointed out that the link capacity in a pure HDV environment is random due
to the influence of exogenous (e.g., weather) and endogenous (e.g., traffic breakdown) random factors.
Chen and Zhou (2010) summarized the sources of uncertainty that contribute to link capacity and travel
time variability in a pure HDV flow environment.
In addition to the upper bound and lower bound of the link capacity, we can compute the mean
(expected) link capacity. Let be the ratio of CAVs in the flow,
. Then, each
vehicle in the mixed traffic flow is a CAV with probability and is an HDV with probability .
For a pair of vehicles, they are a CAV following a CAV with a probability of , a CAV following an
HDV with a probability of , an HDV following a CAV with a probability of , and
an HDV following an HDV with a probability of . Therefore, the expected (mean) time
headway among all orders of HDVs and CAVs in the flow is
(5a)
and the expected link capacity is
(5b)
3. Worst-case mixed traffic assignment model considering uncertain link capacity
3.1 Motivation for the worst-case mixed traffic assignment model
Most of the mixed traffic assignment models proposed in the literature use the BPR function and the
ELC to characterize the link travel time (e.g., Wang et al., 2019b; Liu and Song, 2020; Levin and Boyles,
2015). As a result, the estimated link travel times in these models are deterministic for given HDV and
CAV link flows. These studies overlook the impacts of the uncertain link capacity on link travel time
and corresponding equilibrium network flows. The predicted network performance may be too optimistic,
and the traffic planning strategies (e.g., signal optimization, toll optimization, and AVDL deployment
optimization) developed based on these results can be inefficient.
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To improve the robustness of the traffic planning strategies against uncertain link capacity in a mixed
traffic flow environment, robust network design problems can be developed to improve the network
performance in the worst case, where the network performance is minimized by the link capacities.
Let be the vector of the capacities of all links in the network and be the decision variables (e.g.,
signal splits, link tolls, and location of AVDLs), which will be controlled to improve the network
performance. Let be the network performance indicator, e.g., the total travel time, where
is the equilibrium link flow for given and . Denote as the range of all link capacities at
the equilibrium states of the mixed traffic and as the set of constraints for the decision variables .The
robust network design problem can be formulated as
(6)
In problem (6), the term
finds the input of the link capacity vector within its
range such that the network performance is worst for a given . Problem (6) is to optimize the decision
variables to improve the network performance in the worst case due to uncertain link capacity. Note
that the impacts of uncertain link capacity on equilibrium network flow and network performance are
fully factored into problem (6). The optimal strategies computed based on problem (6) can effectively
improve the robustness of the network performance in a mixed traffic flow environment.
The robust network design problem has been extensively used in the literature to find the optimal
planning strategies to improve the robustness of network performance against uncertain network flows.
For example, Ban et al. (2013) proposed a robust toll pricing problem developed based on the UE
problem with multiple equilibrium solutions due to pseudo-monotonicity of traffic impedance function.
The robust toll pricing problem seeks to optimize the tolling strategies to minimize the maximum (i.e.,
worst-case) total system travel time (TSTT) with uncertain network flows. Similarly, Di et al. (2016)
proposed a robust toll pricing problem to improve the network performance in the worst case due to
uncertain network flows induced by bounded rational route choice behavior. Wang and Szeto (2020)
proposed a so-called reliability-based continuous network design problem. It seeks to optimize the link
capacity expansion stratifies to improve the robustness of network performance against uncertain travel
time and demand. It is worth mentioning that all aforementioned studies formulate the robust network
design problem as the min-max problem (6).
This study does not intend to solve problem (6). Instead, we seek to formulate and solve the
subproblem
to estimate the worst-case network performance and corresponding
equilibrium flow. These results are important prerequisites for designing robust planning strategies to
improve network performance based on problem (6). To do this, in what follows, we will develop a
worst-case mixed traffic assignment problem. It computes the optimal (worst) input of such that the
network performance is minimized. For simplicity, we label the solution of to the problem
as the worst-case input of all link capacities. The outputs of the worst-case mixed
traffic assignment problem include the worst-case input of all link capacities, the worst (minimum)
network performance, and the corresponding worst-case equilibrium flow.
3.2 Bi-level formulation of the worst-case mixed traffic assignment
The worst-case mixed traffic assignment model is formulated as a bilevel optimization problem. The
lower-level problem is a traffic assignment problem that characterizes the equilibrium flows for given
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link capacities. The upper-level problem seeks to find the worst inputs of all link capacities within their
ranges such that the network performance is minimized. The notation below will be used to formulate
the bilevel worst-case mixed traffic assignment problem.
Let subscripts and denote an HDV and CAV, respectively. Let denote the set of vehicle
classes; i.e., . Let be the set of all origin-destination (OD) pairs and be the set of all
routes connecting OD pair , , for vehicle class . Let be the flow of vehicles in class
on link . Let denotes the travel time on link ; let
denotes the travel time on route between
OD pair for vehicle class . Let
denotes the flow of vehicles in class on route between OD pair
. Let denotes the demand between OD pair , for vehicle class z, and let be the vector
of all OD demand for vehicle class . Let denote the set of all links for vehicle class . Denote
the vectors as follows:
;
; ; let and be the link-path and OD-path matrices, respectively,
for vehicle class .
3.2.1 Mixed Traffic assignment problem considering uncertain link capacity
This section proposes a mixed traffic assignment model to characterize the distribution of mixed
traffic flows. Due to uncertain link capacity, the equilibrium link flows are not unique. The proposed
mixed traffic assignment model seeks to compute all possible equilibrium link flow solutions based on
the range of link capacity. This model will be leveraged to develop the worst-case mixed traffic
assignment problem.
Similar to Wang et al. (2019b, 2021) and Li et al. (2018), the logit-based SUE model and the UE
model are used to capture the route choice behavior of HDV travelers and CAV travelers, respectively.
Let be the set of regular links where both HDVs and CAVs can access and
be the set of autonomous
vehicle dedicated lanes. The following BPR function is used to characterize the travel time of all links:
(7)
where
and are positive coefficients; is the capacity of link ; If link is an autonomous vehicle
dedicated lane, its capacity can be computed using Eq. (5b) with being 1. If link is a regular lane,
its capacity is uncertain, which can change between .
Suppose the network contains only HDVs. According to Wang et al. (2019b),
is the equilibrium
path flow of the logit-based SUE if and only if it satisfies
(8)
where is the dispersion parameter.
Suppose the network contains only CAVs. is the UE link flow if and only if it satisfies
(9)
where
is the vector of all link travel times at the equilibrium state.
.
Suppose the network flow is mixed, containing HDVs and CAVs. Let
and be the HDV path
flow and CAV link flow, respectively, at the equilibrium state of the mixed traffic assignment model.
According to variational inequality (VI) problems (8) and (9),
must be a solution to the
following VI problem:
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(10a)
s.t.
(10b)
(10c)
(10d)
(10e)
(10f)
(10g)
(10h)
where inequalities (10e)-(10g) specify the maximum and minimum capacity of each link . The
maximum and minimum capacity of each link are functions of the HDV flow and CAV flow on that link.
The mixed traffic assignment problem (10) characterizes all the possible equilibrium flows under a
mixed traffic flow environment. It should be noted that the decision variables of problem (10) are
while the capacity of all regular links (i.e., ) are inputs whose values are constrained by respective
equilibrium link flows. As there exist many possible inputs of , the solutions of problem (10) are non-
unique. Further, we cannot solve problem (10) directly to obtain the equilibrium flows because the exact
link capacity is unknown. To address this problem, in the following, we will develop an equivalent
formulation of problem (10) by formulating the solution set of problem (10). It also presents a feasible
way to compute a solution of problem (10).
3.2.2 Equivalent formulation for the mixed traffic assignment problem (10)
To compute the solutions of problem (10), this section proposes an equivalent formulation. Suppose
the capacity of all links is given (denoted as ). Let
;
. Consider the following VI-based traffic assignment problem
developed based on the problem (10) where the inequalities (10e)-(10g) are removed
(11)
where .
The VI problem (11) is developed based on fixed link capacity. The proposition below shows an
important property of the VI problem (11).
Proposition 1: If is fixed, then VI problem (11) has a unique solution.
Proof: see appendix A.
Proposition 1 indicates that for given link capacity vector , the solution to the problem (11)
is unique. Thereby, the implicit function is adjective, i.e., it maps the vector to only one
equilibrium flow solution .
The VI problem (11) can be solved using the route-swapping-based solution algorithm proposed by
Wang et al. (2019b). Note that the only difference between problem (11) and problem (10) is the
inequalities (10e)-(10g) are removed in problem (11). Thereby, if the solution to problem (11) for given
(i.e., ) can satisfy the inequalities (10e)-(10g) simultaneously, is then a solution to the
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problem (10). Let
be the set of all equilibrium flow solutions to the mixed traffic assignment model
(10), it can be formulated as
(12)
where is the solution to problem (11) for given ;
;
;
.
Eq. (12) characterizes all the solutions to the problem (10). Thereby, they are equivalent. Eq. (12)
also provides a feasible way to compute the solutions to the problem (10). The steps are as follows: (1)
select an arbitrary input for , compute by solving VI problem (11) for given ; (2) If and
satisfy the three inequalities on the right-hand side of Eq. (12) simultaneously, then is a feasible
equilibrium flow solution to the problem (10), and is a vector of link capacity at an equilibrium state.
By varying , we can compute other feasible equilibrium flow solutions to the problem (10).
Let be the set of all possible link capacity vectors at the equilibrium states of the mixed traffic.
Note that and is one-to-one match. Similar to Eq. (12), can also be formulated as
(13)
Note that there exist uncountably many solutions in
because the feasible solution of to the
problem (10) is continuous with respect to . In section 4, we will design a new algorithm to compute
the worst-case equilibrium flow solution by avoiding enumerating the solutions in
.
3.2.3 Worst-case mixed traffic assignment problem
The worst-case mixed traffic assignment problem seeks to find the minimum (worst) network
performance and corresponding equilibrium flows due to uncertain link capacity. Note that the solutions
of problems
and
are the same. Thereby, the worst-case mixed traffic
assignment problem can be formulated as the following bi-level problem
(14a)
s.t.
(14b)
(14c)
(14d)
where is the performance indicator; inequalities (14b)-(14d) specify the set of all link
capacity vectors at the equilibrium states of mixed traffic flow. is the solution to the lower-level
VI-based traffic assignment problem (11) with fixed link capacity
.
The problem (14) seeks to find the optimal input for in (or equivalently, the equilibrium
solution ) such that the network performance is the worst. Without loss of generality, this study
uses the total travel time of all vehicles in the network as the performance indicator; i.e.,
(15)
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4. A partition-based NRMFD solution algorithm for the bilevel optimization problem (14)
4.1. Introduction of the line search principle
According to inequalities (14c) and (14d), the lower bound of the link capacity contains two
functional forms, which change at the point
. According to Eq. (4), the lower bound of
the link capacity is continuous at the point
. However, it is not first-order differentiable
at this point. Therefore, the traditional solution algorithms proposed in the literature (e.g., Friesz et al., 1990;
Chiou, 2005) are not applicable in solving the bilevel programming problem (14). To address this problem,
this study proposes a partition-based NRMFD solution algorithm based on the line search principle. At
each iteration, it aims to find a feasible descent direction for the decision variables (i.e., all link capacities)
and a corresponding step to minimize the objective function. Denote as the input for obtained by
this algorithm at iteration . The next point found by the NRMFD solution algorithm can be formulated
as
(16)
where is the corresponding step size; is a feasible descent direction found by the NRMFD
solution algorithm at iteration ., i.e., there exists such that
Condition 1:
(17a)
Condition 2:
(17b)
Condition 1 ensures that the direction of is descending, and condition 2 ensures that it is feasible.
One of the most critical steps for implementing the line search principle to solve the bilevel problem (14)
is to find a at .
4.2 Equivalent optimization problems to compute a at
Note that the lower bound formulation of the link capacity changes at the point
.
To find a at the given point , we divide the set of all links into three subsets based on the
equilibrium HDV and CAV link flows at the point . The three link sets are denoted as ,
and . They are formulated as
(18a)
(18b)
(18c)
Note that both
and
are continuous functions of . Let be a small ball centered
at with a radius of , where is sufficiently small. Then,
(18a)
(18b)
Suppose ; then, we can construct the following constraints based on inequalities (14b-
14d):
(19a)
(19b)
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(19c)
(19d)
(19e)
Let
be the feasible set of determined by inequalities (19a)-(19e); then, we have
.
Consider the following bilevel programming problem:
(20a)
s.t.
, i.e., it satisfies inequalities (19a)-(19e).
(20b)
The only difference between problem (20) and problem (14) is that the feasible set of is reduced to
. Let
be a feasible descent direction for problem (20) at . Because
,
must be a
feasible descent direction for problem (14). The proposition below shows another important connection
between problem (20) and problem (14).
Proposition 2: If , then is a feasible descent direction for problem (14) if and only if it is
a feasible descent direction for problem (20).
Proof: we only prove the "if part", the proof of "only if part" can follow the same logic.
Note that the objective functions of problems (20) and (14) are the same; then, if is a descent
direction for problem (20), it must be a descent direction for problem (14) (i.e., it satisfies Eq. (17a)).
Note that ; then,
or
for all links in the network.
Without loss of generality, let and be the set of links defined in Eq. (18a) and Eq. (18b),
respectively. As is a feasible direction for problem (20), it satisfies Eq. (19). Then, for a sufficiently
small positive value ,
(21a)
(21b)
and,
(21c)
(21d)
(21e)
where is the vector of the upper bounds of all link capacities in the network at the
point ;
and
denote the capacities of all links in sets and , respectively;
and
are vectors of the inputs in corresponding to and , respectively;
is the minimum capacity of all links in the set ; and
is the minimum capacity of all links in the set .
According to Eq. (21), . Therefore, is also a feasible direction for problem (14).
The above discussion shows that if and is a feasible descent direction for problem (20),
then it must be a feasible descent direction for problem (14).
Proposition 2 indicates that the feasible descent direction for problem (14) at point can be
obtained by finding a feasible descent direction for problem (20) at point if .
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Now, consider the scenario in which . In this case, when changes by a small value at
, we cannot determine the relationship between the HDV flow and CAV flow on these links; i.e., when
is close to , for both the case in which
and the case in which
may hold. To determine the functional form of the lower bound of each link capacity
in problem (14), we divide the set into two disjoint subsets
and
, where
is
the set of links in which
and
is the set of links in which
.
Consider the following constraints:
(22a)
(22b)
(22c)
(22d)
(22e)
Inequalities (22d) and (22e) are consistent with inequalities (22b) and (22c), respectively. Let
be the feasible set of determined by inequalities (22a)-(22e). Note that an arbitrary link in the set
can belong to either of the sets
and
. There exist different combinations
of
and
. Let
be a subset of determined by inequalities
(21a)-(21e). Note that
. Similar to problem (20), we construct the following problem:
(23a)
s.t.
, i.e., it satisfies inequalities (21a)-(21e).
(23b)
where is the solution to the VI problem (11).
Problem (20) is a special case of problem (23) in which
. Note that we can
construct different problems (23) by enumerating the set
in (23b). For simplicity, we say
that problem (23) is problem (23) constructed upon one of the sets
).
Note that
; then, if
is a feasible descent direction for problem (23) at , it must be a
feasible descent direction for problem (14) at . In cases where no feasible descent direction exists for
problem (23) with
, we need to construct a new problem (23) by replacing the feasible subset
with another subset
, and we need to find a feasible descent direction
based on the new optimization problem. This step continues until a feasible descent direction is found
for one of the problems (23). The proposition below shows the connections between the feasible descent
directions for problem (23) and problem (14).
Proposition 3: Suppose , then is a feasible descent direction for problem (14) if and only
if it is a feasible descent direction for one of the problems (23).
Proof: we first prove the “only if” part i.e., if is a feasible descent direction for one of the
problems (23), then it must be a feasible descent direction for problem (14). Without loss of generality,
let be a feasible descent direction for problem (23) constructed upon
().
Note that the objective functions of problem (23) and problem (14) are the same, then must be a
descent direction for problem (14). Further,
, is also a feasible direction for problem (14).
13
Thereby, if is a feasible descent direction for problem (23) constructed upon
, it is also a feasible
descent direction for problem (14).
Now we prove the “if” part. Note that if is a feasible descent direction for problem (14), it must
be a descent direction for all problems (23). Next, we will find a set
such that is a feasible
direction for problem (23) constructed upon this set. Let be a positive scalar and be sufficiently small.
Let
and
be the sets of links in the set such that
and
, respectively. Let
be the feasible set for constructed
upon
and
. Then,
. This indicates that must be a feasible direction for
problem (23) constructed upon
. This completes the proof.
Proposition 3 also indicates that if no feasible descent directions exist for all problems (23),
then no feasible descent directions exist at point . In this case, is a local optimal solution to problem
(14).
According to Proposition 2 and Proposition 3, to find a feasible descent direction for problem (14)
at the point where , we can find a feasible descent direction for problem (23) at the point
at which
. When , we can find a feasible descent direction for one
of the problems (23) to obtain a feasible descent direction for problem (14).
4.3 Partition-based NRMFD solution algorithm
This section presents the explicit procedures for implementing the partition-based NRMFD solution
algorithm to solve problem (14). Let
and
,
where
is the input for
for the set
. At point , the partition-
based NRMFD solution algorithm will solve the following convex quadratic optimization problem to
find a feasible descent direction
for problem (23) constructed upon the set
:
(24a)
(24b)
(24c)
(24d)
(24e)
(24f)
(24g)
where is an arbitrary positive definite matrix;
is a positive constant with a value between 2 and 10
to increase the convergence speed (Wang et al., 2021). is a decision variable;
is the direction of the
capacity of link ; and is a -dimension column vector with all entries being 1. The terms on the
left-hand sides of inequalities (24c)-(24g) are first-order Taylor approximations of the terms on the left-
hand sides of inequalities (23a)-(23e), respectively.
Let be the solution to problem (24). If , then must be a descent direction for
14
problem (23) constructed upon the set
. Note that problem (24) is a convex quadratic optimization
problem. According to Chen and Kostreva (1994), the solution to problem (24) is unique if it exists.
Problem (24) can be solved using, e.g., the active set algorithm. The following theorem discuss the
existence of a solution to problem (24) at point .
Theorem 1 (Chen and Kostreva, 1999): If a feasible descent direction exists for problem (23) at , then
the solution to problem (24) exists and is unique; furthermore, , i.e., must be a
feasible descent direction for problem (23) (and for problem (14)) at .
For simplicity, we label problem (24) the direction-finding subproblem (DFS problem). If no feasible
descent directions are found by solving the DFS problem (24) constructed upon the subset
, then we
need to solve another DFS problem constructed upon a new subset
.
These steps will be repeated until a feasible descent direction is found. If no feasible descent directions
are found by solving all DFS problems at point , then according to Proposition 3, is the
optimal solution to problem (14).
After a is found, the next step is to find a step size satisfying the following two conditions:
Condition 1:
(25a)
Condition 2:
(25b)
To increase the convergence speed, this study aims to find a step size as large as possible. The method
is as follows: Let , where is a sufficiently small value (e.g., ) and is
a positive integer. Starting from , continue increasing the value of until the computed cannot
satisfy the two conditions in Eq. (25) simultaneously. Let be the maximum integer for which both
conditions in Eq. (25) are satisfied. Then, .
The full steps needed to implement the partition-based NRMFD solution algorithm are summarized
as follows:
Step 1: Set , and find a feasible initial link capacity in the feasible set .
Step 2: Solve the low-level fixed-demand-based traffic assignment problem (11) to obtain the
equilibrium solution
and
at .
Step 3: Let be the set of links such that
. Let
denotes the
feasible subset of satisfying inequalities (22a-22e). Note that there exist different subsets at
based on how the set is divided into two disjoint sets
and
. Construct a problem
(23) and the corresponding DFS problem (24) based on subset
. Solve the DFS problem (24) to obtain
a feasible descent direction . If no feasible descent directions are found by solving all DFS
problems at the point , then go to step 5; otherwise, go to step 4.
Step 4: Compute the step size based on the method introduced above. If , where
is a predetermined threshold, then go to step 5; otherwise, set , and go to step 3.
Step 5: Output , the corresponding equilibrium flow solution and the value of the network
performance indicator at . Stop iteration.
In step 1, we need to find an initial feasible link capacity vector in the set . Note that the upper
and lower bounds of the capacity of a link depend on the equilibrium HDV and CAV flow on this link.
Therefore, finding a feasible link capacity vector is not a trivial task. To address this problem, this
study computes the ELC at the equilibrium state of the VI problem (11) based on the ELC model (5b).
15
The expected capacity of all links at the equilibrium state will be used as . Based on the definition of
ELC, must belong to the set .
Note that is closed and bounded, and is also bounded for . The proposition
below shows that the partition-based NRMFD solution algorithm is globally convergent.
Proposition 4: The NRMFD solution algorithm is globally convergent; i.e., from a feasible starting point
, the sequence must converge to a local solution of problem (14).
Proof: Let be the solution computed by the NRMFD algorithm at iteration . As a step is found at
each iteration such that (1) and (2)
, the sequence is monotonically increasing; i.e.,
. Note that is bounded. Therefore, sequence must converge.
Let . Clearly,
. Suppose the sequence
does not converge to a local solution of problem (14). Then, according to Proposition 2,
proposition 3 and Theorem 1, for there exists a feasible descent direction at the point
and a step such that is feasible and .
This indicates that for . Let . Therefore,
, which contradicts the condition that
. This indicates that the sequence
must converge to a local solution of problem (14).
It should be noted that the computational time of the partition-based NRMFD solution algorithm
depends significantly on the size of the set at each iteration, which determines the number of
DFS problems at this point. Fortunately, with the method of finding the step size in the partition-
based NRMFD solution algorithm, in general. Therefore, this algorithm can solve the
worst-case mixed traffic assignment problem (14) very efficiently.
4.4 Sensitivity analysis of the VI-based traffic assignment problem (11) with fixed link capacity
To implement the partition-based NRMFD solution algorithm, we need to compute the gradients
and
at point . These gradients can be computed using the
sensitivity analysis method for the mixed traffic assignment model with fixed link capacity.
Let denotes the path flows for CAVs. To derive the analytical formulation of these gradients, the
VI-based traffic assignment problem with fixed link capacity (i.e., problem (11)) is reformulated
equivalently as the following expression, which is constructed based on HDV and CAV path flows:
(26)
where and
are vectors of the flows and costs of all paths for CAVs at the equilibrium state
under a given input of link capacity.
.
Let
be a local equilibrium path flow solution to the VI problem (26) with the input of
being . The Karush-Kuhn-Tucker (KKT) conditions of the VI problem (26) at are
(27a)
(27b)
16
(27c)
(27d)
(27e)
(27f)
(27g)
where
and
are vectors of Lagrange multipliers associated with the constraints and
, respectively.
and
are vectors of Lagrange multipliers associated with the nonnegative
path flow constraints for HDVs and CAVs, respectively.
Without loss of generality, assume the equilibrium path flow solution for CAVs of VI problem (26)
at is not degenerate, i.e., that there exists a path flow solution for which the flow on all equilibrated
paths for CAVs is positive. This assumption is not strong, as the possibility of being degenerate is
almost 0 (Wang et al., 2016).
Note that the CAV path flow solution is not unique. To address this problem, let be the set of
equilibrated paths for CAVs at , and let be the flow vector for all equilibrated paths. Let
be the
link-path and OD-path matrix for paths in the set . Divide
into two submatrices,
, where
is a full column matrix constituted by column vectors in
that has the same
rank as
. Let and be the vectors of the flows of paths
corresponding to
and
, respectively. When changes from , only changes, and
remains fixed at . This helps to generate a unique path flow solution
when changes from
. Note that the non-equilibrated paths will remain non-equilibrated for a small perturbation of . Since
the HDV path flow solutions are all positive and , all the Lagrange multipliers in vectors
and
are nonbinding. Therefore, Eq. (27) can be simplified as
(28a)
(28b)
(28c)
Differentiating both sides of Eq. (28) with respect to , we have
(29a)
(29b)
(29c)
17
Therefore,
(30)
can be obtained directly with Eq. (30), can be computed as
(31)
5. Numerical example
In this section, the worst-case mixed traffic assignment method is applied to two numerical networks
to study the impacts of uncertain link capacity on the distribution of network flows. According to Ghiasi
(2017), , and are set to 1.8 seconds, 1 second, and 0.6 seconds, respectively. To make a
comparison, the VI problem (11) with the ELC model (5b) is applied to estimate the equilibrium flows.
5.1 Nguyen-Dupuis network
This section applies the Nguyen-Dupuis network (Figure 2) to test the performance of the partition-
based NRMFD solution algorithm and to compute the worst-case inputs of all link capacities and
corresponding equilibrium network flows. The HDV demands between OD pairs 1-2, 1-3, 4-2, and 4-3
are 1155, 866, 717 and 866, respectively. The CAV demands between these four OD pairs are 2145,
1609, 1332 and 1609, respectively. The free flow travel time of each link is shown in Table 1, and the
number of lanes for all links is set to 1.
Figure 2. Nguyen-Dupuis network
As mentioned in section 4.3, to implement the partition-based NRMFD algorithm to solve the bilevel
worst-case mixed traffic assignment problem (14), the VI problem (11) with ELC is used to find the
initial feasible link capacity vector that satisfies inequalities (14b)-(14d). By applying the route-
swapping-based solution algorithm proposed by Wang et al. (2019), the equilibrium HDV and CAV link
flow of the VI problem (11) with the ELC model (5b) is obtained, as shown in Table 2. Based on Eqs.
(2), (4) and (5), Figure 3 shows the computed expected capacity of all links and their ranges at the
equilibrium state. The computed expected capacities of all links are within their ranges. The expected
1 12
5 6 874
910 11 2
13 3
Origin
Destination
2
3 5
46
12
13 16
19
14
7
15
1110
9
18
1
8
17
18
capacity of all links is used as the starting point of the partition-based NRMFD algorithm.
Table 1 Inputs for the Nguyen-Dupuis network
Link
Link
Link
1
7
8
12
15
9
2
9
9
5
16
8
3
9
10
9
17
2
4
12
11
9
18
14
5
3
12
10
19
8
6
9
13
9
7
5
14
6
Figure 3. Expected capacity of all links and their ranges in the equilibrium state of the VI problem (11)
with the ELC model (5b)
Table 2 Comparison of the exact UE link flows
Links
Worst-case mixed
traffic assignment
model
VI problem (11) with
ELC model (5b)
Links
Worst-case mixed
traffic assignment
model
VI problem (11) with
ELC model (5b)
HDV
CAV
HDV
CAV
HDV
CAV
HDV
CAV
1
1050.6
1606
1048.7
1605
11
899.1
2942.7
904.6
2965.9
2
970.7
2147.7
972.5
2148.8
12
349.1
522.5
336
508.2
3
688.6
818.1
656.5
832.4
13
878.9
2308.6
907.8
2279.6
4
899.5
2131.3
931.6
2117
14
1011.1
552.7
954.5
514
5
1410.7
1724.3
1393
1766.6
15
977.8
542.9
972.3
519.7
6
328.5
699.8
312.3
670.8
16
853.6
908.9
824.7
937.9
7
1340.5
1709.6
1398
1780.8
17
591.8
15.4
623.5
20
8
662
30.2
618.5
5.8
18
378.9
2132.3
349.1
2128.8
9
520.2
810.4
555.5
837.1
19
878.9
2308.6
907.8
2279.6
10
820.3
899.2
842.4
943.7
19
Figure 4 shows the evolution of the total travel time computed by the partition-based NRMFD
algorithm. The total travel time at each iteration takes only 22 iterations to become stable, indicating that
the partition-based NRMFD algorithm can effectively and efficiently solve the worst-case mixed traffic
assignment problem (14) to find the minimum network performance due to uncertain link capacities. The
HDV and CAV link flows in the worst-case equilibrium state are shown in Table 2. They differ
significantly from the results computed by the VI problem (11) with the ELC model (5b). Furthermore,
the total travel time in the worst-case equilibrium state is larger than that in the equilibrium state of the
VI problem with ELC by over minutes (approximately 1%). It should be noted that in a
congested network, the differences in the total travel time computed by the two traffic assignment
methods will be much larger. These results indicate that the uncertain link capacity has dramatic impacts
on network performance and equilibrium flows. It is necessary to study the worst-case network
performance due to uncertain link capacity to develop robust planning strategies to ensure minimum
network performance.
Figure 4. Evolution of the total travel time computed by the partition-based NRMFD algorithm
Figure 5. The capacity of all links and their upper and lower bounds in the worst-case equilibrium state
Figure 5 shows the capacity of all links and their upper and lower bounds in the worst-case
1 4 7 10 13 16 19 22
4.185
4.19
4.195
4.2
4.205
4.21
4.215
4.22
4.225 x 105
Iterations
Total travel time (miniutes)
20
equilibrium state computed based on the equilibrium flows shown in Table 2. As seen from Figure 5, in
the worst-case equilibrium state, the capacities of several links are very close to their lower or upper
bounds, preventing the total travel time from being maximized further. Figure 6 describes the evolution
of the computed link capacity at each iteration. This demonstrates that compared to the initial link
capacity, the capacities of some links in the worst-case equilibrium state increase while those of others
decrease. These results imply that simply reducing the capacities of all links cannot solve the worst-case
mixed traffic assignment problem (14). The proposed partition-based NRMFD algorithm can help us
find the optimal input of link capacity to minimize the network performance.
Figure 6. Evolution of the computed link capacity and its range; (a) evolution of the computed capacity
of link 8 and its range; (b) evolution of the computed capacity of link 17 and its range.
5.2 Sioux Falls network
The Sioux Falls network is used to test the performance of the partition-based NRMFD solution
algorithm on a larger network. It contains 24 nodes, 76 links, and 552 O-D pairs. For simplicity, suppose
there exist three lanes on all the links. The link capacity at the equilibrium state of the VI problem (11)
developed based on ELC model (5b) is used as the starting point for the partition-based NRMFD solution
algorithm.
Figure 7. Evolution of the total travel time computed by the partition-based NRMFD solution
algorithm at each iteration
21
Figure 7 demonstrates that the total travel time computed by the partition-based NRMFD solution
algorithm increases at each iteration. It converges after only 23 iterations. In the worst-case equilibrium
state, the computed capacities of all links are within their ranges (see Figure 8). These results indicate
that the partition-based NRMFD solution algorithm can effectively and efficiently solve the bilevel
worst-case mixed traffic assignment problem (14).
To validate the effectiveness and efficiency of the proposed partition-based NRMFD solution
algorithm, the genetic algorithm (GA) is also used to solve the worst-case mixed traffic assignment
problem (14). The fitness function of the GA is defined as
, where the second term seeks to punish the individuals who violate the
link capacity constraints (14(b)-14(c)). The population size, the crossover probability, the mutation
probability, and the maximum generations are set as 60, 0.8, 0.1, and 300, respectively. To enable
comparison, one individual (link capacity) in the initial population is set the same as the starting point of
the partition-based NRMFD solution algorithm.
Figure 8. The worst-case capacities of all links and their ranges
Figure 9 shows the evolution of the minimum fitness and population mean computed by the GA. As
can be seen, GA converges after about 150 iterations. The worst-case equilibrium flow solution and
respective total travel time (i.e., minutes) computed by the GA are the same as that of the
partition-based NRMFD solution algorithm, which validates the effectiveness of the proposed partition-
based NRMFD solution algorithm. However, GA takes over 23 hours to find the optimal solution of the
worst-case mixed traffic assignment model. In contrast, the partition-based NRMFD solution algorithm
only takes 485 seconds to solve this model. Thereby, the proposed partition-based NRMFD solution
algorithm outperforms the GA significantly.
Figure 7 and Figure 9 indicate that compared with the total travel time computed for the VI problem
(11) with ELC model (5b) (at the first iteration), the total travel time computed for the worst-case mixed
traffic assignment problem (14) (at the 23rd iteration) increases by over 10%. This is because the
equilibrium CAV link flows and HDV link flows computed by the two equilibrium traffic assignment
models are dramatically different. As seen from Figure 10, the relative differences of most of the HDV
and CAV equilibrium links of the two traffic assignment models are over 6%. These results further
demonstrate the importance of evaluating the worst-case traffic performance in a mixed traffic flow
22
environment to improve the robustness of traffic planning strategies to counteract the negative effects of
uncertain link capacity.
Figure 9. Evolution of the minimum fitness and population mean computed by the GA
Figure 10. The relative differences of HDV and CAV equilibrium flow computed by the VI problem
(11) with the ELC model (5b) and the worst-case mixed traffic assignment problem (14)
6. Conclusion
This study proposes a worst-case mixed traffic assignment model to estimate the worst network
performance and corresponding equilibrium flows that may occur due to uncertain link capacity in a
mixed traffic flow environment. The worst-case mixed traffic assignment model is formulated as a bilevel
programming problem. A partition-based NRMFD solution algorithm is proposed to solve the bilevel
programming problem. It can deal well with the noncontinuous differentiable inequality constraints in
the bilevel programming problem. We show analytically that the partition-based NRMFD solution
algorithm is globally convergent. A numerical application indicates that the partition-based NRMFD
solution algorithm can effectively and efficiently solve the bilevel worst-case mixed traffic assignment
problem. Furthermore, dramatic differences exist between the worst-case equilibrium results (i.e., the
23
network performance and equilibrium flows) and the expected equilibrium results computed by the
traditional mixed traffic assignment methods constructed on the basis of expected link capacity models.
Therefore, traditional mixed traffic assignment methods can overestimate the effects of traffic planning
strategies on network performance. The proposed worst-case mixed traffic assignment method can help
traffic managers design robust traffic planning strategies to counteract the negative effects of uncertain
link capacity.
The current research can be extended in a few directions. First, this study only analytically discusses
the theoretical upper bound and lower bound of link capacity in a mixed traffic flow environment by
considering heterogeneous time headway of different pairs of vehicles. It does not consider the impacts
of traffic merging and/or lane change behavior. Note that the traffic merging and/or lane change behavior
can induce perturbations in both the range and probability distribution of link capacity. In the future,
more simulation-based and field-based experiments will be conducted to investigate the range and
probability distribution of link capacity in the actual case by holistically considering these impact factors.
New methods (such as 95 confidence interval) will be proposed to identify the possible maximum and
minimum link capacity in different traffic situations. Second, robust network design problems will be
developed based on the worst-case mixed traffic assignment method to improve the network performance
in a mixed traffic flow environment. Third, a combined modal split and traffic assignment model can be
developed based on the worst-case mixed traffic assignment problem to simultaneously estimate the
network flows and OD demand of both HDVs and CAVs by incorporating the uncertain link capacity.
Acknowledgments
This study is supported by the National Natural Science Foundation of China (52002191, 51878166
52072068, 52072066), and Jiangsu Province Science Fund for Distinguished Young Scholars
(BK20200014). Any errors or omissions remain the sole responsibility of the authors.
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Appendix A: Proof of Proposition 1
Proof: Proposition 1 can be proven if we can show that is a strictly monotonic function of .
Note that
(a1)
As is fixed, . Furthermore, these are all diagonal positive definite
matrices. Note that ; then,
is a symmetric matrix. The theorem below will be useful in
showing that
(the symbol means that the corresponding matrix is positive definite).
Theorem 1 (Proposition 16.1 in Gallier (2011)): Let
be a symmetric matrix; if
and
, then .
According to the logit-based SUE model, the equilibrium path flows are all positive. Therefore,
25
(a2)
Furthermore,
(a3)
Eqs. (a2) and (a3) show that
is a positive definite matrix. Therefore, the solution to the VI problem
(9) is unique if is fixed.
