Path-constrained traffic assignment: Modeling and computing network impacts of stochastic range anxiety

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Path-constrained traffic assignment: Modeling and computing network impacts of stochastic range anxiety✩ Chi Xie a,b,∗, Tong-Gen Wang c, Xiaoting Pu d, Ampol Karoonsoontawong e a

Key Laboratory of Road and Traffic Engineering of the Ministry of Education, Tongji University State Key Laboratory of Ocean Engineering, Shanghai Jiaotong University c Division of City and Transportation Planning, Lin Tung-Yen and Li Guo-Hao Consultants Shanghai Limited d Centre for Transport Studies, Imperial College London e Department of Civil Engineering, King Mongkut’s University of Technology Thonburi b

a r t i c l e

i n f o

Article history: Received 16 August 2016 Revised 21 April 2017 Accepted 24 April 2017 Available online xxx Keywords: Traffic assignment Network equilibrium Electric vehicles Range anxiety Driving ranges Trip chains Combined activity-travel choices

a b s t r a c t It is notoriously known that range anxiety is one of the major barriers that hinder a wide adoption of plug-in electric vehicles, especially battery electric vehicles. Recent studies suggested that if the caused driving range limit makes any impact on travel behaviors, it more likely occurs on the tour or trip chain level than the trip level. To properly assess its impacts on travel choices and traffic congestion, this research is devoted to studying a new network equilibrium problem that implies activity location and travel path choices on the trip chain level subject to stochastic driving ranges. Convex optimization and variational inequality models are respectively constructed for characterizing the equilibrium conditions under both discretely and continuously distributed driving ranges. For deriving the equilibrium flow solutions for these problem cases, we suggested different adaptations of a well-known path-based algorithm—the projected gradient method. While the problem instance with a discrete number of driving ranges can be simply treated as a multi-class version of its deterministic counterpart, the one with continuous driving ranges poses a much more complicated situation. To overcome this arising modeling and algorithmic complication, we introduce a couple of newly defined variables, namely, path-indexed travel subdemand rate and traffic subflow rate, by which the demand and flow rates as well as their corresponding feasible path sets can be dynamically indexed in the solution process with reference to path lengths. An illustrative example with various types and forms of driving range distributions demonstrates the applicability of the proposed modeling and solution methods and various impacts of the heterogeneity of range anxiety on network flows and computational costs. The numerical analysis results from this example show that stochastic driving ranges confine network flows in a different way from deterministic or no driving ranges and the projected gradient procedure relying on dynamically indexed subdemand and subflow rates is generally preferable to its counterpart on pre-indexed ones for both the discrete and continuous driving range cases. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Range anxiety associated with those who drive plug-in electric vehicles, especially battery electric vehicles, is often referred to as their mental distress or fear of being stranded on roads because the battery runs out of charge (Marrow et al., 2008; Mock et al., 2010; Franke and Krems, 2013). This term first appeared in the press in 1997, in a San Diego Business Journal article authored by Acello (1997), who described his worry on the driving range of an example electric vehicle model ✩

Accepted by Transportation Research Part B: Methodological. Corresponding author at: A610 Ruth Mulan Chu Chao Bldg., 800 Dongchuan Rd., Shanghai 200240, China. E-mail addresses: [email protected] (C. Xie), [email protected] (T.-G. Wang), [email protected] (X. Pu), [email protected] (A. Karoonsoontawong). ∗

http://dx.doi.org/10.1016/j.trb.2017.04.018 0191-2615/© 2017 Elsevier Ltd. All rights reserved.

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produced by General Motors. Range anxiety has quickly become a popular topic in the public media in the first decade of the 21st century (Schott, 2009; Rahim, 2010; Malone, 2010; Eberle and von Helmolt, 2010),1 with a climbing sales number of electric vehicles worldwide around that time. As a follow-up of this widespread concern, General Motors soon filed the term of range anxiety as a trademark, stating it was for the purpose of “promoting public awareness of electric vehicle capabilities” (Hyde, 2010). Range anxiety is actually a common issue that harasses people’s trip making and choice behaviors in driving any kind of vehicles when refueling opportunities are scarce (Xie and Jiang, 2016). For electric vehicles, the main technical reasons behind range anxiety are inadequate battery performance and capacity and insufficient public electricity-charging providers (Pearre et al., 2011; Neubauer and Wood, 2014). Even if the battery storage and charging technologies have experienced continuous, significant progress and the number of newly constructed public charging stations climbed at an increasing rate in the past decade, range anxiety is still one of the major concerns and barriers nowadays that impede the wide acceptance and adoption electric vehicles (Kassakian, 2013). Many automobile manufacturers and transportation economists predicted that the range anxiety phenomenon will last and concern the driving community for quite a long time, continuously affecting their travel behaviors, commuting customs and even daily schedules, unless a real breakthrough of relevant electricity storage and charging technologies occurs and the price, stability and durability of onboard batteries reaches a commercially satisfactory level. Range anxiety imposes negligible impacts on the scope and flexibility of trip makings, spatially and temporarily restricting travel choices as well as productivity and life choices in different ways and levels. The aggregate U.S. driving distance distribution data provided by Tamor et al. (2013) on the trip chain level clearly shows that a significant amount of travel demand cannot be satisfied by any electric vehicle model in the current consumer-grade market, due to their insufficient driving ranges even under a full charge. The resulting range anxiety inevitably excludes the possibility of using electric vehicles for those long-distance trips or tours, or forces travelers to seek a multimodal travel solution and consider other travel choice alternatives. Beyond the travel distance supported by a single charge, range anxiety also impacts the total vehicles mile traveled by the entire driving population in a region or country, if a significant number of electric vehicles are injected into the market (Neuhauer and Wood, 2014). To properly reflect these impacts in travel demand forecasting, Jiang et al. (2012, 2013) and Jiang and Xie (2014) first introduced range anxiety, as represented by the maximum driving distance or driving distance limit, into travel choice and network assignment problems. These authors presented a series of network equilibrium models involving spatially constrained travel choices by range anxiety, including destination choice, mode choice and route choice. This modeling concept was further extended by He et al. (2014) and Xie and Jiang (2016) to embrace the recharging requirement of electric vehicles for long-haul trips in congested networks. In all these mentioned studies, researchers hold a rather strict modeling assumption that all drivers in a traffic network are of the same driving distance limit. This simple assumption seems to be largely deviated from the reality. What actually impacts individual travel behaviors is actually the estimated or perceived driving ranges by electric vehicle drivers. Given that this is the result of drivers’ subjective perceptions and judgment on the actual driving range, it is much more reasonable to conjecture that the range anxiety within a driving population could be better represented by a stochastic distance limit, which consists of a diverse number of heterogeneous values instead of a single common value. The diversity is a reflection of multiple explanatory factors, including not only nominal battery capacity, initial state of charge, electricity consumption rate, driving environment and conditions and other physical factors, but also range gauge mechanisms and the drivers’ cognitive, understanding, appraisal, coping, adapting, stress-buffering and risk-taking behaviors (e.g., the well-known “guess-o-meter” confusion on the dashboard reading). Psychological theory suggests that physically identical situations may constitute a fundamentally different psychological and decision making situation for different individuals (Bowers, 1973). A recent psychology experiment by Franke et al. (2012) revealed that the range anxiety with electric vehicle drivers is primarily quantified by their self-perceived comfortable driving ranges, for which personal stress-buffering competence and coping skills play a substantial role. As a result, the aggregate data of their experiment, collected from 40 participants driving electric vehicles for 6 months, showcased a large diversity in perceived driving ranges across the surveyed population. Perceived driving ranges are often lower than what we expect or estimate. As an illustration, the distributional pattern of perceived driving ranges and its derivation process are given in Fig. 1. Note that in this figure the nominal, actual and perceived driving ranges are all represented by discrete distributions, which are the direct results sampled from a limited number of vehicles and drivers. This diagram shows that, for any rational electric vehicle driver, to make himself or herself feel “comfortable” or “not anxious”, his or her perceived driving range is typically set lower than the actual driving range his or her vehicle can make, and in turn lower than the nominal driving range of his or her vehicle. Moreover, the distribution of perceived driving ranges tends to exhibit a more scattered or decentralized pattern, compared to actual and nominal driving ranges, since it is the interactive result from multiple stochastic physical and psychological factors. To assess the impact of range anxiety on individual activity-travel choices and network equilibria, we consider for each driver a specific perceived driving range as the upper bound imposed on the driving distance he or she can drive farthest and assume that no driver would choose a path with its physical length greater than this bound. The aggregation of individual upper bounds over the driving population poses a probability distribution, as shown as the leftmost distribution

1

For a comprehensive review on the range anxiety issue and its measure and mitigation strategies, interested readers may refer to Nilsson (2011).

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Fig. 1. Illustration of heterogeneity and derivation of perceived driving ranges.

in Fig. 1. The type and form of the distribution could be various, depending on the influence mechanism and magnitude of the aforementioned physical and psychological factors and on how we measure and represent perceived driving ranges of individual drivers. From the modeling perspective, the distribution may be specified in either a discrete or continuous type. In these regards, some critical research questions may arise up here, from the perspective of modeling and evaluating traffic networks affected by perceived driving ranges: (1) Can we develop a tractable approach to model and solve a network assignment problem with stochastic perceived driving ranges distributed in any given form? (2) If yes, how does the modeling and solution difficulty increase and how is the resulting network flow solution changed, compared to that of its deterministic counterpart? (3) Moreover, if yes, which type of distributions, discrete or continuous, is more technically suitable and advantageous for implementation, in terms of modeling complexity and solution efficiency? To answer these questions, the network equilibrium problem in both types of driving range distributions will be analyzed and numerically evaluated in this study. The remainder of this paper includes the following parts. In Section 2, we discuss our assumptions and specifications for modeling trip chain structures and traffic equilibrium conditions embedded in the proposed network assignment problem, which are simultaneously applicable for the discrete and continuous driving range cases. Next, in the two subsequent sections, Sections 3 and 4, we construct convex programming and variational inequality models for the problem with both the discrete and continuous cases, and develop different implementable procedures of the projected gradient algorithm for the two cases, respectively. Section 5 then presents a numerical analysis on the results from applying the modeling and solution methods to a synthetic example network with different distribution types (discrete vs. continuous) and forms (uniform vs. triangle vs. quadratic vs. lognormal) of driving ranges. Finally, in Section 6, we summarized the paper with our findings and recommendations. 2. Trip chains and network equilibrium Our discussion in this section starts from a number of modeling assumptions and specifications for the proposed network equilibrium problem. These assumptions and specifications are critical and essential to the methodological development in the paper, reflecting the technological barriers we face and the concerns and motivations for which we initialized this research, and hence paving a behavioral and economic basis for the subsequent construction of the modeling framework. Specifically, two important modeling elements related to perceived driving ranges, namely, individual activity-travel choice behavior (i.e., joint activity location and travel path choices) and aggregate network equilibrium conditions, are conceptually and mathematically described. For discussion simplification, we hereafter refer to perceived driving ranges simply as driving ranges. 2.1. Trip chains It has long been recognized that different types of trips in a chain are generally spatially and temporally interrelated and it is necessary to incorporate trip chain structures into traffic network flow and travel demand analyses, if one wants to properly capture mutual effects among interrelated trips in an individual’s travel itinerary (Adler and Ben-Akiva, 1979; Kitamura, 1984, 1988; Shiftan, 1998; Bowman and Ben-Akiva, 20 0 0; Recker, 1995, 20 01). However, casting trip chains into an analytical network assignment model poses a challenging task. To date, the number of existing analytical network assignment models is still very limited that employ trip chains as their basic modeling and analysis unit, due to the caused modeling complexity and solution intractability. Our literature search results show that trip chain-based models for network Please cite this article as: C. Xie et al., Path-constrained traffic assignment: Modeling and computing network impacts of stochastic range anxiety, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.04.018

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assignment have been developed in both static and dynamic (including quasi-dynamic) paradigms. Static models include Maruyama and Harata (20 05, 20 06) and Higuchi et al. (2011), for example. The former researchers presented a set of convex programming models with different types of trip chains and applied these models for evaluating tolled networks; the latter developed a variational inequality model for a static network assignment with trip chains for mixed traffic and transit networks. In the dynamic paradigm, a set of variational inequality models for quasi-dynamic and dynamic network assignment problems with embedded trip chains appeared in Lam and Yin (2001), Lam and Huang (20 02, 20 03), Ouyang et al. (2011), and Fu and Lam (2014), to name a few, in which time-dependent activity disutilities are explicitly modeled as part of trip chain costs. While a number of recent studies (see Jiang et al., 2012, 2013; He et al., 2014; Jiang and Xie, 2014; Xie and Jiang, 2016) incorporated the impact of range anxiety on travel choices on the trip level, some recent travel behavior studies (e.g., Tamor et al., 2013) suggested that this concern more likely occurs on the trip chain level, where a trip chain here is defined as a series of trips between two possible electricity-charging opportunities. In daily travels, more charging opportunities at the origins or destinations of a series of consecutive trips where drivers park their vehicles for much longer time, such as home or workplace, than at intermediate parking places. Given that most electric vehicles in the current market are of a nominal driving range of 60 miles or higher (Borden and Boske, 2013), the resulting perceived driving range is often well beyond the distance of a typical commuting trip or a trip with other purposes, even if the drivers’ anxiety averseness is taken into account and their onboard batteries are not fully charged. To comply with these behavior findings, He et al. (2015) and Wang et al. (2016) extended the above traffic network equilibrium and optimization models to the trip chain level, in which intermediate activity locations along trip chains are specified exogenously and endogenously, respectively, in these two studies. Specifically, Wang et al. (2016) proposed a trip chain-based, distance-constrained network equilibrium modeling framework, in which every driver’s activity location and travel path choices are restrained by his or her driving range and activity sequence. Following this modeling framework, we assume in a similar way that the driving range imposed by range anxiety limits the maximum driving distance along a driver’s trip chain and the driver makes a decision from a set of discrete choices of activity locations and travel paths to satisfy his or her economic or social need while minimizing his or her total activity-travel cost. The driver’s joint activity location and travel path choice behavior can be mathematically characterized by an optimal path problem, which was coined the distance-constrained, node-sequenced shortest path problem in the aforementioned paper. For a technical description of this problem, let us consider it in a traffic network G = (N, A), where N = {n} and A = {a} are the set of nodes and the set of links, respectively. Among the nodes in N, some of them are origin nodes from which travelers start their activity-travel itineraries, some of them are destination nodes at which travelers end their itineraries, and some others of them are activity nodes at which travelers will stop and stay to conduct planned activities. Let us use R = {r}, S = {s} and P = {p} to represent the set of origin nodes, the set of destination nodes, and the set of activity nodes, respectively, where R⊆N, S⊆N and P⊆N. Moreover, we use M = {m} to represent the set of prespecified activity sequences. In addition to these common supply and demand components, some specific network component definitions are also required. For example, we use n+ and n− to represent the sets of outgoing and incoming links from and to node n, respectively, and use i to represent the ith set of activity nodes (i.e., the set of activity nodes where the ith type of planned activities can be conducted) in the given activity sequence. Following these settings, this distance-constrained, node-sequenced shortest path problem for a driver with driving range dw traveling from origin r to destination s and following activity sequence m = {r, 1 , · · · . . . , i , · · · . . . , imax , s}, where r = 0 and s = imax +1 , can be mathematically expressed as,

min

 i

+1 ta xi,i + a

a

subject to

 a∈

a∈

+1 xi,i − a



+1 xi,i − a

a∈



n = r, i = 0 n = s, i = imax + 1

(2)

∀ p ∈ i , i = 1, . . . , imax ∀ p ∈ i+1 , i = 1, . . . , imax

(3)

∀n ∈ N\(i ∪ i+1 ), i = 1, . . . , imax

(4)

+1 xi,i = a

a∈

 − p



 +1 xi,i a

(1)

≤1 ≥ −1

+1 xi,i =0 a

1 −1

a∈

+ n





v p yip

p

i

− n

a∈

+ p

θ

+1 xi,i − a

+ n



1 

− n

+1 xi,i = yip a

∀ p ∈ i , i = 1, . . . , imax

(5)

a∈ + p



yip = 1

i = 1, . . . , imax

(6)

p∈i

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5

Table 1 Notation. Parameters da ca d (dw ) ta0 cp

Physical length of link a Nominal capacity of link a Driving range (of class w) Free-flow travel cost of traffic link a Nominal capacity of activity node p Free-flow activity cost of node p Physical length of trip chain k for the activity-travel demand with activity sequence m from origin r to destination s,  rs rs = da δa,k,m where dk,m

v0p rs dk,m

a

rs qrs m (qm,w )

δ

Activity-travel demand rate with activity sequence m (and driving range w) from origin r to destination s Link-chain incidence indicator, indicating the number of times a traffic link is incident upon a trip chain, where rs δa,k,m = n and n ≥ 1 is an integer number, if traffic link a is used n times by trip chain k for the activity-travel rs = 0 if traffic link a is not used by trip demand with activity sequence m from origin r to destination s, and δa,k,m chain k Node-chain incidence indicator, where δ rs = 1 if activity node p is on trip chain k for the activity-travel demand p,k,m = 0 otherwise with activity sequence m from origin r to destination s, and δ rs p,k,m Activity-travel cost conversion factor

rs a,k,m

δ rsp,k,m θ Variables ta vp i,i+1 tk,m,rs

Travel cost on traffic link a Activity cost at activity site p Travel cost on a trip from activity node of the ith type to another activity node of the (i+1)th type, where this trip is part of trip chain k for the activity-travel with activity sequence m from origin r to destination s Activity-travel cost on trip chain k for the activity-travel with activity sequence m from origin r to destination s, where   rs rs v p δ rsp,k,m = ta δa,k,m + θ1 tk,m

rs tk,m

a

+1 μi,i m,w,rs

μrsm,w i,i+ 1 i,i+1 fk,m,rs ( f k,m,w,rs )

rs rs f k,m ( f k,m,w ) rs fk,l,m rs ρk,m +1 ) xa (xi,i a yp (yip )

 i

p

Minimum activity-travel cost along all trips from an activity node of the ith type to another activity node of the (i+1)th type for the activity-travel demand with activity sequence m and driving range w from origin r to destination s Minimum activity-travel cost along all trip chains for the activity-travel demand with activity sequence m and driving range w from origin r to destination s Traffic flow rate on a trip from an activity node of the ith type to another activity node of the (i+1)th type, where this trip is part of trip chain k generated from the activity-travel demand with activity sequence m (and driving range w) from origin r to destination s Traffic flow rate on trip chain k generated from the activity-travel demand with activity sequence m (and driving range w) from origin r to destination s Traffic flow rate on trip chain k generated from the activity-travel demand with activity sequence m and driving range rs from origin r to destination s d between dlrs and dl+ 1 Traffic flow density on trip chain k generated from the activity-travel demand with activity sequence m from origin r to destination s Traffic flow rate on traffic link a (along a trip connecting the ith type of activities and (i+1)th type of activities) Traffic flow rate at activity site p (for conducting the ith type of activities)

+1 da xi,i ≤ dw a

(7)

a

+1 xi,i = {0, 1} a

yip = {0, 1}

∀a, i = 1, . . . , imax

∀ p, i = 1, . . . , imax

(8) (9)

where all parameters and variables used in the formulation can be referred to in Table 1. Among them, link flow variable +1 xi,i represents the traffic flow rate traversing link a along a trip between the ith type of activities and i + 1 type of activities a and node flow variable yip represents activity flow rate at node p for conducting the ith type of activities; activity cost vp and travel cost ta are given as input here, although they are flow-dependent variables in the network assignment problem discussed later on. Activity type indexes i and i + 1 here are two consecutive ones in the given activity sequence connecting origin r and destination s. In this way, origin r and destination s can be virtually regarded as two special activity types at the starting and ending points of a trip chain, where no cost is incurred at these two dummy activity nodes and only a single activity node exists within each of the two activity types. It should be noted that the above formulation is used for determining an optimal activity-travel path for a single driver, in which the values of all its traffic and activity flow variables in the optimal solution are equal to 1 (indicating those links and sites used by the driver) or 0 (indicating other unused links and sites). While this formulation for the distance-constrained, node-sequenced shortest path problem can be read following the manner by which those classic link-based shortest path problem formulations are constructed (see, for example, Ahuja et al., 1993), we would like to emphasize that this formulation has not appeared in any previous research. Please cite this article as: C. Xie et al., Path-constrained traffic assignment: Modeling and computing network impacts of stochastic range anxiety, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.04.018

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As shown in the above problem formulation, two types of costs, activity costs, defined as disutilities paid or utilities lost by activity participants (including, for example, monetary cost, activity duration, queuing time, service quality and so on), denoted by vp , incurred at activity nodes, and travel costs (including, for example, travel time, operations cost, road toll and so on), denoted by ta , spent on road links are modeled as variables. To reflect the congestion effect, both activity costs and travel costs are defined as flow-dependent variables. Other flow-independent utilities or disutilities related to the problem appear implicitly only in the form of parameters. The underlying reason is that in such a network equilibrium problem, can only those benefit or cost attributes that are a direct result of individual choices become endogenous variables. Moreover, the flow-dependence of these cost variables is set to be only locally in effect, which means that its value is a function of its on-site flow rate only. As a mathematical modeling requirement, we also presume that both the activity cost and travel cost functions are convex, increasing, and continuously differentiable, with respect to their corresponding activity and traffic flow rates. We further set that activity and travel costs along a trip chain are both additive and mutually commensurable; in other words, the total activity-travel cost along a trip chain is the sum of all at-node activity costs and on-link travel costs along the chain. On the other hand, when evaluating the impact from the distance limit, we only take into account the physical length of traffic links in calculating the total length of a trip chain, since in general no activity needs to consume electricity from vehicle batteries. In other words, the total length of a trip chain is the sum of lengths of all traffic links along the trip chain. Due to the incorporation of prespecified activity sequences, the activity-travel demand between any origin–destination pair can be accordingly classified. Specifically, for each origin–destination pair, its travel demand can be distinguished for each activity sequence such as “school-shopping-dining”, “work-dining-shopping”, and “dining-shopping-entertainment”. As a result, the activity-travel demand over the network is distinguished by origin–destination pairs and activity sequences. Accordingly, a unique feasible choice set of activity nodes and travel paths can be specified for each combination of origin– destination pair and activity sequence. When driving range is taken into account, the demand may be further differentiated; however, how to proportion the demand into different classes in terms of driving range values poses a question with very different answers for the discrete and continuous driving range cases. This issue will be dealt with in detail in the model and algorithm development parts of this paper. Finally, it must be acknowledged that such an exogenous specification of activity sequences for individual travelers is a quite strict behavioral assumption. A more flexible setting that allows the model to endogenously determine individual activity sequences can be readily realized, but it involves solving a distance-constrained version of the generalized traveling salesman problem (Srivastava et al., 1969; Laporte and Nobert, 1983; Laporte et al., 1987) through a prespecified number of sets of activity nodes and hence considerably increases the computing burden.

2.2. Equilibrium conditions Based on the above assumptions and specifications, the main task of this paper is set to developing modeling and solution methods for the proposed network equilibrium problem that explicitly considers the impacts of heterogeneous range anxiety on activity location and travel path choices over trip chains. For this purpose, we define below a set of equilibrium conditions for the proposed problem from the perspective of individual choice behaviors, which will be used as a behavioral basis for constructing and validating our proposed models. Since it is defined on the individual level, this set of equilibrium conditions is applicable for both discrete and continuous driving range cases. For discussion convenience, let us still use here the same network components in the network G defined above, including the set of nodes, N, set of links, A, set of origins, R, set of destinations, S, and set of activity sites, P. For each rs to represent the set of trip chains connecting this origin– origin–destination pair r −s and activity sequence m, we use Km destination pair and following this activity sequence. For other parameters and variables used in the discussion below as well as the subsequent text, their notation is given in Table 1. Following the classic user equilibrium principle proposed by Wardrop (1952), the equilibrium conditions for the proposed problem subject to the driving range and activity sequence constraints can be stated as: For a certain class of activity-travel demand with driving range w, moving between origin–destination pair r −s and through activity sequence m, if the total length of a trip chain is no longer than the driving range and the total cost of this trip chain is equal to the minimum cost of all trip chains, this trip chain may carry a positive amount of traffic flow generated from this demand; otherwise, i.e., either the total length of a trip chain is longer than the driving range or the total cost of this trip chain is higher than the minimum one, this trip chain must carry no flow. Mathematically, the equilibrium conditions can be expressed as:



∗ ∗ ∗ rs rs dk,m ≤ dw and t rs = μrs m,w ⇒ f k,m,w ≥ 0 k,m rs dk,m

>

∗ dw or t rs k,m

>μ

rs ∗ m,w

⇒

∗ f rs k,m,w

=0

∀r, s, m, w, k ∈ Kmrs

(10)

rs , d rs , and μrs where tk,m m,w are the total cost, total length, and minimum cost of trip chain k, experienced by the activityk,m rs travel demand of activity sequence m and driving range w traveling between origin r and destination s, f k,m,w is the traffic flow rate on trip chain k generated from this class of demand, and dw is the driving range of class w. It is noted that the   rs rs = rs total activity-travel cost tk,m is the sum of all its related activity costs and link costs, i.e., tk,m ta δa,k,m + θ1 v p δ rs , p,k,m a

p

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 a

7

rs da δa,k,m . Here we use “∗ ” to indicate the

values of decision variables in the equilibrium solution. rs Meanwhile, traffic flows in the network must satisfy a set of constraints, where two (sub)path flow variables, f k,m,w

i,i+1 and fk,m,w,rs , for the trip chain and trip levels are used. The two path flow variables are defined exactly the same as the definition in Beckmann’s (1956) transformation, in which a path between an origin–destination pair is a combination of a consecutive series of links connecting the origin and destination. These constraints are: i,i+1 fk,m,w,rs ≥0

∀r, s, m, w, i, k ∈ Krsm

(11)

i,i+1 rs fk,m,w = fk,m,w,rs

∀r, s, m, w, i, k ∈ Krsm

(12)

∀r, s, m, w, k ∈ Krsm

(13)



rs fk,m,w = qrs m,w

k

where qrs m,w is the travel demand rate activity sequence m and driving range w from origin r to destination s. It is noted that in a network under our setting, a trip chain is represented by a path, and the trips on this trip chain are represented by subpaths of this path. Constraint (11) simply indicates the nonnegativity of the traffic flow rate on any trip that connects two activity nodes, which belong to two consecutive activity types. Constraint (12) establishes an equivalency relationship of traffic flows between the trip chain and trip levels; it also means that a traffic stream on any trip chain can be spatially decomposed into a set of consecutive traffic streams on individual trips of the trip chain. Constraint (13) indicates the flow conservation between an origin–destination pair. Following the above assumptions and specifications, we then turn to discuss the models and solution algorithms for the proposed network assignment problem, for both discretely and continuously distributed driving ranges, which constitute the main block of this paper. 3. Models In view of the separate cost-flow relationships between different network components and resulting symmetric costflow Jacobian matrices, this section describes how we use convex optimization and variational inequality techniques to construct models for the proposed network assignment problem. Both discretely and continuously distributed driving range cases are considered. For the discrete case, it is readily known that the resulting model can be regarded as a multi-class version extended from the simpler models in, for example, Wang et al. (2016), in which each class of travel demand has a unique driving range value and a separate flow-distance complementary constraint. For the continuous case, however, such a direct extension would result in a model with an infinite number of constraints, which is generally not preferred, from both the modeling and algorithmic perspectives. Thus we resort to a different approach to achieve its formulation and solution algorithm. Without loss of generality, we preset driving ranges to be distributed in the following manner. In the discrete case, driving range is specified by a set of discrete classes of maximum driving distance values {…, dw , …}, where dw is the indexed by its class number w; in the continuous case, it is a continuously distributed parameter denoted by d, where the distribution is generally in a bounded or half bounded interval, i.e., d ∈ [dmin , dmax ] or d ∈ [dmin , + ∞). To ensure the solution feasibility of the proposed problem, we assume that for each origin–destination pair r −s, min dw ≥ dkrs , where dkrs

min

= min{dkrs }, in the discrete driving range case, and dmin ≥ dkrs k∈Krs

min

, where dkrs

min

w

min

= min{dkrs }, in the continuous case. These k∈Krs

settings guarantee that for any origin–destination pair r-s of the network, every driver has at least one feasible path. As a prerequisite for the following modeling work, a complete notation list for all used parameters and variables throughout this text is provided in Table 1. 3.1. Discretely distributed driving ranges Given a set of discrete driving range values, the constraints for the proposed network assignment problem may be described by the following system of equations and inequalities. Specifically, constraints (14)–(16) are the same as those presented in the given equilibrium conditions, which specify the flow conservation on the origin–destination level, flow equivalency between trip and trip chain levels, and flow nonnegativity on the trip level, respectively. Constraint (17) then presents rs rs a complementarity relationship between redundant driving distance dw − dk,m and path flow rate fk,m,w , implying that if rs rs rs rs dw ≥ dk,m , then fk,m,w ≥ 0, and if dw < dk,m , then fk,m,w = 0.



rs fm,k,w = qrs m,w

∀r, s, m, w

(14)

∀r, s, m, w, i, k ∈ Krsm

(15)

k i,i+1 rs fm,k,w = fm,k,w,rs

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i,i+1 fm,k,w,rs ≥0





rs rs dw − dk,m fk,m,w ≥0

∀r, s, m, w, i, k ∈ Krsm

(16)

∀r, s, w, m, k ∈ Krsm

(17)

Please note that constraint (17) is flow-independent, which means that the feasible path set for each class of travel demand could be determined in advance. However, doing this requires path enumeration, which is computationally tractable only for problems of small size. If a large-size problem is given, we are then required to develop a more efficient method that involves a procedure of efficiently generating paths as needed. For discussion convenience below, we denote by  the feasible region confined by constraints (14)–(17). Now we are ready to present the convex programming and variational inequality formulations for the discrete driving range case. Following Beckmann’s (1956) transformation, a convex programming problem is given as:

min z (x, y ) =

 a

where xa =

0

ta (ω )dω +

 rs

yp =

xa

w

 rs w m

k

m

 1

θ

rs rs fk,m,w δa,k,m

p

yp

0

v p (ω )dω

(18)

∀a

(19)

k

rs fk,m,w δ rsp,k,m

∀p (20)

( x, y ) ∈  where constraints (19) and (20) are two definitional constraints for the traffic flow rates on road links and at activity nodes, rs in which δa,k,m and δ rs are link-chain and node-chain incidence indicators, respectively. It should be noted here that linkp,k,m rs chain incidence indicator δa,k,m may be an integer number greater than 1, indicating link a may be used multiple times by trip chain k for the demand of activity sequence m from origin r to destination s. In contrast, node-chain incidence indicator δ rs are still a 0–1 integer number, implying that an activity node is visited at most once by a driver in his or her trip p,k,m chain. Alternatively, a variational inequality in the following mathematical form determines the equilibrium solution (x∗ , y∗ ) ∈  of the proposed network assignment problem for the discrete driving range case:

1

t∗ ( x − x∗ ) + where xa =

θ

v∗ ( y − y ∗ ) =

yp =

 rs w m

k

ta (x∗a )(xa − x∗a ) +

a

 rs



w

m

rs rs fk,m,w δk,a,m

∀a

1

θ

   v p y∗p y p − y∗p ≥ 0

(21)

p

(22)

k

rs rs fk,m,w δk,p,m

∀p (23)

( x, y ) ∈  he solution existence, equivalency and uniqueness of the above two formulations can be obtained by standard nonlinear programming and variational inequality analysis methods (see Sheffi, 1985; Nagurney, 1993), respectively. For the sake of space saving, the details of deriving these solution properties are omitted here. 3.2. Continuously distributed driving ranges There are a couple of particular reasons for which we are more interested in studying the proposed network assignment problem with continuously distributed driving ranges. First, as a result of various physical and psychological influence factors, the distribution of perceived driving ranges is often more appropriately approximated by a continuous distribution rather than a discrete one. Second, even if the distribution naturally falls into a discrete type, when the number of alternative range values is large, a discrete model based on a combination of multiple classes of travel demand with different driving ranges do not seem to be a favorable choice in terms of solution efficiency, since it will require executing a large number of network loadings in the solution process, no matter what kind of solution algorithms is used. As we know, the network loading process, or more accurately, its included path generation process, is the most time-consuming step in any network assignment algorithm. For the continuous driving range case, we believe that two alternative modeling elements are the key for constructing models with only a limited number of constraints. The first element is a redefined path set for each origin–destination pair rs , which indexes all its included paths in the increasing order of path r − −s and each activity sequence m, denoted by K¯m length: rs rs = {1, . . . , |Km K¯m |}

∀r, s

(24)

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rs . which is just an element-ordered counterpart of Km rs (d rs ≤ d < d rs ), which is defined as the The second one is an alternative path flow variable named path subflow rate, f k,m l l+1 path flow rate on path k connecting origin–destination pair r − −s and following activity sequence m, the driving range of which is set between the lengths of paths l and l + 1 in the ordered path set of this origin–destination pair:







rs rs fk,m dlrs ≤ d < dl+1 =

rs dl+1

dlrs

rs ρk,m (θ )dθ

∀r, s, k ∈ K¯mrs , l ∈ K¯mrs

(25)

rs (d rs ≤ d < d rs ) = 0, when l < k, d rs > d rs rs rs rs where it is noted that fk,m max , or dl+1 ≤ dmin ; and f k,m (dl ≤ d < dl+1 ) ≥ 0, if otherl l+1 l rs , the path flow density ρ rs (d ) and path subflow rate wise. It should be emphasized here that like the path flow rate f k,m k,m rs (d rs ≤ d < d rs ) are both variables, where d is a stochastic parameter and d rs and d rs can be regarded as a determinisfk,m l l+1 l l+1 rs (d rs ≤ d < d rs ). For notational convenience, we rewrite f rs (d rs ≤ d < d rs ) in a more concise form as tic parameter of fk,m l l+1 k,m l l+1 rs fk,l,m in the remaining part of this paper. Thanks to the introduction of path subflow rate, the constraint set for the continuous driving range case can then be written as:



rs fk,m = qrs m

∀r, s, m

(26)

∀r, s, m, i, k ∈ K¯mrs , l ∈ K¯mrs

(27)

k i,i+1 rs fk,m = fk,m,rs

rs fk,l,m ≥0

∀r, s, m, k ∈ K¯mrs , l ∈ K¯mrs

(28)

rs ≥0 (l − k ) fk,l,m

∀r, s, m, k ∈ K¯mrs , l ∈ K¯mrs

(29)

where

rs fk,m

=

rs K¯m | |

∀r, s, m, k ∈ K¯mrs

rs fk,l,m

(30)

l=1

 rs fk,l,m =

rs dl+1

rs ρk,m (θ )dθ

dlrs

∀r, s, m, k ∈ K¯mrs , l ∈ K¯mrs

(31)

Compared to the counterpart for the discrete driving range case, the above constraint set presents a very similar structure rs and path subflow rate f rs , the flowand set of relationships. However, with the introduction of ordered path set K¯m k,l,m distance complementarity relationship (i.e., constraint (29)) and path flow rate (i.e., constraint (30)) are defined in a different way. As for the model construction, since the objective function of the convex programming problem and the main inequality of the variational inequality problem only take advantage of link-based and node-based flow variables, they can be specified in the same manner as those in the discrete driving range case. To this end, by incorporating the above constraint set (i.e., constraints (26)–(31)), the convex programming model can be written as:

min z (x, y ) =



0

a

where

xa =

 rs

m

ta (ω )dω +

 rs

yp =

xa

m

 1

θ

rs rs fk,m δa,k,m

p

yp 0

v p (ω )dω

(32)

∀a

(33)

k

rs fk,m δ rsp,k,m

∀p

(34)

k

x, y subject to (26)–(31) and the variational inequality model is written as:

t∗ ( x − x∗ ) + where

1

θ

xa =

v∗ ( y − y ∗ ) =



 rs

m

ta (x∗a )(xa − x∗a ) +

a

rs

yp =



m

rs rs fk,m δa,k,m

∀a

1

θ

   v p y∗p y p − y∗p ≥ 0

(35)

p

(36)

k

rs fk,m δ rsp,k,m

∀p

(37)

k

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x, y subject to (26)–(31) where (x∗ , y∗ ) represents the equilibrium solution of the proposed network assignment problem for the continuous driving range case. Once again, we omitted the proof for the solution existence, equivalency and uniqueness of the above convex programming and variational inequality problems. For the continuous driving range case, these solution properties can also be readily proved. Interested readers may refer to a conference paper authored by the same authors for those technical details (Pu et al., 2016). Remark 1. For any equilibrium solution of the proposed network assignment problem with stochastic driving ranges, if the physical length of an arbitrary used path k is greater than or equal to that of another arbitrary (used or unused) path k , rs∗ > 0, where paths k and k connect the same origin–destination pair r −s and follow the same activity i.e., dkrs ≥ dkrs and fk,m rs∗ ≤ t rs∗ , where t rs∗ and t rs∗ are the activitysequence m, then the activity-travel costs on these two paths must satisfy tk,m  ,m k ,m    ∗ k rs k,m∗ rs 1 rs∗ = rs rs ∗ = travel costs on paths k and k , i.e., tk,m ta∗ δk,a,m +θ v∗p δk,p,m and tkrs ,m ta δk ,a,m + θ1 v p δk ,p,m . a

a

a

a

It is noted that this conclusion is irrelevant to the distribution type and form of driving ranges and can be more concisely written as:







rs∗ ∗ dkrs − dkrs tk,m − tkrs ,m ≤0

∀r, s, m, k ∈ Krsm , k ∈ Krsm

rs∗ > 0, or d rs ≤ d rs and f rs∗ > 0. if dkrs ≥ dkrs and fk,m k k ,m k

4. Solution algorithms The remaining focus of this paper is to develop and implement an efficient algorithmic framework for solving the proposed problem in both the discrete and continuous driving range cases. This section presents such a solution algorithm of the quadratic approximation type, the core algorithmic logic of which is to iteratively make a descent move along the projected gradient direction. This so-called projected gradient method was first proposed by Rosen (1960) for nonlinear programming problems with linear constraints. An adaptation of this method for the prime traffic assignment problem was achieved and tested by Florian et al. (2009), in which for any origin–destination pair the projected descent direction of the problem’s objective function with respect to a path flow rate is the difference between the travel cost on this path and the average travel cost among all used paths of the origin–destination pair. The same idea of making use of projected descent directions for improving the objective function value is adopted here, but its implementation under the constraint of stochastic driving ranges is much more different and complicated. As similar to Florian et al. (2009), our adaption of this method employs the Gauss–Seidel decomposition to separate the path generation and equilibration process for different demand proportions in a sequential manner. However, the decomposition is conducted on a different level from Florian et al. (2009) implementation. Moreover, due to the existence of driving ranges and activity sequences, path generation in our adaptation involves solving a distance-constrained, node-sequenced minimum cost path problem, which adds computing complexity significantly. The following text elaborates the two algorithmic procedures for the discrete and continuous driving range cases and provides detailed explanatory comments for the algorithmic design when needed. 4.1. Discretely distributed driving ranges The projected gradient method implemented for solving the proposed network assignment problem with discrete driving ranges is a relatively easy task, posing such an iterative process as follows until the solution convergence. Specifically, for each combination of any origin–destination pair r −s, activity sequence m, and driving range w in the sequential process, the algorithm performs the following algorithmic steps: Step 0 (Initialization): If this origin–destination pair is dealt with for the first time, find an initial feasible network flow rs pattern {xrs a } for the origin–destination demand qm,w by a k-minimum cost path algorithm in terms of the current travel costs over the network, where k is the minimum number of paths on which qrs m,w can be exhaustively assigned; otherwise, go to step 1. Step 1 (Direction finding): The descent direction for altering the flow rate on path h is: rs ¯rs grs h,m,w = tm,w − th

rs+ ∀h ∈ Hm,w

(38)

rs+ rs where thrs is the travel cost on path h, t¯m,w is the average travel cost of all paths in Hm,w , i.e.,



rs = t¯m,w

rs+

t rs

h∈Hrsm,w h + Hm,w

(39)

rs+ rs+ rs | f rs > 0}. If the convergence criterion is met, i.e., and Hm,w is the set of paths that carry traffic flows, i.e., Hm,w = {h ∈ Hm,w h rs maxh∈H rs+ (|gh | ) <  , go to step 4. m,w

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11

rs∗ that solves the following one-dimensional optimization problem: Step 2 (Line search): Find an optimal step size αm,w

min



rs rs xrs a,m,w +x¯ a +αm,w x˙ a,m,w

0

a

ta (ω )dω



rs subject to 0 ≤ αm,w ≤ min rs+

where (1) x˙ rs a,m,w =

 rs+ h∈Hm,w

h∈Hm,w

−

(40)

rs fh,m,w

grs h,m,w

grsh,m,w < 0

 (41)

grs δ rs is the descent direction of the traffic flow rate on link a that connects origin– h,m,w a,h,m,w

destination pair r-s, and (2) x¯a =

 



r¯s¯ =rs m w h∈H r¯s¯+ m,w

r¯s¯ r¯s¯ fh,m,w δa,h,m,w +

 



¯ =m w h∈H rs+ m ¯ ,w m

rs fh, δ rs ¯ ,w is the existing traffic flow rate ¯ ,w a,h,m m

on link a that connects all origin–destination pairs in the network except r −s. rs+ rs rs rs∗ grs rs rs∗ rs Step 3 (Flow update and path elimination): Set fh,m,w = fh,m,w + αm,w , ∀h ∈ Hm,w and xrs a,m,w = xa,m,w + αm,w x˙ a,m,w , h,m,w

rs+ rs+ ∀a ∈ A. If path flow rate fhrs , where h ∈ Hm,w , diminishes to zero, path h should be eliminated from the path set Hm,w , i.e., rs+ rs+ Hm,w = Hm,w \{h }. Then update xa and ta , ∀a ∈ A.

Step 4 (Path generation): Find the minimum cost path by a path generation algorithm in terms of the updated travel costs over the network, subject to the driving range w of the current demand proportion. Denote this shortest path by k. If the travel cost of this constrainted minimum cost path satisfies tkrs < minh∈H rs+ thrs (where this condition can be equivalently m,w

rs+ rs+ rs+ represented by k ∈ / Hm,w ), set Hm,w = Hm,w ∪ {k} and return to step 1; otherwise, it means that the stopping criterion is satisfied, stop the procedure. The entire algorithmic process will be completed when every combination of origin–destination pair, activity sequence and driving range in the network achieves the stopping criterion in step 4.

Remark 2. Compared to the previous application of the projected gradient algorithm for the prime network assignment problem in Florian et al. (2009), this implementation differs mainly in two aspects: (1) The decomposition is conducted by origin–destination pairs, activity sequences and driving ranges, instead of origin–destination pairs only; (2) Path generation is accomplished by a cascading label-correcting algorithm (Wang et al., 2016) for a distance-constrained, node-sequenced minimum cost path problem, instead of a label-setting algorithm (e.g., Dijkstra’s, 1959 algorithm) for a simple minimum cost path algorithm. 4.2. Continuously distributed driving ranges The projected gradient method implemented for the proposed network assignment problem with continuous driving ranges has the same structure as the above, but its line search and path generation steps involves much more complex operations. Its algorithmic steps can be sketched as follows. Specifically, for each combination of origin–destination pair r −s and activity sequence m in the sequential process, the algorithm performs the following algorithmic steps: rs Step 0 (Initialization): Find an initial feasible network flow pattern {xrs a } for the origin–destination demand qm by a kminimum cost path algorithm in terms of the current travel costs over the network, where k is the minimum number of rs+ paths on which qrs m can be exhaustively assigned. Use Hm to denote the path set that includes all these k paths. rs+ rs+ For each path h ∈ Hm , form such a subset of Hm that the length of each path in the subset is shorter than or equal to dhrs :



rs+ rs+ rs Hh,m = l ∈ Hm |dl ≤ dhrs

∀h ∈ Hmrs+

(42)

rs+ Hh,m

rs+ where we name the h-indexed path subset. It is readily known that Hh,m is a feasible path set for a range of origin– rs rs destination demand rate with its driving range within the interval of [dhrs , dhrs+1 ), qrs m (dh ≤ D < dh+1 ), which we name the rs ≤ d < d rs ) by qrs . Accordingly, the flow rate on link a h-indexed origin–destination subdemand rate. Let us denote qrs ( d m h h+1 h,m contributed by qrs , xrs , is named the h-indexed link subflow rate. h,m a,h,m rs+ rs+ Step 1 (Direction finding): For each path h ∈ Hm , the descent direction for altering the flow rate on path l ∈ Hh,m is: rs ¯rs grs h,l,m = th,m − tl

where

tlrs

+ ∀l ∈ H rsh,m

is the travel cost on path l and



l∈H rs+

(43) rs t¯h,m

is the average of the travel costs on all paths in

tlrs

rsh,m+ H

rs = t¯h,m

rs+ Hh,m ,

i.e.,

(44)

h,m

rs+ rs+ If the convergence criterion is met, i.e., max{|grs ||l ∈ Hh,m , h ∈ Hm } <  , go to step 4. l,h,m

rs+ rs∗ that solves the following one-dimensional Step 2 (Line search): For each path h ∈ Hm , find an optimal step size αh,m optimization problem:

min

 a

rs ˙ rs xrs +x¯a +αh,m xa,h,m a,h,m

0

ta (ω )dω

(45)

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subject to 0 ≤

rs  fh,l,m rs rs gh,l,m < 0 αh,m ≤ min − rs rs+ l∈Hh,m

(46)

gh,l,m

where some used link flow or subflow variables are defined as: (1) xrs is the h-refered subflow rate on link a traveling a,h,m between origin–destination pair r −s and following activity sequence m,

xrs a,h,m =



rs rs fh,l,m δa,l,m

rs+ l∈Hh,m

(2) x˙ rs is the descent direction of xrs , a,h,m a,h,m

x˙ rs a,h =



rs grs h,l,m δa,l,m

rs+ l∈Hh,m

and (3) x¯a is the current flow rate on link a except the h-indexed subflow rate on this link,

x¯a =

  

r¯s¯ r¯s¯ fh,l,m δa,l,m +

  

rs rs fh,l, ¯ δa,l,m ¯ + m

rs+ rs+ ¯ =m h∈Hm m l∈Hh, ¯ ¯ m

r¯s¯+ r¯s¯+ r¯s¯ =rs m h∈Hm l∈Hh,m

 

rs+ h¯ =h l∈Hh¯ ,m

rs fh¯rs,l,m δa,l,m

rs+ rs rs Step 3 (Flow update and path elimination): Set a temporary path set P = ∅. For each path h ∈ Hm , set fh,l,m = fh,l,m + rs+ rs rs rs rs ∗ rs rs  ∀l ∈ H h,m +, and xa,h,m = xa,h,m + αh,m x˙ a,h,m , ∀a ∈ A, where if path subflow rate fh,l  ,m , l ∈ Hh,m , diminishes to

rs∗ grs αh,m , h,l,m

rs+ rs+ rs+ rs+ zero, path l should be eliminated from path subset Hh,m , i.e., Hh,m = Hh,m \{l  }3 , and update P = P ∪ Hh,m .  rs rs+ rs+ Set Hh,m = P . Update xa = x¯a + xa,h,m and ta = ta (xa ), ∀a ∈ A. If any path l is eliminated from path set Hm (i.e., path h∈H rs+

h,m

rs+ rs+ l is eliminated from each path subset Hh,m in Hm , rs+  path l − 1 ∈ Hh,m are reset as,

∀h ∈ Hmrs+ ), the path subset and path subflow rates with reference to

+ + + Hlrs −1 = Hlrs −1 ∪ Hlrs ,m ,m ,m

(47) + + ∀l ∈ Hlrs −1 ∪ Hlrs ,m ,m

flrs −1,l,m = flrs −1,l,m + flrs ,l,m

(48)

Step 4 (Path generation): Find the minimum cost path by a path generation algorithm in terms of the updated travel costs over the network, subject to the maximum driving range across the demand population, dmax . Denote by k this distancers+ constrained minimum cost path, denote by k the path satisfying dkrs = max{dhrs |h ∈ Hm , dhrs < dkrs }, and denote by k  the rs+ path satisfying dkrs = min{dhrs |h ∈ Hm , dhrs > dkrs }. If the travel cost of this path satisfies tkrs > minh∈H rs+ thrs , it means that the m stopping criterion for origin–destination pair r–s is reached and we should stop the algorithmic procedure for this origin– destination pair (i.e., steps 1–4). If tkrs < minh∈H rs+ thrs is satisfied, do one of the following. m

rs+ rs+ (1) If dkrs > dmin , set Hm = Hm ∪ {k}. For path k, its path subset and path subflow rates in this subset are given as:4 rs+ rs+ Hk,m = Hk,m ∪ {k}

 rs fk,l,m

=

(49)

( ) (dkrs ≤d
qrs dkrs ≤d
fkrs ,l qrsm m

0

+ ∀l ∈ Hkrs ,m

l=k

(50)

then, for path k , its path subset is not changed but the path subflow rates in the subset is updated as:

fkrs ,l

=

fkrs ,l

  rs

  rs

rs rs qrs m dk ≤ d < dk

qm dkrs ≤ d < dk

+ ∀l ∈ Hkrs ,m

(51)

rs+ rs+ rs+ and, finally, for any other path h ∈ Hm , if dkrs < dhrs , its path subset is reset as Hh,m = Hh,m ∪ {k} and the link subflow rate with reference to this newly added path k is zero, otherwise their path subsets and corresponding path subflow rates are still kept the same. Return to step 1. rs+ rs+ rs+ rs+ (2) If minh∈H rs+ dhrs < dkrs ≤ dmin , set Hm = Hm ∪ {k} and Hm = Hm \ {k }, where path k satisfies dkrs = minh∈Hmrs+ dhrs . For m path k, its path subset and path subflow rates in this subset are given as: rs+ + Hk,m = Hkrs ,m ∪ {k }

(52)

rs+ 3 . We emphasize here that even if path l is the referred path h of the path subset, i.e., l = h, path l still needs to be eliminated from path subset Hh,m This possible result implies that the referred path of a path subset does not necessarily exist in the subset. 4 rs rs rs+ rs When dk = max{dh |h ∈ Hm }, dk+1 in this function should be replaced by dmax .

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 rs fk,l,m

=

fkrs ,l,m 0

+ ∀l ∈ Hkrs ,m l=k

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(53)

rs+ rs+ rs+ and, for any other path h ∈ Hm , its path subset is reset as Hh,m = Hh,m ∪ {k} and the link subflow rate with reference to this newly added path k is zero. Return to step 1. rs+ rs+ rs+ (3) If dkrs < minh∈H rs+ dhrs , for any path h ∈ Hm , its path subset is reset as Hh,m = Hh,m ∪ {k} and the link subflow rate with m reference to this newly added path k is zero. Return to step 1. The entire algorithmic process will be completed when the path flow pattern generated from the demand proportion of each combination of origin–destination pair and activity sequence in the network reaches the stopping criterion in step 4. While the above algorithmic procedure for the continuous driving range case poses a very complex algorithmic implementation and is worth a further detailed discussion, we would like to remark below a couple of quick facts about its unique features and advantages. rs+ rs , Remark 3. In this algorithmic procedure, we introduced the definitions of path-indexed path subset, Hh,m , ∀r, s, m, h ∈ H¯ m rs rs rs ¯ path-indexed origin–destination subdemand rate, qh,m , ∀r, s, m, h ∈ Hm , and path-indexed link subflow rate, xa,h,m , ∀a, r, s, rs . By using them, we can in each iteration separately identify a descent direction and conduct a line search for each m, h ∈ H¯ m rs+ portion of origin–destination demand with its driving range distributed between dhrs and dhrs+1 , ∀h ∈ Hm , where the demand proportion confined by this length interval is fully homogeneous in terms of the route choice behavior and constraint.

However, compared to the application of the projected gradient algorithm for the prime traffic assignment problem, the increasing complexity in this algorithm, with the use of path-indexed origin–destination subdemand and link subflow rates, is due to the operations of inserting and removing paths from path subsets and the corresponding decomposition and combination of path-indexed origin–destination subdemand and link subflow rates. Remark 4. Compared to the linear approximation algorithm adopted for solving the proposed network assignment problem (see Pu et al., 2016), this algorithmic procedure of the quadratic approximation type runs at an evidently faster convergence speed. The underlying reasons for this are at least twofold: (1) In terms of algorithmic principle, the projected gradientbased flow equilibration process in the quadratic approximation procedure makes use of second-order derivatives, which implies higher convergence efficiency than the gradient-based flow equilibration process in the linear approximation that makes use of first-order derivatives only; (2) in terms of computational complexity, utilizing a constrained shortest path algorithm for path generation in the quadratic approximation procedure is typically more efficient than a k-shortest path algorithm in the linear approximation procedure. 4.3. Path generation From the algorithmic procedures presented above, it has been seen that, for both the discrete and continuous cases, the projected gradient method requires repeatedly solving a distance-constrained, node-sequenced minimum cost path problem for generating new paths with respect to a given activity sequence (see step 4 in either of the procedures). This path-finding problem was formulated and discussed in Section 2.2. It is well known that path generation is in general the most computationally intensive part of a network assignment algorithm. The above path generation process is not an exclusive case. For this algorithmic requirement, as we mentioned earlier, we embedded a cascading label-correcting algorithm into the above algorithmic procedures. Its core algorithmic component is a quadruple label set with each node and its main algorithmic operation is to repeatedly use a bicriterion (i.e., travel cost and travel distance) domination rule to compare and update label sets of nodes across the network until no label set in the network can be updated. Due to the requirement of following an activity sequence, the above labeling, comparing and updating process must be conducted in a cascading manner through the ordered activity node sets specified by the given activity sequence. The technical details of this embedded algorithm can be found in Section 4.1 of Wang et al. (2016). 4.4. Summary By comparing the two procedures for the proposed network assignment problem for the discrete and continuous driving range cases, we identified three major algorithmic differences between them: (1) Although both procedures are implemented in the Gauss–Seidel decomposition framework, their decompositions levels are different: The discrete procedure decomposes the problem by origin–destination pairs, activity sequences, and driving ranges, while the continuous procedure decomposes it by origin–destination pairs and activity sequences only. (2) In the discrete case, path equilibration occurs on each path set specific to a combination of origin–destination pair, activity sequence and driving range, while in the continuous case path equilibration occurs on each path subset, as specified by a combination of origin–destination pair and activity sequence. (3) For the above two reasons, the continuous case has to dynamically determine driving range intervals for proportioning the demand and decomposing the problem, and hence resorts to a set of much more complex algorithmic operations, Please cite this article as: C. Xie et al., Path-constrained traffic assignment: Modeling and computing network impacts of stochastic range anxiety, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.04.018

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Fig. 2. A test network for numerical analysis.

compared to the discrete case, for dealing with the separation and combination of path flows and path subsets to conduct each path elimination and path generation. As for a further comparison in solution efficiency, we will have to wait until analyzing the computational results from a numerical study below. For the subsequent numerical analysis, we coded the two developed algorithmic procedures in C++, in which the common parts of the two procedures share the same code. The following section details the settings and results of this numerical study. 5. Numerical analysis The numerical analysis presented below was powered by applying the developed solution procedures for a synthetic example network under various types and forms of driving range distributions, in which our purpose is to evaluate the impacts of stochastic driving ranges on network flows and computational costs. Whenever appropriate, the analysis results will be explained from the perspective of comparing the discrete and continuous driving range cases. Using a fully synthetic network in this study is simply due to the unavailability of trip chain-related travel demand data. As a result, we emphasize that this analysis is purely for the purpose of methodological illustration, from which we do not expect to derive any activity-travel behavior insight or any transportation management policy. The computing environment for all the network assignment experiments in the numerical analysis is a Microsoft Windows-based desktop computer with an Intel Core i5-4200 U 3.2 GHz CPU and 4 GB RAM. The C++ programs were compiled by Microsoft C/C++ Optimizing Complier and a common convergence criterion of 10−5 was used in all the experiments. 5.1. Example network The hypothetical network owns a typical grid topology of urban road networks, as shown in Fig. 2, consisting 24 nodes and 86 links. There are three types of nodes in the network: Origin or destination nodes (representing residential areas or workplaces), activity nodes (representing dining places, shopping malls, entertainment centers, or other activity sites), and other intermediate nodes (representing interchanges or intersections). For discussion convenience, we use the following capital letters to label these different types of nodes: • • • • •

H: Residential areas (including nodes 1, 5 and 10) W: Workplaces (including nodes 15 and 24) D: Dining places (including nodes 4, 7, 9, 18 and 19) S: Shopping malls (including nodes 16, 21 and 22) E: Entertainment centers (including nodes 12 and 13)

Each line segment connecting a pair of nodes in the network represents a pair of links with counter traffic directions, where any pair of counter links are assumed to own the same link attributes. The numbers beside each line segment indicate its free-flow travel cost, physical length, and capacity, respectively. In addition, the network supply attributes are specified by the following travel link cost function,



ta =

ta0

1+α

 x β  a

ca

∀a

(54)

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Fig. 3. Probability density functions of the assumed driving range distributions.

where α = 0.15 and β = 4, and the following activity node cost function,

γ   x v p = v0p 1 + δ p ∀p cp

(55)

where δ = 0.1 and γ = 5. Moreover, the following activity-related parameter values are applied to different types of activity nodes: v0P = 20 and cp = 400 for dining places D2 , D3 , and D5 (i.e., nodes 7, 9 and 19); v0P = 30 and cp = 300 for dining places D1 and D4 (i.e., nodes 4 and 18); v0p = 30 and cp = 600 for all shopping malls; v0p = 20 and cp = 800 for entertainment center E1 (i.e., node 12); v0p = 30 and cp = 10 0 0 for entertainment center E2 (i.e., node 13). Finally, we arbitrarily set the activity-travel cost conversion factor equal to 1.0, i.e., θ = 1.0. On the other hand, a hypothetical travel demand table for this network is given below, as specified by the combinations of origin–destination pairs and activity sequences: • • • • • • • • • • • •

H1 -D-S-H1 : 300 veh/h H1 -D-E-H1 : 450 veh/h H2 -D-H2 : 300 veh/h H2 -D-S-H2 : 450 veh/h H3 -D-S-E-H3 : 150 veh/h H3 -S-D-E-H3 : 300 veh/h W1 -D-S-H1 : 600 veh/h W1 -D-H1 : 450 veh/h W1 -S-D-H2 : 300 veh/h W1 -D-S-H3 : 300 veh/h W2 -D-S-H1 : 750 veh/h W2 -D-E-H3 : 600 veh/h

where D = {D1 , D2 , D3 , D4 , D5 }, S = {S1 , S2 , S3 }, and E = {E1 , E2 } are the three given sets of activity nodes. Obviously, the list of origin–destination pairs includes two types of trip chains for the peak period, home-based and work-to-home trip chains. This demand setting is given in such an electricity-charging availability assumption that electric vehicle drivers may charge their vehicles at either home or workplace, but not other places in the network. To illustrate the applicability of the modeling and solution methods for diverse driving range distributions, we assumed four different probability distributions for continuous driving ranges, namely, uniform distribution, triangle distribution, quadratic distribution, and lognormal distribution, as shown in Fig. 3. For the purpose of comparison, we set their mean, minimum and maximum values equal5 (except the lognormal distribution6 ). With these parameters, the probability density functions of these distributions are given as:

5

These distributions could be alternatively set with the same mean and standard deviation values. For the lognormal distribution, only its mean and minimum values are set to equal those of other three distributions; its maximum value is the positive infinity by definition. 6

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Uniform distribution (d ∼UNIF(dmin , dmax )):

f (d ) = •

1 2(dmax − dmin )

dmin ≤ d ≤ dmax

Triangle distribution (d ∼TRI(dmin , dmax )):



(x − dmin ) dmin ≤ d ≤ dmean 4 (dmax − x ) d (dmax −dmin )2 mean ≤ d ≤ dmax 4

(dmax −dmin )2

f (d ) = •

Quadratic distribution (d ∼QUAD(dmin , dmax )): 6 f (d ) = 2(dmax3−d ) − (x − dmean ) min (dmax −dmin )3 dmin ≤ d ≤ dmax

•

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2

Lognormal distribution (d ∼LOGN(dmin , dmax )):

f (d ) =

1 √ (x−dmin )dstd 2π

e

−

2

[ln (x−dmin )−dmean ] 2d 2 std

dmin ≤ d < +∞

where dmin , dmax , dmean , and dstd denote the minimum, maximum, mean and standard deviation values of these distributions, respectively. In the numerical analysis presented below, we set dmin = 75, dmax = 95, and dmean = 85 for all problem cases with continuously distributed driving ranges. On the other hand, the corresponding discrete counterparts to these continuous distributions were developed in such a manner: For any continuous distribution described above except the lognormal distribution, we divided its entire range between dmin and dmax into a number of subranges with equal intervals and set the probability of each subrange as the probability associated with the mean value of this subrange; for the lognormal distribution, its discrete counterpart was formed in a similar way but its far right tail is ignored. Specifically, if we divide the entire range of a continuous distribution into n subranges, such as [dτ , dτ + 1 ], where τ = 1, 2, , n and d1 = dmin and dn + 1 = dmax , and dτ + 1 − dτ = (dmax − dmin )/n, the probability of range [dτ , dτ + 1 ] is:



dτ +1

dτ

f (x )dx

τ = 1, 2, . . . , n

which is assigned to the following single driving range value:

 dτ +1 dτ

 dτ +1 dτ

x f (x )dx f (x )dx

τ = 1, 2, . . . , n

The resulting discrete distribution is then composed of n distinct driving range values. 5.2. Network flows To understand the impacts of stochastic driving ranges, we resort to studying the resulting network flow patterns under the four stochastic driving range distributions as well as a deterministic driving range value and no driving range limit. For the purpose of comparison, the common mean values of the four distributions is set equal to the deterministic driving range value and the network flow solution for the deterministic driving range case can be computed by using the above algorithmic procedure for discrete driving range cases with setting the number of driving range classes equal to 1. We start with a discussion of the results from the continuous driving range cases. Fig. 4 shows the link flow results from the aforementioned six different problem cases, in which it is noted that all stochastic driving range cases are set in a continuous distribution form. By comparing these six cases, we found that the link flow results exhibit significantly different patterns, not only between different distribution types of driving ranges (i.e., stochastic, deterministic, and no driving ranges), but also between different forms of stochastic distributions, even if the mean, minimum and maximum values of these distributions are respectively equal. In particular, it is counted that on more than 30 links in the network, the flow rates obtained from either the deterministic or any of the stochastic driving range cases are over 100 veh/h more or less than those in the no driving range case. Among them, for example, the flow rate on link (3, 4) is up to 503 veh/h when no driving range is limited, while it decreases to a number approximately between 0 and 84 veh/h when any of the given deterministic or stochastic driving range limits is imposed. These large changes indicate a fact that any limited driving range, no matter deterministic or stochastic, greatly constrains the number of feasible paths containing this link. Furthermore, by comparing the results under the four stochastic driving range cases, we found that while the overall network flow patterns pose a similar or comparable picture, some link flow rates still show a great variation. For example, the flow rates on link (20, 15) under the uniform, triangle, quadratic and lognormal distributions are 145, 205, 192 and 157 veh/h, respectively. If we use as a reference the flow rate on this link in the deterministic case, these flow rates under the stochastic distributions Please cite this article as: C. Xie et al., Path-constrained traffic assignment: Modeling and computing network impacts of stochastic range anxiety, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.04.018

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Fig. 4. Network flow patterns under different types and forms of driving ranges.

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Fig. 5. Comparison of link flow differences between discrete and continuous driving range cases.

are, respectively, 5–33% lower than the reference. In overall, these preliminary comparison results reveal the manner and magnitude of the impacts of stochastic driving ranges on network flows. We then make a comparison on network flows between the discrete and continuous driving range distributions. The relative link flow differences between each given continuous driving range case and some of its discrete counterparts with selected driving range values (i.e., n = 5, 10, and 20) are given in Fig. 5. By this comparison, we can see a very good approximation of the network flow results under a continuous driving range distribution to be reached by the results produced by its corresponding discrete distribution, especially when the number of driving range values is sufficiently large (i.e., n ≥ 10). Based on this observation, we speculate that if increasing the number of driving range values, we should see a tendency of the link flow results from a continuous distribution to be gradually approached by its discrete counterpart. To validate this speculation, for each distribution, we ran the algorithmic procedure for the discrete driving range case with a wide range of the number of driving range values, from 2 to 30, and compared all obtained results with that from the continuous driving range case. The comparison results from some randomly selected links are depicted in Fig. 6, in which the rightmost points in each diagram are link flow rates from the continuous case. These diagrams clearly show that when the number of driving range values in the discrete case increases, its link flow rates converge to those given by the continuous case. This result justifies that the network flow pattern of the proposed network assignment problem under a continuous driving range distribution can be well approximated by its discrete counterpart, as long as the number of driving range values in the discrete case goes beyond some critical number (e.g., n = 10). 5.3. Computational costs In terms of computing time, we speculated two possible outcomes: (1) first, for the proposed network assignment problem with discrete driving range values, the computing time is expected to increase with the number of discrete driving range values, since each number of driving range values implies a separate network loading process and the network loading process is the most computationally intensive part of the procedure; (2) second, given the above speculation, the computing time required for solving a discrete driving range case should be much higher than that for its corresponding continuous driving range case when the number of driving range values, n, in this discrete case is relatively large. To validate the above two speculations, we conducted the relevant network assignment experiments by running the compiled procedures and recorded the computing times of all the discrete and continuous driving range cases with different forms of distributions and numbers of driving range values. The computing time data are depicted in Fig. 7. The first speculation is readily verified by this figure, for any form of distributions. Moreover, it is observed that the computing time increases approximately linearly with the increase of the number of driving range values. The second speculation can also be proved, by the following observation: When n < 2 or 3, for example, the computing time for solving any discrete case is Please cite this article as: C. Xie et al., Path-constrained traffic assignment: Modeling and computing network impacts of stochastic range anxiety, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.04.018

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Fig. 6. Variation of link flow rates over the number of discrete driving range values.

Fig. 7. Increase of computing times over the number of discrete driving range values.

lower than that for its continuous counterpart; with the number of driving range values increasing, the computing time for solving the former increases and goes far higher than the latter. We further examined the network loading steps of the solution procedures and their computing times for both the discrete and continuous driving range cases. It is found that while the operations of the network loading step for continuous cases look much more complex than those for discrete cases, the computing time for the former is only 2–3 times higher than that for the latter. This is shown in Table 2, which summarizes the average computing times spent for the network Please cite this article as: C. Xie et al., Path-constrained traffic assignment: Modeling and computing network impacts of stochastic range anxiety, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.04.018

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C. Xie et al. / Transportation Research Part B 000 (2017) 1–22 Table 2 Average computing times for executing the network loading step once. × 10−3 s

Discrete driving range distributions

Continuous driving range distributions

n=1

2

3

4

5

6

7

8

9

10

20

30

Uniform

3.02

2.22

3.57

3.00

3.13

2.91

2.99

2.80

2.93

3.41

2.98

2.57

8.50

Triangle

3.02

2.98

3.40

3.03

3.00

2.80

2.97

3.01

2.89

2.76

2.60

2.67

6.23

Quadratic

3.02

3.00

3.54

3.42

2.96

2.84

2.97

2.83

2.79

2.70

2.56

2.57

6.00

Lognormal

3.02

3.27

3.37

3.75

3.24

2.94

3.04

2.90

3.08

2.97

2.80

2.40

5.25

loading step in all discrete and continuous cases with different types of driving range distributions and numbers of discrete driving range values. We can clearly see from the table that the average computing times for executing once the network loading step in all discrete cases are around 2.5–3.5 × 10−3 s, which are lower than those in their continuous counterparts. The fact that the average network loading times in different discrete cases show a very stable pattern well explains why for any discrete driving range case its total computing time increases with the number of discrete driving range values in an approximately linear manner. To this end, only when n is very small (i.e., n < 2 or 3), is the computing time for solving a discrete driving range case lower than that for its continuous counterpart. By rechecking the algorithmic procedure for the problem with continuous driving ranges, we found that this procedure could be also used for solving the problem with discrete driving ranges, as long as some slight modifications are made. These modifications will result in a set of simpler path elimination and generation steps (i.e., steps 3 and 4), compared to those in the solution procedure for the continuous case. In other words, in spite of some minor differences, this procedure is applicable for both the discrete and continuous driving range cases. In this regard, we suggest in terms of solution efficiency that, unless the number of driving range values is very small, we should employ this procedure, even if it is originally designed for the continuous driving range case and consumes more time for implementation (i.e., coding and debugging), for the proposed problem with discrete driving ranges. This conclusion is somehow surprising, but well supported by our computing experiment results. 6. Conclusions With the aim of developing an analytical evaluation tool for traffic networks serving electric vehicles with heterogeneous driving ranges and insufficient charging opportunities, this paper describes a new network assignment problem, in which individuals are assumed to make joint activity location and travel path choices on the trip chain level. The underlying reason for which we embed trip chains into such a network assignment problem is to place and evaluate the impacts of range anxiety on a proper driving distance level to realistically reflect the present situation of limited charging opportunities in most traffic networks. This requirement calls for an integrated consideration of activity location and travel path choices in the network equilibrium paradigm. If the relaxation of activity sequences and dimensions of ride sharing and mode choice are considered, a trip chain-based, activity-oriented modeling framework could be used to better capture and characterize flow distributions. This direction poses a set of new network equilibrium problems. It is widely believed that network flows containing trip chains can be more realistically modeled in a time-dependent network environment, since the combination of travels and activities along a trip chain typically spans a relatively large time frame (up to a few hours) and experiences varying network conditions. The problem proposed in this paper, with either discretely or continuously distributed driving ranges, is not an exclusive case. To comply with this modeling preference, we need to adapt an existing or develop a new dynamic or quasi-dynamic network modeling framework that can achieves the desired network equilibrium and enables the resulting time-dependent distance-constrained, node-sequenced shortest path problem to be conveniently solved. No matter which modeling framework is used, in the face of heterogeneous driving ranges, the resulting dynamic network loading process would be much more complicated than its static counterpart. Such dynamic network models not only better reflect travel and activity dynamics, but also can be extended to embrace the need of modeling time-dependent parking and charging demands of electric vehicles. This latter extension provides a basic building block of evaluating the mutual impacts and interactions between transportation networks and electricity grids through electric vehicles’ parking and charging activities. For describing heterogeneous driving ranges, we considered two different problem settings, which respectively regard driving ranges as discretely and continuously distributed parameters, in the same modeling and solution frameworks. In particular, both convex optimization and variational inequality models were constructed and the projected gradient method is tailored and implemented for the two problem instances. From both the problem modeling and solving perspectives, it is acknowledged that the problem instance with discrete driving ranges can be simply treated as a multi-class version extended from its deterministic counterpart, while the instance with continuous driving ranges results in a much more modeling and solving challenge. To tackle the arising modeling and solution difficulty with the latter problem instance, we introduced the path-indexed subdemand and subflow rates as the core decision variables for the model and algorithm development, which make the proportioned travel demand and traffic flow rates are completely homogeneous in terms of Please cite this article as: C. Xie et al., Path-constrained traffic assignment: Modeling and computing network impacts of stochastic range anxiety, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.04.018

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the route choice behavior and constraint. It is noted that the set of path-indexed subdemand and subflow rates as well as their corresponding path subsets must be dynamically determined in the solution process, which, in our opinion, makes the path elimination and generation steps be most conveniently and efficiently implemented. Nevertheless, this appealing solution efficiency rooted from the use of path-indexed subdemand and subflow rates is also concurrent on the problem case with discrete driving ranges. Our computational results indicate that as long as the number of discrete driving range values is not exceedingly small, the solution procedure based on dynamically classified demand and flow rates in reference to used path lengths is more efficient than the procedure based on statically classified subdemand and subflow rates in terms of given discrete driving range values. In other words, the former procedure is not only applicable for the proposed problem with continuous driving ranges, but also preferable for the problem with discrete driving ranges in most cases. To this end, we conclude that in terms of solution efficiency, this procedure should be a prior choice for the network assignment problem with stochastic driving ranges, no matter the stochasticity is represented by a discrete or continuous distribution. Acknowledgments This study is jointly supported by research grants through the Young Talent Award from the China Recruitment Program of Global Experts, the National Natural Science Foundation of China (Grant no. 71471111), and the Research Fund for the Doctoral Program of Higher Education of China (Grant no. 2013-007312-0069). The last author also gratefully acknowledges the support from the Thailand Research Fund and King Mongkut’s University of Technology Thonburi (Contract no. RSA5980030). References Acello, R., 1997. 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Please cite this article as: C. Xie et al., Path-constrained traffic assignment: Modeling and computing network impacts of stochastic range anxiety, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.04.018