# A Quantitative Method to Delineate the Influence Area of Work Zone Considering Route Choice Inertia and Elastic Demand

Authors: Guanfeng Wang; Hongfei Jia; Jingjing Tian; Yu Lin; Ruiyi Wu; Zhendong Liu; Heyao Gao
Journal: Journal of Advanced Transportation
 (2022)

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## Abstract

The work zone on the urban road network will affect the surrounding road traffic. To represent the influence area of the work zone, the concept of a subnetwork is proposed in this paper. Delineating a suitable subnetwork quantitatively is a challenging problem. To address this issue, the node synthesized indexes (NSI) are deployed as a variability measure that captures both the change of link flow and origin-destination (OD) demand. The inertia-based stochastic user equilibrium with the elastic demand (ISUEED) model is proposed to accurately provide the data of link flow and OD demand for the network with the work zone. Correspondingly, the data of the network without a work zone can be obtained by the stochastic user equilibrium (SUE) model. According to the value of NSI, the initial range of the subnetwork is determined. Finally, the connectivity and compactness can be guaranteed by the modified L-shell algorithm. To demonstrate the performance of the method, two case studies and sensitivity analyses are conducted based on the Braess network and the local road network in Changchun, China. The proposed method is beneficial to reduce the complexity of the traffic model by substituting the entire network with a subnetwork.


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## Body

## 1. Introduction

The long-term presence of a work zone will cause two thorny issues for the traffic network, which are as follows: congestion and safety problems [1]. It induces link saturation and causes the accident rate surrounding the work zone to increase noteworthily [2]. Especially, for urban areas with a high road network density, the effects are even more profound. Hence, traffic agencies usually implement a series of traffic adjustment measures around the work zone, so that the deterioration of road network performance does not exceed the acceptable threshold.Generally speaking, the network performance will be better as the application scope of traffic adjustment measures expands. However, it is not desirable in terms of sustainability and economy. Therefore, to achieve a trade-off between the application effect and expenditure of the measures, the subnetwork should be delineated appropriately. Unfortunately, few studies have concentrated on a quantitative method to determine the influence area of the work zone. The existing literature mainly focuses on judgmental and mathematical methods, which is difficult to meet the requirements of formulating specific construction organization scheme [3, 4]. Meanwhile, the quantitative delineation method of the subnetwork also puts forward higher requirements on the accuracy of data provided by the corresponding traffic models. Although many studies consider route choice inertia and elastic demand separately in traffic assignment models, few researches consider both simultaneously [5–7]. Hence, it is difficult for the model to truly and comprehensively reflect the behavior of traveller, resulting in a reduction in data precision. Furthermore, how to get a connected and compact subnetwork is also a thorny problem. This study solves the aforementioned knowledge gaps by (1) proposing a quantitative delineation method for the subnetwork, (2) building the ISUEED model, and (3) improving the L-shell algorithm [8].To achieve the above objectives, this study takes nodes as the objects of analysis. A more realistic ISUEED model is developed, which describes the behaviors of travellers in response to an incident by tree structure and exponential elastic demand. The ISUEED model is applied to obtain the data of link flow and OD demand for the network with work zone. Correspondingly, the data of the network without work zone can be obtained by the SUE model. The change rates of the two types of data are synthesized into NSI and compared with the threshold value to generate the initial range of the subnetwork. Then the modified L-shell algorithm is used to realize the last step of the subnetwork quantitative delineation method. Meanwhile, connectivity and compactness can be guaranteed naturally. Finally, the validity and practicability of the method are verified by the Braess network and the local road network construction project in Changchun, China. Although some parameters are relevant to the case context, the method proposed in this paper is universal.


## 2. Literature Review

Until now, a large number of pieces of research has concentrated on mitigating the adverse impact of the work zone on traffic operations [9, 10]. However, the literature mainly focuses on the microsimulation at the link-level and the performance optimization at the network-level.At the link-level, the researchers realized the optimization of safety, queuing length, time delay, and accident rate by adjusting the length of the work zone, construction time, number of construction teams, and other factors [11–15]. In addition, there is also a way to further divide the work zone into multiple segments to optimize link performance. However, as the number of segments increases, maintenance becomes more expensive. Hence, a compromise between the two needs to be considered [11, 13, 16, 17]. More deeply, to explore the mechanism of action for the work zone under different traffic conditions or vehicle behavior, some researchers also conducted simulations for traffic scenes near the work zone, in which the variables include the headway of the vehicle, lane-changing behavior, and following behavior [18–23]. In addition, the duality of new technology is also reflected vividly [24]. For example, the application of the floating car and big data technology provides new opportunities for microsimulation data collection, while the emergence of a moving work zone (a fast construction method using advanced machinery) is a new challenge for relevant research [25].From the perspective of the network-level, most researchers adopt a bilevel programming model. The upper level aims to minimize the total travel time, traffic delay, pollutant emissions, etc. In contrast, the lower level generally carries out traffic simulations using software, such as VISSIM or PARAMICS, and some of them acquire the indicator data through traffic assignment [26–29]. Moreover, most of the studies utilize metaheuristic algorithms to solve the problem of obtaining the optimal work zone scheduling [29, 30]. Among them, several researchers also discuss the factors affecting the schedule of the work zone. Lee considered that the number of work zones and construction teams would affect the construction order [27]. On the other hand, Rey and Bar-Gera point out the critical role of budget, the number of construction staff, and pieces of equipment [28]. The length of the construction period and the flexibility of the time window also affect the work zones scheduling [29]. In addition, researchers try distributed algorithms, simplified mathematical distribution models, and other methods to solve the problem of enormous computation burden at the network-level [30].In urban areas with high road network density, the application of subnetwork has more advantages in reducing the complexity of the traffic model and improving the effectiveness of traffic adjustment schemes. However, to our best knowledge, the literature on the subnetwork level is sparse. Firstly, Erath et al. had overcome the challenge of excessive traffic data calculation to assess the vulnerability of the Swiss road network under natural disasters [31]. The subnetwork division of fixed scope is adopted according to the map, which considers the detour distance constraint of travellers. Secondly, Memarian et al. assessed the impact of lane closures on the surrounding road network during road construction [32]. Subnetwork delineation and a new route choice method are integrated simultaneously to speed up the traffic assignment. Compared with the research of Erath et al., this study adopted the mathematical method of linear regression to delineate the subnetwork. It has realized the leap from judgmental approach to quantitative method, however, there are many research gaps.Previous studies have paid little attention to the subnetwork, which also inhibits the development of the relevant research of subnetwork delineation methods to a certain extent. In the past, the judgmental approach based on the experience or urban geographic structure was adopted to delineate the subnetwork when a relevant piece of research needs a local part of the network [33, 34]. These methods are simple and operable, however, they will bring suboptimal problems. Therefore, some researchers have introduced related methods of the graph theory into the delineation of the subnetwork, most of which are adopted for road user charging cordon design [34]. However, with the expansion of the road network, the shortcomings of this method began to appear, and it became extremely difficult to obtain the initial solution. To overcome it, in recent years, data, such as link density, speed, or travel time, have been used for network clustering to realize subnetwork delineation. Gu and Saberi used the weighted average of density similarity measure and distance similarity measure for graph clustering to achieve the bipartition of the network under the premise of ensuring homogeneity, connectivity, and compactness [3]. This method can not only effectively capture the spatial changes of congestion patterns in the network but also provide effective insights for the setting of a double layer or even multilayer charging cordon. To sum up, the literature on the subnetwork determination method is arid, and the methods currently used have their limitations, respectively. Hence, the purpose of this study is to establish a scientific method of quantitative subnetwork delineation and provide the scope basis for related work zone research.


## 3. Methodology

The subnetwork delineation method proposed in this study is displayed in Figure1. Firstly, the SUE model is used for traffic assignment to obtain the flow and cost of each route between each OD pair when the network is without a work zone. At the same time, the travel demand between each OD pair is classified according to the selected route (multiclass user data). Travellers who choose the same route are known as a class of users. Next, the multiclass user data is treated as the input for the ISUEED model. Then, the data of the link flow and OD demand of the network with a work zone is output as a result of the model. The NSI is deployed as a variability measure that captures the changes of the network with and without a work zone. Meanwhile, the initial range of the subnetwork can be obtained by comparing NSI with the threshold value. Finally, the L-shell algorithm is used to further process the comparison results to achieve subnetwork delineation. It should be noted that SUE is a mature model and will not be repeated in this article. Please refer to the literature for details [35–37].Figure 1
The framework of the method.To accurately delineate the subnetwork, it is necessary to precisely estimate the data of the link flow and OD demand when there is a work zone on the network. Therefore, the ISUEED model for traffic assignment is proposed, which fully considers the route choice inertial behavior of the traveller and elastic travel demand. The model takes the multiclass user data obtained from the SUE model as input, represents the route choice inertia behavior of different users using the tree structure, and employs exponential elastic demand to denote demand changes.


### 3.1. Modeling Route Choice Inertia Behavior

The change of network topology caused by the existence of the work zone is a new situation for travellers. Research has shown that travellers prefer to stick with past route choice under the circumstances [6]. A tree structure is adopted to describe this phenomenon in this study, as shown in Figure 2(a). The reasons are chiefly stated as follows: firstly, it is highly flexible and can correspond to the change of route by the select limb. Secondly, it has favorable adaptability, and the model parameters can be adjusted to adapt to different traveller attributes and sensitivity to cost changes. Thirdly, this structure has the advantages of convenience, low data requirements, and accurate calculation results. Specifically, the nests in the upper layer represent the original and currently available set of routes for a specific class of traveller. In contrast, the select limbs in the lower layer represent routes. It is essential to point out that the set of original routes contain only one element because their original routes are determined. The probability calculation processes are expressed as follows:(1)pw,im=pw,in×pw,im|n∀w∈Wi∈Iwn∈Nw,im∈Mw,i,n,(2)pw,in=e−θnμw,in∑n∈Nw,ie−θnμw,in∀w∈Wi∈Iwn∈Nw,i,(3)pw,im|n=e−θmcw,in,m∑m∈Mw,i,ne−θmcw,in,m∀w∈Wi∈Iwn∈Nw,im∈Mw,i,n,where pw,im,pw,in represent the probability of choosing route m and nest n for class i travellers between the OD pair w, respectively. pw,im|n is the conditional probability of choosing route m for class i travellers between OD pair w in the case of choosing nest n. θn,θm are the parameters of the tree structure. μw,in is the expected minimum travel cost of the nest n for class i travellers between the OD pair w. cw,in,m is the actual travel cost of route m in the nest n for class i travellers between the OD pair w. W is the set of OD pairs. Iw is the set of original shortest routes (user classes). Nw,i is the set of nests for class i travellers between the OD pair w. Mw,i,n is the set of currently shortest routes for class i travellers between the OD pair w in the case of choosing nest n.Figure 2
The tree structure of route choice.
(a)(b)(c)The specific mechanism of the model to express the route choice inertia behavior is described below. On the network without work zone, travellers choose the routemold. If the route still exists in the shortest route set of the corresponding OD pair on the network with work zone, mold∈nnew, this class of travellers tend to maintain their original choices. In this case, the route choice model is the nested logit (NL) model, as shown in Figure 2(b). The travel cost of the select limb mold in nest nnew is cnnewmold⟶∞. In nest nold, it can be expressed as the travel cost of the route in the network without a work zone multiplied by the inertia coefficient.(4)cnoldmold=ρcmold,(5)cmold=∑l∈Lcl.The cost of link can be calculated by the Bureau of Public Roads (BPR) function, which is a strictly monotone function of traffic flow.(6)cl=tf×1+αQCβ,where tf is the free-flow time of link. Q is the traffic flow on link. C is the capacity of link. α and β are the parameters of the model.If the routemold no longer exists in the set of the shortest routes because of a topological change, mold∉nnew. The expected minimum travel cost of nold is μnold⟶∞. The NL degenerates into a multinomial logit (MNL) model, as shown in Figure 2(c). The MNL model was first used by Daganzo and Sheffi in the route choice of the stochastic traffic assignment. Subsequently, to overcome the independence of irrelevant alternatives property, the improved versions of the logit model have been developed successively. The reader is referred to Daganzo and Sheffi [35], Prashker and Bekhor [38], and Mai et al. [39]for an in-depth understanding of the logit model in the traffic domain.


### 3.2. Elastic Demand Function

The change of network topology will lead to the variation of the generalized travel cost of relevant travellers. Travellers may switch their travel mode or cancel their trips. For simplicity, only car travel mode is considered in this study, and the travel mode transfer is not involved. As a result, the OD demand will change. Meanwhile, as an important data source of subnetwork delineation, OD demand variation cannot be ignored. Therefore, this phenomenon is expressed as an exponential elastic travel demand, as shown in equations (7) and (8) [7]. The elastic demand function is a strict monotonically decreasing function of the expected minimum travel cost of class i travellers between the OD pair w.(7)qw,i=Dw,iλw,i,(8)qw,i=qw,i0eφλw,icw,i0−1.Correspondingly, the inverse function is defined as follows:(9)λw,i=Dw,i−1qw,i,(10)λw,i=cw,i0+cw,i0φlnqw,iqw,i0,where qw,i is the travel demand of the class i travellers between the OD pair w. qw,i0 is the origin travel demand of the class i travellers between the OD pair w. φ is the elastic coefficient. λw,i is the expected minimum travel cost of the class i travellers between the OD pair w. cw,i0 is the travel cost that corresponds to qw,i0, which is acquired using the SUE model.


### 3.3. Model Formulation

According to equations (1)–(10), the ISUEED model can be described as a variational inequality (VI) problem. It seeks to minimize the total travel time while ensuring that the traffic flow complies with the SUE principle (equations (1)–(3)). The link flow fw,in,m∗ and travel demand qw,in∗, qw,i∗ are defined as the optimal solution of the VI problem:(11)∑w∈W∑i∈Iw∑n∈Nw,i∑m∈Mw,i,ncw,in,mfw,in,m∗+1θmlnfw,in,m∗qw,in∗fw,in,m−fw,in,m∗+∑w∈W∑i∈Iw∑n∈Nw,i1θnlnqw,in∗qw,i∗qw,in−qw,in∗−∑w∈W∑i∈IwDw,i−1qw,i∗qw,i−qw,i∗≥0,s.t.(12)qw,i=∑n∈Nw,iqw,in∀w∈Wi∈Iwλw,i,(13)qw,in=∑m∈Mw,i,nfw,in,m∀w∈Wi∈Iwn∈Nw,iμw,in,(14)xa=∑w∈W∑i∈Iw∑n∈Nw,i∑m∈Mw,i,nfw,in,mδw,i,an,m∀a∈A,(15)fw,in,m≥0∀w∈Wi∈Iwn∈Nw,im∈Mw,i,nτw,in,m,(16)qw,i≥0∀w∈Wi∈Iwψw,i,where fw,in,m is the flow of route m in the nest n for class i travellers between the OD pair w. qw,in is the travel demand of nest n for class i travellers between the OD pair w. xa is the flow of link a. δw,i,an,m represents whether the link a is on the route m in the nest n for class i travellers between the OD pair w. If so, δw,i,an,m = 1, otherwise δw,i,an,m = 0.The constraints given by equations (12) and (13) denote the flow conservation equations. The constraint equation (14) represents the relationship function between the route flow and the link flow. Equations (15) and (16) denote the non-negative constraints. The corresponding dual variables are in parentheses.Proposition 1.
The VI problem defined by equations (11–16) is equivalent to equations (1)–(10).Proof.
Equation (11) is expressed as z1,z2, and z3, respectively, as follows:(17)z1=∑w∈W∑i∈Iw∑n∈Nw,i∑m∈Mw,i,ncw,in,mfw,in,m∗+1θmlnfw,in,m∗qw,in∗fw,in,m−fw,in,m∗,z2=∑w∈W∑i∈Iw∑n∈Nw,i1θnlnqw,in∗qw,i∗qw,in−qw,in∗,z3=∑w∈W∑i∈IwDw,i−1qw,i∗qw,i−qw,i∗.
The Lagrange function is constructed by the Lagrange multiplier method.(18)L=z1+z2+z3+λw,iqw,i−∑n∈Nw,iqw,in+μw,inqw,in−∑m∈Mw,i,nfw,in,m.
Then Karush–Kuhn–Tucker (KKT) conditions can be expressed as follows:(19)fw,in,mcw,in,m+1θmlnfw,in,mqw,in−μw,in=0,cw,in,m+1θmlnfw,in,mqw,in−μw,in≥0,fw,in,m≥0,.(20)qw,in1θnlnqw,inqw,i−λw,i+μw,in=0,1θnlnqw,inqw,i−λw,i+μw,in≥0,qw,in≥0,.(21)qw,i−Dw,i−1qw,i+λw,i=0,−Dw,i−1qw,i+λw,i≥0,qw,i≥0,.
Based on equation (19), equations (22)–(24) can be obtained.(22)cw,in,m+1θmlnfw,in,mqw,in−μw,in=0,(23)lnfw,in,mqw,in=θmμw,in−cw,in,m,(24)fw,in,mqw,in=eθmμw,in×e−θmcw,in,m.
Add the traffic flow ofm routes.(25)1=eθmμw,in∑m∈Mw,i,ne−θmcw,in,m.
Then, we can acquire equation (26), which is equivalent to equation (3).(26)fw,in,mqw,in=pw,im|n=e−θmcw,in,m∑m∈Mw,i,ne−θmcw,in,m.
Similarly, we can obtain equations (27) and (28) from equation (20), where equation (28) is equivalent to equation (2).(27)λw,i=−1θnln∑n∈Nw,ie−θnμw,in.(28)qw,inqw,i=pw,in=e−θnμw,in∑n∈Nw,ie−θnμw,in.
Lastly, based on equation (21), (29) can be obtained, which is equivalent to equation (8).(29)qw,i=Dw,iλw,i.Proposition 2.
The VI problem has a unique optimal solution.Proof.
According to equation (11), the Hessian matrix corresponding to z1,z2, and z3 are positive definite. Thus, the model is strictly convex. Besides, the constraints are all linear. Therefore, this problem is convex programming. As shown in equations (4)–(6), the travel cost of the link is continuous with respect to xa, and the route cost is continuous with the route flow. Then, any local optimal solution must be a global optimal solution, and it is unique. In sum, the VI problem has a unique optimal solution.


### 3.4. Solution Algorithm for the ISUEED Model

The ISUEED model essentially is a multiclass user problem, which can be solved by the diagonalized method of the successive average (DMSA) algorithm [40–42]. It is worth noting that according to the definition of the multiclass user in this study, only one OD pair is involved per user class. Therefore, the travel demand is loaded according to the route choice model, and the equilibrium state is achieved simultaneously in the process of single-class user assignment. As a result, it eliminates the traffic adjustment process within the per class of user. Even though the number of user classes is enormous, the total running time is still acceptable. To meet the needs of this study, the link flow X and OD demand set WE are taken as the output results. Compared to previous literature, the algorithm has been slightly adjusted. Algorithm 1 gives the corresponding solving steps specifically.


### 3.5. Delineating the Subnetwork

This study takes nodes as the analysis object. The change rates of the link flow and OD demand with and without a work zone are used to generate the NSI. Then, we compare the NSI with the threshold value to determine whether a node is in the influence area. Finally, the subnetwork is delineated by the affected nodes, which are selected by the L-shell algorithm.Algorithm 1: DMSA.
Input: the network GN,A with work zone, Iw∀w∈W, set of travel demand for each class travellers between OD pair wQw∀w∈W, convergence accuracy ξ.Output:X, WE(1)
By the k-shortest paths method to generate the set of shortest routesIw′∀w∈W.(2)
Initialize: assignQw∀w∈W according to the SUE model. Acquire the traffic flow(3)
of the routefw,i′1 and the traffic flow of the link xa1. e⟶∞. For n=1.(4)
whilee>ξ(5)
forw in W(6)
Update the link travel time.(7)
Update the route travel time.(8)
Update the expected minimum travel costμw between each OD pair w.(9)
fori in Iw∀w∈W(10)
Updated the ODqw,in of such travellers according to equation (8).(11)
ifi∈Iw′∀w∈W(12)
Correct route travel cost according to equation (6).(13)
Loadqw,in to each route according to the NL shown in equations.(14)
(1)–(3)(15)
else(16)
Loadqw,in to each route according to the MNL.(17)
end if(18)
end for(19)
end for(20)
Update the link flowxan.(21)
Acquire additional traffic flow of the routegw,i′n.(22)
Update the route traffic flowfw,i′n+1=fw,i′n+χngw,i′n−fw,i′n, χn=1/n.(23)
Computational convergence precisione=∑w∈W∑i′∈Iw′fw,i′n+1−fw,i′n2/∑w∈W∑i′∈Iw′fw,i′n.(24)
Forn=n+1(25)
end while(26)
Calculateqwn=∑i∈Iwqw,in∀w∈W.(27)
Store eachxan and each qwn to X and WE, respectively.(28)
OutputX and WE.The change rate of the link flow for nodeu is related to the incoming flow and outgoing flow, which are calculated, respectively, as follows:(30)ΔFuin=∑∈ℳufum−∑m′∈ℳu′fum'∑m∈ℳufum,ΔFuout=∑m∈ℳufum−∑m′∈ℳu′fum'∑m∈ℳufu′m∑,where ΔFuin,ΔFuout are the change rates of the incoming (outgoing) flow at node u. fum,fu'm are the flows that enter (leave) node u through link m in the network without work zone. fum',fu'm' are the flows that enter (leave) node u through link m' in the network with work zone. ℳu,ℳu′ are the sets of links connected by node u when the network is with (without) a work zone.The average value of the two is calculated as the change rate of flow for nodeu and normalized.(31)ΔFuave=ΔFuin+ΔFuout2,ΔFu=ΔFuave−ΔFminaveΔFmaxave−ΔFminave,where ΔFuave is the average value of flow change rate at node u. ΔFmaxave,ΔFminave are the maximum and minimum values of the average flow change rate. ΔFu is the normalized value of ΔFuave.The change rate of OD demand for nodeu is tightly associated with the generating demand and attracting demand, which are calculated, respectively, as follows:(32)ΔQugen=∑n∈Ndun−∑n∈Ndun'∑n∈Ndun,ΔQuatt=∑n∈Ndu'n−∑n∈Ndu'n'∑n∈Ndu'n,where ΔQugen,ΔQuatt are the change rates of the generation (attraction) demand of node u. dun,du'n are the demands generation (attraction) of node u to node n when the network is without a work zone. dun',du'n' are the demands generation (attraction) of node u to node n when the network is with a work zone.Similarly, the change rate of OD demand for nodeu is as follows:(33)ΔQuave=ΔQugen+ΔQuatt2,ΔQu=ΔQuave−ΔQminaveΔQmaxave−ΔQminave,where ΔQuave is the average value of demand change rate at node u. ΔQmaxave,ΔQminave are the maximum and minimum values of the average demand change rate. ΔQu is the normalized value of ΔQuave.Based on equations (31) and (33), we define NSI for each node u in the network as a weighted average of ΔQu and ΔFu.(34)Su=αΔQu+1−αΔFu,where Su is the value of NSI for node u. α is the weight coefficient.The NSI is compared with the threshold to determine whether nodeu is within the area of influence, if so, put it into the set E.(35)state=inSu≥SstaoutSuSsta∀u∈N,where Ssta is the threshold value of NSI, which can be adjusted from 0 to 1 according to various needs to generate different subnetworks.


### 3.6. Modified L-Shell Algorithm

The L-shell algorithm is a method of community division in the complex network. This section only briefly explains the part involved in this study. The specific details and algorithm processes are shown in the literature [8]. In Figure 3(a), nodes 1 and 2 are vital nodes. Nodes 6, 7, and 8 are the adjacent nodes of node 1 in level 1. Nodes 12 and 13 are the level 1 adjacent nodes of node 6 and the level 2 adjacent nodes of node 1. According to the above process, it can be seen that the nodes 3–8 in Figure 3(a) are the level 1 adjacent nodes of vital nodes 1 and 2, denoted as the set l1. The nodes 9–18 are all the level 2 nodes of vital nodes 1 and 2, denoted as the set l2.Figure 3
Examples of related concepts in the L-shell algorithm.
(a)(b)(c)The distribution of NSI is not necessarily a centralized radial but may be sporadic and disconnected. To ensure the connectivity and compactness of the subnetwork, a modified L-shell algorithm is proposed to perform secondary filtering on nodes. The algorithm takes setE as input. It outputs the set F composed of all nodes within the subnetwork. The specific steps are shown in Algorithm 2. The central idea is to use the first “while” loop to basically determine the subnetwork, and then use the second “while” loop to check and fill gaps. In the first loop, it goes through over the elements in set En and places it in set F if its level 1 adjacent nodes already have at least one node in set F. Repeat the process until the size of set En does not change, which represents that all nodes in set E that can connect to a network have been selected. This process can be described as the vital nodes constantly adsorb the eligible nodes in set En layer-by-layer around it. The second loop is equivalent to a tolerance device, which is mainly to ensure the integrity of the subnetwork. Compared with the first loop, it changes both in the value of level and the judgment condition of putting the node into the set F. By traversing all the nodes that are not in set F, if the level 1 or level 2 adjacent nodes of one node are already in set F, put it in set F. For example, as shown in Figure 3(b), node 4 is not taken into set F after the above procedure. However, it is also going to be put in F in this loop. However, if the number of nodes exceeds the threshold, as depicted in Figure 3(c), nodes 1–7 are considered to be outside the subnetwork.Algorithm 2: Modified L-shell algorithm.
Input: the set of nodes E filtered by a threshold, the set of nodes N in the network.Output: the set of nodes F which compose of sub-network.(1)
Initialize: put the vital nodes intoF, E1=∅,E2=E,n=2.(2)
whileEn≠En−1(3)
fore in En(4)
Taking nodee as the starting point, search all level 1 adjacent nodes of node(5)
e according to the L-shell algorithm and store to Ne.(6)
Determine whether there isat least one member of Ne that is already(7)
included inF, if so, put e into F, otherwise, put it into En+1.(8)
end for(9)
Forn=n+1(10)
end while(11)
CalculateY=N−E+En.(12)
Initialize:Y1 = ∅, Y2=Y, n=2.(13)
whileYn≠Yn−1(14)
fory in Yn(15)
Taking nodey as the starting point, search all level 1 adjacent nodes and(16)
level2 adjacent nodes of y according to the L-shell algorithm and store(17)
toNy1,Ny2, respectively.(18)
Determine whether the members ofNy1orNy2are all included in F, if(19)
so, puty into F, otherwise, put it into Yn+1.(20)
end for(21)
Forn=n+1(22)
end while(23)
OutputF.


## 3.1. Modeling Route Choice Inertia Behavior

The change of network topology caused by the existence of the work zone is a new situation for travellers. Research has shown that travellers prefer to stick with past route choice under the circumstances [6]. A tree structure is adopted to describe this phenomenon in this study, as shown in Figure 2(a). The reasons are chiefly stated as follows: firstly, it is highly flexible and can correspond to the change of route by the select limb. Secondly, it has favorable adaptability, and the model parameters can be adjusted to adapt to different traveller attributes and sensitivity to cost changes. Thirdly, this structure has the advantages of convenience, low data requirements, and accurate calculation results. Specifically, the nests in the upper layer represent the original and currently available set of routes for a specific class of traveller. In contrast, the select limbs in the lower layer represent routes. It is essential to point out that the set of original routes contain only one element because their original routes are determined. The probability calculation processes are expressed as follows:(1)pw,im=pw,in×pw,im|n∀w∈Wi∈Iwn∈Nw,im∈Mw,i,n,(2)pw,in=e−θnμw,in∑n∈Nw,ie−θnμw,in∀w∈Wi∈Iwn∈Nw,i,(3)pw,im|n=e−θmcw,in,m∑m∈Mw,i,ne−θmcw,in,m∀w∈Wi∈Iwn∈Nw,im∈Mw,i,n,where pw,im,pw,in represent the probability of choosing route m and nest n for class i travellers between the OD pair w, respectively. pw,im|n is the conditional probability of choosing route m for class i travellers between OD pair w in the case of choosing nest n. θn,θm are the parameters of the tree structure. μw,in is the expected minimum travel cost of the nest n for class i travellers between the OD pair w. cw,in,m is the actual travel cost of route m in the nest n for class i travellers between the OD pair w. W is the set of OD pairs. Iw is the set of original shortest routes (user classes). Nw,i is the set of nests for class i travellers between the OD pair w. Mw,i,n is the set of currently shortest routes for class i travellers between the OD pair w in the case of choosing nest n.Figure 2
The tree structure of route choice.
(a)(b)(c)The specific mechanism of the model to express the route choice inertia behavior is described below. On the network without work zone, travellers choose the routemold. If the route still exists in the shortest route set of the corresponding OD pair on the network with work zone, mold∈nnew, this class of travellers tend to maintain their original choices. In this case, the route choice model is the nested logit (NL) model, as shown in Figure 2(b). The travel cost of the select limb mold in nest nnew is cnnewmold⟶∞. In nest nold, it can be expressed as the travel cost of the route in the network without a work zone multiplied by the inertia coefficient.(4)cnoldmold=ρcmold,(5)cmold=∑l∈Lcl.The cost of link can be calculated by the Bureau of Public Roads (BPR) function, which is a strictly monotone function of traffic flow.(6)cl=tf×1+αQCβ,where tf is the free-flow time of link. Q is the traffic flow on link. C is the capacity of link. α and β are the parameters of the model.If the routemold no longer exists in the set of the shortest routes because of a topological change, mold∉nnew. The expected minimum travel cost of nold is μnold⟶∞. The NL degenerates into a multinomial logit (MNL) model, as shown in Figure 2(c). The MNL model was first used by Daganzo and Sheffi in the route choice of the stochastic traffic assignment. Subsequently, to overcome the independence of irrelevant alternatives property, the improved versions of the logit model have been developed successively. The reader is referred to Daganzo and Sheffi [35], Prashker and Bekhor [38], and Mai et al. [39]for an in-depth understanding of the logit model in the traffic domain.


## 3.2. Elastic Demand Function

The change of network topology will lead to the variation of the generalized travel cost of relevant travellers. Travellers may switch their travel mode or cancel their trips. For simplicity, only car travel mode is considered in this study, and the travel mode transfer is not involved. As a result, the OD demand will change. Meanwhile, as an important data source of subnetwork delineation, OD demand variation cannot be ignored. Therefore, this phenomenon is expressed as an exponential elastic travel demand, as shown in equations (7) and (8) [7]. The elastic demand function is a strict monotonically decreasing function of the expected minimum travel cost of class i travellers between the OD pair w.(7)qw,i=Dw,iλw,i,(8)qw,i=qw,i0eφλw,icw,i0−1.Correspondingly, the inverse function is defined as follows:(9)λw,i=Dw,i−1qw,i,(10)λw,i=cw,i0+cw,i0φlnqw,iqw,i0,where qw,i is the travel demand of the class i travellers between the OD pair w. qw,i0 is the origin travel demand of the class i travellers between the OD pair w. φ is the elastic coefficient. λw,i is the expected minimum travel cost of the class i travellers between the OD pair w. cw,i0 is the travel cost that corresponds to qw,i0, which is acquired using the SUE model.


## 3.3. Model Formulation

According to equations (1)–(10), the ISUEED model can be described as a variational inequality (VI) problem. It seeks to minimize the total travel time while ensuring that the traffic flow complies with the SUE principle (equations (1)–(3)). The link flow fw,in,m∗ and travel demand qw,in∗, qw,i∗ are defined as the optimal solution of the VI problem:(11)∑w∈W∑i∈Iw∑n∈Nw,i∑m∈Mw,i,ncw,in,mfw,in,m∗+1θmlnfw,in,m∗qw,in∗fw,in,m−fw,in,m∗+∑w∈W∑i∈Iw∑n∈Nw,i1θnlnqw,in∗qw,i∗qw,in−qw,in∗−∑w∈W∑i∈IwDw,i−1qw,i∗qw,i−qw,i∗≥0,s.t.(12)qw,i=∑n∈Nw,iqw,in∀w∈Wi∈Iwλw,i,(13)qw,in=∑m∈Mw,i,nfw,in,m∀w∈Wi∈Iwn∈Nw,iμw,in,(14)xa=∑w∈W∑i∈Iw∑n∈Nw,i∑m∈Mw,i,nfw,in,mδw,i,an,m∀a∈A,(15)fw,in,m≥0∀w∈Wi∈Iwn∈Nw,im∈Mw,i,nτw,in,m,(16)qw,i≥0∀w∈Wi∈Iwψw,i,where fw,in,m is the flow of route m in the nest n for class i travellers between the OD pair w. qw,in is the travel demand of nest n for class i travellers between the OD pair w. xa is the flow of link a. δw,i,an,m represents whether the link a is on the route m in the nest n for class i travellers between the OD pair w. If so, δw,i,an,m = 1, otherwise δw,i,an,m = 0.The constraints given by equations (12) and (13) denote the flow conservation equations. The constraint equation (14) represents the relationship function between the route flow and the link flow. Equations (15) and (16) denote the non-negative constraints. The corresponding dual variables are in parentheses.Proposition 1.
The VI problem defined by equations (11–16) is equivalent to equations (1)–(10).Proof.
Equation (11) is expressed as z1,z2, and z3, respectively, as follows:(17)z1=∑w∈W∑i∈Iw∑n∈Nw,i∑m∈Mw,i,ncw,in,mfw,in,m∗+1θmlnfw,in,m∗qw,in∗fw,in,m−fw,in,m∗,z2=∑w∈W∑i∈Iw∑n∈Nw,i1θnlnqw,in∗qw,i∗qw,in−qw,in∗,z3=∑w∈W∑i∈IwDw,i−1qw,i∗qw,i−qw,i∗.
The Lagrange function is constructed by the Lagrange multiplier method.(18)L=z1+z2+z3+λw,iqw,i−∑n∈Nw,iqw,in+μw,inqw,in−∑m∈Mw,i,nfw,in,m.
Then Karush–Kuhn–Tucker (KKT) conditions can be expressed as follows:(19)fw,in,mcw,in,m+1θmlnfw,in,mqw,in−μw,in=0,cw,in,m+1θmlnfw,in,mqw,in−μw,in≥0,fw,in,m≥0,.(20)qw,in1θnlnqw,inqw,i−λw,i+μw,in=0,1θnlnqw,inqw,i−λw,i+μw,in≥0,qw,in≥0,.(21)qw,i−Dw,i−1qw,i+λw,i=0,−Dw,i−1qw,i+λw,i≥0,qw,i≥0,.
Based on equation (19), equations (22)–(24) can be obtained.(22)cw,in,m+1θmlnfw,in,mqw,in−μw,in=0,(23)lnfw,in,mqw,in=θmμw,in−cw,in,m,(24)fw,in,mqw,in=eθmμw,in×e−θmcw,in,m.
Add the traffic flow ofm routes.(25)1=eθmμw,in∑m∈Mw,i,ne−θmcw,in,m.
Then, we can acquire equation (26), which is equivalent to equation (3).(26)fw,in,mqw,in=pw,im|n=e−θmcw,in,m∑m∈Mw,i,ne−θmcw,in,m.
Similarly, we can obtain equations (27) and (28) from equation (20), where equation (28) is equivalent to equation (2).(27)λw,i=−1θnln∑n∈Nw,ie−θnμw,in.(28)qw,inqw,i=pw,in=e−θnμw,in∑n∈Nw,ie−θnμw,in.
Lastly, based on equation (21), (29) can be obtained, which is equivalent to equation (8).(29)qw,i=Dw,iλw,i.Proposition 2.
The VI problem has a unique optimal solution.Proof.
According to equation (11), the Hessian matrix corresponding to z1,z2, and z3 are positive definite. Thus, the model is strictly convex. Besides, the constraints are all linear. Therefore, this problem is convex programming. As shown in equations (4)–(6), the travel cost of the link is continuous with respect to xa, and the route cost is continuous with the route flow. Then, any local optimal solution must be a global optimal solution, and it is unique. In sum, the VI problem has a unique optimal solution.


## 3.4. Solution Algorithm for the ISUEED Model

The ISUEED model essentially is a multiclass user problem, which can be solved by the diagonalized method of the successive average (DMSA) algorithm [40–42]. It is worth noting that according to the definition of the multiclass user in this study, only one OD pair is involved per user class. Therefore, the travel demand is loaded according to the route choice model, and the equilibrium state is achieved simultaneously in the process of single-class user assignment. As a result, it eliminates the traffic adjustment process within the per class of user. Even though the number of user classes is enormous, the total running time is still acceptable. To meet the needs of this study, the link flow X and OD demand set WE are taken as the output results. Compared to previous literature, the algorithm has been slightly adjusted. Algorithm 1 gives the corresponding solving steps specifically.


## 3.5. Delineating the Subnetwork

This study takes nodes as the analysis object. The change rates of the link flow and OD demand with and without a work zone are used to generate the NSI. Then, we compare the NSI with the threshold value to determine whether a node is in the influence area. Finally, the subnetwork is delineated by the affected nodes, which are selected by the L-shell algorithm.Algorithm 1: DMSA.
Input: the network GN,A with work zone, Iw∀w∈W, set of travel demand for each class travellers between OD pair wQw∀w∈W, convergence accuracy ξ.Output:X, WE(1)
By the k-shortest paths method to generate the set of shortest routesIw′∀w∈W.(2)
Initialize: assignQw∀w∈W according to the SUE model. Acquire the traffic flow(3)
of the routefw,i′1 and the traffic flow of the link xa1. e⟶∞. For n=1.(4)
whilee>ξ(5)
forw in W(6)
Update the link travel time.(7)
Update the route travel time.(8)
Update the expected minimum travel costμw between each OD pair w.(9)
fori in Iw∀w∈W(10)
Updated the ODqw,in of such travellers according to equation (8).(11)
ifi∈Iw′∀w∈W(12)
Correct route travel cost according to equation (6).(13)
Loadqw,in to each route according to the NL shown in equations.(14)
(1)–(3)(15)
else(16)
Loadqw,in to each route according to the MNL.(17)
end if(18)
end for(19)
end for(20)
Update the link flowxan.(21)
Acquire additional traffic flow of the routegw,i′n.(22)
Update the route traffic flowfw,i′n+1=fw,i′n+χngw,i′n−fw,i′n, χn=1/n.(23)
Computational convergence precisione=∑w∈W∑i′∈Iw′fw,i′n+1−fw,i′n2/∑w∈W∑i′∈Iw′fw,i′n.(24)
Forn=n+1(25)
end while(26)
Calculateqwn=∑i∈Iwqw,in∀w∈W.(27)
Store eachxan and each qwn to X and WE, respectively.(28)
OutputX and WE.The change rate of the link flow for nodeu is related to the incoming flow and outgoing flow, which are calculated, respectively, as follows:(30)ΔFuin=∑∈ℳufum−∑m′∈ℳu′fum'∑m∈ℳufum,ΔFuout=∑m∈ℳufum−∑m′∈ℳu′fum'∑m∈ℳufu′m∑,where ΔFuin,ΔFuout are the change rates of the incoming (outgoing) flow at node u. fum,fu'm are the flows that enter (leave) node u through link m in the network without work zone. fum',fu'm' are the flows that enter (leave) node u through link m' in the network with work zone. ℳu,ℳu′ are the sets of links connected by node u when the network is with (without) a work zone.The average value of the two is calculated as the change rate of flow for nodeu and normalized.(31)ΔFuave=ΔFuin+ΔFuout2,ΔFu=ΔFuave−ΔFminaveΔFmaxave−ΔFminave,where ΔFuave is the average value of flow change rate at node u. ΔFmaxave,ΔFminave are the maximum and minimum values of the average flow change rate. ΔFu is the normalized value of ΔFuave.The change rate of OD demand for nodeu is tightly associated with the generating demand and attracting demand, which are calculated, respectively, as follows:(32)ΔQugen=∑n∈Ndun−∑n∈Ndun'∑n∈Ndun,ΔQuatt=∑n∈Ndu'n−∑n∈Ndu'n'∑n∈Ndu'n,where ΔQugen,ΔQuatt are the change rates of the generation (attraction) demand of node u. dun,du'n are the demands generation (attraction) of node u to node n when the network is without a work zone. dun',du'n' are the demands generation (attraction) of node u to node n when the network is with a work zone.Similarly, the change rate of OD demand for nodeu is as follows:(33)ΔQuave=ΔQugen+ΔQuatt2,ΔQu=ΔQuave−ΔQminaveΔQmaxave−ΔQminave,where ΔQuave is the average value of demand change rate at node u. ΔQmaxave,ΔQminave are the maximum and minimum values of the average demand change rate. ΔQu is the normalized value of ΔQuave.Based on equations (31) and (33), we define NSI for each node u in the network as a weighted average of ΔQu and ΔFu.(34)Su=αΔQu+1−αΔFu,where Su is the value of NSI for node u. α is the weight coefficient.The NSI is compared with the threshold to determine whether nodeu is within the area of influence, if so, put it into the set E.(35)state=inSu≥SstaoutSuSsta∀u∈N,where Ssta is the threshold value of NSI, which can be adjusted from 0 to 1 according to various needs to generate different subnetworks.


## 3.6. Modified L-Shell Algorithm

The L-shell algorithm is a method of community division in the complex network. This section only briefly explains the part involved in this study. The specific details and algorithm processes are shown in the literature [8]. In Figure 3(a), nodes 1 and 2 are vital nodes. Nodes 6, 7, and 8 are the adjacent nodes of node 1 in level 1. Nodes 12 and 13 are the level 1 adjacent nodes of node 6 and the level 2 adjacent nodes of node 1. According to the above process, it can be seen that the nodes 3–8 in Figure 3(a) are the level 1 adjacent nodes of vital nodes 1 and 2, denoted as the set l1. The nodes 9–18 are all the level 2 nodes of vital nodes 1 and 2, denoted as the set l2.Figure 3
Examples of related concepts in the L-shell algorithm.
(a)(b)(c)The distribution of NSI is not necessarily a centralized radial but may be sporadic and disconnected. To ensure the connectivity and compactness of the subnetwork, a modified L-shell algorithm is proposed to perform secondary filtering on nodes. The algorithm takes setE as input. It outputs the set F composed of all nodes within the subnetwork. The specific steps are shown in Algorithm 2. The central idea is to use the first “while” loop to basically determine the subnetwork, and then use the second “while” loop to check and fill gaps. In the first loop, it goes through over the elements in set En and places it in set F if its level 1 adjacent nodes already have at least one node in set F. Repeat the process until the size of set En does not change, which represents that all nodes in set E that can connect to a network have been selected. This process can be described as the vital nodes constantly adsorb the eligible nodes in set En layer-by-layer around it. The second loop is equivalent to a tolerance device, which is mainly to ensure the integrity of the subnetwork. Compared with the first loop, it changes both in the value of level and the judgment condition of putting the node into the set F. By traversing all the nodes that are not in set F, if the level 1 or level 2 adjacent nodes of one node are already in set F, put it in set F. For example, as shown in Figure 3(b), node 4 is not taken into set F after the above procedure. However, it is also going to be put in F in this loop. However, if the number of nodes exceeds the threshold, as depicted in Figure 3(c), nodes 1–7 are considered to be outside the subnetwork.Algorithm 2: Modified L-shell algorithm.
Input: the set of nodes E filtered by a threshold, the set of nodes N in the network.Output: the set of nodes F which compose of sub-network.(1)
Initialize: put the vital nodes intoF, E1=∅,E2=E,n=2.(2)
whileEn≠En−1(3)
fore in En(4)
Taking nodee as the starting point, search all level 1 adjacent nodes of node(5)
e according to the L-shell algorithm and store to Ne.(6)
Determine whether there isat least one member of Ne that is already(7)
included inF, if so, put e into F, otherwise, put it into En+1.(8)
end for(9)
Forn=n+1(10)
end while(11)
CalculateY=N−E+En.(12)
Initialize:Y1 = ∅, Y2=Y, n=2.(13)
whileYn≠Yn−1(14)
fory in Yn(15)
Taking nodey as the starting point, search all level 1 adjacent nodes and(16)
level2 adjacent nodes of y according to the L-shell algorithm and store(17)
toNy1,Ny2, respectively.(18)
Determine whether the members ofNy1orNy2are all included in F, if(19)
so, puty into F, otherwise, put it into Yn+1.(20)
end for(21)
Forn=n+1(22)
end while(23)
OutputF.


## 4. Case Study

We conduct two case studies to demonstrate the properties of the proposed method. Firstly, the Braess network is used to illustrate the effectiveness of the ISUEED model. Subsequently, we apply the Guilin Road network to test the feasibility and adaptability with respect to a larger scale network.


### 4.1. The Braess Network

The ISUEED model is the crucial component of the method proposed in this paper. Therefore, we select a small example to manifest the improvement of the ISUEED model. As shown in Figure4, the network includes 4 nodes, 6 links, 1 OD pair, and 4 routes. The OD demand is 100 veh and the 4 routes are as follows: route 1: links 1-2; route 2: links 3–5; route 3: links 1-4-5; route 4: link 6. In the original network, the three valid routes are 1, 2, and 4. We assume that link 6 is interrupted. The valid routes in the network are routes 1, 2, and 3. Detailed information about the network is shown in Table 1.Figure 4
The Braess network.Table 1
The properties of links.
LinkLength (m)Capacity (veh/h）Free-flow time(sec)αβ140060200.154240060200.154340060200.154440060200.154540060200.154660060300.154Table2 depicts the flow of each route and link before and after link 6 interruption. To illustrate the improvement of the model, we specially compare the results of the SUE, SUEED, and ISUEED models, which can present the influence of elasticity and inertia on traffic flow. First of all, we can observe that because of the interruption of link 6, the travel demand decreases from 100 to 97.34, with the expected travel cost between node O and node D increasing. Secondly, as travellers have a preference for routes 1 and 2, inertia leads to a slight decrease in the number of travellers choosing route 3, which is confirmed by the results of the SUEED and ISUEED models. To sum up, these findings demonstrate that the ISUEED model proposed in this study has distinguished performance. By introducing elasticity and inertia, it can portray the travel behavior of travellers more realistically and obtain traffic assignment results closer to reality.Table 2
The flow of links and routes.
Before interruptionAfter interruptionOriginal (veh/h)SUE (veh/h)SUEED (veh/h)ISUEED (veh/h)Link 132.2064.5362.8162.41Link 232.2035.4734.5334.93Link 332.2035.4734.5334.93Link 4—29.0628.2827.48Link 532.2064.5362.8162.41Link 635.60———Route 132.2035.4734.5334.93Route 232.2035.4734.5334.93Route 3—29.0628.2827.48Route 435.60———OD demand100 veh100 veh97.34 veh97.34 veh


### 4.2. The Guilin Road Network

To explore the properties of the proposed method in subnetwork delineation, a case study is carried out based on the road maintenance project of the Guilin Road business district in Changchun, China. Since the method in this study takes nodes as the analysis object, the real road network is abstracted into a grid network for the convenience of expression. As shown in Figure5, it has 136 nodes and 482 links. There is a work zone between node 65 and node 66, which is 200 meters in length. It involves the construction of pipeline, cable, and pavement projects. The planned construction period is two months. During this period, the lanes are closed in both directions, and all vehicles are required to bypass the surrounding roads. The road is a branch with one lane in each direction, and its peak hour two-way traffic volume is 1566 veh/h. The link flow of the network with and without work zone are obtained from field traffic surveys. The corresponding travel demand data is derived from the OD estimation by TransCAD software.Figure 5
Case study work zone in the Guilin Road.


### 4.3. Delineation Results

The subnetwork of work zone in the road network is determined by the delineation method proposed in this study. With respect to the increasing threshold, Figure6 depicts an obviously smaller subnetwork. When the threshold value is 0.8, it means that the nodes with extremely drastic changes would be classified into the subnetwork. Hence, it can be seen that the subnetwork is only limited to the adjacent links of the work zone. To verify the feasibility and accuracy of the method, this study obtains the data of the link flow before and after work zone existence from field investigation. Taking the 20% change rate of link saturation as the criterion, the actual area of influence is determined as shown in Figure 7. It is basically consistent with the subnetwork delineated by the proposed method with the threshold value of 0.2. On the other hand, with the decrease of threshold value, the spatial variation of the subnetwork is captured, which reflects the spatial propagation trend of network disturbance to a great extent. It is beneficial to achieve the network congestion prediction and targeted intervention. Furthermore, the subnetwork under different requirements can be realized by adjusting the threshold value. For example, when the traffic management agency has a higher requirement on the performance of the road network, the threshold can be lowered to make the subnetwork larger, and vice versa.Figure 6
Subnetwork with different thresholds.Figure 7
Subnetwork of field survey.Although the change rates of link flow and OD demand are only the subsidiary products of the calculation of the NSI, their appearance provides new perspectives for the subsequent formulation of targeted traffic adjustment strategies. Figure8 comprehensively summarizes the spatial distribution of the change rates of OD demand among nodes, which appears as regional aggregation and cyclical change. As displayed in Table 3, more than 90% of OD pairs slightly decrease by about 1%, which corresponds to the blue part in Figure 8. This phenomenon also reflects that the elastic demand will keep it stable by itself, and the changes only occur in a narrow area. Public transportation and other modes should be prepared in advance to undertake this part of transferred OD demand and avoid greater disturbance to the traffic network. Therefore, the reasonable cooperation of various transportation modes should be an important part of the transportation organization scheme of work zone. On the contrary, there are only a few parts in red, which are OD demands that vary dramatically. It is worth noting that the white part in Figure 8 indicates that OD demand is 0, that is to say, there is no travel between the corresponding two zones. From a spatial point of view, there is an obvious negative correlation between the change rate of link saturation and its distance from the work zone (link ID = 224 and 227), as shown in Figure 9. The adjacent links of the work zone have changed even more dramatically. According to the statistical results in Table 3, the change rate of link saturation is mainly between −100% and 150%, and only one link is close to 200%, presenting a normal distribution with 0 as the expectation on the whole. Traffic managers should focus on these links because slight improvements may significantly boost overall network performance.Figure 8
The change rate of OD matrix.Table 3
Statistics on the change rates of OD demand and link flow.
OD demandLink flowChange rate (%)NumberChange rate (%)Number[−1.05, −0.90]7573[−100, −90]2(−0.9, −0.75]2567(−90, −80]0(−0.75, −0.60]2045(−80, −70]2(−0.60, −0.45]1247(−70, −60]0(−0.45, −0.30]799(−60, −50]0(−0.30, −0.15]529(−50, −40]5(−0.15, 0]882(−40, −30]7(0, 0.15]457(−30, −20]13(0.15, 0.30]448(−20, −10]21(0.30, 0.45]435(−10, 0]63(0.45, 0.60]371(0, 10]57(0.60, 0.75]313(10, 20]75(0.75, 0.90]246(20, 30]76(0.90, 1.05]154(30, 40]47(1.05, 1.30]110(40, 50]41(1.30, 1.45]74(50, 60]31(1.45, 1.60]38(60, 70]17(1.60, 1.75]32(70, 80]8(1.75, 1.90]19(80, 90]5(1.90, 2.05]10(90, 100]6(2.05, 2.30]5≥1008(2.30, 2.45]4(2.45, 2.60]1(2.60, 2.75]1Figure 9
The change rate of link saturation.


### 4.4. Sensitivity Analysis

Table4 shows the statistical results of the three indicators with the subnetwork under different parameter scenarios, which are the number of nodes, the number of links, and the average value of NSI (ANSI). It is interesting to note that ANSI has no obvious trend of change. These results might attribute to the second loop of the modified L-shell algorithm. As the selection criteria is relaxed, there may be an indefinite number of nodes whose NSI is inferior to the threshold to be delineated into the subnetwork. Hence, ANSI is an unpredictable value, which may be related to the OD demand distribution and road network topology. The specific action mechanism of each parameter is displayed in Figures 10–17, respectively.Table 4
Subnetwork constitution under different parameters.
ParameterNumber of nodesNumber of linksANSIInertia coefficient0.1580.20.420520.250.7391160.241561740.22Elastic coefficient−0.15632080.23−0.09491580.27−0.0630900.25−0.017120.22Capacity0491580.350.522620.270.810220.22Scenarios1351100.25223680.263451480.23Coexisting993420.21OD distributionUpper581860.23Right461220.22Surrounding702560.23Figure 10
Box plot of NSI at different inertia.Figure 11
Box plot of NSI at different elasticity.Figure 12
Subnetwork under different inertia.Figure 13
Subnetwork under different elasticity.Figure 14
Subnetwork under different disturbances.Figure 15
Subnetwork under different scenarios.Figure 16
The situation of area division.Figure 17
Subnetwork under different OD distributions.Firstly, we explore the effect of inertial route choice behavior and elastic demand on sub-network delineation. Scenarios with different inertia coefficients and elastic coefficients are discussed. As shown in Figure10, with the inertial coefficient increasing from 0.1 to 1 at step size 0.1, the maximum, minimum, and average values of NSI present an upward trend at first and then increase slowly. Meanwhile, Figure 12 illustrates the subnetworks with inertial coefficients of 0.1, 0.4, 0.7, and 1, respectively, which present a significant tendency to expand. The main reason for this is that travellers are more inclined to choose the fresh route as the coefficient goes up, and the disturbance to the road network is drastic, and vice versa. When the inertia coefficient is close to 0, there will be an extreme scene, where almost all travellers travel on the original route, unless the route no longer exists because of a change of network topology. Hence, the network changes are only limited to a narrow scope. To sum up the aforementioned conclusions, it reminds that traffic managers can guide the travellers to choose the original route as much as possible through media publicity, economic leverage, and road traffic restrictions. After all, it can effectively reduce the disturbance to the network.On the contrary, from Figure11, it can be seen that as the elastic coefficient gradually increases from −0.18 to −0.01, and the box plots of NSI show an obvious trend of decline. The subnetwork (elastic coefficient = −0.15, −0.09, −0.06, and −0.01) will diminish at the same time, as shown in Figure 13. It is well-known that OD demand is tightly connected with the generalized travel costs of related routes. Travellers with greater flexibility are more likely to shift their travel mode or even cancel their trips when they are faced with the situation of work zone existing in the network, which has deep influence on the OD demand. Meanwhile, it suggests that maybe changing the cost of private car or other alternative travel modes is a feasible way to achieve minimal degradation of network performance.To explore the influence of different disturbance degrees on subnetwork delineation, three scenarios, including single-lane interruption, one-way interruption, and two-way interruption, are proposed. They assume that the corresponding capacities are 80%, 50%, and 0% of the original capacity, respectively. As shown in Figure14, with the decrease of construction intensity, its subnetwork will shrink significantly. In view of the above reasons, it is not recommended to completely cut off construction, especially if the construction period is long. Inversely, the gap of the impact caused by different methods may be marginal when the construction period shortens. On the other hand, it also emphasizes the importance of designing individualized construction schemes according to the characteristics of different projects.In some cases, multiple work zones will exist at the same time, which will have a superimposed impact on the road network. To investigate the impact of this situation on the mechanism of subnetwork delineation, we assumed that there are three work zones located between node 29 and node 30, node 55 and node 56, and node 85 and node 86. Figure15 depicts the area of influence when three work zones exist alone and coexist. The different locations of work zone are equivalent to the different space coverages of subnetworks, and they overlap with each other. However, it should be noted that when multiple work zones coexist, their subnetworks are not simply superimposed but redistribute the overall network traffic flow. Therefore, the joint organization and optimization of multiple work zones is the preferred way. It also provides powerful insights for the design of the construction schedule with multiple work zones.Furthermore, the effect of OD distribution on subnetwork is also discussed briefly. We assume the following three OD distributions: firstly, the OD demand of the upper area of the work zone increases twice as its origin (upper). Secondly, the OD demand of the right area of the work zone increases twice as its origin (right). Thirdly, the OD demand in the surrounding area of work zone remains the same, however, the OD demand outside the surrounding area increases twice as its origin (surrounding). The specific area division and corresponding subnetwork are shown in Figures16 and 17. It can be observed that with the change of OD demand, the subnetwork has obvious deviation. Specifically, the subnetwork will extend to the area with a larger OD demand and display distinct asymmetry to the work zone. This phenomenon also presents that it is necessary to fully investigate OD distribution before designing traffic adjustment scheme, which will make the scheme more efficient.


## 4.1. The Braess Network

The ISUEED model is the crucial component of the method proposed in this paper. Therefore, we select a small example to manifest the improvement of the ISUEED model. As shown in Figure4, the network includes 4 nodes, 6 links, 1 OD pair, and 4 routes. The OD demand is 100 veh and the 4 routes are as follows: route 1: links 1-2; route 2: links 3–5; route 3: links 1-4-5; route 4: link 6. In the original network, the three valid routes are 1, 2, and 4. We assume that link 6 is interrupted. The valid routes in the network are routes 1, 2, and 3. Detailed information about the network is shown in Table 1.Figure 4
The Braess network.Table 1
The properties of links.
LinkLength (m)Capacity (veh/h）Free-flow time(sec)αβ140060200.154240060200.154340060200.154440060200.154540060200.154660060300.154Table2 depicts the flow of each route and link before and after link 6 interruption. To illustrate the improvement of the model, we specially compare the results of the SUE, SUEED, and ISUEED models, which can present the influence of elasticity and inertia on traffic flow. First of all, we can observe that because of the interruption of link 6, the travel demand decreases from 100 to 97.34, with the expected travel cost between node O and node D increasing. Secondly, as travellers have a preference for routes 1 and 2, inertia leads to a slight decrease in the number of travellers choosing route 3, which is confirmed by the results of the SUEED and ISUEED models. To sum up, these findings demonstrate that the ISUEED model proposed in this study has distinguished performance. By introducing elasticity and inertia, it can portray the travel behavior of travellers more realistically and obtain traffic assignment results closer to reality.Table 2
The flow of links and routes.
Before interruptionAfter interruptionOriginal (veh/h)SUE (veh/h)SUEED (veh/h)ISUEED (veh/h)Link 132.2064.5362.8162.41Link 232.2035.4734.5334.93Link 332.2035.4734.5334.93Link 4—29.0628.2827.48Link 532.2064.5362.8162.41Link 635.60———Route 132.2035.4734.5334.93Route 232.2035.4734.5334.93Route 3—29.0628.2827.48Route 435.60———OD demand100 veh100 veh97.34 veh97.34 veh


## 4.2. The Guilin Road Network

To explore the properties of the proposed method in subnetwork delineation, a case study is carried out based on the road maintenance project of the Guilin Road business district in Changchun, China. Since the method in this study takes nodes as the analysis object, the real road network is abstracted into a grid network for the convenience of expression. As shown in Figure5, it has 136 nodes and 482 links. There is a work zone between node 65 and node 66, which is 200 meters in length. It involves the construction of pipeline, cable, and pavement projects. The planned construction period is two months. During this period, the lanes are closed in both directions, and all vehicles are required to bypass the surrounding roads. The road is a branch with one lane in each direction, and its peak hour two-way traffic volume is 1566 veh/h. The link flow of the network with and without work zone are obtained from field traffic surveys. The corresponding travel demand data is derived from the OD estimation by TransCAD software.Figure 5
Case study work zone in the Guilin Road.


## 4.3. Delineation Results

The subnetwork of work zone in the road network is determined by the delineation method proposed in this study. With respect to the increasing threshold, Figure6 depicts an obviously smaller subnetwork. When the threshold value is 0.8, it means that the nodes with extremely drastic changes would be classified into the subnetwork. Hence, it can be seen that the subnetwork is only limited to the adjacent links of the work zone. To verify the feasibility and accuracy of the method, this study obtains the data of the link flow before and after work zone existence from field investigation. Taking the 20% change rate of link saturation as the criterion, the actual area of influence is determined as shown in Figure 7. It is basically consistent with the subnetwork delineated by the proposed method with the threshold value of 0.2. On the other hand, with the decrease of threshold value, the spatial variation of the subnetwork is captured, which reflects the spatial propagation trend of network disturbance to a great extent. It is beneficial to achieve the network congestion prediction and targeted intervention. Furthermore, the subnetwork under different requirements can be realized by adjusting the threshold value. For example, when the traffic management agency has a higher requirement on the performance of the road network, the threshold can be lowered to make the subnetwork larger, and vice versa.Figure 6
Subnetwork with different thresholds.Figure 7
Subnetwork of field survey.Although the change rates of link flow and OD demand are only the subsidiary products of the calculation of the NSI, their appearance provides new perspectives for the subsequent formulation of targeted traffic adjustment strategies. Figure8 comprehensively summarizes the spatial distribution of the change rates of OD demand among nodes, which appears as regional aggregation and cyclical change. As displayed in Table 3, more than 90% of OD pairs slightly decrease by about 1%, which corresponds to the blue part in Figure 8. This phenomenon also reflects that the elastic demand will keep it stable by itself, and the changes only occur in a narrow area. Public transportation and other modes should be prepared in advance to undertake this part of transferred OD demand and avoid greater disturbance to the traffic network. Therefore, the reasonable cooperation of various transportation modes should be an important part of the transportation organization scheme of work zone. On the contrary, there are only a few parts in red, which are OD demands that vary dramatically. It is worth noting that the white part in Figure 8 indicates that OD demand is 0, that is to say, there is no travel between the corresponding two zones. From a spatial point of view, there is an obvious negative correlation between the change rate of link saturation and its distance from the work zone (link ID = 224 and 227), as shown in Figure 9. The adjacent links of the work zone have changed even more dramatically. According to the statistical results in Table 3, the change rate of link saturation is mainly between −100% and 150%, and only one link is close to 200%, presenting a normal distribution with 0 as the expectation on the whole. Traffic managers should focus on these links because slight improvements may significantly boost overall network performance.Figure 8
The change rate of OD matrix.Table 3
Statistics on the change rates of OD demand and link flow.
OD demandLink flowChange rate (%)NumberChange rate (%)Number[−1.05, −0.90]7573[−100, −90]2(−0.9, −0.75]2567(−90, −80]0(−0.75, −0.60]2045(−80, −70]2(−0.60, −0.45]1247(−70, −60]0(−0.45, −0.30]799(−60, −50]0(−0.30, −0.15]529(−50, −40]5(−0.15, 0]882(−40, −30]7(0, 0.15]457(−30, −20]13(0.15, 0.30]448(−20, −10]21(0.30, 0.45]435(−10, 0]63(0.45, 0.60]371(0, 10]57(0.60, 0.75]313(10, 20]75(0.75, 0.90]246(20, 30]76(0.90, 1.05]154(30, 40]47(1.05, 1.30]110(40, 50]41(1.30, 1.45]74(50, 60]31(1.45, 1.60]38(60, 70]17(1.60, 1.75]32(70, 80]8(1.75, 1.90]19(80, 90]5(1.90, 2.05]10(90, 100]6(2.05, 2.30]5≥1008(2.30, 2.45]4(2.45, 2.60]1(2.60, 2.75]1Figure 9
The change rate of link saturation.


## 4.4. Sensitivity Analysis

Table4 shows the statistical results of the three indicators with the subnetwork under different parameter scenarios, which are the number of nodes, the number of links, and the average value of NSI (ANSI). It is interesting to note that ANSI has no obvious trend of change. These results might attribute to the second loop of the modified L-shell algorithm. As the selection criteria is relaxed, there may be an indefinite number of nodes whose NSI is inferior to the threshold to be delineated into the subnetwork. Hence, ANSI is an unpredictable value, which may be related to the OD demand distribution and road network topology. The specific action mechanism of each parameter is displayed in Figures 10–17, respectively.Table 4
Subnetwork constitution under different parameters.
ParameterNumber of nodesNumber of linksANSIInertia coefficient0.1580.20.420520.250.7391160.241561740.22Elastic coefficient−0.15632080.23−0.09491580.27−0.0630900.25−0.017120.22Capacity0491580.350.522620.270.810220.22Scenarios1351100.25223680.263451480.23Coexisting993420.21OD distributionUpper581860.23Right461220.22Surrounding702560.23Figure 10
Box plot of NSI at different inertia.Figure 11
Box plot of NSI at different elasticity.Figure 12
Subnetwork under different inertia.Figure 13
Subnetwork under different elasticity.Figure 14
Subnetwork under different disturbances.Figure 15
Subnetwork under different scenarios.Figure 16
The situation of area division.Figure 17
Subnetwork under different OD distributions.Firstly, we explore the effect of inertial route choice behavior and elastic demand on sub-network delineation. Scenarios with different inertia coefficients and elastic coefficients are discussed. As shown in Figure10, with the inertial coefficient increasing from 0.1 to 1 at step size 0.1, the maximum, minimum, and average values of NSI present an upward trend at first and then increase slowly. Meanwhile, Figure 12 illustrates the subnetworks with inertial coefficients of 0.1, 0.4, 0.7, and 1, respectively, which present a significant tendency to expand. The main reason for this is that travellers are more inclined to choose the fresh route as the coefficient goes up, and the disturbance to the road network is drastic, and vice versa. When the inertia coefficient is close to 0, there will be an extreme scene, where almost all travellers travel on the original route, unless the route no longer exists because of a change of network topology. Hence, the network changes are only limited to a narrow scope. To sum up the aforementioned conclusions, it reminds that traffic managers can guide the travellers to choose the original route as much as possible through media publicity, economic leverage, and road traffic restrictions. After all, it can effectively reduce the disturbance to the network.On the contrary, from Figure11, it can be seen that as the elastic coefficient gradually increases from −0.18 to −0.01, and the box plots of NSI show an obvious trend of decline. The subnetwork (elastic coefficient = −0.15, −0.09, −0.06, and −0.01) will diminish at the same time, as shown in Figure 13. It is well-known that OD demand is tightly connected with the generalized travel costs of related routes. Travellers with greater flexibility are more likely to shift their travel mode or even cancel their trips when they are faced with the situation of work zone existing in the network, which has deep influence on the OD demand. Meanwhile, it suggests that maybe changing the cost of private car or other alternative travel modes is a feasible way to achieve minimal degradation of network performance.To explore the influence of different disturbance degrees on subnetwork delineation, three scenarios, including single-lane interruption, one-way interruption, and two-way interruption, are proposed. They assume that the corresponding capacities are 80%, 50%, and 0% of the original capacity, respectively. As shown in Figure14, with the decrease of construction intensity, its subnetwork will shrink significantly. In view of the above reasons, it is not recommended to completely cut off construction, especially if the construction period is long. Inversely, the gap of the impact caused by different methods may be marginal when the construction period shortens. On the other hand, it also emphasizes the importance of designing individualized construction schemes according to the characteristics of different projects.In some cases, multiple work zones will exist at the same time, which will have a superimposed impact on the road network. To investigate the impact of this situation on the mechanism of subnetwork delineation, we assumed that there are three work zones located between node 29 and node 30, node 55 and node 56, and node 85 and node 86. Figure15 depicts the area of influence when three work zones exist alone and coexist. The different locations of work zone are equivalent to the different space coverages of subnetworks, and they overlap with each other. However, it should be noted that when multiple work zones coexist, their subnetworks are not simply superimposed but redistribute the overall network traffic flow. Therefore, the joint organization and optimization of multiple work zones is the preferred way. It also provides powerful insights for the design of the construction schedule with multiple work zones.Furthermore, the effect of OD distribution on subnetwork is also discussed briefly. We assume the following three OD distributions: firstly, the OD demand of the upper area of the work zone increases twice as its origin (upper). Secondly, the OD demand of the right area of the work zone increases twice as its origin (right). Thirdly, the OD demand in the surrounding area of work zone remains the same, however, the OD demand outside the surrounding area increases twice as its origin (surrounding). The specific area division and corresponding subnetwork are shown in Figures16 and 17. It can be observed that with the change of OD demand, the subnetwork has obvious deviation. Specifically, the subnetwork will extend to the area with a larger OD demand and display distinct asymmetry to the work zone. This phenomenon also presents that it is necessary to fully investigate OD distribution before designing traffic adjustment scheme, which will make the scheme more efficient.


## 5. Conclusion

This paper realizes the quantitative delineation of the subnetwork, which provides a scientific scope basis for the evaluation of traffic impact and the formulation of traffic measures. This method takes the NSI as a variability measure to capture both the changes of link flow and OD demand. The ISUEED model fully reflects the response of the network to the incident by route choice behavior and localized demand change. Therefore, it can accurately provide the data of link flow and OD demand for the network with work zone. Correspondingly, the SUE model is applied to obtain the data of the network without a work zone. Then, by comparing NSI with the threshold value, the initial range of the subnetwork can be determined. Finally, the modified L-shell algorithm is employed to ensure the connectivity and compactness of the subnetwork.The results not only show the feasibility and practicability of the proposed subnetwork delineation method but also demonstrate its robustness to parameters and scenario changes. Hence, managers can adjust parameters to obtain different subnetworks that meet the personalized requirements of various projects. At the same time, it also gives the changes of link flow and OD demand in the spatial dimension, which is of great help to the development of traffic management plan. On the other hand, the subnetwork with different interruption modes, the coexistence of multiple work zones, and OD distributions are also conducted. The conclusions can be used to clarify the direction of the construction scheme formulation.The current research can be widened in the following aspects: firstly, day-to-day traffic assignment models can be applied in the framework to investigate the cumulative loss during the construction period. Secondly, since it is expensive for data collection, advanced data acquisition technologies should be introduced. Finally, the popularity of connected autonomous vehicles is an irreversible trend, which will influence the network performance deeply. Thus, there exists the necessity to explore the subnetwork delineation under mixed traffic environment.



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*Source: 2507107-2022-04-20.xml*