Research Article  Open Access
Chenming Jiang, Linjun Lu, Junliang He, Caimao Tan, "A TwoClass Stochastic Network Equilibrium Model under Adverse Weather Conditions", Journal of Advanced Transportation, vol. 2020, Article ID 2626084, 14 pages, 2020. https://doi.org/10.1155/2020/2626084
A TwoClass Stochastic Network Equilibrium Model under Adverse Weather Conditions
Abstract
Adverse weather condition is one of the inducements that lead to supply uncertainty of an urban transportation system, while travelers’ multiple route choice criteria are the nonignorable reason resulting in demand uncertainty. This paper proposes a novel stochastic traffic network equilibrium model considering impacts of adverse weather conditions on roadway capacity and route choice criteria of twoclass mixed roadway travellers on demand modes, in which the twoclass route choice criteria root in travelers’ different network information levels (NILs). The actual route travel time (ARTT) and perceived route travel time (PRTT) are considered as the route choice criteria of travelers with perfect information (TPI) and travelers with bounded information (TBI) under adverse weather conditions, respectively. We then formulate the user equilibrium (UE) traffic assignment model in a variational inequality problem and propose a solution algorithm. Numerical examples including a small triangle network and the Sioux Falls network are presented to testify the validity of the model and to clarify the inner mechanism of the twoclass UE model under adverse weather conditions. Managerial implications and applications are also proposed based on our findings to improve the operation efficiency of urban roadway network under adverse weather conditions.
1. Introduction
Uncertainties of transportation networks are logically derived from supply side (roadway capacity fluctuation) and demand side (travel demand variation) [1–4], which can also be considered as exogenous and endogenous sources [5], respectively. On account of the overwhelming majority of transportation modes being directly exposed in the atmospheric environment, the total process of traffic activity is inevitably more or less subject to adverse weather conditions, such as rainfall, snow, fog, and high winds, etc. [6]. Therefore, adverse weather conditions are widely recognized as a primary inducement markedly generating the uncertainties of roadway capacity. On the other side, travel demand uncertainties mainly come from the temporal factors, such as time of day [7], day of week or season effects. Note that travelers’ heterogeneous route choice criteria are also considered as a nonignorable reason [8, 9] resulting in demand uncertainty. Multiclass route choice behavior throughout population usually root from travelers’ differentiated levels of attainable network information about the present network scenarios. Finally, the interaction existing between traffic supply and demand [10, 11] of the unreliable or uncertain transport system can change traveler’s behavior and then arouse the spatial and temporal fluctuation of traffic demand.
As a vital inducement to the travel time uncertainty, adverse weather condition has been understanding and modeling its relationship with travelers’ route choice decisions [12]. Therefore, in this study, we will focus on the impact of adverse weather conditions on network equilibrium traffic assignment model considering traffic network uncertainty. Lam et al. [13] and Shao et al. [14] are believed to be the first studies who implicitly incorporated the impacts of adverse weather (i.e., rainfall) on both free flow speed and roadway capacity into the Bureau of Public Roads (BPR) function, which was followed and transplanted to multimodal transport network problems under adverse weather conditions by Sumalee et al. [15] and Uchida et al. [8]. However, few works have been conducted in the subdivision of travelers’ various route choice criteria including the impacts of different levels of information penetration under weather conditions. In the SUE model considering the effects of rainfall intensity on the road performance proposed by Lam et al. [13], they assume that travelers are homogeneous that all travelers are considered as undifferentiated ones who choose their routes with the minimum perceived effective travel time based on the occurring probability of weather scenarios (i.e., no rain/light rain, normal rain, amber rainstorm, etc.). Nowadays, however, with the aid of Advanced Traveler Information System (ATIS), some motorists can obtain the realtime traffic information of the current network, and then make their optimal routing choices based on the recommended actual shortest routes. Therefore, in this study, to model the network equilibrium problem more realistically, we intuitively assume that the traveler population are divided into two exclusive classes by their different network information levels (NILs) under adverse weather conditions: (1) with the equipment with ATIS, one part of travelers who can acquire a perfect knowledge of actual, scenariospecific network condition (i.e., travelers with perfect information, TPI) would fully choose the recommended actual shortest route with minimum actual travel cost, which accords with the wisdom of conventional user equilibrium [16], (2) analogous to the traveler feature that depicted in Lam et al. [13], the other part of travelers, who are clustered without the actual information of the network and named as travelers with bounded information (TBI), choose their routes with minimum perceived travel cost according to their individual travelling experiences and perceptions on the information of weather forecast. In Lam et al. [13] and Shao et al. [14], all the travelers considered were those who can acquire the weather forecast information, while those who are armed with ATIS and can acquire the actual link travel time in the network were not considered. This paper contributes to the literature by extending the works of Lam et al. [13] and Shao et al. [14] by simultaneously considering both of the two user classes, i.e., TBI (in [13, 14] and TPI with ATIS (in the traditional Wardrop UE model). Besides, to focus on a traffic planning level, this study is to propose a static UE model without considering travelers’ attitudes towards risk (i.e., travel time reliability), rather than a SUE one.
The remainder of this paper is organized as follows. Section 2 develops a brief problem description of this study. Section 3 elaborately depicts the UE model in a mathematical formulation. Corresponding solution algorithm is proposed in Section 4. In Section 5, two numerical examples are presented to illustrate the validity of the proposed model and solution algorithm. Finally, the conclusions of the study and recommendations for future perspective are given in Section 6.
2. Problem Descriptions
Define a stochastic traffic system consisting of a traffic network subsystem and a twoclass traveler subsystem . For stochastic traffic networks , is the node set, is the link set, is the network scenario set and is the occurring probability set related to network scenarios. For users , is the set of travelers’ network information levels (NILs).
In meteorology, adverse weather conditions refer to various categories of local weather events that are destructive or not conducive to human production and activities, which differentiate with each other in disastercausing mechanism, such as rainfall, snow, fog, high winds, and extreme temperatures etc., While in this transportationrelated study, adverse weather conditions are further defined as those local weather events that can directly degrade roadway capacity and may affect travelers’ route choice behavior. Meanwhile, for a specific adverse weather condition, we define the network scenariosas subdivisions by different intensity levels under a certain adverse weather condition. For example, for a certain adverse weather condition, say, rainfall, there may be five network scenarios in China consisting of light rain, moderate rain, heavy rain, rainstorm, and heavy rainstorm that are classified by different rainfall intensity levels (mm per hour). To model the traffic network equilibrium problem, the foundation is to elaborate the travelers’ route choice behavior (or criterion) under different network scenarios. In the traditional user equilibrium [16], travelers are assumed to know the network information perfectly and choose their routes according to actual minimum disutilities. Nowadays, by being equipped with ATIS (e.g., some prevailing cellphone navigation apps), travelers with heterogeneous traveling experience are elevated to ones of homogeneous, nearlyperfect insight on actual network situation. Without the help of ATIS, however, travelers are generally unrealistic to grasp perfect knowledge of all route travel costs in the network. Therefore, in this paper, the notion of twoclass users consisting of travelers with perfect information (TPI) and bounded information (TBI) is incorporated to our model to describe roadway users’ heterogeneity in NILs. All travelers (TPI and TBI) choose their own travel routes in terms of their individual perceived route travel time (PRTT). To be specific, the PRTT of TPI is the actual route travel time (ARTT) acquired from the ATIS, while the PRTT of TBI is derived from travelers’ perception on the accuracy of weather forecast information and the ARTT under each network scenario.
The majority of notations used throughout the paper are listed in Table 1 unless otherwise specified.

3. Model and Formulation
3.1. Link Cost Function under Adverse Weather Conditions
The impacts of adverse weather conditions on the road performance, which can intuitively be learned as the deterioration of the visibility and pavement friction, would degrade the roadway performance by decreasing the freeflow speed and ideal link capacity. To capture these impacts, a generalized Bureau of Public Roads (GBPR) function [14] is proposed based on the Bureau of Public Roads (BPR) function as follows:
in which , , and is the adverse weather intensity under adverse weather scenario (AWS) s, e.g., the hourly average rainfall intensity (i.e., mm/h) of moderate rain.
are the freeflow travel time and capacity of link under scenario (intensity ), respectively; and denote freeflow travel time and capacity of link under good weather scenario (), respectively; while and are the scaling functions of scenario (intensity ) on freeflow travel time and ideal link capacity under good weather scenario, respectively. The heavier the adverse weather intensity (i.e., a larger ), the higher the freeflow travel time and the lower link capacity. Therefore, the scaling functions should hold the following features [13]: (a) is an increasing function with respect to indicating that the freeflow travel time increases with the adverse weather intensity. (b) is a decreasing function with respect to indicating that the link capacity decreases with the adverse weather intensity. (c) indicates that when the weather is clear (), this link travel time function (1) is equivalent to the conventional BPR function.
The actual travel time of route r between OD pair w in scenario s (intensity ) is:
3.2. Effects of Information from Weather Forecast on TBI
Weather forecast usually is a familiar and useful information source, based on which travelers can make their individual travel arrangements to maximize their expected utilities. Travelers can acquire realtime weather forecast information via many approaches, e.g., vehiclemounted radio broadcasting programs or weather forecast apps, etc. Compared with TPI who can acquire actual route travel time of the network, TBI without ATIS in this study are assumed to have the identical attitude to a piece of weather forecast information. However, weather forecast results cannot always perfectly match the physical truth, which is known as its prediction bias. Therefore, besides the information about adverse weather intensity under adverse weather scenario s, weather forecast also issue corresponding predictive probability , e.g., heavy rainfall ( mm/h) with the occurring probability of 80% ().
Let represent the set of prior occurring probability of each network scenario from the weather forecast. For instant means that, the weather forecast predicts that light rain, moderate rain, heavy rain, rainstorm and heavy rainstorm would occur tomorrow with the chances of 5%, 10%, 70%, 10% and 5%, respectively. Then following Lam et al. [13] and based on Bayes’ formula, we can obtain that , the perceived occurring probability of scenario of TBI under weather forecasting probability set , is:
in which is the perceived conditional probability of given scenario s which represents the perceived average accuracy of the weather forecast according to the previous experience of a TBI. For example, means that based on the previous experience of an arbitrary TBI, when a heavy rain () occurred, this traveler estimates that the average prediction probability that the weather forecast had predicted is 0.8, which hints that he/she considers that the forecast information is acceptably accurate. The subscript set consists of all adverse weather scenarios of this adverse weather condition. For occurring probability of the weather scenario perceived by any TBI, the conservation equation holds:
3.3. Basic Model and Its Characteristics
Let and denote the flow of TPI and TBI under scenario on link , respectively. Assume that and denote the perceived link travel time of TPI under scenario and TBI on link , respectively. This study assumes that the following formulas are tenable:
in which
To be specific, formula (7) assumes that the cost function of link a merely correlates with the flow of j categories of travelers (i.e., TPI and TBI) on link a, and is independent of flows on other links, e.g., . We further assume that the total equivalent flow on link a is a linear combination of TPI flow and TBI flow on this link, with weighting coefficients being , as formulated in formula (8). For example, if the flows of TPI and TBI have even contributions (weight) on travel time of link , (i.e., ), the total flow of link just equals to the flow summation of TPI and TBI, i.e.,
then formula (7) turns to be
For TPI, their perceived link travel time is the actual travel time of link a in scenario (intensity ) acquired from ATIS, shown as (11). While for TBI, their perceived link travel time is the perceived synthetic link travel time combining their personal perception on the accuracy of weather forecast information with kinds of actual link travel time in all scenarios , shown as formula (12):
To make the problem we studied herein prominent and clearly, we assume that the travel demand under a certain network scenario is deterministic. Assume that and denote the travel demand of TPI and TBI traveling between OD pair and under adverse weather scenario , respectively. Then the flow conservation constraint equations can be formulated as
which jointly form the feasible region of link flow pattern , in which .
The route choice criteria of TPI—Actual Route Travel Time (ARTT). Given a specific adverse weather scenario s, the perceived route travel time of TPI on route r connecting OD pair , i.e., ARTT, can be written as
The route choice criteria of TBI—Perceived Route Travel Time (PRTT). Given all AWSs that might occur, the perceived route travel time of TBI on route r connecting OD pair w, i.e., PRTT, can be written as
in which is the perceived occurring probability of TBI under scenario s given in (5).
Assume that N_{w}, denotes the total travel demand between OD pair w, then consists of TPI and TBI. As shown in Table 2, denotes the quantity of TPI, and indicates corresponding proportion, then we have . Analogously, and denote the quantity and proportion of TBI, then we also have , , .

3.4. Equilibrium Conditions
(1) Any TPI chooses a route with the minimum ARTT () given network scenario s. So the following equilibrium conditions must hold for TPI traveling on all routes between OD pair w under an arbitrary network scenario s:
For all TPI who travel between OD pair w, no one can further lower his ARTT by unilaterally altering his travelling route; the ARTT of all utilized routes between OD pair w are equal and minimal (), and will not larger than that of any unutilized routes.
(2) As discussed above, any TBI chooses a route with the minimum PRTT () given all network scenarios. So the following equilibrium conditions must hold for TBI of knowledge level k traveling on all routes between OD pair w under any network scenario.
No matter how the network scenario would be, for any TBI traveling between OD pair , no one can further lower his PRTT by unilaterally altering his travelling route; the PRTT of all utilized routes between OD pair are equal and minimal (), and will not larger than that of any unutilized routes.
The travel demands of both TPI and TBI between OD pair satisfy the flow conservation constraint Equations (13)–(15).
3.5. A Variational Inequality Formulation
The nonlinear complementarity conditions can be formulated as a finitedimentional variational inequality (VI) problem:
in which, , , denotes dimentional Euclidean space. is the feasible region of . is the matrix of optimal solution of . Equation (20) can be also reformulated as
Theorem 1. Given adverse weather scenario , the twoclass user link flow pattern that consists of TPI and TBI accords with the equilibrium solution, i.e., satisfying the equilibrium conditions (18) and (19), if and only if it satisfies the VI problem (20).
Proof. See Appendix.
As with the existence and uniqueness of the equilibrium solution, though inserted by scaling functions that can measure adverse weather intensities, the GBPR function is still continuous in the feasible region . Besides, the feasible region is a typical nonempty compact set and convex set. Therefore, at least one solution of the VI problem (20) exists [17]. Given congestion effects, the GBPR function in this study is a continuous, strictly monotonic increasing function of link flow, and the feasible region is a typical nonempty compact set and convex set. Thus, the solution of VI problem (20) is unique [17].
4. Solution Algorithm
The stochastic network user equilibrium problem given route choice heterogeneity of travelers is a typical multiclass, asymmetric network equilibrium problem. Though the Jacobian of link cost function in this problem is asymmetric, the precondition still requires that the Jacobian is positive definite. Therefore, the Streamlined Diagonalization Algorithm (SDA) [18], is adopted to solve the proposed asymmetric SNUE VI problem. The solution generated by each iteration is input as the initial solution of next iteration. The SDA for solving asymmetric network UE problem consists of the following general steps:
Step 0. Network initialization. Set . Find a feasible linkflow pattern vector ;
Step 1. Link travel time updating. Set .
Step 2. Direction finding. Assign the OD flows, , to the network utilizing allornothing approach based on , which yields a linkflow pattern; :
Step 3. Movesize determination. Find a scalar, , which solves the following program:
Step 4. Linkflow updating. Set ;
Step 5. Convergence test. If , stop. The solution is ; otherwise, set and go to Step 1.
SDA is similar to the FrankWolfe algorithm [19] that solves the traditional UE problem, which mainly differs in that the SDA’s update of each link travel time is based on the whole network flow, i.e., , rather than in FW algorithm. The diagonalization principle reflects in that, in the iteration process of step size , the cost function of each link is merely related to its own link flow , while the flows of other links are fixed in . With respect to the convergence of SDA, please see reference [20].
5. Numerical Examples
In this section, a smallsize triangle network and a middlesize Sioux Falls network are used to testify the validity of the model and solution algorithm proposed in this study. To show the efficiency of our model and algorithm, we test our examples on a PC with a 3.40 GHz CPU, an 8 GB RAM and Windows 10 Enterprise 64bit operating system. The model was coded with Matlab.
5.1. A SmallSize Triangle Network
5.1.1. Network Parameter Settings and Descriptions
A small triangle network consisting of 4 nodes, 5 links (denoted in italics), 2 OD pairs and 4 routes is presented in Figure 1.
The linkroute incidence matrix is shown in Table 3. The functional form and parameters of the scaling functions and are shown in Table 4.


Four network scenarios s, i.e., small rain (), moderate rain (), heavy rain (), rainstorm () and corresponding rainfall intensities are shown in Table 5, respectively.

The perceived conditional probability distribution of the TBI is listed in Table 6.

This example employs the following form of link cost function under adverse weather conditions:
in which , .
As shown in Table 7, the total traffic demands between OD pairs 1–4, 2–4 are set to be 1500 and 2000 units, respectively. The proportion of TPI and TBI between OD pair 1–4 are set to be 30% and 70%, respectively, while these proportions between OD pair 2–4 are 60% and 40%, respectively. OD pair 1–4 and 2–4 can be considered as the developing and developed regions, respectively, for the following reasons: (1) the penetration rate of ATIS in developed regions is usually higher, so is the proportion of the TPI (60% > 30%); (2) the social and economic activity would be more active in the developed region (2000 > 1500), so more traffic demand is expected.

The convergence performances (the total iteration and convergence gap ) of the six cases considered in this example (i.e., Z1–Z6) are listed in Table 8.

Take case Z1, i.e., small rain (, ), as an example, the algorithm converges at iteration and the gap . The blue curve and the red curve in Figure 2 denote the iteration step size and the convergence gap , respectively. As shown in this figure, the values of and e reach to a relatively steady level at iteration n = 20, (, ), but the convergence gap e does not satisfy the convergence criterion () until another 90 iterations (i.e., ), which implies that the SDA converges rapidly in the forepart of the iteration, but may perform tardily in the latter stage.
5.1.2. Optimal Solution (Z1)
The optimal solution of link flows in Z1 (, small rain, ) is
and is further depicted in Figure 3, in which the blue and red numbers marked beside a link denote the flow of TPI and TBI on that link, respectively. According to formula (9), the equilibrium solution of the total link flows (denoted as black numbers in the links) is:
For Figure 3, the proportion (80.02%) of TPI in developed area (OD pair 2–4) choosing route R4 (i.e., link 2) is higher than that (12.6%) of TPI in developing area (OD pair 1–4) choosing route R1 (i.e., link 1). This result is expected because the ATIS penetration rate of the developed areas (60%) is higher than that of the developing areas (30%). When the network has not been heavily worsened by the rainfall (small rain), more travelers in developed areas can capture actual route travel time of the network and then are likely to prefer route R4, which is a highcapacity, “weatherresistant” route compared with R3. Analogously, given that the proportion (70%) of the TBI who are not armed with ATIS in developing area (OD pair 1–4) is higher than that (40%) of those in developed area (OD pair 2–4), in the equilibrium condition, more TBI (41.47%) in developing area (OD pair 1–4) select route R2 (i.e., link 3–5) than those TBI (4.17%) in developed area (OD pair 2–4) who select R3 (i.e., link 4–5).
On the other hand, we can learn from Figure 3 that, for the proportion of TPI is higher in the developed area (OD pair 2–4), more travelers (86.34%) prefer to “weatherresistant” route R4 than that of travelers (44.75%) in the developing area (OD pair 1–4) who select the “weatherresistant” route R1. This result suggests that ATIS makes more TPI turn to highcapacity, “weatherresistant” routes with more probability being a shorter route travel time. While without the help of ATIS, travelers (i.e., TBI) who make their route choices according to their existing travel experience will spread evenly on the “weatherresistant” and “weathersensitive” routes in the whole network.
The iterationconvergence procedures of the link flows of TPI, TBI and the total link flow (TPI+TBI) are presented in Figures 4(a) and 4(b) and Figure 5, respectively.
(a)
(b)
According to the optimal solution of the link flow, the perceived route travel time of TPI and TBI are:
in which the elements in the first and second rows are the ARTT of TPI and the PRTT of TBI between the two OD pairs, respectively. Accordingly, the corresponding iterationconvergence procedures are depicted in Figures 6(a) and 6(b), respectively. According to the route travel time in the equilibrium state of this example (formula (26), Figures 6(a) and 6(b)), no matter which class a traveler belongs to, the route travel time in developed area (OD pair 2–4) (i.e., 0.81 and 0.82 for TPI and TBI, respectively) is higher than that in developing area (OD pair 1–4) (about 0.76), which results from the following reasons: (1) the traffic demand (2000) in developed area (OD pair 2–4) is higher than that (1500) in developing area (OD pair 1–4); (2) high ATIS penetration rate in developed area intensifies the travelers’ aggregation effects on “weatherresistant” routes to seek for lower travel time cost under adverse weather conditions.
(a)
(b)
Moreover, we can learn from (26) that, in the case Z1 (small rain), the PRTT of TBI are always larger than the ARTT of TPI who use that route (i.e., ), which could be proved in mathematics.
The convergence curves of ARTT of TPI (Figure 6(a)) show that, the subnetwork of the developed area (i.e., R3 & R4 between OD pair 2–4) reaches to equilibrium state rapidly within 20 iterations. By contrast, the subnetwork of the developing area (i.e., R1 & R2 between OD pair 1–4) reaches to an approximate equilibrium state with the travel time differential of the two routes is less than 1.2%. Analogous results can be also observed from the convergence curves of PRTT of TBI (Figure 6(b)) that, the subnetwork of the developing area (i.e., R1 & R2 between OD pair 1–4) reaches to equilibrium state rapidly, while the subnetwork of the developed area (i.e., R3 & R4 between OD pair 2–4) reaches to an approximate equilibrium state with the travel time differential of the two routes is about 0.55%. These derive from the fact that the SDA might contribute limitedly to the further improvement of the objective functions when it is not highly demanding on the solution precision.
5.1.3. Equilibrium Solutions of the Link Flow of the TPI and TBI in Cases Z1–Z4
The equilibrium solutions in scenario Z1–Z4 are presented in Figures 7(a)–7(d). From the prospective of the total link flow, with the deteriorating of rainfall scenarios with the highest occurring probability moving from small rain (Z1, 80%) to rainstorm (Z4, 65%), more and more travelers in the developed area (i.e., OD pair 2–4) move from “weathersensitive” route R3 to the “weatherresistant” route R4 (1727—1773—1827—1885, the total traffic demand is 2000). Similar results can be also observed in the developing area (i.e., OD pair 1–4) that increasing travelers incline to prefer “weatherresistant” route R1 to R2 (671—1050—1353—1500, the total traffic demand is 1500). Such results are expected that with the worsening of the adverse weather scenarios, the travel time of “weathersensitive” routes increase more sharply than that of “weatherresistant” ones, and travelers have a preference to those “weatherresistant” routes rather than “weathersensitive” ones to evade higher travel costs. Note that this conclusion is drawn on the premise that it is widely believed by TBI that the weather forecast is relatively accurate (shown as cases Z1–Z4 in Table 6).
(a)
(b)
(c)
(d)
(e)
(f)
5.1.4. Impact of Accuracy of Weather Forecast on Route Choice (Z5/Z6)
Recall that in Table 6 cases Z5 and Z6 are designed to investigate the impact of weather forecast accuracy on travelers’ route choice behavior, and heavy rain ( ) was assumed to be the scenario with the highest probability of occurrence. The equilibrium flow patterns in Z5 & Z6 are presented in Figures 7(e) and 7(f). There are not large differences in the total link flows (denoted as black numbers in the links) of the network between Z5 (when the weather forecast is accurate) and Z6 (when the weather forecast is less accurate). Actually, on one hand, whether the weather forecast is accurate or not will not directly affect the route choice of TPI, because TPI would always acquire the actual route travel time of the network with the aid of ATIS. On the other hand, the calculated posterior probabilities in the hypothetic cases Z5 (0.99) and Z6 (0.90) have approximate values, which implies that in some cases if the occurring probability of some scenario reported from weather forecast is relatively high, the posterior probabilities of TBI calculated via the Bayes formula would have little impact on their PRTT.
5.2. Sioux Falls Network
5.2.1. Network Descriptions
In this section, our model and algorithm are applied and tested to a wellknown middlesize traffic network, i.e., the Sioux Falls network (Figure 8(a)), which has been frequently used as a numerical example in the field of transportation network modeling. The Sioux Falls network has 24 nodes, 76 links, and 552 OD pairs. The general configurations of the network, including the free flow travel time and ideal link capacity, are consistent with reference [21, 22], and further illustrated in Figure 8(b). The OD demand are the values from the Table 1 in [21] multiplied by 100. Given the complexity of the Sioux Falls network and to lower the computational cost, the link cost function in this case adopts the following reduced version, and the parameters are as previously stated:
(a)
(b)
5.2.2. Network Parameter Settings
In this section, to investigate the impact of links with different levels of adverse weather sensitivity on travelers’ route choice behavior, the Sioux Falls network is divided into three parts, i.e., the weathersensitive area (denoted in red dashed lines), weatherneutral area (in blue dashed lines) and weatherresistant area A_{r} (in green dashed lines) as shown in Figure 8(b). The absolute value of link parameters and in the weathersensitive area (0.07 and 0.04, respectively) are larger than those of the other two areas, which indicates that for a given increment of rainfall intensity and all things being equal, the link travel time of links in weathersensitive areas increase more sharply than those of links in other two areas.
Two rainfall scenarios, i.e., moderate rain () and rainstorm (), are assumed in this case, and the corresponding rainfall intensities and the perceived conditional probability distributions of the TBI are shown in Tables 9 and 10, respectively. In this case, the TBI are assumed to hold positive attitudes towards the accuracy of the weather forecast.


5.2.3. Results and Sensitivity Analysis
For scenario Z7 (TPI: TBI = 60% : 40%), the algorithm converged after iterations. The iteration step size and the convergence gap e are shown in Figure 9. The missing points of the convergence gap curve at the beginning of the iterations indicate a positive infinite value of the convergence gap e.
The equilibrium flow pattern of five directed links for each of the three areas (i.e., , , and ) in case Z7 and Z8 (TPI : TBI = 60% : 40%) in the network are reported in Table 11. The five links in one area are selected as spatially multidirectional as possible. To capture the variation tendency of the flows when the adverse weather scenario worsens from moderate rain to rainstorm, two rightward links (i.e., (3, 4), (13, 24)) are selected in the leftmost area A_{s} in Figure 8(b), and for the rightmost area , two leftward links (i.e., (7, 8), (20, 22)) are selected.

We can learn from Table 11 that, when the weather scenario worsens, i.e., varies from moderate rain (case Z7) to rainstorm (case Z8), the flows of both TPI and TBI on the five selected links in the weathersensitive area decrease by 15–46%. While the flows of the twoclass users in the weatherresistant area increase by 9–42%. The OD demand transferred from the weathersensitive area superimpose those local demand of the weatherneutral area , leading to the flow of TPI and TBI on the selected rightward links (1, 2) connecting the and increase by 46.1% and 42%, respectively. In the case of Z8 (i.e., rainstorm), for the increasing travel cost in the areas nd , the link travel costs in the weatherresistant area increase relatively more slowly and more preferred by travelers. Other results we can obtain are, for the weathersensitive area and weatherresistant area , the proportions of flow increment or decrement of TPI are larger than those of TBI. With the help of ATIS, TPI are more sensitive to the increasing actual link travel time in the weathersensitive area and react faster than TBI. However, as a middle traversing area, the mixing effect makes it not always the case. For example, the proportions of four flowdecreasing links (i.e., (15, 22), (10, 9), (10, 11) and (23, 22)) of TPI are smaller than those of TBI.
Besides the aforementioned results, we also care about how the proportion of the two classes of travelers impacts the network travel costs under the adverse weather scenarios. In other words, whether the increasing usage of ATIS, i.e, more percentage of TPI in the network, would increase drivers’ perceived link travel cost. We adjust the share of the TPI in the twoclass network, and depict the link travel time in Figure 10. As the proportion of TPI in the network increases from 20%, 40%, 60% to 80%, the perceived link travel time of TPI rise with varying degrees. If the users in the network are more likely to be smarter and smarter with ATIS, more travelers can learn about the actual link travel cost in the network. Under the adverse weather conditions, when more smart users are guided to the recommended “shortest” routes, the perceived link travel cost also increases, and the advantages of ATIS in a lowproportion smart network could be weakened.
6. Conclusions
This paper focuses on the problem that how the adverse weather affects stochastic traffic network user equilibrium. We proposed a UE model incorporating the impacts of adverse weather scenarios on roadway performance degradation, and the influence of ATIS and weather forecast information on the route choice behavior of travelers with perfect information (TPI) and travelers with bounded information (TBI). Our consideration of the twoclass user system is based on the discrepancy of network information level (NIL) of travelers, which divides the whole travelers into TPI who can obtain the actual network information with the help of ATIS and TBI who can just make their route choices based on their travel experience and perceptions on the accuracy of past weather forecast. The UE model was formulated as a VI problem and solved by a proposed algorithm. Numerical examples based on a simple triangle network and the wellknown Sioux Falls network were adopted to verify the feature of the proposed model and solution algorithm, and to calculate the network equilibrium solutions.
From the results of the modeling and numerical analysis, several conclusions can be drawn as follows. The validity and feasibility of the proposed model and algorithm were testified. In the scenarios of more adverse weather intensities, more travelers (especially for TPI) are inclined to choose those “weatherresistant” routes to lower the travel time cost. From the point of the effect of ATIS on the system benefits, when the weather scenario is relatively good and the network capacity has not been deteriorated, ATIS can make travelers take full advantage of those highcapacity routes to shorten route travel timethe ATIS improve the overall system benefits. In contrast, when the weather scenario turns relatively adverse and the network capacity decreases rapidly, ATIS, however, guides travelers (especially for TPI and those TBI who consider the weather forecast information is accurate) to gather on those weatherresistant routes, in which the ATIS worsens the overall system benefits. In other words, when more smarter users can learn about the actual link travel cost in the network, the advantages of ATIS in a lowproportion smart network could be weakened.
Along with the deep research on stochastic network traffic assignment and equilibrium problem under adverse weather conditions, future research can be extended based on this study as follows. Besides TPI, given that different TBI with discrepant travel experience, homogeneous NILs in this study can be subdivided to heterogeneous levels, in which different TBI may have different perceptions on the accuracy of the same weather forecast information. What’s more, the travel demand could be further considered as a decreasing function of the adverse weather intensity or route travel time, in which the deterministic travel demand problem is expanded to an elastic one. Finally, other expansions can lie in the consideration of travel time uncertainty, or developing more efficient VI solution algorithms.
Appendix
The proof below is given for Theorem 1 that the equilibrium conditions (18), (19) of the network are in accordance with the solutions of the VI problem (20).
Proof. The generalization of this proof referred to [9].
(1) Sufficiency.According to equilibrium conditions (18), (19) for TPI and TBI. Add the two inequations together, then sum for all routes , By using and , we obtain Sum the inequation for all OD pairs and all network scenarios , and by using (13)–(15), we have which is exactly the vector quantity pattern of the VI problem, i.e., (21). The sufficiency is proved.(2) then we consider the necessity. Assume that , are the solutions of the VI problem (i.e., satisfy (21)), but they does not satisfy the equilibrium conditions (i.e., (18), (19)), for those TBI who travel between OD pair , there exists path , so that, Thus, let us shift a small amount of path flow from path to the path () with its cost equal to minimum route travel time ( and ) in equilibrium then i.e., This contradicts with the assumption. So the solutions of the VI problem, i.e., , coincide with the equilibrium conditions of the stochastic traffic network. So the sufficiency is proved.
Data Availability
The data of the smallsize triangle network used in this study is meticulously designed by the authors, while the configurations of the wellknown Sioux Falls network can be obtained via http://www.bgu.ac.il/~bargera/tntp/ and some classical references that have been declared in the paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors are indebted to two anonymous reviewers for their instructive and valuable comments to this paper. This study is supported by the Fund of Cultivation Project for Young University Teachers from Shanghai Municipal Education Commission (grant number: ZZHS18001), the National Natural Science Foundation of China (grant number: 71602114), and Shanghai Science and Technology Committee (grant numbers 17040501700 and 19QA1404200).
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