required service level given variable arc capacities, while accounting for drivers' route choice behavior. Previous papers on this topic provide a comprehensive ...
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Effect of Route Choice Models on Estimating Network Capacity Reliability Anthony Chen, Maya Tatineni, Der-Horng Lee, and Hai Yang The issue of planning for adequate capacity in transportation systems to accommodate growing traffic demand is becoming a serious problem. Recent research has introduced “capacity reliability” as a new network performance index. Capacity reliability is defined as the probability that a network can accommodate a certain volume of traffic demand at a required service level given variable arc capacities, while accounting for drivers’ route choice behavior. Previous papers on this topic provide a comprehensive methodology for assessing capacity reliability along with extensive simulation results. However, an important issue that remains is what type of route choice model should be used to model driver behavior in estimating network capacity reliability. Three different route choice models (one deterministic and two stochastic models) are compared, and the effect of using each of these models on estimating network capacity reliability is examined.
Transportation systems, water supply systems, and communication and power transmission systems are all examples of lifelines (lifelines are systems that extend spatially over large geographic regions carrying vital services to the points of demand). In fact, the transportation system has been identified as the most important lifeline in the event of natural disasters such as earthquakes, floods, hurricanes, and others (1). Restoration of other lifelines depends strongly on the ability to transport people and equipment to damaged sites. An unreliable transportation system would hinder the restoration process and increase not only economic loss but also fatalities, which are difficult to quantify. Thus the importance of developing probabilistic procedures for quantitative evaluation of the reliability of the transportation system cannot be overemphasized.
MEASURES OF RELIABILITY FOR A ROAD NETWORK With increasing demands for better and more reliable services, many systems (e.g., electric power systems, water distribution systems, communication networks) have incorporated reliability analysis as an integral part of their planning, design, and operation (2). In the past few years considerable attention has been given to incorporating reliability analysis in the study of road networks. A general theoretical framework for analysis of degradable transportation systems is described by Du and Nicholson (3). Bell and Iida (4) provide an overview of reliability studies for road networks. Here three reliability measures for road networks are summarized. A. Chen, Department of Civil and Environmental Engineering, Utah State University, Logan, UT 84322-4110. M. Tatineni, TranSmart Technologies, 948 West Bailey Road, Naperville, IL 60565. D.-H. Lee, Department of Civil Engineering, National University of Singapore, BLK E1A 07-16, 1 Engineering Drive 2, Singapore 117576. H. Yang, Department of Civil Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, People’s Republic of China.
Connectivity Reliability Connectivity reliability is used as a measure of the probability that network nodes are connected. A special case of connectivity reliability is terminal reliability, which is concerned with the existence of a path between a specific origin-destination (O-D) pair (5). For each node pair, the network is considered successful if at least one path is operational. A path consists of a set of components (e.g., roadways or arcs), which are characterized by zero-one variables denoting the state of each arc (operating or failed). Capacity constraints on the arcs are not accounted for when determining connectivity reliability. This type of connectivity reliability analysis may be suitable for abnormal situations such as earthquakes, but there is an inherent deficiency in the sense that it allows for only two operating states: those operating at full capacity or complete failure with zero capacity.
Travel Time Reliability Travel time reliability is concerned with the probability that a trip between a given O-D pair can be made successfully within a specified interval of time (6). This measure is useful when evaluating network performance under normal daily flow variations. Bell et al. (7) proposed a sensitivity analysis–based procedure to estimate the variance of travel time arising from daily demand fluctuations. Asakura (8) extended the travel time reliability to consider capacity degradation due to deteriorated roads. He defined travel time reliability as a function of the ratio of travel times under the degraded and nondegraded states. This type of reliability can be used as a criterion to define the level of service that should be maintained despite the deterioration of certain arcs in the network. When the ratio is close to unity, it is essentially operating at ideal capacity, whereas when it approaches infinity the destination is not reachable because certain arcs are severely degraded. This extreme case is consistent with the definition of connectivity reliability.
Capacity Reliability Capacity reliability is concerned with the probability that the network can accommodate a certain volume of traffic demand at a required service level (9, 10). Arc capacities for a road network can change from time to time for various reasons such as traffic incidents that may block one or more lanes, lane blockage due to construction, and so forth. Capacity reliability explicitly considers the uncertainties associated with arc capacities by treating roadway capacities as continuous quantities subject to routine degradation due to physical and operational factors. Readers may note that, when
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the roadway capacities are assumed to take only discrete binary values (zero for total failure and one for operating at ideal capacity), then capacity reliability includes connectivity reliability as a special case. Also, because arc travel times are a function of arc flows and capacities, any measure of network capacity reliability also must involve some measure of network travel time reliability. However, the exact functional form of the relationship between travel time reliability and capacity reliability is yet to be determined and remains a topic of research. Thus capacity reliability has the potential of being useful as a comprehensive measure of reliability for a road network. Basically this measure addresses the issue of whether the available network capacity relative to the required demand is sufficient. This could be particularly useful at a system level, in planning and deciding roadway capacity expansion projects, planning the timing and location of various road improvement projects, and so forth. The potential applications of capacity reliability measures are described in more detail elsewhere (9, 10).
Computing Capacity Reliability Capacity reliability has only recently been introduced by Chen et al. as a performance measure to evaluate the performance of a degradable road network (9, 10). Estimation of the capacity reliability curve is based on the concept of network reserve capacity (11), which is defined as the largest multiplier applied to an existing O-D demand matrix that can be allocated to a transportation network in a user-optimal way without violating the arc capacities. Mathematically stated, the problem is to find the maximum O-D matrix multiplier µ subject to the arc flows resulting from the userequilibrium route choice model not exceeding the arc capacities. This may be written as follows: max µ
(1)
subject to va (µq ) ≤ Ca
∀a ∈ A
(2)
where va(µq) is the equilibrium flow on arc a, with the demands of all O-D pairs being uniformly scaled by µ times the base O-D demands q, obtained by solving the following traffic assignment (TA) problem: min Z = ∑ a ∈ A
∫
va
0
ta ( x, Ca )dx
(3)
∀w ∈ W
( 4)
subject to ∑ r ∈ R w fr = µqw va =
∑fδ r
ar
∀a ∈ A
(5)
r ∈R
fr ≥ 0
∀r ∈ R
where A = set of arcs in the network, W = set of O-D pairs in the network, R = set of routes in the network,
(6)
Rw = set of routes between O-D pair w ∈ W, µ = O-D matrix multiplier for the whole network, Ca = random capacity on arc a ∈ A, Z = user-equilibrium objective function, va = flow on arc a ∈ A, ta(va, Ca) = travel time on arc a ∈ A, qw = existing demand between O-D pair w ∈ W, q = existing O-D demand matrix in vector form, fr = flow on route r ∈ R, and δar = 1 if route r uses arc a; 0 otherwise. The problem of computing µ is treated as a bilevel programming problem. At the upper level (Equations 1 and 2) the value of µ is computed and at the lower level (Equations 3 to 6) a standard TA problem is solved to determine the values of arc flows. Route choice behavior and congestion effects are explicitly considered by the lower-level problem, whereas the upper-level problem determines the maximum O-D matrix multiplier subject to the capacity constraints. Here arc capacities Ca are treated as nonnegative random variables. Because the largest O-D matrix multiplier depends on arc capacities, µ is also a random variable and it can be computed subject to error. However, it is difficult, if not impossible, to derive an analytic expression for µ as a function of random arc capacities. Given that it is mathematically intractable to analytically compute µ, simulations are used to estimate the probability distribution of µ [details are provided elsewhere (9, 10)]. For each sample of arc capacities (. . . , Ca, . . .) and a given µ, the lower-level problem can be solved efficiently. The equilibrium arc flows (. . . , va, . . .) resulting from the TA problem are then transmitted to the upper-level problem to determine the maximum µ. Because the upper-level problem has only one decision variable, it can be treated as a parameter in the lower-level problem. Then, the overall bilevel problem can be solved as a single level by varying the value of µ until at least one of the equilibrium arc flows violates the capacity constraints. For every sample of arc capacities the largest value of µ that can be satisfied without violating the arc capacity constraints is computed. For each sample of arc capacities, all demand levels that are less than the computed value of µ can be accommodated with a probability of 1 and all demand levels that are higher than the maximum computed value of µ have a probability of 0 of being satisfied. Thus for each sample of arc capacities, a 0 or 1 probability can be determined for each incremental value of the network reserve capacity that can be accommodated by the network. The capacity reliability is then computed for each value of the network reserve capacity or µ by averaging the probabilities of each value of µ being accommodated by the network for the entire sample of arc capacities (for details, see the Monte Carlo simulation procedure presented in the implementation issues section). The route choice behavior used by Chen et al. (9, 10) to obtain the equilibrium flows is based on the deterministic route choice model. This model is appropriate only for modeling long-term degradation of a road network in which travelers have sufficient knowledge to adjust their route choices to an eventual equilibrium point. However, the assumption of the deterministic model may be restrictive in certain situations in which the network itself is stochastic, and travelers do not have accurate perceptions of the network travel times to make optimal route choice decisions. The purpose of this paper is to examine the effects of three different route choice models on estimation of network capacity reliability. Specifically, the results obtained from the deterministic route choice model are compared with two stochastic route choice models.
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In the next section, three different route choice models are reviewed. This is followed by a discussion of the implementation of the reliability evaluation framework, and numerical results that examine the effects of different route choice models on the estimation of capacity reliability measure are presented. Finally, some concluding remarks based on the results of this study are presented.
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Stochastic Model
The model most widely used in practice is the deterministic route choice model (i.e., user-equilibrium model). Each traveler is assumed to have perfect knowledge of the network travel times on all possible routes between his or her O-D pair. Each traveler’s route choice criterion is to minimize the known value of the route travel time, which is obtained by adding the travel times on all the arcs belonging to the route. The choices of routes by all travelers result in a network flow allocation so that all used routes between every origin and destination have equal travel times and no unused route has a lower travel time (13). This model was previously used (9, 10) as a route choice behavior to estimate the capacity reliability.
A truly stochastic model should consider both the traveler perception errors as well as the stochasticity of network travel times. Arc travel times are explicitly considered as random variables. For a given set of flows, there is a probability distribution associated with the arc travel times, which describes the variations in the travel times experienced by the travelers on the network. Such variations could result from the differences in the mix of vehicle types on the network for the same flow rates, differences in driver reactions under various weather and driving conditions, differences in delays experienced by different vehicles at intersections, and so forth. A stochastic route choice model that accounts for variable network travel times as well as a method to model different traveler responses to network travel time variability by assuming different risk-taking behaviors has been presented (15). This model also accounts for the variations in each individual traveler’s perception errors. The mean and variance of the travelers’ perception errors are sampled from distributions defined over the population. Thus, the perceived distribution of travel times on the network by each traveler is a function of the actual distribution of the network travel times as well as the distribution of the traveler’s own perception error. Travelers are assumed to know the variable nature of the network travel times and take this into account while choosing among alternative routes. Depending on their behavioral nature, travelers are classified as risk averse, risk prone, or risk neutral. The risk in this case is the variance associated with network travel times. For instance, a risk-averse traveler will trade off a reduction in travel time variability with some increase in expected travel time, whereas a risk-prone traveler may choose a route with a greater perceived variability in order to increase the possibility of a smaller travel time. A risk-neutral traveler would choose a route based on only expected travel times without consideration of its variability. It should be noted that the risk-neutral model is essentially the semistochastic model in which each traveler makes his or her route choice based on the perceived mean arc travel times. Each traveler type is assumed to associate a disutility with each arc that is a function of the arc travel time and the traveler’s risk-taking behavior. The route choice criterion in this model is to minimize the perceived disutility associated with a route, which is obtained by summing the perceived disutilities on all the arcs belonging to the route. The choices of routes by the travelers result in a flow allocation so that no traveler can reduce his or her perceived expected disutility by changing to another route.
Semistochastic Model
RELIABILITY EVALUATION PROCEDURE
Because of the unrealistic assumption that all travelers have perfect knowledge of the network conditions, the semistochastic model relaxes this stringent assumption by introducing a perception error into the route choice decision (14). In this model, each traveler is assumed to have some perception of the mean travel times on each arc of the network, which include a random error term. Each traveler’s route choice criterion is to minimize the perceived value of the route travel time, which is obtained by adding the perceived travel times on all the arcs belonging to the route. The choices of routes by the travelers result in a network flow allocation so that no traveler can reduce his or her perceived travel time by changing to another route. This model is called semistochastic because it accounts for the randomness only in the travelers’ perceived travel times. Arc travel times are still modeled as a deterministic function of the traffic flow on the arc.
The framework of the reliability evaluation procedure is based on a Monte Carlo simulation. As mentioned before, the reliability measures considered in this paper treat arc capacities as random variables of a degradable road network. Assuming that the random variables Ca follow a probability distribution, a random variate generator is used to generate the values of Ca for each arc that preserve the provided distribution properties. For each set of arc capacities generated, a TA algorithm is used to find the equilibrium arc flows. As explained before, three different route choice models are used to find the equilibrium arc flows and O-D travel times. The stochastic route choice model that assumes different risk-taking behaviors is implemented twice. The first time it is assumed that all travelers are risk averse and the second time it is assumed that all travelers are risk prone. By doing so, it is hoped that whether the difference in risk-taking behaviors has any impact on computation of network reliability measures
DESCRIPTION OF ROUTE CHOICE MODELS For a given set of flows between various origins and destinations in a study area, the role of a route choice model is to determine the allocation of these flows to the various arcs of the road network. In general, allocation of the flow between a given origin and destination to the arcs of the road network is based directly on the routes predicted to be chosen by the travelers making the trip between the given O-D pair. To model the travelers’ route choices, therefore, assumptions are needed with regard to (a) characterization of the arc travel times, (b) travelers’ knowledge of the travel times on the network, and (c) route choice criteria of each individual traveler (12). In the route choice literature, several models have been proposed that differ in the assumptions listed above. In general, these models can be classified into three categories: (a) deterministic, (b) semistochastic, and (c) stochastic. In each model, the travel time for every arc on the network is assumed to be an increasing function of the flow of vehicles on the arc. Each traveler is assumed to make route choices to minimize his or her disutility, which is a direct function of the travel time on the network. Deterministic Model
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can be ascertained. Based on the link flows resulting from implementation of the different route choice models [refer to Tatineni et al. (12) for implementation details], the capacity reliability measure is computed. With the advancement of computer technology, it is possible to efficiently analyze many problems by simulation methods. There has been increased interest in particular in using Monte Carlo simulation to estimate reliability measures by simulating the random behavior of the system (16). In fact, Monte Carlo methods are often preferred when complex operating conditions are involved or when the number of component states is large (e.g., many combinations resulting from different levels of capacity degradation). In addition to the convenience of implementation, simulation methods can be used to calculate not only the reliability measures in the form of expected values from the random variables but also the distributions of these measures, which in general cannot be done by analytic techniques. With the Monte Carlo simulation method, a procedure is developed to estimate the distribution for the largest O-D demand multiplier given a set of random arc capacity samples. This procedure is described as follows: Step 0. Initialize the multiplier value µ = µ0. Step 1. Set sample number k : = 1. Step 2. Generate a vector of arc capacities Ck = {C1, . . . ,CA}. Step 3. Perform TA with the scaled demand µ q and arc capacity vector Ck. Step 4. Collect statistics from TA to compute network capacity reliability. Step 5. If sample number k is less than the required sample size kmax, then increment sample number k : = k + 1 and go to Step 2. Otherwise, go to Step 6. Step 6. If the current value µ is the largest multiplier, then stop. Otherwise, increase the value µ by δ µ and go to Step 1. DESCRIPTION OF NUMERICAL EXPERIMENTS In this section, some numerical results are presented from experiments with different route choice models to estimate capacity reliability for a small test network. The test network used is presented in Figure 1 and consists of five nodes, seven arcs, and two O-D pairs. The base demands for O-D pairs (1,4) and (1,5) are 20 and 25, respectively. The free-flow travel times for each arc on the network as well as the capacity values are presented in Table 1. Arc capacities are assumed to vary uniformly between a lower and upper bound as indicated. To model the risk-sensitive traveler’s
Transportation Research Record 1733
TABLE 1 Arc Free-Flow Travel Times and Statistical Properties of Arc Capacities
route choice behavior, all the arcs in the network are assumed to have some variability associated with their travel times. For Arc 1 the variance on the travel time is assumed to be 100 percent of the arc’s free-flow travel time. For all other arcs the travel times are assumed to vary up to 10 percent of the arc’s free-flow travel time. Note that variations in arc capacities and travel times should be related. However, in this study it is assumed that these two distributions are independent. A total of 5,000 values were sampled for arc capacities as part of the Monte Carlo procedure. As indicated in Table 1, the theoretically computed values of the mean and standard deviation for the capacity on each arc match quite closely with the values computed from the sampled capacity values. Therefore, the number of samples drawn is sufficient to generate a representative distribution of arc capacities. Arc travel times are computed by using the standard Bureau of Public Roads function as follows: ta = taf [1 + 0.15(va Ca )4 ]
( 7)
Here va, taf , and Ca are the flow, free-flow travel time, and random capacity on arc a, respectively. Exponential disutility functions are used to represent different risk-taking behaviors. The disutility functions estimated for the different risk-taking behaviors are as follows: risk averse DU (t ) = 0.309[exp(0.289t ) − 1] risk prone DU (t ) = 1.309[1 − exp( −0.289t )]
FIGURE 1
Test network.
Here t is the travel time in minutes for a given route. Note that only the final functional forms of the disutilities for each risk-taking behavior are provided. For details of the derivation, see Tatineni (17). When the capacity of every arc is fixed at the upper bound of the uniform distribution (i.e., nondegraded capacity), the largest multiplier estimated in determining the value of the network’s reserve capacity is 1.0, which means the current network capacity is just enough to accommodate the base travel demand. This value serves as the upper bound for the degradable network; thus, travel demand greater than the base case cannot be satisfied.
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Arc Flow Comparisons for Different Route Choice Models In Figures 2 and 3 the average arc flows for various values of the demand level for each route choice model are presented. Figure 2 indicates the arc flow allocations assuming travelers have perception errors and Figure 3 indicates the arc flow allocations assuming
FIGURE 2
Arc flow comparisons for different route choice models.
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travelers have no perception errors. Note that, under the assumption that travelers do not have any perception errors, the semistochastic (sue) model that assumes risk neutral behavior is the same as the deterministic route choice (due) model. It is apparent that as the demand level is increased (i.e., as the flow on the network is increased) the flow on almost all the arcs increases for all the models. However on Arc 1, which has a higher variability
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FIGURE 3
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Arc flow comparisons without perception error.
associated with its travel time, the model assuming risk-averse travelers (sue-ra) allocates less flow, whereas the model assuming riskprone travelers (sue-rp) allocates a larger proportion of its flow to this arc. Note the opposite effect on Arc 2, which represents an alternative to Arc 1, in that the model assuming risk-averse travelers allocates more flow and the model assuming risk-prone travelers
allocates less flow. When the models are implemented assuming that the risk-sensitive travelers have no perception errors, then the flow differences for Arcs 1 and 2 for the risk-averse and risk-prone travelers are even more apparent. Although arc flows increase in general with the increase in the value of the travel demand, there are some fluctuations associated
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with the flows on Arc 3 for all the route choice models implemented. Apparently, even as the flow allocated to the network is increased, some arcs that may be used in alternative paths may not necessarily see an increase in arc flow. Therefore an increase in network flows in general does not correspond to an increase in flow for all arcs but even results in lower flow levels for some arcs.
Effect of Route Choice Models on Estimation of Capacity Reliability Figure 4 presents capacity reliability curves for each route choice model for different values of the demand level. As might be expected, for lower demands, the capacity reliability is equal to 1 for all the route choice models, because the network can easily accommodate all the flows. However, for higher demands the capacity reliability of the network drops off and eventually becomes close to zero when the demand level is equal to 1. The capacity reliability values also appear to differ for the various route choice models implemented. The deterministic route choice model appears to result in higher values for the capacity reliability than the semistochastic route choice model assuming risk-neutral travelers. The stochastic route choice models that assume travelers are risk sensitive appear to have the lowest values of capacity reliability. It appears that, because travelers in the deterministic model are assumed to act on the basis of perfect information, the resulting network flows are also optimally distributed, making the best use of the network; hence, the capacity reliability of the network under this scenario appears to be higher. However, for the semistochastic models, travelers are assumed to make route choices with imperfect information; hence, the flow allocation over the network may actually be suboptimal, resulting in lower capacity reliabilities. For the stochastic models assuming risk-sensitive travelers, route choices are based on minimizing disutilities, which are based on arc travel time variabilities leading to either overusing or underusing some arcs; hence, flows are likely to be even less optimally distributed across the network. Thus, it appears reasonable that the capacity reliability of the network is lower when route choices are assumed to be based on imperfect information and even more so when they are based on minimizing disutilities that are a function of factors other than arc travel times. Similar results (Figure 5) were obtained when the models were implemented without perception error. However, for the stochastic
FIGURE 5
Capacity reliability without perception error.
models assuming risk-sensitive travelers, the differences in capacity reliabilities for the risk-averse and risk-prone travelers are minor, which suggests that the traveler’s perception error may have a larger effect that overshadows his or her risk-taking behavior.
CONCLUSIONS Based on the results of this study the following main conclusions may be drawn with regard to the effect of using different assumptions in modeling route choice behavior while estimating network capacity reliability. 1. Regardless of which route choice model is used, it appears likely that some arcs will be more fully utilized than others depending on how many routes use each arc. Travelers who exhibit different risk-sensitive behaviors most likely will favor some arcs more than others based on the disutility associated with each arc. 2. In general the stochastic route choice models result in lower values of capacity reliability across the network, which indicates that flow allocation across the network is suboptimal. The value of capacity reliability is even lower when travelers are assumed to be risk sensitive and therefore choosing routes based on criteria other than travel time alone. 3. Most of the differences in the reliability measures for the different route choice models are more apparent at medium to high levels of demand, which indicates that perhaps at low levels of congestion any of the route choice models may produce similar results. Finally, note that all the results and conclusions drawn from this study are relevant for the particular network and parameters used in this study. Although the results appear to be intuitive, it would be pertinent to implement similar studies on different networks and perhaps under different parameter settings to verify the results of this study.
REFERENCES
FIGURE 4 Effect of route choice models on estimation of capacity reliability.
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10. Chen, A., H. Yang, H. K. Lo, and W. H. Tang. Capacity Reliability of a Road Network: An Assessment Methodology and Numerical Results. Transportation Research, in press. 11. Wong, S. C., and H. Yang. Reserve Capacity of a Signal-Controlled Road Network. Transportation Research, Vol. 31B, 1997, pp. 397–402. 12. Tatineni, M., D. E. Boyce, and P. Mirchandani. Comparison of Deterministic and Stochastic Traffic Loading Models. In Transportation Research Record 1607, TRB, National Research Council, Washington, D.C., 1997, pp. 16–23. 13. Wardrop, J. G. Some Theoretical Aspects of Road Traffic Research. Proc., Institution of Civil Engineers, Part II, Vol. 1, 1952, pp. 325–378. 14. Sheffi, Y. Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods. Prentice-Hall, Englewood Cliffs, N.J., 1985. 15. Mirchandani, P., and H. Soroush. Generalized Traffic Equilibrium with Probabilistic Travel Times and Perceptions. Transportation Science, Vol. 21, 1987, pp. 133–152. 16. Billington, R., and W. Li. Reliability Assessment of Electric Power Systems Using Monte Carlo Methods. Plenum Press, New York, 1994. 17. Tatineni, M. Solution Properties of Stochastic Route Choice Models. Ph.D. dissertation, University of Illinois at Chicago, 1996.
