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Stochastic Dynamic Network Design Problem
Abstract
The continuous network design problem that occurs when the origin-destination time-dependent demands are random variables with known probability distributions is described. A chance-constraint and two-stage linear programming formulation are introduced based on a system optimum dynamic traffic assignment model that propagates traffic according to the cell transmission model. The introduced approaches are limited to continuous link improvements and do not provide for new link additions. The chance-constrained model is tested on an example network that resembles a freeway corridor to demonstrate the simplicity and applicability of the approach. Initial results suggest that planning for an inflated demand may produce benefits in terms of system performance and reduced variance.
... The literature on the DTA-based NDP is limited (3)(4)(5)(6)(7). Janson (3) showed that the NDP solution based on static UO assignment might not reduce the total system travel time (TSTT) when evaluating the solution by the dynamic UO assignment. ...
... Ukkusuri and Waller (5) proposed a continuous NDP formulated as an LP model where the users' route choices are based on the UO DTA LP model by Waller and Ukkusuri (10) also using Daganzo's cell transmission model. Further, Waller and Ziliaskopoulos (6 ) introduced the stochastic SO DTA-based NDP with long-term singledestination O-D demand uncertainty, formulated as a two-stage stochastic linear program with recourse and a chance-constrained program (CCP). They tested the individual CCP on a test problem and concluded that planning for an inflated demand might produce benefits in terms of system performance and reduced variance. ...
... Waller (11) solved the SLP2 model by the simplex method; this was possible only because of the considered small number of demand realizations. Ukkusuri et al. (7) introduced the UO versions of SLP2 and CCP models and solved only the CCP model with a conclusion similar to that of Waller and Ziliaskopoulos (6). D. P. Morton (class lectures, spring 2004) discussed the pros and cons of SLP2 and CCP as follows. ...
This paper compares the continuous network design problem formulations using system-optimal (SO) and user-optimal (UO) dynamic traffic assignment when the following independent stochastic parameters with known discrete probability distributions are considered: time-dependent origin–destination demands, time-varying saturation flow rates and jam density, and network improvement unit costs. These models propagate traffic according to Daganzo's cell transmission model. Two Monte Carlo bounding techniques, common random numbers (CRN) and independent random numbers (IRN) strategies, are used to solve the stochastic models. The results show that the CRN strategy outperforms IRN on a simple test network resembling a freeway corridor. The network size is sacrificed to gain higher confidence probabilistic behavior and to understand intuitively the effects of different network improvement policies. Although the findings may not necessarily be generalized, they provide interesting and insightful information. First, the SO and UO models allocate investment differently for certain budgets, and the stochastic models may lead to an erroneous solution (i.e., a bottleneck) for some budgets. Subsequently, the results of three comparison cases are discussed: (a) for the SO models, it should be more valuable to solve the stochastic than the deterministic models, but it is not always the case for the UO models; (b) the SO models appear more desirable than the UO models for single-level analysis; and (c) it should be more valuable to solve the stochastic model accounting for more randomness.
... It is well recognized that network uncertainties with respect to demand fluctuations and road capacity degradation inevitably and inherently exist on transportation networks. Recently, more researchers and practitioners have pointed out that network uncertainty would have significant impacts on network planning and other transportation management decisions, and ignoring this factor might result in suboptimal or even misleading optimization schemes [8,16,17]. Chen et al. [18] provided a comprehensive review of stochastic NDPs on various model formulations (e.g., expected-value model, mean-variance model, chance-constrained model, probability model, and min-max model) and proposed an attractive simulationbased genetic algorithm. To properly characterize the effect of network stochasticity, simulation-based sampling (e.g., Monte-Carlo simulation) approximation technique is often used to estimate probabilistic statistics, including expectation and variance, of various system performance measures [8,[18][19][20][21][22][23][24]. ...
... Some stochastic NDP models had also integrated other traffic factors and/or optimization targets (e.g., sustainability, traffic dynamics, and land use). For example, Waller and Ziliaskopoulos [16] formulated a chance-constrained NDP model that takes into account both uncertain demands and dynamic traffic flows. Wang et al. [26] incorporated multiple indices of sustainability, including traffic emission and social equity, into a stochastic NDP. ...
... Specifically, dynamic NDP models have been developed to properly describe realtime traffic dynamics and/or other unsteady-state traffic conditions (e.g., traffic shockwaves propagation, build-up process of queues) [27]. For example, Waller and Ziliaskopoulos [16] proposed a two-stage linear programming NDP model where a system-optimal dynamic traffic assignment and cell transmission loading technique were used to depict dynamic traffic propagation on the network. Karoonsoontawong and Waller [27] presented a robust dynamic NDP model with the objective of minimizing the weighted sum of the expected total travel time and expected risk. ...
We study transportation network design with stochastic demands and emergency vehicle (EV) lanes. Different from previous studies, this paper considers two groups of users, auto and EV travelers, whose road access rights are differentiated in the network, and addresses the value of incorporating inverse-direction lanes in network design. We formulate the problem as a bilevel optimization model, where the upper-level model aims to determine the optimal design of EV lanes and the lower-level model uses the user equilibrium principle to forecast the route choice of road users. A simulation-based genetic algorithm is proposed to solve the model. With numerical experiments, we demonstrate the value of deploying inverse-direction EV lanes and the computational efficiency of the proposed algorithm. We reach an intriguing finding that both regular and EV lane users can benefit from building EV lanes.
... There is an increasing body of relevant literature on the DTA-based NDP under uncertainty [1,2,10,[14][15][16][17][18]. The common feature of these studies is that the cell transmission model (CTM) [19,20] is adopted to model the time-varying traffic flows propagation, and the traffic demand is assigned to the network by either the dynamic system-optimal (SO) [21] or user-optimal (UO) [10] principle. ...
... They showed that it is more beneficial to solve a stochastic model than a deterministic model. Waller and Ziliaskopoulos [14] formulated the CCP and SLP2 models of single-level CTM-based SO NDP under demand uncertainty. Ukkusuri and Waller [10] introduced the CTM-based single-level UO versions of CCP and SLP2 and compared them with the corresponding SO versions. ...
... The remaining constraints, from (9) to (13), report the initial conditions and nonnegativity conditions. Constraint (14) guarantees that the total cost of expansion is within the available budget. ...
This paper develops an adjustable robust optimization approach for a network design problem explicitly incorporating traffic dynamics and demand uncertainty. In particular, a cell transmission model based network design problem of linear programming type is considered to describe dynamic traffic flows, and a polyhedral uncertainty set is used to characterize the demand uncertainty. The major contribution of this paper is to formulate such an adjustable robust network design problem as a tractable linear programming model and justify the model which is less conservative by comparing its solution performance with the robust solution from the usual robust model. The numerical results using one network from the literature demonstrate the modeling advantage of the adjustable robust optimization and provided strategic managerial insights for enacting capacity expansion policies under demand uncertainty.
... As for the chance-constrained model, Waller and Ziliaskopoulos [36] presented a chance-constrained model for a network design problem (NDP) and specified timedependent random demand as chance constraints. Lo and Tung [37] presented an NDP model by introducing a degradable link capacity as chance constraints. ...
... As it was mentioned in Assumption 1 that the travel demand is a random variable with a known probability distribution. According to Waller and Ziliaskopoulos [36], based on the definition of the distribution function Pr , ...
This paper studied the train stop planning problem with variable train length and stop time under stochastic demand (TSPPWVTLSTSD). As we know, stop time and train length were often considered in a fixed manner in train stop planning for high-speed rail (HSR). One obvious disadvantage was that it often resulted in insufficient time for passengers to get on or get off a train. In addition, fixed train length may cause a waste of train capacity. And travel demand was usually assumed to be deterministic, although travel demand usually was stochastic. In this paper, we presented an optimization model that simultaneously optimized the stop-schedule, the stop time schedule and the train length schedule for each train trip under stochastic demand. By formulating deterministic equivalents to the chance constraints, we obtained a deterministic mixed integer model. To solve the problem, a novel column generation–based heuristic solution technique based on Dantzig–Wolfe decomposition principle and column generation procedure was proposed. Some numerical experiments and a case study based on realworld data were used to demonstrate that the proposed solution method can yield a service plan within a reasonable time compared with ILOG Cplex. Besides, the variable train length and stop time model needed fewer carriages and also gave rise to less total time loss compared with the fixed scenario.
... The general approaches of addressing uncertainty in DTAbased NDP studies can be roughly classified into four categories: chance-constrained programming (CCP) [23][24][25], a two-stage stochastic programming with recourse (SLP2) [23,[26][27][28][29], scenario-based robust optimization [26][27]30] and set-based robust optimization [31][32][33][34][35][36]. The probability distributions of uncertain parameters are assumed to be known explicit in the model with the CCP, SLP and scenario-based robust optimization approach. ...
... The general approaches of addressing uncertainty in DTAbased NDP studies can be roughly classified into four categories: chance-constrained programming (CCP) [23][24][25], a two-stage stochastic programming with recourse (SLP2) [23,[26][27][28][29], scenario-based robust optimization [26][27]30] and set-based robust optimization [31][32][33][34][35][36]. The probability distributions of uncertain parameters are assumed to be known explicit in the model with the CCP, SLP and scenario-based robust optimization approach. ...
This paper presents a globalized robust optimization approach for a network design problem explicit incorporating traffic dynamics and demand uncertainty. In particular, a non-holding back cell transmission model (CTM) based network design problem of linear programming type is considered to describe dynamic traffic flows, and the normal range of the uncertain demand is assumed to be a box set, i.e., the uncertain demand outside box set is allowed. The major contribution of this paper is to formulate such a globalized robust network design problem as a tractable linear programming model and demonstrate the model robustness and flexibility by comparing its solution performance with the robust solution from the usual robust model and the adjustable robust solution from the adjustable robust model, respectively. A numerical experiment is conducted to demonstrate that the modeling advantage of the globalized robust optimization in terms of solution quality. The proposed globalized robust optimization approach may provide useful insights and have broader applicability in traffic management and traffic planning problems under uncertainty.
... First, the decision variables (y ij ) are discrete and not continuous, meaning that additional capacity is defined in units of one lane (Lou et al., 2009). The presence of the discrete or integer variables makes NDP more difficult to solve because the gradient-based approaches are not applicable (Waller and Ziliaskopoulos, 2001;Lou et al., 2009). Adopting discrete decision variables rather than continuous variables better reflects real world conditions because you either build/widen a road or you do not (Jeon et al., 2006;Boyce and Janson, 1980). ...
... An easier, but less accurate, approach is to adopt a system-optimal (SO) traffic flow solution. To calculate the SO flow, the users' shortest path choice behaviour is ignored and instead an artificial flow minimising total system cost is identified (Boyce, 1979;Sheffi, 1985;Waller and Ziliaskopoulos, 2001;Duthie and Waller, 2008). ...
... However, these studies treat expressways as separate entities and do not meet realistic requirements. Waller and Ziliaskopoulos (2001) modeled urban arterials and expressways using the cell transmission model (CTM) to investigate road design issues. Nevertheless, the study examines the traffic operations of individual road links. ...
As urbanization advances, cities are expanding, leading to a more decentralized urban structure and longer average commuting durations. The construction of an urban expressway system emerges as a critical strategy to tackle this challenge. However, the traditional link-level network design method faces modeling and solution challenges when dealing with the large-scale expressway network design problem (ENDP). To address the challenges, this paper proposes an expressway network design method for multiple urban subregions based on the macroscopic fundamental diagram (MFD). Initially, a mixed road network traffic model that describes traffic dynamics of multiple subregions and candidate expressways is developed by integrating the MFD and the cell transmission model (CTM). Then, treating urban subregions and candidate expressways as route nodes in the mixed road network, a route choice model is established based on stochastic user equilibrium. Finally, a decision model for ENDP is proposed to minimize vehicle travel time under the construction budget constraint. The impact of financial investment and traffic demand on expressway network design schemes in the case study is explored separately. The simulation results indicate that during the initial stages of expressway planning, the construction of new expressways can significantly alleviate traffic congestion. However, as the expressway network expands further, the effectiveness of improving traffic conditions through new expressway construction gradually diminishes if traffic demand does not continue to increase. Additionally, variations in traffic demand between subregions result in different construction schemes, emphasizing the importance of adjusting budget allocations based on specific traffic demands.
... Lee [12] extended the CTM framework to encompass interrupted traffic flow modeling. Subsequently, the CTM has found wide-ranging applications in network design [13][14][15] and network traffic assignment [16][17][18][19][20][21]. It is noteworthy for its ability to accurately replicate various traffic flow characteristics, as demonstrated in studies [22][23][24]. ...
Lane-changing prewarning is an active management measure used to mitigate the impact of incidents on expressways. This measure affects the operational efficiency and safety of traffic flow upstream of the incident location. To evaluate the effectiveness of this measure in mitigating incident-induced congestion and enhancing traffic safety, it is essential to model and simulate traffic flow on expressways, followed by analyzing the simulation results. In this study, we use the lane-changing selection cell transmission model (ls-CTM) to investigate the impact of setting lane-changing prewarning information on expressway traffic flow. By enhancing the composition of flow within the diverging cell in CTM, traffic can determine its downstream direction based on lane-changing probability rather than the path. A case study was conducted to investigate the impact of prewarning distances of 50, 100, and 150 m on road operational efficiency and traffic safety under different traffic volumes. The effectiveness and applicability of three measures to improve traffic safety were also summarized. The research results can provide more rigorous and effective decision-making support for expressway traffic management and control.
... SO-DTA quantifies the best possible performance of a traffic network based on the dynamic extension of Wardrop's second principle (Wardrop 1952). For its advantage of exploring the spatial-temporal traffic dynamics to achieve the best performance of traffic networks, the research of the application of SO-DTA problem focuses on many different areas, such as emission reduction (Lu et al. 2016, Ma et al. 2017, Long et al. 2018, Tan et al. 2021, congestion mitigation (Levin 2017, Samaranayake et al. 2018, Liu et al. 2020, parking services (Levin 2019, Qian & Rajagopal 2014, traffic signal control (Lo 2001, Han et al. 2016, emergency evacuation (Liu et al. 2006, Chiu et al. 2007) and network design (Waller & Ziliaskopoulos 2001, Waller et al. 2006. Different from human-driven vehicles (HVs), AVs show great potential in improving control efficiency and may play an important role in achieving SO-DTA (Wang et al. 2018). ...
This study develops the headway control framework in a fully automated road network, as we believe headway of Automated Vehicles (AVs) is another influencing factor to traffic dynamics in addition to conventional vehicle behaviors (e.g. route and departure time choices). Specifically, we aim to search for the optimal time headway between AVs on each link that achieves the network-wide system optimal dynamic traffic assignment (SO-DTA). To this end, the headway-dependent fundamental diagram (HFD) and headway-dependent double queue model (HDQ) are developed to model the effect of dynamic headway on roads, and a dynamic network model is built. It is rigorously proved that the minimum headway could always achieve SO-DTA, yet the optimal headway is non-unique. Motivated by these two findings, this study defines a novel concept of maximin headway, which is the largest headway that still achieves SO-DTA in the network. Mathematical properties regarding maximin headway are analyzed and an efficient solution algorithm is developed. Numerical experiments on both a small and large network verify the effectiveness of the maximin headway control framework as well as the properties of maximin headway. This study sheds light on deriving the desired solution among the non-unique solutions in SO-DTA and provides implications regarding the safety margin of AVs under SO-DTA.
... Such randomness comes from various sources, including demand side (such as random numbers of travelers) and supply side (such as weather, accidents, or events). The existing DTA approaches typically require an explicit functional form of the stochastic sources (Peeta and Zhou, 1999;Waller and Ziliaskopoulos, 2001;Sumalee et al., 2011). In addition, adaptive route choice behavior of other agents may contribute to the inherent stochasticity in traffic evolution, which has not been generally accounted for in the existing DTA paradigm. ...
This paper aims to develop a unified paradigm that models one's learning behavior and the system's equilibrating processes in a routing game among atomic selfish agents. Such a paradigm can assist policymakers in devising optimal operational and planning countermeasures under both normal and abnormal circumstances. To this end, a multi-agent reinforcement learning (MARL) paradigm is proposed in which each agent learns and updates her own en-route path choice policy while interacting with others on transportation networks. This paradigm is shown to generalize the classical notion of dynamic user equilibrium (DUE) to model-free and data-driven scenarios. We also illustrate that the equilibrium outcomes computed from our developed MARL paradigm coincide with DUE and dynamic system optimal (DSO), respectively, when rewards are set differently. In addition, with the goal to optimize some systematic objective (e.g., overall traffic condition) of city planners, we formulate a bilevel optimization problem with the upper level as city planners and the lower level as a multi-agent system where each rational and selfish traveler aims to minimize her travel cost. We demonstrate the effect of two administrative measures, namely tolling and signal control, on the behavior of travelers and show that the systematic objective of city planners can be optimized by a proper control. The results show that on the Braess network, the optimal toll charge on the central link is greater or equal to 25, with which the average travel time of selfish agents is minimized and the emergence of Braess paradox could be avoided. In a large-sized real-world road network with 69 nodes and 166 links, the optimal offset for signal control on Broadway is derived as 4 seconds, with which the average travel time of all controllable agents is minimized.
... Power demand. Traditional models have considered power demand as an exogenous term, which is either deterministically (Benavides et al. 2013, Correa-Florez et al. 2014, Jenabi et al. 2015, Wang et al. 2016, Lara et al. 2018 or stochastically (Waller & Ziliaskopoulos 2001, Atamtürk & Zhang 2007, Lium et al. 2009) specified. ...
In the context of centralized electricity markets, we propose an integrated planning model for power pricing and network expansion, which endogenizes the scaling costs from power losses. While the substitutability pattern between pricing and expansion has been overlooked in the power flow optimization literature, this becomes particularly relevant in centralized electricity markets (where the headquarters are enabled to take decisions over a wide range of operational factors). In this paper, we tailor an optimization model and solution approach, that can be effectively applied to large-scale instances of centralized power systems. Specifically, we develop bounds to the optimal operator profit and use them within a mixed-integer linear programming problem, derived from the linearization of an extended power flow model. On the empirical side, we conduct computational tests on a comprehensive power system data set from the Saudi Electricity Company, uncovering the value of the proposed integrated planning. The results reveal the complex substitutability patterns which appear when deciding about integrated operational factors in centralized power systems and support the correctness and efficiency of the proposed resolution mechanism.
... However, these static assignment strategies have been criticised for their own deficiencies over the course of time, as the distinguishing characteristic of demand is dynamic in nature. In this sense, some researchers, such as Waller and Ziliaskopoulos [36], Lin [37], Lin et al. [38], Chung et al. [39], Sun et al. [40] and Wismans et al. [41], have employed dynamic traffic assignment models to address TNDPs. ...
This study aims to investigate a robust network design problem to maximise the expected space-time accessibility, in which the origin-destination demand matrices are time-varying or unsynchronised under stochastic space-time scenarios. By defining the accessibility measure as the expected amount of accessible demand pairs, a linear 0-1 integer programming model is formulated to characterise the network design process. In order to find the near-optimal solutions for the problem of interest, an effective heuristic algorithm is proposed based on the framework of Lagrangian relaxation that decomposes the primal model into a series of tractable subproblems. Finally, the performance of the developed methods is demonstrated through numerical experiments in two networks with different scales. The results highlight the importance of considering robustness in transportation network design problems.
... These DTA LP models propagate traffic according to the cell transmission model (7). Further, Waller and Ziliaskopoulos (8) and Ukkusuri et al. (9) introduced two-stage stochastic LPs with recourse (SLP2) and chanceconstrained programs based respectively on SO and UO DTA, accounting for demand uncertainty. The SO and UO versions of SLP2 models were comprehensively compared by Karoonsoontawong and Waller (10). ...
A linear bilevel programming model and two analytical solution methods (the Kth-best algorithm and mixed integer programming reformulation) for the continuous network design problem are presented on the basis of the multiorigin, single-destination, user-optimal dynamic traffic assignment (UO DTA) problem. From the test problem, it is shown that the bilevel formulation is more desirable than the two known single-level models based respectively on system-optimal and UO DTA. For the multiorigin, multidestination, larger-size problem, three metaheuristics that can produce solutions beyond local optimality are employed: simulated annealing (SA), genetic algorithm (GA), and random search (RS). These metaheuristics share the same functional evaluation: a simulation-based UO DTA that propagates traffic according to Daganzo's cell transmission model. From computational results, GA outperforms the others for all three test problems in terms of solution quality, convergence speed, and processor time, whereas SA and RS appear nondominated. It is also shown that the appropriate set of algorithm parameters is network-specific and should be recalibrated for each network to achieve the best results.
... CTM was sufficiently validated with field data by Lin and Daganzo (5) and Lin and Ahanotu (6). The model was also used to support dynamic traffic assignment (7)(8)(9) and network design applications (10,11). Other studies applied CTM to evaluate the design of intersection timing plans (12)(13)(14)(15). ...
The cell-transmission model (CTM), developed by Daganzo in 1994, has not been fully exploited as an operations model for analysis of large-scale traffic networks. Because of its macroscopic and mesoscopic features, CTM offers calibration and computational advantages over microscopic models. This paper demonstrates specific improvements to the original form of CTM to increase its accuracy and realism of traffic flow representation. These improvements include modifications to provide flexibility in selection of cell lengths, noninteger movements of vehicles between cells, and algorithmic enhancements in the merging and diverging logics. The effect of such improvements on the performance of CTM was evaluated independently and comparatively. A sample freeway network of the I-10 corridor in Baton Rouge, Louisiana, was used to evaluate and compare the performance of the improved version of CTM versus CORSIM under heavily congested traffic conditions. The results showed comparable performance of both platforms in terms of link occupancy (density) and total network travel time.
... Such transportation network design problems have been extensively studied using STA models in the literature (see Farahani et al., 2013 ). These problems have also been researched using DTA models (e.g., Janson, 1995;Waller and Ziliaskopoulos, 2001;Heydecker, 2002 ) but with less attention. DTA models can account for traffic dynamics and time-variant demand. ...
The fact that road transportation negatively affects the quality of the environment and deteriorates its bearing capacity has drawn a wide range of concerns among researchers. In order to provide more realistic traffic data for estimations of environmental impacts, dynamic traffic assignment (DTA) models have been adopted in transportation planning and traffic management models concerning environmental sustainability. This review summarizes and examines the recent methodological advances of DTA models in environmentally sustainable road transportation applications including traffic signal control concerning vehicular emissions and emission pricing. A classification of emission estimation models and their integration with DTA models are accordingly reviewed as supplementary to the existing reviews. Finally, a variety of future research prospects of DTA for environmentally sustainable road transportation research are discussed. In particular, this review also points out that at present the research about DTA models in conjunction with noise predictive models is relatively deficient.
... Due to the linearity property of this formulation, it is tractable for realistic size network. Waller and Ziliaskopoulos (2001; address the uncertainty of traveller demand for single destination network of the SO-DTA problem. However, none of these contributions include traffic signal control. ...
Dynamic Traffic Assignment (DTA) provides an approach to determine the optimal path and/or departure time based on the transportation network characteristics and user behavior (e.g., selfish or social). In the literature, most of the contributions study DTA problems without including traffic signal control in the framework. The few contributions that report signal control models are either mixed-integer or nonlinear formulations and computationally intractable. The only continuous linear signal control method presented in the literature is the Cycle-length Same as Discrete Time-interval (CSDT) control scheme. This model entails a trade-off between cycle-length and cell-length. Furthermore, this approach compromises accuracy and usability of the solutions. In this study, we propose a novel signal control model namely, Signal Control with Realistic Cycle length (SCRC) which overcomes the trade-off between cycle-length and cell-length and strikes a balance between complexity and accuracy. The underlying idea of this model is to use a different time scale for the cycle-length. This time scale can be set to any multiple of the time slot of the Dynamic Network Loading (DNL) model (e.g. CTM, TTM, and LTM) and enables us to set realistic lengths for the signal control cycles. Results show, the SCRC model not only attains accuracy comparable to the CSDT model but also more resilient against extreme traffic conditions. Furthermore, the presented approach substantially reduces computational complexity and can attain solution faster.
... Due to the linearity property of this formulation, it is tractable for realistic size network. Waller and Ziliaskopoulos (2001; address the uncertainty of traveller demand for single destination network of the SO-DTA problem. However, none of these contributions include traffic signal control. ...
Dynamic Traffic Assignment (DTA) provides an approach to determine the optimal path and/or departure time based on the transportation network characteristics and user behavior (e.g., selfish or social). In the literature, most of the contributions study DTA problems without including traffic signal control in the framework. The few contributions that report signal control models are either mixed-integer or non-linear formulations and computationally intractable. The only continuous linear signal control method presented in the literature is the Cycle-length Same as Discrete Time-interval (CSDT) control scheme. This model entails trade-off between cycle-length and cell-length. Furthermore, this approach compromises accuracy and usability of the solutions. In this study, we propose a novel signal control model namely, Signal Control with Realistic Cycle-length (SCRC) which overcomes the trade-off between cycle-length and cell-length and strikes a balance between complexity and accuracy. The underlying idea of this model is to use a different time scale for the cycle-length. This time scale can be set to any multiple of the time slot of the Dynamic Network Loading (DNL) model (e.g. CTM, TTM, and LTM) and enables us to set realistic lengths for the signal control cycles. Results show, the SCRC model not only attains accuracy comparable to the CSDT model but also more resilient against extreme traffic conditions. Furthermore, the presented approach substantially reduces computational complexity and can attain solution faster.
... Most of the studies considered above use either mathematical programming approaches or algebraic methods. Since the two-stage stochastic programming model, presented in Beale (1955) and Dantzig (1955), is capable to capture uncertainty, it was applied in many transportation and related areas, such as to solve the stochastic vehicle routing problem (Gendreau et al. (1996)), reliable facility location problem (Owen and Mark (1998);Snyder et al. (2006); Shen et al. (2011)), network design and protection problem (Waller and Ziliaskopoulos (2001); Liu et al. (2009)), transit network design problem (An and Lo (2016)), signal control problem (Tong et al. (2015)), and so on. ...
Path flow identification is of particular interest for a number of traffic applications, such as OD demand estimation, link flow inference, and toll freeway revenue management. Optimal positioning of active sensors can
help to identify path flows. Due to the stochastic nature of transportation systems, we propose a scenario based
two stage stochastic programming framework which considers the uncertainty of the link-path matrix. The first
stage model aims to minimize the total traffic sensor installation cost and the expected penalty for uncovered
and undifferentiated paths. The second stage model attempts to minimize uncovered and undifferentiated paths for a given sensor location pattern and a specific scenario. In addition, a mean risk measure is also incorporated into the two stage stochastic programming framework, and consequently a mean risk two stage stochastic programming model is proposed. Both models have the same structure, where the first stage and second stage decision variables are binary. The second stage decision variable can be relaxed to a continuous variable without changing the nature of the model. To solve the two stochastic programming models, a branch and bound based integer L-shaped algorithm is presented. Finite steps convergence is guaranteed for the algorithm. To handle the problem with a large number of scenarios, a sampling technique is introduced, and the confidence bound is analyzed with respect to the scenario size. Extensive numerical experiments are conducted to verify the effectiveness of the proposed models and algorithm. The most important numerical results are as follows: (i) the stochastic programming framework is capable of capturing the reality more efficiently and accurately, (ii) the path differentiation factor is more critical than the path coverage factor in determining the sensor placement pattern, and (iii) in the partial parameter setting case, the mean risk based stochastic programming model results in a significantly different sensor placement pattern compared to the normal stochastic programming model. The study contributes to practical sensor placement design.
... Under uncertain traffic situation, adding more assumptions or constraints to guarantee unique equilibrium is more desirable for a specific case with particular emphasis on ensuring a unique equilibrium. By considering time-dependent demands in a dynamic traffic assignment process as random variables with known probability distributions, Waller and Ziliaskopoulos (2001) offered one of the first modeling frameworks to fully take into the impact of stochastic demand variability for network design applications. Ng and Waller (2010) further presented a fast Fourier transform-based method to characterize travel time variability distributions due to random capacity. ...
In this study, to incorporate realistic discrete stochastic capacity distribution over a large number of sampling days or scenarios (say 30–100 days), we propose a multi-scenario based optimization model with different types of traveler knowledge in an advanced traveler information provision environment. The proposed method categorizes commuters into two classes: (1) those with access to perfect traffic information every day, and (2) those with knowledge of the expected traffic conditions (and related reliability measure) across a large number of different sampling days. Using a gap function framework or describing the mixed user equilibrium under different information availability over a long-term steady state, a nonlinear programming model is formulated to describe the route choice behavior of the perfect information (PI) and expected travel time (ETT) user classes under stochastic day-dependent travel time. Driven by a computationally efficient algorithm suitable for large-scale networks, the model was implemented in a standard optimization solver and an open-source simulation package and further applied to medium-scale networks to examine the effectiveness of dynamic traveler information under realistic stochastic capacity conditions.
... Yin and Ieda (2002) analyzed the NDP with uncertain travel time. Waller and Ziliaskopoulos (2001) studied the problem of a dynamic NDP with demand uncertainty. Chen et al. (2003) introduced a mean-variance model to determine optimal tolls under the build-operate-transfer scheme with uncertain demand. ...
... is an increasing function with respect to [20][21][22][23]. To reflect the uncertain environment of the real world, the lognormal distribution is used in stochastic travel demand modeling in this study. ...
... The use of Benders decomposition to solve network design, improvement, and flow problems, particularly where the data is stochastic in nature, is common. Waller and Ziliaskopoulos (2001) solve a two-stage stochastic version of the continuous network design problem in which origin-destination demands are represented using probability distributions. Their formulation allows for the introduction of a confidence level which, as it is increased, requires the solution to meet demands that are actually greater than their expected values, thereby providing a more robust solution. ...
... A bilevel optimization model is developed that aims to redesign a surviving transportation network. The problem is addressed as a variant of the network design problem (NDP); a surviving network is redesigned (e.g., available resources such as lanes and shoulders are reallocated to operations) so that social welfare is maximized (5)(6)(7)(8). The bilevel structure of the NDP is that of a leader-follower game, with operators acting as leaders and users as followers. ...
Vital lifelines in the event of a disaster, transportation networks support evacuation activities, emergency logistics, and the restoration of daily activities. The rapid recovery of surviving transportation network operations immediately after a disaster is critical for the well-being of communities. The focus here is on planning highway operations in the recovery period after a disaster. A bilevel network design model with choice interdependencies was developed for choosing the strategies necessary to maximize the performance of a surviving highway network. A genetic algorithm was coupled with a traffic assignment procedure to solve the associated problem. Results on a real-size network in Greece under two disaster scenarios indicate both adequate computational performance and significant improvement in network performance indicators.
... The L-CTM adopts a nonconcave flow-density relationship and builds a lagged time method to improve the computation accuracy that may be affected by congestion within the conventional CTM model. The CTM has been utilized and developed within the field of dynamic traffic assignment (DTA) and as a traffic flow simulation tool by a number of authors including: Ziliaskopoulos [18], Waller and Ziliaskopoulos [19], Lo [20], Lo and Szeto [21], [22], Szeto [23], Ukkusuri and Waller [24], Sumalee et al. [25]. There have been a variety of efficient applications of the CTM. ...
The traditional cell transmission model (CTM), a well-known dynamic traffic simulation method, does not cater to the presence of moving bottlenecks, which may be caused by buses traveling within a network. This may affect the dynamics of congestion that is present and may also affect route choice by all vehicles on a network. The main contribution of this paper is to provide an analytical formulation for a mixed traffic system that includes cars and buses, which realistically replicates moving bottlenecks. We modify the CTM model using methods from the lagged CTM to recognize speed differentials between the free-flow speed of buses and cars. In addition, the impact of capacity reduction caused by buses was incorporated. These developments led to the replication of moving bottlenecks caused by buses within the CTM framework. The formulated variant of CTM was utilized to determine a system optimal assignment that minimizes the total passenger travel time across cars and buses. The proposed modified CTM model, defined as the BUS-CTM, has been applied on a road link and a more detailed network to demonstrate the effectiveness of the approach. The numerical results and the depiction of the bottleneck phenomenon within the model suggests that the BUS-CTM obtains more realistic results compared with the application of the traditional CTM in a mixed car-bus transportation system. The sensitivity analysis shows that bus passenger demand, passenger occupancy of bus, and bus free-flow speeds are the key parameters that influence the system performance.
... During the morning and evening peak hours, surging demand may overwhelm a roadway's physical capacity and results in delays (Federal Highway Administration [FHWA], 2009). Waller and Ziliaskopoulos (2001), Chen, Skabardonis, and Varaiya (2003) and Lam, Shao, and Sumalee (2008) have used the normal distribution for modeling travel demand variation. Other researchers have modeled travel demand using the Poisson distribution (Clark & Watling, 2005;Hazelton, 2001) and the uniform distribution (Unnikrishnan, Ukkusuri, & Waller 2005). ...
Trip travel time reliability is an important measure of transportation system performance and a key factor affecting travelers’ choices. This paper explores a method for estimating travel time distributions for corridors that contain multiple bottlenecks. A set of analytical equations are used to calculate the number of queued vehicles ahead of a probe vehicle and further capture many important factors affecting travel times: the prevailing congestion level, queue discharge rates at the bottlenecks, and flow rates associated with merges and diverges. Based on multiple random scenarios and a vector of arrival times, the lane-by-lane delay at each bottleneck along the corridor is recursively estimated to produce a route-level travel time distribution. The model incorporates stochastic variations of bottleneck capacity and demand and explains the travel time correlations between sequential links. Its data needs are the entering and exiting flow rates and a sense of the lane-by-lane distribution of traffic at each bottleneck. A detailed vehicle trajectory data-set from the Next Generation SIMulation (NGSIM) project has been used to verify that the estimated distributions are valid, and the sources of estimation error are examined.
This paper studies how to reduce the overall travel time of commuters in a transportation network by reversing the direction of some lanes in the network using a macroscopic network-wide perspective. Similar to the Network Design Problem, the lane reversal problem has been shown to be NP-hard given the dependence of the users’ route selection on the lane direction decision. Herein, we propose and compare three efficient methods to solve the routing and lane reversal problem jointly. First, we introduce an alternating method that decouples the routing and lane assignment problems. Second, we propose a Frank–Wolfe method that jointly takes gradient steps to adjust both the lane assignment and routing decisions. Third, we propose a convex approximation method that uses a threshold-based approach to convexify the joint routing and lane reversal objective. The convex approximation method is advantageous since it finds a global optimum solution for the approximated problem and it enables the possibility to include linear constraints. Using this method, we extend the main formulation to be able to limit a maximum number of reversed lanes, as well as to incorporate multiple origin–destination (OD) patterns. We test the proposed methods in a case study using the transportation network of Eastern Massachusetts where our results indicate an overall reduction in travel times of 4.7% by selecting the best 15 reversals. Moreover, using a small test network, we investigate the performance of the lane reversal strategies as a function of the OD demand symmetry. As expected, we observe that when the OD demand is very asymmetric (e.g., for a single OD pair, evacuations, large events), the reduction in travel times is larger than the symmetric case, reaching travel time reductions of 60%.
Exclusive bus lanes (XBLs) have been widely discussed and implemented worldwide in recent years. An improved and more flexible transit lane management strategy—intermittent bus lanes (IBLs)—prove to be potentially more efficient and car-friendly than XBLs. The common benefit of XBLs and IBLs arises from the fact that they separate the bus and car traffic and hence eliminate the impacts of slowly moving buses on the car traffic. Compared to XBLs, the application of IBLs has not been broadly adopted in the real world so far except very few cases, because managing an urban IBL segment or area typically involves extra traffic signals and message signs and the switch between its regular and protected statuses could potentially cause unexpected traffic chaos and safety issues. The recent emergence of connected and automated vehicle (CAV) technologies, however, favors relieving these concerns and bring the dawn to a future application of this theoretically sound traffic control strategy. This paper is dedicated to evaluating the network performance of XBLs and IBLs and optimizing the networkwide configurations of these two strategies for future urban traffic networks where CAVs prevail. For this purpose, we formulate a system-optimal dynamic traffic assignment (SO-DTA) problem with car-exclusive lane segments, on the basis of a cell transmission model for separate car and bus traffic (CTM-SCB). By limiting its applicable context to morning commute networks and emergency evacuation networks, this model is set with only one real or virtual destination, which greatly eases its modeling and solution complexity. On the basis of the single-destination SO-DTA model, an XBL-based network design problem and an IBL-based network design problem are then both formulated into mixed integer programming models and solved by a discrete optimization solver, where the former problem statically configures bus lanes while the later one allows a dynamic allocation of bus lanes. Insightful findings obtained by applying the models and solver to a synthetic corridor network and a real commute network illustrate the great promise of these two lane-based strategies in mixed car and bus traffic management.
Transport networks are expected to remain highly reliable to ensure that users can experience smooth travel under both normal and abnormal traffic conditions. To design a reliable transport network properly it is necessary to take into account users reactions to incidents which cause disruption in different parts of the network. In order to design a reliable network to deal with such events, it is necessary to capture the users route choice behavior under degraded conditions. Furthermore, for practical reasons, it is imperative that any improvement scheme should give priority to links in the network which are most vulnerable to failure. Therefore in this paper we formulate a reliable continuous network design problem (RCNDP) in which the link flow is assigned to different routes considering the risk-taking behavior of the network users when confronted by network uncertainty. To capture this behavior, we pose the RCNDP under the multiple network spoiler formulation of risk-averse traffic assignment. The ability of this model to introduce higher capacity expansions on links vulnerable to failure and consider many operational states to cater for the worst-case scenarios of network reliability is demonstrated through two numerical studies.
The urban transportation network design problem involving exclusive bus lanes (XBLs) has been widely discussed and analyzed in recent years. An improved and more flexible transit lane management strategy—intermittent bus lanes (IBLs)—prove to be potentially more efficient and car-friendly than XBLs. The common benefit of XBLs and IBLs arises from the fact that they separate the bus and car traffic and hence can eliminate the impacts of slowly moving buses on the car traffic. This paper proposes a cell transmission model for separate car and bus traffic (CTM-SCB) in a network with some dedicated roadway segments reserved for buses. By encapsulating the CTM-SCB model, an XBL-based network design problem and an IBL-based network design problem are then formulated and solved, respectively, where the former model statically sets bus lanes while the later one allows a dynamic allocation of bus lanes. A synthetic freeway-arterial network and a real-world urban street network (where the latter was extracted from the Harbin South New Industrial City) are used as test networks for evaluating the proposed models and methods. The numerical results show that both XBLs and IBLs enjoy significant operational efficiency benefits compared to the situation of no protected bus lanes. However, we believe that the expected improvement from XBLs to IBLs need further tests and validations.
The focus of this study is on developing solutions for the traffic equilibrium problem (TEP) when origin–destination demand is assumed to be uncertain. Instead of developing and solving an expensive stochastic solver, it proposes single-point approximate measures that provide a reasonably good estimate of the network performance. Seven approximation schemes that utilize the deterministic TEP to account for uncertain demand and illustrate how these approaches can be used to solve the problem are proposed. The performance of the different approximation measures on different demand distributions for the Sioux Falls, South Dakota, network is reported, and a consistent approximation scheme is identified that performs well for the problem. Such an approximation scheme can be used in solving the TEP accounting for stochastic demand when computational resources are scarce or when a stochastic solver is unavailable.
A novel user equilibrium traffic assignment model based on travel time reliability is presented in view of day-to-day demand fluctuation. Because of daily demand variations, path travel times are not constants, and so they can be viewed as random variables. Assuming that travelers are able to learn the variation of path travel time from experience, a demand-driven user equilibrium principle based on travel time reliability is proposed to characterize travelers’ path choice behavior under uncertainty in travel times caused by demand variation. This principle can be formulated as a variational inequality (VI) problem in terms of path flows. The existence of at least one solution for the VI problem, for which a heuristic solution algorithm is adopted, is rigorously proved. Numerical examples are used to illustrate the applications of the proposed model and the solution algorithm.
In the reliable network design problem (RNDP) the main sources of uncertainty are variable demand and route choice. The objective is to maximize network total travel time reliability (TTR), which is defined as the probability that the network total travel time will be less than a threshold. A framework is presented for a stochastic network model with Poisson-distributed demand and uncertain route choice. The travelers are assumed to choose their routes to minimize their perceived expected travel cost following the probit stochastic user equilibrium condition. An analytical method is presented for approximation of the first and second moments of the total travel time. These moments are then fitted with a log-normal distribution. Then the design problem is tackled in which the analytical derivative of the TTR is derived with the sensitivity analysis of the equilibrated path choice probability. This derivative is then supplied to a gradient-based optimization algorithm to solve the RNDP. The algorithm is tested with a small network example.
Abstract Dynamic Traffic Assignment formulations are mathematical frameworks for optimal
transport system planning, assessing, and strategic traffic management. In this thesis several
novel Dynamic Traffic Assignment frameworks are proposed. These models can offer
following benefits to the society: 1. The proposed models can help us to reduce traffic
congestion. 2. They can enable us to manage traffic strategically which would reduce
emission and fuel consumption. 3. The proposed Bus Rapid Transit with Transit Signal
Priority can help us to design and analyse public transport system which would eventually
reduce passenger travel-time.
Real-time traffic network management systems are envisioned to provide network operators with decision support capabilities to alleviate recurrent and non-recurrent congestion. These capabilities involve predicting the network congestion dynamics and facilitating the development of proactive traffic management schemes that integrate traffic control and demand management strategies. However, traffic networks are subject to numerous sources of stochasticity that make it difficult to accurately predict their operational conditions and generate effective traffic management schemes to cope with these conditions. This paper presents a decision support system for proactive-robust traffic network management, which accounts for uncertainty in the network operational conditions. The objective is to develop robust traffic management schemes such that the network overall performance remains close to optimality under all possible future operational conditions. The modeling framework of the system is presented, which adopts a rolling horizon framework that integrates a meta-heuristic search algorithm and a dynamic traffic assignment simulation-based methodology. The system performance is examined through application to the traffic network of the US-75 corridor in Dallas, TX, USA.
In this paper, we formulate a dynamic transportation network design model in which traffic dynamics are modeled by the cell transmission model. In the formulation, transportation planners decide on the optimal capacity expansion policies of existing transportation network infrastructure with limited resources, while road users react to the capacity changes by selfishly choosing routes to maximize their own profit. Owing to the problem complexity, a majority of the research efforts have focused on tackling this problem using meta-heuristics. In this study, we incorporate a series of dual-variable approximation techniques into the paradigm of a quantum-inspired genetic algorithm (QIGA) and devise an efficient evaluation function based on these techniques. The proposed QIGA contains a series of enhancements compared to conventional genetic algorithms (GAs) and can be considered as a better alternative when solving problems with a complex solution space. The QIGA is applied to a synthetic network, a subnetwork of a real-world road network, and a realistic network to justify its theoretical and practical value. From the numerical results, it is found that in the same computational time, the QIGA outperforms the conventional GA by 3.86–5.63% in terms of the objective value, which can be significant, especially when network expansion of a large urban area is considered. Technical, computational, and practical issues involved in developing the QIGA are investigated and discussed.
Maintaining air-quality standards has been a priority for transportation planners and policy makers worldwide. However, most existing system optimum dynamic traffic assignment (SO-DTA) models do not accommodate environmental objectives. In this paper, we use the link transmission model (LTM) to develop SO-DTA models that minimize total system emissions (TSE) in single destination networks. We use step functions to approximate cumulative flow curves for individual links, and to decompose link inflow into sub-flows according to time intervals at which they leave the link. The decomposed link inflows are used to estimate link emissions. Dynamic network constraints, non-vehicle holding constraints and link inflow decomposition constraints are considered, and SO-DTA problems with environmental objectives are formulated as mixed integer linear programming (MILP) problems. Any average speed based emission functions can be used for our models. Finally, numerical examples are provided to demonstrate the performance of the proposed models.
This paper proposes a bi-level risk averse network design model for transportation networks with heterogeneous link travel time distributions. The objective of the network design is to minimize the total system travel time budget, which consists of the mean total system travel time and a safety margin. The design is achieved by selecting optimal link capacity expansions subject to a fixed expansion budget. Users’ selfish behavior and risk attitude are captured in the lower-level traffic assignment constraints, in which travelers select routes to minimize their own path travel time budget. The properties of the design problem are analyzed analytically and numerically. The analysis shows that despite the lack of knowledge of travel time distributions, the probabilities that the actual total system travel time and the actual path travel time are respectively within the optimal total system travel time budget and the minimum path travel time budget under optimal design have lower bounds. The lower bounds are related to the system manager's and travelers’ risk aversion. The optimal total system travel time budget is proven to be bounded below even when the link expansion budget is unlimited.
Under transportation network supply and demand uncertainty, travel time reliability emerges from nonlinear interactions among numerous travelers who are driven by self-interest, learn and adapt in changing situations, and therefore can be modeled as an emergent network property. This paper proposes a novel theoretical framework for the study of complexity regarding travel time reliability on the basis of empirically-derived individual decision rules, and develops an agent-based evolutionary model for travel time reliability analysis. Findings show that actual route choice behaviors related to network reliability are often non-optimal, and that these behaviors themselves are important determinants of travel time reliability under network supply and demand uncertainty. While many travelers search for alternative routes under uncertainty, few travelers actually change routes. Route choice rules employed by travelers can successfully improve travel time reliability in most tested uncertainty scenarios, but they do not necessarily reduce absolute travel time. While reducing demand uncertainty improves travel time and planning under uncertainty, the model provides a theoretically sound tool for analyzing the impact of traveler information systems on individual travel behavior and network performance. It can also serve as an empirically-estimated router for microscopic traffic simulators or act as a behavioral assignment algorithm for activity-based travel demand models.
Dynamic user optimal simultaneous route and departure time choice (DUO-SRDTC) problems are usually formulated as variational inequality (VI) problems whose solution algorithms generally require continuous and monotone route travel cost functions to guarantee convergence. However, the monotonicity of the route travel cost functions cannot be ensured even if the route travel time functions are monotone. In contrast to traditional formulations, this paper formulates a DUO-SRDTC problem (that can have fixed or elastic demand) as a system of nonlinear equations. The system of nonlinear equations is a function of generalized origin-destination (OD) travel costs rather than route flows and includes a dynamic user optimal (DUO) route choice subproblem with perfectly elastic demand and a quadratic programming (QP) subproblem under certain assumptions. This study also proposes a solution method based on the backtracking inexact Broyden-Fletcher-Goldfarb-Shanno (BFGS) method, the extragradient algorithm, and the Frank-Wolfe algorithm. The BFGS method, the extragradient algorithm, and the Frank-Wolfe algorithm are used to solve the system of nonlinear equations, the DUO route choice subproblem, and the QP subproblem, respectively. The proposed formulation and solution method can avoid the requirement of monotonicity of the route travel cost functions to obtain a convergent solution and provide a new approach with which to solve DUO-SRDTC problems. Finally, numeric examples are used to demonstrate the performance of the proposed solution method.
Transportation investments have profound indirect effects on urban development. Improving transportation infrastructure reduces travel times, and thus, enhances accessibility of the population to employment and vice versa. In this procedure, the cities whose accessibilities mostly benefit from the transportation improvements would attract more population and capital. This paper attempts to quantify these impacts in the megaregions, and to prioritize the transportation investments according to their potential effects on the sustainability of the future development in these regions. For this purpose, the problem is formulated as a bilevel programming model. At the upper level, subject to a budget and environmental constraints, the optimal subset of projects is chosen from the candidate set. The objective is to maximize the total production of the megaregion at the end of the planning horizon. The final product of each city is determined according to its resultant population and physical capital. Under the network investment decisions of the upper level problem, the lower level represents the route choice behavior of users. Because the focus of the study is interurban road networks, which have sparse connectivity, a simple all-or-nothing trip assignment technique is employed at this level. To illustrate the application of the proposed model, the Northeast Megaregion of the United States is used as the case study. The exact solution is obtained for this example and compared to the regional development without improving the transportation network.
Conventional transportation network design problems treat origin-destination (OD) demand as fixed, which may not be true in reality. Some recent studies model fluctuations in OD demand by considering the first and the second moment of the system travel time, resulting in stochastic and robust network design models, respectively. Both of these models need to solve the traffic equilibrium problem for a large number of demand samples and are therefore computationally intensive. In this paper, three efficient solution-approximation approaches are identified for addressing demand uncertainty by solving for a small sample size, reducing the computational effort without much compromise on the solution quality. The application and the performance of these alternative approaches are reported. The results from this study will help in deciding suitable approximation techniques for network design under demand uncertainty. DOI: 10.1061/(ASCE)CP.1943-5487.0000091. (C) 2011 American Society of Civil Engineers.
This paper provides an approach to solve the system optimal dynamic traffic assignment problem for networks with multiple O-D pairs. The path-based cell transmission model is embedded as the underlying dynamic network loading procedure to propagate traffic. We propose a novel method to fully capture the effect of flow perturbation on total system cost and accurately compute path marginal cost for each path. This path marginal cost pattern is used in the projection algorithm to equilibrate the departure rate pattern and solve the system optimal dynamic traffic assignment. We observe that the results from projection algorithm are more reliable than those from method of successive average algorithm (MSA). Several numerical experiments are tested to illustrate the benefits of the proposed model.
Capturing the impact of uncertain events in emergency evacuation time estimation is an important issue for public officials to avoid unexpected delays and related losses of life and property. However, most of the current studies in evacuation planning only focus on exogenous uncertainties, such as flooding damage in a hurricane, but ignore uncertainties caused by endogenously determined risks, or so called flow-related risks. This paper proposes an analytical framework along with an efficient solution methodology to evaluate the impact of endogenously determined risks in order to estimate evacuation time. We incorporate the probability function of endogenously determined risks in a cell-based macroscopic evacuation model. A network flow algorithm based on the sample average approximation approach is used as part of the solution procedure. Finally, a sample network is used to demonstrate the salient features of the proposed stochastic model and solution procedure.
This paper develops a distributionally robust joint chance constrained optimization model for a dynamic network design problem (NDP) under demand uncertainty. The major contribution of this paper is to propose an approach to approximate a joint chance-constrained Cell Transmission Model (CTM) based System Optimal Dynamic Network Design Problem with only partial distributional information of uncertain demand. The proposed approximation is tighter than two popular benchmark approximations, namely the Bonferroni's inequality and second-order cone programming (SOCP) approximations. The resultant formulation is a semidefinite program which is computationally efficient. A numerical experiment is conducted to demonstrate that the proposed approximation approach is superior to the other two approximation approaches in terms of solution quality. The proposed approximation approach may provide useful insights and have broader applicability in traffic management and traffic planning problems under uncertainty.
This paper reviews our recent developments on applying a robust optimization approach to transportation network design problems where future travel demands are uncertain and traffic flows on the underlying network are in user equilibrium. We assume that the travel demands belong to an uncertainty set instead of having them follow some probability distributions and then design the network against the worst-case scenario realized in the set. The problems are formulated as mathematical programs with complementarity constraints, which are efficiently solvable by a cutting-plane scheme. Numerical examples are provided to demonstrate that the designs from the robust optimization approach perform more stably and guard better against worst-case scenarios than those from traditional deterministic approach.
Many group-living animals construct transportation networks of trails, galleries and burrows by modifying the environment to facilitate faster, safer or more efficient movement. Animal transportation networks can have direct influences on the fitness of individuals, whereas the shape and structure of transportation networks can influence community dynamics by facilitating contacts between different individuals and species. In this review, we discuss three key areas in the study of animal transportation networks: the topological properties of networks, network morphogenesis and growth, and the behaviour of network users. We present a brief primer on elements of network theory, and then discuss the different ways in which animal groups deal with the fundamental trade-off between the competing network properties of travel efficiency, robustness and infrastructure cost. We consider how the behaviour of network users can impact network efficiency, and call for studies that integrate both network topology and user behaviour. We finish with a prospectus for future research.
In the present paper we are concerned with developing more realistic dynamic models of route choice and departure time decisions of transportation network users than have been proposed in the literature heretofore. We briefly review one class of models that is a dynamic generalization of the static Wardropian user equilibrium, the so-called Boston traffic equilibrium. In contrast, we then propose a new class of models that is also a dynamic generalization of the static Wardropian user equilibrium. In particular, we show for the first time that there is a variational inequality formulation of dynamic user equilibrium with simultaneous route choice and departure time decisions which, when appropriate regularity conditions hold, preserves the first in, first out queue discipline.
The equilibrium network design problem can be formulated as a mathematical program with variational inequality constraints. We know this problem is nonconvex; hence, it is difficult to solve for a globally optimal solution. In this paper we propose a simulated annealing algorithm for the equilibrium network design problem. We demonstrate the ability of this algorithm to determine a globally optimal solution for two different networks. One of these describes an actual city in the midwestern United States.
Recently, Daganzo introduced the cell transmission model--a simple approach for modeling highway traffic flow consistent with the hydrodynamic model. In this paper, we use the cell transmission model to formulate the single destination System Optimum Dynamic Traffic Assignment (SO DTA) problem as a Linear Program (LP). We demonstrate that the model can obtain insights into the DTA problem, and we address various related issues, such as the concept of marginal travel time in a dynamic network and system optimum necessary and sufficient conditions. The model is limited to one destination and, although it can account for traffic realities as they are captured by the cell transmission model, it is not presented as an operational model for actual applications. The main objective of the paper is to demonstrate that the DTA problem can be modeled as an LP, which allows the vast existing literature on LP to be used to better understand and compute DTA. A numerical example illustrates the simplicity and applicability of the proposed approach.
The papers in this special issue on transportation network design were first presented at a workshop held at the University of Illinois in September 1977 under the auspices of the Mathematical Social Science Board with the support of the National Science Foundation. The participants in the workshop included most of the leading research workers on formal models of transportation network design. The extensive discussions of the papers that ensued during the two-day workshop led to several suggestions that may be useful in stimulating new research directions and increased research activity on design models.
The Network Design Problem (NDP) has long been recognized to be one of the most difficult and challenging problems in transport. In the past two decades, we have witnessed the development of a vast, growing body of research focused on formulations and solution procedures for the NDPs, which deal with the selection of either link improvements or link additions to an existing road network, with given demand from each origin to each destination. The objective is to make an optimal investment decision in order to minimize the total travel cost in the network, while accounting for the route choice behaviour of network users. In this paper, we present a general survey of existing literature in this area, and present some new developments in model formulations. We incorporate the elasticity of travel demand into the NDP and seek the economic‐based objective function for optimization. We also pose the mixed network design problem involving simultaneous choice of link addition and capacity improvement which is considered more sensible for road networks. In addition, we introduce the network reserve capacity concept for a capacity improvement plan, and raise and clarify some interesting issues relating to NDP and Braess's paradoxes. Finally, from the survey and the new proposal made herein, we offer some perspectives on future research.
Chance constrained programming admits random data variations and permits constraint violations up to specified probability limits. Different kinds of decision rules and optimizing objectives may be used so that, under certain conditions, a programming problem not necessarily linear can be achieved that is deterministic---in that all random elements have been eliminated. Existence of such “deterministic equivalents” in the form of specified convex programming problems is here established for a general class of linear decision rules under the following 3 classes of objectives 1 maximum expected value “E model”, 2 minimum variance “V model”, and 3 maximum probability “P model”. Various explanations and interpretations of these results are supplied along with other aspects of chance constrained programming. For example, the “P model” is interpreted so that H. A. Simon's suggestions for “satisficing” can be studied relative to more traditional optimizing objectives associated with “E” and “V model” variants.
This paper examines the effects of dynamic user-equilibrium (DUE) traffic assignment with scheduled trip arrival times on network design outcomes in comparison to outcomes with steady-state travel demands. The objective is to minimize systemwide travel cost by considering alternative link improvements to an existing network (e.g., select among budget-constrained subsets of link-improvement candidates). DUE is a temporal generalization of static user-equilibrium (SUE) assignment with additional constraints to insure temporally continuous trip paths and first-in first-out (FIFO) trip ordering between all origin-destination pairs. Previous research has not investigated the effects of dynamic travel demands and schedule delay (i.e., shifts by trips to earlier or later arrival times) on network design with multiple trip origins and destinations. DUE is formulated as a bilevel program of two subproblems solved successively by an iterative algorithm that consistently converges to solutions that closely satisfy the necessary optimality conditions of this problem. Examples show the impacts of alternative combinations of network changes affecting capacities and/or free-flow travel times (e.g., ramp metering or road widening) to depend on temporal travel demands and schedule delay distributions.
In this paper we propose application of multiple criteria decision making to problems of a metropolitan network improvement plan. Initially, a bilevel multiple objective network design model is considered in two objectives which are minimal government budget and minimal total travel time of road users. We seek feasible improvement alternatives among those bottleneck links in an existing road network structure and travel demand. We present an effective heuristic algorithm to obtain noninferior solutions; then ELECTRE III multiple criteria decision making and group decision making are used to evaluate and to select a compromise solution among those noninferior solutions. From the design phase in multiple criteria decision making, multiple objective mathematical programming is used to formulate a continuous network design model. However, from the phase of evaluation, multiple criteria decision making to solve the discrete network design problem. The network of metropolitan Taipei is taken as an example to illustrate the operation of this model.
The paper considers two network optimization problems which have the following characteristics: control parameters vary continuously and network users behave according to Wardrop's first principle of traffic equilibrium ( ″user-optimization″ ). An exact algorithm based on constraint accumulation and a heuristic algorithm previously proposed in the literature.
Numerical tests are reported which indicate that, for networks with significant congestion, the heuristic is markedly more efficient than the Hooke-Jeeves algorithm which has been employed previously. The efficiency of the heuristic results from decomposition of the original problem into a set of interacting optimization subproblems. This decomposition is such that, at each iteration of the algorithm, only one user equilibrium needs to be calculated in order to update the improvement variables of all arcs of the network. This contrasts sharply with the Hooke-Jeeves algorithm which can require that a new user equilibrium be calculated each time an individual arc improvement variable is updated.
This paper gives an algorithm for L-shaped linear programs which arise naturally in optimal control problems with state constraints and stochastic linear programs (which can be represented in this form with an infinite number of linear constraints). The first section describes a cutting hyperplane algorithm which is shown to be equivalent to a partial decomposition algorithm of the dual program. The two last sections are devoted to applications of the cutting hyperplane algorithm to a linear optimal control problem and stochastic programming problems.
Numerous transportation applications as diverse as capital investment decision-making, vehicle fleet planning, and traffic light signal setting all involve some form of (discrete choice) network design. The authors review some of the uses and limitations of integer programming-based approaches to network design, and describe several discrete and continuous choice models and algorithms. The objectives are threefold - to provide a unifying view for synthesizing many network design models, to propose a unifying framework for deriving many network design algorithms, and to summarize computational experience in solving design problems. The authors also show that many of the most celebrated combinatorial problems that arise in transportation planning are specializations and variations of a generic design model.
This paper addresses the problem of determining which links should be improved in an urban road network so that total congestion in the city is minimized. A nonlinear mixed integer programming model is developed, and strategies for a branch-and-bound algorithm are presented. Particular attention is paid to the computational aspects of large-scale problems, and numerical results are reported.
In this paper we present a computationally efficient technique for determining the optimal design of an urban road network. The procedure involves the assignment of network flows and the determination of improved link parameter values so that congestion is minimized subject to a budget constraint. The resulting problem is a very large nonconvex minimization program. It is shown that by dualizing with respect to a single constraint the resulting dual objective function can be evaluated by solving a traffic assignment problem. Since the dual objective function is a concave function of one variable, effective one-dimensional search techniques based on subgradients can be utilized to solve the dual (and thus the primal network design) problem. Since this network design problem reduces to solving several traffic assignment problems, it should be efficient for realistically large networks. Computational results for several problems with up to 553 constraints and 1,862 variables are reported.
This paper is concerned with characterizing decision rules for the sequential E-model of chance-constrained programming. A key feature of our characterization will be a detailed discussion of various interpretations of the probability operator in the chance constraints.
Specifically we define two new classes of decision rules by exhibiting those sets of constraints which locally support the corresponding probability requirements. The question of how the probabilistic constraints for future periods are affected by previous decisions and realizations of the random variables is considered in detail.
Since we are primarily concerned with the feasibility of decision rules, we deal mainly with the constraints of the model. The procedure for selecting the optimum rule from among a particular class of feasible rules depends on the objective function and is briefly discussed in the final section along with some implications concerning the form of the optimum rule.
The application of our proposed rules to a two-period example previously appearing in the literature concludes the paper.
This chapter originally appeared in Management Science, April–July 1955, Vol. 1, Nos. 3 and 4, pp. 197–206, published by The
Institute of Management Sciences. This article was also reprinted in a special issue of {\em Management Science}, edited by
Wallace Hopp, featuring the “Ten Most Influential Papers of Management Science´s First Fifty Years,” Vol. 50, No. 12, Dec.,
2004, pp. 1764–1769. For this special issue George B. Dantzig provided the following commentary: “I am very pleased that my
first paper on planning under uncertainty is being republished after all these years. It is a fundamental paper in a growing
field.”
The continuous Network Design Problem (NDP) deals with determining optimal expansions for the capacities of a street network, subject to the constraint that the street traffic volumes must be the outcome of a user-optimal equilibrium assignment. Although the use of deterministic equilibrium methods tends to produce computationally intractable problems, in this paper it is shown that a stochastic user equilibrium based on the logit model leads to a differentiable and large-scale, but tractable, version of the NDP. A procedure for computing the derivatives of the stochastic user equilibrium (SUE) assignment without having to first compute the route choice probabilities is given, and this procedure is coupled with two standard algorithms for solving nonlinear programs, the generalized reduced gradient method and sequential quadratic programming. These algorithms are tested on several example networks, and the results of these tests suggest that the SUE-constrained version of the NDP offers both a promising heuristic for solving DUE-constrained problems as well as a viable procedure in its own right.
This paper is concerned with the system optimum-dynamic traffic assignment (SO-DTA) problem when the time-dependent demands are random variables with known probability distributions. The model is a stochastic extension of a deterministic linear programming formulation for SO-DTA introduced by Ziliaskopoulos (Ziliaskopoulos, A.K., 2000. A linear programming model for the single destination system optimum dynamic traffic assignment problem, Transportation Science, 34, 1–12). The proposed formulation is chance-constrained based and we demonstrate that it provides a robust SO solution with a user specified level of reliability. The model provides numerous insights and can be a useful tool in producing robust control and management strategies that account for uncertainty in applications where SO-DTA is relevant (e.g. evacuation modeling, computing alternate routes around freeway incidents and establishing lower bounds on network performance).
This article shows how the evolution of multi-commodity traffic flows over complex networks can be predicted over time, based on a simple macroscopic computer representation of traffic flow that is consistent with the kinematic wave theory under all traffic conditions. The method does not use ad hoc procedures to treat special situations. After a brief review of the basic model for one link, the article describes how three-legged junctions can be modeled. It then introduces a numerical procedure for networks, assuming that a time-varying origin-destination (O-D) table is given and that the proportion of turns at every junction is known. These assumptions are reasonable for numerical analysis of disaster evacuation plans. The results are then extended to the case where, instead of the turning proportions, the best routes to each destination from every junction are known at all times. For technical reasons explained in the text, the procedure is more complicated in this case, requiring more computer memory and more time for execution. The effort is estimated to be about an order of magnitude greater than for the static traffic assignment problem on a network of the same size. The procedure is ideally suited for parallel computing. It is hoped that the results in the article will lead to more realistic models of freeway flow, disaster evacuations and dynamic traffic assignment for the evening commute.
The optimal transportation network design problem is formulated as a convex nonlinear programming problem and a solution method based on standard traffic assignment algorithms is presented. The technique can deal with network improvements which introduce new links, which increase the capacity of existing links, or which decrease the free-flow (uncongested) travel time on existing links (with or without simultaneously increasing link capacity). Preliminary computational experience with the method demonstrates that it is capable of solving very large problems with reasonable amounts of computer time.
This article is concerned with the continuous network design problem on traffic networks, assuming system optimum traffic flow conditions and time-dependent demand. A linear programming formulation is introduced based on a dynamic traffic assignment (DTA) model that propagates traffic according to the cell transmission model. The introduced approach is limited to continuous link improvements and does not provide for new link additions. The main contribution of the article is to provide an analytical formulation for network design that accounts for DTA conditions that can be used for further analysis and extensions. The model is tested on a single destination example network, resembling a freeway corridor, for various congestion levels, loading patterns and budget sizes, to demonstrate the simplicity and effectiveness of the approach.
This paper presents a simple representation of traffic on a highway with a single entrance and exit. The representation can be used to predict traffic's evolution over time and space, including transient phenomena such as the building, propagation, and dissipation of queues. The easy-to-solve difference equations used to predict traffic's evolution are shown to be the discrete analog of the differential equations arising from a special case of the hydrodynamic model of traffic flow. The proposed method automatically generates appropriate changes in density at locations where the hydrodynamic theory would call for a shockwave; i.e., a jump in density such as those typically seen at the end of every queue. The complex side calculations required by classical methods to keep track of shockwaves are thus eliminated. The paper also shows how the equations can mimic the real-life development of stop-and-go traffic within moving queues.
In this report, we compare the computational efficiency and results of solving two alternative models for the problem of determining improvements to an urban road network. Using a 1462 link, 584 node test network of the north Dallas area, we compare a model which assumes user-optimum behavior of travelers with a model which assumes system-optimum flows. Both of these models allow improvements to the road network to take on any nonnegative value, rather than requiring discrete improvement values. Investment costs are modeled by functions with decreasing marginal costs. Unfortunately, the user-optimum model, which is much more realistic than the system-optimum one, normally cannot be solved optimally. However, the simpler system-optimum model can be optimally solved, provided that investment costs are approximated by linear functions. Thus, for this network design problem we compare an accurate representation which can be solved only approximately with an approximate representation which can be solved optimally. Our computational testing showed that the system-optimum model produces solutions as good as those from the user-optimum model, and thus seems justified when favored by other considerations, such as ease of coding, availability of "canned" programs, etc.
It is known that the network design problem with the assumption of user optimal flows can be modeled as a 0-1 mixed integer programming problem. Instead, we formulate the network design problem with continuous investment variables subject to equilibrium assignment as a nonlinear optimization problem. We show that this optimization problem is equivalent to an unconstrained problem which we solve by direct search techniques. For convex investment cost functions, the performance of both Powell's method and the method of Hooke and Jeeves is approximately the same with respect to computational requirements for a 24 node, 76 arc network. For the case of concave investment functions, Hooke and Jeeves was superior. The solution to the concave continuous model was very similar to that of the 0-1 model. Furthermore, the required solution time was far less than that required by the corresponding discrete model of the same network. The advantages and disadvantages of the continuous approach as well as the computational requirements are discussed.
A class of network topological optimization problems is formulated as a nonlinear mixed integer programming model, which can be used to design transportation and computer communication networks subject to a budget constraint. The approach proposed for selecting an optimal network consists of separating the continuous part of the model from the discrete part by generalized Benders decomposition. One then solves a sequence of master and subproblems. The subproblems of the minimal convex cost multicommodity flow type are used to generate cutting planes for choosing potential topologies by means of the master problems. Computational techniques suited to solving the master and subproblems are suggested, and very encouraging experimental results are reported.
A Taboo-Search Based Approach for Network Design
- Mouskosk
Mouskos, K. A Taboo-Search Based Approach for Network Design. Ph.D. dissertation. University of Texas, Austin, 1992.
The contents of the paper are the sole responsibility of the authors
- Wiley
Wiley, New York, 1964. The contents of the paper are the sole responsibility of the authors. FIGURE 3 Expected system performance.
- Dantzig G. C.