Production, Manufacturing and Logistics
Robust improvement schemes for road networks under demand uncertainty

https://doi.org/10.1016/j.ejor.2008.09.008Get rights and content

Abstract

This paper is concerned with development of improvement schemes for road networks under future travel demand uncertainty. Three optimization models, sensitivity-based, scenario-based and min–max, are proposed for determining robust optimal improvement schemes that make system performance insensitive to realizations of uncertain demands or allow the system to perform better against the worst-case demand scenario. Numerical examples and simulation tests are presented to demonstrate and validate the proposed models.

Introduction

As the gap between investment needs and available funds for highway network improvements continues to grow, a critical issue facing transportation authorities today is how to allocate limited resources so as to obtain the best return for their expenditure. Often, the decisions are made without adequately taking into account uncertainty associated with future demand and supply (e.g., capacity loss due to natural disasters). As a result, road network performance may not be as good as expected or deteriorate substantially under certain realizations of future demand and supply. In some cases, in order to address the possible adverse impacts of uncertain demand and supply, sensitivity analysis is performed to evaluate the sensitivities of the decisions to the uncertainty (FHWA, 2003). However, such practices are intrinsically a posteriori or reactive, and thus provide no direct mechanism for controlling these sensitivities.

This paper is concerned with development of improvement schemes for road networks under demand uncertainty. Demand uncertainty here refers to the uncertainty of travel demand in the future, which could be attributed to uncertain developments of the socioeconomic system or prediction errors in travel demand modeling (e.g., Zhao and Kockelman, 2002) etc. The problem of interest is the following: subject to a given budget, determine which links from a network need improvement (capacity increase) and decide how much budget should be allocated to the links so as to maximize the improvements in both efficiency (total system travel time) and robustness (to be defined) of the network.

Solving for improvement schemes is essentially a network design problem. Although a vast and growing body of research on network design has been developed in the past two decades [see Yang and Bell (1998) for a survey of this topic], most of the studies were conducted without considering the impacts of uncertainty. Waller et al. (2001) pointed out that evaluation of network performance using expected demand tends to overestimate and could lead to erroneous choice of improvements. Several potential actions were discussed to deal with the problem, the simplest of which is demand inflation that yields benefits not only selecting improvements with lower expected total system travel time but also significant reductions in the variance associated with these measures. They suggested that a well-defined theory is needed for selecting a suitable inflation level. Waller and Ziliaskopoulos (2001) applied the demand inflation method in a dynamic network design problem to address demand uncertainty. They also suggested a two-stage stochastic formulation with recourse for the problem.

A few recent studies have applied the notion of reliability-based design to address the network design problem under uncertainty. Their objective is to obtain a design characterized by a low probability of failure; the approaches adopted are either to maximize a reliability measure or to introduce reliability as chance constraints while maximizing the efficiency. Examples of the former approach include Yin and Ieda (2002) for user satisfaction reliability maximization, Chootinan et al. (2005) for capacity reliability maximization, and Sumalee et al. (2006) for total travel time reliability maximization. Lo and Tung (2003) is one example of the latter approach where the reserved network capacity is maximized subject to reliability constraints on degradable networks.

In contrast to the above reliability-based design approach, this paper aims to seek for a robust design that continues to function reasonably well in the presence of demand surges or disruptive events. Robustness is a broad concept and is used frequently with different meanings. In this paper, we call an improvement scheme robust if the resulting system performance (total system travel time) is insensitive to realizations of uncertain demands (Fowlkes and Creveling, 1995) or the system performs better against the worst-case or high-consequence demand scenarios (Kouvelis and Yu, 1997). The following is an illustrative example for our definition: assuming a network confronted with only two demand levels: high and low, and each scenario has 50% probability to occur. Given two improvement schemes: A and B, with the resulting total travel times of 90 (under high demand) and 40 (under low demand) for scheme A, and 80 (high) and 50 (low) for scheme B. Although both schemes achieve the same average performance of 65, scheme B is considered to be more robust and thus more preferable, because it leads to a more stable system performance or better worst-case performance.

Robust design methods have been widely applied in other fields, such as mechanical engineering and process systems (e.g., Zang et al., 2005, Park et al., 2006). To obtain a robust optimal design, one of the prevalent approaches is to use variance (or its approximation) as a proxy for robustness of the system performance and then establish a mean–variance tradeoff. In the transportation literature, Chen et al. (2003) applied this approach to determine the optimal toll and capacity in a build-operate-transfer roadway subject to demand uncertainty. Applying the same notion, this paper proposes two new models, sensitivity-based and scenario-based, for network design under demand uncertainty. We further note that decision makers tend to be risk averse and may be more concerned with the worst-case performance. Unfortunately, variance gives equal weight to deviations above and below the mean without addressing the risks associated with extreme outcomes (List et al., 2003). As a remedy, asymmetric robustness measures, such as semivariance, could be used. Another plausible approach is to design the system using high-consequence scenarios, given that optimizing against the worst case may lead to an overly-conservative design. For example, Chen et al. (2007) adopted a risk measure called as α-value-at-risk and essentially optimized the performance against a set of demand scenarios whose collective probability of occurrence is α. In contrast, this paper applies the robust optimization approach to develop a new min–max network design model. The fundamental idea is to seek a robust design that tolerates changes of travel demand, up to a given bound known a priori. The resulting robust network will perform much better against the worst-case scenario while ensuring a near-optimal average performance.

In summary, this paper proposes three alternate models, sensitivity-based, scenario-based and min–max, for determining robust optimal improvement schemes for road networks under demand uncertainty. These models have simple structures and are computationally tractable. The models represent different perspectives or philosophies on robustness and apply different techniques to model uncertainty, thereby requiring different amount of input data.

The remaining of the paper presents the three models in sequence covering model formulation and solution algorithm, followed by another section of numerical examples for model validation and justification. Conclusions and recommendations for further research are offered in the last section.

Section snippets

Formulation preparation

Consider a network G = (N, A), where N is the set of nodes, and A is the set of links. Let W be the set of all origin–destination (O–D) pairs in the network, Rw be the set of routes between O–D pair w  W and qw be the demand between O–D pair w. To consider demand uncertainty, we assume that there exists a perturbation in the demand and the perturbed demand q(ε) is given by the following equation:qw(ε)=qw+εwwhere εw is a perturbation associated with the nominal travel demand qw. Note that the

Scenario-based model

The shortcoming of the sensitivity-based model with high levels of demand uncertainty motivates us to develop another scenario-based robust network design model. Scenario-based optimization represents uncertainty via a limited number of discrete scenarios associated with strictly positive probabilities of occurrence, and then attempts to solve the optimization problem across these scenarios for solutions that are near-optimal with respect to the population of all possible realizations of

Min–max model

Both the sensitivity-based and the scenario-based models use variance (or approximation of variance) as a proxy for robustness and attempt to establish a mean–variance tradeoff. However, variance gives equal weights to deviations above and below the mean, and the focus on the mean–variance tradeoff often fails to address the risks associated with extreme outcomes (List et al., 2003). In real-world applications, decision makers tend to be risk averse and may be more concerned with the worst

Numerical examples

This section presents numerical examples to illustrate the proposed models. The examples are based on a road network shown in Fig. 1, which is adopted from Nguyen and Dupuis (1984) and consists of 13 nodes, 19 links and 4 O–D pairs. The Bureau of Public Road link travel time function was usedta(va,ca)=ta0·1+0.15·vaca4,where ta0 is the free-flow travel time for link a.

The network characteristics and O–D demand are given in Table 1, Table 2 respectively. In these examples, the link construction

Conclusions

Three alternate models, sensitivity-based, scenario-based and min–max, respectively, have been developed to determine robust optimal improvement schemes for road networks under demand uncertainty. Numerical examples and simulation tests have been presented to demonstrate and validate the proposed models.

These three models represent different perspectives or philosophies on robustness and apply different techniques to model and address uncertainty. The models should be chosen to use based on

Acknowledgements

This research was partly funded by the California Department of Transportation. The contents of this paper reflect the views of the authors, who are responsible for the facts and the accuracy of the data presented herein. The contents do not necessarily reflect the official views or policies of the State of California. The authors would like to thank three anonymous reviewers for constructive comments that helped to improve the paper.

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