Journal of Advanced Transportation

Volume 2018 (2018), Article ID 2738930, 18 pages

https://doi.org/10.1155/2018/2738930

## Time-Dependent Transportation Network Design considering Construction Impact

Correspondence should be addressed to Lihui Zhang; nc.ude.ujz@gnahziuhil

Received 21 August 2017; Revised 27 November 2017; Accepted 12 December 2017; Published 15 January 2018

Academic Editor: Martin Trépanier

Copyright © 2018 Yi He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A traditional discrete network design problem (DNDP) always assumes transportation infrastructure projects to be one-time events and ignores travelers’ delays caused by construction work. In fact, infrastructure construction usually lasts for a long time, and the impact on traffic can be substantial. In this paper, we introduce time dimension into the traditional DNDP to explicitly consider the impact of road construction and adopt an overtime policy to add flexibility to construction duration. We address the problem of selecting road-widening projects in an urban network, determining the optimal link capacity, and designing the schedule of the selected projects simultaneously. A time-dependent DNDP (T-DNDP) model is developed with the objective of minimizing total weighted net user cost during the entire planning horizon. An active-set algorithm is applied to solve the model. A simple example network is first utilized to demonstrate the necessity of considering the construction process in T-DNDP and to illustrate the trade-off between the construction impact and the benefit realized through capacity extension. We also solve the T-DNDP model with data from the Sioux Falls network, which contains 24 nodes, 76 links, and 528 origin-destination (O-D) pairs. Computational results for the problem are also presented.

#### 1. Introduction

Transportation plays an essential role in economic development and in the overall improvement of people’s living standard. Unfortunately, rapid population growth and the accompanying accelerated economic development in urban areas are usually accompanied by sharp increases in traffic demand, which may induce all sorts of traffic-related issues, such as congestion, air pollution, and noise pollution. Among these issues, traffic congestion is particularly a prominent problem, especially in large cities. According to the 2015 Urban Mobility Scorecard, the cumulative travel delay caused by congestion was about 7 billion hours and the wasted fuel totaled almost 3 billion gallons in 2014 alone in the United States [1]. To maintain a reasonable level of service and alleviate current and future traffic congestion, transportation agencies are often faced with the need to expand the existing road network. However, generally they must work with a limited amount of resources. Therefore, given all of the potential candidate road-widening projects, selecting appropriate projects becomes a crucial decision in maximizing the benefit of expansion projects. This kind of problem is usually referred to in the literature as the network design problem (NDP).

Over the past few decades, NDP has been widely studied. Most of the literature related to NDP has focused on either modeling or new algorithms for network design models. However, these early studies regarded road construction work as a one-time event and did not consider the gradual improvement of the network until researchers introduced the time dimension to the traditional NDP (see [2–4]). Lo and Szeto [4] claimed that the road network is improved year by year before the completion of the improvement project, which makes the NDP model more realistic. Nevertheless, even though they considered network improvement to be gradual in their model, they still assumed the construction process to be a one-off procedure. Actually, capacity expansion work usually involves work zones and lane closures, which may reduce the current link capacity during construction and result in congestion and delays for road users. Furthermore, road infrastructure construction generally lasts for months or even years, and the impact of construction may greatly affect planners’ decisions. For example, when multiple projects are simultaneously underway, planners may choose to adjust the schedule of some projects to avoid excessive delays in a region. Therefore, the impact of construction work should not simply be ignored.

This study explicitly considers construction impact in conjunction with the benefits brought about by capacity expansion as the two primary factors that govern the network design problem. Furthermore, in light of the fact that the construction process may have a tremendous impact on the road network, shortening the construction period represents a possible method for mitigating the impact. Thus, the proposed model also allows the construction period to be flexible, which means the planners can choose to speed up construction to shorten its duration by paying overtime to construction personnel.

Compared with existing NDP models, the proposed model has the following advantages:(1)The construction impact is clearly evaluated so that the selection and schedule of road infrastructure projects will be optimized.(2)This model adopts an overtime policy in the candidate projects, which allows planners to choose whether or not to accelerate a project by paying overtime. Thus, the construction durations of the candidate projects are flexible.(3)This model is able to address the problem of selecting road-widening projects from several candidate projects, simultaneously determining the optimal amount of increased capacity and designing the optimal schedule for the chosen projects.

#### 2. Background

##### 2.1. Network Design Problem

The transportation NDP aims to achieve certain objectives, for example, reducing traffic congestion, energy consumption, and environmental pollution, by choosing improvements or additions to an existing network [5]. A common methodology used to formulate the NDP is bilevel programming. The upper level is the system level, which optimizes the system benefits subject to limited resources, while the lower level is the users’ level, which models users’ route choice behavior in the network. The upper level can be formulated with different decision variables and objective functions. The decision variables can be merely continuous or discrete or can contain both continuous and discrete elements. Based on the types of decision variables, network design problems are generally divided into three categories. The network design problem with only continuous variables is called the continuous network design problem (CNDP) (see [6–11]). In road network design problems, continuous variables are usually introduced in order to simplify computation. For example, the capacity expansion of a roadway can be continuous (see [12, 13]). However, continuous variables do not necessarily indicate the changes that are practical, because road capacity is normally measured by the number of lanes. Hence, despite the fact that it may be more computationally expensive, the discrete network design problem (DNDP) with solely discrete variables (see [14–22]) and the mixed network design problem (MNDP) with both continuous and discrete variables (see [23–26]) are still worth investigating.

Previous studies have made substantial contributions to the understanding and applications of DNDP. Some have studied various applications associated with DNDP. For instance, Drezner and Wesolowsky [18] formulated a DNDP for the purpose of selecting the best distribution of one-way and two-way routes in a road network. Wu et al. [27] solved the DNDP of choosing the location of pedestrian-only streets in a multimodel network. Song et al. [28] developed a DNDP model that settled the problems of selecting locations for high-occupancy vehicle (HOV)/high-occupancy vehicle (HOT) lanes and determining toll rates on HOT lanes. Miandoabchi and Farahani [29] determined street orientations and expansions, as well as lane allocations, based on the reserve capacity concept in a DNDP model. Others have developed different kinds of approaches to solve DNDP. It is well known that solving a bilevel network design problem is very difficult because the problem is NP-hard and nonconvex. After Leblanc [15] proposed a branch-and-bound algorithm to solve this bilevel problem, many researchers began to seek better approaches to assess the trade-off between computation of speed and solution accuracy. For example, Dantzig et al. [6] transformed the nonconvex programming problem to a convex problem using system equilibrium flow to replace user equilibrium flow. Poorzahedy and Turnquist [30] utilized approximation to transform the bilevel problem into a single-level problem. Solanki et al. [31] decomposed the highway network design problem in a sequence of small subproblems and limited the search using heuristics to reduce computation time. Poorzahedy and Abulghasemi [20] adapted metaheuristic algorithms to solve NDP for the Sioux Falls network. Poorzahedy and Rouhani [32] improved the metaheuristic algorithm and designed the hybrid metaheuristic algorithm. A genetic algorithm is also widely used (see [19, 33, 34]). Gao et al. [21] transformed the upper-level programming of the traditional DNDP to a nonlinear problem based on the support function concept. Zhang et al. [35] developed the active-set algorithm, which eliminates complementarity constraints in the DNDP by assigning initial values and solving binary knapsack problems. Farvaresh and Sepehri [36] revised the branch-and-bound algorithm proposed by Leblanc [15] for bilevel DNDP.

##### 2.2. Time-Dependent Network Design Problem

In recent years, the time varying evolution of road networks began to gain interest in transportation network design problems. Different time scales were studied in the literature, ranging from the smallest day-to-day dynamics [2, 3, 37] to network upgrades spanning years [38–40]. Lo and Szeto [4] introduced the time dimension to CNDP and built a comprehensive and practical model that considered not only user equilibrium (UE), but also travel demand and land-use patterns as time-dependent. In conjunction with other researchers, they further studied a series of time-dependent NDP problems, including budget sensitivity analysis among users, private toll road operators, and the government [41]; the trade-off between the social and financial aspects of three possible network improvement strategies under demand and the value of time uncertainty [42]; the trade-off between social benefit and intergeneration equity [38]; cost recovery issues over time [40, 43]; land-use transport interaction over time [44]; sustainability with land-use transport interaction over time [45]; health impacts attributable to network construction [46]; and a multiobjective time-dependent model to determine the sequence of link expansion projects and link construction projects [47]. Time dimension was also introduced in other studies. For instance, Kim et al. [48] formulated a time-dependent DNDP framework to address the project scheduling problem; Ukkusuri and Patil [49] developed a multiperiod flexible network design model with demand uncertainty and demand elasticity; Hosseininasab and Shetab-Boushehri [50] integrated project selection and scheduling into a single time-dependent DNDP model.

However, in the literature referenced above, the road network is optimized for a certain future time without considering the construction impact. In practice, modifications to a network are gradual processes rather than one-off events. Hence, the construction process, which results in a negative impact to traffic, should also be considered. The construction process is explicitly modeled in this paper.

#### 3. Basic Considerations

This paper considers the problem of simultaneously determining the selection and scheduling of road expansion projects for a transportation network. The evaluation of a design is based on system performance throughout a given planning horizon, which includes the construction process. Below, we summarize our basic considerations and assumptions for the modeling and analysis of the construction process of road expansion projects.(1)Within the planning horizon, a road segment has at most one expansion project. This consideration is not overly restrictive, as we can always divide a road segment into several parallel links and assign each link with a project.(2)The construction procedure of an expansion project spans a continuous period of time.(3)Throughout the planning horizon, the route choice behaviors of drivers in the network follow the UE principle. Considering the construction process, the traffic network will change, as will the UE pattern.(4)The potential demand growth over time is known.(5)The interest and inflation rates are constant within the planning horizon.

For the convenience of readers, below we list some notations frequently used in the paper.

*Sets* : set of nodes : set of links : set of links with a potential expansion project : set of links without a potential expansion project : set of O-D pairs

*Parameters* : link : O-D pair : the total number of unit time intervals for the planning horizon : the total number of unit time intervals for the construction time window : time interval : travel demand between O-D pair in time interval : fixed time cost for the expansion project on link : variable time cost per additional lane for the expansion project on link : average cost per time interval during construction for the expansion project on link without overtime work

*Variables* : traffic flow on link for O-D pair in time interval : aggregate traffic flow on link in time interval : travel time of link in time interval : capacity of link in time interval : a binary variable, representing whether link is under construction in time interval . If yes, ; otherwise, : a binary variable, representing whether construction has been finished on link in time interval . If yes, ; otherwise, : a binary variable, representing whether time interval is the start date of construction on link . If yes, ; otherwise, : a binary variable, representing whether time interval is the end date of construction on link . If yes, ; otherwise, : number of newly added lanes on link : the estimated construction duration for the expansion project on link without overtime work : reduced construction duration for the expansion project on link through overtime work.

#### 4. Problem Formulation

Consider a general transportation network , where and are the set of nodes and the set of directed links, respectively. The latter are represented as a node pair , where and , or a single letter . There are two types of links in the network, the links with a potential road-widening project, and the links without a potential project, denoted as and , respectively. In this study, the planning horizon is equally divided into unit intervals. The unit interval could be a month, a season, or another reasonable time interval. Note that the unit interval is the unit of measurement of the time cost of the construction process. The planning horizon includes a construction time window and a nonconstruction time window. All construction projects are supposed to be completed within the construction time window; the nonconstruction time window is designed to evaluate the continuing benefits realized through the finished road expansion projects. Planners determine the lengths of these two time windows. Approximately, the duration of the nonconstruction time window represents the service life of the improved roads before requiring extensive renovation. For an individual project, the benefit period begins immediately after the completion of the project. Therefore, the benefit period should be at least as long as the nonconstruction time window. Let denote the number of intervals in the construction time window,

Figure 1 shows an example of the timeline of one road expansion project. In this example, the planning horizon is divided into 10 intervals, among which the former 5 intervals belong to the construction time window, and the latter 5 intervals belong to the nonconstruction time window. This project is scheduled to start at the beginning of the second time interval, and the estimated construction duration is four unit intervals. The planner decides to shorten the construction duration by one interval through overtime work. Therefore, the actual construction duration is reduced to 3 unit intervals, and the benefit lasts for 6 unit intervals (the detailed description of the flexible construction duration will be presented in the following model).