Optimizing reserve capacity of urban road networks in a discrete Network Design Problem
Highlights
► We address designing of street directions and lane additions in urban road networks. ► We study lane allocation in two-way streets with and without symmetry restrictions. ► We develop a bi-level mathematical model which maximizes network reserve capacity. ► We propose two hybrid metaheuristic algorithms to solve the problem. ► Computational results indicate that hybrid genetic algorithm performs better.
Introduction
Planning to cope with the continuous growth of travel demand in urban transportation networks is one of the most challenging and important issues that decision makers often face with. It covers a variety of decisions in transportation planning, ranging from expansion and configuration of network streets, to adjustment of traffic lights. Traditionally, this problem has been referred to as Network Design Problem (NDP). NDP by definition is the optimal decision on the expansion of a street and highway system in response to a growing demand for travel [1]. NDP ideally fits to the class of Stackelberg leader–follower games, as network authorities act as the leaders at one side and network users respond to the results of the decision made by leaders at the other side. The problem can be expressed mathematically as a bi-level programming model, where the upper and lower level problems represent decision making to maximize the social welfare and the evaluation of network performance based on user responses to network design scenarios respectively.
NDP has been extensively reviewed by Steenbrink [2], Magnanti and Wong [3], Freisz [4], Migdalas [5], and Yang and Bell [1]. As proposed in Magnanti and Wong [3], NDP can be classified into strategic, tactical and operational decision levels. The strategic level is concerned with the expansion or construction of network streets. Setting the street orientations is a typical tactical decision and finally, setting the traffic lights is an operational level decision. NDP with link expansion or construction has received a significant interest among other Network Design Problems. Classically this class of problems is considered in two forms; Discrete Network Design Problem (DNDP) which deals with adding new streets or lanes to the existing streets, and Continuous Network Design Problem (CNDP) which deals with the continuous capacity expansion of the existing streets. Another less considered problem is Mixed Network Design Problem (MNDP) which deals with both discrete and continuous network design variables. Although DNDP, CNDP and MNDP usually refer to only street construction or expansion decisions in the literature, the definitions can be broadened to include any decision with discrete or continuous variables. As an instance, signal setting or toll setting decisions can be considered as types of CNDPs. Up to now, a substantial number of NDP studies have focused on development of solutions procedures for CNDPs. Instances of some earlier studies in CNDP are Steenbrink [2], Abdulaal and LeBlanc [6], and Dantzig et al. [7]. Researches in CNDP are still being continued in the current decade (e.g. [8], [9], [10], [11], [12], [13], [14], [15]). Also DNDP has been studied by some authors (e.g. [16], [17], [18], [19], [20], [21], [22], [23]). Finally, MNDP has received less attention than CNDP and DNDP (e.g. [24], [25], [26], [27]).
Almost all of the studies in DNDP and CNDP by some exception are based on the minimization of total travel cost or travel time of users of the network. Design of transportation networks based on the concept of reserve capacity as an alternative objective function (and also problem formulation) has been recently incorporated to CNDP by some authors. Reserve capacity can be defined by the largest multiplier applied to a given existing demand matrix, that can be allocated to a network without violating the arc capacities which depend on capacity constraints, cycle times, green constraints and others. The concept of reserve capacity has been investigated previously for individual network intersections (e.g. [28], [29], [30]), and for a signal controlled network (e.g. [31]).
The use of reserve capacity as the objective function of NDPs was first suggested by Yang and Bell [1], [32]. Yang and Bell [32] studied the occurrence of capacity paradox in traditional NDPs and indicated that reserve capacity concept can be applied to avoid it. They argued that the results of total travel time minimization problems depend on the level of demand selected, and the Braess’s paradox is not revealed until a particular travel demand occurs. In contrast, reserve capacity is not sensitive to demand level, and therefore it is more favorable when uncertainty exists in traffic demands. Yang and Wang [33] investigated the relationship between the total travel cost minimization and reserve capacity maximization. They concluded that in low congested networks, the effect of reserve capacity maximization and travel cost minimization is the same, and in high congested networks, maximizing reserve capacity can only minimize the total travel cost to a certain extent.
Ziyou and Yifan [34] embedded the concept of reserve capacity in a CNDP with street expansions and traffic signal settings. Chen et al. [35] studied the concept of capacity reliability of road networks, based on reserve capacity. Ceylan and Bell [36] investigated the reserve capacity of road networks in the case of optimized fixed time signal controls. And recently, Chiou [12] presented a solution approach for simultaneously solving the maximization of reserve capacity and minimization of total travel costs in a network, by the use of continuous street expansions and signal settings.
Turning some of two-way streets into one-way streets in urban road networks is a common approach to reduce the congestion, by increasing the capacity of some streets at one direction. Network users benefit from the increased speeds and reduced travel times, if the combination of one-way and two-way streets is selected carefully. Thus, selecting a proper combination of two-way and one-way streets can contribute to the increase of the reserve capacity of the network. An NDP related line of research deals with this type of decision, i.e. finding the optimum orientations in a network with all one-way streets (e.g. [37], [38], [39], [40]). The problem of designing mixed one-way and two-way streets was studied by Lee and Yang [41], Drezner and Wesolowsky [42], and Drezner and Salhi [43], [44]. Cantarella et al. [24] considered the direction finding and the signal setting for a set of predetermined sequences of contiguous streets in terms of capacity, where all streets of each sequence must have the same lane layout i.e. all one-way in one direction or all two-way.
On the other hand, some authors have investigated the problem of street orientations with other decisions. Drezner and Wesolowsky [45] studied the simultaneous design of street constructions and street orientations. Cantarella and Vitetta [25] investigated the problem of simultaneous design of street orientations, lane allocations, and traffic signal settings. In the experimental results, they considered sequences of network links to be oriented as in [24]. The combination of street orientations, lane allocations, existing street expansions, and new street constructions, with exclusive bus lane allocations was studied in [46], and with route design for bus lines in [47].
An overview of the literature reveals that reserve capacity optimization has been only studied in CNDPs (i.e. traffic signal settings with or without street capacity expansions), and no research has been carried out to employ the capabilities of various discrete type decisions in optimizing the reserve capacity. A variety of decisions exists in DNDP which has potentials to be used for the reserve capacity optimization. As mentioned earlier, turning some two-way streets to one-way streets can help to increase the capacity of some streets. Expanding the capacity of the exiting streets is another way to directly improve the street capacities. Besides, if unequal numbers of lanes are allowed on the opposite directions of two-way streets, additional flexibility can be provided for adjusting the capacity of both sides of the streets.
The study in this paper, attempts to contribute to the literature, by embedding the above decisions, and especially determining street orientations to the notion of reserve capacity to employ their capabilities and their synergic effects to enhance it. This paper investigates a DNDP for reserve capacity maximization which simultaneously considers three decisions: (1) Determining the orientation of streets (be one-way or two-way), (2) Expanding the capacity of the existing streets by adding new lanes, and (3) Determining lane allocations in two-way streets.
Two variations are defined for the problem addressed in this paper, and are modeled using bi-level programming technique as often is used in Network Design Problems. The upper level problems are to determine the maximum reserve capacity of the network, considering street expansions, street orientations, street lane allocations, and the lower level problems are the common deterministic user equilibrium assignment problems, which compute equilibrium traffic flows for each network design scenario. Because of the intrinsic complexity and non-convexity of the mixed integer bi-level models proposed here, a hybrid genetic algorithm and an evolutionary simulated annealing solution procedure are developed to solve the problems, and their ability to achieve favorable solutions are compared using a set of test problems. To the knowledge of the authors this is the first time that evolutionary simulated annealing is being used to solve the Network Design Problem.
The rest of the paper is organized as follows. In the next section the notations and mathematical formulations of the problems are given. In Section 3, the two hybrid metaheuristics are proposed to solve the Network Design Problem under study. Section 4 contains the computational results and a comparison of two problem variations. Finally, conclusions and future research suggestions are made in Section 5.
Section snippets
Problem definition
The problem under consideration is to determine the street expansions, street orientations, and the lane allocations of an existing network, based on the reserve capacity notion. Two-way links are considered in two forms resulting in two variations of the problem: (1) two-way links have no symmetricity restrictions on lane allocations in each direction, and (2) two-way links are symmetric in terms of lane numbers in each direction. These problems are referred to as DNDP1 and DNDP2. DNDP1 is a
Mathematical models and notations
The following are the notations used in the model formulation.
Sets
N: set of network nodes.
A: set of existing network arcs.
L: set of existing network links.
Sl: set of two arcs corresponding to network link l.
W: set of all OD pairs.
Rpq: set of auto routes between OD pair (p, q).
Variables
μ: demand matrix multiplier.
yl: number of lanes added to link l.
zij: binary variable, which equals 1 if arc (i, j) is built or present, and zero otherwise.
kij: number of lanes allocated to arc (i, j).
: user
Solution procedures for the bi-level problems
In general, bi-level programming problems are NP-hard. A study conducted by Ben-Ayed et al. [49], showed that even a simple linear bi-level programming problem is still NP-hard. On the other hand, many bi-level programming problems are non-convex, and even when both upper and lower level problems are convex, there is no guarantee for the whole problem to be convex. The upper level problems (U) in Section 3 are both mixed-integer programming problems which are non-convex by their nature. The
Computational results
Since no similar problems exist in the urban network design literature to compare the computational results with, a number of commonly used test problems in NDP or traffic assignment references were selected, and after applying modifications were used to test the performance HGA and ESA. Any required missing data such as number of lanes, free flow travel times and capacities in Bureau of Public Roads (BPR) travel time function form, allowable lane additions, budget levels and etc. was provided
Conclusions and future research
In this paper, a discrete urban road Network Design Problem is investigated, which involves the concurrent design of street capacity expansions, street directions, and lane allocations for two-way streets, based on the reserve capacity maximization. Having an existing urban road network with all two-way streets, the problem is to maximize the demand multiplier that can be applied to the current travel demand matrix without violating capacity limits of links, by deciding to turn which streets to
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