Multi-objective optimization of a road diet network design

https://doi.org/10.1016/j.tra.2011.03.005Get rights and content

Abstract

The present study focuses on the development of a model for the optimal design of a road diet plan within a transportation network, and is based on rigorous mathematical models. In most metropolitan areas, there is insufficient road space to dedicate a portion exclusively for cyclists without negatively affecting existing motorists. Thus, it is crucial to find an efficient way to implement a road diet plan that both maximizes the utility for cyclists and minimizes the negative effect on motorists. A network design problem (NDP), which is usually used to find the best option for providing extra road capacity, is adapted here to derive the best solution for limiting road capacity. The resultant NDP for a road diet (NDPRD) takes a bi-level form. The upper-level problem of the NDPRD is established as one of multi-objective optimization. The lower-level problem accommodates user equilibrium (UE) trip assignment with fixed and variable mode-shares. For the fixed mode-share model, the upper-level problem minimizes the total travel time of both cyclists and motorists. For the variable mode-share model, the upper-level problem includes minimization of both the automobile travel share and the average travel time per unit distance for motorists who keep using automobiles after the implementation of a road diet. A multi-objective genetic algorithm (MOGA) is mobilized to solve the proposed problem. The results of a case study, based on a test network, guarantee a robust approximate Pareto optimal front. The possibility that the proposed methodology could be adopted in the design of a road diet plan in a real transportation network is confirmed.

Highlights

► Optimal network design for road diet is derived based on rigorous mathematical models. ► Competition between motorists and cyclists constitutes multi-objective optimization problems. ► Braess’s paradox is confirmed for the fixed mode-share model. ► Model parameters have a great effect on the variable mode-share model.

Introduction

Many communities have recently considered reducing automobile road space and using it for safer, more robust, vital, and economically sound purposes (Burden and Lagerwey, 1999). Great attention has also been paid to bicycling as a non-motorized travel mode, since sustainability has become the most important issue in urban transportation. According to these trends, the phrase “road diet” is used in the present study to refer to a reduction in the number of vehicular lanes on a roadway cross-section, and dedicating the freed-up space to cyclists. Of course, such implementations can be found in many real locations (Ronkin, 2007, Rosales, 2007). The South Korean government is also preparing a master plan to provide a nation-wide bicycle-way network that includes exclusive bicycle lanes made possible by limiting the width of roadways (Korean Ministry of Knowledge and Economy, 2010). To justify the implementation of the project, the Korean government is now seeking information about the benefits of a road diet. In previous studies, some researchers have focused on livability enhancement due to road diets (Rosales, 2005, Rosales, 2007, Saak, 2007), while others have emphasized the safety increase after qualitative or quantitative analysis (Gates et al., 2007, Stout et al., 2006, Pawlovich et al., 2006). On the other hand, it is difficult to quantify these benefits and incorporate them into a rigorous mathematical form. Thus, the present study focuses on the benefits and risks of a road diet plan with respect to travel utility, which is easily quantified. The negative impact of a road diet plan on motorists and the positive impact on cyclists and on society as a whole are taken into account. A plausible way to assure the appropriate implementation of road diets and to minimize the possible harm to motorists is presented.

Unfortunately, in most metropolitan areas existing road space is insufficient to accommodate non-motorized traffic without negatively affecting existing car traffic. It appears very difficult to find a way to satisfy both motorists and cyclists when implementing a road diet. In such a case, it is common to inform decision-makers of the trade-off relationship, so that they can choose the preferred option on a rational basis. However, the benefit-cost trade-off relationship in the implementation of road diets has rarely been investigated in a systematic manner. The present study focuses on the question of how to allocate limited road space to different road users who compete with one another, as well as the need to provide decision-makers with as many non-dominated solutions as possible. Rigorous mathematical models are proposed to derive optimal design patterns for road diets.

The issue of finding an optimal road design pattern is frequently discussed in transportation studies. It is referred to as the network design problem (NDP), and it is used to determine the optimal sub-network, the improvement of which will optimize the entire network. Since the early stages of the NDP, most researchers have focused on a single objective function: only the benefit of motorists was taken into account with budget constraints for construction costs (Friesz, 1985, Friesz et al., 1992, Davis, 1994, Ferrari, 1995, Bell and Iida, 1997, Meng et al., 2004). As computation capability advanced and many heuristic algorithms emerged, NDP researchers turned their attention to multi-objective optimization. The viewpoints of various stakeholders with differing interests are now included in NDP formulations. For example, Chen et al. (2010) minimized both the total travel time for road users and the infrastructure cost to the government. Yin (2002) included minimization of total travel time and maximization of the revenue from private investors. Yang and Wang (2002) considered minimization of total travel time and maximization of reserve capacity. Chen and Subprasom (2007) also established a model with three objective functions, to maximize the expected social welfare of the government, maximize the expected profit of private investors, and minimize the expected inequity for road users in a build–operate–transfer project.

Although the present study adopts the basic structure of a NDP, the issue of finding the optimal road diet design is fundamentally different from that of a typical NDP, in that an optimal road diet design finds the best sub-network, each link of which has reduced capacity, while a NDP traditionally deals with capacity enhancement. Also, the network design problem for a road diet (NDPRD) takes the form of a bi-level formulation. The upper-level problem of the NDPRD is comprised of multi-objective formulations, and the lower-level problem corresponds to the user equilibrium (UE) trip assignment with either fixed or variable mode-shares.

The present study considered the objectives of different parties in the upper-level problem of NDPRD, i.e., motorists vs. cyclists and motorists vs. the government (or the entire society). Cyclists gain advantages and motorists suffer damage if a portion of road space is allocated to cyclists. The trade-off relationship between motorists and cyclists, when travel demand for each user group is assumed to be fixed, can be identified by minimizing the total travel times of each user group. The minimization of the total travel time of motorists should be regarded as the minimization of the damage that would be done to motorists by the implementation of a road diet plan. It is possible to evaluate the damage as the difference in total automobile travel times between “Do Action” and “Do Nothing” cases. If the constant value of total travel time in a “Do Nothing” case is omitted, the objective function is simply reduced to the total travel time of motorists that varies according to design variables.

The situation changes if suffering motorists switch their travel mode to the bicycle due to the limited space for automobiles. The minimization of total travel times for each traveler group is meaningless when the travel demand for each group is not constant. In this case, the effort to minimize the motorists’ share can be an objective on the part of the government or the entire society. Moreover, instead of minimizing the total travel time of motorists, maximizing the average travel speed of motorists who keep using automobiles after implementation of the road diet is a more adequate objective, and is equivalent to minimizing the average travel time per unit distance. The latter objective can be understood as guaranteeing an acceptable level of service for the remaining motorists. In summary, the variable mode-share NDPRD minimizes both the motorists’ average travel time per unit distance and the automobile mode-share. The variable mode-share model has rarely been investigated in NDP studies. Although Chen et al. (2006) adopted a variable-demand model, they did not consider a varying mode-share according to design variables. The differences between two types of trip assignment models were accommodated in the lower-level problem in the NDPRD. The variable mode-share model adopted a combined trip assignment with the mode choice, while the fixed mode-share model adopted a simple UE trip assignment.

The general NDP is known to be NP-hard, which means the problem cannot be solved within a practical computation time if the problem size grows larger (Roughgarden, 2004). Magnanti and Wong (1984) described the NDP problem as “essentially unsolvable” from a practical perspective. Furthermore, non-linearity in UE assignment makes it difficult to solve the bi-level problem as a single problem by converting the lower-level problem into constraints of the upper-level problem. This conversion entails non-convexity of the bi-level problem, which cannot be solved by standard optimization methods (Yang and Bell, 1998). Many researchers have recently focused on the merits of the evolutionary algorithm in solving a multi-objective NDP (Sharma et al., 2009, Cree et al., 1998, Yin, 2000). A genetic algorithm is often used to represent an evolutionary algorithm. This meta-heuristic potentially guarantees a global optimum by providing a means to escape local optima. That is, neighborhood solutions from an incumbent solution are allowed to move in the direction of the worse objective. The genetic algorithm is applicable for any form of problem that may be present: multi-modal objective problems, non-differentiable objective problems, or problems of a non-convex feasible region (Goldberg, 1989). Moreover, what makes the genetic algorithm superior to other rigorous algorithms is its simplicity, which requires only repetitive evaluations of an objective function. Thus, no further calculations, such as gradient or Hessian, are needed.

A diversity of solutions should be secured in the multi-objective optimization, so that a wide range of choices can be given to a decision-maker. The genetic algorithm, when applied to multi-objective problems, is very successful in approximating the optimal Pareto front and providing solutions that are as diverse as possible (Chen et al., 2006, Chen et al., 2010). A multi-objective genetic algorithm (MOGA) is adopted in the present study to solve the NDPRD. Details of the algorithm are described in the third section.

Regarding the issue of computation time, it is true that, with the exception of a few studies, network design models based on a genetic algorithm have been tested only on very small toy networks. Aggarwal and Mathew (2004) applied a genetic algorithm to solve the problem of very large transit route network design. Mathew and Sharma (2009) also applied it to solve a capacity expansion problem for an actual city (Pune, India). The city network consisted of 1131 links and 370 nodes. The two studies confirmed the potential of a genetic algorithm to obtain a high-quality solution for large network design problems. There is another merit in adopting a genetic algorithm to large network problems. The genetic algorithm is the best fit for parallel computing. In the present network design problem, computations can be divided into parts with relative ease, since the objective of a set of design variables can be evaluated independently from computing the objective of another set.

The next section presents the formulation of a NDPRD and introduces two types of models: fixed and variable mode-share. The third section describes the details of the genetic algorithm that are used to solve the NDPRD. Results from the experimental tests of the proposed models are addressed in the fourth section. In the last section, the results of the present study are summarized and conclusions are provided.

Section snippets

Formulation of the NDPRD

The NDPRD takes the form of a bi-level problem, in which the upper- and lower-levels are linked through variables. Deterministic variables of the upper-level problem are taken as fixed in the lower-level problem, and vice versa. The upper-level problem determines road diet design patterns, indicating which links should be given to cyclists when the traffic volume of each mode in every link is fixed. The traffic volume of each mode in every link is then determined in the lower-level problem,

Algorithm for solving the NDPRD

With the NDPRD, an optimal solution in the view of one party may yield unacceptable results with respect to the other. That is, it is impossible to derive a perfect multi-objective solution that simultaneously satisfies the needs of both parties. The most desirable answer for the NDPRD would be to find the Pareto optimal solution set, each solution of which is not dominated by any other solution in the solution space. For example, with Model 1, a feasible solution (x, y) is said to dominate

Experimental test

The experimental test was performed based on a test network proposed by Nguyen and Dupuis (1984), which has been widely used in NDP studies (Sharma et al., 2009, Ukksuri et al., 2007). The network was composed of 13 nodes and 19 links, as shown in Fig. 2. A fourth-order BPR function was chosen to evaluate link performance with respect to traffic volume. The link capacity was set as 2200 (vehicles/hour) and reduced by 25% on a link with a road diet. The bicycle speed was set as 8 km/h and it was

Conclusion

The present study proposed a methodology to support a decision-maker who is in charge of the design and implementation of a road diet plan. The methodology adopted the framework of the conventional multi-objective NDP, wherein two parties with different objectives competed with one another. Two models were established to accommodate the trade-off between motorists and cyclists and the trade-off between motorists and the government (or the entire society), respectively. As a solution algorithm

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