Journal of Advanced Transportation

Volume 2018, Article ID 4539324, 11 pages

https://doi.org/10.1155/2018/4539324

## Road Maintenance Optimization Model Based on Dynamic Programming in Urban Traffic Network

^{1}School of Transportation, Southeast University, Nanjing, China^{2}Department of Engineering, National University of Singapore, E1, No. 08-20, 3 Engineering Drive 2, Singapore^{3}Jiangsu Key Laboratory of Urban ITS, Southeast University, Nanjing, China^{4}Collaborative Innovation Center of Modern Urban Traffic, School of Transportation, Southeast University, Nanjing, China

Correspondence should be addressed to Lin Cheng; nc.ude.ues@tsig and Dawei Li; nc.ude.ues@iewadil

Received 11 December 2017; Revised 2 May 2018; Accepted 29 May 2018; Published 28 June 2018

Academic Editor: Paola Pellegrini

Copyright © 2018 Jie Ma 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

Urban road maintenance is an important part of urban traffic management. However, in modern cities, road maintenance work needs to occupy some traffic resources; therefore, unreasonable road maintenance schemes often lead traffic networks to unexpected large-scale congestion. In this paper, a dynamic programming model is proposed in order to minimize the delay caused by road maintenance scheme. This model can obtain a globally optimal maintenance scheme which contains the decisions and sequence for every stage of maintenance. Each stage of this model can be boiled down to a discrete network design problem. This model helps make suggestions for the traffic managers with the request of minimizing the delay caused by the maintenance scheme. This paper uses two examples to illustrate this method, one is a small-scale Nguyen-Dupuis network, and the other one is a larger scale Sioux-Falls network.

#### 1. Introduction

After the construction of urban network, the maintenance of urban roads is an important part of urban traffic management. As urban roads are often combined by a large number of travel demands and urban road maintenance takes up a lot of road resources, it is reasonable to develop urban road maintenance schemes.

Many departments of transportation (DOTs) use pavement management systems (PMS) to determine when to maintain the pavements. Using PMS can reduce the total financial costs of the agency; however, it still cannot deal with the delay problems efficiently. In addition, many researchers have begun to focus on other constraints of road maintenance, such as reducing environment pollutes, like greenhouse gas (GHG) [1–3].

The design of road maintenance concerns about multiple objectives. Multicriteria optimization in pavement management systems (PMS) has been used to incorporate heterogeneous objectives simultaneously [4–7]. The studies of road maintenance usually involves many fields, for example, (a) pavement planning [8]; (b) road infrastructure quality [9, 10]; (c) budgets and investments [11, 12]; (d) transportation infrastructure vulnerability [13]; (e) pavement management [14]; (f) link restriction [15]; (g) fuel efficiency [16–19]; (h) GHG emissions [20, 21]; (i) pavement design [22]; (j) separating trucks from passenger vehicles [23–25]; (k) road maintenance in extreme climate [26]; and (l) maintenance equipment [27].

With the development of cities, the urban traffic network becomes more and more complex, and the scale of urban traffic network is also growing. The urban traffic network provides daily services for a large number of travel demand and traffic users. Therefore, an unreasonable road maintenance scheme may result in large-scale traffic congestion and bring huge losses to the traffic network users. In addition, the budget of road maintenance is an important aspect of limiting the progress of road maintenance. Urban traffic network managers often need to develop optimal maintenance schemes according to the limited budget and thus need a more flexible optimization model.

To solve this problem, we propose a dynamic programming model and integrate in the proposed model a subproblem of identifying the best combination of roads to be maintained simultaneously. This subproblem considers the limited budget and the number of links to be maintained simultaneously. This subproblem is indeed a discrete network design problem (DNDP) which has been discussed in many researches. DNDP is usually formulated as a bilevel mixed-integer nonlinear programming (MINLP) which is difficult to solve. The difficulty derives from both the nonlinearity and the nonconvexity which lead it to an NP-hard problem. LeBlanc [28] proposed a branch-and-bound based method to solve DNDP. However, this method can hardly solve large-scale problems. Many heuristic algorithms were proposed to trade off the accuracy and the computational time [29–31]. In order to reduce the computational time, by using a substitution for cost function, Poorzahedy and Turnquist [32] developed a single-level model to approximate the original DNDP. Due to the property of nonlinearity and nonconvexity, the global optimal solution of DNDP is difficult to obtain. Recently, Wang DZW and Lo HK [33] transformed DNDP into a single-level programming and then linearized it into a mixed-integer linear programming (MILP). The global optimum can be obtained and the computational time relies on the linearization scheme. Wang et al. [34] proposed two global optimization solution methods for DNDP; one is SO-relaxation method and the other one is UE-reduction method. Both of them can figure out global optimum and reduced the computational time greatly. However, few published researches on DNDP concerned about the optimal maintenance scheme, although there is strong connection between DNDP and the subproblem of optimal maintenance scheme which will be illustrated in this paper.

Conventionally, DNDP is employed to optimize the topology of a traffic network [28, 34–36]. It can be used to describe the problem whether to add a new link or whether to expand the capacity of existing links in the network. However, in the problem of optimal maintenance scheme, DNDP can be used to temporarily close the best combination of link(s) in every stage of the optimal maintenance scheme. Different from the most general DNDP in literature, the subproblem, which is in the framework of DNDP, does not improve the capacity of the network at all. In fact, it reduces both the capacity of certain link(s) and the accessibility of the network temporarily for the purpose of maintenance. Besides, after each stage of maintenance is completed, the capacity of the maintained link(s) restores as before. In other words, as long as the links are not being maintained, their capacity remains unchanged. The special property differentiates our problem from the general DNDP. What is more, actually, when formulating the proposed model and solving the problem in this paper, we wisely make use of this property. We interpret this property as a guarantee of the non-aftereffect requirement of dynamic programming (DP) so that we can propose a DP for obtaining the optimal maintenance scheme. If this property was ignored, an optimal maintenance scheme could not be obtained like that. DP was first proposed and systematically illustrated by Bellman [37, 38]. Obviously, since DP provides recurrence formulas for obtaining the globally optimal solution, the global optimum of problem can be guaranteed.

This paper provides a reliable model to optimize the maintenance schemes for urban traffic network. This model allows the managers of urban networks to freely set the number of links to be maintained simultaneously according to some limits, for example, the financial budget and the schedule of maintenance. Through this model, we can obtain a road maintenance strategy aiming at minimizing the total generalized costs of the urban traffic network and minimize the influence of the maintenance on road users and provide guidance for urban road network managers. All in all, the main innovations of this paper are as follows:

This paper presents a dynamic programming model to optimize the urban road maintenance scheme, which aims at minimizing the total cost of the network and the degree of influence of road maintenance on users. By doing this, the decisions and sequence of each stage of the maintenance can be obtained.

This paper describes three critical components of the dynamic programming model: optimal substructure, state transition equation, and terminal condition. By utilizing the property of this problem, the paper splits each stage of this model into a subproblem which is a discrete network design problem (DNDP). By doing this, this model allows traffic managers to freely set the number of roads to be simultaneously maintained according to their financial budget, schedule of maintenance, etc.

Two global optimization methods are introduced to solve the subproblem. The algorithm of the subproblem is optimized based on the SO-relaxation method. This algorithm can improve the computational efficiency. With its help, the optimal maintenance scheme can be obtained quickly. Two example networks are tested.

The rest of this article is as follows. Section 2 introduces the principle of user equilibrium (UE), which is widely used in traffic assignment problem (TAP) and reminds us that the system cost at UE state cannot be lower than that at System Optimization (SO) state. Section 3 describes the method of optimizing urban road maintenance schemes and its algorithm. This method is validated by testing two examples in Section 4. We use a small-scale Nguyen-Dupuis network to verify the feasibility of this method. A larger scale Sioux-Falls network shows the efficiency of the algorithm.

#### 2. User Equalization

In this section, we introduce the principle of user equilibrium (UE). Besides, in this paper, for exposure reasons, we also employ UE in the traffic assignment problem (TAP).

All users in the network want to select the shortest path to reach the destination as far as possible, which leads the users with the same origin and destination (OD pair) to choose the same routes and results in link congestion and travel time increasing. Therefore, when the travel time increases to a certain extent, this path will no longer be the shortest path, and the users will turn to other feasible paths. As a result of choice behavior, eventually the entire network achieves a stable state, that all used paths have the same travel costs. No one can reduce their travel costs by unilaterally change his/her choice of path. Such state is called the user equilibrium state. Obviously, according to the definition of user equilibrium, the user equilibrium state has the property of spontaneity.

Besides, we should note that, when the maintenance happens, the users will respond to this change of the network within a certain time. The time depends on the level of information and perception. In modern traffic networks, we can use many existing techniques, such as information center, navigators, Connected Vehicles (CV), and Intelligent Transportation System (ITS), to obtain the real-time information about the road circumstance of a network. If the level of information and perception is not high enough, only a small part of users will change their route choice. Others may change after a certain time afterwards. But at last, maybe after a long time, all the users will reach the UE state again. Otherwise, if the level of information and perception is high enough, most users can receive the information within a very short time so that they can respond within a short time. In this case, the users can reach the UE state again quickly and probably before the maintenance is finished. Because of the increasing level of information and perception in modern cities, the UE state will be reached more and more quickly and UE models will also work better and better.

For a general urban network, because of its large scale, it is generally a complex network with lots of users. In general, the UE state keeps the users in an efficient state. However, due to the existence of the Braess paradox in networks, a poor traffic state may come out. It makes the total travel time longer than that of an SO state and all users need to spend equal but longer travel time.

#### 3. Methodology

In this section, we propose a model to solve the problem of optimal maintenance scheme.

Before proposing the model, we first introduce some notations that may be involved in the model proposed in this paper.

The following notations are used in the formulation: : the set of all link in the traffic network : the travel time on path* p* between OD pair : a vector defined as , , : the flow on path* p* between OD pair : the number of links in the traffic network : the traffic demand between OD pair , which is given as a constant : the set of all paths between OD pair : the travel time on link : a vector defined as the network design decision, , : a binary variable which indicates whether a link is a potential link which needs to be maintained: if link* a* needs to be maintained; otherwise : the number of links that manager wants to maintain simultaneously in the network : the set of all OD pairs in the traffic network : a vector defined as , : the traffic flow of link : the set of all feasible path flows** f**, : the set of all feasible network design decisions** u**, : the set of all feasible link flows** x**, *: *the link-path incidence matrix, , , , , where if link* a* belongs to path* p* between OD pair , and otherwise : a binary variable which indicates whether a path is used: path* p* is used if and is not used if

The purpose of our maintenance schemes is to minimize the delay caused by the maintenance work. Thus, we first propose a subproblem which is an optimization problem as follows. This optimization problem is to find out the best combination of link(s) to be maintained simultaneously. The objective function of the optimization is

where represents the flow on link* a* before being maintained and represents its travel time.

One should note that, in (1), the second term is a constant. Thus, (1) can be transformed into

When making maintenance schemes, due to the maintenance costs and duration, etc., it is often needed to limit the number of the links simultaneously maintained. In addition, too many links maintained at the same time may cause a sharp increase in travel time. Therefore, it is sometimes important to appropriately select the number of links to be maintained at the same time. These factors are taken into account in the design of our model so that the traffic managers can set the number of links to be maintained at the same time according to their own needs. By doing this, it can also help managers make decisions by comparing the results of setting different numbers of links simultaneously maintained.

Constraint (3) allows the traffic manager to freely set the number of links to be maintained simultaneously.* V* represents the number of simultaneously maintained links, and this value can be freely set by the manager. For example, if the number of links that manager wants to maintain simultaneously is 2, we set . is a binary variable, which represents whether link* a *needs to be maintained (e.g., if link* a* needs to be maintained, ; otherwise, ).

Note: (3) actually indicates that all links in the network are needed to be maintained. However, the proposed model is also available for the case where a scope of links is given.

In most cases, the links needed to be maintained are only a small part of the whole network. Thus, we do not need to consider all the links. Accordingly, we modify (3) into :where denotes the set of links needed to be maintained; denotes the number of links needed to be maintained.

To illustrate more clearly, here, we make use of the example of Nguyen-Dupuis (ND) network shown in later section. For example, there are 5 links needed to be maintained (e.g., in the ND network, 5-9, 7-8, 9-10, 10-11, and 12-8), and the manager wants 2 links to be maintained simultaneously due to the funds and schedule limit. Then, we set =5, and* V=*2. The other constraints and objective function keep unchanged.

is an adaptation for the case where a scope of links to be maintained is given. Details can be seen in later section. where is the capacity of link* a*;* m* is a parameter which can be set according to actual cases and often be set as 4 in general;* E* is the parameter which represents the performance change due to the maintenance work; the other parameters are the same as above.

Constraint (4) indicates that when link* a *is maintained, its cost is increased due to the change of its capacity. For example, when link* a* needs to be fully enclosed due to the maintenance scheme so that no vehicles can pass through, in this case,* E* is an infinite large number. When only parts of link* a* need to be closed, although it can still meet some parts of the traffic demand, but its capacity is limited, which results in the increased travel time. In this case,* E* can be set as a corresponding positive number in practice.

Constraint (5) is a classical representation of Wardrop’s principle which is also known as the UE principle in traffic assignment problems (TAP) [33].

Constraint (6) is the nonnegative constraint and the flow conservation constraint.

We rewrite the above mixed-integer nonlinear programming (MINLP) as follows:

*[MINLP]*

Note that the only thing which makes [MINLP] a nonlinear model is (9). Once the nonlinearity of (9) is tackled, both of the constraints and the objective function of [MINLP] are linearized. Then, [MINLP] becomes a mixed-integer nonlinear programming (MILP) which is known as a global optimization model. Here, we employ a linearization method which is proposed by Wang et al. [33]. For each link* a*, we choose a series of to partition the feasible domains of into many small regions. In each region , . Then, we linearize the first term on the right of (9) (the second term is already a linear one) by the following linear constraints:where which is a known coefficient, since is prespecified.

By doing this, [MINLP] becomes an MILP which has globally optimal solution. However, the computing efficiency of this model strongly relies on the linearization scheme. The details and the proofs can be seen in literature [33, 39]. If the high resolution of the linearization scheme is required, the computing time will become very long. This is very uneconomic especially when there are only a few links needed to be maintained. Thus, we introduce another efficient method which is also a global optimization.

We note that some of the constraints of [MINLP] are for solving TAP. Therefore, taking the advantage of them, we transform [MINLP] into a bilevel model for the road maintenance problems.

The upper-level optimization indicates that traffic managers aim to minimize the total travel time in maintenance problems by setting appropriate binary decision variables** u**.

*[OP: Original Problem]*Subject to

Constraints are equal to constraint (8).

The lower-level optimization is user equilibrium traffic assignment problem. When the network design decision changes, the traffic flow over the network may change. We assume that the demand for travel is given and fixed, and users follow the user equilibrium principle:

*[UE]*Subject to

Formulas (15)– (18) are equal to constraint (10).

One can note that both the constraints and the objective function of [UE] are convex, which means [UE] always has a unique solution. After obtaining the solution of [UE] for each decision** u**, we take it into [OP] and compare the value of objective function. After enumerating all feasible decisions, we can figure out the best solution of** u** that makes the travel time of the network minimum. Obviously, this is an enumeration method. Thus, the global optimum can be inherently guaranteed (although the solution may be not unique).

Since this is an enumeration method, it is efficient enough only in the cases where a small number of links need to be maintained. However, going through this enumeration process to calculate all the combination of more links may be absurd and occupy large amount of calculating resource. So, we present an approach to avoid the fussy calculation.

Since the number of iterations of [UE] depends on [OP] and the number of combinations of links is exponential to the number of links in [OP], [OP] is the key factor that leads to the huge computational complexity. Considering that [OP] is an MINLP problem, the calculation of [OP] itself is also a complex problem. Therefore, we try to modify the algorithm through optimizing [OP].

We use SO-relaxation method to optimize [OP]. The SO-relaxation method was first mentioned by Wang et al. [34]. This method works because of the Braess paradox which makes the total cost of a network at UE not lower than that at SO. A research of using this method to analyze Braess paradox can be seen in Ma [15].

We propose the following relaxed problem (RP) to relax the range of the variable** x** of [OP]:

*[RP]*where and other constraints of [RP] are the same as [OP].

However, in [OP]:where

Obviously, ; that is the reason why we call [RP] a relaxed problem of [OP].

One can note that, since [RP] is a relaxed problem of [OP], which means the range of variables in [RP] is larger than that in [OP]. Therefore, the optimal solution of [OP] should be the optimal solution of [RP] first. That is to say, the objective function value (total cost) of [RP] of the maintenance decision provides a lower bound for that of [OP] under this decision. The objective function value of [OP] of any maintenance decision cannot be lower than that of [RP] of the same decision. Therefore, we can first minimize the objective function of [RP] to obtain the best solution for [RP] and calculate the objective function value of [OP] of . Then we compare it with the objective function value of other solutions for [RP] (i.e., other maintenance decisions). If the objective function value of [RP] of other decisions is larger than that of [OP] of , then is the optimal solution, and [OP] of other decisions does not need to be solved. What is more, we note that [RP] is actually a single-level model. That means we can obtain its optimal solution easily. Thus, we can sort its solutions for different decisions orderly from minimum to maximum by kicking out former solutions from (with the following algorithm). By doing this, once the objective function value of a specific solution for [RP] is larger that of , all later solutions also need not to be tested, orderly from minimum to maximum

By the above method, we convert a large number of calculations of [OP] into one calculation of [RP]. The [RP] problem can be transformed into a mixed-integer linear problem by multicommodity flow (MCF) [34], which is easy to calculate by personal computer.

Based on above analysis, we propose an algorithm to solve this bilevel problem as follows:

Step 0: define a set which will include all the generated solutions for generating the cuts through (21). Define the upper bound: , incumbent optimal solution .

Step 1: solve [RP] based on the following constraint.

Constraint (22) excludes all the solutions in , so that the feasible region becomes . Constraint (22) ensures the calculating solution** u** is not a calculated one . Case 1.1: if this problem has no solutions; that is to say we have listed all feasible solutions and hence is the optimal solution and stop. Case 1.2: otherwise, obtain the provisional optimal solution denoted by , and the provisional optimal value denoted by under constraint (22). Case 1.2.1: if , is the optimal solution and stop. Case 1.2.2: otherwise (if ), go to Step 2.

Step 2: fix the value of** u** at and solve [UE] and obtain the UE link flow solution denoted by . Case 2.1: if then is not a better solution than . Then we set , and go to Step 1. (Case 2.2: otherwise (is a better solution than ), set(Case 2.2.1: if , is the optimal solution and stop. Case 2.2.2: otherwise (), set and go to step 1.

By employing [RP], the computational efficiency has been greatly improved. Lots of repeated calculations of UE are replaced by one SO process which is actually a single-level programming whose running time is negligible compared with that of [OP].

So far, we have introduced two methods whose solutions are globally optimal. The former one is efficient when links in the network are not too many. This is because more links lead to more binary variables for the linearization scheme. And the latter one is extremely suitable for the cases where there are only a few links to be maintained in a large network.

One should note that both of the above methods only have dealt with one problem: given the number of links that would be maintained at the same time, properly selecting links to make best choice according to the generalized costs (monetary costs and traffic congestion). However, our purpose is to obtain the optimal maintenance scheme (i.e., given links to be maintained, find the optimal* order* of maintains). The above two methods cannot achieve this goal directly but play an important role in achieving it.

We find that, given links to be maintained, the optimal maintenance scheme establishes on both the optimal maintenance scheme when given links to be maintained and the best combination of* V *links. Take the advantage of this rule, we propose a dynamic programming (DP) to describe our problem and then achieve our goal.

Before we give out the [DP], we first explain our problem with a diagram. Suppose we have =6 links to maintain, and we can maintain* V*=2 links at a stage. We brief the links as . A three-stage maintenance scheme can be simplified as a directed acyclic graph as shown in Figure 1.