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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.

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) [

The design of road maintenance concerns about multiple objectives. Multicriteria optimization in pavement management systems (PMS) has been used to incorporate heterogeneous objectives simultaneously [

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 [

Conventionally, DNDP is employed to optimize the topology of a traffic network [

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:

The rest of this article is as follows. Section

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.

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 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

One should note that, in (

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 (

Note: (

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 (

To illustrate

Constraint (

Constraint (

Constraint (

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

Note that the only thing which makes [MINLP] a nonlinear model is (

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 [

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

Constraints

The lower-level optimization is user equilibrium traffic assignment problem. When the network design decision

Formulas (

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

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. [

We propose the following relaxed problem (RP) to relax the range of the variable

However, in [OP]:

Obviously,

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

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) [

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

Step 0: define a set

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

Constraint (

Step 2: fix the value of

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

We find that, given

Before we give out the [DP], we first explain our problem with a diagram. Suppose we have

A directed acyclic graph of the three-stage maintenance scheme.

In Figure

From the above explanation, we transform the problem into a classic shortest path problem which can be easily solved by many existing algorithms [

Now we have done all the preliminary work for proposing a dynamic programming for our problem. Since

The state transition equation is shown as follows:

The optimal substructure of [DP] is

Note, due to the definition of

(

The calculation of

To illustrate [DP] more clearly, we set

In this example, there are 5 links to be maintained. We can obtain the optimal maintenance scheme with a bottom-up process (from (

One should also note that

In the next section, we test the proposed algorithm through two numerical examples; one is for the Nguyen-Dupuis network and the other is for a larger scale network, the well-known Sioux-Falls network.

In order to clearly illustrate the proposed method, we use two different networks as the examples.

We use a personal computer with an Intel(R) Core(TM) i5 4210M @ 2.60 GHz CPU, an 8GB RAM and Windows 8.1 Enterprise operating system (64-bit) to do the numerical test. The model was coded with MATLAB and the lower-level model [UE] is dealt with Convex Combinations Method.

We testify the proposed method in Nguyen-Dupuis network. It has 13 nodes, 19 links, and 4 OD pairs. The diagram of it can be seen in Figure

The Nguyen-Dupuis network with its OD pairs and capacities.

In this example, the link performance function of each link is the BPR function, and the parameters _{a,0} and their capacities are listed in Table

Free-flow travel time and capacities of the links in Nguyen-Dupuis network.

| | _{ a } | _{ a,0 } | | | _{ a } | _{ a,0 } |
---|---|---|---|---|---|---|---|

1-5 | 1 | 900 | 7 | 8-2 | 11 | 700 | 10 |

1-12 | 2 | 700 | 8 | 9-10 | 12 | 700 | 10 |

4-5 | 3 | 700 | 9 | 9-13 | 13 | 600 | 9 |

4-9 | 4 | 900 | 14 | 10-11 | 14 | 700 | 8 |

5-6 | 5 | 800 | 5 | 11-2 | 15 | 700 | 9 |

5-9 | 6 | 600 | 9 | 11-3 | 16 | 700 | 8 |

6-7 | 7 | 900 | 5 | 12-6 | 17 | 300 | 7 |

6-10 | 8 | 500 | 13 | 12-8 | 18 | 700 | 15 |

7-8 | 9 | 300 | 5 | 13-3 | 19 | 700 | 11 |

7-11 | 10 | 400 | 9 |

We apply the proposed method to the Nguyen-Dupuis network. The test results can be seen in Table

Feasible solutions of the first stage of the optimal maintenance scheme.

| | ||||
---|---|---|---|---|---|

| | | | | |

1 | (9-10) | 9.6664E+03 | 1 | (10-11) | 1.0548E+04 |

2 | (10-11) | 1.0548E+04 | 2 | (12-8) | 1.7192E+04 |

3 | (12-8) | 1.7192E+04 | 3 | (5-9) | 2.4606E+04 |

4 | (5-9) | 2.4606E+04 | 4 | (7-8) | 2.8514E+04 |

5 | (7-8) | 2.8514E+04 | |||

| |||||

| |||||

| | | | | |

| |||||

1 | (7-8,12-8) | 1.0110E+04 | 6 | (5-9, 12-8) | 1.7224E+04 |

2 | (9-10, 10-11) | 1.0548E+04 | 7 | (9-10, 12-8) | 2.1207E+04 |

3 | (5-9, 9-10) | 1.2044E+04 | 8 | (7-8, 9-10) | 2.5950E+04 |

4 | (12-8, 10-11) | 1.6357E+04 | 9 | (5-9, 7-8) | 3.7239E+04 |

5 | (5-9, 10-11) | 1.6828E+04 | 10 | (7-8, 10-11) | 4.7624E+04 |

For exposure reasons, we show details of the first stage of our maintenance. However, since both of [MINLP] and [RP] are models which have already been optimized, they can only give out the best selection of

According to the first column

Optimal maintenance scheme of the Nguyen-Dupuis network.

| | | |
---|---|---|---|

1 | (7-8, 12-8) | 1.0110E+04 | 1.0110E+04 |

2 | (5-9, 9-10) | 1.2044E+04 | 2.2154E+04 |

3 | (10-11) | 1.0548E+04 | 3.2702E+04 |

From the test results, we can see that the optimal maintenance scheme is (7-8, 12-8), (5-9, 9-10), and (10-11) in order. One may note that the cost of the

We test the computational efficiency of the proposed method with Sioux-Falls network, which has 24 nodes, 76 links, and 528 OD pairs as Figure

Sioux-Falls network.

As analyzed in the former section, the time complexity of [DP] is

The computational cost of

For a large

In this example, the link travel time functions are BPR functions; the parameters

Test results of the calculation of

| | | |
---|---|---|---|

1 | 76 | (51) | 152 |

2 | 2850 | (30, 51) | 166 |

3 | 70300 | (30, 31, 51) | 502 |

Along with

In the case where 10 links are needed to be maintained, even if

In this paper, we propose a dynamic programming model [DP] to optimize the urban road maintenance scheme. This model allows maintenance managers to freely set the number of links to be simultaneously maintained in each stage of the maintenance scheme. Through this model, the decisions and sequence of each stage of the maintenance can be obtained.

We describe 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).

The subproblem can be solved by two global optimization methods [MINLP] and [OP] whose computational efficiency matters the running time of [DP]. The computational efficiency of [MINLP] relies on the linearization scheme employed. And [OP] can be optimized by SO-relaxation method. With its help, the running time of the proposed model is reduced sharply and the optimal maintenance scheme can be obtained quickly.

We test this model in two networks, one is a small-scale Nguyen-Dupuis network, and the other one is a larger scale Sioux-Falls network. Through these two examples, both the validity and the efficiency of this method are verified. This method can be applied in large-scale networks.

In this paper, the maintenance time is described as stages. In future research, we will consider unsynchronized maintenance time which will definitely make the problem more complex but more practical. However, since maintenance is a simple work in our modern cities. In general, only a few days are needed. And the times consumed for maintaining different links usually do not differ too much. Thus, the proposed model is also practical to some degree. It can help the urban traffic managers to figure out optimal road maintenance schemes effectively and efficiently.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research is supported by the National Natural Science Foundation of China (nos. 51608115, 51578150, and 51378119), Projects of International Cooperation and Exchange of the National Natural Science Foundation of China (no. 51561135003), the Natural Science Foundation of Jiangsu Province (no. BK20150613), and the Scientific Research Foundation of the Graduate School of Southeast University (no. YBJJ1840).