# Modeling the Optimal Maintenance Scheduling Strategy for Bridge Networks

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## Abstract

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## Featured Application

**This work proposes a method framework to improve the efficiency of maintenance scheduling for bridge networks under resource constraints, which may provide a potential aid for bridge management.**

## Abstract

## 1. Introduction

## 2. Literature Review

## 3. Methodology

#### 3.1. Assumption

- (1)
- Every single bridge is closed when its maintenance activity is ongoing, i.e., the traffic capacity of the bridge that is being maintained drops to zero.
- (2)
- Users can obtain the maintenance schedule of the bridge network in advance according to which users choose their routes.
- (3)
- All the traffic demands are fixed during the period of maintenance activities.

#### 3.2. Notation

**Sets and indices**

A | Set of all links in the highway network, indexed by $a\in A$. |

I | Set of bridges to be maintained or set of maintenance activities, indexed by $i\in I$. |

S | Set of all crews, indexed by $s\in S$. |

W | Set of origin-destination (OD) pairs, indexed by $w\in W$. |

${K}_{w}$ | Set of paths that connect OD pair w, indexed by $k\in {K}_{w}$. |

**Parameters**

${d}_{i}$ | Duration of the maintenance activity i. |

${c}_{i}$ | Maintenance cost of bridge i. |

B | Availability of budget. |

T | Length of a discrete time period. |

t | Time period, $t=1,2,\cdots ,T$. |

${q}_{w}$ | Traffic demands of OD pair w. |

${l}_{a}$ | Length of link a. |

${V}_{0a}$ | Pre-maintenance speed limit of link a. |

$\alpha ,\beta $ | Parameters of BRP function. BRP function proposed by U.S. Bureau of Public Roads is used to formulate the relationship between travel time and traffic flows. $\alpha $ is the correction factor and $\beta $ indicates the growth rate of travel time |

${C}_{0a}$ | Pre-maintenance traffic capacity of link a. |

${q}_{a}^{*}$ | Pre-maintenance traffic flow on link a under user equilibrium (UE) state. UE state is reached when, for each OD pair, the actual route travel time experienced by travelers within a traffic network is equal and minimal. |

**Variables**

Z | Cumulative traffic delays in the network during the makespan. |

${u}_{0a}$ | Pre-maintenance free-flow travel time of link a. |

M | Makespan of the maintenance scheduling scheme. |

${U}_{t}$ | Travel time in the network at time t. |

N | Number of bridges maintained. |

${U}^{*}$ | Pre-maintenance travel time in the network. |

c | Total maintenance costs. |

${q}_{a}^{t}$ | Traffic flow on link a at time t. |

${C}_{a}^{t}$ | Traffic capacity of link a at time t. If the bridge i maintained at time t lies on link a, ${C}_{a}^{t}=0$; otherwise, ${C}_{a}^{t}{=C}_{0a}$. |

${u}_{a}^{t}\left(q\right)$ | Travel time of link a at time t. |

${h}_{k}^{w,t}$ | Traffic flow on the path k that connects OD pair w at time t. |

${x}_{\mathit{ist}}$ | Binary variable, which means if the maintenance activity of bridge i is implemented by crew s and starts at time t, ${x}_{\mathit{ist}}=1$; otherwise, ${x}_{\mathit{ist}}=0$. |

${\delta}_{a,k}^{w,t}$ | Binary variable, which is defined as ${\delta}_{a,k}^{w,t}=$1, if link a lies on path k that connects OD pair w; otherwise, ${\delta}_{a,k}^{w,t}=$0. |

#### 3.3. Modeling

**The upper-level model**

**The lower-level model**

#### 3.4. Model Solution

#### 3.4.1. Single-Level Model

#### 3.4.2. Simulated Annealing Algorithm

## 4. Case Study

#### 4.1. Testing Network and Basic Data

#### 4.2. Results

## 5. Discussion

#### 5.1. Traffic Demand

#### 5.2. Number of Crews

#### 5.3. Availability of Budget

#### 5.4. Decision Maker’s Preference

## 6. Conclusions

- (1)
- Compared with the two empirical strategies, i.e., FFSS and WFSS, the optimal maintenance scheduling strategy generated by the proposed method has an advantage in saving maintenance cost, reducing traffic delays, minimizing makespan and can provide a potential aid for transportation agencies in making efficient maintenance decisions.
- (2)
- Traffic demand has a significant impact on the optimal maintenance scheduling strategy including time sequence and job sequence. For the bridge network with heavy traffic, avoiding simultaneous maintenance activities of bridges on two parallel links can reduce traffic delays during the maintenance period.
- (3)
- More crews can shorten the entire maintenance period apparently, but cannot always guarantee minimal traffic delays. Because more crews mean more bridges being maintained simultaneously, which aggravate traffic congestion. Only when the manpower and funds match can the optimal maintenance scheduling strategy be improved.
- (4)
- Decision maker’s preference also affects both the time sequence and job sequence. Hence, decision makers should consider a reasonable tradeoff between the traffic delays and the maximum number of bridges to be maintained.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Results of the OMSS. (

**a**) Time sequence of maintenance activities; (

**b**) Job sequence of crews.

**Figure 4.**Results of the FFSS. (

**a**) Time sequence of maintenance activities; (

**b**) Job sequence of crews.

**Figure 5.**Results of the WFSS. (

**a**) Time sequence of maintenance activities; (

**b**) Job sequence of crews.

a | ${\mathit{l}}_{\mathit{a}}\left(\mathbf{km}\right)$ | ${\mathit{V}}_{0\mathit{a}}(\mathbf{km}/\mathbf{h})$ | ${\mathit{C}}_{0\mathit{a}}(\mathbf{veh}/\mathbf{day})$ | a | ${\mathit{l}}_{\mathit{a}}\left(\mathbf{km}\right)$ | ${\mathit{V}}_{0\mathit{a}}(\mathbf{km}/\mathbf{h})$ | ${\mathit{C}}_{0\mathit{a}}(\mathbf{veh}/\mathbf{day})$ |
---|---|---|---|---|---|---|---|

1–2 | 5.7 | 80 | 3200 | 2–1 | 5.7 | 80 | 3200 |

1–14 | 9.3 | 70 | 2800 | 14–1 | 9.3 | 70 | 2800 |

2–3 | 6.5 | 60 | 2400 | 3–2 | 6.5 | 60 | 2400 |

2–13 | 6.4 | 70 | 2800 | 13–2 | 6.4 | 70 | 2800 |

3–4 | 10.5 | 80 | 3200 | 4–3 | 10.5 | 80 | 3200 |

3–10 | 5.2 | 70 | 3000 | 10–3 | 5.2 | 70 | 3000 |

3–11 | 4.8 | 60 | 2400 | 11–3 | 4.8 | 60 | 2400 |

4–5 | 6.3 | 70 | 2600 | 5–4 | 6.3 | 70 | 2600 |

4–6 | 1.3 | 60 | 2600 | 6–4 | 1.3 | 60 | 2600 |

5–7 | 5.9 | 80 | 3200 | 7–5 | 5.9 | 80 | 3200 |

5–8 | 6.1 | 60 | 2600 | 8–5 | 6.1 | 60 | 2600 |

6–7 | 1.1 | 70 | 2800 | 7–6 | 1.1 | 70 | 2800 |

6–10 | 6.8 | 60 | 2600 | 10–6 | 6.8 | 60 | 2600 |

7–8 | 4.2 | 60 | 2400 | 8–7 | 4.2 | 60 | 2400 |

8–9 | 6.3 | 80 | 3600 | 9–8 | 6.3 | 80 | 3600 |

9–10 | 6.7 | 60 | 2200 | 10–9 | 6.7 | 60 | 2200 |

9–12 | 5.8 | 80 | 3600 | 12–9 | 5.8 | 80 | 3600 |

11–12 | 5.5 | 60 | 2200 | 12–11 | 5.5 | 60 | 2200 |

11–13 | 6.3 | 60 | 2200 | 13–11 | 6.3 | 60 | 2200 |

12–14 | 7.4 | 80 | 3600 | 14–12 | 7.4 | 80 | 3600 |

13–14 | 5.6 | 70 | 3000 | 14–13 | 5.6 | 70 | 3000 |

OD | ${\mathit{q}}_{\mathit{w}}\left(\mathbf{veh}\right)$ | OD | ${\mathit{q}}_{\mathit{w}}\left(\mathbf{veh}\right)$ | OD | ${\mathit{q}}_{\mathit{w}}\left(\mathbf{veh}\right)$ | OD | ${\mathit{q}}_{\mathit{w}}\left(\mathbf{veh}\right)$ |
---|---|---|---|---|---|---|---|

1–5 | 560 | 3–8 | 1200 | 5–11 | 840 | 13–4 | 920 |

1–9 | 750 | 3–14 | 550 | 6–14 | 620 | 14–5 | 610 |

1–11 | 680 | 4–1 | 1160 | 8–2 | 780 | 11–5 | 1300 |

2–5 | 1260 | 4–14 | 950 | 8–11 | 1450 | 12–1 | 1240 |

2–8 | 1120 | 5–1 | 1100 | 9–1 | 750 | 12–4 | 780 |

i | ${\mathit{d}}_{\mathit{i}}\left(\mathbf{day}\right)$ | ${\mathit{c}}_{\mathit{i}}(\mathbf{Fund}-\mathbf{Unit})$ | i | ${\mathit{d}}_{\mathit{i}}\left(\mathbf{day}\right)$ | ${\mathit{c}}_{\mathit{i}}(\mathbf{Fund}-\mathbf{Unit})$ |
---|---|---|---|---|---|

A | 15 | 260 | H | 13 | 230 |

B | 12 | 240 | I | 15 | 120 |

C | 11 | 170 | J | 7 | 190 |

D | 17 | 280 | K | 12 | 230 |

E | 9 | 260 | L | 5 | 210 |

F | 6 | 190 | N | 16 | 140 |

G | 11 | 180 |

i | a | ${\mathit{q}}_{\mathit{a}}^{*}(\mathbf{veh}/\mathbf{day})$ | R | i | a | ${\mathit{q}}_{\mathit{a}}^{*}(\mathbf{veh}/\mathbf{day})$ | R |
---|---|---|---|---|---|---|---|

A | 1–2 | 3647 | 7 | H | 8–9 | 1377 | 9 |

2–1 | 2752 | 9–8 | 1491 | ||||

B | 2–3 | 1231 | 10 | I | 9–10 | 1973 | 1 |

3–2 | 1127 | 10–9 | 1681 | ||||

C | 3–4 | 1486 | 3 | J | 11–13 | 3007 | 6 |

4–3 | 1265 | 13–11 | 2461 | ||||

D | 3–4 | 2065 | 13 | K | 12–14 | 2091 | 2 |

4–3 | 2329 | 14–12 | 2177 | ||||

E | 4–5 | 1863 | 12 | L | 1–14 | 4210 | 11 |

5–4 | 1463 | 14–1 | 2806 | ||||

F | 5–7 | 2149 | 4 | N | 1–14 | 3157 | 8 |

7–5 | 1432 | 14–1 | 3859 | ||||

G | 6–10 | 2852 | 5 | ||||

10–6 | 2530 |

Maintenance Scheduling Strategy | OMSS | FFSS | WFSS |
---|---|---|---|

c | 1450 | 1480 | 1490 |

Z | 21,978 | 23,243 | 22,366 |

M | 25 | 32 | 28 |

N | 7 | 7 | 6 |

τ | −30% | 0 | 30% | 62% | 86% | 100% |
---|---|---|---|---|---|---|

c | 1450 | 1450 | 1450 | 1450 | 1430 | 1430 |

Z | 18,022 | 21,978 | 26,374 | 32,962 | 41,758 | 52,945 |

M | 25 | 25 | 25 | 25 | 24 | 24 |

N | 7 | 7 | 7 | 7 | 6 | 6 |

OMSS | Crew 1: B, H Crew 2: L, E, G Crew 3: N, J | Crew 1: B, H Crew 2: G, L, E Crew 3: N, J | Crew 1: B, K Crew 2: E, H Crew 3: D, J |

S | 1 | 3 | 5 |

c | 1450 | 1450 | 1450 |

Z | 45,153 | 21,978 | 22,537 |

M | 73 | 25 | 16 |

N | 7 | 7 | 7 |

OMSS | Crew 1: B, L, N, E, H, G, J | Crew 1: B, H Crew 2: L, E, G Crew 3: N, J | Crew 1: B Crew 2: L, G Crew 3: N Crew 4: E, J Crew 5: H |

B | 1500 | 2000 | 2500 |

c | 1450 | 1970 | 1970 |

Z | 21,978 | 39,158 | 39,158 |

M | 25 | 37 | 37 |

N | 7 | 10 | 10 |

OMSS | Crew 1: B, H Crew 2: L, E, G Crew 3: N, J | Crew 1: B, E, N Crew 2: C, G, I Crew 3: H, J, K, L | Crew 1: B, E, N Crew 2: C, G, I Crew 3: H, J, K, L |

ρ. | 0.3 | 0.5 | 0.7 |
---|---|---|---|

c | 1450 | 1450 | 1260 |

Z | 29,851 | 21,978 | 19,657 |

M | 29 | 25 | 22 |

N | 7 | 7 | 6 |

OMSS | Crew 1: B, N Crew 2: C, D Crew 3: G, H, L | Crew 1: B, H Crew 2: L, E, G Crew 3: N, J | Crew 1: C, G Crew 2: E, K Crew 3: H, J |

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## Share and Cite

**MDPI and ACS Style**

Mao, X.; Jiang, X.; Yuan, C.; Zhou, J.
Modeling the Optimal Maintenance Scheduling Strategy for Bridge Networks. *Appl. Sci.* **2020**, *10*, 498.
https://doi.org/10.3390/app10020498

**AMA Style**

Mao X, Jiang X, Yuan C, Zhou J.
Modeling the Optimal Maintenance Scheduling Strategy for Bridge Networks. *Applied Sciences*. 2020; 10(2):498.
https://doi.org/10.3390/app10020498

**Chicago/Turabian Style**

Mao, Xinhua, Xiandong Jiang, Changwei Yuan, and Jibiao Zhou.
2020. "Modeling the Optimal Maintenance Scheduling Strategy for Bridge Networks" *Applied Sciences* 10, no. 2: 498.
https://doi.org/10.3390/app10020498