# An Improved Mathematical Model for Green Lock Scheduling Problem of the Three Gorges Dam

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## Abstract

**:**

## 1. Introduction

- How can the carbon emissions related to the lockage process be reduced through an improved and green navigation scheduling at the Three Gorges Dam?
- Whether a green navigation scheduling can reduce the total waiting time and improve the overall efficiency.

## 2. Literature Review

#### 2.1. Vessel Scheduling

#### 2.2. Lock Scheduling

#### 2.3. Summary

- We proposed a new mathematical model for the green lock scheduling problem of the Three Gorges Dam in order to reduce the total carbon emissions and improve the lockage efficiency, fairness and economic performance.
- In the model formulation, we incorporate both vessel scheduling problem and lock scheduling problem in order to fully address the characteristics of the green lock scheduling problem of the Three Gorges Dam.
- We obtain managerial implications that can be adopted in order to improve the navigation scheduling at the Three Gorges Dam.

## 3. Problem Description

## 4. Mathematical Model

#### 4.1. Fuel Consumption

- v
_{i}—Speed of vessel i (kilometer per hour); - w
_{i}—Payload of vessel i (tonnes); - a
_{i}—The weight of the vessel i if empty (tonnes); - k, p, q—Constants, and k$>$ 0, p $\ge $ 0 and q $\ge $ 3.

- ${F}_{i}^{1}$—The daily fuel consumption of vessel i in step 1 (tonnes/day).

- ${F}_{if}^{2}$—The daily fuel consumption of vessel i in step 2 (tonnes/day);

#### 4.2. Carbon Emissions

- CE
_{i}—the carbon emissions of vessel i (tonnes/day); - α
_{co2}—carbon coefficient.

#### 4.3. Chamber Capacity and Vessel Arrangement

- ${G}_{f}^{i}$—Binary variable, if ${G}_{f}^{i}$ = 1, vessel i is in chamber f, and ${G}_{f}^{i}$ = 0 otherwise;
- x
_{i}, y_{i}—Variables that define the x and y position of vessel i in the chamber; - l
_{i}, d_{i}—Length and width of vessel i (meter); - D
_{2}, L_{2}—Length and width of the lock chamber (meter).

_{f}is used to represent the number of vessels in chamber f, which decides the dimension of the particle in particle swarm optimization (PSO).

#### 4.4. Expected Cost

- P
_{1}—Cost of fuel per unit (CNY/tonne); - P
_{2}—Multiplier of the additional cost in waiting (CNY/hour); - C
_{i1}—Expected fuel cost of carrier i (CNY); - C
_{i2}—Expected additional cost of carrier i in waiting (CNY); - ${RT}_{i}^{A}$—Real arrival time of vessel i at the anchorage;
- ${RT}_{i}^{D}$—Real departure time of vessel i from the anchorage.

_{D}) is implemented for the delay on departure of vessels parking at the anchorage after an order has been given by the TGNA, and this may influence the planning of the lockage operations. Compared with the arrival delay, the influence of the departure delay on the overall efficiency of the navigation scheduling and the lockage operations is much more serious, so the penalty on the departure delay should be higher than that of the arrival delay.

- P
_{A}—Arrival delay penalty per unit (CNY/hour); - P
_{D}—Departure delay penalty per unit (CNY/hour); - C
_{i3}—Expected arrival delay of carrier i (CNY); - C
_{i4}—Expected departure delay of carrier i (CNY); - ${PT}_{i}^{A}$—Estimated arrival time of vessel i at the anchorage;
- ${DT}_{I}^{D}$—Scheduled departure time of vessel i from the anchorage
- Expected cost
_{i}—Total expected cost of carrier i (CNY).

#### 4.5. The Mathematical Model for Green Lock Scheduling Problem

**Decision variables:**

${v}_{i}$ | Speed of vessel i. (km/hour) |

${K}_{ij}$ | Binary variable, if K_{ij} = 1, vessel i directly follows vessel j, and K_{ij} = 0Otherwise. |

${G}_{f}^{i}$ | Binary variable, if ${G}_{f}^{i}$ = 1, vessel i is in chamber f, and ${G}_{f}^{i}$ = 0 otherwise. |

$D{T}_{i}^{D}$ | Scheduled departure time of vessel i from the anchorage. (hour) |

$S{C}_{1}^{i}$ | Scheduled time for vessel i in the first chamber. (hour) |

**Sets and Parameters:**

$S$ | Set of vessels scheduled on a daily basis, i, j ∈ S |

${N}_{s}$ | Number of elements in set S |

$F$ | Set of chambers, f ∈ F |

${W}_{i}$ | The total weight of vessel i. (tonnes) |

${F}_{3}^{i}$ | The daily fuel consumption of vessel i in chamber. (tonnes/day) |

$T{C}_{f}^{i}$ | Real time for vessel i in chamber f. (hour) |

${T}_{start}$ | Starting time of navigation scheduling. (hour) |

${T}_{end}$ | Required end time of the scheduling period. (hour) |

${t}_{r}$ | Upper limit of waiting time at anchorage. (hours) |

${t}_{i}^{ber}$ | The waiting time of vessel i at the anchorage. (hours) |

${t}_{1}$ | Overall operation time in a chamber. (hours) |

${L}_{1}$ | Distance from the anchorage to the pier. (km) |

${v}_{2}$ | Limited speed in the lock chamber. (km/hour) |

$U$ | A constant specifying the relationship between weight and speed of a vessel |

$Safet{y}_{dis}$ | Safety departure interval for two connecting vessels in sequence. (hours) |

## 5. Solution Approach

#### 5.1. Greedy Algorithm

#### 5.2. Particle Swarm Optimization (PSO)

_{i}= (x

_{i,1}, x

_{i,2},…, x

_{i,Qf}) and V

_{i}= ( v

_{i,1}, v

_{i,2},…, v

_{i,Qf}). Each particle has their fitness that is determined by the objective function. The fitness measures the distance between the position of the particle and the current best position in the search space identified by other particles. In each iteration, a particle knows the best position in the search space (a local optimum) and its current position, based on which it determines the next movement towards the local optimum. The mathematical model depicting the movement of particles is given as follows [42]:

- p
_{i,j}—Maximum fitness value of the individual i; - p
_{g,j}—Maximum fitness value of the whole population; - w—Inertia weight;
- c
_{1}, c_{2}—Positive learning factor; - r
_{1}, r_{2}—Uniformly distributed random numbers between 0 and 1.

#### 5.3. Greedy-Particle Swarm Optimization (G-PSO)

## 6. Numerical Experiments

_{1}, c

_{2}) are 2; the number of iterations (t) is 100; and the inertia weight (w) is 0.5.

## 7. Result and Discussion

#### 7.1. Speed Adjustment

#### 7.2. The Correlation between Vessel Weight and Carbon Emission

#### 7.3. Reduction on Waiting Time and Improvement of the Fairness

#### 7.4. Managerial Implications

- The carbon emissions from the vessels in passing through the Three Gorges Dam may be greatly reduced by optimizing the navigation scheduling. The reduction on carbon emissions of the overall lockage process is mainly contributed by the speed adjustment in the sailing from the anchorage to the gate of the ship lock.
- In the optimized planning, the overall efficiency of the navigation scheduling is not affected by the speed adjustment of the vessels due to the reduction on the average waiting time at the anchorage.
- Due to the proportionality between the cost and fuel consumption given in Section 4.4. The speed adjustment of vessels may also reduce the cost related to fuel consumptions, and the reduction on waiting time may minimize the risk of good damage at the anchorage.
- The fairness of the navigation scheduling may also be improved through the implementation of the rule of FCFS and the reduction on the difference between the waiting time of different vessels.

## 8. Conclusions

- The average area utilization rate (AUR) and lockage times of lock chamber problem in step 3 has not explicitly formulated in the current model. Besides, the cooperation between the ship lift and the ship lock is not taken into account. Thus, for further improvement, the modeling efforts may be done in order to improve the formulation in Step 3.
- Even if the delay penalty is formulated in this paper, it has not been thoroughly tested in the experiments due to the lack of relevant information. Thus, future research may be conducted in order to test different economic means (e.g., different levels of delay penalty) for ensuring the efficiency of the navigation scheduling.
- In addition, the navigation scheduling may be disrupted by unexpected events, i.e., accidents happened in the lockage process, equipment malfunction, etc., so the development of a decision-support tool for reactive strategy is of interest.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**A simplified instance of vessel sequence and lockage process at the Three Gorges Dam (Northern ship lock).

**Figure 5.**Comparison of the carbon emissions between original navigation scheduling and optimized navigation scheduling.

**Figure 6.**Comparison of carbon emission between the original navigation scheduling and optimized navigation scheduling in: (

**a**) Step 1; (

**b**) Step 2.

**Figure 7.**The correlation between the speed and normalized weight of each vessel in the optimized navigation scheduling.

**Figure 9.**Comparison of vessel sequence in both original navigation scheduling and optimized navigation scheduling.

Number | Arrival Time (hh:mm:ss) | Weight (Tonnes) | Length (Meter) | Width (Meter) |
---|---|---|---|---|

1 | 00:33:00 | 3878 | 56 | 25 |

2 | 01:55:00 | 4894 | 56 | 26 |

3 | 02:03:00 | 6496 | 60 | 38 |

4 | 02:22:00 | 3985 | 56 | 27 |

5 | 02:26:00 | 6317 | 60 | 32 |

6 | 04:08:00 | 4019 | 54 | 28 |

7 | 04:24:00 | 4906 | 57 | 23 |

8 | 04:29:00 | 6465 | 59 | 39 |

9 | 05:51:00 | 4881 | 56 | 20 |

10 | 05:54:00 | 6392 | 56 | 30 |

11 | 06:14:00 | 6457 | 60 | 38 |

12 | 07:21:00 | 6489 | 65 | 49 |

13 | 07:49:00 | 3709 | 56 | 22 |

14 | 08:08:00 | 3739 | 56 | 32 |

15 | 08:10:00 | 3599 | 56 | 31 |

16 | 09:16:00 | 3983 | 56 | 23 |

17 | 10:07:00 | 3972 | 57 | 21 |

18 | 10:25:00 | 4917 | 59 | 24 |

19 | 11:11:00 | 3853 | 56 | 32 |

20 | 12:11:00 | 6322 | 58 | 42 |

21 | 12:20:00 | 6421 | 65 | 41 |

22 | 12:31:00 | 6496 | 58 | 26 |

23 | 13:00:00 | 3672 | 55 | 33 |

24 | 13:46:00 | 5884 | 65 | 38 |

25 | 13:48:00 | 3883 | 56 | 29 |

26 | 14:05:00 | 4904 | 57 | 32 |

27 | 14:52:00 | 5740 | 57 | 19 |

28 | 15:07:00 | 3658 | 55 | 29 |

29 | 16:11:00 | 4948 | 57 | 27 |

30 | 17:05:00 | 4746 | 56 | 31 |

31 | 17:14:00 | 5854 | 66 | 44 |

32 | 17:24:00 | 6265 | 68 | 44 |

33 | 18:16:00 | 5918 | 58 | 38 |

34 | 19:37:00 | 4899 | 56 | 36 |

35 | 19:41:00 | 5848 | 58 | 25 |

36 | 19:58:00 | 3727 | 65 | 44 |

37 | 21:09:00 | 3620 | 55 | 21 |

38 | 21:49:00 | 4919 | 58 | 30 |

39 | 21:57:00 | 3986 | 56 | 24 |

40 | 22:04:00 | 4997 | 58 | 33 |

**Table 2.**Comparison of the vessel sequence, the departure time from the anchorage, the arrival time at the pier and the waiting time at the anchorage of each vessel in both original navigation scheduling and optimized navigation scheduling.

Number | Original Navigation Scheduling | Optimized Navigation Scheduling | ||||||
---|---|---|---|---|---|---|---|---|

Sequence | Departure Time (hh:mm:ss) | Arrival Time at the Pier (hh:mm:ss) | Waiting Time (Hour) | Sequence | Departure Time (hh:mm:ss) | Arrival Time at the Pier (hh:mm:ss) | Waiting Time (Hour) | |

1 | 1 | 01:03:00 | 02:08:00 | 0.5 | 1 | 01:03:00 | 02:35:00 | 0.5 |

2 | 2 | 02:45:00 | 03:50:00 | 0.8 | 2 | 02:00:00 | 03:37:00 | 0.1 |

3 | 8 | 07:13:00 | 08:18:00 | 5.2 | 3 | 02:08:00 | 03:57:00 | 0.1 |

4 | 3 | 03:42:00 | 04:47:00 | 1.3 | 4 | 02:28:00 | 04:04:00 | 0.1 |

5 | 7 | 07:07:00 | 08:12:00 | 4.7 | 5 | 02:33:00 | 04:25:00 | 0.1 |

6 | 4 | 04:38:00 | 05:43:00 | 0.5 | 6 | 04:18:00 | 05:55:00 | 0.2 |

7 | 5 | 05:21:00 | 06:26:00 | 1.0 | 7 | 04:36:00 | 06:11:00 | 0.2 |

8 | 12 | 08:52:00 | 09:57:00 | 4.4 | 8 | 04:41:00 | 06:31:00 | 0.2 |

9 | 6 | 06:07:00 | 07:12:00 | 0.3 | 9 | 06:01:00 | 07:38:00 | 0.2 |

10 | 14 | 10:34:00 | 11:39:00 | 4.7 | 10 | 06:08:00 | 07:59:00 | 0.2 |

11 | 16 | 10:50:00 | 11:55:00 | 4.6 | 11 | 06:28:00 | 08:30:00 | 0.2 |

12 | 18 | 12:09:00 | 13:14:00 | 4.8 | 12 | 07:26:00 | 09:21:00 | 0.1 |

13 | 9 | 08:09:00 | 09:14:00 | 0.3 | 13 | 08:07:00 | 09:41:00 | 0.3 |

14 | 10 | 08:18:00 | 09:23:00 | 0.2 | 14 | 08:33:00 | 10:10:00 | 0.4 |

15 | 11 | 08:27:00 | 09:32:00 | 0.3 | 15 | 08:43:00 | 10:40:00 | 0.6 |

16 | 13 | 09:56:00 | 11:01:00 | 0.7 | 16 | 09:41:00 | 11:17:00 | 0.4 |

17 | 15 | 10:41:00 | 11:46:00 | 0.6 | 17 | 10:40:00 | 12:14:00 | 0.6 |

18 | 21 | 14:15:00 | 15:41:00 | 3.8 | 18 | 11:12:00 | 12:49:00 | 0.8 |

19 | 17 | 11:31:00 | 12:36:00 | 0.3 | 19 | 12:11:00 | 13:48:00 | 1.0 |

20 | 23 | 15:51:00 | 16:56:00 | 3.7 | 20 | 13:16:00 | 15:04:00 | 1.1 |

21 | 25 | 17:17:00 | 18:22:00 | 5.0 | 21 | 13:26:00 | 15:18:00 | 1.1 |

22 | 24 | 16:47:00 | 17:52:00 | 4.3 | 22 | 13:41:00 | 15:34:00 | 1.2 |

23 | 19 | 13:05:00 | 14:10:00 | 0.1 | 23 | 14:12:00 | 15:48:00 | 1.2 |

24 | 26 | 17:46:00 | 18:51:00 | 4.0 | 24 | 15:00:00 | 16:30:00 | 1.2 |

25 | 20 | 13:56:00 | 15:01:00 | 0.1 | 25 | 15:05:00 | 16:42:00 | 1.3 |

26 | 27 | 17:56:00 | 19:01:00 | 3.9 | 26 | 15:34:00 | 17:11:00 | 1.5 |

27 | 28 | 18:41:00 | 19:46:00 | 3.8 | 27 | 16:23:00 | 17:58:00 | 1.5 |

28 | 22 | 15:17:00 | 16:22:00 | 0.2 | 28 | 17:02:00 | 18:35:00 | 1.9 |

29 | 29 | 19:40:00 | 20:45:00 | 3.5 | 29 | 18:06:00 | 19:42:00 | 1.9 |

30 | 30 | 19:50:00 | 20:55:00 | 2.8 | 30 | 19:03:00 | 20:40:00 | 2.0 |

31 | 32 | 20:36:00 | 21:41:00 | 3.4 | 31 | 19:19:00 | 20:51:00 | 2.1 |

32 | 36 | 22:15:00 | 23:20:00 | 4.9 | 32 | 19:24:00 | 21:14:00 | 2.0 |

33 | 34 | 21:35:00 | 22:40:00 | 3.3 | 33 | 20:16:00 | 21:47:00 | 2.0 |

34 | 35 | 21:59:00 | 23:04:00 | 2.4 | 34 | 21:36:00 | 23:13:00 | 2.0 |

35 | 38 | 23:45:00 | 01:02:00 | 4.3 | 35 | 21:57:00 | 23:31:00 | 2.3 |

36 | 31 | 20:04:00 | 21:09:00 | 0.1 | 36 | 22:16:00 | 23:53:00 | 2.3 |

37 | 33 | 21:24:00 | 22:29:00 | 0.3 | 37 | 23:28:00 | 01:02:00 | 2.3 |

38 | 40 | 01:23:00 | 02:28:00 | 3.6 | 38 | 00:14:00 | 01:46:00 | 2.4 |

39 | 37 | 22:21:00 | 23:53:00 | 0.4 | 39 | 00:24:00 | 01:59:00 | 2.5 |

40 | 39 | 00:59:00 | 02:04:00 | 2.9 | 40 | 01:16:00 | 02:52:00 | 3.2 |

**Table 3.**Comparison of the vessel groups and the starting time of the lockage process in both original navigation scheduling and optimized navigation scheduling.

Group | Original Navigation Scheduling | Optimized Navigation Scheduling | ||
---|---|---|---|---|

Vessel Number | Starting Time of the Lockage Process | Vessel Number | Starting Time of the Lockage Process | |

1 | 1, 2, 4, 6, 7, 9 | 07:12:00 | 1, 2, 3, 4, 5 | 04:25:00 |

2 | 5, 3, 13, 14 | 09:23:00 | 6, 7, 8, 9, 10 | 07:59:00 |

3 | 15, 8, 16, 10, 17 | 11:46:00 | 11, 12, 13, 14 | 10:10:00 |

4 | 11, 19, 12, 23 | 14:10:00 | 15, 16, 17, 18, 19 | 13:48:00 |

5 | 25, 18, 28, 20, 22 | 17:52:00 | 20, 21, 22, 23 | 15:48:00 |

6 | 21, 24, 26, 27 | 19:46:00 | 24, 25, 26, 27, 28 | 18:35:00 |

7 | 29, 30, 36, 31 | 21:41:00 | 29, 30, 31, 32 | 21:14:00 |

8 | 37, 33, 34, 32 | 23:20:00 | 33, 34, 35, 36 | 23:53:00 |

9 | 39, 35, 40, 38 | 02:28:00 | 37, 38, 39, 40 | 02:52:00 |

**Table 4.**The vessel sequence, weight and RCESA (reduction on carbon emissions in step 2/reduction on vessel speed) in the optimized navigation scheduling.

Number | Vessel Weight (Tonnes) | RCESA |
---|---|---|

1 | 3878 | 26.9 |

2 | 4894 | 30.8 |

3 | 6496 | 35.6 |

4 | 3985 | 27.5 |

5 | 6317 | 34.5 |

6 | 4019 | 27.8 |

7 | 4906 | 31.1 |

8 | 6465 | 35.3 |

9 | 4881 | 30.7 |

10 | 6392 | 34.9 |

11 | 6457 | 35.7 |

12 | 6489 | 34.9 |

13 | 3709 | 25.9 |

14 | 3739 | 25.7 |

15 | 3599 | 25.7 |

16 | 3983 | 26.9 |

17 | 3972 | 27.1 |

18 | 4917 | 30.9 |

19 | 3853 | 26.3 |

20 | 6322 | 35.0 |

21 | 6421 | 35.0 |

22 | 6496 | 35.0 |

23 | 3672 | 25.5 |

24 | 5884 | 35.8 |

25 | 3883 | 26.4 |

26 | 4904 | 30.9 |

27 | 5740 | 34.5 |

28 | 3658 | 25.7 |

29 | 4948 | 31.1 |

30 | 4746 | 30.2 |

31 | 5854 | 35.3 |

32 | 6265 | 34.6 |

33 | 5918 | 35.8 |

34 | 4899 | 30.8 |

35 | 5848 | 35.0 |

36 | 3727 | 25.7 |

37 | 3620 | 25.4 |

38 | 4919 | 31.5 |

39 | 3986 | 27.0 |

40 | 4997 | 31.3 |

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## Share and Cite

**MDPI and ACS Style**

Zhao, X.; Lin, Q.; Yu, H.
An Improved Mathematical Model for Green Lock Scheduling Problem of the Three Gorges Dam. *Sustainability* **2019**, *11*, 2640.
https://doi.org/10.3390/su11092640

**AMA Style**

Zhao X, Lin Q, Yu H.
An Improved Mathematical Model for Green Lock Scheduling Problem of the Three Gorges Dam. *Sustainability*. 2019; 11(9):2640.
https://doi.org/10.3390/su11092640

**Chicago/Turabian Style**

Zhao, Xu, Qianjun Lin, and Hao Yu.
2019. "An Improved Mathematical Model for Green Lock Scheduling Problem of the Three Gorges Dam" *Sustainability* 11, no. 9: 2640.
https://doi.org/10.3390/su11092640