Distribution-Free Approaches for an Integrated Cargo Routing and Empty Container Repositioning Problem with Repacking Operations in Liner Shipping Networks
Abstract
:1. Introduction
- (1)
- This work studies a new integrated cargo routing and empty container repositioning problem with repacking operations and uncertain laden and empty container demands in liner shipping networks.
- (2)
- For the problem, a chance-constrained programming model based on moment-based ambiguous sets is proposed.
- (3)
- To solve the problem, four distribution-free solution methods including Sample Average Approximation (SAA), enhanced SAA (eSAA), Approximation based on Markov’s Inequality (AMI) and Mixed Integer Second-Order Cone Program (MI-SOCP) are adopted to solve the problem.
2. Literature Review
2.1. Cargo Routing and Empty Container Repositioning Problems
2.2. Moment-Based Ambiguity Set and Chance Constrained Programming
2.3. Distribution-Free Approach
3. Problem Statement
- (1)
- The repacking operations can only be executed at container yards. Therefore, before TEUs are repackaged into FEUs, containers need to be discharged from the liner ship, and then unload costs are incurred.
- (2)
- Each TEU and FEU are repacked at most once for a cargo route .
- (3)
- The uncertain demand of each cargo route is considered, and it is assumed to be not over the capacity of a cargo route.
- (4)
- Repacking operations can occur at transshipment ports, origin and destination ports.
- (5)
- The capacity of a shipping route is limited. That is, the numbers of transported TEUs and FEUs at each leg are less than or equal to the capacity of the shipping route. Note that we consider the capacity of the shipping route but not the capacity of the liner ship, because a shipping route has a group of liner ships to provide services.
4. Mathematical Formulation
4.1. Notations and Moment-Based Ambiguity Set
4.2. Formulation
- :
- the capacity of both empty and laden TEUs in shipping route ;
- :
- the capacity of both empty and laden FEUs in shipping route ;
- :
- the laden TEU demand of cargo route , random parameter;
- :
- the empty TEU demand of cargo route , random parameter;
- :
- the empty FEU demand of cargo route , random parameter;
- :
- the maximum probability of failing to meet the demand of the laden TEUs in cargo route ; Note that laden FEUs derived from repacking operations are finally unpacked to laden TEUs in cargo route s;
- :
- the maximum probability of failing to meet the demand of the empty TEUs in cargo route ;
- :
- the maximum probability of failing to meet the demand of the empty FEUs in cargo route .
- :
- the number of laden TEUs that are repacked at port p () and unpacked at port q () while transshipping the laden containers, where are the origin and destination ports of cargo route , respectively;
- :
- the number of laden TEUs without repacking operations from o to d in cargo route ;
- :
- the number of empty TEUs from o to d in cargo route ;
- :
- the number of empty FEUs from o to d in cargo route ;
- :
- TEU flow of leg () of shipping route j, where ;
- :
- FEU flow of leg () of shipping route j, where ;
- :
- TEU flow of leg () of cargo route s, where ;
- :
- FEU flow of leg () of cargo route s, where ;
- :
- the quantity of laden and empty TEUs transshipped at port ;
- :
- the quantity of laden and empty FEUs transshipped at port ;
- :
- the quantity of laden TEUs repacked into FEUs at port ;
- :
- the quantity of laden FEUs unpacked into TEUs at port ;
- :
- the transshipment cost of all ports;
- :
- the total packing cost incurred at all ports in the transshipment port set H;
- :
- the cost for loading, discharge, repacking and unpacking operations at the origin and destination ports;
- :
- the costs of both empty TEU and FEU containers for loading, discharge, repacking and unpacking operations at the origin and destination ports.
5. Distribution-Free Approximation Approaches
5.1. Sample Average Approximation (SAA)
- :
- a sufficiently large positive number;
- :
- the laden TEU demand of cargo route s in scenario ;
- :
- the empty TEU demand of cargo route s in scenario ;
- :
- the empty FEU demand of cargo route s in scenario ;
- :
- binary variable, equal to 0 if the number of transshipment laden TEU containers meets the demand of cargo route s in scenario , 1 otherwise;
- :
- binary variable, equal to 0 if the number of empty TEU containers meets the demand of cargo route s in scenario , 1 otherwise;
- :
- binary variable, equal to 0 if the number of empty FEU containers meets the demand of cargo route s in scenario , 1 otherwise.
5.2. Enhanced Sample Average Approximation (eSAA)
Algorithm 1: SAA with K-means clustering (eSAA with K-means) |
Step 1: Set parameters: the number of scenarios; number of clusters; set of points that belongs to cluster k, where k = 1, 2, 3, …, K. Step 2: Generate N scenarios: , where n = 1, 2, 3, …, N. Step 3: Randomly select K initial cluster centers recorded as from , where n = 1, 2, 3, …, N, k = 1, 2, 3, …, K, . Step 4: Attribute the nearest cluster to each data point: , where k = 1, 2, 3, …, K. Step 5: Fix the position of each cluster by calculating the mean of all points belonging to that cluster: . Step 6: Repeat Step 4 and Step 5 until convergence. Step 7: Randomly select one sample of each cluster as input. Step 8: Solve the model by CPLEX. |
Algorithm 2: SAA with K-means++ clustering (eSAA with K-means++) |
Step 1: The same as Step 1 of Algorithm 1. Step 2: The same as Step 2 of Algorithm 1. Step 3: Randomly select one cluster center, recorded as , from , where n = 1, 2, 3, …, N; set ; Step 4: Calculate the core of m cluster center(s) recorded as : Step 5: Select one sample () apart from as cluster center recorded as with probability: . Step 6: Set , repeat Step 4 and Step 5 when . Step 7: The same as Step 4 of Algorithm 1. Step 8: The same as Step 5 of Algorithm 1. Step 9: Repeat Step 7 and Step 8 until convergence. Step 10: The same as Step 7 of Algorithm 1. Step 11: The same as Step 8 of Algorithm 1. |
5.3. An Approximation Based on Markov’s Inequality (AMI)
5.4. Mixed Integer Second-Order Cone Program (MI-SOCP)
6. Computational Experiments
6.1. Parameter Settings
6.2. Experiment Results
7. Discussion
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Literature | Problem Setting | Modeling Method | Solution Method | |||
---|---|---|---|---|---|---|
Demand | Cargo Routing | Empty Container Repositioning | Repacking Operations | Chance Constrained | ||
Meng et al. [1] | Known distribution | √ | √ | SAA | ||
Ng [13] | Distribution-free | √ | √ | AMI | ||
Dong et al. [11] | Known distribution | √ | SAA, PHA | |||
Li et al. [8] | Known distribution | √ | DPA | |||
Wang et al. [6] | Deterministic | √ | √ | Cplex | ||
Liu et al. [9] | Distribution-free | √ | √ | √ | AMI, MI-SOCP | |
Song and Dong [15] | Deterministic | √ | Cplex | |||
Jeong et al. [17] | Deterministic | √ | HOP | |||
Kuzmicz and Pesch [18] | Deterministic | √ | Cplex | |||
Song and Dong [10] | Deterministic | √ | √ | TSP, THR | ||
Our work | Distribution-free | √ | √ | √ | √ | SAA, eSAA, AMI, MI-SOCP |
Modes | Origin Port | Transshipment Port | Destination Port |
---|---|---|---|
2TEUs→1FEU | None | 1FEU→2TEUs | |
None | 2TEUs→1FEU | 1FEU→2TEUs | |
None | 2TEUs→1FEU 1FEU→2TEUs | None | |
2TEUs→1FEU | 1FEU→2TEUs | None | |
2TEUs | None | 2TEUs |
Port Type | Ship Type | Loading | Discharge | Transship |
---|---|---|---|---|
A | TEU | 248 | 324 | 183 |
FEU | 256 | 332 | 198 | |
51.61% | 51.23% | 54.10% | ||
B | TEU | 118 | 118 | 145 |
FEU | 148 | 148 | 145 | |
62.71% | 62.71% | 50.00% | ||
C | TEU | 110 | 110 | 71 |
FEU | 156 | 156 | 106 | |
70.91% | 70.91% | 74.65% |
Method | Comb | Mixed | Normal | Uniform | |||
---|---|---|---|---|---|---|---|
obj() | Time(s) | obj() | Time(s) | obj() | Time(s) | ||
SAA | 4–8 | 198.24 | 211.97 | 191.60 | 216.52 | 187.86 | 211.30 |
5–8 | 201.40 | 213.83 | 195.25 | 215.42 | 185.72 | 213.96 | |
6–8 | 205.63 | 217.55 | 195.63 | 219.41 | 187.55 | 217.70 | |
4–10 | 181.36 | 243.33 | 166.52 | 246.48 | 171.20 | 243.40 | |
5–10 | 182.18 | 244.89 | 169.95 | 244.69 | 172.06 | 245.39 | |
6–10 | 184.44 | 249.54 | 170.69 | 248.22 | 177.12 | 249.96 | |
4–12 | 185.86 | 274.51 | 172.46 | 274.58 | 176.90 | 275.71 | |
5–12 | 186.42 | 276.23 | 176.78 | 275.01 | 179.87 | 278.28 | |
6–12 | 188.90 | 280.43 | 177.22 | 278.32 | 183.67 | 281.55 | |
average | 190.49 | 245.81 | 179.57 | 246.52 | 180.22 | 246.36 | |
eSAA with K-means | 4–8 | 195.28 | 211.42 | 191.86 | 212.56 | 188.92 | 211.45 |
5–8 | 194.42 | 202.73 | 202.14 | 213.70 | 192.03 | 213.75 | |
6–8 | 194.68 | 217.37 | 196.97 | 217.77 | 191.15 | 216.51 | |
4–10 | 185.10 | 243.07 | 171.01 | 243.43 | 178.58 | 242.60 | |
5–10 | 183.25 | 245.00 | 174.71 | 245.48 | 178.83 | 245.31 | |
6–10 | 185.58 | 248.75 | 169.91 | 249.58 | 177.05 | 248.61 | |
4–12 | 187.11 | 274.53 | 179.98 | 274.49 | 183.42 | 274.75 | |
5–12 | 189.64 | 276.88 | 179.90 | 276.69 | 180.63 | 276.92 | |
6–12 | 188.56 | 280.16 | 182.75 | 280.18 | 183.29 | 280.37 | |
average | 189.29 | 244.43 | 183.25 | 245.98 | 183.77 | 245.59 | |
eSAA with K-means++ | 4–8 | 193.86 | 214.26 | 202.48 | 213.98 | 190.28 | 214.03 |
5–8 | 192.53 | 217.14 | 206.22 | 216.10 | 188.98 | 216.79 | |
6–8 | 194.77 | 220.81 | 197.26 | 219.19 | 188.99 | 220.78 | |
4–10 | 186.13 | 246.49 | 169.87 | 245.24 | 175.56 | 245.83 | |
5–10 | 184.07 | 248.71 | 170.01 | 248.62 | 179.66 | 248.92 | |
6–10 | 185.27 | 252.48 | 171.99 | 252.07 | 178.86 | 252.15 | |
4–12 | 188.44 | 279.57 | 178.99 | 278.24 | 188.16 | 278.61 | |
5–12 | 186.04 | 281.97 | 182.16 | 281.79 | 179.76 | 282.15 | |
6–12 | 186.38 | 285.79 | 184.43 | 285.12 | 184.29 | 286.61 | |
average | 188.61 | 249.69 | 184.82 | 248.93 | 183.84 | 249.54 |
Method | Comb | Mixed | Normal | Uniform | |||
---|---|---|---|---|---|---|---|
obj() | Time(s) | obj() | Time(s) | obj() | Time(s) | ||
SAA | 4–8 | 198.24 | 214.78 | 191.60 | 214.60 | 187.86 | 214.94 |
5–8 | 201.40 | 216.93 | 195.25 | 216.61 | 185.72 | 217.10 | |
6–8 | 205.63 | 221.11 | 195.63 | 219.91 | 187.55 | 221.15 | |
4–10 | 200.96 | 246.24 | 191.10 | 245.80 | 188.59 | 246.83 | |
5–10 | 210.19 | 248.91 | 204.54 | 249.30 | 190.16 | 249.98 | |
6–10 | 210.14 | 252.95 | 201.35 | 253.02 | 193.67 | 253.50 | |
4–12 | 206.53 | 279.40 | 199.37 | 278.01 | 191.64 | 278.19 | |
5–12 | 211.06 | 281.31 | 205.81 | 280.17 | 194.84 | 281.10 | |
6–12 | 212.42 | 284.80 | 206.74 | 284.22 | 196.29 | 285.07 | |
average | 206.28 | 249.60 | 199.04 | 249.07 | 190.70 | 249.76 | |
eSAA with K-means | 4–8 | 195.28 | 213.67 | 191.86 | 214.28 | 188.92 | 213.96 |
5–8 | 194.42 | 216.58 | 202.14 | 216.26 | 192.03 | 216.67 | |
6–8 | 194.68 | 220.67 | 196.97 | 220.16 | 191.15 | 219.81 | |
4–10 | 199.26 | 245.22 | 195.15 | 244.81 | 192.59 | 245.42 | |
5–10 | 196.15 | 248.88 | 204.85 | 247.75 | 195.35 | 248.41 | |
6–10 | 197.35 | 252.50 | 204.21 | 251.40 | 192.99 | 251.60 | |
4–12 | 200.64 | 276.89 | 209.17 | 276.85 | 197.47 | 276.94 | |
5–12 | 200.00 | 281.18 | 207.64 | 280.20 | 194.05 | 279.99 | |
6–12 | 198.56 | 284.04 | 210.73 | 283.97 | 196.50 | 283.24 | |
average | 197.37 | 248.85 | 202.52 | 248.41 | 193.45 | 248.45 | |
eSAA with K-means++ | 4–8 | 193.86 | 215.45 | 202.48 | 214.67 | 190.28 | 216.01 |
5–8 | 192.53 | 218.59 | 206.22 | 218.00 | 188.98 | 219.14 | |
6–8 | 194.77 | 222.08 | 197.26 | 221.88 | 188.99 | 223.08 | |
4–10 | 197.94 | 248.09 | 200.11 | 248.10 | 192.36 | 249.60 | |
5–10 | 196.47 | 251.15 | 202.55 | 251.32 | 192.76 | 251.38 | |
6–10 | 199.88 | 254.81 | 201.69 | 254.49 | 193.02 | 254.96 | |
4–12 | 202.06 | 282.66 | 205.74 | 281.21 | 198.24 | 282.34 | |
5–12 | 198.54 | 285.05 | 209.39 | 284.70 | 193.27 | 284.07 | |
6–12 | 197.71 | 288.22 | 210.50 | 287.91 | 196.31 | 287.87 | |
average | 197.08 | 251.79 | 203.99 | 251.36 | 192.69 | 252.05 |
Method | Comb | Mixed | Normal | Uniform | |||
---|---|---|---|---|---|---|---|
obj() | Time(s) | obj() | Time(s) | obj() | Time(s) | ||
SAA | 4–8 | 198.24 | 217.07 | 191.60 | 219.01 | 187.86 | 217.41 |
5–8 | 201.40 | 219.08 | 195.25 | 222.83 | 185.72 | 220.20 | |
6–8 | 205.63 | 223.00 | 195.63 | 224.05 | 187.55 | 223.98 | |
4–10 | 200.96 | 249.08 | 191.10 | 249.77 | 188.59 | 250.02 | |
5–10 | 210.19 | 252.23 | 204.54 | 251.51 | 190.16 | 252.09 | |
6–10 | 210.14 | 255.49 | 201.35 | 254.90 | 193.67 | 256.27 | |
4–12 | 206.53 | 281.55 | 199.37 | 281.35 | 191.64 | 282.02 | |
5–12 | 211.06 | 284.30 | 205.81 | 283.87 | 194.84 | 284.54 | |
6–12 | 212.42 | 288.10 | 206.74 | 287.80 | 196.29 | 288.10 | |
average | 206.28 | 252.21 | 199.04 | 252.79 | 190.70 | 252.74 | |
eSAA with K-means | 4–8 | 195.28 | 217.79 | 191.86 | 217.55 | 188.92 | 217.83 |
5–8 | 194.42 | 219.97 | 202.14 | 220.07 | 192.03 | 219.74 | |
6–8 | 194.68 | 223.77 | 196.97 | 223.66 | 191.15 | 224.19 | |
4–10 | 199.26 | 251.05 | 195.15 | 250.20 | 192.59 | 250.25 | |
5–10 | 196.15 | 252.38 | 204.85 | 252.55 | 195.35 | 252.76 | |
6–10 | 197.35 | 255.75 | 204.21 | 255.76 | 192.99 | 256.30 | |
4–12 | 200.64 | 283.32 | 209.17 | 283.22 | 197.47 | 282.78 | |
5–12 | 200.00 | 286.21 | 207.64 | 285.17 | 194.05 | 284.71 | |
6–12 | 198.56 | 288.37 | 210.73 | 288.76 | 196.50 | 288.38 | |
average | 197.37 | 253.18 | 202.52 | 252.99 | 193.45 | 252.99 | |
eSAA with K-means++ | 4–8 | 193.86 | 233.29 | 202.48 | 219.24 | 190.28 | 217.61 |
5–8 | 192.53 | 221.91 | 206.22 | 221.48 | 188.98 | 219.89 | |
6–8 | 194.77 | 225.41 | 197.26 | 225.41 | 188.99 | 223.24 | |
4–10 | 197.94 | 252.25 | 200.11 | 252.57 | 192.36 | 250.84 | |
5–10 | 196.47 | 255.03 | 202.55 | 254.98 | 192.76 | 252.23 | |
6–10 | 199.88 | 259.06 | 201.69 | 259.50 | 193.02 | 256.72 | |
4–12 | 202.06 | 286.34 | 205.74 | 285.73 | 198.24 | 283.85 | |
5–12 | 198.54 | 287.83 | 209.39 | 288.63 | 193.27 | 286.25 | |
6–12 | 197.71 | 290.15 | 210.50 | 292.30 | 196.31 | 289.45 | |
average | 197.08 | 256.81 | 203.99 | 255.54 | 192.69 | 253.34 |
obj() | Time(s) | obj() | Time(s) | obj() | Time(s) | |
---|---|---|---|---|---|---|
361.53 | 107.83 | 419.23 | 116.85 | 547.00 | 128.89 | |
423.76 | 107.88 | 521.85 | 115.65 | 718.53 | 131.81 | |
476.12 | 107.24 | 600.11 | 117.19 | 847.34 | 135.03 | |
355.10 | 108.46 | 406.18 | 118.74 | 517.14 | 131.82 | |
422.45 | 107.67 | 515.40 | 118.34 | 702.00 | 134.81 | |
476.85 | 109.00 | 596.82 | 119.51 | 836.13 | 134.01 | |
344.53 | 111.09 | 387.99 | 121.43 | 479.07 | 170.19 | |
418.71 | 110.72 | 505.86 | 120.47 | 681.74 | 355.96 | |
475.08 | 112.32 | 590.85 | 122.33 | 821.95 | 188.90 | |
323.10 | 104.34 | 363.25 | 125.78 | 495.63 | 153.27 | |
413.35 | 114.07 | 494.13 | 124.89 | 658.56 | 335.38 | |
471.85 | 114.87 | 583.32 | 126.16 | 805.79 | 253.89 |
obj() | Time(s) | Rel | obj() | Time(s) | Rel | obj() | Time(s) | Rel | |
---|---|---|---|---|---|---|---|---|---|
AMI | 398.05 | 107.41 | 100.00% | 407.54 | 99.95 | 100.00% | 568.82 | 119.21 | 100.00% |
MI-SOCP | 326.91 | 106.55 | 100.00% | 385.51 | 121.28 | 100.00% | 483.68 | 177.94 | 100.00% |
SAA-M | 185.07 | 284.86 | 87.05% | 211.42 | 284.16 | 91.68% | 212.68 | 287.54 | 94.77% |
SAA-N | 174.45 | 253.17 | 81.55% | 205.16 | 248.56 | 92.71% | 201.52 | 266.71 | 92.47% |
SAA-U | 171.07 | 246.93 | 78.44% | 203.08 | 251.37 | 85.76% | 200.43 | 260.11 | 87.06% |
eSAA-K-M | 181.09 | 256.94 | 85.92% | 204.10 | 252.64 | 89.64% | 195.29 | 242.73 | 89.64% |
eSAA-K-N | 185.77 | 268.92 | 87.08% | 204.38 | 268.95 | 91.93% | 216.03 | 269.07 | 91.93% |
eSAA-K-U | 174.47 | 271.45 | 81.56% | 191.15 | 279.49 | 86.88% | 198.49 | 271.75 | 86.88% |
eSAA-K+M | 180.31 | 287.82 | 84.71% | 198.52 | 244.01 | 89.04% | 204.47 | 258.39 | 89.69% |
eSAA-K+N | 180.56 | 284.39 | 85.06% | 206.82 | 265.94 | 92.86% | 226.51 | 274.01 | 92.26% |
eSAA-K+U | 173.75 | 269.15 | 81.70% | 196.57 | 263.59 | 86.77% | 195.28 | 255.31 | 88.74% |
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Liu, M.; Liu, Z.; Liu, R.; Sun, L. Distribution-Free Approaches for an Integrated Cargo Routing and Empty Container Repositioning Problem with Repacking Operations in Liner Shipping Networks. Sustainability 2022, 14, 14773. https://doi.org/10.3390/su142214773
Liu M, Liu Z, Liu R, Sun L. Distribution-Free Approaches for an Integrated Cargo Routing and Empty Container Repositioning Problem with Repacking Operations in Liner Shipping Networks. Sustainability. 2022; 14(22):14773. https://doi.org/10.3390/su142214773
Chicago/Turabian StyleLiu, Ming, Zhongzheng Liu, Rongfan Liu, and Lihua Sun. 2022. "Distribution-Free Approaches for an Integrated Cargo Routing and Empty Container Repositioning Problem with Repacking Operations in Liner Shipping Networks" Sustainability 14, no. 22: 14773. https://doi.org/10.3390/su142214773