# A Simulated Annealing Algorithm for Solving Two-Echelon Vehicle Routing Problem with Locker Facilities

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

_{2}emissions per ton-kilometer of the linehaul-delivery system by adjusting the central depot location or developing the loaded-semitrailer demand among O-D pairs to eliminate empty-running of tractors. Eitzen et al. [15] applied 2E-VRP in transporting goods in urban logistics to reduce traffic congestion that might occur.

## 3. Two-Echelon Vehicle Routing Problem with Locker Facilities

_{ij}represents distance travelled from node i to j where $i,j\in M$. The second echelon consists of the locker facilities and the customer node. The customer node are represented in set N = {1 … |N|} and also includes the distance between pairs of customers and lockers facilities represented as Cc

_{ij}represent distance travelled from node i to j where $i\in N,j\in M$. Each customer is associated with demand D

_{n}where $n\in N$. The customer assignment in the second echelon is restricted to the parcel locker facility Qs

_{m}where $m\in \{2\dots |M\left|\right\}$. The routing decision in the first echelon is limited by the capacity of the available vehicle Qv

_{v}where v is one of the vehicles available represented in set V = {1 … |V|}. The models developed are using the decision variables as follows.

- X
_{i,j,v} - 1; if there is a route from i to j using vehicle v0; Otherwise
- Z
_{i} - 1; if the locker i is opened0; Otherwise
- S
_{i,j} - 1; if customer i is served by parcel locker j
- Q
_{i} - Total amount of demand in parcel locker i
- a
_{i,v} - Number of demand in parcel locker i using vehicle v

## 4. Simulated Annealing for Solving 2EVRP-LF

#### 4.1. Solution Representation

#### 4.2. Parameters Used

#### 4.3. SA Procedure

#### 4.4. Neighborhood

## 5. Computational Experiments

#### 5.1. Test Instances

#### 5.2. Parameter Settings

#### 5.3. Computational Results

## 6. Conclusions and Future Research Directions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Node | X | Y | Capacity | Demand |
---|---|---|---|---|

0 | 7 | 8 | - | - |

1 | 3 | 10 | 20 | - |

2 | 4 | 5 | 20 | - |

3 | 9 | 12 | 20 | - |

4 | 12 | 8 | 20 | - |

5 | 2 | 2 | - | 10 |

6 | 1 | 4 | - | 10 |

7 | 1 | 12 | - | 10 |

8 | 3 | 14 | - | 10 |

9 | 15 | 10 | - | 10 |

10 | 14 | 3 | - | 10 |

Nodes | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

0 | 0.00 | 4.47 | 4.24 | 4.47 | 5.00 | 7.81 | 7.21 | 7.21 | 7.21 | 8.25 | 8.60 |

1 | 4.47 | 0.00 | 5.10 | 6.32 | 9.22 | 8.06 | 6.32 | 2.83 | 4.00 | 12.00 | 13.04 |

2 | 4.24 | 5.10 | 0.00 | 8.60 | 8.54 | 3.61 | 3.16 | 7.62 | 9.06 | 12.08 | 10.20 |

3 | 4.47 | 6.32 | 8.60 | 0.00 | 5.00 | 12.21 | 11.31 | 8.00 | 6.32 | 6.32 | 10.30 |

4 | 5.00 | 9.22 | 8.54 | 5.00 | 0.00 | 11.66 | 11.70 | 11.70 | 10.82 | 3.61 | 5.39 |

5 | 7.81 | 8.06 | 3.61 | 12.21 | 11.66 | 0.00 | 2.24 | 10.05 | 12.04 | 15.26 | 12.04 |

6 | 7.21 | 6.32 | 3.16 | 11.31 | 11.70 | 2.24 | 0.00 | 8.00 | 10.20 | 15.23 | 13.04 |

7 | 7.21 | 2.83 | 7.62 | 8.00 | 11.70 | 10.05 | 8.00 | 0.00 | 2.83 | 14.14 | 15.81 |

8 | 7.21 | 4.00 | 9.06 | 6.32 | 10.82 | 12.04 | 10.20 | 2.83 | 0.00 | 12.65 | 15.56 |

9 | 8.25 | 12.00 | 12.08 | 6.32 | 3.61 | 15.26 | 15.23 | 14.14 | 12.65 | 0.00 | 7.07 |

10 | 8.60 | 13.04 | 10.20 | 10.30 | 5.39 | 12.04 | 13.04 | 15.81 | 15.56 | 7.07 | 0.00 |

Instances | VRP | 2EVRPLF − GUROBI | 2Phase (P-Median + VRP) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

No | Name | m | Obj | NV | Echelon1 | Echelon2 | NV | Time (s) | Echelon1 | Echelon2 | NV | Time (s) | Saving (%) | 2Echelon vs. 2Phase (%) |

1 | A-n33-k5 | 5 | 661 | 5 | 213.85 | 904.33 | 2 | 0.39 | 256.37 | 1015.09 | 2 | 0.08 | 67.65 | 13.71 |

2 | A-n46-k7 | 7 | 914 | 7 | 379.91 | 716.53 | 3 | 4.06 | 415.33 | 856.45 | 3 | 0.19 | 58.43 | 15.99 |

3 | A-n60-k9 | 9 | 1354 | 9 | 470.14 | 1016.99 | 3 | 312.06 | 581.80 | 1135.83 | 3 | 1.41 | 65.28 | 15.50 |

4 | B-n35-k5 | 6 | 955 | 5 | 298.68 | 590.15 | 2 | 0.83 | 329.13 | 725.82 | 2 | 1.06 | 68.73 | 18.69 |

5 | B-n45-k5 | 7 | 751 | 5 | 282.90 | 693.53 | 2 | 1.59 | 327.97 | 770.10 | 2 | 0.27 | 62.33 | 12.46 |

6 | B-n68-k9 | 11 | 1272 | 9 | 414.52 | 821.60 | 3 | 2127.36 | 569.14 | 906.45 | 3 | 23.94 | 67.41 | 19.37 |

7 | B-n78-k10 | 12 | 1221 | 10 | 431.41 | 1028.23 | 4 | 28,437.00 | 481.58 | 1188.09 | 4 | 9.19 | 64.67 | 14.39 |

8 | E-n51-k5 | 5 | 521 | 5 | 131.30 | 479.62 | 2 | 1.09 | 178.69 | 552.14 | 2 | 0.22 | 74.80 | 19.63 |

9 | E-n76-k7 | 12 | 682 | 7 | 245.70 | 705.72 | 3 | 4843.98 | 795.60 | 806.87 | 3 | 0.08 | 63.97 | 68.43 |

10 | F-n135-k7 | 21 | 1162 | 7 | 476.92 | 1633.86 | 1 | 19,274.10 | 478.38 | 1746.42 | 1 | 7.44 | 58.96 | 5.40 |

11 | M-n101-k10 | 16 | 820 | 10 | 361.84 | 982.33 | 4 | 29,188.80 | 404.22 | 1097.12 | 4 | 185.72 | 55.87 | 11.69 |

12 | P-n76-K4 | 12 | 593 | 4 | 236.17 | 683.95 | 2 | 36.67 | 236.17 | 795.60 | 2 | 0.98 | 60.17 | 12.13 |

13 | P-n101-k4 | 16 | 681 | 4 | 285.63 | 836.32 | 2 | 1668.06 | 301.27 | 980.71 | 2 | 24.55 | 58.06 | 14.26 |

Average | 11 | 837 | 6.3 | 325.30 | 853.32 | 2.5 | 6607.38 | 411.97 | 967.44 | 2.5 | 15.94 | 70.40 | 18.95 |

Instances | 2EVRPLF − GUROBI | SA | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Name | m | Echelon1 | Echelon2 | NV | Time | Echelon1 | Echelon2 | NV | Time (s) | Difference | |

1 | A-n33-k5 | 5 | 213.85 | 904.33 | 2 | 0.39 | 213.85 | 904.33 | 2 | 2.25 | 0.00 |

2 | A-n46-k7 | 7 | 379.91 | 716.53 | 3 | 4.06 | 379.91 | 716.53 | 3 | 1.12 | 0.00 |

3 | A-n60-k9 | 9 | 470.14 | 1016.99 | 3 | 312.06 | 470.14 | 1016.99 | 3 | 2.79 | 0.00 |

4 | B-n35-k5 | 6 | 298.68 | 590.15 | 2 | 0.83 | 298.68 | 590.15 | 2 | 3.67 | 0.00 |

5 | B-n45-k5 | 7 | 282.90 | 693.53 | 2 | 1.59 | 282.90 | 693.53 | 2 | 1.09 | 0.00 |

6 | B-n68-k9 | 11 | 414.52 | 821.60 | 3 | 2127.36 | 414.52 | 721.60 | 3 | 2.67 | −8.09 |

7 | B-n78-k10 | 12 | 431.41 | 1028.23 | 4 | 28,437.00 | 431.41 | 921.43 | 4 | 3.78 | −7.32 |

8 | E-n51-k5 | 5 | 131.30 | 479.62 | 2 | 1.09 | 131.30 | 479.62 | 2 | 1.71 | 0.00 |

9 | E-n76-k7 | 12 | 245.70 | 705.72 | 3 | 4843.98 | 235.70 | 692.22 | 3 | 3.24 | −2.47 |

10 | F-n135-k7 | 21 | 476.92 | 1633.86 | 1 | 19,274.10 | 473.52 | 1521.26 | 1 | 3.68 | −5.50 |

11 | M-n101-k10 | 16 | 361.84 | 982.33 | 4 | 29,188.80 | 352.43 | 972.53 | 4 | 2.09 | −1.43 |

12 | P-n76-K4 | 12 | 236.17 | 683.95 | 2 | 36.67 | 236.17 | 683.95 | 2 | 3.65 | 0.00 |

13 | P-n101-k4 | 16 | 285.63 | 836.32 | 2 | 1668.06 | 271.43 | 835.32 | 2 | 1.65 | −1.35 |

10.69 | 325.30 | 853.32 | 2.54 | 6607.38 | 322.46 | 826.88 | 2.54 | 2.57 | −2.01 |

Instances | SA | |||||||
---|---|---|---|---|---|---|---|---|

Best | Avg | Std. Dev | ||||||

Name | m | Echelon1 | Echelon2 | Echelon1 | Echelon2 | Echelon1 | Echelon2 | |

1 | A-n33-k5 | 5 | 213.85 | 904.33 | 215.71 | 904.27 | 3.34 | 1.78 |

2 | A-n46-k7 | 7 | 379.91 | 716.53 | 379.60 | 716.48 | 2.22 | 3.18 |

3 | A-n60-k9 | 9 | 470.14 | 1016.99 | 470.76 | 1017.82 | 3.16 | 3.73 |

4 | B-n35-k5 | 6 | 298.68 | 590.15 | 298.61 | 589.17 | 2.24 | 3.31 |

5 | B-n45-k5 | 7 | 282.90 | 693.53 | 281.93 | 693.28 | 1.32 | 4.25 |

6 | B-n68-k9 | 11 | 414.52 | 721.60 | 413.39 | 719.97 | 2.35 | 1.11 |

7 | B-n78-k10 | 12 | 431.41 | 921.43 | 432.70 | 921.28 | 2.10 | 5.06 |

8 | E-n51-k5 | 5 | 131.30 | 479.62 | 131.08 | 479.36 | 2.93 | 4.05 |

9 | E-n76-k7 | 12 | 235.70 | 692.22 | 237.02 | 689.88 | 3.52 | 3.16 |

10 | F-n135-k7 | 21 | 473.52 | 1521.26 | 472.48 | 1521.61 | 0.74 | 1.71 |

11 | M-n101-k10 | 16 | 352.43 | 972.53 | 354.20 | 970.77 | 2.01 | 4.95 |

12 | P-n76-K4 | 12 | 236.17 | 683.95 | 236.33 | 682.46 | 1.74 | 1.78 |

13 | P-n101-k4 | 16 | 271.43 | 835.32 | 272.18 | 833.91 | 8.37 | 6.70 |

10.69 | 322.46 | 826.88 | 322.77 | 826.17 | 2.77 | 3.44 |

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## Share and Cite

**MDPI and ACS Style**

Redi, A.A.N.P.; Jewpanya, P.; Kurniawan, A.C.; Persada, S.F.; Nadlifatin, R.; Dewi, O.A.C.
A Simulated Annealing Algorithm for Solving Two-Echelon Vehicle Routing Problem with Locker Facilities. *Algorithms* **2020**, *13*, 218.
https://doi.org/10.3390/a13090218

**AMA Style**

Redi AANP, Jewpanya P, Kurniawan AC, Persada SF, Nadlifatin R, Dewi OAC.
A Simulated Annealing Algorithm for Solving Two-Echelon Vehicle Routing Problem with Locker Facilities. *Algorithms*. 2020; 13(9):218.
https://doi.org/10.3390/a13090218

**Chicago/Turabian Style**

Redi, A. A. N. Perwira, Parida Jewpanya, Adji Candra Kurniawan, Satria Fadil Persada, Reny Nadlifatin, and Oki Anita Candra Dewi.
2020. "A Simulated Annealing Algorithm for Solving Two-Echelon Vehicle Routing Problem with Locker Facilities" *Algorithms* 13, no. 9: 218.
https://doi.org/10.3390/a13090218