# Optimization of Location-Routing Problem in Emergency Logistics Considering Carbon Emissions

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Algorithm for LRP in Emergency Logistics

#### 2.2. Sustainability Issues in LRP

## 3. Mathematical Model

#### 3.1. Problem Description

#### 3.2. Notations

#### 3.3. Model Construction

#### 3.3.1. Analysis of Objectives Function

#### 3.2.2. FLCOLRP Model Setting

## 4. Design of PSO-TS Algorithm

## 5. Experimental Design and Results Analysis

#### 5.1. Algorithm Experiment

#### 5.2. Model Experiment

#### 5.2.1. Experimental Design

#### 5.2.2. Experimental Results

- The minimum delivery time is better when the weight of total costs becomes smaller, and the weight of the carbon emissions becomes bigger. When the weight of delivery time ${w}_{1}=1/3$ the minimum delivery time goes up gradually with the increase of the weight of total costs ${w}_{2}$; when the weight of total costs ${w}_{2}=1/3$, the minimum rises initially, then goes up as the weight of delivery time ${w}_{1}$ go up; when the weight of carbon emissions ${w}_{3}=1/3$, this figure goes up, then declines to the lowest value 851.21, finally rises slightly with the increase of ${w}_{1}$. The range of the minimum delivery time is from 851.21 to 1139.40.
- The minimum of total costs is better when the weight of delivery time and the weight of carbon emissions become smaller. When the weight of delivery time ${w}_{1}=1/3$ , the minimum of total costs levels out initially and declines afterwards with the increase of the weight of total cost ${w}_{2}$; when the weight of total costs ${w}_{2}=1/3$, the minimum rises slightly, then drops dramatically, after that remains unchanged as the weight of delivery time ${w}_{1}$ goes up; when the weight of carbon emissions ${w}_{3}=1/3$, this figure remains stable, then declines slowly, finally rises to the highest value 757,801.72 with the increase of ${w}_{1}$. In brief, the minimum of total costs is between 757,801.72 and 458,997.82.
- The minimum of carbon emissions is better when the weight of delivery time becomes bigger, and the weight of total costs becomes smaller. When the weight of delivery time ${w}_{1}=1/3$ , the minimum delivery time drops a little, then rises steadily with the increase of the weight of total costs ${w}_{2}$; when the weight of total costs ${w}_{2}=1/3$, the minimum gradually increases reaching the highest value 504.65 as the weight of delivery time ${w}_{1}$ goes up; when the weight of carbon emissions ${w}_{3}=1/3$, this figure rises and falls, finally decreases to the lowest value 504.65 with the increase of ${w}_{1}$.
- The changing of the weight of ${w}_{1},{\text{}w}_{2}$ and ${w}_{3}$ has an influence on the optimal result of the total emergency logistics system. From Table 9 and Figure 4, we can see that the optimal result rises, then declines when ${w}_{1}=1/3,$ and ${w}_{2}$ = $1/3$; while this result remains stable, then drops when ${w}_{3}$ = $1/3$. Although all of them have a downward trend, the result of ${w}_{3}$ = $1/3$ decreases more quickly than others. What’s more, the optimal result of the total emergency logistics system reaches the highest value 1.23 when ${w}_{2}=1/3,$ ${w}_{1}$ = $5/12,$ ${w}_{3}=1/4$. It reaches the lowest value, 1.06, when ${w}_{3}=1/3,$ ${w}_{1}$ = $7/12,$ ${w}_{2}=1/12$.

#### 5.3. Analysis of Results

- The minimum of the multi-objective function is closely related to the value of weight (${w}_{1},{w}_{2},{w}_{3}$. When ${w}_{2}=1/3,$ ${w}_{1}$ = $5/12,$ ${w}_{3}=1/4$, the emergency logistics system obtains the overall optimality.
- Though setting the different weight of objective functions and adding several candidate distribution centers, it is proved that the built FLCOLRP model in this study is applicable for LRP in emergency logistic.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**(

**a**) The changing trends of the minimum of the delivery time, the total costs and the carbon emissions, when w

_{1}= 1/3. (

**b**) The changing trends of the minimum of the delivery time, the total costs and the carbon emissions, when w

_{2}= 1/3. (

**c**) The changing trends of the minimum of the delivery time, the total costs and the carbon emissions, when w

_{3}= 1/3.

Notations | Explanation |
---|---|

${D}_{c}$ | Set of candidate distribution centers $\left\{i|i=1,\dots ,C\right\}$. |

${D}_{o}$ | Set of open distribution centers ${D}_{o}\subseteq {D}_{c}$. |

${N}_{d}$ | Set of demand points $\left\{i|i=C+1,C+2,\dots ,C+D\right\}$. |

${V}_{h}$ | Set of vehicles $\left\{h|h=1,2,\dots ,H\right\}$. |

${P}_{m}$ | Set of sub-paths $\left\{m|m=1,2,\dots ,M\right\}$. |

${t}_{ij}$ | Transportation time from node I to node j. |

${C}_{i}$ | Construction and operation costs of the distribution center i. |

${H}_{h}$ | Operation costs of the vehicle h. |

${S}_{1}$ | Transportation costs of per unit distance. |

${S}_{2}$ | Penalty costs of per unit unmet need. |

${\left[E{D}_{i}\right]}^{o}$ | Represents the most optimistic demand for demand point i. |

${\left[E{D}_{i}\right]}^{l}$ | Represents the most likely demand for demand point i. |

${\left[E{D}_{i}\right]}^{p}$ | Represents the most pessimistic demand for demand point i. |

${\omega}_{1}$ | Weight coefficient of the most optimistic demand. |

${\omega}_{2}$ | Weight coefficient of the most likely demand. |

${\omega}_{3}$ | Weight coefficient of the most pessimistic demand. |

${Q}_{i}$ | Demand for relief supplies of demand point i. |

$\mu $ | Fuel consumption rate when the vehicle is full-load. Consumption Rate |

${u}_{0}$ | Fuel consumption rate when the vehicle is no-load. |

${M}_{h}$ | Maximal weight the vehicle h could carry. |

${M}_{ijh}$ | Carried load of vehicle h from node i to node j. |

$\tau $ | Conversion factor for carbon dioxide and fuel consumption. |

${W}_{i}$ | Maximum capacity of candidate distribution center i. |

${O}_{h}$ | Longest travel distance of the vehicle h. |

${y}_{i}$ | ${y}_{i}=1$ represents the candidate distribution center i are employed. |

Otherwise, ${y}_{i}=0$. | |

${x}_{ij}^{h}$ | ${x}_{ij}^{h}=1$ represents the vehicle h visit the node j from the node i. |

Otherwise,${x}_{ij}^{h}=0$. | |

${z}_{mhi}^{c}$ | ${z}_{mhi}^{c}=1$ represents the sub-path m of candidate distribution center c. |

includes node i served by the vehicle h. Otherwise, ${z}_{mhi}^{c}=0$. |

Part 1 | 3 | 4 | 2 | 3 | 4 | 3 | 2 | 4 |

Part 2 | 1 | 2 | 3 | 3 | 4 | |||

Part 3 | 10 | 7 | 5 | 8 | 12 | 9 | 6 | 11 |

Case | Number of DCs | Number of DPs | PSO | PSO-TS | ||
---|---|---|---|---|---|---|

Number of DCs | Total Distance | Number of DCs | Total Distance | |||

Gaspelle1 | 5 | 21 | 3 | 544.57 | 1 | 545.01 |

Gaspelle2 | 5 | 22 | 2 | 892.69 | 1 | 898.07 |

Christ50 | 5 | 50 | 3 | 1462.01 | 3 | 1401.17 |

Christ75 | 10 | 75 | 7 | 2448.57 | 6 | 2316.46 |

Christ100 | 10 | 100 | 6 | 3027.12 | 6 | 2895.13 |

Min27 | 5 | 27 | 3 | 5744.55 | 1 | 5206.01 |

Min134 | 8 | 134 | 6 | 31,933.38 | 6 | 30,361.27 |

Das88 | 8 | 88 | 3 | 2411.84 | 3 | 2341.46 |

Das150 | 10 | 150 | 8 | 166,473.80 | 6 | 161,141.65 |

Or117 | 14 | 117 | 5 | 60,203.40 | 4 | 56,399.91 |

$\mathbf{Distribution}\text{}\mathbf{Centers}\text{}{\mathit{D}}_{\mathit{i}}$ | X Coordinate | Y Coordinate | $\mathbf{Maximum}\text{}\mathbf{Capacity}\text{}{\mathit{W}}_{\mathit{i}}$ | $\mathbf{Construction}\text{}\mathbf{Cos}\mathbf{ts}\text{}{\mathit{C}}_{\mathit{i}}\text{}\left(\mathbf{CNY}\right)$ |
---|---|---|---|---|

1 | 40 | 5 | 1500 | 200,000 |

2 | 70 | 60 | 2000 | 250,000 |

3 | 20 | 50 | 1800 | 300,000 |

$\mathbf{Demand}\text{}\mathbf{Points}\text{}{\mathit{N}}_{\mathit{i}}$ | X Coordinate | Y Coordinate | $\mathbf{Optimistic}\text{}\mathbf{Demand}\text{}{\left[\mathit{E}{\mathit{D}}_{\mathit{i}}\right]}^{\mathit{o}}$ | $\mathbf{Likely}\text{}\mathbf{Demand}\text{}{\left[\mathit{E}{\mathit{D}}_{\mathit{i}}\right]}^{\mathit{l}}$ | $\mathbf{Pessimistic}\text{}\mathbf{Demand}\text{}{\left[\mathit{E}{\mathit{D}}_{\mathit{i}}\right]}^{\mathit{p}}$ |
---|---|---|---|---|---|

4 | 25 | 85 | 129 | 135 | 150 |

5 | 5 | 45 | 369 | 375 | 387 |

6 | 42 | 15 | 66 | 75 | 84 |

7 | 38 | 5 | 141 | 153 | 165 |

8 | 95 | 35 | 128 | 135 | 150 |

9 | 85 | 25 | 69 | 75 | 82 |

10 | 62 | 80 | 180 | 195 | 212 |

11 | 58 | 75 | 129 | 135 | 150 |

12 | 50 | 50 | 64 | 75 | 82 |

13 | 18 | 80 | 269 | 275 | 280 |

14 | 25 | 30 | 63 | 69 | 72 |

15 | 15 | 10 | 129 | 135 | 143 |

16 | 45 | 65 | 60 | 69 | 78 |

17 | 65 | 20 | 245 | 251 | 257 |

18 | 31 | 52 | 165 | 177 | 186 |

19 | 2 | 60 | 39 | 45 | 52 |

20 | 5 | 5 | 105 | 111 | 117 |

21 | 57 | 29 | 119 | 123 | 138 |

22 | 4 | 18 | 159 | 165 | 171 |

23 | 26 | 35 | 296 | 305 | 308 |

$\mathbf{Number}\text{}\mathbf{of}\text{}\mathbf{Vehicles}\text{}{\mathit{V}}_{\mathit{h}}$ | $\mathbf{Maximal}\text{}\mathbf{Weight}\text{}{\mathit{M}}_{\mathit{h}}$ | $\mathbf{Operation}\text{}\mathbf{Cos}\mathbf{ts}\text{}{\mathit{H}}_{\mathit{h}}\text{}\left(\mathbf{CNY}\right)$ | $\mathbf{Longest}\text{}\mathbf{Distance}\text{}{\mathit{O}}_{\mathit{h}}\text{}\left(\mathbf{km}\right)$ |
---|---|---|---|

1–3 | 100 | 500 | 350 |

4–6 | 150 | 700 | 360 |

7–9 | 200 | 900 | 370 |

Parameters | Value |
---|---|

${S}_{1}$ | 8 CNY/km |

${S}_{2}$ | 150 CNY/kg |

$\mu $ | 0.165 L/km |

${u}_{0}$ | 0.377 L/km |

$\tau $ | 2.63 kg/L |

Value | $\mathbf{M}\mathbf{i}\mathbf{n}\mathit{F}\text{}\left(\mathbf{min}\right)$ | $\mathbf{M}\mathbf{i}\mathbf{n}\mathit{C}\text{}\left(\mathbf{NCY}\right)$ | $\mathbf{M}\mathbf{i}\mathbf{n}{\mathit{E}}_{\mathit{c}{\mathit{o}}_{2}}$ (kg) |
---|---|---|---|

${w}_{1}=1,$${w}_{2}$ = ${w}_{3}=0$ | 851.21 | - | - |

${w}_{2}=1,$${w}_{1}$ = ${w}_{3}=0$ | - | 458,997.82 | - |

${w}_{3}=1,$${w}_{1}$ = ${w}_{2}=0$ | - | - | 365.76 |

Value | $\mathbf{M}\mathbf{i}\mathbf{n}\mathit{F}\text{}\left(\mathbf{min}\right)$ | $\mathbf{M}\mathbf{i}\mathbf{n}\mathit{C}\text{}\left(\mathbf{NCY}\right)$ | $\mathbf{M}\mathbf{i}\mathbf{n}{\mathit{E}}_{\mathit{c}{\mathit{o}}_{2}}\text{}\left(\mathbf{kg}\right)$ | Optimal | ||
---|---|---|---|---|---|---|

${w}_{1}=1/3$ | ${w}_{2}$ = $1/12,$ ${w}_{3}=7/12$ | 1 | 851.21 | 757,509.68 | 378.39 | 1.07 |

${w}_{2}$ = $1/6,$ ${w}_{3}=1/2$ | 2 | 913.09 | 757,804.73 | 368.25 | 1.14 | |

${w}_{2}$ = $5/12,$ ${w}_{3}=1/4$ | 3 | 1088.24 | 509,605.92 | 451.42 | 1.19 | |

${w}_{2}$ = $1/2,$ ${w}_{3}=1/6$ | 4 | 1088.24 | 509,605.92 | 451.42 | 1.18 | |

${w}_{2}$ = $7/12,$ ${w}_{3}=1/12$ | 5 | 1120.93 | 486,267.49 | 453.73 | 1.15 | |

${w}_{2}=1/3$ | ${w}_{1}$ = $1/12,$ ${w}_{3}=7/12$ | 1 | 1097.92 | 574,483.42 | 435.06 | 1.20 |

${w}_{1}$ = $1/6,$ ${w}_{3}=1/2$ | 2 | 1129.94 | 509,939.58 | 461.62 | 1.22 | |

${w}_{1}$ = $5/12,$ ${w}_{3}=1/4$ | 3 | 1094.38 | 509,655.09 | 484.96 | 1.23 | |

${w}_{1}$ = $1/2,$ ${w}_{3}=1/6$ | 4 | 1058.95 | 508,971.64 | 504.65 | 1.21 | |

${w}_{1}$ = $7/12,$ ${w}_{3}=1/12$ | 5 | 1058.95 | 508,971.64 | 504.65 | 1.20 | |

${w}_{3}=1/3$ | ${w}_{1}$ = $1/12,$ ${w}_{2}=7/12$ | 1 | 1129.57 | 509,536.58 | 458.94 | 1.17 |

${w}_{1}$ = $1/6,$ ${w}_{2}=1/2$ | 2 | 1139.40 | 509,615.18 | 431.38 | 1.17 | |

${w}_{1}$ = $5/12,$ ${w}_{2}=1/4$ | 3 | 1037.23 | 458,997.82 | 457.41 | 1.17 | |

${w}_{1}$ = $1/2,$ ${w}_{2}=1/6$ | 4 | 851.21 | 757,509.68 | 378.39 | 1.11 | |

${w}_{1}$ = $7/12,$ ${w}_{2}=1/12$ | 5 | 862.71 | 757,801.72 | 365.76 | 1.06 |

Number of Vehicle | Service Order | Number of Vehicle | Service Order |
---|---|---|---|

2 | DC3-11-DC2 | 7 | DC3-18-23-DC3-DC1-6-DC1 |

3 | DC3-4-5-19-DC3-DC2-10-DC2 | 8 | DC3-20-DC1-DC2-8-16-13-DC3-DC2-9-DC2 |

4 | DC3-14-DC3 | 9 | DC1-15-DC1-DC3-12-DC2 |

6 | DC1-7-22-17-21-DC1 |

$\mathbf{Distribution}\text{}\mathbf{Centers}\text{}{\mathit{D}}_{\mathit{i}}$ | X Coordinate | Y Coordinate | $\mathbf{Maximum}\text{}\mathbf{Capacity}\text{}{\mathit{W}}_{\mathit{i}}$ | $\mathbf{Construction}\text{}\mathbf{Cos}\mathbf{ts}\text{}{\mathit{C}}_{\mathit{i}}\text{}\left(\mathbf{CNY}\right)$ |
---|---|---|---|---|

4 | 5 | 20 | 1900 | 240,000 |

5 | 90 | 80 | 1500 | 300,000 |

Number of Vehicle | Service Order | Number of Vehicle | Service Order |
---|---|---|---|

2 | DC4-22-DC4 | 6 | DC1-9-DC1 |

4 | DC3-6-DC4 | 7 | DC3-7-13-16-DC3-DC2-10-19-12-14-DC2 |

5 | DC4-25-DC3-DC1-8-11-20-DC3-DC1-17-23-DC1 | 8 | DC3-21-24-DC4-DC3-15-18-DC2 |

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## Share and Cite

**MDPI and ACS Style**

Shen, L.; Tao, F.; Shi, Y.; Qin, R.
Optimization of Location-Routing Problem in Emergency Logistics Considering Carbon Emissions. *Int. J. Environ. Res. Public Health* **2019**, *16*, 2982.
https://doi.org/10.3390/ijerph16162982

**AMA Style**

Shen L, Tao F, Shi Y, Qin R.
Optimization of Location-Routing Problem in Emergency Logistics Considering Carbon Emissions. *International Journal of Environmental Research and Public Health*. 2019; 16(16):2982.
https://doi.org/10.3390/ijerph16162982

**Chicago/Turabian Style**

Shen, Ling, Fengming Tao, Yuhe Shi, and Ruiru Qin.
2019. "Optimization of Location-Routing Problem in Emergency Logistics Considering Carbon Emissions" *International Journal of Environmental Research and Public Health* 16, no. 16: 2982.
https://doi.org/10.3390/ijerph16162982