# Optimization of a Capacitated Vehicle Routing Problem for Sustainable Municipal Solid Waste Collection Management Using the PSO-TS Algorithm

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## Abstract

**:**

## 1. Introduction

_{2}is the main source of greenhouse gases (GHGs) [11].

## 2. Literature Review

#### 2.1. Research on Sustainable Development in MSW Collection

_{2}emissions. In their follow-up study, this method had an excellent optimization effect for week-long scheduling [18]. By using multi-objective decision-making approaches, comprehensive optimization was often performed so as to keep a balance between various objectives [9]. Tamás Bányai et al. [20] introduced a cyber-physical system to simulate the waste collection process of downtown areas and guarantee energy usage cost-efficiency and environmental awareness of GHG emissions simultaneously. Bektas and Laporte [19] put forward the pollution-routing problem, and designed vehicle routes aimed at minimizing the fixed costs of drivers, fuel consumption, and CO

_{2}emission costs. Maurizio Faccioa et al. [21] proposed a comprehensive multi-objective VRP (vehicle routing problem) model aimed at minimizing the total covered distance, the necessary number of vehicles, and the environmental impact.

#### 2.2. Research about Algorithms for the CVRP Model

_{2}emissions from transportation activities and the emissions are transformed to the costs by per unit carbon cost. The social factor is considered by defining a balanced solution in terms of the number of trips received among disposal facilities.

## 3. Mathematical Model

#### 3.1. Problem Description

#### 3.2. Problem Assumptions

- (1)
- Only one depot is considered in this model. All vehicles start from the depot at the same time, and return there eventually.
- (2)
- The vehicles start and end their trips with an empty load.
- (3)
- All vehicles are homogeneous with the same capacity limit.
- (4)
- The collection points are also homogeneous with the same capacity limit. Each point should be served once by one vehicle.
- (5)
- The vehicles may take multiple trips.

#### 3.3. Parameters and Variables

#### 3.4. Model Construction

#### 3.4.1. Objectives Function

- (1)
- Vehicles’ Fixed Costs

- (2)
- Fuel Consumption Costs

- (3)
- Carbon Emission Costs

- (4)
- Penalty Costs

#### 3.4.2. Model Setting

## 4. Algorithm Description

#### 4.1. Algorithm Step Design

- (a)
- The length of particle code VarSize, the number of population nPop, and maximum number of iterations MaxIt are initialized.
- (b)
- PSO parameters are set: maximum value of inertia weight w
_{max}, minimum value of inertia weight w_{min}, variance of random inertia weight $\sigma $, random value of R_{1}, R_{2}, and acceleration factors C_{1}, C_{2}. - (c)
- TS parameters are set: tabu length TL, neighborhood size NS, and candidate size CS.
- (d)
- For each particle, initial position X
_{i}and velocity V_{i}are determined as per Equation (16).

- (e)
- A set of vehicle routes ${K}_{i}\left(t\right)$ is determined by decoding ${X}_{i}\left(t\right)$.
- (f)
- The fitness value of ${K}_{i}\left(t\right)$ is determined by the objective function $\phi \left({X}_{i}\left(t\right)\right)$.
- (g)
- The personal best position of particle i is identified as ${P}_{i}^{best}={X}_{i}\left(t\right)$.
- (h)
- The global best position ${G}^{best}$ of all particles is identified. If $\phi \left({P}_{i}^{best}\right)<\phi \left({G}^{best}\right)$, ${G}^{best}={P}_{i}^{best}$. Otherwise, ${G}^{best}$ remains unchanged.

- (i)
- The velocity and position of particle i according to Equation (17) are updated. N(0,1) represents the standard normally distributed random numbers.

- (j)
- A set of vehicle routes ${K}_{i}$(t+1) is updated by decoding ${X}_{i}$(t+1).
- (k)
- ${P}_{i}^{best}$: ${P}_{i}^{best}={X}_{i}$(t+1) is updated, if $\phi \left({X}_{i}\left(t+1\right)\right)<\phi \left({P}_{i}^{best}\right)$.
- (l)
- ${G}^{best}$: ${G}^{best}={P}_{i}^{best}$ is updated, if $\phi \left({P}_{i}^{best}\right)<\phi \left({G}^{best}\right)$.
- (m)
- When the number of iterations is greater than the number of population nPop, the current partial optimization solution ${G}^{best}$ calculated by the PSO is regarded as the initial solution of TS: Y = ${G}^{best}$.
- (n)
- Three kinds of neighborhood search algorithms, swap, reversion, and insertion, are randomly selected to improve the partial optimization solution Y.
- (o)
- The tabu list is renewed based on the special rules. Thus, the final selected solution is taken as the optimal solution ${Y}^{*}$.
- (p)
- Return to step (i) until the maximum number of iteration MaxIt is met.
- (q)
- ${Y}^{*}$ as the best set of vehicle routes ${K}^{*}$ is decoded, with its corresponding fitness value $\phi \left({Y}^{*}\right)$.

#### 4.2. Solution Representation and Decoding Method

## 5. Experimental Design and Results Analysis

#### 5.1. Algorithm Experiment

#### 5.2. Model Experiment

#### 5.2.1. Experimental Design

#### 5.2.2. Experimental Results

- (1)
- When minimized penalty costs are added to the objective function in model 2, the values of SV obtained by model 2 are smaller than the values in model 1 every single day. Therefore, model 2 is efficient for improving social equity by acquiring balanced trip assignments of disposal facilities.
- (2)
- After accumulating for a whole week, the SV is 2.97 in model 2, while the value is 55.97 in model 1. However, for each day, the values of SV are between 0 and 3.5 in the two models. Therefore, the imbalanced phenomenon can be more severe in the long-term in model 1.
- (3)
- In the meantime, the distance, carbon emissions, and operational costs of model 2 all increase in the results of model 1. Thus, we infer that there is a trade-off between economic costs, environmental benefits, and social equity.

- (1)
- The change trends of distance, carbon emissions, and operational costs coincide every day, that is, they ascend and descend simultaneously at each turning point. Therefore, we infer that there is a positive correlation between economic and environmental benefits.
- (2)
- The change rates of distance, carbon emissions, and operational costs are all situated in the interval between 1% and 15%, while the change rate of SV varies from −60% and −110%. Therefore, compared with the increase of social equity, the decrease in economic and environmental benefits is much smaller.

#### 5.3. Analysis of Results

- The proposed CVRP model can simultaneously take into account economic cost, environmental benefits (carbon emissions), and social equity (balanced workload of disposal facilities), resulting in a sustainable solution.
- There is a certain trade-off between economic costs, environmental benefits, and social equity. Social equity can be increased between 60% and 110% when economic and environmental benefits only decrease between 1% and 15%.
- There is a positive correlation between economic costs and environmental benefits, which can be combined into one objective.

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

PSO Parameters | Explanation |
---|---|

t | Iteration index: $t=1,2,\dots ,MaxIt$ |

i | Population index: $i=1,2,\dots ,nPop$ |

VarSize | Length of particle code |

w(t) | Random inertia weight in the itth iteration |

w_{max} | Maximum value of inertia weight |

w_{min} | Minimum value of inertia weight |

$\sigma $ | Variance of random inertia weight |

R_{1} | Random number in the interval [0, 1] |

R_{2} | Random number in the interval [0, 1] |

C_{1} | Personal acceleration factor |

C_{2} | Global acceleration factor |

VarMin | Lower bound of the position for each particle |

VarMax | Upper bound of the position for each particle |

VelMin | Lower bound of the velocity for each particle |

VelMax | Upper bound of the velocity for each particle |

Vi(t) | Velocity of particle i in the tth iteration |

Xi(t) | Position of particle i in the tth iteration |

Ki(t) | Set of vehicle routes corresponding to particle i in the tth iteration |

${P}_{i}^{best}$ | Personal best position of particle i |

${G}^{best}$ | Global best position of all particles |

$\phi \left({X}_{i}\left(t\right)\right)$ | Fitness value of Xi(t) |

TS Parameters | Explanation |

TL | Tabu length |

NS | Neighborhood size |

CS | Candidate size |

Day | Model 1 | Model 2 |
---|---|---|

Monday | 6-17-4-${R}_{1}$, 42-39-37-18-${R}_{1}$, 13-38-${R}_{1}$, 33-32-20-31-9-${R}_{2}$, 16-34-12-5-46-${R}_{2}$, 3-19-47-27-10-${R}_{2}$, 44-41-30-36-${R}_{2}$, 43-35-11-29-1-${R}_{4}$, 15-26-25-28-23-8-${R}_{5}$, 24-2-${R}_{5}$, 22-14-${R}_{6}$, 45-21-7-40-${R}_{6}$ | 6-17-4-${R}_{1}$, 42-39-37-18-${R}_{1}$, 16-34-12-5-46-${R}_{2}$, 3-19-47-27-10-${R}_{2}$, 33-32-20-31-9-${R}_{3}$, 24-2-${R}_{3}$, 43-35-11-29-1-${R}_{4}$, 44-41-30-36-${R}_{4}$, 15-26-25-28-23-8-${R}_{5}$, 22-14-${R}_{5}$, 13-38-${R}_{6}$, 45-21-7-40-${R}_{6}$ |

Tuesday | 16-41-39-37-18-${R}_{1}$, 44-17-${R}_{1}$, 13-38-${R}_{1}$, 14-22-4-${R}_{1}$, 1-34-10-36-27-9-${R}_{2}$, 5-46-${R}_{2}$, 33-32-26-20-31-${R}_{2}$, 12-42-30-${R}_{4}$, 43-11-29-3-25-${R}_{5}$, 6-45-8-28-${R}_{5}$, 19-${R}_{5}$, 35-47-15-2-${R}_{5}$, 24-23-7-21-40-${R}_{6}$ | 44-17-${R}_{1}$, 13-38-${R}_{1}$, 5-46-${R}_{2}$, 1-34-10-36-27-9-${R}_{2}$, 33-32-26-20-31-${R}_{2}$, 6-45-8-28-${R}_{3}$, 19-${R}_{3}$, 12-42-30-${R}_{4}$, 14-22-4-${R}_{4}$, 35-47-15-2-${R}_{5}$, 43-11-29-3-25-${R}_{5}$, 16-41-39-37-18-${R}_{6}$, 24-23-7-21-40-${R}_{6}$ |

Wednesday | 22-13-18-${R}_{1}$, 39-37-38-${R}_{1}$, 12-34-16-4-${R}_{1}$, 44-17-${R}_{1}$, 35-15-47-${R}_{2}$, 20-31-27-${R}_{2}$, 36-10-46-9-${R}_{2}$, 5-${R}_{2}$, 30-42-41-${R}_{4}$, 43-11-29-1-${R}_{4}$, 32-33-3-25-${R}_{5}$, 19-26-2-${R}_{5}$, 24-8-23-28-${R}_{5}$, 14-6-21-${R}_{6}$, 45-7-40-${R}_{6}$ | 5-${R}_{1}$, 22-13-18-${R}_{1}$, 12-34-16-4-${R}_{1}$, 20-31-27-${R}_{2}$, 35-15-47-${R}_{2}$, 36-10-46-9-${R}_{2}$, 24-8-23-28-${R}_{3}$, 39-37-38-${R}_{4}$, 30-42-41-${R}_{4}$, 32-33-3-25-${R}_{5}$, 14-6-21-${R}_{5}$, 19-26-2-${R}_{5}$, 45-7-40-${R}_{6}$, 43-11-29-1-${R}_{6}$, 44-17-${R}_{6}$ |

Thursday | 4-${R}_{1}$, 41-39-18-${R}_{1}$, 44-17-${R}_{1}$, 13-37-38-${R}_{1}$, 36-10-31-${R}_{2}$, 15-27-${R}_{2}$,12-16-46-${R}_{2}$, 11-35-9-${R}_{2}$, 5-47-26-20-${R}_{2}$, 2-1-${R}_{4}$, 43-34-30-42-${R}_{4}$, 29-3-25-${R}_{5}$, 45-23-28-${R}_{5}$, 19-32-33-${R}_{5}$, 6-${R}_{6}$, 24-8-21-${R}_{6}$, 7-40-${R}_{6}$, 22-14-${R}_{6}$ | 22-14-${R}_{1}$, 41-39-18-${R}_{1}$, 4-${R}_{1}$, 12-16-46-${R}_{2}$, 11-35-9-${R}_{2}$, 36-10-31-${R}_{2}$, 19-32-33-${R}_{3}$, 45-23-28-${R}_{3}$, 5-47-26-20-${R}_{3}$, 43-34-30-42-${R}_{4}$, 15-27-${R}_{4}$, 2-1-${R}_{4}$, 7-40-${R}_{5}$, 6-${R}_{5}$, 29-3-25-${R}_{5}$, 44-17-${R}_{6}$, 13-37-38-${R}_{6}$, 24-8-21-${R}_{6}$ |

Friday | 39-41-34-16-${R}_{1}$, 4-12-${R}_{1}$, 13-38-${R}_{1}$, 37-18-${R}_{1}$, 17-${R}_{1}$, 10-46-5-${R}_{2}$, 30-42-36-${R}_{2}$, 27-${R}_{2}$, 47-31-9-${R}_{2}$, 11-29-35-${R}_{2}$, 15-26-${R}_{3}$, 32-33-3-${R}_{3}$, 43-24-1-${R}_{4}$, 2-${R}_{5}$, 25-28-${R}_{5}$, 45-23-8-${R}_{5}$, 20-19-${R}_{5}$, 40-${R}_{6}$,44-6-${R}_{6}$, 22-14-${R}_{6}$, 21- 7-${R}_{6}$ | 39-41-34-16-${R}_{1}$, 17-${R}_{1}$, 13-38-${R}_{1}$, 37-18-${R}_{1}$, 10-46-5-${R}_{2}$, 30-42-36-${R}_{2}$, 27-${R}_{2}$, 47-31-9-${R}_{2}$, 20-19-${R}_{3}$, 15-26-${R}_{3}$, 32-33-3-${R}_{3}$, 43-24-1-${R}_{4}$, 4-12-${R}_{4}$, 2-${R}_{5}$, 25-28-${R}_{5}$, 11-29-35-${R}_{5}$, 40-${R}_{5}$, 44-6-${R}_{6}$, 22-14-${R}_{6}$, 21-7-${R}_{6}$, 45-23-8-${R}_{6}$ |

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**Figure 2.**Basic procedures of the proposed particle swarm algorithm (PSO)-tabu search (TS) algorithm.

**Figure 3.**The optimal vehicle routes with minimized operational costs and carbon emissions on Monday (model 1).

**Figure 4.**The optimal vehicle routes with minimized operational costs, carbon emissions, and penalty costs on Monday (model 2).

**Figure 9.**Workload of six disposal facilities in two models and the average workload on Monday (Average workload = 125 t).

Variables | Explanation |
---|---|

${\mathrm{x}}_{\mathrm{ijh}}$ | ${\mathrm{x}}_{\mathrm{ijh}}=1$, if vehicle h visits from point i to point j, Otherwise, ${\mathrm{x}}_{\mathrm{ijh}}=0$ |

${\mathrm{y}}_{\mathrm{ih}}$ | ${\mathrm{y}}_{\mathrm{ih}}=1,$ if vehicle h visits point i, Otherwise, ${\mathrm{y}}_{\mathrm{ih}}=0$ |

${\mathrm{z}}_{\mathrm{mhi}}^{\mathrm{r}}$ | ${\mathrm{z}}_{\mathrm{mhi}}^{\mathrm{r}}=1,$ if sub-path m of vehicle h unloads waste at disposal facility r, includes point i served by the vehicle h, Otherwise, ${\mathrm{z}}_{\mathrm{mhi}}^{\mathrm{r}}=0$ |

${\mathrm{f}}_{\mathrm{m}}^{\mathrm{r}}$ | ${\mathrm{f}}_{\mathrm{m}}^{\mathrm{r}}=1,$ if sub-path m assigns to disposal facility r causing overload of facility r, Otherwise, ${\mathrm{f}}_{\mathrm{m}}^{\mathrm{r}}=0$ |

Parameters | Explanation |

$\mathrm{G}$ | Set of all the nodes in the graph network, $\mathrm{G}=\left\{\mathrm{V},\mathrm{K}\right\}$ |

$\mathrm{K}$ | Set of vehicles $\left\{\mathrm{h}|\mathrm{h}=1,2,\dots ,\mathrm{H}\right\}$ |

$\mathrm{V}$ | Set of collection points $\left\{\mathrm{i}|\mathrm{i}=0,1,2,\dots ,\mathrm{N}\right\}$, 0 is the depot |

$\mathrm{R}$ | Set of disposal facilities $\left\{\mathrm{r}|\mathrm{r}=1,2,\dots ,\mathrm{S}\right\}$ |

$\mathrm{T}$ | Set of sub-paths $\left\{\mathrm{m}|\mathrm{m}=1,2,\dots ,\mathrm{M}\right\}$ |

${\mathrm{Q}}_{\mathrm{ijh}}$ | Carried load of vehicle h visit from point i to point j |

$\mathrm{Q}$ | Maximal load capacity of the vehicle |

${\mathrm{q}}_{\mathrm{i}}$ | Waste collection demand of collection point i |

${\mathrm{U}}_{\mathrm{r}}$ | Workload limit of disposal facility r |

${\mathrm{d}}_{\mathrm{ij}}$ | Transportation distance from point i to point j |

${\mathrm{c}}_{\mathrm{v}}$ | Fixed costs of per unit vehicle |

${\mathrm{c}}_{\mathrm{f}}$ | Cost of per unit fuel consumption |

${\mathrm{c}}_{\mathrm{e}}$ | Cost of per unit carbon emission |

$\mathsf{\eta}$ | Fuel consumption rate when vehicle is full-loadConsumption Rate |

${\mathsf{\eta}}_{0}$ | Fuel consumption rate when vehicle is empty |

$\mathsf{\lambda}$ | Conversion factor for carbon dioxide and fuel consumption |

$\mathrm{p}$ | Penalty cost of overload disposal facility for per sub-path |

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

Part 2 | 3 | 6 | 5 | 1 | 8 | 4 | 2 | 10 | 7 | 9 |

Part 3 | 11 | 13 | 11 | 12 | ||||||

Part 4 | 15 |

Vehicle Routes | |
---|---|

1 | 15-3-6-2-10-11-15 |

2 | 15-1-4-9-13-15 |

3 | 15-5-8-11-15 |

4 | 15-7-12-15 |

Parameters of the PSO | Values | Parameters of the TS | Values |
---|---|---|---|

MaxIt | 1000 | TL | 20 |

nPop | 50 | NS | ${C}_{Varsize}^{2}$ |

w_{max} | 0.8 | CS | (0.1*${C}_{Varsize}^{2}$) |

w_{min} | 0.5 | ||

$\sigma $ | 0.2 | ||

R_{1}, R_{2} | rand (Varsize) | ||

C_{1}, C_{2} | 1.5 | ||

VarMin | 0 | ||

VarMax | 1 |

Instance | Collection Point | Depot | Disposal Facility | Workload Limit | Vehicle | Capacity |
---|---|---|---|---|---|---|

p01 | 50 | 1 | 3 | 4 | 16 | 80 |

p02 | 50 | 1 | 3 | 2 | 8 | 160 |

p03 | 70 | 1 | 4 | 3 | 15 | 140 |

p06 | 100 | 1 | 2 | 6 | 18 | 100 |

p07 | 100 | 1 | 3 | 4 | 16 | 100 |

p15 | 160 | 1 | 3 | 5 | 20 | 60 |

Instance | PSO | PSO-TS | Optimization Rate (%) | ||
---|---|---|---|---|---|

Number of Sub-Paths | Distance | Number of Sub-Paths | Distance | ||

p01 | 14 | 1517.24 | 12 | 1175.85 | 22.50% |

p02 | 6 | 1183.91 | 6 | 904.23 | 23.62% |

p03 | 13 | 1874.15 | 12 | 1369.59 | 26.92% |

p06 | 19 | 2940.19 | 18 | 2445.96 | 16.81% |

p07 | 18 | 2701.45 | 18 | 2196.23 | 18.70% |

p15 | 18 | 14,376.75 | 17 | 11,528.92 | 19.81% |

Average | - | - | - | - | 21.39% |

Disposal Facilities | X Coordinate | Y Coordinate |
---|---|---|

1 | 20 | 20 |

2 | 50 | 30 |

3 | 60 | 50 |

4 | 36 | 16 |

5 | 42 | 57 |

6 | 8 | 52 |

Depot | X Coordinate | Y Coordinate | Number of Vehicles | Maximal Weight/t |
---|---|---|---|---|

1 | 30 | 40 | 16 | 80 |

Collection Points | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

X | 37 | 49 | 52 | 20 | 40 | 21 | 17 | 31 | 52 | 51 | 42 | 31 |

Y | 52 | 49 | 64 | 26 | 30 | 47 | 63 | 62 | 33 | 21 | 41 | 32 |

Waste Load/t | 7 | 30 | 16 | 9 | 21 | 15 | 19 | 23 | 11 | 5 | 19 | 29 |

Collection Points | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

X | 5 | 12 | 52 | 27 | 17 | 13 | 57 | 62 | 16 | 7 | 27 | 30 |

Y | 25 | 42 | 41 | 23 | 33 | 13 | 58 | 42 | 57 | 38 | 68 | 48 |

Waste Load/t | 23 | 21 | 15 | 3 | 41 | 9 | 28 | 8 | 16 | 28 | 7 | 15 |

Collection Points | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 |

X | 43 | 58 | 58 | 37 | 38 | 46 | 61 | 62 | 63 | 32 | 45 | 59 |

Y | 67 | 48 | 27 | 69 | 46 | 10 | 33 | 63 | 69 | 22 | 35 | 15 |

Waste Load/t | 14 | 6 | 19 | 11 | 12 | 23 | 26 | 17 | 6 | 9 | 15 | 14 |

Collection Points | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | |

X | 5 | 10 | 21 | 5 | 30 | 39 | 32 | 25 | 25 | 48 | 56 | |

Y | 6 | 17 | 10 | 64 | 15 | 10 | 39 | 32 | 55 | 28 | 37 | |

Waste Load/t | 7 | 27 | 13 | 11 | 16 | 10 | 5 | 25 | 17 | 18 | 10 |

Day | Upper Limits for the Number of Sub-Paths | |||||

${\mathit{U}}_{1}$ | ${\mathit{U}}_{2}$ | ${\mathit{U}}_{3}$ | ${\mathit{U}}_{4}$ | ${\mathit{U}}_{5}$ | ${\mathit{U}}_{6}$ | |

Monday | 2 | 2 | 2 | 2 | 2 | 2 |

Tuesday | 2 | 2 | 2 | 2 | 2 | 2 |

Wednesday | 3 | 3 | 3 | 3 | 3 | 3 |

Thursday | 3 | 3 | 3 | 3 | 3 | 3 |

Friday | 4 | 4 | 4 | 4 | 4 | 4 |

Parameters | Values |
---|---|

${c}_{v}$ | 300 CNY (Chinese Yuan) |

${c}_{f}$ | 7 CNY/L |

${c}_{e}$ | 0.64 CNY/kg |

$\eta $ | 0.377 L/km |

${\eta}_{0}$ | 0.165 L/km |

$\lambda $ | 2.32 kg/L |

$p$ | 150 CNY |

Day | Sub-Paths | Distance (km) | Carbon Emissions (kg) | Operational Costs (CNY) | Sub-Path Assignments of Disposal Facilities | SV | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{R}}_{1}$ | ${\mathit{R}}_{2}$ | ${\mathit{R}}_{3}$ | ${\mathit{R}}_{4}$ | ${\mathit{R}}_{5}$ | ${\mathit{R}}_{6}$ | ||||||

Monday | 12 | 1023.50 | 351.46 | 5608.39 | 3 | 4 | 0 | 1 | 2 | 2 | 2.00 |

Tuesday | 13 | 1039.31 | 364.43 | 5982.53 | 4 | 3 | 0 | 1 | 4 | 1 | 2.97 |

Wednesday | 15 | 1105.62 | 371.89 | 6625.14 | 4 | 4 | 0 | 2 | 3 | 2 | 2.30 |

Thursday | 18 | 1222.04 | 421.18 | 7206.82 | 4 | 5 | 0 | 2 | 3 | 4 | 3.20 |

Friday | 21 | 1290.33 | 425.17 | 7229.59 | 5 | 5 | 2 | 1 | 4 | 4 | 2.70 |

Week | 79 | 5680.80 | 1934.13 | 32,652.48 | 20 | 21 | 2 | 7 | 16 | 13 | 55.77 |

Day | Sub-Paths | Distance (km) | Carbon Emissions (kg) | Operational Costs (CNY) | Sub-Path Assignments of Disposal Facilities | SV | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{R}}_{1}$ | ${\mathit{R}}_{2}$ | ${\mathit{R}}_{3}$ | ${\mathit{R}}_{4}$ | ${\mathit{R}}_{5}$ | ${\mathit{R}}_{6}$ | ||||||

Monday | 12 | 1136.56 | 398.41 | 5876.69 | 2 | 2 | 2 | 2 | 2 | 2 | 0.00 |

Tuesday | 13 | 1118.65 | 398.53 | 6177.38 | 2 | 3 | 2 | 2 | 2 | 2 | 0.17 |

Wednesday | 15 | 1207.62 | 418.57 | 6891.89 | 3 | 3 | 1 | 2 | 3 | 3 | 0.70 |

Thursday | 18 | 1354.04 | 474.80 | 7513.22 | 3 | 3 | 3 | 3 | 3 | 3 | 0.00 |

Friday | 21 | 1368.75 | 453.50 | 7391.49 | 4 | 4 | 3 | 2 | 4 | 4 | 0.70 |

Week | 79 | 6185.62 | 2143.81 | 33,850.68 | 14 | 15 | 11 | 11 | 14 | 14 | 2.97 |

**Table 14.**Detailed route assignments and workload of disposal facilities on Monday in the two models.

Model | Detailed Route Assignments of Disposal Facilities on Monday | |||||
---|---|---|---|---|---|---|

${\mathit{R}}_{1}$ | ${\mathit{R}}_{2}$ | ${\mathit{R}}_{3}$ | ${\mathit{R}}_{4}$ | ${\mathit{R}}_{5}$ | ${\mathit{R}}_{6}$ | |

Model 1 | 6-17-4-${R}_{1}$ 42-39-37-18-${R}_{1}$ 13-38-${R}_{1}$ | 33-32-20-31-9-${R}_{2}$ 16-34-12-5-46-${R}_{2}$ 3-19-47-27-10-${R}_{2}$ 44-41-30-36-${R}_{2}$ | -- | 43-35-11-29-1-${R}_{4}$ | 15-26-25-28-23-8-${R}_{5}$ 24-2-${R}_{5}$ | 22-14-${R}_{6}$ 45-21-7-40-${R}_{6}$ |

Workload/t | 154 | 304 | 0 | 58 | 121 | 112 |

Model 2 | 6-17-4-${R}_{1}$ 42-39-37-18-${R}_{1}$ | 16-34-12-5-46-${R}_{2}$ 3-19-47-27-10-${R}_{2}$ | 33-32-20-31-9-${R}_{3}$ 24-2-${R}_{3}$ | 43-35-11-29-1-${R}_{4}$ 44-41-30-36-${R}_{4}$ | 15-26-25-28-23-8-${R}_{5}$ 22-14-${R}_{5}$ | 13-38-${R}_{6}$ 45-21-7-40-${R}_{6}$ |

Workload/t | 104 | 158 | 113 | 136 | 125 | 113 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Qiao, Q.; Tao, F.; Wu, H.; Yu, X.; Zhang, M.
Optimization of a Capacitated Vehicle Routing Problem for Sustainable Municipal Solid Waste Collection Management Using the PSO-TS Algorithm. *Int. J. Environ. Res. Public Health* **2020**, *17*, 2163.
https://doi.org/10.3390/ijerph17062163

**AMA Style**

Qiao Q, Tao F, Wu H, Yu X, Zhang M.
Optimization of a Capacitated Vehicle Routing Problem for Sustainable Municipal Solid Waste Collection Management Using the PSO-TS Algorithm. *International Journal of Environmental Research and Public Health*. 2020; 17(6):2163.
https://doi.org/10.3390/ijerph17062163

**Chicago/Turabian Style**

Qiao, Qingqing, Fengming Tao, Hailin Wu, Xuewei Yu, and Mengjun Zhang.
2020. "Optimization of a Capacitated Vehicle Routing Problem for Sustainable Municipal Solid Waste Collection Management Using the PSO-TS Algorithm" *International Journal of Environmental Research and Public Health* 17, no. 6: 2163.
https://doi.org/10.3390/ijerph17062163