# Minimizing the Carbon Footprint for the Time-Dependent Heterogeneous-Fleet Vehicle Routing Problem with Alternative Paths

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## Abstract

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## 1. Introduction

_{2}emission quantity produced (directly or indirectly) throughout the entire life cycle of a service or a product. Compared with the greenhouse gas emission term commonly used by the public, carbon footprint differs in that it includes all the CO

_{2}emission produced from the extraction and production of the product’s raw materials, production and assembly of the product, and product use, disposal and recycling. Thus, a carbon footprint covers the entire life cycle of a product. Contemporary life is full of environmental pollution, and carbon footprint reduction is necessary to improve the environment for future generations.

_{2}are being emitted into the environment, and the numbers are continuing to grow. Therefore, the design of logistics route planning to reduce vehicle exhaust is an effective method for reducing the carbon footprint. The focus of logistics operating system is how effectively to use vehicles. Reducing the number of vehicles through effective vehicle route planning to transport raw materials, semi-finished products, or finished products to their destinations can reduce vehicle exhaust and vehicle transportation costs. Thus, effective vehicle route planning is an important topic in delivery problems. Conventionally, logistics delivery centers determine vehicle routing manually. However, simple manual vehicle route planning methods cannot determine ideal route planning in a short time as a large number of customers and different vehicle fleets are involved. Effective vehicle route planning prior to vehicle assignment can result in the maximal resource utilization effectiveness and drastically reduce vehicle transportation costs. This naturally reduces vehicle exhaust and lowers carbon footprints.

_{2}emission, speed, load, distance, and time. Xiao et al. [5] developed an optimization model that adds a new factor, fuel consumption rate (FCR), and uses the simulated annealing method to resolve the model. Their results showed that the fuel consumption for 27 VRPs is reduced by an average of 5%. Although the previous works have addressed the basic VRP, real life VRPs often contain additional variables (e.g., road situations and vehicle types). Since it is unclear whether the basic VRP can be applied to the reality of multiple vehicle types and alternative path selection problems, and the factors in calculating carbon foot prints depend upon different vehicles, vehicle travel speeds, and road situations, most previous works on VRP often only considered a single characteristic, such as vehicle types [6], time-dependent [7], and multiple alternative path selection [8]. However, practical problems often include multiple variables, and applying the VRP that only consider one variable to practical problems may pose difficulties.

_{2}emission. Compared with the conventional VRPs, the MTHVRPP can be broadly applied to practical situations, which is more in line with real life.

- This paper proposes a GA for minimizing the carbon footprint for the time-dependent heterogeneous-fleet vehicle routing problem with alternative paths (MTHVRPP).
- In solution representation, time is divided into numerous time steps so that the average vehicle speed on different alternative paths and in different time zones can be expressed.
- Compared with previous VRPs, the factors that influence carbon footprints such as vehicle type, alternative path selection, vehicle load, and time zone speed are considered in this paper, in order to conform to real life situations.

## 2. Literature Review

#### 2.1. VRP

#### 2.2. Description of VRP

**Figure 1.**Illustration of the traveling salesman problem (TSP) and vehicle route problem (VRP) route patterns.

#### 2.3. Approaches for VRP

## 3. Our Genetic Algorithm Approach for MTHVRPP

#### 3.1. Problem Description

- The objective of the MTHVRPP is to minimize the total carbon footprint, not distance or time.
- The MTHVRPP has more than one vehicle type, and the capacity and basic costs of each vehicle are different.
- In the MTHVRPP, multiple paths exist between customer nodes; the distance of each path and the speed of different time zones of a day and different vehicles are also different.
- Because multiple vehicle types exist, the basic costs for each vehicle are different. The carbon emissions also change based on the load and travel speed during the routing trip.

#### 3.1.1. Problem Assumptions

- (1)
- Route information:
- There is only a single depot, and its location is fixed.
- The distance between customers, the customer locations, the customer demand quantity, and the vehicle speed on routes between customer nodes in different time zones are known and fixed.
- Vehicles leave from the depot and return to the depot after servicing all the customers.
- The capacities for different vehicles are fixed and known.
- The total customer demand requirements on each route cannot exceed the vehicle capacity.
- More than one path between nodes can be selected.
- Each customer node must be visited, and is only visited once.
- Different vehicle types can be used for delivery according to the number of the required routes and the routing distance.

- (2)
- Costs and time:
- There is a positive proportional relationship between the vehicle routing distance and the vehicle carbon emission.
- There is a positive proportional relationship between the vehicle load and the vehicle carbon emission.
- The carbon emission from servicing customers is not considered.
- The vehicle types and vehicle number limitations are not considered.
- The routing time of each vehicle is limited.
- The vehicle fixed costs are not considered.
- The time window limitations are not considered.

#### 3.1.2. Notations

- Node information:
I : I = {V _{1}, V_{2}, V_{3}... V_{S}}, where v_{i}is a customer nodes for i ϵ {1, 2, …, S}.v _{0}: v _{0}is the depot node.I _{0}: I _{0}= I ∪ {v_{0}} - Vehicle information
U : The number of vehicle types. A _{u}: The surface area of each vehicle type. W _{u}: The empty weight of each vehicle type. - Constant
q _{i}: The cargo demand requirement of node i. Z _{ij}: The number of selectable paths from node i to node j. Q _{u}: The maximum load capacity of vehicle type u. : The speed in time zone m on path z between nodes i and j. D _{ijz}: The distance of path z between nodes i and j. S : The total number of customer nodes. N : The number of vehicle routes. T _{m}: The starting time point for time zone m. L : The vehicle routing time limitation. - Label
i, j, k : Node label. z : Route label. m : Time zone label. u : Vehicle type label. - Variables:
s : The number of customers’ nodes that have already been serviced. : The cargo load of vehicle type u on route n after passing node i. : The time point when the vehicle reaches node i on route n. : The time point when the vehicle leaves node i on route n. : The time consumed by traveling on route n in time zone m on path z between i and j. : The total routing time between nodes i and j on route n. : The routing distance taken on route n in time zone m on path z between nodes i and j. L _{n}: The time traveling on route n. : The total carbon emission of vehicle u between node i and j on route n. g _{n}: The total carbon emission on route n. TG : The total carbon emission for the entire route. TD : The total routing distance. TT : The total routing time. - Variables:

_{ij}is the energy (J) required by the vehicle from customer nodes i to j; w represents the weight of an empty vehicle (kg); f

_{ij}represents the vehicle load (kg) between customer nodes i and j; d

_{ij}represents the distance (m) between customer nodes i and j; v

_{ij}represents the travel speed (m/s) of the vehicle between customer nodes i and j; α

_{ij}is the specific route-related constant and β is the vehicle-related constant, respectively calculated as follows:

_{ij}= a + gsinθ

_{ij}+ gC

_{r}cosθ

_{ij}

_{d}Aρ

^{2}); g represents the gravitational constant of 9.81 (m/s

^{2}); θ

_{ij}represents the road inclination angle between customer nodes i and j; C

_{r}represents the rolling resistance coefficient; C

_{d}represents the road resistance coefficient; A represents the surface area of the vehicle (m

^{2}); ρ represent the air viscosity coefficient. However, the aforementioned equation is only appropriate for the ordinary VRP and is not completely suitable for the MTHVRPP proposed in this paper. Thus, we improve the previously described equation and enable the equation to be used in MTHVRPP with different vehicle speeds at different time zones, as shown below:

_{u}is the empty weight of vehicle type u (kg); is the load (kg) of vehicle type u on route n after passing through node i; is the vehicle speed (m/s) in time zone m on path z between nodes i and j; is the distance (m) traveled by the vehicle on route n in time zone m on path z between customer nodes i and j. Furthermore, because the surface area of each vehicle type is different, Equation (3) is modified to:

_{d}A

_{u}ρ

#### 3.1.3. Mathematical Model

- Objective equation:

^{2}/s

^{2}, our requirement is that this unit must be converted to a kilowatt-hour. One liter of gasoline can produce approximately 8.8 kilowatt-hours of energy [9] and approximately 2.32 kg of CO

_{2}[9]. Thus, this objective equation is multiplied with a coefficient of 2.32/8.8/360000.

- The constraints of vehicle flow rate conservation:

- Vehicle load capacity constraint:

_{i}represents the demand quantity requirement of each node.

- Vehicle routing time constraint:

_{n}represents the routing time of each route n and L is the maximum routing time.

_{n}≤ L, ∀n = 1, 2, …, N.

- Definition constraint:

- Calculation of parameters:

_{i,j}is the path selected between i and j.

_{n}must be smaller than L.

#### 3.1.4. Example Description

^{2}), the vehicle front surface area A = 5 (m

^{2}), the vehicle acceleration speed a = 0 (m/s

^{2}), the road inclination angle between customer nodes is θ

_{ij}= 0, the rolling resistance coefficient C

_{r}= 0.01, the road resistance coefficient C

_{d}= 0.7, and the air viscosity coefficient ρ = 1.2041(kg/m

^{3}). Detailed calculation for this route configuration is shown in Table 1.

#### 3.2. Genetic Algorithm Design

#### 3.2.1. Solution Representation

**Table 1.**The detailed spreadsheet for the problem in Figure 1.

Time Zone | 1 | 2 | 3 | 4 | 5 | 6 | ||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Route | Vehicle type | Path | Distance | Weight | Speed | Routing distance | CO_{2} | Speed | Routing distance | CO_{2} | Speed | Routing distance | CO_{2} | Speed | Routing distance | CO_{2} | Speed | Routing distance | CO_{2} | Speed | Routing distance | CO_{2} | Total | |||

n = 1 | u = 1 | 0-1 | 20 | 3+3 = 6 | 40 | 6.67 | 0.41 | 50 | 8.33 | 0.60 | 60 | 5 | 0.42 | 1.45 | ||||||||||||

1-3 | 10 | 3+1 = 4 | 60 | 5 | 0.35 | 50 | 5 | 0.29 | 0.65 | |||||||||||||||||

3-0 | 15 | 3+0 = 3 | 50 | 3.33 | 0.17 | 40 | 6.67 | 0.27 | 50 | 5 | 0.25 | 0.7 | ||||||||||||||

n = 2 | u = 2 | 0-2 | 15 | 5+6 = 10 | 50 | 8.33 | 0.84 | 50 | 6.67 | 0.67 | 1.52 | |||||||||||||||

2-4 | 10 | 5+3 = 8 | 40 | 1.34 | 0.10 | 40 | 6.67 | 0.51 | 40 | 2.00 | 0.15 | 0.77 | ||||||||||||||

4-5 | 5 | 5+2 = 7 | 60 | 5 | 0.46 | 0.46 | ||||||||||||||||||||

5-0 | 15 | 5+0 = 5 | 45 | 1.33 | 0.08 | 45 | 7.50 | 0.45 | 45 | 6.17 | 0.37 | 0.9 | ||||||||||||||

n = 3 | u = 2 | 0-6 | 20 | 5+5 = 10 | 60 | 10.00 | 1.14 | 50 | 8.33 | 0.84 | 40 | 1.67 | 0.15 | 2.14 | ||||||||||||

6-0 | 20 | 5+0 = 5 | 40 | 5 | 0.27 | 50 | 8.33 | 0.54 | 60 | 6.67 | 0.52 | 1.35 | ||||||||||||||

Total | 9.95 |

#### 3.2.2. Our Algorithm

- (1)
- Initial population

- (2)
- Capacity check

- (3)
- Alternative path selection

- (4)
- Fitness evaluation

- (5)
- Population selection

- (6)
- Crossover and mutation

- (7)
- Termination condition

## 4. Experimental Design and Results

^{®}Core™ (Santa Clara, CA, USA) i7-3610QM CPU@ 2.30GHz 2.30 GHz and memory of 8 GB.

#### 4.1. Description of Experimental Problems and Parameter Setting

#### 4.1.1. Parameter Settings

_{bm}for each time zone m is as follows:

Vehicle type | Type 1 | Type 2 | Type 3 | Type 4 | Type 5 |
---|---|---|---|---|---|

Capacity (ton) | 1 | 1.5 | 2 | 2.5 | 3 |

Empty vehicle weight (ton) | 0.75 | 1 | 1.5 | 2 | 2.5 |

_{ij}, vehicle-related constant β, vehicle acceleration a, road inclination angle θ

_{ij}between customer nodes, rolling resistance coefficient C

_{r}, road resistance coefficient C

_{d}, and air viscosity coefficient ρ, our experiments refer the experimental settings in [4] (as shown in Table 3).

Parameter | Value |
---|---|

Vehicle acceleration a | 0 |

Road inclination angle θ_{ij} | 0.01 |

Rolling resistance coefficient C_{r} | 0.7 |

Road resistance coefficient C_{d} | 0 |

Air viscosity coefficient ρ | 1.2041 |

#### 4.1.2. Parameter Setting for our GA

#### 4.2. Influence of Different Objectives on Carbon Footprint Emission

**Figure 6.**Experimental results with different combinations of numbers of generations and mutation rates.

#### 4.2.1. Experimental Results

Objective | Minimum Carbon Footprint | Minimum Routing Time | Minimum Routing Distance | ||||||
---|---|---|---|---|---|---|---|---|---|

Number of Times | Carbon Footprint (kg) | Routing Time (minutes) | Routing Distance (km) | Carbon Footprint (kg) | Routing Time (minutes) | Routing Distance (km) | Carbon Footprint (kg) | Routing Time (minutes) | Routing Distance (km) |

1 | 201.39 | 3522.89 | 3466.02 | 221.64 | 3118.71 | 3286.00 | 222.70 | 3045.60 | 3162.49 |

2 | 202.85 | 3496.71 | 3454.11 | 210.79 | 2984.85 | 3124.42 | 213.57 | 3121.18 | 3196.92 |

3 | 197.75 | 3440.53 | 3385.34 | 240.77 | 3095.69 | 3329.09 | 201.78 | 3217.58 | 3236.93 |

4 | 198.05 | 3415.39 | 3355.70 | 224.88 | 3019.47 | 3173.51 | 213.09 | 3215.16 | 3290.20 |

5 | 203.92 | 3552.83 | 3498.39 | 233.80 | 3093.27 | 3291.10 | 209.70 | 3249.87 | 3278.01 |

6 | 198.69 | 3421.67 | 3363.89 | 231.07 | 3015.33 | 3200.99 | 208.12 | 3147.69 | 3210.50 |

7 | 203.42 | 3494.83 | 3484.16 | 219.76 | 3123.51 | 3256.96 | 218.14 | 3156.72 | 3228.00 |

8 | 198.72 | 3480.55 | 3409.40 | 236.94 | 3089.65 | 3293.46 | 225.73 | 3141.88 | 3257.11 |

9 | 203.53 | 3358.02 | 3353.18 | 249.57 | 3142.63 | 3378.40 | 224.07 | 3125.42 | 3235.01 |

10 | 187.91 | 3194.74 | 3130.70 | 231.80 | 3136.67 | 3349.25 | 217.34 | 3092.43 | 3185.39 |

Average | 199.62 | 3437.82 | 3390.09 | 230.10 | 3081.98 | 3268.32 | 215.42 | 3151.35 | 3228.06 |

Standard deviation | 4.79 | 102.87 | 106.12 | 11.23 | 55.68 | 80.40 | 7.65 | 61.94 | 40.45 |

#### 4.2.2. Comparison of Results Obtained with and without Alternative Path Selection Considerations

Instance | c20-3 | c50-3 | c100-3 | c20-5 | c50-5 | c100-5 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Route | Yes | No | Yes | No | Yes | No | Yes | No | Yes | No | Yes | No |

1 | 22.53 | 22.93 | 82.51 | 89.51 | 195.82 | 202.29 | 21.30 | 23.80 | 83.00 | 88.07 | 201.39 | 212.50 |

2 | 20.53 | 23.27 | 80.11 | 90.28 | 193.65 | 206.02 | 20.00 | 23.53 | 84.56 | 91.24 | 202.85 | 210.07 |

3 | 21.90 | 24.18 | 84.32 | 87.23 | 196.88 | 207.01 | 20.70 | 23.81 | 84.23 | 87.15 | 197.75 | 210.66 |

4 | 21.63 | 24.85 | 82.77 | 85.62 | 197.28 | 207.64 | 20.48 | 22.40 | 79.64 | 89.66 | 198.05 | 212.05 |

5 | 22.66 | 23.62 | 83.07 | 88.28 | 197.74 | 207.53 | 20.86 | 24.51 | 82.23 | 89.99 | 203.92 | 210.04 |

6 | 20.48 | 25.29 | 83.56 | 85.97 | 196.42 | 208.03 | 19.95 | 22.36 | 80.30 | 88.96 | 198.69 | 209.65 |

7 | 22.01 | 24.04 | 82.32 | 90.34 | 196.82 | 207.68 | 21.81 | 22.87 | 83.85 | 91.33 | 203.42 | 203.56 |

8 | 22.83 | 25.73 | 82.84 | 89.35 | 199.21 | 210.20 | 20.87 | 23.20 | 86.17 | 88.93 | 198.72 | 208.24 |

9 | 21.26 | 24.10 | 79.85 | 88.11 | 192.69 | 204.05 | 21.82 | 21.96 | 83.44 | 90.15 | 203.53 | 210.41 |

10 | 21.56 | 24.20 | 79.28 | 87.71 | 197.23 | 201.44 | 21.70 | 22.77 | 81.13 | 91.70 | 187.91 | 211.31 |

Average | 21.74 | 24.22 | 82.06 | 88.24 | 196.37 | 206.19 | 20.95 | 23.12 | 82.86 | 89.72 | 199.62 | 209.85 |

Standard deviation | 0.82 | 0.87 | 1.71 | 1.66 | 1.92 | 2.76 | 0.70 | 0.79 | 2.04 | 1.48 | 4.79 | 2.52 |

#### 4.2.3. Analysis of the Experimental Results Parameters for All Sample Problems

_{0}: The variances of the carbon footprint results yielded under the three objectives are equal”, is not rejected when the level of significance is set at α = 0.05. Thus, considering the homogeneity of the samples, they can be used to conduct a single-factor ANOVA. The results show that the p-valueis0.000, which is smaller than 0.05. Hence, the null hypothesis, that is, “H

_{0}: The carbon footprint results yielded under the three objectives are equal”, is rejected when the level of significance is set at α = 0.05. Subsequently, Scheffé’s method is employed to conduct post hoc multiple comparisons. The results show the means of the carbon footprint results obtained under various objectives differ. Finally, a t-test is conducted between the carbon footprint results obtained using minimal carbon footprint as the objective and those obtained using minimal routing time or minimal routing distance as the objectives. The results show that the p-value is 0.000, which is smaller than α = 0.05. Thus, we conclude that when the level of significance is set at α = 0.05, the carbon footprint results obtained using minimal carbon footprint as the objective are significantly smaller than those obtained using minimal routing time or minimal routing distance as the objectives.

#### 4.2.4. Analysis of the Decreases in the Carbon Footprint Results for All Sample Problems

Problem Instance | Performance | Using the Objective of Minimal Carbon Footprint | Comparison to the Results with Objective of Minimal Routing Time | Comparison to the Results with Objective of Minimal Routing Distance | ||
---|---|---|---|---|---|---|

Mean | Mean | Decrease (%) | Mean | Decrease (%) | ||

c20-3 | Carbon footprint (kg) | 21.74 | 26.31 | 17.38 | 23.91 | 9.06 |

Routing time (min) | 564.09 | 484.47 | 16.43 | 513.84 | −9.78 | |

Routing distance (km) | 465.82 | 454.27 | −2.54 | 445.96 | −4.45 | |

c50-3 | Carbon footprint (kg) | 82.06 | 96.11 | 14.62 | 85.79 | 4.34 |

Routing time (min) | 1580.70 | 1439.02 | −9.85 | 1461.75 | −8.14 | |

Routing distance (km) | 1480.49 | 1476.98 | −0.24 | 1426.82 | −3.76 | |

c100-3 | Carbon footprint (kg) | 196.37 | 219.13 | 10.39 | 203.09 | 3.31 |

Routing time (min) | 3441.72 | 3129.40 | −9.98 | 3222.66 | −6.80 | |

Routing distance (km) | 3401.05 | 3288.08 | −3.44 | 3254.69 | −4.50 | |

c20-5 | Carbon footprint (kg ) | 20.95 | 27.07 | 22.60 | 23.68 | 11.52 |

Routing time (min) | 538.85 | 461.29 | −16.81 | 482.37 | −11.71 | |

Routing distance (km) | 439.12 | 433.76 | −1.23 | 418.20 | −5.00 | |

c50-5 | Carbon footprint (kg ) | 82.86 | 100.85 | 17.84 | 88.95 | 6.85 |

Routing time (min) | 1575.54 | 1387.64 | −13.54 | 1433.68 | −9.89 | |

Routing distance (km) | 1477.79 | 1441.01 | −2.55 | 1398.84 | −5.64 | |

c100-5 | Carbon footprint (kg) | 199.62 | 230.10 | 13.25 | 215.42 | 7.33 |

Routing time (min) | 3437.82 | 3081.98 | −11.55 | 3151.35 | −9.09 | |

Routing distance (km) | 3390.09 | 3268.32 | −3.73 | 3228.06 | −5.02 |

**Figure 7.**Comparison of the decreases in carbon footprints obtained using various vehicle types and objectives.

#### 4.2.5. Analysis of the Computational Time Required for Executing our GA

## 5. Conclusions and Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Liu, W.-Y.; Lin, C.-C.; Chiu, C.-R.; Tsao, Y.-S.; Wang, Q.
Minimizing the Carbon Footprint for the Time-Dependent Heterogeneous-Fleet Vehicle Routing Problem with Alternative Paths. *Sustainability* **2014**, *6*, 4658-4684.
https://doi.org/10.3390/su6074658

**AMA Style**

Liu W-Y, Lin C-C, Chiu C-R, Tsao Y-S, Wang Q.
Minimizing the Carbon Footprint for the Time-Dependent Heterogeneous-Fleet Vehicle Routing Problem with Alternative Paths. *Sustainability*. 2014; 6(7):4658-4684.
https://doi.org/10.3390/su6074658

**Chicago/Turabian Style**

Liu, Wan-Yu, Chun-Cheng Lin, Ching-Ren Chiu, You-Song Tsao, and Qunwei Wang.
2014. "Minimizing the Carbon Footprint for the Time-Dependent Heterogeneous-Fleet Vehicle Routing Problem with Alternative Paths" *Sustainability* 6, no. 7: 4658-4684.
https://doi.org/10.3390/su6074658