A Sustainable Multi-Objective Model for Capacitated-Electric-Vehicle-Routing-Problem Considering Hard and Soft Time Windows as Well as Partial Recharging
<p>The conceptual model of the research.</p> "> Figure 2
<p>An example of the solution chromosome.</p> "> Figure 3
<p>An example of the solution chromosome.</p> "> Figure 4
<p>An example of a crossover operator.</p> "> Figure 5
<p>An example of a crossover operator.</p> "> Figure 6
<p>An example of crossover operator.</p> "> Figure 7
<p>Flowchart of the NSGA-II-TLBO algorithm.</p> "> Figure 8
<p>Three-dimensional Pareto front obtained by FPGP, MOSA, MOGW, MOPSO, and NSGA-II-TLBO.</p> "> Figure 9
<p>Relationship between economic, environmental, and social objectives.</p> "> Figure 10
<p>Route of EVs for the selected Pareto point 4.</p> "> Figure 11
<p>Comparison of algorithms in terms of the four indexes.</p> "> Figure 12
<p>The results of the statistical analysis.</p> ">
Abstract
:1. Introduction
- What are the indicators of pollution reduction in the studied CVRP problem in this research?
- What is the effect of the time window on reducing the emission of pollution by the green vehicle?
- How is it possible to recharge green vehicles along the way?
- What is the effect of using green vehicles with different capacities on reducing pollution?
2. Literature Review
- Presenting a multi-objective mathematical programming model for the Capacitated Electric Vehicle Routing Problem (CEVRP), taking both soft and hard time windows as well as the possibility of EV charging into consideration,
- Proposing a hybrid of Non-dominated Sorting Genetic Algorithm (NSGA-II) and Teaching–Learning-Based Optimization (TLBO) meta-heuristic algorithms to solve this problem,
- Comparing the performance of the proposed hybrid NSGA-II-TLBO meta-heuristic algorithm with other well-known meta-heuristic algorithms by solving different test problems based on six criteria,
- Performing statistical analysis for determining the meaningful difference between the performance of the algorithms according to six criteria.
3. Problem Definition
4. Multi-Objective Mathematical Model
- A group of customers that need to receive goods are included in a distribution network.
- The EV fleet is heterogeneous.
- The tour of each EV starts at the depot with the total amount of goods that must be delivered to the customers, and each EV has a full charge while departing the depot.
- Partial delivery is not allowed.
- Each customer must be serviced (receive goods) once with a certain fleet of identical EVs.
- Each tour terminates in the depot, and a time window is assumed for the time to reach the depot and can go beyond the allowable time; however, this amount is taken into consideration in the environmental (second) objective function.
- Every EV is chosen for one route.
- A hard time window is assumed for the customer’s visiting time for rendering service, which must not go beyond the time interval.
- A certain time interval is assumed for the vehicles’ accessibility.
- There are various types of EVs with a defined capacity.
- The vehicle load cannot exceed the vehicle capacity at any time during the tour.
- The capacity of the depot is large enough.
- Charging stations may be visited more than once by an EV.
- Partial charging is also possible.
- EVs are assumed to be charged at a fixed rate at charging stations.
- The battery charge of the EV and its load must be zero when it returns to the depot.
- The load flow is calculated for each arc.
5. Solution Approaches
- : The degree of achievement of the goal
- : The upper limit for the goal
- : The lower limit for the goal
- : Positive changes from the expected level
- : Negative changes from the expected level
- : The expected goal of the objective function
5.1. Solution Representation Scheme
- The first vehicle: Customer 3, Customer 1, and Customer 4
- The second vehicle: Customer 5 and Customer 2.
5.2. Multi-Objective Simulated Annealing (MOSA) Algorithm
5.3. Multi-Objective Grey Wolf Optimization (MOGWO) Algorithm
5.4. Non-Dominated Sorting Genetic Algorithm II (NSGA-II)
- Generating the initial population,
- Calculating the fitness values (according to the objective functions),
- Sorting non-dominated population and computing the swarm distance,
- Implementing the crossover and mutation operators for generating the next population (offspring),
- Combining the initial population with the next population generated by the crossover and mutation operators,
- Swapping the initial population with the fittest (best) population members generated in the previous steps,
- All the aforementioned steps are iterated to achieve the optimal conditions.
5.5. Teaching–Learning-Based Optimization (TLBO) Algorithm
5.6. Hybrid of Non-Dominated Sorting Genetic Algorithm (NSGA-II) and Teaching–Learning-Based Optimization (TLBO) Algorithms
5.7. Multi-Objective Particle Swarm Optimization (MOPSO) Algorithm
6. A Numerical Example for Validating the Proposed Model and Methodology
6.1. Comparison of the Proposed Methods for Solving the Mathematical Programming Model
6.2. The Criteria for the Performance Comparison of the Meta-Heuristic Algorithms
7. Performance Evaluation of the Proposed Meta-Heuristic Algorithms
8. Statistical Analysis
9. Discussion
10. Conclusions
- Considering multiple depots in EVRP,
- Considering the conditions of uncertainty to achieve results similar to reality,
- Using artificial intelligence to predict system costs in the future,
- Solving the mathematical model with exact methods such as the Augmented Epsilon Constraint (AEC) method.
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Conflicts of Interest
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e1 | e2 | |||||
D0 | 0 | 0 | 0 | 4 | 5 | |
1 | 2 | 12 | 20 | 25 | 48 | |
2 | 5 | 10 | 47 | 55 | 65 | |
3 | 7 | 13 | 21 | |||
4 | 9 | 18 | 29 | g | 0.001 | |
5 | 5 | 12 | 41 | |||
6 | 3 | 15 | 49 | |||
7 | 8 | 12 | 29 | |||
F1 | 0 | 0 | 0 | |||
F2 | 0 | 0 | 0 | |||
Objective | PayoffMat | Min | Max | ||||||
---|---|---|---|---|---|---|---|---|---|
O1 | O2 | O3 | O1 | O2 | O3 | O1 | O2 | O3 | |
O1 | 7125.3 | 11,256.9 | 31,247.1 | 7125.3 | 22.054 | 23 | 31,247.1 | 41.08 | 117 |
O2 | 25.02 | 22.054 | 41.08 | ||||||
O3 | 108 | 117 | 23 |
O1 | O2 | O3 | |
---|---|---|---|
1 | 7125.3 | 23.07 | 61 |
2 | 7051.07 | 30.18 | 32 |
3 | 7949.2 | 21.12 | 79 |
4 | 7091.7 | 25.24 | 34 |
5 | 7401.2 | 31.14 | 29 |
6 | 6941.7 | 27.09 | 65 |
Problem | Number of Customers | Number of Vehicles | Number of Charging Stations |
---|---|---|---|
1 | 5 | 2 | 1 |
2 | 10 | 2 | 2 |
3 | 15 | 2 | 3 |
4 | 20 | 2 | 4 |
5 | 25 | 2 | 5 |
6 | 30 | 3 | 6 |
7 | 35 | 3 | 6 |
8 | 40 | 3 | 6 |
9 | 45 | 3 | 6 |
10 | 50 | 3 | 6 |
11 | 55 | 4 | 7 |
12 | 60 | 4 | 7 |
13 | 65 | 4 | 7 |
14 | 70 | 4 | 7 |
15 | 75 | 4 | 7 |
16 | 80 | 6 | 8 |
17 | 85 | 6 | 8 |
18 | 90 | 8 | 8 |
19 | 95 | 8 | 8 |
20 | 100 | 10 | 10 |
Problem | FPGP | MOPSO | MOGWO | MOSA | NSGAII-TLBO |
---|---|---|---|---|---|
O1 | |||||
1 | 4617.53 | 4945.46 | 4675.20 | 4863.44 | 4817.52 |
2 | 5656.48 | 6055.20 | 5724.30 | 5905.74 | 5837.88 |
3 | 7272.62 | 7796.81 | 7370.72 | 8108.23 | 7440.87 |
FPGP | MOPSO | MOGWO | MOSA | NSGAII-TLBO | |
O2 | |||||
1 | 15.290 | 15.491 | 15.987 | 15.686 | 15.316 |
2 | 17.208 | 17.645 | 18.210 | 17.645 | 17.430 |
3 | 26.765 | 26.806 | 27.665 | 27.309 | 27.042 |
FPGP | MOPSO | MOGWO | MOSA | NSGAII-TLBO | |
O3 | |||||
1 | 46.78 | 47.36 | 46.85 | 47.96 | 46.84 |
2 | 52.63 | 53.94 | 53.36 | 53.95 | 53.28 |
3 | 81.86 | 81.97 | 81.08 | 83.52 | 82.69 |
MOSA | NSGA-II-TLBO | |||||||
---|---|---|---|---|---|---|---|---|
Time | MID | HV | MOCV | Time | MID | HV | MOCV | |
1 | 65.09 | 919.02 | 13,231.02 | 11,549.16 | 386.27 | 633.68 | 17,791.52 | 319.97 |
2 | 65.7 | 1712.6 | 16,977.4 | 14,819.32 | 389.94 | 1180.86 | 40,493.75 | 728.26 |
3 | 75.1 | 4656.17 | 44,545 | 38,882.66 | 445.68 | 3210.51 | 142,673.26 | 2565.9 |
4 | 80.66 | 5311.5 | 48,515.08 | 42,348.07 | 478.73 | 3662.38 | 183,035.93 | 3291.8 |
5 | 85.07 | 2978.64 | 31,523.15 | 27,516.08 | 504.83 | 2053.83 | 90,933.26 | 1635.39 |
6 | 91.23 | 4288.6 | 49,247.5 | 42,987.39 | 541.38 | 2957.07 | 82,993.78 | 1492.6 |
7 | 97.32 | 3087.81 | 52,539.83 | 45,861.22 | 577.52 | 2129.11 | 116,355.12 | 2092.59 |
8 | 111.77 | 4714.77 | 53,777.57 | 46,941.63 | 663.32 | 3250.92 | 179,499.66 | 3228.21 |
9 | 127.86 | 2813.16 | 57,346.39 | 50,056.79 | 758.76 | 1939.73 | 125,680.74 | 2260.3 |
10 | 152.51 | 2337.07 | 69,921.12 | 61,033.09 | 905.11 | 1611.46 | 128,844.43 | 2317.2 |
11 | 182.54 | 4548.1 | 33,858.19 | 29,554.3 | 1083.32 | 3136 | 177,516.44 | 3192.54 |
12 | 191.64 | 2915.71 | 86,182.17 | 75,227.11 | 1137.31 | 2010.44 | 143,633.2 | 2583.17 |
13 | 208.55 | 4087.96 | 73,498.75 | 64,155.94 | 1237.68 | 2818.72 | 115,628.28 | 2079.51 |
14 | 215.32 | 6498.74 | 124,063.35 | 108,293.02 | 1277.86 | 4481.01 | 112,985.93 | 2031.99 |
15 | 224.73 | 6998.01 | 85,199.74 | 74,369.56 | 1333.72 | 4825.26 | 174,065.79 | 3130.48 |
16 | 230.16 | 6609.68 | 137,592.96 | 120,102.8 | 1365.96 | 4557.5 | 217,056.48 | 3903.65 |
17 | 254.68 | 6846.62 | 162,772.63 | 142,081.76 | 1511.43 | 4720.87 | 238,465.43 | 4288.67 |
18 | 269.82 | 4731.44 | 113,299.81 | 98,897.69 | 1601.3 | 3262.42 | 233,045.34 | 4191.2 |
19 | 276.01 | 11,529.54 | 197,777.95 | 172,637.37 | 1638.02 | 7949.83 | 366,747.57 | 6595.76 |
20 | 292.38 | 8221.25 | 175,970.2 | 153,601.72 | 1735.16 | 5668.71 | 216,963.08 | 3901.97 |
AV | 164.9 | 4790.32 | 81,391.99 | 71,045.83 | 978.67 | 3303.01 | 155,220.44 | 2791.56 |
MOGWO | MOPSO | |||||||
---|---|---|---|---|---|---|---|---|
Time | MID | HV | MOCV | Time | MID | HV | MOCV | |
1 | 112.08 | 833.57 | 10,615.48 | 8916.37 | 87.64 | 926.83 | 17,041.83 | 6736.64 |
2 | 125.59 | 1988.58 | 12,214.20 | 10,259.20 | 98.20 | 2211.06 | 19,608.38 | 7751.20 |
3 | 130.49 | 3778.24 | 32,419.18 | 27,230.18 | 102.03 | 4200.94 | 52,044.95 | 20,573.38 |
4 | 147.60 | 3208.35 | 33,951.44 | 28,517.18 | 115.41 | 3567.30 | 54,504.80 | 21,545.75 |
5 | 151.44 | 2643.60 | 24,686.61 | 20,735.28 | 118.41 | 2939.37 | 39,631.29 | 15,666.25 |
6 | 154.69 | 3398.37 | 43,113.38 | 36,212.66 | 120.96 | 3778.58 | 69,213.16 | 27,359.97 |
7 | 162.27 | 4477.97 | 42,312.53 | 35,540.00 | 126.89 | 4978.96 | 67,927.49 | 26,851.75 |
8 | 176.17 | 3802.77 | 41,583.71 | 34,927.83 | 137.75 | 4228.22 | 66,757.46 | 26,389.23 |
9 | 186.16 | 8056.84 | 46,576.49 | 39,121.47 | 145.56 | 8958.23 | 74,772.76 | 29,557.68 |
10 | 229.68 | 7127.11 | 55,443.25 | 46,569.02 | 179.59 | 7924.48 | 89,007.22 | 35,184.57 |
11 | 267.72 | 6650.01 | 50,001.34 | 41,998.14 | 209.34 | 7394.00 | 80,270.92 | 31,731.10 |
12 | 287.23 | 3541.16 | 48,298.59 | 40,567.93 | 224.60 | 3937.34 | 77,537.37 | 30,650.53 |
13 | 303.83 | 3844.19 | 36,491.05 | 30,650.30 | 237.57 | 4274.27 | 58,581.82 | 23,157.40 |
14 | 319.05 | 10,205.72 | 61,709.78 | 51,832.52 | 249.47 | 11,347.52 | 99,067.36 | 39,161.33 |
15 | 329.02 | 13,316.20 | 64,525.86 | 54,197.87 | 257.27 | 11,518.73 | 103,588.23 | 40,948.44 |
16 | 343.25 | 5356.00 | 63,463.48 | 53,305.53 | 268.39 | 5955.22 | 101,882.70 | 40,274.24 |
17 | 363.45 | 9095.35 | 63,350.50 | 53,210.63 | 284.19 | 10,112.93 | 101,701.32 | 40,202.54 |
18 | 370.43 | 12,242.12 | 37,840.02 | 31,783.36 | 289.65 | 13,611.76 | 60,747.45 | 24,013.47 |
19 | 388.43 | 14,496.30 | 93,281.12 | 78,350.57 | 303.72 | 16,118.13 | 149,751.22 | 59,196.68 |
20 | 419.89 | 6828.57 | 79,495.31 | 66,771.32 | 328.32 | 7592.54 | 127,619.82 | 50,448.13 |
AV | 248.43 | 9244.55 | 47,068.66 | 39,534.87 | 194.25 | 10,278.82 | 75,562.88 | 29,870.02 |
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Azadi, A.H.S.; Khalilzadeh, M.; Antucheviciene, J.; Heidari, A.; Soon, A. A Sustainable Multi-Objective Model for Capacitated-Electric-Vehicle-Routing-Problem Considering Hard and Soft Time Windows as Well as Partial Recharging. Biomimetics 2024, 9, 242. https://doi.org/10.3390/biomimetics9040242
Azadi AHS, Khalilzadeh M, Antucheviciene J, Heidari A, Soon A. A Sustainable Multi-Objective Model for Capacitated-Electric-Vehicle-Routing-Problem Considering Hard and Soft Time Windows as Well as Partial Recharging. Biomimetics. 2024; 9(4):242. https://doi.org/10.3390/biomimetics9040242
Chicago/Turabian StyleAzadi, Amir Hossein Sheikh, Mohammad Khalilzadeh, Jurgita Antucheviciene, Ali Heidari, and Amirhossein Soon. 2024. "A Sustainable Multi-Objective Model for Capacitated-Electric-Vehicle-Routing-Problem Considering Hard and Soft Time Windows as Well as Partial Recharging" Biomimetics 9, no. 4: 242. https://doi.org/10.3390/biomimetics9040242