Improved Ant Colony Algorithm for the Split Delivery Vehicle Routing Problem
Abstract
:1. Introduction
- For the state transition process of the ant colony algorithm, we propose an adaptive selection threshold to replace the commonly used fixed threshold, which enables the algorithm to better balance convergence speed and global search ability.
- We designed new pheromone updating rules by referring to the max-min ant colony algorithm. To improve the existing methods, we added a pheromone backtracking mechanism and an adaptive pheromone volatile factor to improve the ability of the algorithm to jump out of a local optimum.
- Unlike the common single strategy of customer selection, ours divides customers into initial customers and noninitial customers. We also designed selection strategies for each type of customer, so that the selection of customer nodes is more targeted.
- We applied the proposed improved ant colony algorithm to SDVRP instances of various scales for a comprehensive evaluation. The experiments show that our method can effectively solve problems of different sizes and its performance is competitive compared with other heuristic algorithms.
2. Related Work
2.1. Exact Algorithms
2.2. Heuristic Algorithms
3. Modeling the SDVRP
- (1)
- The distance between any two customer points is symmetric, which can be expressed as .
- (2)
- The distance between the customer points conforms to the triangle inequality and can be expressed as .
- (3)
- The demand of each customer should be met, and this demand can be met by one or more vehicles.
- (4)
- All vehicles start from the depot and return to the depot after completing the task, and the vehicles cannot be overloaded.
4. Improved Ant Colony Algorithm
4.1. Customer Selection Method
4.1.1. Initial Customer Selection
- (1)
- All coverage methods. The vehicle can search the customer points as the initial customer one by one, thereby expanding the search scope. However, when handling larger-scale problems, the algorithm running time may increase dramatically.
- (2)
- Random placement. This method will randomly select the initial customer of the vehicle according to a probability distribution. The advantage of this method is that the running time of the algorithm can be reduced. However, due to the random selection, the best path length generated by each iteration may greatly differ, thus reducing the convergence speed of the algorithm.
4.1.2. Noninitial Customer Selection
4.2. Pheromone Processing
4.2.1. Pheromone Initialization
4.2.2. Pheromone Update Method
4.2.3. Pheromone Restriction
4.3. Search Method Based on the 2-Opt Strategy
- 1.
- Vehicle path self-search
Algorithm 1. Vehicle path self-search |
1: Input solution s; |
2: Set vehicle_num = number of vehicles used in s; |
3: Set old_length = path length in s; |
4: for vehicle = 1 to vehicle_num do |
5: Set c_path = path of current vehicle; |
6: while not all customer pairs in c_path have been exchanged do |
7: Exchange the customer points in c_path in pairs to obtain new solution s’; |
8: Calculate the path length new_length of the new solution s’; |
9: if new_length < old_length do |
10: old_length = new_length; s = s’; |
11: end if |
12: end while |
13: end for |
14: Output s; |
- (1)
- Search should be performed every p iterations. After experimental verification, the best effect is achieved when the value of p is 10.
- (2)
- If the current iterative best solution is better than the global best solution, the search is performed.
- 2.
- Route search is performed between vehicles.
Algorithm 2. Route search between vehicles |
1: Input solution s, vehicle capacity Q; |
2: Set vehicle_num = number of vehicles used in s; |
3: Set f_path = path of the last vehicle in s; |
4: Set old_length = path length in s; |
5: for vehicle = 1 to (vehicle_num − 1) do |
6: Set c_path = path of current vehicle; |
7: while not all customer pairs in c_path and f_path have been swapped do |
8: Swap customer point i in c_path and customer point j in f_path in pairs for new solution s’; |
9: Set demand_c = total demand of customer points on c_path; |
10: Set demand_f = total demand of customer points on f_path; |
11: if demand_c ≤ Q and demand_f ≤ Q do |
12: Calculate the path length new_length of the new solution s’; |
13: if new_length < old_length do |
14: old_length = new_length; s = s’; |
15: end if |
16: end if |
17: end while |
18: end for |
19: Output s; |
4.4. Steps of the IACA
Algorithm 3. IACA |
1: Initialize parameters α, β, ρ, ρmax, Z, λ, ω, A, ant_num, max_iter; |
2: Initialize τij(0) according to Equation (14) for all node pairs; |
3: Set iteration counter iter = 0, Set N = number of customers; |
4: for iter = 1 to max_iter do |
5: for each ant k = 1 to ant_num do |
6: Set demand = demand of nodes; |
7: Set capacity v = Q; |
8: while do |
9: if v > 0 then |
10: Put nodes with demandj > 0 into allow; |
11: end if |
12: if allow is not empty then |
13: if the node to be selected is the first node then |
14: Select the next node j according to Equation (12), then add j into the path; |
15: else |
16: Select the next node j according to Equation (13), then add j into the path; |
17: end if |
18: if v − demandj ≥ 0 then |
19: demandj = 0; v = v − demandj; |
20: else |
21: demandj = demandj − v; v = 0; |
22: end if |
23: else |
24: Return to depot 0; Reload set v = Q; |
25: end if |
26: end while |
27: end for |
28: Save the shortest paths of all ants in this iteration; |
29: Update the pheromone on the path according to Equations (15), (16) and (18); |
30: if iter MOD 10 = 0 or pathib < pathgb then |
31: Execute Algorithm 1; Execute Algorithm 2; |
32: end if |
33: if pathgb has not changed for A consecutive iterations then |
34: Set the pheromone on the path to the value before A iterations; |
35: Update ρ according to Equation (17); |
36: end if |
37: iter = iter + 1; |
38: end for |
39: Output the shortest paths; |
5. Experimental Results
5.1. Datasets
5.2. Setup of Experimental Parameters
5.3. Small-Scale Dataset Experiments
5.3.1. Experiment 1
5.3.2. Experiment 2
5.3.3. Experiment 3
5.4. Medium-Scale Dataset Experiments
5.5. Large-Scale Dataset Experiments
5.6. Exploratory Ability of the Algorithm
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Type | Algorithm | Author(s) |
---|---|---|
Exact algorithm | Branch-and-cut algorithm | Archetti et al. [28] |
Ferreira et al. [29] | ||
Wolfinger et al. [30] | ||
Hernández-Pérez et al. [31] | ||
Branch-and-price algorithm | Casazza et al. [32] | |
Li et al. [33] | ||
Branch-price-and-cut algorithm | Gschwind et al. [34] | |
Li et al. [35] | ||
Heuristic algorithm | Tabu search algorithm | Qiu et al. [8] |
Aleman et al. [9] | ||
Xia et al. [36] | ||
Genetic algorithm | Jiang et al. [10] | |
Wilck et al. [11] | ||
Fan et al. [37] | ||
Particle swarm algorithm | Shi et al. [12] | |
Qing et al. [13] | ||
Variable neighborhood search algorithm | Chen et al. [14] | |
Han et al. [15] | ||
Large neighborhood search algorithm | Wolfinger [16] | |
Gasque et al. [17] | ||
Gu et al. [38] | ||
Ant colony algorithm | Tang et al. [18] | |
Mcnabb et al. [19] | ||
Yang et al. [20] | ||
Simulated annealing algorithm | Moshref-Javadi et al. [39] | |
Guo et al. [40] |
Symbol | Definition |
---|---|
N | The number of customers. |
M | The minimum number of vehicles, the value of which is shown in Equation (1). |
V | The set of depot and customer points, including 0 and Vc, where 0 represents the depot and Vc = {1, 2, …, N} represents the customer set. |
dij | The distance between customers i and j. |
qi | The demand of customer i. |
Q | The capacity of each vehicle. |
The indicator of whether edge (i, j) has been traversed by vehicle m, as shown by Equation (2). | |
Customer i’s demand supplied by vehicle m. |
Name 1 | N 2 | D 3 | Q 4 | M 5 | BKS 6 |
---|---|---|---|---|---|
S51D1 | 50 | [⌈0.01Q⌉, ⌊0.1Q⌋] | 160 | 3 | 458 |
S51D2 | 50 | [⌈0.1Q⌉, ⌊0.3Q⌋] | 160 | 9 | 726 |
S51D3 | 50 | [⌈0.1Q⌉, ⌊0.5Q⌋] | 160 | 15 | 972 |
S51D4 | 50 | [⌈0.1Q⌉, ⌊0.9Q⌋] | 160 | 27 | 1677 |
S51D5 | 50 | [⌈0.3Q⌉, ⌊0.7Q⌋] | 160 | 23 | 1440 |
S51D6 | 50 | [⌈0.7Q⌉, ⌊0.9Q⌋] | 160 | 41 | 2327 |
S76D1 | 75 | [⌈0.01Q⌉, ⌊0.1Q⌋] | 160 | 4 | 594 |
S76D2 | 75 | [⌈0.1Q⌉, ⌊0.3Q⌋] | 160 | 15 | 1147 |
S76D3 | 75 | [⌈0.1Q⌉, ⌊0.5Q⌋] | 160 | 23 | 1474 |
S76D4 | 75 | [⌈0.1Q⌉, ⌊0.9Q⌋] | 160 | 37 | 2257 |
S101D1 | 100 | [⌈0.01Q⌉, ⌊0.1Q⌋] | 160 | 5 | 716 |
S101D2 | 100 | [⌈0.1Q⌉, ⌊0.3Q⌋] | 160 | 20 | 1393 |
S101D3 | 100 | [⌈0.1Q⌉, ⌊0.5Q⌋] | 160 | 31 | 1975 |
S101D5 | 100 | [⌈0.3Q⌉, ⌊0.7Q⌋] | 160 | 48 | 2915 |
Name 1 | N | Q | D(avg) 2 | D(std) 3 | BKS |
---|---|---|---|---|---|
kelly01 | 240 | 550 | 20 | 10 | 5646.46 |
kelly02 | 320 | 700 | 20 | 10 | 8566.04 |
kelly03 | 400 | 900 | 20 | 10 | 11,649.06 |
kelly04 | 480 | 1000 | 20 | 10 | 14,639.32 |
α | β | λ | ρs | ρmax | ant_num |
---|---|---|---|---|---|
2 | 8 | 0.5 | 0.2 | 0.8 | 40 |
Algorithm | Best Solution | Average Solution | Time/s 1 | NV 2 | Iterations 3 |
---|---|---|---|---|---|
CA | 1721.7 | 1728.9 | 1.33 | 10 | 500 |
PSO | 1722.1 | - | - | 10 | 200 |
ACO | 1836.0 | - | - | 10 | 1000 |
ABC | 1757.6 | 1785.7 | 1.71 | 10 | 50 |
PSA | 1827.7 | 1832.3 | 2.11 | 10 | 500 |
IPES | 1705.8 | - | 4.60 | 10 | 100 |
IACA | 1695.8 | 1713.1 | 1.30 | 10 | 60 |
Algorithm | Best Solution | Average Solution | Time/s | NV | Iterations |
---|---|---|---|---|---|
CA | 172.5 | 173.9 | 1.35 | 8 | 500 |
PSO | 171.1 | - | - | 8 | 200 |
IPES | 170.5 | - | - | 8 | 100 |
TSA | 186.4 | 192.3 | 168.60 | 8 | 1000 |
IACA | 170.5 | 170.8 | 1.40 | 8 | 60 |
Algorithm | Best Solution | Average Solution | Time/s | NV | Iterations |
---|---|---|---|---|---|
PSO | 208.24 | 213.19 | - | 7 | 200 |
IACA | 199.58 | 208.14 | 3.75 | 7 | 60 |
Dataset | BKS | Best Solution | Average Solution | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
TACA | ASGCT * | ACGCT * | MMGCT * | IACA | TACA | ASGCT * | ACGCT * | MMGCT * | IACA | ||
S51D1 | 458 | 545.84 | 520.94 | 516.26 | 492.34 | 466.25 | 551.32 | 526.55 | 525.63 | 507.75 | 480.30 |
S51D2 | 726 | 841.20 | 833.25 | 816.11 | 808.92 | 727.00 | 847.13 | 838.63 | 825.65 | 817.81 | 741.47 |
S51D3 | 972 | 1116.38 | 1075.30 | 1065.46 | 1060.03 | 994.01 | 1126.52 | 1083.42 | 1075.53 | 1070.38 | 1024.40 |
S51D4 | 1677 | 1729.55 | 1705.87 | 1697.93 | 1678.50 | 1639.69 | 1741.37 | 1709.36 | 1702.94 | 1690.17 | 1659.02 |
S51D5 | 1440 | 1470.59 | 1431.80 | 1425.05 | 1421.95 | 1385.01 | 1482.30 | 1440.47 | 1438.78 | 1429.32 | 1407.64 |
S51D6 | 2327 | 2409.84 | 2325.30 | 2310.76 | 2293.69 | 2261.11 | 2429.45 | 2330.89 | 2324.90 | 2308.66 | 2273.79 |
S76D1 | 594 | 686.80 | 681.42 | 665.83 | 661.45 | 611.84 | 695.27 | 688.55 | 681.83 | 676.22 | 631.47 |
S76D2 | 1147 | 1249.53 | 1241.94 | 1233.37 | 1228.70 | 1151.66 | 1252.24 | 1245.00 | 1240.26 | 1238.11 | 1190.20 |
S76D3 | 1474 | 1655.48 | 1619.65 | 1611.79 | 1568.74 | 1540.38 | 1680.31 | 1627.20 | 1619.21 | 1607.64 | 1567.79 |
S76D4 | 2257 | 2374.50 | 2306.21 | 2292.60 | 2279.72 | 2229.19 | 2391.05 | 2312.63 | 2302.74 | 2296.79 | 2268.51 |
S101D1 | 716 | 883.14 | 868.54 | 853.58 | 793.77 | 735.65 | 890.21 | 877.22 | 876.39 | 842.56 | 760.50 |
S101D2 | 1393 | 1642.72 | 1622.76 | 1606.94 | 1546.38 | 1510.85 | 1647.67 | 1631.83 | 1627.15 | 1602.27 | 1553.99 |
S101D3 | 1975 | 2127.84 | 2110.99 | 2096.58 | 2082.30 | 2026.97 | 2137.85 | 2118.74 | 2103.45 | 2096.16 | 2063.99 |
S101D5 | 2915 | 3088.61 | 3036.29 | 3025.76 | 3014.29 | 2957.87 | 3094.72 | 3053.90 | 3038.33 | 3029.32 | 2990.59 |
Dataset | TACA/s | ASGCT/s | ACGCT/s | MMGCT/s | IACA/s |
---|---|---|---|---|---|
S51D1 | 22.28 | 21.84 | 19.20 | 19.06 | 19.59 |
S51D2 | 24.25 | 22.58 | 20.13 | 20.05 | 20.09 |
S51D3 | 26.31 | 23.19 | 21.63 | 20.87 | 20.86 |
S51D4 | 30.77 | 24.41 | 22.23 | 21.90 | 22.50 |
S51D5 | 29.43 | 23.62 | 21.84 | 21.01 | 21.53 |
S51D6 | 35.78 | 25.87 | 24.15 | 23.21 | 24.22 |
S76D1 | 45.50 | 43.13 | 38.05 | 37.76 | 39.59 |
S76D2 | 47.87 | 44.11 | 39.36 | 41.42 | 40.67 |
S76D3 | 51.13 | 45.81 | 41.08 | 42.47 | 41.19 |
S76D4 | 56.60 | 46.92 | 42.44 | 44.58 | 43.10 |
S101D1 | 73.53 | 72.02 | 65.24 | 66.15 | 66.95 |
S101D2 | 80.54 | 73.81 | 65.36 | 68.58 | 67.93 |
S101D3 | 85.66 | 75.20 | 67.11 | 67.49 | 69.53 |
S101D5 | 94.52 | 76.83 | 69.51 | 69.93 | 71.19 |
Dataset | BKS | Best Solution | Average Solution | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
TACA | ASGCT | ACGCT | MMGCT | IACA | TACA | ASGCT | ACGCT | MMGCT | IACA | ||
kelly01 | 5646.46 | 5879.58 | 5794.95 | 5769.36 | 5754.20 | 5662.98 | 5893.62 | 5841.34 | 5837.55 | 5824.72 | 5684.91 |
kelly02 | 8566.04 | 9195.57 | 9176.28 | 9117.09 | 9077.50 | 8654.37 | 9257.48 | 9208.78 | 9170.32 | 9152.06 | 8735.00 |
kelly03 | 11,649.06 | 12,428.92 | 12,282.97 | 11,935.64 | 11,843.65 | 11,356.66 | 12,813.61 | 12,361.17 | 12,345.98 | 12,078.94 | 11,486.70 |
kelly04 | 14,639.32 | 15,909.32 | 15,731.11 | 15,553.70 | 15,065.53 | 14,433.08 | 16,356.85 | 15,932.12 | 15,777.64 | 15,593.20 | 14,679.80 |
Dataset | ASGCT/% | ACGCT/% | MMGCT/% | IACA/% |
---|---|---|---|---|
kelly01 | 1.44 | 1.87 | 2.13 | 3.68 |
kelly02 | 0.21 | 0.85 | 1.28 | 5.89 |
kelly03 | 1.17 | 3.97 | 4.71 | 8.63 |
kelly04 | 1.12 | 2.24 | 5.31 | 9.28 |
Dataset | TACA/s | ASGCT/s | ACGCT/s | MMGCT/s | IACA/s |
---|---|---|---|---|---|
kelly01 | 1157.84 | 1104.85 | 993.33 | 989.58 | 994.36 |
kelly02 | 2039.38 | 1943.89 | 1733.78 | 1757.55 | 1747.29 |
kelly03 | 3251.28 | 3100.41 | 2860.84 | 2912.49 | 2904.04 |
kelly04 | 4111.77 | 4008.17 | 3978.55 | 4004.51 | 3923.41 |
Dataset | ASGCT/% | ACGCT/% | MMGCT/% | IACA/% |
---|---|---|---|---|
kelly01 | 4.58 | 14.21 | 14.53 | 14.12 |
kelly02 | 4.68 | 14.98 | 13.82 | 14.32 |
kelly03 | 4.64 | 12.01 | 10.42 | 10.68 |
kelly04 | 2.52 | 3.24 | 2.61 | 4.58 |
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Ma, X.; Liu, C. Improved Ant Colony Algorithm for the Split Delivery Vehicle Routing Problem. Appl. Sci. 2024, 14, 5090. https://doi.org/10.3390/app14125090
Ma X, Liu C. Improved Ant Colony Algorithm for the Split Delivery Vehicle Routing Problem. Applied Sciences. 2024; 14(12):5090. https://doi.org/10.3390/app14125090
Chicago/Turabian StyleMa, Xiaoxuan, and Chao Liu. 2024. "Improved Ant Colony Algorithm for the Split Delivery Vehicle Routing Problem" Applied Sciences 14, no. 12: 5090. https://doi.org/10.3390/app14125090