Differential evolution algorithm integrating mutation strategy and adjacency information

doi:10.11772/j.issn.1001-9081.2025010068

Journal of Computer Applications ›› 2026, Vol. 46 ›› Issue (1): 188-197.DOI: 10.11772/j.issn.1001-9081.2025010068

• Advanced computing • Previous Articles Next Articles

Differential evolution algorithm integrating mutation strategy and adjacency information

Min RAN¹^,², Dazhi PAN¹^,²()

^1.School of Mathematical Sciences，China West Normal University，Nanchong Sichuan 637009，China
^2.Key Laboratory of Optimization Theory and Application at Sichuan Provincial Universities （China West Normal University），Nanchong Sichuan 637002，China

Received:2025-01-17 Revised:2025-04-10 Accepted:2025-04-10 Online:2026-01-10 Published:2026-01-10
Contact: Dazhi PAN
About author:RAN Min， born in 1998， M. S. candidate. Her research interests include intelligent computing， combinatorial optimization.
Supported by:
National Natural Science Foundation of China(11871059);Natural Science Foundation of Sichuan Provincial Department of Education(18ZA0469);Graduate Education Reform Research Project of China West Normal University(2024XM05)

融合变异策略与邻接信息的差分进化算法

冉敏¹^,², 潘大志¹^,²()

^1.西华师范大学数学科学学院，四川南充 637009
^2.最优化理论与应用四川省高校重点实验室（西华师范大学），四川南充 637002

通讯作者: 潘大志
作者简介:冉敏（1998—），女（苗族），重庆人，硕士研究生，主要研究方向：智能计算、组合优化
基金资助:
国家自然科学基金资助项目(11871059);四川省教育厅自然科学基金资助项目(18ZA0469);西华师范大学研究生教育改革研究项目(2024XM05)

Abstract

Abstract:

Aiming at multi-objective Vehicle Routing Problem （VRP） with time windows， a Differential Evolution algorithm integrating Mutation Strategy and Adjacency Information （DE-MSAI） was proposed. Firstly， four mutation operators were designed by employing elite sampling strategy， so as to increase the algorithm search breadth. Secondly， customer adjacency information matrix was combined to guide the neighborhood search of the individuals， thereby improving local optimization efficiency. Finally， simulated annealing criterion was adopted to accept inferior solutions with certain probability. If the number of Pareto non-dominated solution set remained unimproved was beyond the threshold during optimization， elite fragment protection strategy would be activated to perturb a randomly selected solution from non-dominated solution set， thereby maintaining the population diversity. Simulation results based on Solomon standard library instances show that the proposed algorithm controls the solving error within 0.07% compared to Hybrid Crow Search Algorithm （HCSA）， and outperforms K-means clustering algorithm and Improved Large Neighborhood Search Algorithm （K-means-ILNSA） in most cases， achieving an average reduction of 4.51% in route deviation metric， verifying the effectiveness of the algorithm.

Key words: Vehicle Routing Problem (VRP), multi-objective optimization, Differential Evolution (DE) algorithm, adjacency information matrix, elite fragment protection strategy

摘要：

针对多目标带时间窗的车辆路径问题（VRP），提出一种融合变异策略与邻接信息的差分进化算法（DE-MSAI）。首先，利用精英抽样策略设计4种变异操作，增加算法搜索的广度；其次，结合客户邻接信息矩阵引导个体进行邻域搜索，提升局部优化效率；最后，基于模拟退火准则以一定的概率接受劣解。在迭代过程中，如果Pareto非支配解集连续未被改善的次数超过阈值，则启动精英碎片保护策略随机选择一个非支配解集中的解进行扰动，以维持种群的多样性。基于Solomon标准库中算例的仿真实验结果表明，所提算法相较于混合乌鸦算法（HCSA）的求解误差控制在0.07%以内；相较于基于聚类的混合大邻域搜索算法（K-means-ILNSA），所提算法在绝大多数算例中表现更优，路线偏差指标平均降低了4.51%，验证了算法的有效性。

关键词: 车辆路径问题, 多目标优化, 差分进化算法, 邻接信息矩阵, 精英碎片保护策略

CLC Number:

TP301.6

Min RAN, Dazhi PAN. Differential evolution algorithm integrating mutation strategy and adjacency information[J]. Journal of Computer Applications, 2026, 46(1): 188-197.

冉敏, 潘大志. 融合变异策略与邻接信息的差分进化算法[J]. 《计算机应用》唯一官方网站, 2026, 46(1): 188-197.

Figures/Tables 11

Tab. 1 Symbol description

符号	含义
$N$	客户与仓库的集合
$Q$	车辆的最大容量
$K$	车辆集合
$D i$	客户i的需求量
$d i j$	客户i与客户j之间的距离
$a i$	客户i接受货物的开始时间
$e i$	客户i接受货物的结束时间
$s i$	客户i卸载货物的服务时间
w_i	车辆在客户i的等待时间
$t i$	车辆到达客户i的时间
$t i j$	车辆从客户i到客户j的运输时间
$L$	限制车辆最晚回到仓库的时间
$C u s$	客户的总数目

Tab. 1 Symbol description

符号	含义
$N$	客户与仓库的集合
$Q$	车辆的最大容量
$K$	车辆集合
$D i$	客户i的需求量
$d i j$	客户i与客户j之间的距离
$a i$	客户i接受货物的开始时间
$e i$	客户i接受货物的结束时间
$s i$	客户i卸载货物的服务时间
w_i	车辆在客户i的等待时间
$t i$	车辆到达客户i的时间
$t i j$	车辆从客户i到客户j的运输时间
$L$	限制车辆最晚回到仓库的时间
$C u s$	客户的总数目

Fig. 1 Schematic diagram of MOVRPTW

Fig. 2 Decoding method

Fig. 3 Greedy in fusion clustering

Fig. 4 Random mutation

Fig. 5 Two-point perturbation mutation

Fig. 6 Pareto solution sets of some instances

Fig. 7 Route maps of minimum total cost for some instances

Tab. 2 Performance test results of three operators in DE-MSAI

算例	DE-MSAI		Vare1		Vare2		Vare3
算例	f₁	f₂	Er1/%	Er2/%	Er1/%	Er2/%	Er1/%	Er2/%
C101	828.94	0.00	0.00	0.00	0.00	0.00	0.00	0.00
C102	828.94	0.00	0.00	0.00	0.00	0.00	0.00	0.00
C201	591.56	0.00	2.04	0.00	0.00	0.00	6.32	0.00
C202	591.56	0.00	5.27	0.00	6.57	100.00	10.84	100.00
R101	1 643.16	683.20	0.58	1.04	1.47	1.44	5.31	2.36
R103	1 213.80	494.30	1.50	-15.01	3.37	-11.86	7.60	2.23
R104	987.13	263.70	0.83	-45.37	3.34	12.10	6.48	20.98
R202	1044.40	3 258.50	0.62	0.17	4.64	-18.34	8.36	-3.38
R203	885.50	2 747.50	0.88	-1.65	3.72	-0.27	8.40	0.80
R204	764.16	1 477.50	2.16	-1.85	10.10	-8.75	10.38	-8.15
RC101	1 643.80	191.50	0.47	-2.24	3.45	-1.06	6.69	11.63
RC102	1 461.33	286.64	2.59	-19.05	6.15	-10.68	6.23	-4.19
RC104	1 144.85	81.76	0.10	0.00	1.68	-47.21	19.27	31.48
RC202	1 097.26	2 751.67	0.42	-0.17	1.39	0.06	5.75	-7.29
RC203	935.12	1 959.11	0.64	1.51	4.44	-2.83	5.85	-1.14
RC205	1 157.66	2 853.42	1.72	1.39	3.69	-2.99	4.33	0.29

Tab. 3 Test results of Solomon instances

算例	MOLNS^［27］		HGA^［28］		DE-MSAI
算例	TD	NV	TD	NV	TD	NV	TM	DET1/%	DET2/%	DEC1/%	DEC2/%
C101	828.94	10	828.94	10	828.94	10	0.00	0.00	0.00	0.00	0.00
C102	828.94	10	828.94	10	828.94	10	0.00	0.00	0.00	0.00	0.00
C103	828.94	10	830.77	10	828.94	10	0.00	0.00	-0.22	0.00	0.00
C104	828.94	10	864.22	10	828.94	10	0.00	0.00	-4.26	0.00	0.00
C105	828.94	10	828.94	10	828.94	10	0.00	0.00	0.00	0.00	0.00
C106	828.94	10	828.94	10	828.94	10	0.00	0.00	0.00	0.00	0.00
C107	828.94	10	828.94	10	828.94	10	0.00	0.00	0.00	0.00	0.00
C108	828.94	10	854.31	10	828.94	10	0.00	0.00	-3.06	0.00	0.00
C109	828.94	10	854.78	10	828.94	10	0.00	0.00	-3.12	0.00	0.00
C201	591.56	3	591.56	3	591.56	3	0.00	0.00	0.00	0.00	0.00
C202	591.56	3	591.56	3	591.56	3	0.00	0.00	0.00	0.00	0.00
C203	591.56	3	591.17	3	591.56	3	0.00	0.00	0.07	0.00	0.00
C204	590.60	3	594.51	3	591.56	3	0.00	0.16	-0.50	0.00	0.00
C205	588.88	3	588.88	3	588.88	3	0.00	0.00	0.00	0.00	0.00
C206	588.49	3	588.49	3	588.49	3	0.00	0.00	0.00	0.00	0.00
C207	588.29	3	588.29	3	593.81	3	0.00	0.93	0.93	0.00	0.00
C208	588.32	3	588.32	3	588.32	3	0.00	0.00	0.00	0.00	0.00
C平均								0.06	-0.60	0.00	0.00
R101	1 654.93	19	1 656.55	19	1643.16	19	683.20	-0.72	-0.81	0.00	0.00
R102	1475.33	18	1 476.85	18	1 482.26	18	1 173.05	0.47	0.36	0.00	0.00
R103	1 240.44	14	1 230.07	14	1213.62	14	494.13	-2.21	-1.36	0.00	0.00
R104	1 010.72	10	1 010.55	10	987.13	11	263.70	-2.39	-2.37	9.09	9.09
R105	1 389.85	15	1 395.68	15	1360.12	15	1 179.74	-2.19	-2.61	0.00	0.00
R106	1 269.14	13	1 261.61	13	1249.75	13	174.92	-1.55	-0.95	0.00	0.00
R107	1 102.72	11	1 103.77	11	1078.85	11	1 431.21	-2.21	-2.31	0.00	0.00
R108	991.57	10	966.99	10	975.25	10	109.34	-1.67	0.85	0.00	0.00
R109	1 177.76	12	1 200.20	12	1 155.28	13	84.98	-1.95	-3.89	7.69	7.69
R110	1 129.60	12	1 127.28	12	1098.50	12	118.64	-2.83	-2.62	0.00	0.00
R111	1 108.70	12	1 096.38	12	1056.10	12	105.79	-4.98	-3.81	0.00	0.00
R112	964.15	10	982.14	10	966.48	10	45.71	0.24	-1.62	0.00	0.00
R201	1 156.73	7	1 442.28	7	1 238.29	8	3 555.28	6.59	-16.47	12.50	12.50
R202	1 065.73	5	1 212.49	5	1044.40	5	3 258.50	-2.04	-16.09	0.00	0.00
R203	901.72	5	1 197.67	5	885.50	5	2 747.50	-1.83	-35.25	0.00	0.00
R204	750.23	4	854.31	4	764.16	4	1 477.50	1.82	-11.80	0.00	0.00
R205	964.23	5	1 337.65	5	956.60	5	1 257.20	-0.80	-39.83	0.00	0.00
R206	907.35	5	1 114.76	5	884.59	5	1 736.42	-2.57	-26.02	0.00	0.00
R207	851.89	3	1 055.71	3	807.56	3	1 093.62	-5.49	-30.73	0.00	0.00
R208	731.84	3	860.53	3	736.36	3	823.89	0.61	-16.86	0.00	0.00
R平均								-1.29	-10.71	1.46	1.46
RC101	1 662.56	15	1 656.59	15	1643.80	15	191.5	-1.14	-0.78	0.00	0.00
RC102	1 486.35	14	1 532.25	14	1461.33	14	286.64	-1.71	-4.85	0.00	0.00
RC103	1 291.95	12	1 344.03	12	1277.68	11	129.57	-1.12	-5.19	-9.09	-9.09
RC104	1 162.53	10	1 184.90	11	1144.85	10	81.76	-1.54	-3.50	0.00	-10.00
RC105	1 604.53	16	1 662.01	14	1530.17	15	1 269.06	-4.86	-8.62	-6.67	6.67
RC106	1 400.09	13	1344.03	12	1 388.97	12	98.73	-0.80	3.24	-8.33	0.00
RC107	1 259.55	12	1 263.07	12	1243.06	12	94.53	-1.33	-1.61	0.00	0.00
RC108	1 205.13	11	1 144.94	11	1139.97	11	89.82	-5.72	-0.44	0.00	0.00
RC201	1 281.81	8	1 442.28	4	1 326.48	8	3 232.53	3.37	-8.73	0.00	50.00
RC202	1 104.94	8	1 212.49	4	1 097.26	8	2 751.67	-0.70	-10.50	0.00	50.00
RC203	938.04	5	1 197.67	3	935.12	5	1 959.11	-0.31	-28.08	0.00	40.00
RC204	805.46	3	854.31	10	790.37	3	820.70	-1.91	-8.09	0.00	-233.33
RC205	1 162.43	7	1 337.65	4	1 157.66	7	2 853.42	-0.41	-15.55	0.00	42.86
RC206	1 097.07	5	1 114.76	4	1 059.54	7	1 737.64	-3.54	-5.21	28.57	42.86
RC207	970.78	5	1 055.71	4	970.82	6	1 636.75	0.00	-8.74	16.67	33.33
RC208	810.99	4	860.53	3	782.93	4	288.82	-3.58	-9.91	0.00	25.00
RC平均								-1.58	-7.29	1.32	2.39

Tab. 3 Test results of Solomon instances

算例	MOLNS^［27］		HGA^［28］		DE-MSAI
算例	TD	NV	TD	NV	TD	NV	TM	DET1/%	DET2/%	DEC1/%	DEC2/%
C101	828.94	10	828.94	10	828.94	10	0.00	0.00	0.00	0.00	0.00
C102	828.94	10	828.94	10	828.94	10	0.00	0.00	0.00	0.00	0.00
C103	828.94	10	830.77	10	828.94	10	0.00	0.00	-0.22	0.00	0.00
C104	828.94	10	864.22	10	828.94	10	0.00	0.00	-4.26	0.00	0.00
C105	828.94	10	828.94	10	828.94	10	0.00	0.00	0.00	0.00	0.00
C106	828.94	10	828.94	10	828.94	10	0.00	0.00	0.00	0.00	0.00
C107	828.94	10	828.94	10	828.94	10	0.00	0.00	0.00	0.00	0.00
C108	828.94	10	854.31	10	828.94	10	0.00	0.00	-3.06	0.00	0.00
C109	828.94	10	854.78	10	828.94	10	0.00	0.00	-3.12	0.00	0.00
C201	591.56	3	591.56	3	591.56	3	0.00	0.00	0.00	0.00	0.00
C202	591.56	3	591.56	3	591.56	3	0.00	0.00	0.00	0.00	0.00
C203	591.56	3	591.17	3	591.56	3	0.00	0.00	0.07	0.00	0.00
C204	590.60	3	594.51	3	591.56	3	0.00	0.16	-0.50	0.00	0.00
C205	588.88	3	588.88	3	588.88	3	0.00	0.00	0.00	0.00	0.00
C206	588.49	3	588.49	3	588.49	3	0.00	0.00	0.00	0.00	0.00
C207	588.29	3	588.29	3	593.81	3	0.00	0.93	0.93	0.00	0.00
C208	588.32	3	588.32	3	588.32	3	0.00	0.00	0.00	0.00	0.00
C平均								0.06	-0.60	0.00	0.00
R101	1 654.93	19	1 656.55	19	1643.16	19	683.20	-0.72	-0.81	0.00	0.00
R102	1475.33	18	1 476.85	18	1 482.26	18	1 173.05	0.47	0.36	0.00	0.00
R103	1 240.44	14	1 230.07	14	1213.62	14	494.13	-2.21	-1.36	0.00	0.00
R104	1 010.72	10	1 010.55	10	987.13	11	263.70	-2.39	-2.37	9.09	9.09
R105	1 389.85	15	1 395.68	15	1360.12	15	1 179.74	-2.19	-2.61	0.00	0.00
R106	1 269.14	13	1 261.61	13	1249.75	13	174.92	-1.55	-0.95	0.00	0.00
R107	1 102.72	11	1 103.77	11	1078.85	11	1 431.21	-2.21	-2.31	0.00	0.00
R108	991.57	10	966.99	10	975.25	10	109.34	-1.67	0.85	0.00	0.00
R109	1 177.76	12	1 200.20	12	1 155.28	13	84.98	-1.95	-3.89	7.69	7.69
R110	1 129.60	12	1 127.28	12	1098.50	12	118.64	-2.83	-2.62	0.00	0.00
R111	1 108.70	12	1 096.38	12	1056.10	12	105.79	-4.98	-3.81	0.00	0.00
R112	964.15	10	982.14	10	966.48	10	45.71	0.24	-1.62	0.00	0.00
R201	1 156.73	7	1 442.28	7	1 238.29	8	3 555.28	6.59	-16.47	12.50	12.50
R202	1 065.73	5	1 212.49	5	1044.40	5	3 258.50	-2.04	-16.09	0.00	0.00
R203	901.72	5	1 197.67	5	885.50	5	2 747.50	-1.83	-35.25	0.00	0.00
R204	750.23	4	854.31	4	764.16	4	1 477.50	1.82	-11.80	0.00	0.00
R205	964.23	5	1 337.65	5	956.60	5	1 257.20	-0.80	-39.83	0.00	0.00
R206	907.35	5	1 114.76	5	884.59	5	1 736.42	-2.57	-26.02	0.00	0.00
R207	851.89	3	1 055.71	3	807.56	3	1 093.62	-5.49	-30.73	0.00	0.00
R208	731.84	3	860.53	3	736.36	3	823.89	0.61	-16.86	0.00	0.00
R平均								-1.29	-10.71	1.46	1.46
RC101	1 662.56	15	1 656.59	15	1643.80	15	191.5	-1.14	-0.78	0.00	0.00
RC102	1 486.35	14	1 532.25	14	1461.33	14	286.64	-1.71	-4.85	0.00	0.00
RC103	1 291.95	12	1 344.03	12	1277.68	11	129.57	-1.12	-5.19	-9.09	-9.09
RC104	1 162.53	10	1 184.90	11	1144.85	10	81.76	-1.54	-3.50	0.00	-10.00
RC105	1 604.53	16	1 662.01	14	1530.17	15	1 269.06	-4.86	-8.62	-6.67	6.67
RC106	1 400.09	13	1344.03	12	1 388.97	12	98.73	-0.80	3.24	-8.33	0.00
RC107	1 259.55	12	1 263.07	12	1243.06	12	94.53	-1.33	-1.61	0.00	0.00
RC108	1 205.13	11	1 144.94	11	1139.97	11	89.82	-5.72	-0.44	0.00	0.00
RC201	1 281.81	8	1 442.28	4	1 326.48	8	3 232.53	3.37	-8.73	0.00	50.00
RC202	1 104.94	8	1 212.49	4	1 097.26	8	2 751.67	-0.70	-10.50	0.00	50.00
RC203	938.04	5	1 197.67	3	935.12	5	1 959.11	-0.31	-28.08	0.00	40.00
RC204	805.46	3	854.31	10	790.37	3	820.70	-1.91	-8.09	0.00	-233.33
RC205	1 162.43	7	1 337.65	4	1 157.66	7	2 853.42	-0.41	-15.55	0.00	42.86
RC206	1 097.07	5	1 114.76	4	1 059.54	7	1 737.64	-3.54	-5.21	28.57	42.86
RC207	970.78	5	1 055.71	4	970.82	6	1 636.75	0.00	-8.74	16.67	33.33
RC208	810.99	4	860.53	3	782.93	4	288.82	-3.58	-9.91	0.00	25.00
RC平均								-1.58	-7.29	1.32	2.39

Tab. 4 Comparative analysis of route cost solution results for C1 and C2 instances

算例	NV	TD			路线偏差/%
算例	NV	DE-MSAI	HCSA^［30］	K-means-ILNSA^［29］	相较于HCSA路线偏差1	相较于K-means-ILNSA路线偏差2
平均					0.07	-4.51
C101	10	828.94	828.94	828.94	0.00	0.00
C102	10	828.94	828.94	828.94	0.00	0.00
C103	10	828.94	828.94	828.94	0.00	0.00
C104	10	828.94	828.39	828.94	0.07	0.00
C105	10	828.94	828.94	828.94	0.00	0.00
C106	10	828.94	828.94	828.94	0.00	0.00
C107	10	828.94	828.94	828.94	0.00	0.00
C108	10	828.94	828.94	828.94	0.00	0.00
C109	10	828.94	828.94	828.94	0.00	0.00
C201	3	591.56	591.56	626.81	0.00	-5.96
C202	3	591.56	591.56	678.53	0.00	-14.70
C203	3	591.56	591.17	705.29	0.07	-19.23
C204	3	591.56	590.60	650.57	0.16	-9.98
C205	3	588.88	588.88	629.94	0.00	-6.97
C206	3	588.49	588.49	620.67	0.00	-5.47
C207	3	593.81	588.32	647.19	0.92	-8.99
C208	3	588.32	588.49	620.27	-0.03	-5.43

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Differential evolution algorithm integrating mutation strategy and adjacency information

融合变异策略与邻接信息的差分进化算法

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