Research on Multi-objective Hybrid Flow-shop Scheduling with Order Splitting

doi:10.16182/j.issn1004731x.joss.25-0745

Abstract

Abstract:

To address the order splitting multi-objective hybrid flow-shop scheduling problem (OSMOHFSP), a dual-objective optimization model was formulated with the objectives of minimizing makespan and total order tardiness. Fixed sub-batch specification constraints were incorporated to reflect common splitting limitations in actual production. A sub-batch generation strategy based on a candidate set of sub-batch specifications was designed to efficiently filter feasible splitting combinations, effectively reducing the search space dimensionality and computational complexity. A hybrid multi-objective metaheuristic algorithm integrating NSGA-II and SA was proposed. The global search capability was enhanced by constructing two crossover and four mutation operators. An improved SA embedding multiple neighborhood strategies, a Pareto-adaptive acceptance criterion, and a Pareto archiving mechanism was incorporated to explore the co-optimization potential of splitting schemes and scheduling sequences. Furthermore, an improved crowding distance calculation and an elite retention strategy were introduced to ensure the diversity of the solution set and the inheritance of high-quality individuals. Simulation experiment results indicate that the proposed algorithm exhibits better convergence, diversity, and dominance advantage than the comparative algorithms in solving the OSMOHFSP.

Key words: order splitting, multi-objective optimization, hybrid flow-shop scheduling, makespan, total order tardiness, NSGA-II, SA

CLC Number:

TP301.6

Yu Junjie, Ji Weixi, Chen Chen, Lu Jingyu, Zhang Chaoyang. Research on Multi-objective Hybrid Flow-shop Scheduling with Order Splitting[J]. Journal of System Simulation, 2026, 38(7): 2020-2036.

Figures/Tables 13

Table 1

Model symbol definition

符号	定义
i, a	订单索引
j, b	子批索引
k	阶段、工序索引
l	机器索引
h, c	速度索引
n	订单总数
p_i	订单i的子批总数
Q_i	订单i的总需求量
s	阶段总数
m_k	阶段k的机器数量
v_l	机器l的速度总数
q_ij	订单i的子批j的批量
O_ijk	订单i的子批j的第k道工序
$T i C$	订单i的完工时间
$T i j k S$	O_ijk 的加工开始时间
$T i j k C$	O_ijk 的加工结束时间
D_i	订单i的截止日期
t_ijklh	工序O_ijk 在机器l上以h速度的加工时间
w_ijkl	0-1变量，若工序O_ijk 能在机器l上加工，其值为1，否则为0
u_ijkl	0-1变量，若工序O_ijk 在机器l上加工，其值为1，否则为0
x_ijklh	0-1变量，若工序O_ijk 在机器l上以h的速度加工，其值为1，否则为0
y_ijkabkl	0-1变量，若工序O_ijk 先于工序O_abk 在机器l上加工，其值为1，否则为0
z_ij	0-1变量，若订单i的子批j的批量非0(q_ij ≥1)，其值为1，否则为0
E	一个很大的正数

Table 1

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Table 2

Fig. 7

Table 3

Table 4

RDI and HV results of each algorithm on test set

组号	算例规模	NSGA-II		NSTLBO		MOPSO		NSLSO		NSGA-II-SA
组号	算例规模	RDI/%	HV	RDI/%	HV	RDI/%	HV	RDI/%	HV	RDI/%	HV
平均		26.30	4.56×10⁸	163.21	1.07×10⁷	150.97	2.03×10⁷	110.75	8.41×10⁷	8.40	5.64×10⁸
1	10×2×3×3	23.67	4.75×10⁶	137.95	4.55×10⁵	113.36	8.72×10⁵	91.03	1.69×10⁶	5.97	5.76×10⁶
2	10×2×3×5	20.63	2.01×10⁶	135.33	1.79×10⁵	121.57	3.16×10⁵	81.49	8.35×10⁵	7.05	2.38×10⁶
3	10×2×5×3	23.32	6.24×10⁶	142.44	3.65×10⁵	126.23	8.90×10⁵	89.40	2.19×10⁶	8.87	7.28×10⁶
4	10×2×5×5	21.52	4.49×10⁶	148.21	3.27×10⁵	131.33	6.67×10⁵	87.06	1.82×10⁶	8.89	5.05×10⁶
5	10×4×3×3	32.07	9.22×10⁶	150.99	6.35×10⁵	138.49	1.06×10⁶	104.60	2.96×10⁶	8.81	1.19×10⁷
6	10×4×3×5	27.26	5.42×10⁶	147.50	4.68×10⁵	139.64	5.99×10⁵	101.13	1.78×10⁶	6.56	6.82×10⁶
7	10×4×5×3	22.77	2.71×10⁷	162.79	1.37×10⁶	148.92	2.45×10⁶	97.02	9.55×10⁶	9.69	3.11×10⁷
8	10×4×5×5	24.43	1.12×10⁷	158.69	5.86×10⁵	146.10	1.04×10⁶	95.39	4.16×10⁶	9.49	1.33×10⁷
9	10×8×3×3	33.57	2.27×10⁷	166.20	9.95×10⁵	151.71	1.97×10⁶	111.76	6.96×10⁶	8.02	3.08×10⁷
10	10×8×3×5	24.52	9.95×10⁶	151.65	7.83×10⁵	136.72	1.26×10⁶	94.84	3.49×10⁶	5.84	1.22×10⁷
11	10×8×5×3	24.62	2.97×10⁷	168.31	1.08×10⁶	155.46	2.06×10⁶	110.48	7.67×10⁶	9.15	3.53×10⁷
12	10×8×5×5	23.99	2.28×10⁷	163.66	1.21×10⁶	148.57	1.99×10⁶	99.60	7.55×10⁶	9.24	2.70×10⁷
13	20×2×3×3	24.21	3.27×10⁷	148.38	2.51×10⁶	139.71	3.02×10⁶	99.93	1.08×10⁷	6.97	3.90×10⁷
14	20×2×3×5	19.53	2.04×10⁷	151.68	1.22×10⁶	130.96	2.92×10⁶	91.54	7.29×10⁶	5.39	2.35×10⁷
15	20×2×5×3	21.14	6.70×10⁷	159.77	3.13×10⁶	151.21	5.30×10⁶	109.27	1.73×10⁷	7.49	7.73×10⁷
16	20×2×5×5	21.07	2.82×10⁷	158.81	1.56×10⁶	143.09	3.01×10⁶	105.18	8.13×10⁶	8.91	3.17×10⁷
17	20×4×3×3	30.15	6.65×10⁷	156.04	4.33×10⁶	153.18	4.65×10⁶	106.05	2.02×10⁷	8.50	8.55×10⁷
18	20×4×3×5	26.59	4.00×10⁷	160.47	2.73×10⁶	149.58	3.48×10⁶	106.75	1.18×10⁷	7.96	5.00×10⁷
19	20×4×5×3	26.25	1.78×10⁸	168.90	6.03×10⁶	154.70	1.32×10⁷	115.24	4.38×10⁷	10.93	2.11×10⁸
20	20×4×5×5	22.19	8.10×10⁷	170.07	2.66×10⁶	155.96	5.58×10⁶	107.86	2.22×10⁷	9.77	9.29×10⁷
21	20×8×3×3	33.70	1.92×10⁸	173.62	5.68×10⁶	154.37	1.70×10⁷	122.69	4.44×10⁷	10.76	2.45×10⁸
22	20×8×3×5	31.95	8.27×10⁷	172.72	3.35×10⁶	157.93	6.34×10⁶	115.22	2.24×10⁷	8.44	1.06×10⁸
23	20×8×5×3	31.89	2.79×10⁸	179.46	4.85×10⁶	169.86	9.55×10⁶	120.30	6.39×10⁷	10.23	3.53×10⁸
24	20×8×5×5	26.22	1.67×10⁸	172.16	5.25×10⁶	162.64	8.64×10⁶	119.69	3.79×10⁷	8.98	2.01×10⁸
25	50×2×3×3	27.11	3.31×10⁸	157.69	1.77×10⁷	147.00	2.93×10⁷	111.81	8.61×10⁷	6.55	4.09×10⁸
26	50×2×3×5	29.03	1.57×10⁸	162.92	7.86×10⁶	152.31	1.21×10⁷	112.58	4.30×10⁷	8.47	1.97×10⁸
27	50×2×5×3	21.74	8.28×10⁸	171.63	2.08×10⁷	159.28	4.67×10⁷	124.17	1.56×10⁸	9.04	9.46×10⁸
28	50×2×5×5	16.94	3.32×10⁸	168.76	1.06×10⁷	160.00	1.72×10⁷	122.46	6.20×10⁷	7.46	3.69×10⁸
29	50×4×3×3	31.78	1.22×10⁹	175.26	2.69×10⁷	163.66	6.09×10⁷	125.10	2.56×10⁸	6.99	1.59×10⁹
30	50×4×3×5	28.90	4.18×10⁸	169.56	1.48×10⁷	162.55	2.01×10⁷	127.17	7.95×10⁷	8.14	5.21×10⁸
31	50×4×5×3	28.84	1.76×10⁹	174.21	3.82×10⁷	169.18	5.87×10⁷	128.43	3.07×10⁸	10.61	2.12×10⁹
32	50×4×5×5	27.06	9.26×10⁸	176.02	2.18×10⁷	164.65	4.03×10⁷	124.32	1.86×10⁸	9.25	1.12×10⁹
33	50×8×3×3	34.51	1.95×10⁹	178.12	3.92×10⁷	169.20	6.98×10⁷	133.34	3.29×10⁸	7.70	2.61×10⁹
34	50×8×3×5	32.93	9.67×10⁸	176.38	2.39×10⁷	169.37	3.79×10⁷	130.96	1.72×10⁸	10.47	1.23×10⁹
35	50×8×5×3	28.19	4.11×10⁹	180.90	7.00×10⁷	171.39	1.46×10⁸	132.39	6.67×10⁸	8.78	5.06×10⁹
36	50×8×5×5	22.43	2.05×10⁹	178.43	4.12×10⁷	164.96	9.52×10⁷	130.91	3.24×10⁸	7.04	2.42×10⁹

Table 4

Table 5

Fig. 8

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算例	算法①	算法②	算法③	算法④	算法⑤
5	121.09	39.18	44.04	36.98	30.64
9	143.16	48.03	37.44	33.30	28.78
10	155.04	38.02	41.02	35.48	26.80
11	126.84	76.51	74.04	63.71	61.89
12	174.41	92.15	63.72	67.16	59.15
18	139.52	76.99	73.71	73.23	37.16
22	131.65	49.70	57.95	27.50	24.77
23	141.94	79.49	59.69	63.90	29.85
26	95.11	49.84	48.97	29.62	9.38
29	143.49	42.13	60.83	41.52	18.16

组号	算例规模	Cov1	Cov1'	Cov2'	Cov3'	Cov4'	T₁/s	T₂/s	T₃/s	T₄/s	T₅/s
平均		0.04	0.91	1	1	1	456	1 025	303	683	533
1	10×2×3×3	0.02	0.95	1	1	1	76	168	45	106	92
2	10×2×3×5	0.07	0.87	1	1	1	65	145	39	91	79
3	10×2×5×3	0.09	0.83	1	1	1	106	245	66	154	127
4	10×2×5×5	0.14	0.73	1	1	1	134	306	83	193	166
5	10×4×3×3	0	0.97	1	1	1	99	224	61	143	120
6	10×4×3×5	0.02	0.98	1	1	1	95	215	59	136	113
7	10×4×5×3	0.09	0.82	1	1	1	189	431	118	274	228
8	10×4×5×5	0.07	0.88	1	1	1	169	380	104	252	204
9	10×8×3×3	0.02	0.95	1	1	1	178	397	110	259	213
10	10×8×3×5	0	0.99	1	1	1	172	395	110	253	205
11	10×8×5×3	0.08	0.86	1	1	1	282	639	177	407	340
12	10×8×5×5	0.08	0.80	1	1	1	329	744	210	479	398
13	20×2×3×3	0	1	1	1	1	94	207	56	131	114
14	20×2×3×5	0	0.93	1	1	1	68	152	42	96	81
15	20×2×5×3	0.05	0.88	1	1	1	95	213	59	135	114
16	20×2×5×5	0.09	0.81	1	1	1	81	180	50	115	98
17	20×4×3×3	0.01	0.95	1	1	1	95	214	60	137	114
18	20×4×3×5	0.01	0.95	1	1	1	91	206	58	130	109
19	20×4×5×3	0.10	0.85	1	1	1	177	397	111	254	212
20	20×4×5×5	0.09	0.88	1	1	1	159	361	102	232	190
21	20×8×3×3	0	1	1	1	1	388	864	246	559	463
22	20×8×3×5	0	0.99	1	1	1	383	860	247	560	446
23	20×8×5×3	0.05	0.89	1	1	1	633	1 421	408	926	747
24	20×8×5×5	0.10	0.88	1	1	1	727	1 643	482	1 088	864
25	50×2×3×3	0	0.99	1	1	1	207	464	129	295	248
26	50×2×3×5	0	1	1	1	1	224	499	144	320	267
27	50×2×5×3	0.11	0.85	1	1	1	484	1 063	296	680	581
28	50×2×5×5	0.13	0.71	1	1	1	575	1 294	364	829	685
29	50×4×3×3	0	1	1	1	1	746	1 664	479	1 092	887
30	50×4×3×5	0	0.98	1	1	1	744	1 648	478	1 085	876
31	50×4×5×3	0	0.96	1	1	1	1 163	2 586	755	1 701	1 379
32	50×4×5×5	0	0.98	1	1	1	641	1 436	423	950	755
33	50×8×3×3	0	0.94	1	1	1	1 102	2 467	746	1 677	1 290
34	50×8×3×5	0.03	0.95	1	1	1	897	2 043	628	1 399	1 045
35	50×8×5×3	0.06	0.93	1	1	1	2 227	5 012	1 560	3 472	2 611
36	50×8×5×5	0.01	0.96	1	1	1	2 509	5 704	1 797	3 981	2 745