Mathematical modeling and a discrete artificial bee colony algorithm for the welding shop scheduling problem

Abstract

The welding process which is one of the most important assembly processes is widespread in the modern manufacturing industry, including aerospace, automotive and engineering machinery. The welding shop scheduling greatly impacts the efficiency of whole production system. However, few studies on the welding shop scheduling problem (WSSP) were reported. In this paper, a mathematical model and an improved discrete artificial bee colony algorithm (DABC) are proposed for the WSSP. Firstly, it is defined where multi-machine can process one job at the same time in the WSSP. Secondly, the mathematical models of WSSP have been constructed. Thirdly, an effective DABC is proposed to solve the WSSP, considering job permutation and machine allocation simultaneously. To improve the performance of proposed DABC algorithm, the effective operators have been designed. Three instances with different scales are used to evaluate the effectiveness of proposed algorithm. The comparisons with other two algorithms including genetic algorithm and grey wolf optimizer are also provided. Experimental results show that the proposed model and algorithm achieve good performance. Finally, the proposed model and DABC algorithm are applied in a real-world girder welding shop from a crane company in China. The results show that proposed model and algorithm reduces 55.17% production time comparing with the traditional algorithm and the scheduled machine allocation provides more reasonable arrangements for workers and machine loads.

Appendix

Instance 1 (10 × 5)

Stage 1		Stage 2		Stage 3		Stage 4		Stage 5
Setup/processing		Setup/processing		Setup/processing		Setup/processing		Setup/processing
5	9	4	5	6	24	7	5	4	12
8	13	11	14	7	28	10	7	5	18
10	17	18	22	8	33	12	8	7	24
12	20	25	31	9	35	15	10	8	29
15	25	32	40	10	40	18	12	10	35
18	30	39	49	11	45	21	14	12	41
21	36	46	58	13	51	24	16	13	47
24	40	51	63	14	56	26	18	15	52
8	12	7	8	9	27	10	8	7	15
12	17	15	18	11	32	14	11	9	22

Instance 2 (30 × 5)

Stage 1		Stage 2		Stage 3		Stage 4		Stage 5
Setup/processing		Setup/processing		Setup/processing		Setup/processing		Setup/processing
13	20	21	25	11	36	15	11	10	27
15	24	28	35	12	39	18	14	11	33
19	29	36	44	14	44	22	16	14	39
21	33	42	52	14	48	24	17	15	44
25	39	50	61	17	54	28	19	17	50
30	48	57	71	20	64	32	26	21	60
13	17	12	13	14	32	15	13	12	20
14	21	17	22	13	36	16	15	11	26
18	23	26	28	16	39	20	14	15	30
20	28	33	39	17	43	23	18	16	37
23	33	40	48	18	48	26	20	18	43
26	36	47	55	19	51	29	20	20	47
29	44	54	66	21	59	32	24	21	55
33	49	60	72	23	65	35	27	24	61
14	18	13	14	15	33	16	14	13	21
17	25	20	26	16	40	19	19	14	30
22	29	30	34	20	45	24	20	19	36
21	32	34	43	18	47	24	22	17	41
27	37	44	52	22	52	30	24	22	47
27	39	48	58	20	54	30	23	21	50
33	48	58	70	25	63	36	28	25	59
36	56	63	79	26	72	38	34	27	68
17	25	16	21	18	40	19	21	16	28
20	29	23	30	19	44	22	23	17	34
26	33	34	38	24	49	28	24	23	40
24	32	37	43	21	47	27	22	20	41
27	41	44	56	22	56	30	28	22	51
34	42	55	61	27	57	37	26	28	53
37	52	62	74	29	67	40	32	29	63
44	55	71	78	34	71	46	33	35	67

Instance 3 (60 × 5)

Stage 1		Stage 2		Stage 3		Stage 4		Stage 5
Setup/processing		Setup/processing		Setup/processing		Setup/processing		Setup/processing
5	9	4	5	6	24	7	5	4	12
8	13	11	14	7	28	10	7	5	18
10	17	18	22	8	33	12	8	7	24
12	20	25	31	9	35	15	10	8	29
15	25	32	40	10	40	18	12	10	35
18	30	39	49	11	45	21	14	12	41
21	36	46	58	13	51	24	16	13	47
24	40	51	63	14	56	26	18	15	52
9	13	8	9	10	28	11	9	8	16
11	17	14	18	10	32	13	11	8	22
13	20	21	25	11	36	15	11	10	27
15	24	28	35	12	39	18	14	11	33
19	29	36	44	14	44	22	16	14	39
21	33	42	52	14	48	24	17	15	44
25	39	50	61	17	54	28	19	17	50
30	48	57	71	20	64	32	26	21	60
13	17	12	13	14	32	15	13	12	20
14	21	17	22	13	36	16	15	11	26
18	23	26	28	16	39	20	14	15	30
20	28	33	39	17	43	23	18	16	37
23	33	40	48	18	48	26	20	18	43
26	36	47	55	19	51	29	20	20	47
29	44	54	66	21	59	32	24	21	55
33	49	60	72	23	65	35	27	24	61
14	18	13	14	15	33	16	14	13	21
17	25	20	26	16	40	19	19	14	30
22	29	30	34	20	45	24	20	19	36
21	32	34	43	18	47	24	22	17	41
27	37	44	52	22	52	30	24	22	47
27	39	48	58	20	54	30	23	21	50
33	48	58	70	25	63	36	28	25	59
36	56	63	79	26	72	38	34	27	68
17	25	16	21	18	40	19	21	16	28
20	29	23	30	19	44	22	23	17	34
26	33	34	38	24	49	28	24	23	40
24	32	37	43	21	47	27	22	20	41
27	41	44	56	22	56	30	28	22	51
34	42	55	61	27	57	37	26	28	53
37	52	62	74	29	67	40	32	29	63
44	55	71	78	34	71	46	33	35	67
25	24	24	20	26	39	27	20	24	27
23	28	26	29	22	43	25	22	20	33
30	32	38	37	28	48	32	23	27	39
32	40	45	51	29	55	35	30	28	49
30	45	47	60	25	60	33	32	25	55
38	45	59	64	31	60	41	29	32	56
36	56	61	78	28	71	39	36	28	67
42	58	69	81	32	74	44	36	33	70
29	27	28	23	30	42	31	23	28	30
32	31	35	32	31	46	34	25	29	36
34	41	42	46	32	57	36	32	31	48
30	44	43	55	27	59	33	34	26	53
39	43	56	58	34	58	42	30	34	53
36	48	57	67	29	63	39	32	30	59
39	54	64	76	31	69	42	34	31	65
45	68	72	91	35	84	47	46	36	80
26	37	25	33	27	52	28	33	25	40
36	41	39	42	35	56	38	35	33	46
31	38	39	43	29	54	33	29	28	45
40	48	53	59	37	63	43	38	36	57