Simulated annealing and genetic algorithm based method for a bi-level seru loading problem with worker assignment in seru production systems

Lan Luo; Zhe Zhang; Yong Yin

doi:10.3934/jimo.2019134

Article Contents

2021, Volume 17, Issue 2: 779-803. Doi: 10.3934/jimo.2019134

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Simulated annealing and genetic algorithm based method for a bi-level seru loading problem with worker assignment in seru production systems

1.
School Economics and Management, Nanjing University of Science and Technology, Nanjing 210094, China
2.
Graduate School of Business, Doshisha University, Karasuma-Imadegawa, Kamigyo-ku, Kyoto, 602-8580, Japan

^* Corresponding author: Zhe Zhang
^* Corresponding author: Zhe Zhang

Received: March 31, 2019

Revised: June 30, 2019

Early access: October 2019

Published: March 2021

This research was sponsored by Natural Science Foundation of China (NSFC Grant no. 71401075). We thank Professor Xiuli Wang and Ding Zhang for their valuable discussions and comments, and we would like to express our appreciation to all the reviewers and editors who contributed to this research

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Abstract

Seru production is one of the latest manufacturing modes arising from Japanese production practice. Seru can achieve efficiency, flexibility, and responsiveness simultaneously. To accommodate the current business environment with volatile demands and fierce competitions, seru has attracted more and more attention both from researchers and practitioners. A new planning management system, just-in-time organization system (JIT-OS), is used to manage and control a seru production system. The JIT-OS contains two decisions: seru formation and seru loading. By seru formation, a seru system with one or multiple appropriate serus is configured; by seru loading, customer ordered products are allocated to serus to implement production plans. In the process of seru formation, workers have to be assigned to serus. In this paper, a seru loading problem with worker assignment is constructed as a bi-level programming model, and the worker assignment on the upper level is to minimize total idle time while the lower level is to minimize the makespan by finding out optimal product allocation. A product lot can be splitted and allocated to different serus. The problem of this paper is shown to be NP-hard. Therefore, a simulated annealing and genetic algorithm (SA-GA) is developed. The SA is for the upper level programming and the GA is for the lower level programming. The practicality and effectiveness of the model and algorithm are verified by two numerical examples, and the results show that the SA-GA algorithm has good scalability.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Three types seru

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Figure 2. Whole bi-level decision procedure

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Figure 3. The outline of SA-GA algorithm

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Figure 4. An example of SA encoding

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Figure 5. The genetic encoding based on allocation ratios

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Figure 6. The flowchart of SA-GA algorithm

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Figure 7. The minimum idle time in each iteration of SA

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Figure 8. Idle time and makespan

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Figure 9. The worker assignment decision

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Figure 10. Loading results

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Figure 11. The worker assignment decision

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Figure 12. Loading results

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Table 1. The parameter setting of SA-GA algorithm

Level	Algorithm	Parameters
Upper	SA	$ T\_max=10000 $	$ T\_min=0.1 $
Upper	SA	$ II=20 $	$ \alpha=0.9 $
Lower	GA	$ pop\_size=300 $	$ GEN=500 $
		$ p\_zero1=0.75 $	$ p\_zero2=0.25 $
		$ p\_cross=0.9 $	$ p\_muta=0.1 $

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Table 2. Data about products

Product	Worker's processing time (min)															Demand	Setup (min)
Product	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	Demand	Setup (min)
1	23	23	21	22	21	24	22	–	21	24	22	–	24	24	23	95	4
2	–	32	37	32	–	34	37	31	34	31	–	31	36	36	37	100	9
3	41	43	–	–	44	47	42	42	–	41	47	45	44	42	–	130	8
4	29	28	29	28	26	27	26	27	27	28	26	31	31	–	28	105	6
5	17	–	17	16	19	17	–	18	16	16	20	20	18	16	17	120	5
6	42	23	20	33	38	33	27	29	–	34	33	29	30	36	19	145	6
7	–	68	48	63	43	71	49	21	66	59	53	–	–	70	83	50	4
8	14	15	14	20	–	19	19	17	22	19	17	18	–	15	10	115	1
¹ The '-' means that the worker cannot produce the product.

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Table 3. Data about products

	1	2	3	4	5	6	7	8
1	–	100	–	–	80	–	50	115
2	–	–	130	–	–	116	–	–
3	95	–	–	105	40	29	–	–

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Table 4. Production timetable

Product	1	2	3	4	5
Seru	3	1	2	3	1
Starting time	Monday 8:00	Monday 8:00	Monday 8:00	Tuesday 9:13	Tuesday10:22
Finishing time	Tuesday 9:07	Tuesday 10:17	Wednesday 11:30	Wednesday 15:54	Tuesday 16:09
Product	5	6	6	7	8
Seru	3	2	3	1	1
Starting time	Wednesday 15:59	Wednesday 11:36	Thursday 9:25	Tuesday 16:13	Thursday 8:05
Finishing time	Thursday 9:19	Thursday 17:12	Thursday 16:29	Thursday 8:04	Thursday 17:07

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Table 5. Results of the small case

No.	Idle time (min)	Makespan (min)	CPU time (s)
1	2381.7	1910	8242.5
2	2629.4	1907.8	8193.5
3	2438.6	1932.3	8222.2
4	2446	1936	8228
5	2723.1	1933	8228.6
6	2022.3	1893	8064.8
7	2461.2	1895	8095.2
8	2300	1906.3	8098.5
9	2819	1859.5	8051.9
10	2566.8	1874.1	8135
Average	2478.81	1904.7	8156.02
SD	214.16	24.07	70.94

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Table 6. Results of GA-GA algorithm for small case

No.	Idle time (min)	Makespan (min)	CPU time (s)
1	2519.5	1913.8	8220.8
2	nonconvergent
3	2384.4	1851.8	8278.8
4	2375.6	1898.3	8230.3
5	nonconvergent
6	2409.5	1829	8284.8
7	2882	1894.2	9085.2
8	2213.2	1941.3	8244.5
9	2526	1941	8190.9
10	2526.9	1918	8289.3
Average	2479.64	1898.43	8353.08
SD	181.55	37.55	278.59

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Table 7. Workers' processing time for each product

Worker	Workers' processing time for each product (min)
Worker	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
1	22	39	47	29	-	34	64	23	50	71	20	21	11	32	39	15	20	27	29	24
2	23	37	47	29	20	35	54	21	55	77	24	20	12	30	38	19	21	21	26	20
3	22	38	46	-	22	35	50	26	56	77	24	15	11	34	43	-	17	23	29	21
4	-	37	46	32	-	35	56	-	50	78	22	19	12	34	42	17	21	21	25	23
5	23	40	47	27	-	34	59	20	50	81	-	17	10	34	42	15	17	22	28	-
6	22	38	47	29	23	33	61	25	-	-	21	20	10	32	39	18	17	27	28	21
7	21	37	-	28	19	32	59	-	52	73	23	21	11	30	35	16	22	22	25	19
8	23	38	49	27	18	-	-	22	48	78	21	20	10	31	44	18	-	23	26	22
9	21	40	50	-	21	33	56	26	53	87	-	18	10	32	40	17	22	21	27	20
10	24	-	49	28	19	38	58	26	52	-	24	18	10	37	41	-	17	25	26	23
11	22	38	47	30	18	33	61	27	53	77	21	19	-	33	41	-	19	21	29	23
12	23	36	49	29	21	34	57	22	53	86	21	20	10	33	36	17	21	21	27	24
13	23	39	47	29	22	-	65	21	53	86	22	19	12	30	38	15	19	22	27	21
14	23	39	-	31	18	30	50	29	57	85	24	-	10	37	36	19	21	27	27	21
15	21	36	49	30	-	36	59	24	50	81	24	17	11	37	35	18	19	26	26	22
16	25	36	49	31	19	37	58	22	54	82	24	19	12	39	39	17	18	23	29	22
17	-	37	45	30	22	38	60	23	55	-	21	18	-	37	44	-	21	22	27	-
18	23	37	49	31	21	37	61	-	48	-	21	21	12	34	-	17	17	-	26	20
19	21	35	48	-	19	37	61	28	48	69	20	19	10	33	36	18	22	25	25	20
20	21	38	-	30	23	32	55	29	53	72	22	16	10	-	35	16	17	21	29	23
21	21	36	50	30	22	35	64	29	53	86	22	17	11	39	-	17	18	24	26	23
22	22	38	50	29	-	33	61	22	48	69	23	17	11	40	-	15	-	22	28	21
23	25	39	47	29	19	37	59	26	-	-	24	16	12	39	-	17	22	25	25	23
24	20	37	49	29	22	30	62	22	47	71	21	18	12	40	42	19	21	27	25	21
25	23	39	47	30	21	38	63	-	55	-	23	15	11	31	38	19	22	27	27	24
26	-	37	-	32	23	32	58	28	50	72	24	16	11	32	44	17	19	22	25	-
27	24	37	49	30	22	-	56	30	51	78	24	19	11	34	40	-	19	-	29	21
28	24	39	47	-	18	37	-	24	47	85	23	16	10	39	35	17	22	20	26	20
29	25	-	47	29	19	36	54	20	49	79	24	16	11	35	41	18	-	23	25	-
30	20	37	47	28	22	40	51	22	51	78	-	21	11	37	39	16	18	-	25	-
31	23	38	48	29	19	35	53	20	56	72	22	16	11	-	37	16	20	25	25	20
32	23	36	45	29	-	37	63	29	50	79	20	16	11	37	-	18	21	27	29	25
33	23	38	47	30	23	40	59	26	56	78	23	18	11	41	40	-	21	22	-	23
34	21	35	50	27	23	38	65	22	47	71	24	16	10	38	36	16	20	27	27	24
35	20	35	48	32	21	33	61	25	-	-	21	21	10	38	-	19	18	24	29	19
36	-	38	45	30	19	31	63	24	56	85	23	-	10	41	37	19	19	24	26	21
37	24	36	-	30	22	38	55	24	50	87	23	19	12	-	39	17	20	26	-	25
38	24	39	49	32	21	37	52	-	-	71	24	19	12	35	38	15	19	23	25	22
39	21	39	49	31	19	35	57	29	55	77	21	19	11	40	43	15	19	-	28	19
40	25	35	47	30	20	34	59	25	48	72	23	-	10	41	35	18	20	20	26	18
41	22	36	48	32	20	-	55	25	49	71	23	19	12	31	43	17	19	22	26	24
42	23	39	47	27	19	39	64	24	53	74	24	21	12	32	-	19	18	-	29	22
43	23	36	46	32	20	35	-	21	55	80	21	16	12	32	40	18	20	23	26	22
44	-	-	45	31	22	37	52	-	57	72	22	16	11	37	-	18	22	20	27	24
45	20	40	47	29	21	32	52	29	55	82	21	15	10	33	-	16	20	27	-	19
46	20	40	-	30	20	38	58	23	50	82	23	20	11	31	38	19	-	-	25	19
47	23	39	49	-	21	39	61	25	-	78	24	19	11	38	39	17	22	27	27	25
48	22	38	49	28	18	33	-	25	49	74	23	18	12	30	43	16	20	21	29	-
49	20	39	47	29	21	-	60	22	52	81	23	21	12	30	40	16	20	24	25	19
50	22	-	47	28	20	32	64	27	49	77	-	18	10	33	37	17	20	20	25	24

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Table 8. Setup time and demand of products

Product	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Setup time (min)	4	9	8	6	5	6	4	1	10	12	24	2	5	7	11	3	15	4	2	7
Demand	145	107	134	105	140	145	115	87	145	126	125	150	118	106	75	80	132	83	65	89

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Table 9. Results of large case

No.	Idle time (min)	Makespan (min)	CPU time (s)
1	5587.3	1932	16236
2	5457.4	1848.30	16376
3	5164.9	1865.4	16221
4	5258.9	1919	15754
5	5757.2	1806	15743
Average	5445.14	1874.14	16066
SD	214.92	46.36	264.84

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