# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2019134

## 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

Received  April 2019 Revised  July 2019 Published  October 2019

Fund Project: 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

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.

Citation: Lan Luo, Zhe Zhang, Yong Yin. Simulated annealing and genetic algorithm based method for a bi-level seru loading problem with worker assignment in seru production systems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019134
##### References:

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##### References:
Three types seru
Whole bi-level decision procedure
The outline of SA-GA algorithm
An example of SA encoding
The genetic encoding based on allocation ratios
The flowchart of SA-GA algorithm
The minimum idle time in each iteration of SA
Idle time and makespan
The worker assignment decision
The worker assignment decision
The parameter setting of SA-GA algorithm
 Level Algorithm Parameters Upper SA $T\_max=10000$ $T\_min=0.1$ $II=20$ $\alpha=0.9$ Lower GA $pop\_size=300$ $GEN=500$ $p\_zero1=0.75$ $p\_zero2=0.25$ $p\_cross=0.9$ $p\_muta=0.1$
 Level Algorithm Parameters Upper SA $T\_max=10000$ $T\_min=0.1$ $II=20$ $\alpha=0.9$ Lower GA $pop\_size=300$ $GEN=500$ $p\_zero1=0.75$ $p\_zero2=0.25$ $p\_cross=0.9$ $p\_muta=0.1$
 Product Worker's processing time (min) Demand Setup(min) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 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.
 Product Worker's processing time (min) Demand Setup(min) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 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.
 1 2 3 4 5 6 7 8 1 – 100 – – 80 – 50 115 2 – – 130 – – 116 – – 3 95 – – 105 40 29 – –
 1 2 3 4 5 6 7 8 1 – 100 – – 80 – 50 115 2 – – 130 – – 116 – – 3 95 – – 105 40 29 – –
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
 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
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
 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
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
 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
Workers' processing time for each product
 Worker Workers' processing time for each product (min) 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
 Worker Workers' processing time for each product (min) 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
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
 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
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
 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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