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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

  • * Corresponding author: Zhe Zhang

    * Corresponding author: Zhe Zhang 

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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  • 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.

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

    Citation:

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  • Figure 1.  Three types seru

    Figure 2.  Whole bi-level decision procedure

    Figure 3.  The outline of SA-GA algorithm

    Figure 4.  An example of SA encoding

    Figure 5.  The genetic encoding based on allocation ratios

    Figure 6.  The flowchart of SA-GA algorithm

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

    Figure 8.  Idle time and makespan

    Figure 9.  The worker assignment decision

    Figure 10.  Loading results

    Figure 11.  The worker assignment decision

    Figure 12.  Loading results

    Table 1.  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 $
     | Show Table
    DownLoad: CSV

    Table 2.  Data about products

    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.
     | Show Table
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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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    Table 7.  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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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