China Mechanical Engineering ›› 2025, Vol. 36 ›› Issue (8): 1811-1823.DOI: 10.3969/j.issn.1004-132X.2025.08.016
Guohui ZHANG(), Yihao CAI, Zhixiao LI, Shenghui GUO, Haijun ZHANG
Received:
2024-08-11
Online:
2025-08-25
Published:
2025-09-18
Contact:
Guohui ZHANG
通讯作者:
张国辉
基金资助:
CLC Number:
Guohui ZHANG, Yihao CAI, Zhixiao LI, Shenghui GUO, Haijun ZHANG. Research on Flexible Job Shop Scheduling Problems Considering Limited AGV Transportation Resources[J]. China Mechanical Engineering, 2025, 36(8): 1811-1823.
张国辉, 蔡翌豪, 李志霄, 郭胜会, 张海军. 考虑有限AGV运输资源的柔性作业车间调度研究[J]. 中国机械工程, 2025, 36(8): 1811-1823.
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URL: https://www.cmemo.org.cn/EN/10.3969/j.issn.1004-132X.2025.08.016
工件 | 工序 | 加工机器和加工时间 | |||
---|---|---|---|---|---|
M1 | M2 | M3 | M4 | ||
J1 | O11 | 3 | 10 | — | 3 |
O12 | — | 4 | 5 | 2 | |
J2 | O21 | — | 7 | 5 | — |
O22 | 6 | 2 | — | 8 | |
O23 | — | 6 | 4 | 2 | |
J3 | O31 | 4 | — | 2 | — |
O32 | — | 3 | 6 | — | |
AGV编号 | 1、2、3 |
Tab.1 FJSP-LAT instance
工件 | 工序 | 加工机器和加工时间 | |||
---|---|---|---|---|---|
M1 | M2 | M3 | M4 | ||
J1 | O11 | 3 | 10 | — | 3 |
O12 | — | 4 | 5 | 2 | |
J2 | O21 | — | 7 | 5 | — |
O22 | 6 | 2 | — | 8 | |
O23 | — | 6 | 4 | 2 | |
J3 | O31 | 4 | — | 2 | — |
O32 | — | 3 | 6 | — | |
AGV编号 | 1、2、3 |
参数 | L/U区 | M1 | M2 | M3 | M4 |
---|---|---|---|---|---|
L/U区 | 0 | 2 | 3 | 2 | 4 |
M1 | 2 | 0 | 4 | 3 | 2 |
M2 | 3 | 4 | 0 | 3 | 4 |
M3 | 2 | 3 | 3 | 0 | 4 |
M4 | 4 | 2 | 4 | 4 | 0 |
Tab.2 AGV transportation time
参数 | L/U区 | M1 | M2 | M3 | M4 |
---|---|---|---|---|---|
L/U区 | 0 | 2 | 3 | 2 | 4 |
M1 | 2 | 0 | 4 | 3 | 2 |
M2 | 3 | 4 | 0 | 3 | 4 |
M3 | 2 | 3 | 3 | 0 | 4 |
M4 | 4 | 2 | 4 | 4 | 0 |
参数 | 定义 |
---|---|
i | 工件号; i=1,2,…,n |
j | 工序号; j=1,2,…,li |
k | 机器号; k=1,2,…,m |
v | AGV号; v=1,2,…,s |
li | 工件i的工序总数 |
Oij | 工件i的第j道工序 |
Ci | 工件i的完工时间 |
tsij | 工件i的第j道工序的开工时间 |
tcij | 工件i的第j道工序的完工时间 |
Tijk | 工件i的第j道工序在机器k上的加工时间 |
Sijks | 工件i的第j道工序在机器k上的开始加工时间 |
Eijks | 工件i的第j道工序在机器k上的开始终止时间 |
TSE(i,j,v) | v运输Oi,j-1时空载行程开始时间 |
ETE(i,j,v) | v运输Oi,j-1时空载行程结束时间 |
TLB(i,j,v) | v运输Oi,j-1时负载行程开始时间 |
ETL(i,j,v) | v运输Oi,j-1时负载行程结束时间 |
PMLijk | 工件i的第j道工序在机器k上的负载功率 |
PMNijk | 工件i的第j道工序在机器k上的空载功率 |
PALijv | v的负载功率 |
PANijv | v的空载功率 |
EP | 加工能耗 |
EI | 空闲能耗 |
ET | 运输能耗 |
DEi | 最早交货期 |
DLi | 最晚交货期 |
Xijkp | 0-1决策变量,工件i的第j道工序在机器k上加工为1,否则为0 |
Rijkv | 0-1决策变量,当工件i的第j道工序在机器k上加工选择AGVs完成运输任务为1,否则为0 |
Tab.3 Notation description
参数 | 定义 |
---|---|
i | 工件号; i=1,2,…,n |
j | 工序号; j=1,2,…,li |
k | 机器号; k=1,2,…,m |
v | AGV号; v=1,2,…,s |
li | 工件i的工序总数 |
Oij | 工件i的第j道工序 |
Ci | 工件i的完工时间 |
tsij | 工件i的第j道工序的开工时间 |
tcij | 工件i的第j道工序的完工时间 |
Tijk | 工件i的第j道工序在机器k上的加工时间 |
Sijks | 工件i的第j道工序在机器k上的开始加工时间 |
Eijks | 工件i的第j道工序在机器k上的开始终止时间 |
TSE(i,j,v) | v运输Oi,j-1时空载行程开始时间 |
ETE(i,j,v) | v运输Oi,j-1时空载行程结束时间 |
TLB(i,j,v) | v运输Oi,j-1时负载行程开始时间 |
ETL(i,j,v) | v运输Oi,j-1时负载行程结束时间 |
PMLijk | 工件i的第j道工序在机器k上的负载功率 |
PMNijk | 工件i的第j道工序在机器k上的空载功率 |
PALijv | v的负载功率 |
PANijv | v的空载功率 |
EP | 加工能耗 |
EI | 空闲能耗 |
ET | 运输能耗 |
DEi | 最早交货期 |
DLi | 最晚交货期 |
Xijkp | 0-1决策变量,工件i的第j道工序在机器k上加工为1,否则为0 |
Rijkv | 0-1决策变量,当工件i的第j道工序在机器k上加工选择AGVs完成运输任务为1,否则为0 |
算例 | INSGA-Ⅱ | IGA | NSGA-Ⅱ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
f1 | f2 | f3 | f1均值 | f1 | f2 | f3 | f1均值 | f1 | f2 | f3 | f1均值 | |
MK01 | 51 | 2.3471×105 | 164 | 53.4 | 55 | 2.5902×105 | 301 | 60.9 | 53 | 2.2215×105 | 258 | 57.3 |
MK02 | 41 | 1.8822×105 | 77 | 43.2 | 45 | 2.0447×105 | 46 | 47.3 | 44 | 1.9734×105 | 40 | 47 |
MK03 | 217 | 1.68×106 | 434 | 229.6 | 235 | 1.8015×106 | 480 | 252.6 | 218 | 2.132×106 | 488 | 237.7 |
MK04 | 89 | 6.0619×105 | 1275 | 92.8 | 97 | 6.168×105 | 1189 | 102.4 | 91 | 5.2505×105 | 1023 | 98 |
MK05 | 199 | 7.9841×105 | 940 | 202.5 | 205 | 9.4963×105 | 1554 | 211.9 | 200 | 8.7233×105 | 1263 | 206.7 |
MK06 | 143 | 7.4185×105 | 113 | 148.4 | 170 | 9.397×105 | 431 | 182.8 | 151 | 8.6657×105 | 137 | 165.7 |
MK07 | 197 | 1.1921×106 | 1281 | 202.1 | 220 | 1.0403×106 | 1013 | 227 | 209 | 1.1754×106 | 1084 | 214.5 |
MK08 | 529 | 5.4534×106 | 1574 | 535.9 | 544 | 4.92×106 | 2683 | 554.3 | 534 | 5.3394×106 | 2181 | 539.9 |
MK09 | 407 | 4.2215×106 | 3964 | 408.8 | 458 | 3.7298×106 | 3208 | 467 | 418 | 5.4472×106 | 3897 | 437 |
MK10 | 334 | 3.3109×106 | 2449 | 343 | 361 | 3.093×106 | 2146 | 377.9 | 355 | 5.6291×106 | 2202 | 367.3 |
Tab.4 Comparison of algorithm optimization objectives
算例 | INSGA-Ⅱ | IGA | NSGA-Ⅱ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
f1 | f2 | f3 | f1均值 | f1 | f2 | f3 | f1均值 | f1 | f2 | f3 | f1均值 | |
MK01 | 51 | 2.3471×105 | 164 | 53.4 | 55 | 2.5902×105 | 301 | 60.9 | 53 | 2.2215×105 | 258 | 57.3 |
MK02 | 41 | 1.8822×105 | 77 | 43.2 | 45 | 2.0447×105 | 46 | 47.3 | 44 | 1.9734×105 | 40 | 47 |
MK03 | 217 | 1.68×106 | 434 | 229.6 | 235 | 1.8015×106 | 480 | 252.6 | 218 | 2.132×106 | 488 | 237.7 |
MK04 | 89 | 6.0619×105 | 1275 | 92.8 | 97 | 6.168×105 | 1189 | 102.4 | 91 | 5.2505×105 | 1023 | 98 |
MK05 | 199 | 7.9841×105 | 940 | 202.5 | 205 | 9.4963×105 | 1554 | 211.9 | 200 | 8.7233×105 | 1263 | 206.7 |
MK06 | 143 | 7.4185×105 | 113 | 148.4 | 170 | 9.397×105 | 431 | 182.8 | 151 | 8.6657×105 | 137 | 165.7 |
MK07 | 197 | 1.1921×106 | 1281 | 202.1 | 220 | 1.0403×106 | 1013 | 227 | 209 | 1.1754×106 | 1084 | 214.5 |
MK08 | 529 | 5.4534×106 | 1574 | 535.9 | 544 | 4.92×106 | 2683 | 554.3 | 534 | 5.3394×106 | 2181 | 539.9 |
MK09 | 407 | 4.2215×106 | 3964 | 408.8 | 458 | 3.7298×106 | 3208 | 467 | 418 | 5.4472×106 | 3897 | 437 |
MK10 | 334 | 3.3109×106 | 2449 | 343 | 361 | 3.093×106 | 2146 | 377.9 | 355 | 5.6291×106 | 2202 | 367.3 |
算例 | HV(望大) | IGD(望小) | ||||
---|---|---|---|---|---|---|
INSGA-Ⅱ | IGA | NSGA-Ⅱ | INSGA-Ⅱ | IGA | NSGA-Ⅱ | |
MK01 | 0.034 92 | 0.000 946 85 | 0.010 643 743 | 0.097 84 | 0.138 57 | 0.129 38 |
MK02 | 0.1022 | 0.039 72 | 0.084 83 | 0.209 96 | 0.249 59 | 0.221 76 |
MK03 | 0.143 02 | 0.004 42 | 0.037 91 | 0.0886 | 0.169 53 | 0.122 01 |
MK04 | 0.125 03 | 0.020 989 356 | 0.052 15 | 0.222 15 | 0.280 81 | 0.2482 |
MK05 | 0.167 92 | 0.002 18 | 0.004 01 | 0.123 89 | 0.296 01 | 0.252 23 |
MK06 | 0.014 11 | 0.002 83 | 0.003 44 | 0.120 68 | 0.153 97 | 0.147 99 |
MK07 | 0.321 15 | 0.044 77 | 0.101 21 | 0.063 52 | 0.113 49 | 0.079 31 |
MK08 | 0.075 49 | 0.006 87 | 0.015 81 | 0.121 99 | 0.174 75 | 0.150 98 |
MK09 | 0.4067 | 0.143 | 0.204 99 | 0.0964 | 0.117 31 | 0.106 13 |
MK10 | 0.2959 | 0.014 43 | 0.141 47 | 0.029 32 | 0.177 79 | 0.091 69 |
Tab.5 HV and IGD comparison table.
算例 | HV(望大) | IGD(望小) | ||||
---|---|---|---|---|---|---|
INSGA-Ⅱ | IGA | NSGA-Ⅱ | INSGA-Ⅱ | IGA | NSGA-Ⅱ | |
MK01 | 0.034 92 | 0.000 946 85 | 0.010 643 743 | 0.097 84 | 0.138 57 | 0.129 38 |
MK02 | 0.1022 | 0.039 72 | 0.084 83 | 0.209 96 | 0.249 59 | 0.221 76 |
MK03 | 0.143 02 | 0.004 42 | 0.037 91 | 0.0886 | 0.169 53 | 0.122 01 |
MK04 | 0.125 03 | 0.020 989 356 | 0.052 15 | 0.222 15 | 0.280 81 | 0.2482 |
MK05 | 0.167 92 | 0.002 18 | 0.004 01 | 0.123 89 | 0.296 01 | 0.252 23 |
MK06 | 0.014 11 | 0.002 83 | 0.003 44 | 0.120 68 | 0.153 97 | 0.147 99 |
MK07 | 0.321 15 | 0.044 77 | 0.101 21 | 0.063 52 | 0.113 49 | 0.079 31 |
MK08 | 0.075 49 | 0.006 87 | 0.015 81 | 0.121 99 | 0.174 75 | 0.150 98 |
MK09 | 0.4067 | 0.143 | 0.204 99 | 0.0964 | 0.117 31 | 0.106 13 |
MK10 | 0.2959 | 0.014 43 | 0.141 47 | 0.029 32 | 0.177 79 | 0.091 69 |
算例 | SC (A,B) | SC (B,A) | SC (A,C) | SC (C,A) | SC (B,C) | SC (C,B) |
---|---|---|---|---|---|---|
MK01 | 0.946 43 | 0 | 0.705 36 | 0.080 86 | 0.150 59 | 0.655 96 |
MK02 | 0.664 73 | 0.040 99 | 0.684 79 | 0.186 16 | 0.226 33 | 0.482 08 |
MK03 | 0.834 06 | 0 | 0.762 18 | 0.032 33 | 0.001 27 | 0.603 87 |
MK04 | 0.651 09 | 0.0934 | 0.545 83 | 0.140 22 | 0.13 | 0.597 62 |
MK05 | 0.781 43 | 0 | 0.724 53 | 0.0225 | 0.016 67 | 0.713 34 |
MK06 | 0.971 43 | 0 | 0.8963 | 0.004 08 | 0.02 | 0.523 57 |
MK07 | 0.563 72 | 0.0029 | 0.773 76 | 0.074 94 | 0.016 54 | 0.138 61 |
MK08 | 0.462 12 | 0 | 0.337 06 | 0.018 29 | 0 | 0.766 67 |
MK09 | 0.495 61 | 0.026 92 | 0.671 15 | 0.120 13 | 0.048 04 | 0.386 36 |
MK10 | 0.558 83 | 0.000 68 | 0.567 04 | 0.097 33 | 0.0245 | 0.6322 |
Tab.6 SC Comparison
算例 | SC (A,B) | SC (B,A) | SC (A,C) | SC (C,A) | SC (B,C) | SC (C,B) |
---|---|---|---|---|---|---|
MK01 | 0.946 43 | 0 | 0.705 36 | 0.080 86 | 0.150 59 | 0.655 96 |
MK02 | 0.664 73 | 0.040 99 | 0.684 79 | 0.186 16 | 0.226 33 | 0.482 08 |
MK03 | 0.834 06 | 0 | 0.762 18 | 0.032 33 | 0.001 27 | 0.603 87 |
MK04 | 0.651 09 | 0.0934 | 0.545 83 | 0.140 22 | 0.13 | 0.597 62 |
MK05 | 0.781 43 | 0 | 0.724 53 | 0.0225 | 0.016 67 | 0.713 34 |
MK06 | 0.971 43 | 0 | 0.8963 | 0.004 08 | 0.02 | 0.523 57 |
MK07 | 0.563 72 | 0.0029 | 0.773 76 | 0.074 94 | 0.016 54 | 0.138 61 |
MK08 | 0.462 12 | 0 | 0.337 06 | 0.018 29 | 0 | 0.766 67 |
MK09 | 0.495 61 | 0.026 92 | 0.671 15 | 0.120 13 | 0.048 04 | 0.386 36 |
MK10 | 0.558 83 | 0.000 68 | 0.567 04 | 0.097 33 | 0.0245 | 0.6322 |
参数 | L/U 区 | M1 | M2 | M3 | M4 | M5 | M6 |
---|---|---|---|---|---|---|---|
L/U区 | 0 | 1 | 3 | 1 | 2 | 2 | 3 |
M1 | 1 | 0 | 1 | 3 | 1 | 2 | 1 |
M2 | 3 | 1 | 0 | 2 | 1 | 1 | 3 |
M3 | 1 | 3 | 2 | 0 | 1 | 3 | 2 |
M4 | 2 | 1 | 1 | 1 | 0 | 1 | 2 |
M5 | 2 | 2 | 1 | 3 | 1 | 0 | 1 |
M6 | 3 | 1 | 3 | 2 | 2 | 1 | 0 |
Tab.7 AGV transportation time of Mk01 instance
参数 | L/U 区 | M1 | M2 | M3 | M4 | M5 | M6 |
---|---|---|---|---|---|---|---|
L/U区 | 0 | 1 | 3 | 1 | 2 | 2 | 3 |
M1 | 1 | 0 | 1 | 3 | 1 | 2 | 1 |
M2 | 3 | 1 | 0 | 2 | 1 | 1 | 3 |
M3 | 1 | 3 | 2 | 0 | 1 | 3 | 2 |
M4 | 2 | 1 | 1 | 1 | 0 | 1 | 2 |
M5 | 2 | 2 | 1 | 3 | 1 | 0 | 1 |
M6 | 3 | 1 | 3 | 2 | 2 | 1 | 0 |
工件 | 工序1 | 工序2 | 工序3 | 工序4 | 工序5 | 工序6 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
名称 | 时间 | 名称 | 时间 | 名称 | 时间 | 名称 | 时间 | 名称 | 时间 | 名称 | 时间 | |
J1 | 数控立车 | 8 | 车端面 | 5 | 镗孔 | 1 | 车槽 | 6 | 钳 | 3 | 铣平面 | 4 |
J2 | 车端面 | 3 | 车外圆 | 10 | 镗孔 | 1 | 数控立车 | 8 | 车槽 | 3 | 铣平面 | 4 |
J3 | 车外圆 | 6 | 铣平面 | 4 | 镗孔 | 7 | 表面处理 | 9 | 数控立车 | 2 | 车端面 | 0.75 |
J4 | 热处理 | 3 | 车槽 | 1 | 铣平面 | 2 | 车外圆 | 3 | 车端面 | 0.75 | — | — |
J5 | 车外圆 | 2 | 铣平面 | 4 | 表面处理 | 3 | 镗孔 | 1.25 | — | — | — | — |
J6 | 车外圆 | 3 | 铣平面 | 5 | 车槽 | 4 | 数控立车 | 5 | 车端面 | 3 | 镗孔 | 3 |
J7 | 车外圆 | 1 | 检验 | 3 | 数控立车 | 6 | 铣平面 | 7 | 镗孔 | 3 | 车端面 | 6 |
J8 | 钳 | 7 | 数控立车 | 10 | 车端面 | 9 | 铣平面 | 3 | 镗孔 | 4 | 车槽 | 10 |
Tab.8 Jobs and operations information
工件 | 工序1 | 工序2 | 工序3 | 工序4 | 工序5 | 工序6 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
名称 | 时间 | 名称 | 时间 | 名称 | 时间 | 名称 | 时间 | 名称 | 时间 | 名称 | 时间 | |
J1 | 数控立车 | 8 | 车端面 | 5 | 镗孔 | 1 | 车槽 | 6 | 钳 | 3 | 铣平面 | 4 |
J2 | 车端面 | 3 | 车外圆 | 10 | 镗孔 | 1 | 数控立车 | 8 | 车槽 | 3 | 铣平面 | 4 |
J3 | 车外圆 | 6 | 铣平面 | 4 | 镗孔 | 7 | 表面处理 | 9 | 数控立车 | 2 | 车端面 | 0.75 |
J4 | 热处理 | 3 | 车槽 | 1 | 铣平面 | 2 | 车外圆 | 3 | 车端面 | 0.75 | — | — |
J5 | 车外圆 | 2 | 铣平面 | 4 | 表面处理 | 3 | 镗孔 | 1.25 | — | — | — | — |
J6 | 车外圆 | 3 | 铣平面 | 5 | 车槽 | 4 | 数控立车 | 5 | 车端面 | 3 | 镗孔 | 3 |
J7 | 车外圆 | 1 | 检验 | 3 | 数控立车 | 6 | 铣平面 | 7 | 镗孔 | 3 | 车端面 | 6 |
J8 | 钳 | 7 | 数控立车 | 10 | 车端面 | 9 | 铣平面 | 3 | 镗孔 | 4 | 车槽 | 10 |
工序名称 | 设备组 | 对应机床设备 | 对应AGV编号 |
---|---|---|---|
数控立车 | 数车组 | M1、M2 | 1、2、3、4 |
车端面 | 普车组 | M3、M4、M5 | 1、2、3、4 |
镗孔 | 镗床组 | M6 | 1、2、3、4 |
车退刀槽 | 普车组 | M3、M4、M5 | 1、2、3、4 |
铣平面 | 数铣组 | M7、M8 | 1、2、3、4 |
车外圆 | 普车组 | M3、M4、M5 | 1、2、3、4 |
Tab.9 Available machines and AGVs of each operation
工序名称 | 设备组 | 对应机床设备 | 对应AGV编号 |
---|---|---|---|
数控立车 | 数车组 | M1、M2 | 1、2、3、4 |
车端面 | 普车组 | M3、M4、M5 | 1、2、3、4 |
镗孔 | 镗床组 | M6 | 1、2、3、4 |
车退刀槽 | 普车组 | M3、M4、M5 | 1、2、3、4 |
铣平面 | 数铣组 | M7、M8 | 1、2、3、4 |
车外圆 | 普车组 | M3、M4、M5 | 1、2、3、4 |
参数 | L/U 区 | M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 |
---|---|---|---|---|---|---|---|---|---|
L/U区 | 0 | 2 | 3 | 2 | 4 | 4 | 2 | 3 | 2 |
M1 | 2 | 0 | 4 | 3 | 2 | 5 | 3 | 6 | 4 |
M2 | 3 | 4 | 0 | 3 | 4 | 5 | 6 | 4 | 3 |
M3 | 2 | 3 | 3 | 0 | 4 | 2 | 6 | 4 | 5 |
M4 | 4 | 2 | 4 | 4 | 0 | 3 | 2 | 2 | 5 |
M5 | 4 | 5 | 5 | 2 | 3 | 0 | 2 | 4 | 6 |
M6 | 2 | 3 | 6 | 6 | 2 | 2 | 0 | 6 | 4 |
M7 | 3 | 6 | 4 | 4 | 2 | 4 | 6 | 0 | 2 |
M8 | 2 | 4 | 3 | 5 | 5 | 6 | 4 | 2 | 0 |
Tab.10 AGV transportation time
参数 | L/U 区 | M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 |
---|---|---|---|---|---|---|---|---|---|
L/U区 | 0 | 2 | 3 | 2 | 4 | 4 | 2 | 3 | 2 |
M1 | 2 | 0 | 4 | 3 | 2 | 5 | 3 | 6 | 4 |
M2 | 3 | 4 | 0 | 3 | 4 | 5 | 6 | 4 | 3 |
M3 | 2 | 3 | 3 | 0 | 4 | 2 | 6 | 4 | 5 |
M4 | 4 | 2 | 4 | 4 | 0 | 3 | 2 | 2 | 5 |
M5 | 4 | 5 | 5 | 2 | 3 | 0 | 2 | 4 | 6 |
M6 | 2 | 3 | 6 | 6 | 2 | 2 | 0 | 6 | 4 |
M7 | 3 | 6 | 4 | 4 | 2 | 4 | 6 | 0 | 2 |
M8 | 2 | 4 | 3 | 5 | 5 | 6 | 4 | 2 | 0 |
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