Journal of Computer Applications ›› 2022, Vol. 42 ›› Issue (8): 2617-2627.DOI: 10.11772/j.issn.1001-9081.2021061071
• Frontier and comprehensive applications • Previous Articles Next Articles
Kuineng CHEN1,2(), Xiaofang YUAN2
Received:
2021-06-22
Revised:
2021-12-10
Accepted:
2021-12-17
Online:
2022-03-16
Published:
2022-08-10
Contact:
Kuineng CHEN
About author:
CHEN Kuineng, born in 1987, M. S., lecturer. His research interests include intelligent optimization algorithm, intelligent manufacturing, control theory.Supported by:
通讯作者:
陈揆能
作者简介:
陈揆能(1987—),男,湖南娄底人,讲师,硕士,主要研究方向:智能优化算法、智能制造、控制理论;基金资助:
CLC Number:
Kuineng CHEN, Xiaofang YUAN. Multi-objective hybrid evolutionary algorithm for solving open-shop scheduling problem with controllable processing time[J]. Journal of Computer Applications, 2022, 42(8): 2617-2627.
陈揆能, 袁小芳. 多目标混合进化算法求解加工时间可控的开放车间调度问题[J]. 《计算机应用》唯一官方网站, 2022, 42(8): 2617-2627.
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URL: https://www.joca.cn/EN/10.11772/j.issn.1001-9081.2021061071
符号 | 含义 |
---|---|
n | 加工工件数量 |
m | 加工机器数量 |
i, i' | 工件指数 |
j, j' | 机器指数 |
Oij | 工件i由机器j负责加工的工序 |
Oij 的加工时间上界,将其视为Oij 的正常处理时间 | |
Oij 的加工时间下界,通过分配更多的资源使Oij 能达到的最小加工时间 | |
tij | Oij 的实际加工时间 |
Eij | Oij 的额外加工能耗 |
Sij | Oij 的开始加工时间 |
Cij | Oij 的完工时间 |
Cmax | 完工时间 |
Rc | 总额外能耗 |
A | 足够大的正整数 |
αij | 能耗系数 |
机器j紧邻且先于机器j'加工工件i则 | |
工件i紧邻且先于工件i'在机器j上加工则 |
Tab. 1 Symbols and their meanings
符号 | 含义 |
---|---|
n | 加工工件数量 |
m | 加工机器数量 |
i, i' | 工件指数 |
j, j' | 机器指数 |
Oij | 工件i由机器j负责加工的工序 |
Oij 的加工时间上界,将其视为Oij 的正常处理时间 | |
Oij 的加工时间下界,通过分配更多的资源使Oij 能达到的最小加工时间 | |
tij | Oij 的实际加工时间 |
Eij | Oij 的额外加工能耗 |
Sij | Oij 的开始加工时间 |
Cij | Oij 的完工时间 |
Cmax | 完工时间 |
Rc | 总额外能耗 |
A | 足够大的正整数 |
αij | 能耗系数 |
机器j紧邻且先于机器j'加工工件i则 | |
工件i紧邻且先于工件i'在机器j上加工则 |
工件机器 | M1 | M2 | M3 |
---|---|---|---|
J1 | [ | [ | [50,54]/19 |
J2 | [ | [75,89]/49 | [55,70]/43 |
J3 | [30,38]/15 | [ | [ |
Tab. 2 Processing information of example
工件机器 | M1 | M2 | M3 |
---|---|---|---|
J1 | [ | [ | [50,54]/19 |
J2 | [ | [75,89]/49 | [55,70]/43 |
J3 | [30,38]/15 | [ | [ |
参数 | 第1水平 | 第2水平 | 第3水平 | 第4水平 | 第5水平 |
---|---|---|---|---|---|
c1 | 0.1 | 0.3 | 0.5 | 0.7 | 0.9 |
c2 | 0.1 | 0.3 | 0.5 | 0.7 | 0.9 |
T | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 |
Tab. 3 Values of parameters with different levels
参数 | 第1水平 | 第2水平 | 第3水平 | 第4水平 | 第5水平 |
---|---|---|---|---|---|
c1 | 0.1 | 0.3 | 0.5 | 0.7 | 0.9 |
c2 | 0.1 | 0.3 | 0.5 | 0.7 | 0.9 |
T | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 |
实验组 | c1 | c2 | T | IGD |
---|---|---|---|---|
1 | 1 | 1 | 1 | 0.164 |
2 | 1 | 2 | 2 | 0.155 |
3 | 1 | 3 | 3 | 0.140 |
4 | 1 | 4 | 4 | 0.137 |
5 | 1 | 5 | 5 | 0.178 |
6 | 2 | 1 | 2 | 0.163 |
7 | 2 | 2 | 3 | 0.157 |
8 | 2 | 3 | 4 | 0.143 |
9 | 2 | 4 | 5 | 0.117 |
10 | 2 | 5 | 1 | 0.145 |
11 | 3 | 1 | 3 | 0.146 |
12 | 3 | 2 | 4 | 0.143 |
13 | 3 | 3 | 5 | 0.140 |
14 | 3 | 4 | 1 | 0.129 |
15 | 3 | 5 | 2 | 0.102 |
16 | 4 | 1 | 4 | 0.110 |
17 | 4 | 2 | 5 | 0.115 |
18 | 4 | 3 | 1 | 0.100 |
19 | 4 | 4 | 2 | 0.116 |
20 | 4 | 5 | 3 | 0.092 |
21 | 5 | 1 | 5 | 0.172 |
22 | 5 | 2 | 1 | 0.166 |
23 | 5 | 3 | 2 | 0.152 |
24 | 5 | 4 | 3 | 0.137 |
25 | 5 | 5 | 4 | 0.148 |
Tab. 4 Orthogonal experiment table and corresponding average IGD values
实验组 | c1 | c2 | T | IGD |
---|---|---|---|---|
1 | 1 | 1 | 1 | 0.164 |
2 | 1 | 2 | 2 | 0.155 |
3 | 1 | 3 | 3 | 0.140 |
4 | 1 | 4 | 4 | 0.137 |
5 | 1 | 5 | 5 | 0.178 |
6 | 2 | 1 | 2 | 0.163 |
7 | 2 | 2 | 3 | 0.157 |
8 | 2 | 3 | 4 | 0.143 |
9 | 2 | 4 | 5 | 0.117 |
10 | 2 | 5 | 1 | 0.145 |
11 | 3 | 1 | 3 | 0.146 |
12 | 3 | 2 | 4 | 0.143 |
13 | 3 | 3 | 5 | 0.140 |
14 | 3 | 4 | 1 | 0.129 |
15 | 3 | 5 | 2 | 0.102 |
16 | 4 | 1 | 4 | 0.110 |
17 | 4 | 2 | 5 | 0.115 |
18 | 4 | 3 | 1 | 0.100 |
19 | 4 | 4 | 2 | 0.116 |
20 | 4 | 5 | 3 | 0.092 |
21 | 5 | 1 | 5 | 0.172 |
22 | 5 | 2 | 1 | 0.166 |
23 | 5 | 3 | 2 | 0.152 |
24 | 5 | 4 | 3 | 0.137 |
25 | 5 | 5 | 4 | 0.148 |
参数 | 第1水平 | 第2水平 | 第3水平 | 第4水平 | 第5水平 |
---|---|---|---|---|---|
c1 | 0.775 | 0.725 | 0.660 | 0.533 | 0.774 |
c2 | 0.755 | 0.736 | 0.675 | 0.635 | 0.664 |
T | 0.689 | 0.687 | 0.672 | 0.681 | 0.722 |
Tab. 5 Average IGD values corresponding to parameter levels
参数 | 第1水平 | 第2水平 | 第3水平 | 第4水平 | 第5水平 |
---|---|---|---|---|---|
c1 | 0.775 | 0.725 | 0.660 | 0.533 | 0.774 |
c2 | 0.755 | 0.736 | 0.675 | 0.635 | 0.664 |
T | 0.689 | 0.687 | 0.672 | 0.681 | 0.722 |
规模 | 收敛时间/s | 规模 | 收敛时间/s |
---|---|---|---|
4×4 | 30/40/60/80 | 10×10 | 400/380/300/320 |
5×5 | 80/90/90/90 | 15×15 | 380/380/400/450 |
7×7 | 120/160/100/200 | 20×20 | 400/500/400/450 |
Tab. 6 Convergence times of other examples
规模 | 收敛时间/s | 规模 | 收敛时间/s |
---|---|---|---|
4×4 | 30/40/60/80 | 10×10 | 400/380/300/320 |
5×5 | 80/90/90/90 | 15×15 | 380/380/400/450 |
7×7 | 120/160/100/200 | 20×20 | 400/500/400/450 |
算法 | 种群数 | 交叉概率 | 变异概率 | 分区数 | 外部档案库大小 |
---|---|---|---|---|---|
NSGA-Ⅱ | 100 | 0.7 | 0.1 | — | — |
NSGA-Ⅲ | 100 | 0.7 | 0.1 | 5 | — |
SPEA2 | 100 | 0.7 | 0.1 | — | 100 |
Tab. 7 Parameter setting of different algorithms
算法 | 种群数 | 交叉概率 | 变异概率 | 分区数 | 外部档案库大小 |
---|---|---|---|---|---|
NSGA-Ⅱ | 100 | 0.7 | 0.1 | — | — |
NSGA-Ⅲ | 100 | 0.7 | 0.1 | 5 | — |
SPEA2 | 100 | 0.7 | 0.1 | — | 100 |
算例 | GD | |||||||
---|---|---|---|---|---|---|---|---|
Mean | Std | |||||||
MOHEA | NSGA-Ⅱ | NSGA-Ⅲ | SPEA2 | MOHEA | NSGA-Ⅱ | NSGA-Ⅲ | SPEA2 | |
4×4_1 | 0.021 | 0.038 | 0.040 | 0.051 | 0.008 | 0.013 | 0.016 | 0.019 |
4×4_2 | 0.039 | 0.056 | 0.066 | 0.070 | 0.016 | 0.034 | 0.024 | 0.030 |
4×4_3 | 0.010 | 0.024 | 0.027 | 0.028 | 0.010 | 0.030 | 0.021 | 0.021 |
4×4_4 | 0.034 | 0.041 | 0.051 | 0.039 | 0.022 | 0.024 | 0.027 | 0.026 |
5×5_1 | 0.040 | 0.092 | 0.089 | 0.081 | 0.025 | 0.035 | 0.050 | 0.037 |
5×5_2 | 0.039 | 0.045 | 0.093 | 0.070 | 0.020 | 0.026 | 0.026 | 0.020 |
5×5_3 | 0.064 | 0.091 | 0.081 | 0.110 | 0.030 | 0.028 | 0.023 | 0.040 |
5×5_4 | 0.059 | 0.083 | 0.117 | 0.094 | 0.033 | 0.041 | 0.043 | 0.037 |
7×7_1 | 0.061 | 0.114 | 0.107 | 0.108 | 0.028 | 0.037 | 0.039 | 0.033 |
7×7_2 | 0.075 | 0.141 | 0.117 | 0.112 | 0.024 | 0.043 | 0.042 | 0.043 |
7×7_3 | 0.056 | 0.127 | 0.105 | 0.156 | 0.024 | 0.031 | 0.026 | 0.042 |
7×7_4 | 0.072 | 0.137 | 0.149 | 0.146 | 0.034 | 0.046 | 0.048 | 0.033 |
10×10_1 | 0.072 | 0.137 | 0.169 | 0.186 | 0.028 | 0.040 | 0.043 | 0.025 |
10×10_2 | 0.076 | 0.168 | 0.173 | 0.176 | 0.028 | 0.044 | 0.039 | 0.032 |
10×10_3 | 0.096 | 0.160 | 0.223 | 0.182 | 0.030 | 0.047 | 0.046 | 0.038 |
10×10_4 | 0.085 | 0.164 | 0.181 | 0.180 | 0.020 | 0.027 | 0.026 | 0.028 |
15×15_1 | 0.076 | 0.207 | 0.207 | 0.187 | 0.032 | 0.035 | 0.028 | 0.038 |
15×15_2 | 0.074 | 0.202 | 0.216 | 0.229 | 0.035 | 0.035 | 0.040 | 0.034 |
15×15_3 | 0.088 | 0.228 | 0.206 | 0.207 | 0.029 | 0.043 | 0.045 | 0.030 |
15×15_4 | 0.061 | 0.190 | 0.203 | 0.166 | 0.030 | 0.035 | 0.049 | 0.033 |
20×20_1 | 0.054 | 0.196 | 0.227 | 0.209 | 0.018 | 0.042 | 0.043 | 0.036 |
20×20_2 | 0.078 | 0.233 | 0.226 | 0.224 | 0.022 | 0.053 | 0.042 | 0.041 |
20×20_3 | 0.054 | 0.203 | 0.212 | 0.233 | 0.017 | 0.032 | 0.030 | 0.035 |
20×20_4 | 0.067 | 0.215 | 0.194 | 0.224 | 0.018 | 0.044 | 0.033 | 0.036 |
算例 | IGD | |||||||
Mean | Std | |||||||
MOHEA | NSGA-Ⅱ | NSGA-Ⅲ | SPEA2 | MOHEA | NSGA-Ⅱ | NSGA-Ⅲ | SPEA2 | |
4×4_1 | 0.039 | 0.089 | 0.097 | 0.122 | 0.016 | 0.029 | 0.042 | 0.051 |
4×4_2 | 0.045 | 0.092 | 0.103 | 0.105 | 0.018 | 0.026 | 0.028 | 0.035 |
4×4_3 | 0.033 | 0.136 | 0.145 | 0.137 | 0.018 | 0.036 | 0.041 | 0.040 |
4×4_4 | 0.056 | 0.112 | 0.119 | 0.109 | 0.031 | 0.033 | 0.033 | 0.024 |
5×5_1 | 0.075 | 0.164 | 0.153 | 0.151 | 0.023 | 0.041 | 0.041 | 0.050 |
5×5_2 | 0.066 | 0.082 | 0.127 | 0.099 | 0.019 | 0.028 | 0.023 | 0.030 |
5×5_3 | 0.088 | 0.153 | 0.169 | 0.187 | 0.041 | 0.039 | 0.058 | 0.052 |
5×5_4 | 0.068 | 0.160 | 0.156 | 0.130 | 0.031 | 0.032 | 0.047 | 0.038 |
7×7_1 | 0.110 | 0.158 | 0.145 | 0.136 | 0.029 | 0.029 | 0.037 | 0.028 |
7×7_2 | 0.083 | 0.159 | 0.132 | 0.152 | 0.016 | 0.033 | 0.028 | 0.043 |
7×7_3 | 0.106 | 0.159 | 0.150 | 0.184 | 0.040 | 0.022 | 0.032 | 0.035 |
7×7_4 | 0.097 | 0.141 | 0.175 | 0.162 | 0.032 | 0.034 | 0.032 | 0.028 |
10×10_1 | 0.086 | 0.141 | 0.174 | 0.179 | 0.018 | 0.025 | 0.031 | 0.022 |
10×10_2 | 0.114 | 0.211 | 0.216 | 0.203 | 0.019 | 0.038 | 0.027 | 0.029 |
10×10_3 | 0.111 | 0.166 | 0.217 | 0.166 | 0.028 | 0.031 | 0.031 | 0.026 |
10×10_4 | 0.133 | 0.184 | 0.196 | 0.204 | 0.034 | 0.024 | 0.024 | 0.024 |
15×15_1 | 0.123 | 0.237 | 0.238 | 0.217 | 0.031 | 0.029 | 0.021 | 0.024 |
15×15_2 | 0.138 | 0.228 | 0.233 | 0.252 | 0.045 | 0.027 | 0.032 | 0.027 |
15×15_3 | 0.142 | 0.250 | 0.238 | 0.230 | 0.025 | 0.027 | 0.027 | 0.019 |
15×15_4 | 0.126 | 0.222 | 0.238 | 0.219 | 0.026 | 0.022 | 0.026 | 0.018 |
20×20_1 | 0.104 | 0.312 | 0.317 | 0.311 | 0.016 | 0.021 | 0.020 | 0.024 |
20×20_2 | 0.139 | 0.301 | 0.281 | 0.289 | 0.022 | 0.024 | 0.023 | 0.025 |
20×20_3 | 0.073 | 0.318 | 0.308 | 0.341 | 0.013 | 0.018 | 0.021 | 0.020 |
20×20_4 | 0.128 | 0.354 | 0.337 | 0.355 | 0.021 | 0.018 | 0.021 | 0.018 |
Tab. 8 Computational results of different algorithms
算例 | GD | |||||||
---|---|---|---|---|---|---|---|---|
Mean | Std | |||||||
MOHEA | NSGA-Ⅱ | NSGA-Ⅲ | SPEA2 | MOHEA | NSGA-Ⅱ | NSGA-Ⅲ | SPEA2 | |
4×4_1 | 0.021 | 0.038 | 0.040 | 0.051 | 0.008 | 0.013 | 0.016 | 0.019 |
4×4_2 | 0.039 | 0.056 | 0.066 | 0.070 | 0.016 | 0.034 | 0.024 | 0.030 |
4×4_3 | 0.010 | 0.024 | 0.027 | 0.028 | 0.010 | 0.030 | 0.021 | 0.021 |
4×4_4 | 0.034 | 0.041 | 0.051 | 0.039 | 0.022 | 0.024 | 0.027 | 0.026 |
5×5_1 | 0.040 | 0.092 | 0.089 | 0.081 | 0.025 | 0.035 | 0.050 | 0.037 |
5×5_2 | 0.039 | 0.045 | 0.093 | 0.070 | 0.020 | 0.026 | 0.026 | 0.020 |
5×5_3 | 0.064 | 0.091 | 0.081 | 0.110 | 0.030 | 0.028 | 0.023 | 0.040 |
5×5_4 | 0.059 | 0.083 | 0.117 | 0.094 | 0.033 | 0.041 | 0.043 | 0.037 |
7×7_1 | 0.061 | 0.114 | 0.107 | 0.108 | 0.028 | 0.037 | 0.039 | 0.033 |
7×7_2 | 0.075 | 0.141 | 0.117 | 0.112 | 0.024 | 0.043 | 0.042 | 0.043 |
7×7_3 | 0.056 | 0.127 | 0.105 | 0.156 | 0.024 | 0.031 | 0.026 | 0.042 |
7×7_4 | 0.072 | 0.137 | 0.149 | 0.146 | 0.034 | 0.046 | 0.048 | 0.033 |
10×10_1 | 0.072 | 0.137 | 0.169 | 0.186 | 0.028 | 0.040 | 0.043 | 0.025 |
10×10_2 | 0.076 | 0.168 | 0.173 | 0.176 | 0.028 | 0.044 | 0.039 | 0.032 |
10×10_3 | 0.096 | 0.160 | 0.223 | 0.182 | 0.030 | 0.047 | 0.046 | 0.038 |
10×10_4 | 0.085 | 0.164 | 0.181 | 0.180 | 0.020 | 0.027 | 0.026 | 0.028 |
15×15_1 | 0.076 | 0.207 | 0.207 | 0.187 | 0.032 | 0.035 | 0.028 | 0.038 |
15×15_2 | 0.074 | 0.202 | 0.216 | 0.229 | 0.035 | 0.035 | 0.040 | 0.034 |
15×15_3 | 0.088 | 0.228 | 0.206 | 0.207 | 0.029 | 0.043 | 0.045 | 0.030 |
15×15_4 | 0.061 | 0.190 | 0.203 | 0.166 | 0.030 | 0.035 | 0.049 | 0.033 |
20×20_1 | 0.054 | 0.196 | 0.227 | 0.209 | 0.018 | 0.042 | 0.043 | 0.036 |
20×20_2 | 0.078 | 0.233 | 0.226 | 0.224 | 0.022 | 0.053 | 0.042 | 0.041 |
20×20_3 | 0.054 | 0.203 | 0.212 | 0.233 | 0.017 | 0.032 | 0.030 | 0.035 |
20×20_4 | 0.067 | 0.215 | 0.194 | 0.224 | 0.018 | 0.044 | 0.033 | 0.036 |
算例 | IGD | |||||||
Mean | Std | |||||||
MOHEA | NSGA-Ⅱ | NSGA-Ⅲ | SPEA2 | MOHEA | NSGA-Ⅱ | NSGA-Ⅲ | SPEA2 | |
4×4_1 | 0.039 | 0.089 | 0.097 | 0.122 | 0.016 | 0.029 | 0.042 | 0.051 |
4×4_2 | 0.045 | 0.092 | 0.103 | 0.105 | 0.018 | 0.026 | 0.028 | 0.035 |
4×4_3 | 0.033 | 0.136 | 0.145 | 0.137 | 0.018 | 0.036 | 0.041 | 0.040 |
4×4_4 | 0.056 | 0.112 | 0.119 | 0.109 | 0.031 | 0.033 | 0.033 | 0.024 |
5×5_1 | 0.075 | 0.164 | 0.153 | 0.151 | 0.023 | 0.041 | 0.041 | 0.050 |
5×5_2 | 0.066 | 0.082 | 0.127 | 0.099 | 0.019 | 0.028 | 0.023 | 0.030 |
5×5_3 | 0.088 | 0.153 | 0.169 | 0.187 | 0.041 | 0.039 | 0.058 | 0.052 |
5×5_4 | 0.068 | 0.160 | 0.156 | 0.130 | 0.031 | 0.032 | 0.047 | 0.038 |
7×7_1 | 0.110 | 0.158 | 0.145 | 0.136 | 0.029 | 0.029 | 0.037 | 0.028 |
7×7_2 | 0.083 | 0.159 | 0.132 | 0.152 | 0.016 | 0.033 | 0.028 | 0.043 |
7×7_3 | 0.106 | 0.159 | 0.150 | 0.184 | 0.040 | 0.022 | 0.032 | 0.035 |
7×7_4 | 0.097 | 0.141 | 0.175 | 0.162 | 0.032 | 0.034 | 0.032 | 0.028 |
10×10_1 | 0.086 | 0.141 | 0.174 | 0.179 | 0.018 | 0.025 | 0.031 | 0.022 |
10×10_2 | 0.114 | 0.211 | 0.216 | 0.203 | 0.019 | 0.038 | 0.027 | 0.029 |
10×10_3 | 0.111 | 0.166 | 0.217 | 0.166 | 0.028 | 0.031 | 0.031 | 0.026 |
10×10_4 | 0.133 | 0.184 | 0.196 | 0.204 | 0.034 | 0.024 | 0.024 | 0.024 |
15×15_1 | 0.123 | 0.237 | 0.238 | 0.217 | 0.031 | 0.029 | 0.021 | 0.024 |
15×15_2 | 0.138 | 0.228 | 0.233 | 0.252 | 0.045 | 0.027 | 0.032 | 0.027 |
15×15_3 | 0.142 | 0.250 | 0.238 | 0.230 | 0.025 | 0.027 | 0.027 | 0.019 |
15×15_4 | 0.126 | 0.222 | 0.238 | 0.219 | 0.026 | 0.022 | 0.026 | 0.018 |
20×20_1 | 0.104 | 0.312 | 0.317 | 0.311 | 0.016 | 0.021 | 0.020 | 0.024 |
20×20_2 | 0.139 | 0.301 | 0.281 | 0.289 | 0.022 | 0.024 | 0.023 | 0.025 |
20×20_3 | 0.073 | 0.318 | 0.308 | 0.341 | 0.013 | 0.018 | 0.021 | 0.020 |
20×20_4 | 0.128 | 0.354 | 0.337 | 0.355 | 0.021 | 0.018 | 0.021 | 0.018 |
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