求解柔性流水车间调度问题的高效分布估算算法

王芳; 唐秋华; 饶运清; 张超勇; 张利平

doi:10.16383/j.aas.2017.c150873

求解柔性流水车间调度问题的高效分布估算算法

doi: 10.16383/j.aas.2017.c150873

王芳^1,2,,
唐秋华^3, ,,
饶运清^2,,
张超勇^2,,
张利平^3,

1.
武汉科技大学管理学院武汉 430081
2.
华中科技大学数字制造装备与技术国家重点实验室武汉 430074
3.
武汉科技大学机械自动化学院武汉 430081

基金项目:

国家重点基础研究发展计划（973计划） 2014CB046705

湖北省教育厅科研项目 Q20151104

国家自然科学基金 51275366

国家自然科学基金 51305311

湖北省教育厅科研项目 15Q027

国家自然科学基金国际合作项目 51561125002

详细信息

作者简介:
王芳武汉科技大学管理学院副教授, 华中科技大学机械科学与工程学院博士研究生.分别于2002年和2005年获得西北工业大学学士和硕士学位.主要研究方向为决策理论与方法, 调度优化与智能算法.E-mail:wangfang79@wust.edu.cn

饶运清华中科技大学机械科学与工程学院教授.1989年获得华中科技大学机械制造专业学士学位, 1999年获得该校机械制造及其自动化专业工学博士学位.主要研究方向为制造执行系统与数字化车间, 制造系统建模与运行优化.E-mail:ryq@mail.hust.edu.cn

张超勇华中科技大学机械科学与工程学院副教授.1993年获得天津科技大学学士学位, 1999年获得北京科技大学硕士学位, 2007年获得华中科技大学博士学位.主要研究方向为制造系统运行优化, 可持续制造.Email:zcyhust@hust.edu.cn

张利平武汉科技大学机械自动化学院讲师.2006年获得三峡大学学士学位, 2013年获得华中科技大学博士学位.主要研究方向为绿色制造, 生产调度, 智能算法.E-mail:zhangliping@wust.edu.cn

通讯作者:
唐秋华武汉科技大学机械自动化学院教授.1992年获得东北大学学士学位, 2000年获得武汉科技大学硕士学位, 2005年获得武汉理工大学博士学位.主要研究方向为现代制造系统, 制造业信息化和工业工程.本文通信作者.E-mail:tangqiuhua@wust.edu.cn

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出版历程
- 收稿日期: 2015-12-24
- 录用日期: 2016-08-15
- 刊出日期: 2017-02-01

Efficient Estimation of Distribution for Flexible Hybrid Flow Shop Scheduling

1.
School of Management, Wuhan University of Science and Technology, Wuhan 430081
2.
State Key Lab of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074
3.
School of Machinery and Automation, Wuhan University of Science and Technology, Wuhan 430081

Funds:

National Basic Research Program of China (973 Program) 2014CB046705

Projects Supported by Hubei Provincial Department of Education Q20151104

National Natural Science Foundation of China 51275366

National Natural Science Foundation of China 51305311

Projects Supported by Hubei Provincial Department of Education 15Q027

International Cooperation and Exchange Program of National Natural Science Foundation of China 51561125002

More Information

Author Bio:
Associate professor at the School of Management, Wuhan University of Science and Technology, Ph. D. candidate at the School of Mechanical Science and Technology, Huazhong University of Science and Technology. She received her bachelor and master degrees from Northwestern Polytechnical University in 2002 and 2005. Her research interest covers decision-making theory and method, production scheduling and intelligent algorithm

Professor at the School of Mechanical Science and Engineering, Huazhong University of Science and Technology (HUST). He obtained his bachelor degree in mechanical engineering from HUST in 1992, and Ph. D. in mechanical engineering and automation from HUST in 1999. His research interest covers manufacturing execution systems (MES) and digital workshops, modelling and running optimization of manufacturing systems

Associate professor at the School of Mechanical Science and Engineering, Huazhong University of Science & Technology. He obtained his bachelor degree from Tianjin University of Science and Technology, master degree from University of Science and Technology Beijing and Ph. D. degree from the Huazhong University of Science and Technology in 1993, 1999 and 2007, respectively. His research interest covers modeling, optimization and scheduling for production manufacturing systems, and sustainable manufacturing

Lecturer at the School of Machinery and Automation, Wuhan University of Science and Technology. She received her bachelor degree from the Three Gorges University and Ph. D. degree from Huazhong University of Science and Technology in 2006 and 2013. Her research interest covers green manufacturing, production scheduling, and intelligent algorithm

Corresponding author: TANG Qiu-Hua Professor at the School of Machinery and Automation, Wuhan University of Science and Technology. She received her bachelor degree from Northeastern University in 1992, master degree from Wuhan University of Science and Technology in 2000, and Ph. D. degree from Wuhan University of Technology in 2005. Her research interest covers modern manufacturing system, and informatization of manufacturing industry, and industrial engineering. Corresponding author of this paper

摘要

摘要: 针对最小化最大完工时间的柔性流水车间调度，利用事件建模思想，线性化0-1混合整数规划模型，使得小规模调度问题通过Cplex可以准确求解，同时设计了高效分布估算算法来求解大规模调度问题.该算法采用的是一种新颖的随机规则解码方式，工件排序按选定的规则安排而机器按概率随机分配.针对分布估算算法中的概率模型不能随种群中个体各位置上工件的更新而自动调整的缺点，提出了自适应调整概率模型，该概率模型能提高分布估算算法的收敛质量和速度.同时为提高算法局部搜索能力和防止算法陷入局部最优，设计了局部搜索和重启机制.最后，采用实验设计方法校验了高效分布估算算法参数的最佳组合.算例和实例测试结果都表明本文提出的高效分布估算算法在求解质量和稳定性上均优于遗传算法、引力搜索算法和经典分布估算算法.
- 柔性流水车间调度 /
- 分布估计算法 /
- 局部搜索 /
- 最小化最大完工时间
Abstract: For flexible flow shop scheduling which minimizes the maximum completion time, a 0-1 mixed integer linear programming model is established by using event modeling method, and small-scale scheduling problems can be accurately solved through any linear solver. At the same time, an efficient estimation of distribution algorithm is designed to solve large-scale problems. A novel decoding way with random probability and rules is adopted by the new algorithm, and workpiece sequencing is based on rule while assignment of machines is based on random probability. Since the original probability model does not automatically adjust sampling probability, an improved probability model is put forward. And local search and restart mechanism are designed and adopted to improve the ability of local search and to avoid falling into local optimum. Finally, optimal combination of parameters is decided by using experimental design method, and experimental results show that the new algorithm outperforms genetic algorithm, gravitational search algorithm, and classical estimation of distribution algorithm in terms of quality and stability.
- Flexible flow shop scheduling /
- estimation of distribution algorithm /
- local search /
- minimizing makespan
注释:

1) 本文责任编委宋士吉

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图 1 不同水平值下的平均值与方差

Fig. 1 Mean values and variances of levels

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图 2 求解FFSP的高效EDA流程图

Fig. 2 Flow chart of efficient EDA to solve FFSP

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图 3 不同水平值下的结果比较

Fig. 3 The results under different levels

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图 4 边际平均值

Fig. 4 The average marginal

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图 5 实例L1的最优解

Fig. 5 The optimal solution of case L1

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表 1 解码方法的伪代码和复杂度

Table 1 Pseudo-code and complexity for decoding method

Algorithm 1. Template of decoding	O (max{Snlog (n), nMlog (m)})
Input: π as the coding, and π_t as the t-th processed job number, P_m as the probability,	O (C)
j=1 (j is the index of the stages);	O (1)
Repeat	O (max{Snlog (n), nM log (m)})
t=1;	O (1)
Repeat	O (nm_jlog (m_j))
Randomly generating a number of [0, 1] which is noted as P m;	O (1)
Calculating the completion time of πt processed on each machine (Tπt, j, k) according to equation (12);	O (m_j)
If P_m≤P_m, the machine of the minimum T_{π_t, j, k} is chosen and is noted as k^*;	O (m_jlog (m_j))
Else, a machine is randomly chosen and is noted as k^*;	O (m_jlog (m_j))
Setting E_{π_t, j}=T_{π_t, j, k}, B_{π_t, j}=T_{π_t, j, k^} -P_{π_t, k^};	O (1)
t + +	O (1)
Until t=n	O (1)
Reorganizing the sequence of jobs according to the ascending order of E_{π_t, j};	O (nlog (n))
Updating π as the sequence of jobs;	O (n)
j + +	O (1)
Until j=S	O (1)
Output: Final solution found.	O (C)

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表 2 概率矩阵构建与种群更新的伪代码和复杂度

Table 2 Pseudo-code and complexity for constructing probability matrix and updating population

Algorithm 2. Template of construction probability matrix and updating population	O (GP_size(n² -n)/2)
Input: Sp as dominant population. P_t.i(g) as the probability matrix;	0(n²)
g as the index evolutional generation, g=0;	0(1)
Repeat	O (GP_size(n² -n)/2)
Determining the values of IS_{t, i}^l (g) according to Sp;	O (n²\|Sp\|)
Calculating P_{t, i}(g + 1) by equation (13) and setting P_{t, i}⁰ (g + 1)=P_{t, i}(g + 1)；	0(n²)
l=l;	0(1)
Setting I as the sequence of jobs in the individual l;	O (n)
Repeat	O (P_size(n² -n)/2)
t=1；	0(1)
Updating the job of individual l which is arranged at position t according to P_{t, i}⁰ (g + 1) by the roulette approach;	0(n -1)
Noting the job as i^*	0(1)
Repeat	O ((n²-n)/2)
I'={i, i ∈ I/i^*}	O (n -t)
Calculating P_{t, i}⁰ (g + 1) by equation (14).	O (n -t)
Dynamically updating P_{t', i}(g + 1), as P_{t', i}'(g + 1),	O (n -t)
t + +	O (1)
I=[]; I=I';	O (n -t)
Updating the job of individual l which is arranged at position t according to (3 + 1) by the roulette approach;	O (n -1)
Until t==n	O (1)
l + +	O (1)
Until l==P_size	O (1)
Calculating the target values for all of updated individuals by decoding method;	O (P_sizesnlog (n))
Sequencing the individuals in ascending order of target values;	O (P_sixesnlog (n))
Selecting the top η % of individuals as sp;	O (\|sp\|)
g + +	O (1)
Until the termination criteria are achieved	O (1)
Output: Final solution found.	O (C)

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表 3 改进概率模型的性能

Table 3 The performance of improved probability model

代数	EDA_I			EDA_W
代数	最优值	多样性	时间(s)	最优值	多样性	时间(s)
10代	249.0	0.83	0.93	251.0	0.89	0.97
50代	237.1	0.35	4.78	241.2	0.60	4.80
100代	230.0	0.09	9.71	239.8	0.23	10.02
200代	226.7	0.01	19.38	235.5	0.03	20.72
500代	223.4	0.00	48.31	230.9	0.00	52.60

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表 4 重启操作的正交实验结果

Table 4 Results for orthogonal test of restart operation

参数组合	水平			AVG
参数组合	$Div{\_}h$	G_max	Res	AVG
1	2	2	1	222.30
2	2	1	2	223.75
3	4	1	1	223.90
4	3	2	1	222.40
5	3	1	3	219.60
6	1	3	1	223.90
7	1	1	1	223.55
8	1	1	2	223.10
9	3	1	1	221.25
10	1	2	3	222.25
11	4	2	2	222.55
12	4	1	3	219.65
13	2	1	1	223.00
14	3	3	2	222.40
15	4	3	1	224.40
16	2	3	3	219.95

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表 5 重启操作的极差表

Table 5 Rang table of restart operation

水平	$Div{\_}\, h$	G_max	Res
1	223.200	222.263	223.088
2	222.150	222.375	222.950
3	221.487	222.563	220.338
4	222.625	-	-
极差	1.713	0.300	2.750
等级	2	3	1

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表 6 重启操作中三种影响因素的方差分析

Table 6 ANOVA for three factors of restart operation

Source	Type Ⅲ sum of squares	df	Mean square	F	Sig.	Partial eta squared
Corrected model	28.553^a	7	4.079	4.510	0.025	0.798
Intercept	646 099.042	1	646 099.042	714 322.413	0.000	1.000
$Div{\_}h$	6.324	3	2.108	2.331	0.151	0.466
G_max	0.240	2	0.120	0.133	0.877	0.032
Res	21.988	2	10.994	12.155	0.004	0.752

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表 7 高效EDA算法各操作复杂度

Table 7 The complexity of efficient EDA

操作名称	复杂度
初始化	O$(nP_{size})$
评价种群	O$(P_{size} Mn\log (n))$
选择优势种群并计算概率	O$(n^2SP_{size})$
按概率生产新种群	O$(n^2P_{size})$
邻域搜索	O$Mn^{3})$或O$Mn^{2})$
重启操作	O$(P_{size}${$Mn$log($n))$

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表 8 算法参数正交实验结果

Table 8 Results for orthogonal test of algorithm parameters

参数组合	水平			AVG
参数组合	$Div{\_}h$	G_max	Res	AVG
1	2	2	4	193.40
2	2	1	2	193.94
3	4	1	4	192.66
4	3	2	1	197.29
5	3	1	3	192.51
6	1	3	4	195.14
7	1	1	1	195.54
8	1	4	2	197.54
9	3	4	4	195.40
10	1	2	3	194.26
11	4	2	2	195.29
12	4	4	3	195.97
13	2	4	1	198.71
14	3	3	2	196.29
15	4	3	1	197.14
16	2	3	3	194.46

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表 9 高效EDA各参数的极差

Table 9 Rang for the parameters of efficient EDA

水平	$Div{\_}\, h$	G_max	Res
1	195.620	193.663	197.170
2	195.128	195.060	195.765
3	195.373	195.758	194.300
4	195.265	196.905	194.150
极差	0.492	3.242	3.020
等级	3	1	2

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表 10 高效EDA的三种参数的方差分析

Table 10 ANOVA for three parameters of efficient EDA

Source	Type Ⅲ sum of squares	df	Mean square	F	Sig.	Partial eta squared
Corrected model	46.692^a	9	5.188	39.490	0.000	0.983
Intercept	610 562.518	1	610 562.518	4 647 478.731	0.000	1.000
$P_{size}$	0.520	3	0.173	1.320	0.352	0.398
$\eta $	22.063	3	7.354	55.980	0.000	0.966
a	24.108	3	8.036	61.169	0.000	0.968

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表 11 四种算法的均值及标准差

Table 11 The mean and standard deviation of these four algorithms

	解码方法	Mean	Std. deviation	N
GA	规则	200.8500	13.46325	40
	随机规则	197.2000	13.49112	40
	Total	199.0250	13.51696	80
GSA	规则	198.5750	13.51331	40
	随机规则	189.7000	12.44928	40
	Total	194.1375	13.66020	80
EDA	规则	199.43	13.454	40
	随机规则	193.20	12.113	40
	Total	196.31	13.100	80
EDA_H	规则	188.68	13.234	40
	随机规则	184.42	13.042	40
	Total	186.55	13.229	80

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表 12 组间因素测试结果

Table 12 Test results of between-subjects

Source	Square sum	df	Mean square	F	Sig.	Partial eta squared
Intercept	12 044 296.013	1	12 044 296.013	19 397.610	0.000	0.996
Decode	2 645.000	1	2 645.000	4.260	0.042	0.052
Error	48 431.488	78	620.917	-	-	-

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表 13 多元变量测试结果

Table 13 Test results of multivariate

Effect		Value	F	Hypoth-esis df	Error df	Sig.	Partial eta squared
算法	Pillai's trace	0.882	190.021^a	3.000	76.000	0.000	0.882
	Wilks' lambda	0.118	190.021^a	3.000	76.000	0.000	0.882
	Hotelling's trace	7.501	190.021^a	3.000	76.000	0.000	0.882
	Roy's largest root	7.501	190.021^a	3.000	76.000	0.000	0.882
算法加解码	Pillai's trace	0.314	11.617^a	3.000	76.000	0.000	0.314
	Wilks' lambda	0.686	11.617^a	3.000	76.000	0.000	0.314
	Hotelling's trace	0.459	11.617^a	3.000	76.000	0.000	0.314
	Roy's largest root	0.459	11.617^a	3.000	76.000	0.000	0.314

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表 14 三个实例在四种算法及原文献中的结果对比

Table 14 Comparison for the results of three cases under these four algorithms and original documents

次数	文献结果			GA			GSA			EDA_W			EDA_I
次数	L1	L2	L3	L1	L2	L3	L1	L2	L3	L1	L2	L3	L1	L2	L3
1	23	297	14	22	298	13.5	22	297	13.5	23	298	13.5	21	297	13
2	24	297	14	22	297	13.5	22	297	13.5	22	297	14	22	297	13
3	23	297	15	22	298	13.5	22	297	13.5	21	300	13.5	21	297	13
4	23	297	14	22	298	13.5	21	298	13.5	22	298	13.5	21	297	13.5
5	23	298	14	22	297	13.5	22	297	13.5	22	297	14	21	297	13
6	23	297	14.5	22	300	13.5	22	298	13.5	21	297	13.5	21	297	13
7	24	297	14	22	298	13.5	21	298	13.5	23	300	14	21	298	13
8	24	298	14.5	22	298	13.5	22	297	13.5	22	197	13.5	21	297	13.5
9	23	298	14	22	298	13.5	22	298	13.5	23	298	13.5	21	297	13
10	24	298	14	22	299	13.5	22	298	13.5	22	298	14	21	297	13

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参考文献(45)

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