GPU-accelerated evolutionary optimization of multi-objective flow shop scheduling problems

doi:10.11772/j.issn.1001-9081.2024010028

Journal of Computer Applications ›› 2024, Vol. 44 ›› Issue (5): 1364-1371.DOI: 10.11772/j.issn.1001-9081.2024010028

Special Issue: 进化计算专题（2024年第5期“进化计算专题”导读，全文即将上线）

• Special issue on evolutionary calculation • Previous Articles

GPU-accelerated evolutionary optimization of multi-objective flow shop scheduling problems

Tao JIANG¹^,², Zhenyu LIANG¹, Ran CHENG¹(), Yaochu JIN³

^1.Department of Computer Science and Engineering，Southern University of Science and Technology，Shenzhen Guangdong 518055，China
^2.Peng Cheng Laboratory，Shenzhen Guangdong 518055，China
^3.Westlake University，Hangzhou Zhejiang 310012，China

Received:2024-01-15 Revised:2024-02-20 Accepted:2024-02-21 Online:2024-04-26 Published:2024-05-10
Contact: Ran CHENG
About author:JIANG Tao， born in 1997， Ph. D. candidate. His research interests include intellligent scheduling， evolutionary computation.
LIANG Zhenyu， born in 1997， M. S. candidate. His research interests include multi-objective optimization.
JIN Yaochu， born in 1966， Ph. D.， chair professor. His research interests include artificial intelligence.

GPU加速的演化算法求解多目标流水车间调度问题

姜涛¹^,², 梁振宇¹, 程然¹(), 金耀初³

^1.南方科技大学计算机科学与工程系，广东深圳 518055
^2.鹏城实验室，广东深圳 518055
^3.西湖大学，杭州 310012

通讯作者: 程然
作者简介:姜涛（1997—），男，安徽马鞍山人，博士研究生，主要研究方向：智能调度、演化计算
梁振宇（1997—），男，广东阳江人，硕士研究生，主要研究方向：多目标优化
金耀初（1966—），男，江苏吴江人，讲席教授，博士，主要研究方向：人工智能。
第一联系人：程然（1987—），男，江苏徐州人，副教授，博士，主要研究方向：计算智能、演化计算、表征学习

Abstract

Abstract:

In the realms of intelligent manufacturing and environmental sustainability， the significance of multi-objective scheduling in orchestrating a balance among production efficiency， cost management， and environmental conservation is paramount. Contemporary research indicates that CPU-based scheduling solutions are constrained by suboptimal efficiency and responsiveness， particularly when managing tasks of considerable scale. Consequently， the parallel computational prowess of GPUs heralds a novel avenue for the refinement of extensive flow shop scheduling challenges. For the multi-objective No-Wait Flow shop Scheduling Problem （NWFSP）， with the concurrent objectives of minimizing both the makespan and the Total Energy Consumption （TEC）， a Mixed-Integer Linear Programming model （MILP） was formulated to delineate the problem， and a bespoke GPU-accelerated tensorized evolutionary algorithm named Tensor-GPU-NSGA-Ⅱ was introduced for problem-solving. The ingenuity of Tensor-GPU-NSGA-Ⅱ resides in its tensorized algorithm for the computation of the makespan and TEC within the NWFSP framework， as well as in converting the conventional CPU-based serial individual updating to a GPU-driven parallel population renewal process. Empirical results demonstrate that for a scenario involving 500 jobs and 20 machines， Tensor-GPU-NSGA-Ⅱ realizes an enhancement in computational efficiency by a speedup of 9 761.75 over the traditional NSGA-Ⅱ algorithm. Furthermore， this acceleration efficacy markedly surges as the population scale expands.

Key words: intelligent manufacturing, multi-objective optimization, flow shop scheduling, GPU acceleration, tensorization method

摘要：

智能制造和环境可持续性研究中，多目标调度问题对于协调生产效率、成本管理与环境保护之间的平衡具有至关重要的意义，但现有基于CPU的调度解决方案在处理大规模生产任务时仍面临效率和时效性的限制，而GPU的并行计算能力可为优化大规模流水车间调度问题提供新的解决途径。针对多目标零等待流水车间调度问题（NWFSP），以同时最小化最大完成时间和总能耗（TEC）为优化目标，构建了混合整数线性规划模型（MILP）表征该调度问题，并提出一种基于GPU加速的张量化演化算法（Tensor-GPU-NSGA-Ⅱ）求解该问题。Tensor-GPU-NSGA-Ⅱ的主要创新在于对NWFSP关于最小化最大完成时间和TEC的计算过程的张量化处理，并提出了一种基于GPU的并行种群更新方法。实验结果表明，在500工件和20机器的问题规模下，Tensor-GPU-NSGA-Ⅱ在计算效率上相较于传统NSGA-Ⅱ算法取得了9 761.75的加速比；且随着种群规模的增加，它的加速性能有显著提升。

关键词: 智能制造, 多目标优化, 流水车间调度, GPU加速, 张量化方法

CLC Number:

TP18

Tao JIANG, Zhenyu LIANG, Ran CHENG, Yaochu JIN. GPU-accelerated evolutionary optimization of multi-objective flow shop scheduling problems[J]. Journal of Computer Applications, 2024, 44(5): 1364-1371.

姜涛, 梁振宇, 程然, 金耀初. GPU加速的演化算法求解多目标流水车间调度问题[J]. 《计算机应用》唯一官方网站, 2024, 44(5): 1364-1371.

Figures/Tables 7

Tab. 1 Variable symbols and definitions

变量	定义	变量	定义	变量	定义
$m$	机器数目	$e$	机器加工速度等级数目	$R P E C$	总加工能耗（total Processing Energy Consumption， PEC）
$n$	工件数目	$p i, j$	工件 $j$ 在 $机器 i$ 上的标准加工时间	$R I E C$	总空闲能耗（total Idle Energy Consumption， IEC）
$i$	机器索引号	$v i, j$	工件 $j$ 在 $机器 i$ 上的加工速度	$R P I$	机器空闲时的平均功率（average Power at Idle， PI）
$j$	工件索引号	$C k, i$	第 $k$ 个加工工件在机器 $i$ 上的完成时间	$R P P$	机器加工时的平均功率（average Power during Processing， PP）
$k$	工件序列中工件的索引号	$t i i, j$	工件 $j$ 在机器 $i$ 后的实际空闲时间	$X j, k$	工件 $j$ 在加工序列中的第 $k$ 个位置加工时值为 $1$ ，否则为 $0$
$π$	指派工件的加工序列	$t p i, j$	工件 $j$ 在机器 $i$ 后的实际加工时间	$Y j, i, c$	工件 $j$ 在第 $i$ 台机器上的加工速度等级为 $c$ 时值为 $1$ ，否则为 $0$
$c$	机器加工速度等级	$R T E C$	总能耗（Total Energy Consumption， TEC）

Tab. 1 Variable symbols and definitions

变量	定义	变量	定义	变量	定义
$m$	机器数目	$e$	机器加工速度等级数目	$R P E C$	总加工能耗（total Processing Energy Consumption， PEC）
$n$	工件数目	$p i, j$	工件 $j$ 在 $机器 i$ 上的标准加工时间	$R I E C$	总空闲能耗（total Idle Energy Consumption， IEC）
$i$	机器索引号	$v i, j$	工件 $j$ 在 $机器 i$ 上的加工速度	$R P I$	机器空闲时的平均功率（average Power at Idle， PI）
$j$	工件索引号	$C k, i$	第 $k$ 个加工工件在机器 $i$ 上的完成时间	$R P P$	机器加工时的平均功率（average Power during Processing， PP）
$k$	工件序列中工件的索引号	$t i i, j$	工件 $j$ 在机器 $i$ 后的实际空闲时间	$X j, k$	工件 $j$ 在加工序列中的第 $k$ 个位置加工时值为 $1$ ，否则为 $0$
$π$	指派工件的加工序列	$t p i, j$	工件 $j$ 在机器 $i$ 后的实际加工时间	$Y j, i, c$	工件 $j$ 在第 $i$ 台机器上的加工速度等级为 $c$ 时值为 $1$ ，否则为 $0$
$c$	机器加工速度等级	$R T E C$	总能耗（Total Energy Consumption， TEC）

Fig. 1 Gantt diagram of NWFSP

Fig. 2 Schematic diagram of OX crossover operator

Fig. 3 Schematic diagram of tensorized population evolution

Tab. 2 Time consumption of NSGA-Ⅱ in various environments

$(n, m)$	时间开销					加速比
$(n, m)$	CPU	GPU	Tensor-noJIT- CPU	Tensor-CPU	Tensor-GPU	加速比
合计/均值	78 998.97	143 611.21	210.29	81.01	43.29	1 825.06
$(20,5)$	412.15	829.02	15.75	5.93	3.85	107.07
$(20,10)$	752.93	1 449.78	16.25	5.73	3.42	220.18
$(20,20)$	1 498.14	2 657.13	16.41	5.85	3.37	445.15
$(50,5)$	931.54	1 760.22	17.59	5.65	3.72	250.32
$(50,10)$	1 775.66	3 293.61	17.31	5.77	3.49	508.19
$(50,20)$	3 717.38	6 541.58	17.77	5.95	3.47	1 071.90
$(100,5)$	1 888.71	3 447.26	16.31	6.29	3.87	487.51
$(100,10)$	3 675.49	6 609.47	16.54	5.97	3.63	1 012.74
$(100,20)$	7 134.09	13 131.29	16.68	6.33	3.51	2 032.36
$(200,10)$	6 978.79	13 135.42	17.38	7.63	3.71	1 878.73
$(200,20)$	14 026.84	25 956.58	19.15	7.82	3.53	3 972.75
$(500,20)$	36 207.25	64 799.84	23.17	12.09	3.71	9 761.75

Tab. 2 Time consumption of NSGA-Ⅱ in various environments

$(n, m)$	时间开销					加速比
$(n, m)$	CPU	GPU	Tensor-noJIT- CPU	Tensor-CPU	Tensor-GPU	加速比
合计/均值	78 998.97	143 611.21	210.29	81.01	43.29	1 825.06
$(20,5)$	412.15	829.02	15.75	5.93	3.85	107.07
$(20,10)$	752.93	1 449.78	16.25	5.73	3.42	220.18
$(20,20)$	1 498.14	2 657.13	16.41	5.85	3.37	445.15
$(50,5)$	931.54	1 760.22	17.59	5.65	3.72	250.32
$(50,10)$	1 775.66	3 293.61	17.31	5.77	3.49	508.19
$(50,20)$	3 717.38	6 541.58	17.77	5.95	3.47	1 071.90
$(100,5)$	1 888.71	3 447.26	16.31	6.29	3.87	487.51
$(100,10)$	3 675.49	6 609.47	16.54	5.97	3.63	1 012.74
$(100,20)$	7 134.09	13 131.29	16.68	6.33	3.51	2 032.36
$(200,10)$	6 978.79	13 135.42	17.38	7.63	3.71	1 878.73
$(200,20)$	14 026.84	25 956.58	19.15	7.82	3.53	3 972.75
$(500,20)$	36 207.25	64 799.84	23.17	12.09	3.71	9 761.75

Fig. 4 Interval figures for HV on Tensor-CPU and Tensor-GPU

Tab. 3 Time consumption comparison on Tensor-CPU and Tensor-GPU

$(n, m)$	时间开销		加速比	时间开销		加速比	时间开销		加速比	时间开销		加速比
$(n, m)$	500-CPU	500-GPU	加速比	1 000-CPU	1 000-GPU	加速比	5 000-CPU	5 000-GPU	加速比	10 000-CPU	10 000-GPU	加速比
$(20,5)$	11.74	4.12	2.85	12.70	4.28	2.97	106.10	11.31	9.38	511.90	37.52	13.64
$(20,10)$	8.90	3.67	2.42	12.13	3.74	3.24	129.28	13.28	9.73	690.14	50.52	13.66
$(20,20)$	8.64	3.73	2.32	11.33	3.87	2.93	169.27	17.84	9.49	967.98	72.57	13.34
$(50,5)$	10.55	3.98	2.65	17.23	4.14	4.17	111.28	10.53	10.57	506.69	33.98	14.91
$(50,10)$	9.67	3.86	2.50	12.68	3.96	3.20	130.83	12.48	10.48	629.94	45.30	13.91
$(50,20)$	9.86	3.92	2.52	13.00	4.04	3.21	172.34	16.82	10.25	933.98	66.14	14.12
$(100,5)$	16.84	4.26	3.95	18.36	4.22	4.35	124.41	10.47	11.88	524.37	33.14	15.82
$(100,10)$	12.46	3.99	3.12	16.82	4.02	4.18	145.15	12.32	11.79	669.89	43.84	15.28
$(100,20)$	12.26	4.11	2.98	18.02	4.09	4.40	193.41	16.55	11.68	946.25	64.61	14.65
$(200,10)$	17.98	4.36	4.12	28.43	4.37	6.51	187.21	12.61	14.84	742.74	43.91	16.92
$(200,20)$	18.45	4.18	4.41	30.72	4.24	7.24	235.95	17.01	13.87	1027.14	64.76	15.86
$(500,20)$	36.10	4.47	8.08	67.12	4.48	14.97	400.18	18.95	21.12	1360.48	68.82	19.77

Tab. 3 Time consumption comparison on Tensor-CPU and Tensor-GPU

$(n, m)$	时间开销		加速比	时间开销		加速比	时间开销		加速比	时间开销		加速比
$(n, m)$	500-CPU	500-GPU	加速比	1 000-CPU	1 000-GPU	加速比	5 000-CPU	5 000-GPU	加速比	10 000-CPU	10 000-GPU	加速比
$(20,5)$	11.74	4.12	2.85	12.70	4.28	2.97	106.10	11.31	9.38	511.90	37.52	13.64
$(20,10)$	8.90	3.67	2.42	12.13	3.74	3.24	129.28	13.28	9.73	690.14	50.52	13.66
$(20,20)$	8.64	3.73	2.32	11.33	3.87	2.93	169.27	17.84	9.49	967.98	72.57	13.34
$(50,5)$	10.55	3.98	2.65	17.23	4.14	4.17	111.28	10.53	10.57	506.69	33.98	14.91
$(50,10)$	9.67	3.86	2.50	12.68	3.96	3.20	130.83	12.48	10.48	629.94	45.30	13.91
$(50,20)$	9.86	3.92	2.52	13.00	4.04	3.21	172.34	16.82	10.25	933.98	66.14	14.12
$(100,5)$	16.84	4.26	3.95	18.36	4.22	4.35	124.41	10.47	11.88	524.37	33.14	15.82
$(100,10)$	12.46	3.99	3.12	16.82	4.02	4.18	145.15	12.32	11.79	669.89	43.84	15.28
$(100,20)$	12.26	4.11	2.98	18.02	4.09	4.40	193.41	16.55	11.68	946.25	64.61	14.65
$(200,10)$	17.98	4.36	4.12	28.43	4.37	6.51	187.21	12.61	14.84	742.74	43.91	16.92
$(200,20)$	18.45	4.18	4.41	30.72	4.24	7.24	235.95	17.01	13.87	1027.14	64.76	15.86
$(500,20)$	36.10	4.47	8.08	67.12	4.48	14.97	400.18	18.95	21.12	1360.48	68.82	19.77

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GPU-accelerated evolutionary optimization of multi-objective flow shop scheduling problems

GPU加速的演化算法求解多目标流水车间调度问题

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