Hybrid social spider optimization algorithm with differential mutation operator for the job-shop scheduling problem

Guo Zhou; Yongquan Zhou; Ruxin Zhao

doi:10.3934/jimo.2019122

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2021, Volume 17, Issue 2: 533-548. Doi: 10.3934/jimo.2019122

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Hybrid social spider optimization algorithm with differential mutation operator for the job-shop scheduling problem

1.
Department of Science and Technology Teaching, China University of Political Science and Law, Beijing 100088, China
2.
College of Information Science and Engineering, Guangxi University for Nationalities, Key Laboratories of Guangxi High Schools Complex System and Computational Intelligence, Nanning 530006, China
3.
School of Computer Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China
4.
College of Information Science and Engineering, Guangxi University for Nationalities, Nanning 530006, China

^* Corresponding author: Yongquan Zhou
^* Corresponding author: Yongquan Zhou

Received: June 30, 2018

Revised: June 30, 2019

Early access: October 2019

Published: March 2021

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Abstract

The job-shop scheduling problem is one of the well-known hardest combinatorial optimization problems. The problem has captured the interest of a significant number of researchers, but no efficient solution algorithm has been found yet for solving it to optimality in polynomial time. In this paper, a hybrid social-spider optimization algorithm with differential mutation operator is presented to solve the job-shop scheduling problem. To improve the exploration capabilities of the social spider optimization algorithm (SSO), we incorporate the DM operator (a mutation operator taken from the deferential evolutionary (DE) algorithm) into the framework of the female cooperative operator. The experimental results show that the proposed method effectiveness in solving job-shop scheduling compared to other optimization algorithms in the literature.

Keywords:

Mathematics Subject Classification: Primary: 78M50, 90C27.

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Figure 1. FT20, Size = 100

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Figure 2. LA40, Size = 225

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Figure 3. ORB10, Size = 100

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Figure 4. YN04, Size = 400

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Figure 5. FT20, Size = 100

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Figure 6. LA40, Size = 225

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Figure 7. ORB10, Size = 100

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Figure 8. YN04, Size = 400

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Table 1. Notations for the JSSP

$ n $	the number of jobs
$ m $	the number of operations for one job
$ O_i $	the completion time of operation $ i $($ i=\{0, 1, 2, \dots, n*m+1\} $)
$ t_i $	the processing time of operation $ i $ on a given machine
$ \omega_{im} $	the flag of operation $ i $ initiated by machine $ m $
$ P_i $	all the predecessor operations of operation $ i $
$ A(t) $	the set of operations processed at time $ t $
$ o_{ji} $	the $ i $th operation of job $ j $
$ C_{\max} $	the makespan

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Table 2. Simulation results for FT, LA, ORB and YN

Name	Size($ n*m $)	Algorithm	Best	Worst	Mean	Std.
FT20	20*5	PSO	1374.00	1521.00	1442.50	42.02
		IGA	1744.00	2527.00	2025.50	198.95
		DE	1456.00	1554.00	1506.00	27.64
		SSO	1527.00	1527.00	1527.00	0
		SSO-DM	1374.00	1374.00	1374.00	0
LA40	15*15	PSO	1498.00	1732.00	1576.05	59.79
		IGA	2154.00	2803.00	2340.25	155.90
		DE	1691.00	1824.00	1767.05	36.46
		SSO	1834.00	1834.00	1834.00	0
		SSO-DM	1528.00	1528.00	1528.00	0
ORB10	10*10	PSO	1039.00	1263.00	1150.05	48.84
		IGA	1431.00	2121.00	1761.25	158.12
		DE	1190.00	1293.00	1244.40	25.04
		SSO	1345.00	1345.00	1345.00	0
		SSO-DM	1114.00	1114.00	1114.00	0
YN4	20*20	PSO	1340.00	1607.00	1425.15	64.84
		IGA	1826.00	2192.00	1997.90	116.48
		DE	1486.00	1601.00	1570.75	26.15
		SSO	1583.00	1583.00	1583.00	0
		SSO-DM	1492.00	1492.00	1492.00	0

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Table 3. Experimental results of running time of the algorithm

	IGA	PSO	DE	SSO	SSO-DM
FT20	4.7350	1.3626	0.6521	0.7993	0.8725
LA40	4.2764	2.6175	1.0695	1.2537	1.2961
ORB10	2.7547	1.8100	1.1091	1.3127	1.0230
YN04	6.8851	3.0504	1.9461	2.3814	2.5842

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Table 4. $ p $-values produced by Wilcoxon's test comparing SSO-DM vs. PSO, SSO-DM vs. IGA, SSO-DM vs. DE and SSO-DM vs. SSO, over the "average best-so-far" (Mean) values from Table 1 to Table 4

	PSO	IGA	DE	SSO
FT20	7.9772E-09	8.0065E-09	4.0136E-03	4.6826E-10
LA40	7.9918E-09	7.9918E-09	8.0065E-09	4.6826E-10
ORB10	7.9918E-09	8.0065E-09	7.9772E-09	4.6826E-10
YN04	2.0993E-07	8.0065E-09	4.0289E-02	4.6826E-10

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