Bi-objective unrelated parallel machines scheduling problem with worker allocation and sequence dependent setup times considering machine eligibility and precedence constraints

Reza Alizadeh Foroutan; Javad Rezaeian; Milad Shafipour

doi:10.3934/jimo.2021190

Article Contents

2023, Volume 19, Issue 1: 402-436. Doi: 10.3934/jimo.2021190

This issue Previous Article A new dynamic model to optimize the reliability of the series-parallel systems under warm standby components Next Article Diagonally scaled memoryless quasi–Newton methods with application to compressed sensing

Bi-objective unrelated parallel machines scheduling problem with worker allocation and sequence dependent setup times considering machine eligibility and precedence constraints

Department of Industrial Engineering and Management, Mazandaran University of Science and Technology, Babol, Iran

^* Corresponding author: Javad Rezaeian
^* Corresponding author: Javad Rezaeian

Received: January 2021

Revised: September 2021

Early access: November 2021

Published: January 2023

Abstract / Introduction Full Text(HTML) Figure(21) / Table(16) Related Papers Cited by

Abstract

In today's competitive world, scheduling problems are one of the most important and vital issues. In this study, a bi-objective unrelated parallel machine scheduling problem with worker allocation, sequence dependent setup times, precedence constraints, and machine eligibility is presented. The objective functions are to minimize the costs of tardiness and hiring workers. In order to formulate the proposed problem, a mixed-integer quadratic programming model is presented. A strategy called repair is also proposed to implement the precedence constraints. Because the problem is NP-hard, two metaheuristic algorithms, a multi-objective tabu search (MOTS) and a multi-objective simulated annealing (MOSA), are presented to tackle the problem. Furthermore, a hybrid metaheuristic algorithm is also developed. Finally, computational experiments are carried out to evaluate different test problems, and analysis of variance is done to compare the performance of the proposed algorithms. The results show that MOTS is doing better in terms of objective values and mean ideal distance (MID) metric, while the proposed hybrid algorithm outperforms in most cases, considering other employed comparison metrics.

Keywords:

Mathematics Subject Classification: Primary: 90B10, 90C20; Secondary: 90C59.

Citation:

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Figure 1. Results of the test problem with $n = 8, m = 2$, and $w = 2$

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Figure 2. The results of validation of solutions quality

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Figure 3. The flowchart of the MOTS algorithm [4]

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Figure 4. Representation of a chromosome with $n = 10$, $m = 3$ and $k = 2$

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Figure 5. An example of the precedence constraints

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Figure 6. The implementation steps of the correcting algorithm

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Figure 7. The swap operator

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Figure 8. The reversion operator

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Figure 9. The workers relocation operator

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Figure 10. Pareto fronts of all three proposed metaheuristics, a medium test problem ($n = 20, m = 6, w = 5$)

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Figure 11. Pareto fronts of all three proposed metaheuristics, a large test problem ($n = 30, m = 10, w = 6$)

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Figure 12. Comparison of the hypervolume indicator for the problems of all sizes

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Figure 13. Hypervolumes comparison, a medium test problem $(n = 20, m = 6, w = 5)$

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Figure 14. Hypervolumes comparison, a large test problem $(n = 30, m = 10, w = 6)$

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Figure 15. Pareto fronts of MOTS in previous iterations

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Figure 16. Pareto fronts of MOSA in previous iterations

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Figure 17. Pareto fronts of hybrid algorithm in previous iterations

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Figure 18. The convergence curve of the proposed MOTS

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Figure 19. The convergence curve of the proposed MOSA

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Figure 20. The convergence curve of the proposed hybrid algorithm

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Figure 21. The last significant change in the Pareto front of the proposed hybrid algorithm

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Table 1. A comparison of some most recent studies existing in the literature

		Constraints
Reference	Unrelated Parallel machines	Worker flexibility/ allocation	Machine eligibility	Sequence dependent setup times	Precedence constraints	Objective function(s)	Solution approach(es)
Cota et al. [7]	Yes	No	No	Yes	No	Makespan; Total energy consumption	ALNS, LA, MO-ALNS, MO-ALNS/D
Zhu and Zhou [39]	No	No	No	No	Yes	Makespan; Total workload; Maximum machine workload	EMOGWO
Munoz-Villamizar et al. [26]	No	No	No	Yes	No	Makespan; Total earliness and tardiness; Cost minimization; Effectiveness optimization	GAMS solver
Arık and Toksarı [3]	No	No	No	No	No	Total tardiness penalty cost; Earliness penalty cost; Cost of setting due dates	A fuzzy local search algorithm
Gong et al. [17]	No	Yes	No	No	No	Makespan; Total worker cost; Green production indicator	A hybrid evolutionary algorithm
Zhang et al. [36]	No	No	No	No	Yes	Total penalties; Makespan; Total completion time	Approximation algorithms
Gong et al. [16]	No	Yes	No	No	No	Makespan	A hybrid artificial bee colony
Wang et al. [33]	No	No	No	No	No	Total energy consumption; Makespan	An augmented $ \varepsilon $-constraint; A heuristic method; NSGA-II
Zhang et al. [38]	Yes	No	No	No	No	Total energy consumption; Makespan	A heuristic evolutionary algorithm
Lei et al. [24]	Yes	No	No	No	No	Makespan	An imperialist competitive algorithm
Khanh Van and Van Hop [21]	Yes	No	No	Yes	No	Total weighted earliness and tardiness; Makespan	A hybrid algorithm based on GA and ISETP
Kim et al. [22]	No	No	No	Yes	Yes	Total tardiness	SA, GA
This paper	Yes	Yes	Yes	Yes	Yes	Cost of tardiness; Cost of hiring workers	MOTS, MOSA, A hybrid evolutionary algorithm

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Table 2. Processing time ($ T_{il} $) of the jobs

	$ J_1 $	$ J_2 $	$ J_3 $	$ J_4 $	$ J_5 $	$ J_6 $	$ J_7 $	$ J_8 $
$ M_1 $	$ 0 $	$ 4 $	$ 2 $	$ 1 $	$ 5 $	$ 3 $	$ 5 $	$ 4 $
$ M_2 $	$ 0 $	$ 3 $	$ 5 $	$ 8 $	$ 2 $	$ 2 $	$ 6 $	$ 3 $
$ M_3 $	$ 0 $	$ 1 $	$ 5 $	$ 4 $	$ 4 $	$ 2 $	$ 3 $	$ 3 $

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Table 3. Delivery time and tardiness penalty of the jobs

	$ J_1 $	$ J_2 $	$ J_3 $	$ J_4 $	$ J_5 $	$ J_6 $	$ J_7 $	$ J_8 $
$ D_j $	$ 10 $	$ 13 $	$ 8 $	$ 15 $	$ 6 $	$ 16 $	$ 14 $	$ 18 $
$ \beta_j $	$ 4 $	$ 3 $	$ 7 $	$ 5 $	$ 9 $	$ 8 $	$ 6 $	$ 2 $

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Table 4. Setup time for each machine

	$ (j, l)\in A $	$ J_{l=1} $	$ J_2 $	$ J_3 $	$ J_4 $	$ J_5 $	$ J_6 $	$ J_7 $	$ J_8 $
	$ J_{j=1} $	$ 3 $	$ 3 $	$ 1 $	$ 3 $	$ 3 $	$ 1 $	$ 3 $	$ 1 $
	$ J_2 $	$ 3 $	$ 2 $	$ 3 $	$ 1 $	$ 3 $	$ 1 $	$ 3 $	$ 2 $
	$ J_3 $	$ 3 $	$ 1 $	$ 1 $	$ 2 $	$ 3 $	$ 3 $	$ 2 $	$ 1 $
	$ J_4 $	$ 3 $	$ 2 $	$ 2 $	$ 3 $	$ 3 $	$ 3 $	$ 1 $	$ 3 $
$ M_1 $	$ J_5 $	$ 2 $	$ 2 $	$ 1 $	$ 2 $	$ 3 $	$ 1 $	$ 1 $	$ 3 $
	$ J_6 $	$ 1 $	$ 3 $	$ 1 $	$ 3 $	$ 1 $	$ 1 $	$ 2 $	$ 2 $
	$ J_7 $	$ 2 $	$ 1 $	$ 2 $	$ 3 $	$ 2 $	$ 1 $	$ 1 $	$ 1 $
	$ J_8 $	$ 2 $	$ 2 $	$ 1 $	$ 3 $	$ 2 $	$ 1 $	$ 2 $	$ 1 $
	$ (j, l)\in A $	$ J_{l=1} $	$ J_2 $	$ J_3 $	$ J_4 $	$ J_5 $	$ J_6 $	$ J_7 $	$ J_8 $
	$ J_1 $	$ 3 $	$ 2 $	$ 2 $	$ 1 $	$ 2 $	$ 2 $	$ 1 $	$ 3 $
	$ J_2 $	$ 3 $	$ 2 $	$ 1 $	$ 1 $	$ 1 $	$ 2 $	$ 3 $	$ 2 $
	$ J_3 $	$ 3 $	$ 2 $	$ 2 $	$ 2 $	$ 1 $	$ 3 $	$ 1 $	$ 3 $
	$ J_4 $	$ 1 $	$ 1 $	$ 2 $	$ 2 $	$ 1 $	$ 2 $	$ 1 $	$ 3 $
$ M_2 $	$ J_5 $	$ 2 $	$ 1 $	$ 2 $	$ 1 $	$ 3 $	$ 1 $	$ 3 $	$ 1 $
	$ J_6 $	$ 3 $	$ 3 $	$ 2 $	$ 2 $	$ 1 $	$ 3 $	$ 1 $	$ 2 $
	$ J_7 $	$ 1 $	$ 1 $	$ 1 $	$ 2 $	$ 3 $	$ 3 $	$ 2 $	$ 3 $
	$ J_8 $	$ 1 $	$ 3 $	$ 1 $	$ 2 $	$ 1 $	$ 3 $	$ 2 $	$ 3 $
	$ (j, l)\in A $	$ J_{l=1} $	$ J_2 $	$ J_3 $	$ J_4 $	$ J_5 $	$ J_6 $	$ J_7 $	$ J_8 $
	$ J_1 $	$ 1 $	$ 1 $	$ 3 $	$ 3 $	$ 1 $	$ 3 $	$ 2 $	$ 3 $
	$ J_2 $	$ 3 $	$ 1 $	$ 1 $	$ 3 $	$ 3 $	$ 2 $	$ 1 $	$ 2 $
	$ J_3 $	$ 1 $	$ 1 $	$ 3 $	$ 1 $	$ 2 $	$ 3 $	$ 1 $	$ 1 $
	$ J_4 $	$ 3 $	$ 1 $	$ 2 $	$ 2 $	$ 3 $	$ 2 $	$ 2 $	$ 1 $
$ M_3 $	$ J_5 $	$ 1 $	$ 3 $	$ 1 $	$ 2 $	$ 2 $	$ 3 $	$ 2 $	$ 2 $
	$ J_6 $	$ 1 $	$ 3 $	$ 1 $	$ 3 $	$ 1 $	$ 2 $	$ 3 $	$ 2 $
	$ J_7 $	$ 1 $	$ 1 $	$ 3 $	$ 2 $	$ 2 $	$ 2 $	$ 1 $	$ 3 $
	$ J_8 $	$ 1 $	$ 1 $	$ 2 $	$ 2 $	$ 3 $	$ 1 $	$ 3 $	$ 3 $

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Table 5. The availability of $ Y_{il} $s

	$ J_1 $	$ J_2 $	$ J_3 $	$ J_4 $	$ J_5 $	$ J_6 $	$ J_7 $	$ J_8 $
$ M_1 $	$ \star $	$ \star $	$ \star $		$ \star $	$ \star $		$ \star $
$ M_2 $	$ \star $	$ \star $		$ \star $			$ \star $	$ \star $
$ M_3 $	$ \star $		$ \star $		$ \star $	$ \star $

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Table 6. Results of the first test problem

Job	$ S_{ikjl} $	$ CS_l $	$ C_l $	$ T_{il} $	$ T_l $	$ X_{ikjl}=1 $
$ 1 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ 0 $	$ X_{1213} $
$ 2 $	$ 1.2 $	$ 10 $	$ 14 $	$ 4 $	$ 1 $	$ X_{1228} $
$ 3 $	$ 1.2 $	$ 6 $	$ 8 $	$ 2 $	$ 0 $	$ X_{1232} $
$ 4 $	$ 2.4 $	$ 17 $	$ 25 $	$ 8 $	$ 10 $	$ X_{2217} $
$ 5 $	$ 1.2 $	$ 2 $	$ 6 $	$ 4 $	$ 0 $	$ X_{2274} $
$ 6 $	$ 3.6 $	$ 14 $	$ 16 $	$ 2 $	$ 0 $	$ X_{3215} $
$ 7 $	$ 1.2 $	$ 4 $	$ 14 $	$ 6 $	$ 0 $	$ X_{3256} $
$ 8 $	$ 2.4 $	$ 20 $	$ 24 $	$ 4 $	$ 6 $
$ F=0.6\star f_1+0.4\star f_2=39+448=487 $

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Table 7. Test problems used to validate the quality of solutions

Test Problem	$ 1 $	$ 2 $	$ 3 $	$ 4 $	$ 5 $	$ 6 $	$ 7 $	$ 8 $	$ 9 $	$ 10 $
$ (n, m, w) $	$ (4, 2, 2) $	$ (5, 3, 2) $	$ (5, 2, 2) $	$ (6, 3, 2) $	$ (6, 2, 2) $	$ (6, 2, 2) $	$ (7, 2, 2) $	$ (7, 2, 3) $	$ (7, 2, 3) $	$ (8, 2, 2) $
with constraints	√	√	√	√		√			√
without constraints					√		√	√		√

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Table 8. Input data for test problems

Parameter	Generating function
Jobs processing time	$ T_{il}\sim U[5\;\;15] $
Jobs delivery time	$ D_j\sim [20\;\;40] $
Setup times	$ S_{ijl}\sim [1\;\;7] $
Jobs tardiness penalties	$ \beta_j\sim [1\;\;10] $
Worker's skill	$ \pi_k\sim rand (0.5, 1.5) $

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Table 9. Levels of the controller parameters

Algorithm	Parameter	Levels
MOTS	$ Tabu_{list} $	$ 2, \;4, \;6 $
	$ N_{new} $	$ 10, \;20, \;40 $
	$ max_{it} $	$ 30, \;50, \;70 $
MOSA	$ max_{it} $	$ 100, \;200, \;300 $
	$ Sub_{it} $	$ 20, \;25, \;30 $
	$ T_0 $	$ 60, \;100, \;150 $
	$ \alpha $	$ 0.93, \;0.95, \;0.99 $
Hybrid	$ Tabu_{list} $	$ 2, \;4, \;5 $
	$ max_{it} $	$ 70, \;150, \;200 $
	$ Sub_{it} $	$ 20, \;25, \;30 $
	$ T_0 $	$ 80, \;100, \;150 $
	$ \alpha $	$ 0.93, \;0.95, \;0.99 $

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Table 10. Ranking of factors (based on $ SN $ ratios and mean of responses) and their best values for each algorithm

Algorithm	Parameter	Rank ($ S/N $ ratios)	Rank (Means)	Best value
MOTS	$ Tabu_{list} $	$ 3 $	$ 3 $	$ 4 $
	$ N_{new} $	$ 2 $	$ 2 $	$ 40 $
	$ max_{it} $	$ 1 $	$ 1 $	$ 70 $
MOSA	$ max_{it} $	$ 1 $	$ 1 $	$ 200 $
	$ Sub_{it} $	$ 2 $	$ 2 $	$ 25 $
	$ T_0 $	$ 4 $	$ 4 $	$ 100 $
	$ \alpha $	$ 3 $	$ 3 $	$ 0.93 $
Hybrid	$ Tabu_{list} $	$ 1 $	$ 2 $	$ 5 $
	$ max_{it} $	$ 3 $	$ 1 $	$ 70 $
	$ Sub_{it} $	$ 2 $	$ 3 $	$ 25 $
	$ T_0 $	$ 4 $	$ 4 $	$ 150 $
	$ \alpha $	$ 5 $	$ 5 $	$ 0.93 $

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Table 11. Computational results for the problems of all sizes: MOTS & MOSA

Problem $ (n, m, w) $	Obj. functions	MOTS				MOSA
Problem $ (n, m, w) $	Obj. functions	P1*	P2	P3	P4	P1	P2	P3
$ (8, 2, 2) $	f1	710	826			1661	1968
$ (8, 2, 2) $	f2	4853	4769			4769	4769
$ (8, 3, 2) $	f1	856				898	1550
$ (8, 3, 2) $	f2	3656				4418	4418
$ (10, 3, 2) $	f1	1063				1433
$ (10, 3, 2) $	f2	3662				4880
$ (10, 3, 3) $	f1	731				4762
$ (10, 3, 3) $	f2	3065				3438
$ (10, 4, 2) $	f1	983	1252			6937	6937
$ (10, 4, 2) $	f2	5719	5314			2055	2229
$ (12, 3, 2) $	f1	1460				3990
$ (12, 3, 2) $	f2	6532				5869
$ (12, 3, 3) $	f1	681	815			3837
$ (12, 3, 3) $	f2	6358	5880			5557
$ (12, 4, 2) $	f1	1207	1233			5487
$ (12, 4, 2) $	f2	6445	6184			5399
$ (15, 4, 3) $	f1	480	527	924		4346	4560
$ (15, 4, 3) $	f2	10760	10542	10092		9807	9807
$ (15, 6, 3) $	f1	2156	2176	2196	2579	5527
$ (15, 6, 3) $	f2	9484	9224	8963	8703	9475
$ (20, 4, 3) $	f1	6317				13816
$ (20, 4, 3) $	f2	6943				9658
$ (20, 6, 5) $	f1	2894	3466	3719		11392	12469
$ (20, 6, 5) $	f2	8410	7044	6702		9826	9826
$ (25, 4, 3) $	f1	12610				20565
$ (25, 4, 3) $	f2	7749				12725
$ (25, 6, 5) $	f1	3715				16032	16037
$ (25, 6, 5) $	f2	8823				11685	11685
$ (30, 8, 4) $	f1	4133	4560			19549	21010	22219
$ (30, 8, 4) $	f2	14096	13584			15081	15081	15081
$ (30, 10, 6) $	f1	3190	3444	3961		23057	25436
$ (30, 10, 6) $	f2	13499	12942	12632		14132	14132
$ (40, 10, 6) $	f1	8993	9944			38190	40236
$ (40, 10, 6) $	f2	17455	17182			21838	21838
$ (40, 12, 8) $	f1	6334	6421	7055		44881	31799
$ (40, 12, 8) $	f2	18491	18383	18187		19731	19731
$ (50, 12, 8) $	f1	13801	14655			73855	78444	80185
$ (50, 12, 8) $	f2	19918	19450			19905	19905	19905
$ (50, 15, 10) $	f1	11484	13550			47375	51987	59331
$ (50, 15, 10) $	f2	22310	21220			26906	26906	26906
$ P_z, (z = 1, 2, …, Z) $ denotes the $ z $th pareto solution.

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Table 12. Computational results for the problems of all sizes: The proposed hybrid

Problem $ (n, m, w) $	Obj. functions	Hybrid
Problem $ (n, m, w) $	Obj. functions	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11
$ (8, 2, 2) $	f1	1524	1553	1617
$ (8, 2, 2) $	f2	5191	5107	5022
$ (8, 3, 2) $	f1	1216	1400	1624	1961
$ (8, 3, 2) $	f2	4266	4266	4113	3961
$ (10, 3, 2) $	f1	989	1405	1677	1711	1791	1867	1870	1896
$ (10, 3, 2) $	f2	6097	5793	5793	5488	5184	5184	4880	4575
$ (10, 3, 3) $	f1	2443	2586	2671	2695	2702	2704
$ (10, 3, 3) $	f2	3295	3295	3208	3152	3152	3065
$ (10, 4, 2) $	f1	858	1047	1196	1330	1452	1601
$ (10, 4, 2) $	f2	6531	6531	6328	6125	5922	5719
$ (12, 3, 2) $	f1	2106	2400	2491	2841	2946
$ (12, 3, 2) $	f2	7526	7195	6863	6532	6201
$ (12, 3, 3) $	f1	1428	2331	2669	2690	3088	3109	3258	3279	3334	3438
$ (12, 3, 3) $	f2	8268	7790	7313	7046	6835	6568	6358	6091	5880	5613
$ (12, 4, 2) $	f1	3812	3829
$ (12, 4, 2) $	f2	6707	6184
$ (15, 4, 3) $	f1	2871	3015	3589	3610	3617	3618	3619
$ (15, 4, 3) $	f2	1080	1080	1078	1077	1074	1054	1052
$ (15, 6, 3) $	f1	2100	2313	2576	2600
$ (15, 6, 3) $	f2	9739	9739	9484	9483
$ (20, 4, 3) $	f1	6887	6979	7519
$ (20, 4, 3) $	f2	11292	10078	9755
$ (20, 6, 5) $	f1	5712	7395	7406	7445	9537	9551	9718	9740	10162	10176	10365
$ (20, 6, 5) $	f2	9718	9718	9587	9518	9518	9386	9309	9253	9044	8782	8779
$ (25, 4, 3) $	f1	11769	11826	12126	13470	17244	17665
$ (25, 4, 3) $	f2	10769	10769	10598	10598	10598	10237
$ (25, 6, 5) $	f1	5851	6530	6679	6684	8501	8534	8715	9107	9122
$ (25, 6, 5) $	f2	10561	10561	10389	10266	10240	10117	10114	10114	9991
$ (30, 8, 4) $	f1	13183	13805	13942	14068	15660	16033	17741	17785
$ (30, 8, 4) $	f2	16567	16411	16391	16211	16191	16010	15498	15142
$ (30, 10, 6) $	f1	10899	10951	10968	13811	17298	18530	18552
$ (30, 10, 6) $	f2	15173	15031	14762	14762	14566	14084	13899
$ (40, 10, 6) $	f1	21130	22146	23254
$ (40, 10, 6) $	f2	25930	25229	24604
$ (40, 12, 8) $	f1	18544	18551	20567	20619	20629	21045
$ (40, 12, 8) $	f2	21022	20914	20914	20910	20866	20866
$ (50, 12, 8) $	f1	47230	47348	60202	62672
$ (50, 12, 8) $	f2	23776	23729	23456	21764
$ (50, 15, 10) $	f1	18791	23480	26830	28710	28254	29460
$ (50, 15, 10) $	f2	28995	28995	28637	28376	28302	28202

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Table 13. Hypervolume indicator for the problems of all sizes

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Table 14. Comparisons results for the problems of all sizes

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Table 15. One-way ANOVA for all test problems (between groups)

Metric	Problem size	Sum of Squares	df	Mean Square	F.	Sig.
S-metric	Small	8758922.903	2	4379461.452	5.519	0.012
	Medium	8.541E7	2	4.271E7	3.688	0.050
	Large	1.348E9	2	6.739E8	6.716	0.008
NP	Small	130.083	2	65.042	24.500	0.000
	Medium	81.333	2	40.667	7.562	0.005
	Large	42.333	2	21.167	15.744	0.000
MID	Small	6748717.583	2	3374358.792	2.312	0.124
	Medium	1.932E8	2	9.662E7	4.538	0.029
	Large	2.856E9	2	1.428E9	11.064	0.001
DM	Small	8430960.250	2	4215480.125	10.568	0.001
	Medium	7.217E7	2	3.608E7	7.737	0.005
	Large	1.954E8	2	9.768E7	5.840	0.013
SNS	Small	122196.583	2	61098.292	5.399	0.013
	Medium	1.564E7	2	7817526.167	2.349	0.130
	Large	1.236E7	2	6179340.222	0.094	0.911
Time	Small	20371.750	2	10185.875	21.221	0.000
	Medium	43694.111	2	21847.056	6.425	0.010
	Large	177652.111	2	88826.056	10.736	0.001

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Table 16. Results of post-hoc comparisons

	Small		Medium		Large
Metrics	Order of best performances	Significant difference	Order of best performances	Significant difference	Order of best performances	Significant difference
S-metric	MOTS-Hybrid-MOSA	MOTS-MOSA	MOTS-Hybrid-MOSA	MOTS-MOSA	MOTS-Hybrid-MOSA	MOTS-MOSA
NPS	Hybrid-MOTS-MOSA	Hybrid-MOSA, Hybrid-MOTS	Hybrid-MOTS-MOSA	-	Hybrid-MOSA-MOTS	Hybrid-MOSA, Hybrid-MOTS
MID	Hybrid-MOTS-MOSA	-	MOTS-Hybrid-MOSA	MOTS-MOSA	MOTS-Hybrid-MOSA	MOTS-MOSA
DM	Hybrid-MOTS-MOSA	Hybrid-MOSA, Hybrid-MOTS	Hybrid-MOTS-MOSA	-	Hybrid-MOSA-MOTS	Hybrid-MOTS
SNS	Hybrid-MOTS-MOSA	Hybrid-MOSA	Hybrid-MOTS-MOSA	-	Hybrid-MOSA-MOTS	-
Time	Hybrid-MOSA-MOTS	Hybrid-MOSA, Hybrid-MOTS	Hybrid-MOSA-MOTS	Hybrid-MOTS	Hybrid-MOSA-MOTS	Hybrid-MOSA, Hybrid-MOTS

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[1]	M. Afzalirad and J. Rezaeian, Resource-constrained unrelated parallel machine scheduling problem with sequence dependent setup times, precedence constraints and machine eligibility restrictions, Computers & Industrial Engineering, 98 (2016), 40-52. doi: 10.1016/j.cie.2016.05.020.
[2]	M. Afzalirad and M. Shafipour, Design of an efficient genetic algorithm for resource-constrained unrelated parallel machine scheduling problem with machine eligibility restrictions, Journal of Intelligent Manufacturing, 29 (2018), 423-437. doi: 10.1007/s10845-015-1117-6.
[3]	O. A. Arik and M. D. Toksari, Multi-objective fuzzy parallel machine scheduling problems under fuzzy job deterioration and learning effects, International Journal of Production Research, 56 (2018), 2488-2505. doi: 10.1080/00207543.2017.1388932.
[4]	A. Baykasoglu, Applying multiple objective tabu search to continuous optimization problems with a simple neighbourhood strategy, International Journal for Numerical Methods in Engineering, 65 (2006), 406-424. doi: 10.1002/nme.1455.
[5]	T. Çakar, R. Köker and Y. Sari, Parallel robot scheduling to minimize mean tardiness with unequal release date and precedence constraints using a hybrid intelligent system, International Journal of Advanced Robotic Systems, 9 (2012), 252. doi: 10.5772/54381.
[6]	C. L. Chen, Iterated hybrid metaheuristic algorithms for unrelated parallel machines problem with unequal ready times and sequence-dependent setup times, The International Journal of Advanced Manufacturing Technology, 60 (2012), 693-705. doi: 10.1007/s00170-011-3623-9.
[7]	L. P. Cota, F. G. Guimarães, R. G. Ribeiro, I. R. Meneghini, F. B. de Oliveira, M. J. Souza and P. Siarry, An adaptive multi-objective algorithm based on decomposition and large neighborhood search for a green machine scheduling problem, Swarm and Evolutionary Computation, 51 (2019), 100601. doi: 10.1016/j.swevo.2019.100601.
[8]	M. Dhiflaoui, H. E. Nouri and O. B. Driss, Dual-resource constraints in classical and flexible job shop problems: A state-of-the-art review, Procedia Computer Science, 126 (2018), 1507-1515. doi: 10.1016/j.procs.2018.08.123.
[9]	E. C. H. Dorion, J. C. F. Guimarães, E. A. Severo, Z. C. Reis and P. M. Olea, Innovation and production management through a just in sequence strategy in a multinational brazilian metal-mechanic industry, 2014 IEEE International Conference on Management of Innovation and Technology, (2014), 54–60. doi: 10.1109/ICMIT.2014.6942400.
[10]	P. Engrand, A multi-objective optimization approach based on simulated annealing and its application to nuclear fuel management, 1998.
[11]	A. E. Ezugwu, O. J. Adeleke and S. Viriri, Symbiotic organisms search algorithm for the unrelated parallel machines scheduling with sequence-dependent setup times, PloS One, 13 (2018), e0200030. doi: 10.1371/journal.pone.0200030.
[12]	A. E. Ezugwu and F. Akutsah, An improved firefly algorithm for the unrelated parallel machines scheduling problem with sequence-dependent setup times, IEEE Access, 6 (2018), 54459-54478. doi: 10.1109/ACCESS.2018.2872110.
[13]	C. M. Fonseca, L. Paquete and M. López-Ibánez, An improved dimension-sweep algorithm for the hypervolume indicator, 2006 IEEE International Conference on Evolutionary Computation, (2006), 1157–1163. doi: 10.1109/CEC.2006.1688440.
[14]	F. Glover, Future paths for integer programming and links to artificial intelligence, Comput. Oper. Res., 13 (1986), 533-549. doi: 10.1016/0305-0548(86)90048-1.
[15]	F. W. Glover and G. A. Kochenberger. (Eds.), Handbook of Metaheuristics, International Series in Operations Research & Management Science, 57. Kluwer Academic Publishers, Boston, MA, 2003.
[16]	G. Gong, R. Chiong, Q. Deng and X. Gong, A hybrid artificial bee colony algorithm for flexible job shop scheduling with worker flexibility, International Journal of Production Research, 58 (2020), 4406-4420. doi: 10.1080/00207543.2019.1653504.
[17]	G. Gong, R. Chiong, Q. Deng, W. Han, L. Zhang, W. Lin and K. Li, Energy-efficient flexible flow shop scheduling with worker flexibility, Expert Systems with Applications, 141 (2020), 112902. doi: 10.1016/j.eswa.2019.112902.
[18]	A. Hamzadayi and G. Yildiz, Modeling and solving static m identical parallel machines scheduling problem with a common server and sequence dependent setup times, Computers & Industrial Engineering, 106 (2017), 287-298. doi: 10.1016/j.cie.2017.02.013.
[19]	M. P. Hansen, Tabu search for multiobjective optimization: MOTS, Proceedings of the 13th International Conference on Multiple Criteria Decision Making, (1997), 574–586.
[20]	O. Hurdies, Just in time (JIT) production: An effective approach to efficiency, Logistics & Supply Chain Review, 1 (2020), 32-39.
[21]	B. Van Khanh and N. Van Hop, Genetic algorithm with initial sequence for parallel machines scheduling with sequence dependent setup times based on earliness-tardiness, Journal of Industrial and Production Engineering, 38 (2021), 18-28. doi: 10.1080/21681015.2020.1829111.
[22]	J. G. Kim, S. Song and B. Jeong, Minimising total tardiness for the identical parallel machine scheduling problem with splitting jobs and sequence-dependent setup times, International Journal of Production Research, 58 (2020), 1628-1643. doi: 10.1080/00207543.2019.1672900.
[23]	S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi, Optimization by simulated annealing, Science, 220 (1983), 671-680. doi: 10.1126/science.220.4598.671.
[24]	D. Lei, Y. Yuan, J. Cai and D. Bai, An imperialist competitive algorithm with memory for distributed unrelated parallel machines scheduling, International Journal of Production Research, 58 (2020), 597-614. doi: 10.1080/00207543.2019.1598596.
[25]	H. M. Md, Lead time reduction and process cycle improvement of an ice-cream manufacturing factory in bangladesh by using value stream map and kanban board, Australian Journal Of Basic and Applied Sciences, (2016).
[26]	A. Munoz-Villamizar, J. Santos, J. Montoya-Torres and M. Alvaréz, Improving effectiveness of parallel machine scheduling with earliness and tardiness costs: A case study, International Journal of Industrial Engineering Computations, 10 (2019), 375-392. doi: 10.5267/j.ijiec.2019.2.001.
[27]	J. Rezaeian, N. Derakhshan, I. Mahdavi and R. A. Foroutan, Due date assignment and JIT scheduling problem in blocking hybrid flow shop robotic cells with multiple robots and batch delivery cost, International Journal of Industrial Mathematics, 13 (2021), 145-162.
[28]	M. Savsar, Simulation analysis of a pull-push system for an electronic assembly line, International Journal of Production Economics, 51 (1997), 205-214. doi: 10.1016/S0925-5273(97)00055-8.
[29]	G. Taguchi, Introduction to quality engineering: Designing quality into products and processes, No. 658.562 T3, (1986).
[30]	E. Vallada and R. Ruiz, A genetic algorithm for the unrelated parallel machine scheduling problem with sequence dependent setup times, European J. Oper. Res., 211 (2011), 612-622. doi: 10.1016/j.ejor.2011.01.011.
[31]	B. Wang and H. Wang, Multiobjective order acceptance and scheduling on unrelated parallel machines with machine eligibility constraints, Math. Probl. Eng., 2018 (2018), 12pp. doi: 10.1155/2018/6024631.
[32]	I. L. Wang, Y. C. Wang and C. W. Chen, Scheduling unrelated parallel machines in semiconductor manufacturing by problem reduction and local search heuristics, Flexible Services and Manufacturing Journal, 25 (2013), 343-366. doi: 10.1007/s10696-012-9150-7.
[33]	S. Wang, X. Wang, J. Yu, S. Ma and M. Liu, Bi-objective identical parallel machine scheduling to minimize total energy consumption and makespan, Journal of Cleaner Production, 193 (2018), 424-440. doi: 10.1016/j.jclepro.2018.05.056.
[34]	J. Xu, X. Xu and S. Q. Xie, Recent developments in Dual Resource Constrained (DRC) system research, European Journal of Operational Research, 215 (2011), 309-318. doi: 10.1016/j.ejor.2011.03.004.
[35]	K. C. Ying and S. W. Lin, Unrelated parallel machine scheduling with sequence-and machine-dependent setup times and due date constraints, International Journal of Innovative Computing, 8 (2012), 3279-3297.
[36]	A. Zhang, X. Qi and G. Li, Machine scheduling with soft precedence constraints, European J. Oper. Res., 282 (2020), 491-505. doi: 10.1016/j.ejor.2019.09.041.
[37]	J. R. Zeidi and S. MohammadHosseini, Scheduling unrelated parallel machines with sequence-dependent setup times, The International Journal of Advanced Manufacturing Technology, 81 (2015), 1487-1496. doi: 10.1007/s00170-015-7215-y.
[38]	L. Zhang, Q. Deng, G. Gong and W. Han, A new unrelated parallel machine scheduling problem with tool changes to minimise the total energy consumption, International Journal of Production Research, 58 (2020), 6826-6845. doi: 10.1080/00207543.2019.1685708.
[39]	Z. Zhu and X. Zhou, An efficient evolutionary grey wolf optimizer for multi-objective flexible job shop scheduling problem with hierarchical job precedence constraints, Computers & Industrial Engineering, 140 (2020), 106280. doi: 10.1016/j.cie.2020.106280.