Comparison of lot streaming division methodologies for multi-objective hybrid flowshop scheduling problem by considering limited waiting time

Beren Gürsoy Yılmaz; Ömer Faruk Yılmaz; Fatma Betül Yeni

doi:10.3934/jimo.2024058

Article Contents

2024, Volume 20, Issue 11: 3373-3414. Doi: 10.3934/jimo.2024058

This issue Previous Article Proximal decomposition of convex optimization via an alternating linearization algorithm with inexact oracles Next Article Uniform asymptotics for a renewal risk model with a random number of delayed claims

Comparison of lot streaming division methodologies for multi-objective hybrid flowshop scheduling problem by considering limited waiting time

1.
Karadeniz Technical University, Department of Industrial Engineering, Turkiye
2.
University of Florida, Department of Industrial and Systems Engineering, USA

^*Corresponding author: Beren Gürsoy Yılmaz
^*Corresponding author: Beren Gürsoy Yılmaz

Received: January 2024

Revised: March 2024

Early access: April 2024

Published: November 2024

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Abstract

In this paper, a multi-objective hybrid flowshop scheduling problem (HFSP) with limited waiting time and machine capability constraints is addressed. Given its importance, the implementation of lot streaming division methodologies with the problem is investigated through a design of experiment (DoE) setting based on real data extracted from a leading tire manufacturer in Gebze, Turkey. By doing so, specific characteristics of the addressed HFSP can be further explored to provide insights into its complexity and suggest recommendations for improving the operational efficiency of such systems resembling it. Based on the problem specifications and constraints, a novel generic multi-objective optimization model with objectives including the makespan, the average flow time, and the total workload imbalance is formulated. Since the studied problem is NP-hard in the strong sense, several algorithms based on the non-dominated sorting genetic algorithm-Ⅱ (NSGA-Ⅱ) are proposed according to the division methodologies, i.e., consistent sublots and equal sublots. Since the main aim of this problem is to further analyze the implementation of lot streaming on the HFSP problem, the developed algorithms are compared with each other to gain remarkable insights into the problem. Four different comparison metrics are employed to assess the solution quality of the proposed algorithms in terms of intensification and diversification aspects. Computational results demonstrate that employing the consistent sublot methodology leads to significant improvements in all metrics compared to the equal sublot methodology.

Keywords:

Mathematics Subject Classification: Primary: 90B35, 90C11; Secondary: 90C27.

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Figure 1. The facility layout showing supermarkets, machines, and stages

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Figure 2. Gantt chart with objective function values

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Figure 3. The proposed algorithms along with strategies and operators

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Figure 4. The chromosome structure and decoding procedure: (a) chromosome structure, (b) first phase of, (c) second phase of, (d) third phase of decoding procedure

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Figure 5. The NSGA-Ⅱ operators: (a) one-point crossover, (b) two-point crossover, and (c) swapping mutation operator

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Figure 6. The box plots of the proposed algorithms with respect to $ D1_R $ metric for all levels of $ L $: (a) $ L $ = 10. (b) $ L $ = 20. (c)$ L $ = 30. (d) $ L $ = 40

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Figure 7. The average C metric values of the proposed algorithms for all levels of L

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Figure 8. The box -plots of the proposed algorithms with respect to $ OS $ metric for all levels of $ L $: (a) $ L $ = 10. (b) $ L $ = 20. (c)$ L $ = 30. (d) $ L $ = 40

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Figure 9. The average OS metric values for pairwise comparisons of the proposed algorithms for all levels of L

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Figure 10. The average number of non-dominated solutions achieved by the proposed algorithms for all levels of $ L $

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Figure 11. Main effects and interaction plots for $ L $ = 10

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Figure 12. Main effects and interaction plots for $ L $ = 20

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Figure 13. Main effects and interaction plots for $ L $ = 30

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Figure 14. Main effects and interaction plots for $ L $ = 40

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Table 1. The number of variables and constraints in the optimization model

Variables	Count	Constraints	Count
$ CT_{bs} $	$ B\times S $	(4), (14-16), (18)	$ L\times S\times B $
$ PT_{bs} $	$ B\times S $	(5), (6)	1
$ L1_{bsm} $	$ B\times S\times M $	(7), (8)	$ S\times M $
$ Y_{lb} $	$ L\times B $	(9), (13)	$ S\times B $
$ XW_s $	$ S $	(10), (12)	$ L\times S\times M\times B $
$ NW_s $	$ S $	(11)	L
$ N_{bs} $	$ B\times S $	(17), (21-23)	$ L\times (L-1) \times S\times B\times (B-1)\times M $
$ W_b $	$ B $	(19), (20)	$ L\times B $
$ X_{bsm} $	$ B\times S\times M $
$ Z_{bb^{'}sm} $	$ B\times (B-1) \times S\times M $

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Table 2. The controlled factors and parameters

	Levels
Factors	1	2	3
M	4	6	8
S	2	6	10
B	2	6	10
L	10-20-30-40
Parameters	Data pattern
LS	Uniform [30-100]
PT	Uniform [10-20]
WT	1000 w.r.t L=10
	3000 w.r.t L=20
	6000 w.r.t L=20
	9000 w.r.t L=30

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Table 3. Performance of NSGA-Ⅱ-CSC1 algorithm for different input parameter settings

	$ cp $	$ mp $	$ ps $	$ D1_R $		$ cp $	$ mp $	$ ps $	$ D1_R $
L=10	0.8	0.05	L×6	0.012	L=20	0.8	0.05	L×6	0.018
	0.8	0.05	L×12	0.012		0.8	0.05	L ×12	0.017
	0.8	0.1	L×6	0.018		0.8	0.1	L ×6	0.028
	0.8	0.1	L×12	0.016		0.8	0.1	L ×12	0.026
	0.9	0.05	L×6	0.014		0.9	0.05	L ×6	0.022
	0.9	0.05	L×12	0.013		0.9	0.05	L ×12	0.019
	0.9	0.1	L×6	0.021		0.9	0.1	L ×6	0.030
	0.9	0.1	L×12	0.018		0.9	0.1	L ×12	0.027
	$ cp $	$ mp $	$ ps $	$ D1_R $		$ cp $	$ mp $	$ ps $	$ D1_R $
L=30	0.8	0.05	L×6	0.030	L=40	0.8	0.05	L×6	0.035
	0.8	0.05	L×12	0.029		0.8	0.05	L ×12	0.033
	0.8	0.1	L×6	0.036		0.8	0.1	L ×6	0.062
	0.8	0.1	L×12	0.030		0.8	0.1	L ×12	0.046
	0.9	0.05	L×6	0.051		0.9	0.05	L ×6	0.040
	0.9	0.05	L×12	0.048		0.9	0.05	L ×12	0.038
	0.9	0.1	L ×6	0.036		0.9	0.1	L ×6	0.056
	0.9	0.1	L×12	0.033		0.9	0.1	L ×12	0.048

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Table 4. The $ D1_R $ metric values of the proposed algorithms for L = 10 and L = 20

			L=10				L=20
B	S	M	P1	P2	P3	P4	P1	P2	P3	P4
1	1	1	0.29	0.40	0.03	0.02	0.42	0.58	0.11	0.02
1	1	2	0.32	1.12	0.20	0.00	0.10	0.24	0.02	0.02
1	1	3	0.28	2.58	0.70	0.01	0.18	0.34	0.04	0.03
1	2	1	0.36	0.76	0.09	0.00	0.34	0.54	0.13	0.00
1	2	2	0.05	0.06	0.02	0.00	0.09	0.29	0.05	0.00
1	2	3	0.22	0.32	0.05	0.03	0.09	0.11	0.04	0.01
1	3	1	0.55	1.67	0.21	0.01	0.29	0.64	0.12	0.00
1	3	2	0.19	0.59	0.12	0.02	0.16	0.22	0.00	0.03
1	3	3	0.13	0.24	0.02	0.03	0.24	0.37	0.04	0.07
2	1	1	0.84	1.13	0.73	0.00	0.09	0.21	0.02	0.00
2	1	2	0.25	0.20	0.09	0.00	0.51	0.91	0.20	0.00
2	1	3	0.20	0.33	0.02	0.00	0.47	0.57	0.15	0.02
2	2	1	0.75	1.08	0.11	0.01	0.50	1.07	0.09	0.00
2	2	2	0.26	0.45	0.08	0.00	1.10	2.17	0.07	0.03
2	2	3	0.14	0.39	0.02	0.01	0.14	0.22	0.06	0.00
2	3	1	0.37	0.25	0.05	0.00	0.43	0.41	0.05	0.00
2	3	2	0.07	0.16	0.04	0.01	0.03	0.23	0.02	0.02
2	3	3	0.17	0.49	0.06	0.04	0.69	0.80	0.26	0.00
3	1	1	3.58	8.17	0.81	0.00	0.09	0.41	0.13	0.00
3	1	2	0.66	2.22	0.11	0.00	0.39	0.52	0.07	0.00
3	1	3	0.42	0.53	0.10	0.00	0.09	0.51	0.21	0.02
3	2	1	1.91	2.68	1.05	0.00	0.19	0.96	0.07	0.00
3	2	2	0.22	0.29	0.19	0.01	0.65	0.82	0.17	0.00
3	2	3	0.08	0.32	0.04	0.01	0.03	0.21	0.11	0.00
3	3	1	0.32	0.61	0.26	0.00	0.21	0.47	0.00	0.01
3	3	2	0.68	0.92	0.08	0.03	0.33	0.64	0.07	0.01
3	3	3	0.28	0.58	0.20	0.01	0.41	0.96	0.17	0.00
Average			0.50	1.06	0.20	0.01	0.31	0.57	0.09	0.01

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Table 5. The C metric values of the proposed algorithms for L = 10

			L=10
B	S	M	P1-P2	P2-P1	P1-P3	P3-P1	P1-P4	P4-P1
1	1	1	0.22	0.00	0.00	1.00	0.00	1.00
1	1	2	1.00	0.00	0.03	0.75	0.00	1.00
1	1	3	1.00	0.00	0.65	0.00	0.00	1.00
1	2	1	1.00	0.00	0.05	0.22	0.00	1.00
1	2	2	1.00	0.00	0.02	0.15	0.00	0.83
1	2	3	1.00	0.00	0.03	0.70	0.42	0.92
1	3	1	1.00	0.00	0.00	0.72	0.00	1.00
1	3	2	1.00	0.00	0.88	0.00	0.08	0.57
1	3	3	0.25	0.02	0.00	1.00	0.10	0.58
2	1	1	0.87	0.00	0.00	1.00	0.00	1.00
2	1	2	0.00	0.88	0.00	0.88	0.00	0.88
2	1	3	0.43	0.00	0.00	1.00	0.00	0.42
2	2	1	0.45	0.00	0.00	1.00	0.00	1.00
2	2	2	0.65	0.00	0.00	1.00	0.00	1.00
2	2	3	1.00	0.00	0.00	1.00	0.00	0.38
2	3	1	0.00	0.98	0.00	1.00	0.00	1.00
2	3	2	0.05	0.03	0.03	0.78	0.00	0.87
2	3	3	1.00	0.00	0.00	0.40	0.53	0.40
3	1	1	0.07	0.02	0.00	1.00	0.00	1.00
3	1	2	1.00	0.00	0.00	1.00	0.00	1.00
3	1	3	1.00	0.00	0.00	0.98	0.00	1.00
3	2	1	0.00	0.00	0.00	1.00	0.00	1.00
3	2	2	0.52	0.00	0.03	0.67	0.00	0.67
3	2	3	1.00	0.00	0.10	0.10	0.00	1.00
3	3	1	0.32	0.02	0.00	0.05	0.00	1.00
3	3	2	0.03	0.00	0.00	1.00	0.00	1.00
3	3	3	0.00	0.00	0.55	0.88	0.00	1.00
Average			0.59	0.07	0.09	0.71	0.04	0.87

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Table 6. The OS metric values of the proposed algorithms for L = 10

			L=10
B	S	M	P1-P2	P1-P3	P1-P4	P2-P3	P2-P4	P3-P4
1	1	1	0.17	0.05	0.43	0.31	2.55	6.34
1	1	2	0.07	0.05	0.09	0.61	1.21	1.97
1	1	3	0.04	0.05	0.13	1.33	3.20	2.42
1	2	1	0.59	0.02	0.02	0.03	0.03	0.89
1	2	2	0.45	0.11	0.03	0.23	0.06	0.28
1	2	3	6.77	0.39	0.23	0.06	0.03	0.60
1	3	1	0.29	0.30	0.41	1.05	1.45	1.39
1	3	2	0.22	0.03	0.20	0.13	0.94	7.26
1	3	3	2.30	0.15	0.10	0.07	0.04	0.63
2	1	1	0.16	0.01	0.31	0.05	1.93	1.33
2	1	2	0.28	0.42	0.22	1.48	0.77	0.52
2	1	3	1.89	0.36	0.01	0.19	0.01	0.04
2	2	1	0.00	0.00	0.00	0.75	0.93	1.24
2	2	2	6.80	0.86	1.23	0.13	0.18	1.43
2	2	3	0.52	0.05	0.07	0.10	0.13	1.33
2	3	1	0.88	0.03	0.06	0.03	0.07	2.02
2	3	2	2.90	0.71	0.28	0.25	0.10	0.39
2	3	3	2.72	0.50	0.45	0.19	0.16	0.89
3	1	1	3.45	1.91	0.23	0.55	0.07	0.12
3	1	2	0.82	0.14	0.32	0.17	0.39	2.23
3	1	3	3.09	0.91	0.35	0.29	0.11	0.38
3	2	1	0.73	0.21	0.19	0.29	0.26	0.88
3	2	2	0.81	1.47	0.25	1.81	0.31	0.17
3	2	3	5.45	0.88	0.18	0.16	0.03	0.21
3	3	1	3.96	4.98	0.94	1.26	0.24	0.19
3	3	2	0.01	0.00	0.01	0.16	0.61	3.71
3	3	3	2.89	6.49	6.34	2.94	2.54	0.86
Average			1.79	0.78	0.48	0.54	0.68	1.47

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Table 7. Three-way ANOVA with respect to $ D1_R $ metric for L = 10

Three-way ANOVA: $ D1_R $ versus $ B $, $ S $, and $ M $ for L=10
Source	DF	Adj SS	Adj MS	F-Value	P-Value	Partial eta squared
$ B $	2	5.534	2.7669	3.84	0.025^*	0.609
$ S $	2	4.606	2.3028	3.2	0.046^*	0.511
$ M $	2	7.204	3.6019	5	0.009^*	0.77
$ B*S $	4	3.594	0.8985	1.25	0.298	0.406
$ B*M $	4	7.212	1.803	2.5	0.049^*	0.77
$ S*M $	4	2.332	0.583	0.81	0.523	0.273
$ BSM $	8	9.391	1.1738	1.63	0.129	0.673
Error	81	58.335	0.7202
Total	107	98.207
R-sq	R-sq(adj)
0.806	0.7153
P-Value $ \le $ 0.05 means a significant difference

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Table 8. Three-way ANOVA with respect to $ D1_R $ metric for L = 20

Three-way ANOVA: $ D1_R $ versus $ B $, $ S $, and $ M $ for L=20
Source	DF	Adj SS	Adj MS	F-Value	P-Value	Partial eta squared
$ B $	2	0.4335	0.2167	2.09	0.13	0.392
$ S $	2	0.1059	0.0539	0.51	0.602	0.096
$ M $	2	0.0907	0.0453	0.44	0.647	0.088
$ B*S $	4	0.2643	0.066	0.64	0.637	0.248
$ B*M $	4	0.435	0.1087	1.05	0.387	0.384
$ S*M $	4	1.0026	0.2506	2.42	0.055	0.848
$ BSM $	8	0.5916	0.0739	0.71	0.678	0.528
Error	81	8.3886	0.1035
Total	107	11.3122
R-sq	R-sq(adj)
0.7584	0.6753
P-Value $ \le $ 0.05 means a significant difference

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Table 9. Three-way ANOVA with respect to $ D1_R $ metric for L = 30

Three-way ANOVA: $ D1_R $ versus $ B $, $ S $, and $ M $ for L=30
Source	DF	Adj SS	Adj MS	F-Value	P-Value	Partial eta squared
$ B $	2	1.0238	0.5119	1.61	0.207	0.266
$ S $	2	0.7697	0.3849	1.21	0.304	0.203
$ M $	2	0.6919	0.346	1.09	0.342	0.182
$ B*S $	4	1.7575	0.4394	1.38	0.249	0.441
$ B*M $	4	2.8963	0.7241	2.27	0.068	0.707
$ S*M $	4	2.2108	0.5527	1.73	0.15	0.553
$ BSM $	8	5.493	0.6866	2.16	0.04^*	0.875
Error	81	25.8065	0.3186
Total	107	40.6495
R-sq	R-sq(adj)
0.7651	0.7106

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Table 10. Three-way ANOVA with respect to $ D1_R $ metric for L = 40

Three-way ANOVA: $ D1_R $ versus $ B $, $ S $, and $ M $ for L=40
Source	DF	Adj SS	Adj MS	F-Value	P-Value	Partial eta squared
$ B $	2	0.8396	0.4198	1.25	0.292	0.21
$ S $	2	0.9187	0.4594	1.37	0.261	0.231
$ M $	2	1.4274	0.7137	2.12	0.126	0.343
$ B*S $	4	2.5902	0.6476	1.93	0.114	0.609
$ B*M $	4	1.0508	0.2627	0.78	0.541	0.259
$ S*M $	4	2.9402	0.7351	2.19	0.078	0.679
$ BSM $	8	5.4532	0.6817	2.03	0.053	0.769
Error	81	27.2359	0.3362
Total	107	42.4562
R-sq	R-sq(adj)
0.8585	0.7526

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