A better dominance relation and heuristics for Two-Machine No-Wait Flowshops with Maximum Lateness Performance Measure

Muberra Allahverdi; Harun Aydilek; Asiye Aydilek; Ali Allahverdi

doi:10.3934/jimo.2020054

Article Contents

2021, Volume 17, Issue 4: 1973-1991. Doi: 10.3934/jimo.2020054

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A better dominance relation and heuristics for Two-Machine No-Wait Flowshops with Maximum Lateness Performance Measure

1.
Department of Mathematical Sciences, Kean University, New Jersey, USA
2.
Mathematics & Natural Sciences Department, Gulf University for Science and Technology, Kuwait
3.
Economics & Finance Department, Gulf University for Science and Technology, Kuwait
4.
Department of Industrial and Management Systems Engineering, Kuwait University, Kuwait

^* Corresponding author: Muberra Allahverdi
^* Corresponding author: Muberra Allahverdi

Received: July 31, 2019

Revised: August 31, 2019

Early access: March 2020

Published: July 2021

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Abstract

In this paper, we consider a manufacturing system with two-machine no-wait flowshop scheduling problem where setup times are uncertain. The problem with the performance measure of maximum lateness was addressed in the literature (Computational and Applied Mathematics 37, 6774-6794) where dominance relations were proposed. We establish a new dominance relation and show that the new dominance relation is, on average, about 90$ \% $ more efficient than the existing ones. Moreover, since it is highly unlikely to find optimal solutions for problems of reasonable size by utilizing dominance relations and since there exist no heuristic in the literature for the problem, we propose constructive heuristics to solve real life problems. We conduct extensive computational experiments to evaluate the proposed heuristics. Computational experiments indicate that the performance of the worst proposed heuristic is at least 20$ \% $ better than a benchmark solution. Furthermore, they also indicate that the best proposed heuristic is about 130$ \% $ better than the worst one. The average CPU time of the heuristics is significantly less than a second.

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Mathematics Subject Classification: 90B36.

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Figure 1. The average errors of the heuristics with respect to n

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Figure 2. The average errors of the heuristics with respect to R

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Figure 3. The average errors of the heuristics with respect to T

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Figure 4. The average errors of the heuristics with respect to $ \Delta $

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Figure 5. The average errors of the heuristics with respect to n (Positive Exponential)

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Figure 6. The average errors of the heuristics with respect to n (Negative Exponential)

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Figure 7. The average errors of the heuristics with respect to n (Uniform)

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Figure 8. The average errors of the heuristics with respect to n (Normal)

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Figure 9. The average errors of the heuristics with respect to n (Positive Linear)

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Figure 10. The average errors of the heuristics with respect to n (Negative Linear)

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Figure 11. The average improvements of the dominance relation with respect to n

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Table 1. Evaluation of the newly established dominance relation

			T=0.25			T=0.5			T=0.75
n	D	R=0.25	R=0.5	R=0.75	R=0.25	R=0.5	R=0.75	R=0.25	R=0.5	R=0.75
	5	122.1	139	69.1	133.2	85.5	112.1	80.8	108.7	96.3
30	10	152	156.2	133.6	153.8	93.2	131.8	125	85.3	96.7
	20	137.9	96	125	172	183	174.7	135.2	75.7	110.6

	5	106.6	72.1	98.6	55.5	94.6	84.2	94.4	103.4	80.7
40	10	100	66.8	90.2	116.8	112.4	91.7	68.5	86	106
	20	124.2	141.9	140	90.5	56.1	103.7	74.5	68.6	82.5

	5	75.7	88.2	80.5	84.4	91.2	95.6	88.1	59.7	95.1
50	10	66.5	68	107.4	118.3	87.8	91.2	76.6	61.6	83.4
	20	63.5	121.4	106.3	74.2	40	140	69.2	75.7	76.8

	5	100	106.1	107.7	64	71.3	71.1	78	56.5	57.8
60	10	66.2	84.3	79.7	63.5	143	80.9	46.8	94.7	70.8
	20	106.5	89.1	62.1	95.9	81.1	76.4	65.1	58.4	75.7

	5	78.4	52.2	54.4	65.9	78.3	62.5	70.3	110.8	68.2
70	10	77.1	81.9	133.5	57.8	52.4	74.7	66.6	72.6	89.7
	20	57.9	47.6	108.3	88.3	96.2	40.2	77	54.9	61.8

	5	96.6	91.5	82.1	80.6	84.2	85.1	82.3	87.8	79.6
Avg	10	92.4	91.4	108.9	102	97.7	94.1	76.7	80	89.3
	20	98	99.2	108.3	104.2	91.3	107	84.2	66.7	81.5

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Table 2. Errors of Heuristics when $ \Delta = 5 $

			T=0.25			T=0.5			T=0.75
n	Heuristic	R=0.25	R=0.5	R=0.75	R=0.25	R=0.5	R=0.75	R=0.25	R=0.5	R=0.75
	CH1	1.21	1.81	6.67	0.4	1.09	1.47	0.32	0.55	0.58
	CH2	1.2	1.54	6.58	0.32	1.26	0.82	0.28	0.64	0.37
30	CH3	0.91	2.4	3.05	0.51	1.08	1.44	0.12	0.67	0.26
	CH4	0.81	1.9	2.07	0.54	0.74	1.3	0.17	0.31	0.23
	CH5	0.99	2.13	1.98	0.55	0.63	1.33	0.26	0.48	0.52

	CH1	0.99	5.3	13.96	0.48	0.96	2.86	0.26	0.56	0.92
	CH2	0.78	4.4	13.05	0.4	1.06	1.88	0.4	0.34	0.87
40	CH3	0.76	4.55	10.53	0.5	1.27	1.53	0.26	0.46	0.37
	CH4	0.59	2.72	3.95	0.33	0.85	1.03	0.27	0.39	0.34
	CH5	0.97	2.9	3.7	0.4	0.86	0.96	0.32	0.63	0.47

	CH1	1.19	4.49	11.86	0.33	1.76	2.28	0.29	0.69	1.2
	CH2	0.92	3.04	11.18	0.39	1.91	1.69	0.24	0.47	1.3
50	CH3	0.77	4	9.17	0.31	1.36	1.07	0.27	0.52	0.94
	CH4	0.61	2.76	4.17	0.3	0.61	0.66	0.23	0.46	0.33
	CH5	1.01	2.68	4.27	0.53	0.65	0.8	0.36	0.59	0.34

	CH1	1.23	4.26	6.22	0.71	1.37	2.1	0.38	0.69	1.07
	CH2	1.26	3.57	4.93	0.59	1.11	1.76	0.34	0.71	0.82
60	CH3	0.98	3.85	4.68	0.59	0.82	1.05	0.4	0.72	0.91
	CH4	0.69	2.54	4.17	0.35	0.79	0.61	0.21	0.4	0.7
	CH5	0.72	2.64	4.8	0.47	0.96	0.64	0.23	0.54	0.71

	CH1	1.28	3.95	9.17	0.48	0.76	1.49	0.33	0.56	0.77
	CH2	1.08	3.7	4.04	0.53	0.78	1.56	0.3	0.5	0.57
70	CH3	1.18	2.62	3.2	0.36	0.79	1.32	0.3	0.38	0.57
	CH4	0.87	1.9	3.11	0.3	0.61	0.64	0.28	0.34	0.38
	CH5	0.95	2.75	2.58	0.37	0.66	0.67	0.26	0.67	0.62

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Table 3. Errors of Heuristics when $ \Delta = 10 $

			T=0.25			T=0.5			T=0.75
n	Heuristic	R=0.25	R=0.5	R=0.75	R=0.25	R=0.5	R=0.75	R=0.25	R=0.5	R=0.75
	CH1	1.48	4.64	15.59	0.79	1.46	2.05	0.43	1.02	2.82
	CH2	1.36	4.14	10.67	0.77	0.92	1.52	0.33	0.97	2.73
30	CH3	1.39	3.33	8.59	0.54	0.72	1.47	0.37	0.69	2.23
	CH4	0.8	2.64	4.42	0.56	0.51	1.1	0.35	0.83	0.81
	CH5	1.64	2.53	3.96	0.7	0.73	0.99	0.42	0.91	0.82

	CH1	1.44	3.81	23.96	0.59	1.91	2.04	0.51	1.12	1.41
	CH2	1.58	2.74	17.86	0.62	1.4	2.22	0.4	0.77	1.27
40	CH3	1.06	2.16	13.1	0.51	1.56	1.97	0.18	0.83	1.01
	CH4	0.85	2.31	1.92	0.56	0.65	1.4	0.28	0.57	0.44
	CH5	1.42	3.14	3.68	0.69	0.87	1.7	0.36	0.53	0.55

	CH1	1.47	5.64	15.22	0.68	1.97	2.68	0.42	0.89	2.04
	CH2	1.32	5.29	11.23	0.4	1.41	2.37	0.32	0.59	1.64
50	CH3	0.85	4.06	7.54	0.42	1.27	1.85	0.27	0.74	1.3
	CH4	0.74	3.81	4.92	0.43	0.89	1.26	0.27	0.37	0.92
	CH5	1.23	4.04	9.89	0.63	0.84	1.29	0.52	0.45	0.91

	CH1	1.4	4.32	11.36	0.56	2.63	2.47	0.43	0.84	1.95
	CH2	1.24	5.41	10.53	0.57	1.87	2.23	0.34	0.62	1.41
60	CH3	1.21	2.73	7.78	0.38	1.34	0.97	0.29	0.49	0.96
	CH4	1.27	2.63	2.8	0.37	1.17	0.78	0.32	0.33	0.61
	CH5	1.37	2.43	5.1	0.51	1.16	0.9	0.4	0.65	0.9

	CH1	1.23	5.9	11.68	0.77	1.81	2.68	0.31	1.22	1.2
	CH2	1.05	3.84	8.2	0.59	1.33	2.24	0.25	0.78	0.98
70	CH3	1.08	3.58	5.97	0.44	0.95	1.46	0.24	0.57	0.74
	CH4	0.9	3.1	4.29	0.37	0.69	0.9	0.28	0.54	0.54
	CH5	1.78	3.13	3.71	0.55	1.45	1.58	0.4	0.65	0.69

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Table 4. Errors of Heuristics when $ \Delta = 20 $

			T=0.25			T=0.5			T=0.75
n	Heuristic	R=0.25	R=0.5	R=0.75	R=0.25	R=0.5	R=0.75	R=0.25	R=0.5	R=0.75
	CH1	2.47	6.96	24.94	0.87	3.03	4.50	0.78	1.22	2.51
	CH2	1.38	3.93	8.76	0.54	2.46	3.79	0.66	0.88	1.80
30	CH3	1.22	3.42	12.21	0.56	1.47	3.14	0.47	0.49	1.52
	CH4	0.91	4.07	10.30	0.69	1.51	1.66	0.46	0.69	1.28
	CH5	1.73	4.18	14.58	1.19	1.42	1.33	0.65	0.92	1.23

	CH1	1.71	7.57	19.56	0.94	1.86	4.27	0.64	1.43	2.98
	CH2	1.16	4.79	14.49	0.66	1.06	4.13	0.47	1.00	2.43
40	CH3	1.13	5.53	11.18	0.62	0.74	3.01	0.36	0.88	1.35
	CH4	1.07	4.63	7.06	0.63	0.82	1.75	0.37	0.55	0.65
	CH5	1.67	5.90	7.55	0.84	1.67	1.77	0.57	0.89	0.85

	CH1	2.39	4.18	14.61	1.07	2.78	4.73	0.82	1.63	2.65
	CH2	1.51	4.17	12.04	0.73	2.86	3.57	0.50	1.17	2.11
50	CH3	1.32	2.37	7.56	0.71	1.55	2.27	0.31	0.76	1.16
	CH4	1.30	3.05	4.35	0.67	0.83	1.52	0.41	0.65	1.00
	CH5	2.24	4.15	8.13	0.81	1.43	1.65	0.77	0.95	1.17

	CH1	1.97	7.43	20.36	1.11	2.23	4.04	0.56	1.64	2.99
	CH2	1.38	5.07	15.71	0.62	1.66	2.71	0.32	1.16	2.34
60	CH3	1.25	5.21	6.99	0.47	1.20	2.20	0.39	0.67	1.75
	CH4	1.13	2.82	6.56	0.48	0.82	1.33	0.33	0.76	0.68
	CH5	1.77	4.61	13.20	1.00	1.61	1.81	0.71	1.06	0.85

	CH1	2.06	6.87	17.54	0.87	2.87	4.90	0.69	1.32	2.97
	CH2	1.32	4.18	12.66	0.38	1.90	2.90	0.40	0.83	1.85
70	CH3	1.11	2.76	7.95	0.42	1.61	2.17	0.37	0.69	1.20
	CH4	1.01	2.74	2.49	0.45	1.31	1.37	0.41	0.78	0.79
	CH5	2.21	4.74	6.85	0.94	1.75	2.02	0.58	0.86	0.98

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