Live fitting of process data within digital twins of manufacturing to use simulation and optimisation

Figures & data

Figure 1. Production scenario with three machines in stage 1 and one machine in stage 2.

Figure 2. Developed framework for combining simulation and optimization models (Schumacher, 2023).

Table 1. Product parameters data concept for optimisation and simulation (Schumacher & Buchholz, 2020).

	Processing times $p_{\mathit{ilj}}$ and Set-up times		Scrap rate $r_{\mathit{ij}}$
Scenario	Optimization	Simulation	Optimization	Simulation
Machine-related data per product is available	Median per product and machine	Random value from the distribution per product and machine	Average or quantile value per product and machine	Random value from the distribution per product for all machines
Machine-related data per product is not sufficient but product-related data per stage is available	Median for the product overall machines	Random value from the distribution for the product for all machines	Average or quantile value for the product for all machines	Random value from the distribution for the product for all machines
No product-related and machine-related data is available but product-related planned data of the product per stage is available	Planned data of the product per stage	Planned data of the product per stage	–	–
No product-related and machine-related data is available but data for all products of the relevant stage is available	Median of all machines and products	Random value from the distribution of all machines and products	Average or quantile value for all machines and products	Random value from the distribution for all machines and products

Figure 3. Data processing.

Figure 4. Data modeling.

Figure 5. Fitted distributions on a sample data set of machine scrap rates.

Figure 6. Fitted distributions on a sample data set of human worker scrap rates.

Figure 7. Fitted distributions on a sample data set of machine processing times.

Figure 8. Model fitting process on a sample data set of machine processing times with multiple clusters.

Figure 9. Example PP- and QQ-Plots for machine scrap data.

Table 2. Evaluation of KS-test for example scrap rate data sets.

	Machine stage	Human stage
	(Figure 5)	(Figure 6)
Distribution	$\mathit{p}-\mathit{value}$	$\mathit{p}-\mathit{value}$
Exponential	0.38458	0.23483
Log-normal	0.58433	0.42257
Wald	0.48264	0.32111
Fatigue-life	0.55839	0.37582
Gumbel-R	0.33428	0.35412

Table 3. Fitted distributions for scrap rates.

Distribution	Machines Stage 1	Human Stage 1	Machine Stage 2	Human Stage 2
Exponential	2	0	1	1
Log-normal	8	16	31	6
Wald	10	2	9	0
Fatigue-life	10	1	16	0
Gumbel-R	1	0	0	1
Insufficient data	83	20	57	6

Table 4. Evaluation of KS-test for example processing time data sets.

	Machine stage 1	Machine stage 2	Machine stage 2
	(Figure 7)	part 1 (Figure 8)	part 2 (Figure 8)
Distribution	$\mathit{p}-\mathit{value}$	$\mathit{p}-\mathit{value}$	$\mathit{p}-\mathit{value}$
Normal	0.07036	0.03559	0.3989
Log-normal	0.10744	0.03357	0.4211
Triangular	0.01450	0.00096	0.18197
Moyal	0.10228	0.03243	0.30571
Gamma	0.10803	0.02653	0.39121
GMM $\mathcal{K}=2$	0.32781	-	0.51161
GMM $\mathcal{K}=3$	0.32682	-	0.44708

Table 5. Fitted Distributions for Processing Times.

Distribution	Machine stage 1	Machine stage 2
Normal	23	18
Log-normal	1	11
Triangular	12	19
Moyal	6	9
Gamma	14	2
GMM $\mathcal{K}=2$	89	16
GMM $\mathcal{K}=3$	21	2
Insufficient data	1	1

Table 6. Change point detection for varying mean and variance values of the processing times (machine A, article 1).

type of variation	factor	TP	TN	FP	FN	steps 1	steps 2
mean, ${\mu }_j=$	1.01	1	1099800	0	199	–	362.0
	1.02	200	1099799	1	0	39.81	42.58
	1.03	200	1099800	0	0	24.56	24.56
	1.04	200	1099798	2	0	14.22	17.14
	1.05	200	1099799	1	0	13.23	15.89
	1.06	200	1099799	1	0	11.22	12.18
std, ${\sigma }_k=$	1.01	3	1099800	0	197	57.67	–
	1.03	4	1099800	0	196	233.0	–
	1.05	162	1099798	2	38	89.44	70.89
	1.10	200	1099800	0	0	45.14	49.66

Figure 10. Histograms of processing times for different change points (Machine A, Article 1).

Figure 11. Histograms of Processing Times for different Change Points (Machine B, Article 2).

Table 7. Change point detection for varying mean and variance values of the processing times (Machine B, Article 2).

type of variation	factor	TP	TN	FP	FN	steps 1	steps 2
mean, ${\mu }_j=$	1.01	98	1099783	17	102	55.81	529.69
	1.02	198	1099765	35	2	40.13	46.43
	1.03	200	1099772	28	0	27.08	25.7
	1.04	200	1099769	31	0	21.92	20.94
	1.05	200	1099770	30	0	19.26	16.86
	1.06	199	1099761	39	1	19.04	14.49
std, ${\sigma }_k=$	1.20	115	1099779	21	85	87.3	185.41
	1.24	148	1099771	29	52	91.15	191.57
	1.30	155	1099777	23	45	74.4	156.96
	1.36	195	1099773	27	5	43.62	56.80

Figure 12. Histograms of Processing Times for different Change Points (Machine C, Article 3).

Table 8. Change point detection for varying mean and variance values of the processing times (Machine C, Article 3).

type of variation	factor	TP	TN	FP	FN	steps 1	steps 2
mean, ${\mu }_j=$	1.01	107	1099781	19	93	131.78	121.18
	1.02	190	1099776	24	10	42.28	68.49
	1.03	195	1099777	23	5	32.65	35.94
	1.04	200	1099767	33	0	19.92	22.23
	1.05	200	1099777	23	0	16.78	18.74
	1.06	200	1099770	30	0	15.66	16.7
std, ${\sigma }_k=$	1.02	2	1099800	0	198	71.0	–
	1.05	168	1099800	0	32	84.29	70.85
	1.1	200	1099798	2	0	46.19	47.67
	1.2	200	1099800	0	0	20.64	15.8

Figure 13. Average log-likelihood for processing times (Machine B, Article 2).

Figure 14. Comparison of average log-likelihood for processing times (Machine A, Article 1) using the EM algorithm for GMMs.

Figure 15. Histograms of scrap rates for different change points (Machine D, Article 4).

Figure 16. Histograms of scrap rates for different change points (Machine E, Article 5).

Table 9. Change point detection for varying mean and variance values of the scrap rates (Machine D, Article 4).

type of variation	factor	TP	TN	FP	FN	steps 1	steps 2
mean, ${\mu }_j=$	1.02	72	1099767	33	128	99.67	1028.05
	1.04	167	1099755	45	33	44.45	69.58
	1.06	192	1099755	45	8	32.70	40.59
	1.08	197	1099744	56	3	29.6	78.79
	1.10	196	1099743	57	4	27.64	40.57
std, ${\sigma }_k=$	1.02	38	1099766	34	162	139.00	2137.60
	1.06	52	1099772	28	148	111.06	410.94
	1.1	152	1099749	51	48	43.75	250.80
	1.2	187	1099748	52	13	31.19	160.30

Table 10. Change point detection for varying mean and variance values of the scrap rates (Machine E, Article 5).

type of variation	factor	TP	TN	FP	FN	steps 1	steps 2
mean, ${\mu }_j=$	1.01	6	1099796	4	194	158.0	3366.4
	1.02	4	1099792	8	196	-	2698.25
	1.03	28	1099794	6	172	144.08	1730.8
	1.04	137	1099791	9	63	55.15	209.34
	1.05	144	1099784	16	56	58.87	138.02
	1.1	193	1099786	14	7	32.93	45.58
std, ${\sigma }_k=$	1.03	9	1099791	9	191	270.5	2935.0
	1.05	147	1099790	10	53	72.83	242.17
	1.1	199	1099784	16	1	34.8	92.22

Figure 17. Average log-likelihood for scrap rates (Machine D, Article 4).

Figure 18. Modified schedule due to change point detection.

Schumacher, C. (2023). Anpassungsfähige Maschinenbelegungsplanung eines praxisorientierten hybriden Flow Shops (1st ed.). Springer Fachmedien Wiesbaden and Imprint Springer Vieweg. https://doi.org/10.1007/978-3-658-41170-1 ISBN 9783658411701

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Schumacher, C., & Buchholz, P. (2020). Scheduling algorithms for a hybrid flow shop under uncertainty. Algorithms, 130(11), 277. https://doi.org/10.3390/a13110277 PII: a13110277.

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