Performance of a concurrent parallel production system through new operating curves of Six Sigma metrics

Fontalvo Herrera, Tomás José; Banquez Maturana, Ana Gabriela; Mendoza Villero, Katherin

doi:10.1590/0103-6513.20230040

Tomás José Fontalvo Herrera ^* * tfontalvoh@unicartagena.edu.co

University of Cartagena, Cartagena, Colombia

https://orcid.org/0000-0003-4642-9251

Ana Gabriela Banquez Maturana

University of Cartagena, Cartagena, Colombia

https://orcid.org/0000-0002-8354-6396

Katherin Mendoza Villero

University of Cartagena, Cartagena, Colombia

https://orcid.org/0009-0005-4974-1242

* tfontalvoh@unicartagena.edu.co

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Figure 1
Diagram of the parallel productive system. U*: units entering externally to the process. Where: * is a simple symbology that does not alter the process. Source: The authors.

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Figure 2
Diagram of the system productive in parallel with values. Source: The authors.

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Figure 3
Curve of operation of quantity of defects per million opportunities (DPMO) for each level sigma (Z). Source: The authors.

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Figure 4
Operating curve percentage of yield (Y) for each sigma level (Z). Source: The authors.

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Figure 5
Defect percentage operation curve (n)for each sigma level (Z). Source: The authors.

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Figure 6
Operating curve of the number of defects per million opportunities (DPMO) for each sigma level (Z). Source: The authors.

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Figure 7
Performance percentage operation curve (Y) for each sigma level (Z). Source: The authors.

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Figure 8
Defect percentage operation curve (n)for each sigma level (Z). Source: The authors.

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Figure 9
Curve of operation of quantity of defects per million opportunities (DPMO) for each level sigma (Z). Source: The authors.

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Figure 10
Yield percentage operation curve (Y) for each sigma level (Z). Source: The authors.

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Figure 11
Defect percentage operation curve (n)for each sigma level (Z). Source: The authors.

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Figure 12
Parts per million defect quantity (DPMO) operating curve for each sigma level (Z). Source: The authors.

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Figure 13
Performance percentage operation curve (Y) for each sigma level (Z). Source: The authors.

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Figure 14
Defect percentage operation curve (n)for each sigma level (Z). Source: The authors.

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Figure 15
Operating limit of parts per million defects (DPMO) for each sigma level (Z). Source: The authors.

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Figure 16
Yield percentage operation curve (Y) for each sigma level (Z). Source: The authors.

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Figure 17
Defect percentage operation curve (n) for each sigma level (Z). Source: The authors.

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Figure 15
Operating limit of parts per million defects (DPMO) for each sigma level (Z). Source: The authors.

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Figure 16
Yield percentage operation curve (Y) for each sigma level (Z). Source: The authors.

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Figure 17
Defect percentage operation curve (n) for each sigma level (Z). Source: The authors.

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Table 1
Six Sigma metrics used to compare system performance.

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Table 2
Performance criteria for serial and parallel systems.

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Table 3
Result of the Six Sigma metrics applied to process 1 in the production system in parallel.

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Table 4
Result of the metrics Seis Sigma applied to process 2 in the production system in parallel.

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Table 5
Result of the metrics Seis Sigma applied to process 3 in the production system in parallel.

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Table 6
Result of the metrics Six Sigma applied to process 4 in the production system in parallel.

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Table 7
Overall system result.

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Table 8
Reduced process performance.

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Table 9
Improved process performance.

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(1)

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(2)

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(3)

Metric	Definition
DPMO	Number of defects per million opportunities
U	Number of critical units reviewed
O	Opportunity for error per unit
Z	Sigma level
n	Number of nonconformities or faults present in the process
And	Process performance

Sigma level ( $Z)$	Performance
< 3.0	Deficient
3.0 ≤ . $Z$ . ≤ $Z$ 3.5	Acceptable
3.5 <. $Z$ . ≤ 4.6	Well
$Z$ > 4.6	Excellent

Process 1
$U_{1}$	DPMO	$n_{1}$	$Z_{1}$	$Y_{1}$
175,171	3,996	700	4.150	99.60%
176,376	3,969	700	4.152	99.60%
177,146	3,952	700	4.154	99.60%
177,789	3,937	700	4.155	99.61%
178,320	3,926	700	4.156	99.61%
179,105	3,908	700	4.166	99.61%
179,609	3,897	700	4.157	99.61%
179,784	3,894	700	4.158	99.61%
179,920	3,891	700	4.159	99.61%
179,954	3,890	700	4.159	99.61%
179,976	3,889	700	4.159	99.61%
179,996	3,889	700	4.159	99.61%

Process 2
$U_{2}$	DPMO	$n_{2}$	$Z_{2}$	$Y_{2}$
350,000	27,592	9,657	3.421	97.24%
350,000	20,709	7,248	3.542	97.93%
350,000	16,308	5,708	3.638	98.37%
350,000	12,634	4,422	3.738	98.74%
350,000	9,599	3,360	3.841	99.04%
350,000	5,113	1,789	4.066	99.49%
350,000	2,233	782	4.340	99.78%
350,000	1,234	432	4.523	99.88%
350,000	458	160	4.811	99.95%
350,000	262	92	4.964	99.97%
350,000	140	49	5.130	99.99%
350,000	24	8	5.563	99.999%

Process 3
$U_{3}$	DPMO	$n_{3}$	$Z_{3}$	$Y_{3}$
176,171	3,406	600	4.203	99.66%
177,376	3,383	600	4.205	99.66%
178,146	3,368	600	4.207	99.66%
178,789	3,356	600	4.208	99.66%
179,320	3,346	600	4.209	99.67%
180,105	3,331	600	4.210	99.67%
180,609	3,322	600	4.211	99.67%
180,784	3,319	600	4.212	99.67%
180,920	3,316	600	4.212	99.67%
180,954	3,316	600	4.212	99.67%
180,976	3,315	600	4.212	99.67%
180,996	3,315	600	4.212	99.67%

Process 4
$U_{4}$	DPMO	. $n_{4}$ .	$Z_{4}$	$Y_{4}$
360,043	26,823	9,657	3.433	97.32%
362,452	19,997	7,248	3.556	98.00%
363,992	15,681	5,708	3.654	98.43%
365,278	12,105	4,422	3.754	98.79%
366,340	9,171	3,360	3.858	99.08%
367,911	4,864	1,789	4.083	99.51%
368,918	2,119	782	4.356	99.79%
369,268	1,170	432	4.539	99.88%
369,540	434	160	4.826	99.96%
369,608	248	92	4.978	99.98%
369,651	133	49	5.144	99.99%
369,692	23	8	5.576	99.999%

Overall result
$Z_{G}$	$D P M O_{G}$	$n_{G}$	$Y_{G}$	$U_{s}$
3.1	55,565	20,657	94.44%	350.185
3.2	45,132	15,796	95.49%	355.204
3.3	36,329	12,715	96.37%	358.285
3.4	28,982	10,144	97.10%	360.856
3.5	22,913	8,019	97.71%	362.981
3.7	13,939	4,879	98.61%	366.121
3.9	8,180	2,863	99.18%	368.137
4	6,183	2,164	99.38%	368.836
4.1	4,631	1,621	99.54%	369.379
4.13	4,239	1,484	99.58%	369.516
4.15	3,994	1,398	99.60%	369.602
4.17	3,762	1,317	99.62%	369.683

Process	U	DPMO	n	Z	Y
1	175,171	3,996	700	4.150	99.60%
2	350,000	27,592	9,657	3.421	97.24%
3	176,171	3,406	600	4.203	99.66%
4	360,043	26.823	9.657	3.433	97.32%

Process	U	DPMO	n	Z	Y
1	179,996	3,889	700	4.159	99.61%
2	350,000	27,592	9,657	3.421	97.24%
3	180,996	3,315	600	4.212	99.67%
4	369,692	23	8	5.576	99.999%

[1] Source: The authors.

Performance of a concurrent parallel production system through new operating curves of Six Sigma metrics

Abstract