Monitoring bivariate processes with synthetic control charts based on sample ranges

Machado, Marcela; Costa, Antonio; Simões, Felipe Domingues

doi:10.1590/1806-9649-2022v29e6822

Marcela Machado

presented idea

developed the theory

performed the computations

wrote the paper

Universidade Estadual Paulista - UNESP, Faculdade de Engenharia e Ciências - FEG, Departamento de Produção, Guaratinguetá, SP, Brasil. E-mail: marcela.freitas@unesp.br; fdsimoes94@yahoo.com.br

http://orcid.org/0000-0002-0621-6932

Antonio Costa

presented idea

developed the theory

performed the computations

Universidade Federal de Itajubá - UNIFEI, Instituto de Engenharia de Produção e Gestão - IEPG, Itajubá, MG, Brasil. E-mail: fbrancosta@gmail.com

http://orcid.org/0000-0001-6620-4573

Felipe Domingues Simões

wrote the paper

Universidade Estadual Paulista - UNESP, Faculdade de Engenharia e Ciências - FEG, Departamento de Produção, Guaratinguetá, SP, Brasil. E-mail: marcela.freitas@unesp.br; fdsimoes94@yahoo.com.br

http://orcid.org/0000-0001-8876-8302

Authors contribution

Marcela Machado and Antonio Costa conceived of the presented idea, developed the theory and performed the computations. Marcela Machado and Felipe Simões wrote the paper in consultation with Antonio Costa.

	Chart		\|S\|	Std	Syn	PSyn	Syn	PSyn	Syn	PSyn	Syn	PSyn
	L				2	2	3	3	5	5	7	7
	CL				5.50		5.50		5.50		5.50
	WL		4.72	5.37	4.40	4.14	4.54	4.30	4.66	4.44	4.74	4.52
$a_{1}$	$a_{2}$
1.00		1.00	370.40	370.38	370.37	370.28	370.35	370.32	370.34	370.27	370.34	370.29
1.25		1.00	74.13	47.12	41.74	49.24	40.29	43.23	39.23	39.20	38.87	38.12
1.50		1.00	26.70	11.85	10.35	13.12	10.00	11.35	9.83	10.43	9.85	10.52
1.75		1.00	13.32	5.22	4.70	6.06	4.60	5.42	4.60	5.18	4.64	5.48
2.00		1.00	8.11	3.15	2.93	3.78	2.90	3.50	2.92	3.44	2.95	3.82
2.50		1.00	4.23	1.82	1.78	2.24	1.78	2.17	1.79	2.17	1.80	2.68
1.25		1.25	21.90	26.15	21.18	22.54	20.37	19.93	19.94	18.52	19.90	18.34
1.50		1.50	5.61	6.64	5.49	6.36	5.35	5.75	5.37	5.57	5.44	5.77
1.75		1.75	2.74	3.10	2.76	3.38	2.74	3.18	2.78	3.18	2.82	3.45
2.00		2.00	1.85	2.00	1.89	2.38	1.89	2.31	1.91	2.32	1.92	2.65
2.50		2.50	1.28	1.31	1.30	1.63	1.31	1.62	1.31	1.62	1.31	2.11

	Chart		\|S\|	Std	Syn	PSyn	Syn	PSyn	Syn	PSyn	Syn	PSyn
	L				2	2	3	3	5	5	7	7
	CL				5.50		5.50		5.50		5.50
	WL		6.56	4.94	3.68	3.62	3.84	3.78	4.00	3.94	4.08	4.03
$a_{1}$	$a_{2}$
1.00		1.00	370.40	370.32	370.29	370.37	370.29	370.26	370.33	370.40	370.30	370.37
1.25		1.00	113.48	64.71	63.15	75.94	58.34	66.60	54.70	59.88	53.18	57.08
1.50		1.00	51.57	19.00	18.53	25.31	16.99	21.03	15.98	18.47	15.68	17.62
1.75		1.00	29.36	8.81	8.67	12.65	8.09	10.46	7.78	9.33	7.73	9.04
2.00		1.00	19.25	5.32	5.31	8.05	5.05	6.74	4.94	6.15	4.96	6.05
2.50		1.00	10.65	2.92	2.99	4.75	2.92	4.12	2.91	3.90	2.93	3.89
1.25		1.25	44.05	36.72	31.53	36.39	28.91	31.70	27.25	28.81	26.74	27.84
1.50		1.50	13.86	10.57	9.19	11.49	8.53	9.89	8.25	9.14	8.26	9.02
1.75		1.75	6.90	5.05	4.63	6.11	4.41	5.36	4.38	5.11	4.43	5.12
2.00		2.00	4.39	3.17	3.05	4.21	2.97	3.78	2.98	3.68	3.02	3.72
2.50		2.50	2.58	1.88	1.92	2.86	1.91	2.66	1.92	2.65	1.93	2.67

	Chart	\|S\|	Std	Syn	PSyn	Syn	PSyn	Syn	PSyn	Syn	PSyn
	L			2	2	3	3	5	5	7	7
	CL			6.00		6.00		6.00		6.00
	WL	5.47	5.60	4.37	4.29	4.52	4.45	4.66	4.59	4.74	4.67
$a_{1}$	$a_{2}$
1.00	1.00	700.00	699.93	699.80	699.76	699.80	699.71	699.81	699.83	699.99	699.77
1.25	1.00	120.46	71.33	62.76	75.77	57.62	65.16	53.79	57.81	52.25	54.88
1.50	1.00	39.28	15.61	13.52	18.04	12.44	14.98	11.83	13.33	11.70	12.88
1.75	1.00	18.28	6.33	5.71	8.01	5.40	6.79	5.29	6.28	5.33	6.22
2.00	1.00	10.56	3.62	3.41	4.99	3.31	4.35	3.30	4.16	3.34	4.18
2.50	1.00	5.13	1.97	1.97	3.08	1.96	2.82	1.97	2.79	1.98	2.81
1.25	1.25	31.60	38.83	28.99	32.54	26.53	28.17	25.05	25.62	24.67	24.86
1.50	1.50	7.02	8.59	6.70	8.14	6.30	7.13	6.19	6.76	6.27	6.78
1.75	1.75	3.17	3.69	3.21	4.18	3.12	3.79	3.15	3.74	3.21	3.80
2.00	2.00	2.03	2.24	2.14	2.97	2.12	2.78	2.14	2.79	2.17	2.83
2.50	2.50	1.33	1.38	1.42	2.23	1.42	2.17	1.43	2.17	1.43	2.18

	Chart		\|S\|	Std	Syn		Syn		Syn		Syn
	L				2	2	3	3	5	5	7	7
	CL				5.50		5.50		5.50		5.50
	WL		8.05	5.18	3.91	3.78	4.06	3.95	4.21	4.10	4.30	4.19
$a_{1}$	$a_{2}$
1.00		1.00	700.00	699.98	699.97	699.77	699.87	699.82	699.87	699.80	699.98	699.74
1.25		1.00	188.79	99.36	92.17	117.71	86.27	101.87	81.45	90.12	79.29	85.03
1.50		1.00	78.82	25.53	23.23	34.05	21.56	27.74	20.38	23.86	19.98	22.50
1.75		1.00	42.24	10.94	10.05	15.62	9.46	12.67	9.12	11.09	9.06	10.66
2.00		1.00	26.45	6.28	5.88	9.43	5.63	7.76	5.52	6.97	5.53	6.82
2.50		1.00	13.74	3.25	3.16	5.24	3.09	4.48	3.09	4.21	3.11	4.20
1.25		1.25	66.19	55.16	45.10	53.22	41.76	45.69	39.41	40.80	38.57	39.01
1.50		1.50	18.38	13.97	11.37	14.64	10.62	12.37	10.24	11.23	10.22	10.98
1.75		1.75	8.49	6.18	5.29	7.20	5.06	6.21	5.01	5.84	5.06	5.83
2.00		2.00	5.14	3.68	3.33	4.73	3.24	4.19	3.26	4.05	3.30	4.09
2.50		2.50	2.85	2.06	2.00	3.05	1.99	2.82	2.00	2.80	2.02	2.83

case	(a₁;a₂)	case	(a₁;a₂)	case	(a₁;a₂)
a	(1.25;1.00)	e	(2.50;1.00)	h	(1.75;1.75)
b	(1.50;1.00)	f	(1.25;1.25)	i	(2.00;2.00)
c	(1.50;1.00)	g	(1.50;1.50)	j	(2.50;2.50)
d	(2.00;1.00)	-	-	-	-

				3			5
	CL		5.40	5.50	5.60	5.40	5.50	5.60
	WL		4.02	4.00	3.98	4.89	4.66	4.58
$a_{1}$	$a_{2}$
1.00		1.00	370.40	370.33	370.40	370.4	370.34	370.40
1.25		1.00	54.43	54.70	55.10	43.47	39.23	37.76
1.50		1.00	15.86	15.98	16.14	10.75	9.83	9.58
1.75		1.00	7.70	7.78	7.86	4.85	4.60	4.55
2.00		1.00	4.89	4.94	5.00	3.00	2.92	2.92
2.50		1.00	2.87	2.91	2.95	1.79	1.79	1.80
1.25		1.25	27.3	27.25	27.29	23.07	19.94	18.86
1.50		1.50	8.22	8.25	8.30	5.90	5.37	5.22
1.75		1.75	4.34	4.38	4.42	2.90	2.78	2.76
2.00		2.00	2,95	2.98	3.02	1.93	1.91	1.92
2.50		2.50	1.90	1.92	1.95	1.30	1.31	1.32

Sample		Observations					RMAX
Sample		1	2	3	4	5	RMAX
1	$X_{1}$	0.53	-1.83	0.20	0.89	-0.80	2.71
	$X_{2}$	-0.27	-1.71	-0.10	-1.30	-1.55
2		-1.63	-0.86	-0.25	0.78	-1.30	2.42
		-0.54	-0.51	-0.93	0.14	-0.93
3		-0.27	0.12	-0.10	-0.73	-1.05	1.37
		-0.66	0.71	-0.18	0.22	-0.43
4		-0.27	-0.68	-0.59	-1.60	-0.23	1.57
		-1.01	-0.31	-0.17	-1.38	0.18
5		-0.07	0.77	2.02	-0.56	0.15	2.58
		0.48	-0.27	0.07	-1.66	-0.39
6		2.00	-1.44	0.45	-2.00	-0.04	3.99
		0.70	-1.25	-0.49	0.31	0.26
7		-0.31	1.96	2.66	-0.87	0.21	3.52
		0.34	0.07	0.96	-0.54	-0.57
8		-1.22	-1.02	-1.18	0.02	2.00	3.46
		-0.78	0.10	-1.90	1.56	-0.20
9		-0.96	-0.40	0.71	3.59	0.97	4.54
		-0.17	-2.00	-0.90	0.63	-0.09
10		0.46	1.52	0.28	0.31	0.51	1.24
		-0.76	0.20	-0.24	0.38	0.33
11		-0.73	-3.28	0.87	-0.62	-0.97	4.15
		-2.36	-1.56	0.38	-0.98	-0.93
12		0.84	0.48	2.81	-2.02	-0.23	4.83
		1.07	-0.09	0.04	-0.04	-0.53
13		-1.40	2.30	4.41	-0.58	3.10	5.82
		-1.90	0.79	0.63	-1.20	-0.15

\begin{matrix} 00..00 & 0..001 & 0..010 & 0..100 & ... & 010..0 & 100..0 & S i g n a l \\ 00...00 & A & B & 0 & 0 & ... & 0 & 0 & C \\ 0..001 & 0 & 0 & A & 0 & ... & 0 & 0 & B + C \\ 0..010 & 0 & 0 & 0 & A & ... & 0 & 0 & B + C \\ ... & .. & ... & ... & ... & ... & ... & ... & ... \\ 001..0 & 0 & 0 & 0 & 0 & ... & A & 0 & B + C \\ 010..0 & 0 & 0 & 0 & 0 & ... & 0 & A & B + C \\ 100..0 & A & 0 & 0 & 0 & ... & 0 & 0 & B + C \\ S i g n a l & 0 & 0 & 0 & 0 & ... & 0 & 0 & 1 \end{matrix}

\begin{matrix}  \end{matrix}

(\begin{matrix} A / (A + B) & B / (A + B) & 0 & 0 & ... & 0 & 0 \\ 0 & 0 & 1 & 0 & ... & 0 & 0 \\ 0 & 0 & 0 & 1 & ... & 0 & 0 \\ ... & ... & ... & ... & ... & ... & ... \\ 0 & 0 & 0 & 0 & ... & 1 & 0 \\ 0 & 0 & 0 & 0 & ... & 0 & 1 \\ 1 & 0 & 0 & 0 & ... & 0 & 0 \end{matrix})

F_{n} (w_{1}, w_{2}) = I_{1} + I_{2} = n \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} D_{1}^{n - 1} d F (y_{1}, y_{2}) + n \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} (n - 1) D_{1}^{n - 2} D_{2} d G (y_{1}, y_{2})

F (y_{1}, y_{2}) = \frac{1}{2 π \sqrt{1 - ρ^{2}}} \int_{- \infty}^{y_{1}} \int_{- \infty}^{y_{2}} \exp (\frac{- y_{1}^{2} - 2 ρ y_{1} y_{2} + y_{2}^{2}}{2 (1 - ρ^{2})}) d y_{1} d y_{2}

G (y_{1}, y_{2}) = \frac{1}{2 π} \int_{- \infty}^{y_{1}} \int_{- \infty}^{y_{2}} \exp (\frac{- y_{1}^{2} + y_{2}^{2}}{2}) d y_{1} d y_{2}

[1] Authors contribution

Marcela Machado and Antonio Costa conceived of the presented idea, developed the theory and performed the computations. Marcela Machado and Felipe Simões wrote the paper in consultation with Antonio Costa.

Brasil

Brasil

Monitoring bivariate processes with synthetic control charts based on sample ranges

Gráficos de controle baseado em amplitudes amostrais e regras especiais de decisão para o monitoramento de processos bivariados

Abstract