Models of Two Types of Data in QM Based on Control Chart and PCA
备份一下质量管理的笔记，顺便练习一下Markdown与KaTeX排版。
关键词： 质量管理 计量值 属性值 控制图 过程能力分析
Keywords: Quality Management, Variable Data, Attribute Data, Control Chart, Process Capability Analysis
Data Type & Fundamental Models
Variable Date
Examples
- Cable wire diameter data
- Piston ring diameter data
- Connector pin data
Statistics Models
X
∼
N
(
μ
,
σ
)
X \sim N \left( \mu , \sigma \right)
X∼N(μ,σ)
μ
^
=
x
‾
σ
^
=
{
S
s
k
e
w
e
d
  
d
i
s
t
r
u
b
u
t
i
o
n
R
‾
/
d
2
S
/
c
4
}
u
n
b
i
a
s
e
d
  
d
i
s
t
r
u
b
u
t
i
o
n
\hat{\mu} = \overline{x} \quad \hat{\sigma} = \left\{ \begin{array}{lr} S & skewed \; distrubution \\ \left. \begin{array}{l} \overline{R} / d_2 \\ S / c_4 \end{array} \right\} & unbiased \; distrubution \end{array}\right.
μ^=xσ^=⎩⎨⎧SR/d2S/c4}skeweddistrubutionunbiaseddistrubution
What if non-normal
- Central limit theorems
X ‾ = x 1 + x 2 + ⋯ + x n n \overline{X}=\frac{x_1+x_2+\dots+x_n}{n} X=nx1+x2+⋯+xn
n ≥ 5 X ‾ ∼ N ( μ X ‾ , σ X ‾ ) n \ge 5 \quad \overline{X} \sim N \left( \mu_{\overline{X}} , \sigma_{\overline{X}} \right) n≥5X∼N(μX,σX)
w h e r e μ X ‾ = μ X , σ X ‾ = σ X / n where \quad \mu_{\overline{X}} = \mu_X , \sigma_{\overline{X}} = \sigma_X / \sqrt{n} whereμX=μX,σX=σX/n - Box-Cox Transformation
- Johnson Transformation
Number of defective units - Attribute Data
Examples
- Fat content data (Number of overproof)
- Engine winding reliability data
- Muffler reliability data
Statistics Models
X
∼
B
(
n
,
p
)
X \sim B \left( n , p \right)
X∼B(n,p)
P
(
X
=
d
)
=
C
n
k
p
d
q
n
−
d
E
(
X
)
=
n
p
‾
D
(
X
)
=
n
p
‾
(
1
−
p
‾
)
P(X=d)=C_n^k p^d q^{n-d} \quad E(X)=n \overline{p} \quad D(X)=n \overline{p} (1-\overline{p})
P(X=d)=Cnkpdqn−dE(X)=npD(X)=np(1−p)
What if non-normal
n p ≥ 5 np \ge 5 np≥5
Number of defects - Attribute Data
Examples
- Customer complaints data
- Television defects data
- Post office customer visits data
Statistics Models
X
∼
P
(
λ
)
X \sim P(\lambda)
X∼P(λ)
P
(
X
=
d
)
=
λ
d
d
​
exp
−
λ
E
(
X
)
=
λ
D
(
X
)
=
λ
P(X=d)=\frac{\lambda^d}{d\!}\exp^{-\lambda} \quad E(X)=\lambda \quad D(X)=\lambda
P(X=d)=dλdexp−λE(X)=λD(X)=λ
What if non-normal
λ ≥ 5 \lambda \ge 5 λ≥5
- Redefine ``Unite’’ :
- DPU ⇒ \Rightarrow ⇒ DPHU ⇒ \Rightarrow ⇒ DPKU
Control Chart
Variable Data
X ‾ − R \overline{X}-R X−R Control Chart
C
L
X
‾
=
X
‾
‾
=
X
‾
1
+
X
‾
2
+
X
‾
3
+
⋯
+
X
‾
K
K
CL_{\overline{X}}=\overline{\overline{X}} =\frac{\overline{X}_1+\overline{X}_2+\overline{X}_3+\dots+\overline{X}_K}{K}
CLX=X=KX1+X2+X3+⋯+XK
σ
X
‾
=
σ
n
=
R
‾
/
d
2
n
A
2
=
3
1
d
2
n
\sigma_{\overline{X}}=\frac{\sigma}{\sqrt{n}} =\frac{\overline{R}/d_2}{\sqrt{n}} \quad A_2=3\frac{1}{d_2\sqrt{n}}
σX=n
σ=n
R/d2A2=3d2n
1
U
C
L
X
‾
=
X
‾
‾
+
3
σ
X
‾
=
X
‾
‾
+
3
1
d
2
n
R
‾
=
X
‾
‾
+
A
2
R
‾
UCL_{\overline{X}}=\overline{\overline{X}} +3\sigma_{\overline{X}} =\overline{\overline{X}}+3\frac{1}{d_2\sqrt{n}}\overline{R} =\overline{\overline{X}}+A_2\overline{R}
UCLX=X+3σX=X+3d2n
1R=X+A2R
L
C
L
X
‾
=
X
‾
‾
−
3
σ
X
‾
=
X
‾
‾
−
3
1
d
2
n
R
‾
=
X
‾
‾
−
A
2
R
‾
LCL_{\overline{X}}=\overline{\overline{X}} -3\sigma_{\overline{X}} =\overline{\overline{X}}-3\frac{1}{d_2\sqrt{n}}\overline{R} =\overline{\overline{X}}-A_2\overline{R}
LCLX=X−3σX=X−3d2n
1R=X−A2R
C
L
R
=
R
‾
=
R
1
+
R
2
+
R
3
+
⋯
+
R
K
K
CL_{R}=\overline{R} =\frac{R_1+R_2+R_3+\dots+R_K}{K}
CLR=R=KR1+R2+R3+⋯+RK
D
4
=
1
+
3
d
2
d
3
D
3
=
1
−
3
d
2
d
3
D_4=1+3\frac{d_2}{d_3} \quad D_3=1-3\frac{d_2}{d_3}
D4=1+3d3d2D3=1−3d3d2
U
C
L
R
=
R
‾
+
3
σ
R
=
R
‾
+
3
d
3
σ
=
R
‾
+
3
d
3
R
‾
d
2
=
(
1
+
3
d
2
d
3
)
R
‾
=
D
4
R
‾
UCL_R=\overline{R}+3\sigma_R =\overline{R}+3d_3\sigma =\overline{R}+3d_3\frac{\overline{R}}{d_2} =\left(1+3\frac{d_2}{d_3}\right)\overline{R} =D_4\overline{R}
UCLR=R+3σR=R+3d3σ=R+3d3d2R=(1+3d3d2)R=D4R
L
C
L
R
=
R
‾
−
3
σ
R
=
R
‾
−
3
d
3
σ
=
R
‾
−
3
d
3
R
‾
d
2
=
(
1
−
3
d
2
d
3
)
R
‾
=
D
3
R
‾
LCL_R=\overline{R}-3\sigma_R =\overline{R}-3d_3\sigma =\overline{R}-3d_3\frac{\overline{R}}{d_2} =\left(1-3\frac{d_2}{d_3}\right)\overline{R} =D_3\overline{R}
LCLR=R−3σR=R−3d3σ=R−3d3d2R=(1−3d3d2)R=D3R
X ‾ − S \overline{X}-S X−S Control Chart
When the sample size is large(n>6 or n
≥
\ge
≥ 10(other terms)) or varible, estimate
S
2
S^2
S2 with
σ
2
\sigma^2
σ2 :
S
2
=
∑
i
=
1
n
(
x
i
−
x
‾
)
2
n
−
1
S^2= \frac{\sum_{i=1}^{n}\left(x_i-\overline{x}\right)^2}{n-1}
S2=n−1∑i=1n(xi−x)2, E(S)=
c
4
c_4
c4
σ
\sigma
σ, Std(S)=
σ
\sigma
σ
1
−
c
4
2
\sqrt{1-c_4^2}
1−c42
C
L
X
‾
=
X
‾
‾
CL_{\overline{X}}=\overline{\overline{X}}
CLX=X
U
C
L
X
‾
=
X
‾
‾
+
3
σ
X
‾
=
X
‾
‾
+
3
σ
n
=
X
‾
‾
+
3
S
‾
c
4
n
=
X
‾
‾
+
A
3
S
‾
UCL_{\overline{X}}=\overline{\overline{X}} +3\sigma_{\overline{X}} =\overline{\overline{X}} +3\frac{\sigma}{\sqrt{n}} =\overline{\overline{X}} +3\frac{\overline{S}}{c_4\sqrt{n}} =\overline{\overline{X}}+A_3\overline{S}
UCLX=X+3σX=X+3n
σ=X+3c4n
S=X+A3S
L
C
L
X
‾
=
X
‾
‾
−
3
σ
X
‾
=
X
‾
‾
−
3
σ
n
=
X
‾
‾
−
3
S
‾
c
4
n
=
X
‾
‾
−
A
3
S
‾
LCL_{\overline{X}}=\overline{\overline{X}} -3\sigma_{\overline{X}} =\overline{\overline{X}} -3\frac{\sigma}{\sqrt{n}} =\overline{\overline{X}} -3\frac{\overline{S}}{c_4\sqrt{n}} =\overline{\overline{X}}-A_3\overline{S}
LCLX=X−3σX=X−3n
σ=X−3c4n
S=X−A3S
U
C
L
S
=
S
‾
+
3
σ
1
−
c
4
2
=
S
‾
+
3
S
‾
c
4
1
−
c
4
2
=
B
6
S
‾
UCL_S=\overline{S}+3\sigma\sqrt{1-c_4^2} =\overline{S}+3\frac{\overline{S}}{c_4}\sqrt{1-c_4^2}=B_6\overline{S}
UCLS=S+3σ1−c42
=S+3c4S1−c42
=B6S
L
C
L
S
=
S
‾
−
3
σ
1
−
c
4
2
=
S
‾
−
3
S
‾
c
4
1
−
c
4
2
=
B
5
S
‾
LCL_S=\overline{S}-3\sigma\sqrt{1-c_4^2} =\overline{S}-3\frac{\overline{S}}{c_4}\sqrt{1-c_4^2}=B_5\overline{S}
LCLS=S−3σ1−c42
=S−3c4S1−c42
=B5S
X i − M R i X_i-MR_i Xi−MRi Control Chart (n=1)
M
R
i
=
∣
x
i
−
x
i
−
1
∣
MR_i=\vert x_i-x_{i-1}\vert
MRi=∣xi−xi−1∣
U
C
L
X
‾
=
X
‾
+
3
σ
=
X
‾
+
3
M
R
‾
d
2
=
X
‾
+
2.66
M
R
‾
UCL_{\overline{X}}=\overline{X}+3\sigma =\overline{X}+3\frac{\overline{MR}}{d_2} =\overline{X}+2.66\overline{MR}
UCLX=X+3σ=X+3d2MR=X+2.66MR
U
C
L
X
‾
=
X
‾
−
3
σ
=
X
‾
−
3
M
R
‾
d
2
=
X
‾
−
2.66
M
R
‾
UCL_{\overline{X}}=\overline{X}-3\sigma =\overline{X}-3\frac{\overline{MR}}{d_2} =\overline{X}-2.66\overline{MR}
UCLX=X−3σ=X−3d2MR=X−2.66MR
C
L
M
R
‾
=
M
R
‾
CL_{\overline{MR}}=\overline{MR}
CLMR=MR
U
C
L
M
R
‾
=
D
4
M
R
‾
=
3.267
M
R
‾
UCL_{\overline{MR}}=D_4\overline{MR}=3.267\overline{MR}
UCLMR=D4MR=3.267MR
L
C
L
M
R
‾
=
0
LCL_{\overline{MR}}=0
LCLMR=0
Number of defective units - Attribute Data
p-chart
C
L
=
p
‾
CL=\overline{p}
CL=p
U
C
L
=
p
‾
+
3
p
‾
(
1
−
p
‾
)
n
UCL=\overline{p}+3\sqrt{\frac{\overline{p} (1-\overline{p})}{n}}
UCL=p+3np(1−p)
L
C
L
=
p
‾
−
3
p
‾
(
1
−
p
‾
)
n
LCL=\overline{p}-3\sqrt{\frac{\overline{p} (1-\overline{p})}{n}}
LCL=p−3np(1−p)
np-chart
C
L
=
n
p
‾
CL=n\overline{p}
CL=np
U
C
L
=
n
p
‾
+
3
n
p
‾
(
1
−
p
‾
)
UCL=n\overline{p}+3\sqrt{n\overline{p}(1-\overline{p})}
UCL=np+3np(1−p)
L
C
L
=
n
p
‾
+
3
n
p
‾
(
1
−
p
‾
)
LCL=n\overline{p}+3\sqrt{n\overline{p}(1-\overline{p})}
LCL=np+3np(1−p)
(
L
C
L
=
0
,
i
f
  
L
C
L
<
0
)
(LCL=0, if \; LCL<0)
(LCL=0,ifLCL<0)
Number of defects - Attribute Data
c-chart
Where n is fixed,
C
L
=
c
‾
CL=\overline{c}
CL=c
U
C
L
=
c
‾
+
3
c
‾
UCL=\overline{c}+3\sqrt{\overline{c}}
UCL=c+3c
L
C
L
=
c
‾
−
3
c
‾
LCL=\overline{c}-3\sqrt{\overline{c}}
LCL=c−3c
u-chart
Where n is varible, and defects in units,
C
L
=
u
‾
CL=\overline{u}
CL=u
U
C
L
=
u
‾
+
3
u
‾
n
UCL=\overline{u}+3\sqrt{\frac{\overline{u}}{n}}
UCL=u+3nu
L
C
L
=
u
‾
−
3
u
‾
n
LCL=\overline{u}-3\sqrt{\frac{\overline{u}}{n}}
LCL=u−3nu
Process Capability Analysis
Variable Data
σ
^
=
S
c
4
=
∑
i
=
1
n
(
x
i
−
x
‾
)
2
n
−
1
/
c
4
\hat{\sigma}=\frac{S}{c_4} =\left. \sqrt{\frac{\sum_{i=1}^n(x_i-\overline{x})^2}{n-1}} \middle/ c_4 \right.
σ^=c4S=n−1∑i=1n(xi−x)2
/c4
σ
^
=
R
‾
d
2
 
(
n
<
7
)
\hat{\sigma}=\frac{\overline{R}}{d_2}\:(n<7)
σ^=d2R(n<7)
Or for simplicity, use biased estimates
σ
^
=
S
\hat{\sigma}=S
σ^=S.
C
p
=
U
S
L
−
L
S
L
6
σ
C_p=\frac{USL-LSL}{6\sigma}
Cp=6σUSL−LSL
C
p
^
=
U
S
L
−
L
S
L
6
σ
^
\hat{C_p}=\frac{USL-LSL}{6\hat{\sigma}}
Cp^=6σ^USL−LSL
C
P
U
=
U
S
L
−
μ
3
σ
C_{PU}=\frac{USL-\mu}{3\sigma}
CPU=3σUSL−μ
C
P
U
^
=
U
S
L
−
X
^
3
σ
^
\hat{C_{PU}}=\frac{USL-\hat{X}}{3\hat{\sigma}}
CPU^=3σ^USL−X^
P
U
=
1
−
Φ
(
U
S
L
−
μ
σ
)
=
1
−
Φ
(
3
C
P
U
)
=
Φ
(
−
3
C
P
U
)
P_U=1-\Phi\left(\frac{USL-\mu}{\sigma}\right) =1-\Phi(3C_{PU})=\Phi(-3C_{PU})
PU=1−Φ(σUSL−μ)=1−Φ(3CPU)=Φ(−3CPU)
C
P
L
=
μ
−
L
S
L
3
σ
C_{PL}=\frac{\mu-LSL}{3\sigma}
CPL=3σμ−LSL
C
P
U
^
=
X
^
−
L
S
L
3
σ
^
\hat{C_{PU}}=\frac{\hat{X}-LSL}{3\hat{\sigma}}
CPU^=3σ^X^−LSL
P
U
=
Φ
(
U
S
L
−
μ
σ
)
=
Φ
(
3
C
P
U
)
=
Φ
(
−
3
C
P
U
)
P_U=\Phi\left(\frac{USL-\mu}{\sigma}\right) =\Phi(3C_{PU})=\Phi(-3C_{PU})
PU=Φ(σUSL−μ)=Φ(3CPU)=Φ(−3CPU)
C
P
K
=
min
{
C
P
U
,
C
P
L
}
=
U
S
L
−
L
S
L
−
2
Δ
6
σ
=
C
P
(
1
−
k
)
C_{PK}=\min\{C_{PU},C_{PL}\}=\frac{USL-LSL-2\Delta}{6\sigma} =C_P(1-k)
CPK=min{CPU,CPL}=6σUSL−LSL−2Δ=CP(1−k)
where
k
=
2
Δ
T
  
Δ
=
∣
σ
−
M
∣
  
M
=
U
S
L
−
L
S
L
2
  
T
=
U
S
L
−
L
S
L
k=\frac{2\Delta}{T}\;\Delta=\vert\sigma-M\vert\; M=\frac{USL-LSL}{2}\;T=USL-LSL
k=T2ΔΔ=∣σ−M∣M=2USL−LSLT=USL−LSL
When process mean coincides with the central tolerance,
D
e
f
e
c
t
i
v
e
 
r
a
t
e
 
P
=
Φ
(
L
S
L
−
μ
σ
)
+
[
1
−
Φ
(
U
S
L
−
μ
σ
)
]
=
2
×
Φ
(
L
S
L
−
μ
σ
)
=
2
×
Φ
(
−
T
/
2
σ
)
=
2
×
Φ
(
−
3
C
P
)
\begin{array}{rl} Defective\:rate\:P&=\Phi\left(\frac{LSL-\mu}{\sigma}\right)+\left[1-\Phi\left(\frac{USL-\mu}{\sigma}\right)\right]\\ &=2\times\Phi\left(\frac{LSL-\mu}{\sigma}\right)\\ &=2\times\Phi\left(\frac{-T/2}{\sigma}\right)\\ &=2\times\Phi\left(-3C_P\right) \end{array}
DefectiverateP=Φ(σLSL−μ)+[1−Φ(σUSL−μ)]=2×Φ(σLSL−μ)=2×Φ(σ−T/2)=2×Φ(−3CP)
When process doesn’t coincides with the central tolerance,
D
e
f
e
c
t
i
v
e
 
r
a
t
e
 
P
=
2
−
Φ
[
3
C
P
(
1
−
k
)
]
−
Φ
[
3
C
P
(
1
+
k
)
]
Defective\:rate\:P=2-\Phi\left[3C_P(1-k)\right]-\Phi\left[3C_P(1+k)\right]
DefectiverateP=2−Φ[3CP(1−k)]−Φ[3CP(1+k)]
where
k
=
∣
μ
−
M
∣
T
/
2
k=\frac{\vert\mu-M\vert}{T/2}
k=T/2∣μ−M∣
C
P
M
=
U
S
L
−
L
S
L
6
(
μ
−
M
)
2
+
σ
2
C_PM=\frac{USL-LSL}{6\sqrt{(\mu-M)^2+\sigma^2}}
CPM=6(μ−M)2+σ2
USL−LSL
M is the target value of the process.
Attribute Data
The DPU measures does not directly take the complexity of the unit into account. Opportunities are the number of potential chances within a unit for a defect to occur Eg. It is different to plug 100 cards and 10 cards in PCA, the defect opportunities are definitely different, the former has 10 defect opportunities and the latter has 100 defect opportunities. Defect Per Million Opportunities.(DPMO)is a standardized index considering taking the complexity of the unit into account. The equation of DPMO can be expressed as follows:
D
P
M
O
=
D
P
U
×
1
0
6
n
u
m
b
e
r
 
o
f
 
d
e
f
e
c
t
 
o
p
p
o
r
t
u
n
i
t
i
e
s
 
a
p
p
e
a
r
 
i
n
 
u
n
i
t
 
p
r
o
d
u
c
t
DPMO=\frac{DPU\times10^6}{number\:of\:defect \:opportunities\: appear\:in\: unit\: product}
DPMO=numberofdefectopportunitiesappearinunitproductDPU×106
D
P
M
O
=
N
u
m
b
e
r
 
o
f
 
D
e
f
e
c
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DPMO=\frac{Number\: of \:Defects}{Number\: of\: Units\times Number\: of\: Opportunities\: per\:unit}\times 10^6
DPMO=NumberofUnits×NumberofOpportunitiesperunitNumberofDefects×106
This is the end of this article, thanks for reading.
Tips: I found that in the CSDN markdown compiler, adding tab keys before paragraphs defaults to code fragments.
