xshewhartrunsrules.arl {spc}

Compute ARLs of Shewhart control charts with and without runs rules

Description

Computation of the (zero-state and steady-state) Average Run Length (ARL) for Shewhart control charts with and without runs rules monitoring normal mean.

Usage

xshewhartrunsrules.arl(mu, c = 1, type = "12")

xshewhartrunsrules.crit(L0, mu = 0, type = "12")

xshewhartrunsrules.ad(mu1, mu0 = 0, c = 1, type = "12")

xshewhartrunsrules.matrix(mu, c = 1, type = "12")

Arguments

mu

true mean.

L0

pre-defined in-control ARL, that is, determine c so that the mean number of observations until a false alarm is L0.

mu1, mu0

for the steady-state ARL two means are specified, mu0 is the in-control one and usually equal to 0 , and mu1 must be given.

c

normalizing constant to ensure specific alarming behavior.

type

controls the type of Shewhart chart used, seed details section.

Details

xshewhartrunsrules.arl determines the zero-state Average Run Length (ARL) based on the Markov chain approach given in Champ and Woodall (1987). xshewhartrunsrules.matrix provides the corresponding transition matrix that is also used in xDshewhartrunsrules.arl (ARL for control charting drift). xshewhartrunsrules.crit allows to find the normalization constant c to attain a fixed in-control ARL. Typically this is needed to calibrate the chart. With xshewhartrunsrules.ad the steady-state ARL is calculated. With the argument type certain runs rules could be set. The following list gives an overview.

"1"

The classical Shewhart chart with +/- 3 c sigma control limits (c is typically equal to 1 and can be changed by the argument c).

"12"

The classic and the following runs rule: 2 of 3 are beyond +/- 2 c sigma on the same side of the chart.

"13"

The classic and the following runs rule: 4 of 5 are beyond +/- 1 c sigma on the same side of the chart.

"14"

The classic and the following runs rule: 8 of 8 are on the same side of the chart (referring to the center line).

Value

Returns a single value which resembles the zero-state or steady-state ARL. xshewhartrunsrules.matrix returns a matrix.

Author(s)

Sven Knoth

References

C. W. Champ and W. H. Woodall (1987), Exact results for Shewhart control charts with supplementary runs rules, Technometrics 29, 393-399.

See Also

xDshewhartrunsrules.arl for zero-state ARL of Shewhart control charts with or without runs rules under drift.

Examples

## Champ/Woodall (1987)
## Table 1
mus <- (0:15)/5
Mxshewhartrunsrules.arl <- Vectorize(xshewhartrunsrules.arl, "mu")
# standard (1 of 1 beyond 3 sigma) Shewhart chart without runs rules
C1 <- round(Mxshewhartrunsrules.arl(mus, type="1"), digits=2)
# standard + runs rule: 2 of 3 beyond 2 sigma on the same side
C12 <- round(Mxshewhartrunsrules.arl(mus, type="12"), digits=2)
# standard + runs rule: 4 of 5 beyond 1 sigma on the same side
C13 <- round(Mxshewhartrunsrules.arl(mus, type="13"), digits=2)
# standard + runs rule: 8 of 8 on the same side of the center line
C14 <- round(Mxshewhartrunsrules.arl(mus, type="14"), digits=2)

## original results are
#  mus     C1    C12    C13    C14                                    
#  0.0 370.40 225.44 166.05 152.73                                    
#  0.2 308.43 177.56 120.70 110.52                                    
#  0.4 200.08 104.46  63.88  59.76                                    
#  0.6 119.67  57.92  33.99  33.64                                    
#  0.8  71.55  33.12  19.78  21.07                                    
#  1.0  43.89  20.01  12.66  14.58                                    
#  1.2  27.82  12.81   8.84  10.90                                    
#  1.4  18.25   8.69   6.62   8.60                                    
#  1.6  12.38   6.21   5.24   7.03                                    
#  1.8   8.69   4.66   4.33   5.85                                    
#  2.0   6.30   3.65   3.68   4.89                                    
#  2.2   4.72   2.96   3.18   4.08                                    
#  2.4   3.65   2.48   2.78   3.38                                    
#  2.6   2.90   2.13   2.43   2.81                                    
#  2.8   2.38   1.87   2.14   2.35                                    
#  3.0   2.00   1.68   1.89   1.99

data.frame(mus, C1, C12, C13, C14)

## plus calibration, i. e. L0=250 (the maximal value for "14" is 255!
L0 <- 250
c1  <- xshewhartrunsrules.crit(L0, type = "1")
c12 <- xshewhartrunsrules.crit(L0, type = "12")
c13 <- xshewhartrunsrules.crit(L0, type = "13")
c14 <- xshewhartrunsrules.crit(L0, type = "14")
C1  <- round(Mxshewhartrunsrules.arl(mus, c=c1,  type="1"), digits=2)
C12 <- round(Mxshewhartrunsrules.arl(mus, c=c12, type="12"), digits=2)
C13 <- round(Mxshewhartrunsrules.arl(mus, c=c13, type="13"), digits=2)
C14 <- round(Mxshewhartrunsrules.arl(mus, c=c14, type="14"), digits=2)
data.frame(mus, C1, C12, C13, C14)

## and the steady-state ARL
Mxshewhartrunsrules.ad <- Vectorize(xshewhartrunsrules.ad, "mu1")
C1  <- round(Mxshewhartrunsrules.ad(mus, c=c1,  type="1"), digits=2)
C12 <- round(Mxshewhartrunsrules.ad(mus, c=c12, type="12"), digits=2)
C13 <- round(Mxshewhartrunsrules.ad(mus, c=c13, type="13"), digits=2)
C14 <- round(Mxshewhartrunsrules.ad(mus, c=c14, type="14"), digits=2)
data.frame(mus, C1, C12, C13, C14)