phat.ewma.arl {spc}

Compute ARLs of EWMA phat control charts

Description

Computation of the (zero-state) Average Run Length (ARL), upper control limit (ucl) for given in-control ARL, and lambda for minimal out-of control ARL at given shift.

Usage

phat.ewma.arl(lambda, ucl, mu, n, z0, sigma=1, type="known", LSL=-3, USL=3, N=15,
qm=25, ntype="coll")

phat.ewma.crit(lambda, L0, mu, n, z0, sigma=1, type="known", LSL=-3, USL=3, N=15, qm=25)

phat.ewma.lambda(L0, mu, n, z0, sigma=1, type="known", max_l=1, min_l=.001, LSL=-3, USL=3,
qm=25)

Arguments

lambda

smoothing parameter of the EWMA control chart.

ucl

upper control limit of the EWMA phat control chart.

L0

pre-defined in-control ARL (Average Run Length).

mu

true mean or mean where the ARL should be minimized (then the in-control mean is simply 0).

n

subgroup size.

z0

so-called headstart (gives fast initial response).

type

choose whether the standard deviation is given and fixed ("known") or estimated and potentially monitored ("estimated").

sigma

actual standard deviation of the data – the in-control value is 1.

max_l, min_l

maximal and minimal value for optimal lambda search.

LSL, USL

lower and upper specification limit, respectively.

N

size of collocation base, dimension of the resulting linear equation system is equal to N.

qm

number of nodes for collocation quadratures.

ntype

switch between the default method coll (collocation) and the classic one markov (Markov chain approximation) for calculating the ARL numerically.

Details

The three implemented functions allow to apply a new type control chart. Basically, lower and upper specification limits are given. The monitoring vehicle then is the empirical probability that an item will not follow these specification given the sequence of sample means. If the related EWMA sequence violates the control limits, then the alarm indicates a significant process deterioration. For details see the paper mentioned in the references. To be able to construct the control charts, see the first example.

Value

Return single values which resemble the ARL, the critical value, and the optimal lambda, respectively.

Author(s)

Sven Knoth

References

S. Knoth and S. Steinmetz (2013), EWMA p charts under sampling by variables, International Journal of Production Research 51, 3795-3807.

See Also

sewma.arl for a further collocation based ARL calculation routine.

Examples

## Simple example to demonstrate the chart.

# some functions
h.mu <- function(mu) pnorm(LSL-mu) + pnorm(mu-USL)
ewma <- function(x, lambda=0.1, z0=0) filter(lambda*x, 1-lambda, m="r", init=z0)

# parameters
LSL <- -3       # lower specification limit
USL <-  3	# upper specification limit
n <- 5		# batch size
lambda <- 0.1	# EWMA smoothing parameter
L0 <- 1000	# in-control Average Run Length (ARL)
z0 <- h.mu(0)	# start at minimal defect level
ucl <- phat.ewma.crit(lambda, L0, 0, n, z0, LSL=LSL, USL=USL)

# data
x0 <- matrix(rnorm(50*n), ncol=5)	# in-control data
x1 <- matrix(rnorm(50*n, mean=0.5), ncol=5)# out-of-control data
x <- rbind(x0,x1)			# all data

# create chart
xbar <- apply(x, 1, mean)
phat <- h.mu(xbar)
z <- ewma(phat, lambda=lambda, z0=z0)
plot(1:length(z), z, type="l", xlab="batch", ylim=c(0,.02))
abline(h=z0, col="grey", lwd=.7)
abline(h=ucl, col="red")


## S. Knoth, S. Steinmetz (2013)

# Table 1

lambdas <- c(.5, .25, .2, .1)
L0 <- 370.4
n <- 5
LSL <- -3
USL <- 3

phat.ewma.CRIT <- Vectorize("phat.ewma.crit", "lambda")
p.star <- pnorm( LSL ) + pnorm( -USL ) ## lower bound of the chart
ucls <- phat.ewma.CRIT(lambdas, L0, 0, n, p.star, LSL=LSL, USL=USL)
print(cbind(lambdas, ucls))

# Table 2

mus <- c((0:4)/4, 1.5, 2, 3)
phat.ewma.ARL <- Vectorize("phat.ewma.arl", "mu")
arls <- NULL
for ( i in 1:length(lambdas) ) {
  arls <- cbind(arls, round(phat.ewma.ARL(lambdas[i], ucls[i], mus,
                n, p.star, LSL=LSL, USL=USL), digits=2))
}
arls <- data.frame(arls, row.names=NULL)
names(arls) <- lambdas
print(arls)

# Table 3

## Not run: 
mus <- c(.25, .5, 1, 2)
phat.ewma.LAMBDA <- Vectorize("phat.ewma.lambda", "mu")
lambdas <- phat.ewma.LAMBDA(L0, mus, n, p.star, LSL=LSL, USL=USL)
print(cbind(mus, lambdas))
## End(Not run)