Western Electric Company Rules (WECO) for Shewhart Control Chart

Simulation

We simulate normally distributed data as follows:

require(weco);

## Loading required package: weco

set.seed(10000);
n.sim     <- 50000;
sdx       <- 2;
simu.data <- rnorm(n.sim, sd = sdx);
quants    <- c(0.25, 0.5, 0.75);
xmax      <- 3000;
ymax      <- 0.015;

Single rule

Rule 1

1 point \(>L\) standard deviations from center line

l          <- 3;
rst.1      <- weco.rule(rule=1, x=simu.data, l=l, sdx=sdx, mux=0);
simu.arl   <- weco.rl(rst.1);
hist(simu.arl, breaks = 100, freq = F,
     xlim=c(0, xmax), ylim=c(0,ymax),
     xlab="Running Length", ylab="Probability", main="Rule 1");

Rule 2

\(K\) points in a row on the same side of the center line

k          <- 9;
rst.2      <- weco.rule(rule=2, x=simu.data, k=k, sdx=sdx, mux=0);
simu.arl   <- weco.rl(rst.2);
hist(simu.arl, breaks = 100, freq = F,
     xlim=c(0, xmax), ylim=c(0,ymax),
     xlab="Running Length", ylab="Probability", main="Rule 2");

Rule 3

\(K\) points in a row, all increasing or decreasing

k          <- 6;
rst.3      <- weco.rule(rule=3, x=simu.data, k=k, sdx=sdx, mux=0);
simu.arl   <- weco.rl(rst.3);
hist(simu.arl, breaks = 100, freq = F,
     xlim=c(0, xmax), ylim=c(0,ymax),
     xlab="Running Length", ylab="Probability", main="Rule 3");

Rule 4

\(K\) points in a row, alternating up and down

k          <- 14;
rst.4      <- weco.rule(rule=4, x=simu.data, k=k, sdx=sdx, mux=0);
simu.arl   <- weco.rl(rst.4);
hist(simu.arl, breaks = 100, freq = F,
     xlim=c(0, xmax), ylim=c(0,ymax),
     xlab="Running Length", ylab="Probability", main="Rule 3");

Rule 5

\(K\) out \(K+1\) points out of 2 standard deviations from center line

k          <- 2;
rst.5      <- weco.rule(rule=5, x=simu.data, k=k, l=2, sdx=sdx, mux=0);
simu.arl   <- weco.rl(rst.5);
hist(simu.arl, breaks = 100, freq = F,
     xlim=c(0, xmax), ylim=c(0,ymax),
     xlab="Running Length", ylab="Probability", main="Rule 5");

Rule 6

\(K\) out \(K+1\) points out of 1 standard deviations from center line

k          <- 4;
rst.6      <- weco.rule(rule=6, x=simu.data, k=k, l=1, sdx=sdx, mux=0);
simu.arl   <- weco.rl(rst.6);
hist(simu.arl, breaks = 100, freq = F, 
     xlim=c(0, xmax), ylim=c(0,ymax),
     xlab="Running Length", ylab="Probability", main="Rule 6");

Rule 7

\(K\) points in a row within 1 standard deviations from center line (either side)

k          <- 15;
rst.7      <- weco.rule(rule=7, x=simu.data, k=k, l=1, sdx=sdx, mux=0);
simu.arl   <- weco.rl(rst.7);
hist(simu.arl, breaks = 100, freq = F, 
     xlim=c(0, xmax), ylim=c(0,ymax),
     xlab="Running Length", ylab="Probability", main="Rule 7");

Rule 8

\(K\) points in a row > 1 standard deviations from center line (either side)

k          <- 8;
rst.8      <- weco.rule(rule=8, x=simu.data, k=k, l=1, sdx=sdx, mux=0);
simu.arl   <- weco.rl(rst.8);
hist(simu.arl, breaks = 100, freq = F, 
     xlim=c(0, xmax), ylim=c(0,ymax),
     xlab="Running Length", ylab="Probability", main="Rule 8");

Multiple rules

Multiple rules can be combined as a list.

##rules 1 and 2
lst.rules  <- list(list(1, l=3),
                   list(2, k=9));
rst.m      <- weco.combine(simu.data, lst.rules=lst.rules);
simu.arl   <- weco.rl(rst.m);
hist(simu.arl, breaks = 100, freq = F, 
     xlim=c(0, xmax), ylim=c(0,ymax),
     xlab="Running Length", ylab="Probability", main="Multiple Rules");

Plot results

A trace plot may be plotted as follows

plot(rst.m, start=1000, end=1500);

Graphical user interface

The package provides a graphical user interface based on Shiny, which can be brought up by

run.weco();