Before going into production, many manufacturers run a capability study to determine if their process will run within specifications enough of the time. Capability indices produced by such a study are used to estimate expected percentages of defective parts. Capability studies are conducted with the capability function. The following capability indices are produced:

mu Sample mean

sigma Sample standard deviation

P Estimated probability of being within the lower (L) and upper (U) specification limits

Pl Estimated probability of being below L

Pu Estimated probability of being above U

Cp (U-L)/(6*sigma)

Cpl (mu-L)./(3.*sigma)

Cpu (U-mu)./(3.*sigma)

Cpk min(Cpl,Cpu)

As an example, simulate a sample from a process with a mean of 3 and a standard deviation of 0.005 by typing the following MATLAB code:

rng(SEED); % use any number for SEED. A different number should be used for each simulation. data = normrnd(3,0.005,100,1);

Compute capability indices if the process has an upper specification limit of 3.01 and a lower specification limit of 2.99 by typing the following MATLAB code:

S = capability(data,[2.99 3.01])

Visualize the specification and process widths by typing the following MATLAB code: capaplot(data,[2.99 3.01]); grid on

Alternatively, you can use R software to obtain the same results by following the instructions below:

1. Download and install R to your computer: https://cran.r-project.org 2. Download qcc package in R (by adding all dependencies) 3. Load qcc package 4. Launch R and type the following in the console: data = rnorm(100,3.0,0.005) # generate the random data with mean 3 and std 0.005 q = qcc(data,type="xbar.one") # create an object of type xbar process.capability(q, spec.limits=c(2.99,3.01)) # produce the capability results with the given limits

Repeat the same code three times and comment on the obtained capability plots by considering the probability of producing products within the specification limits. Why is there variation? Also provide the screenshots of the capability plots.