Operations Management question (Control Limits)
HVN Nursery sells “U-Fill” topsoil where patrons fill their own bags of soil and pay by the bag. The nursery wishes to find out the upper and lower control limits for the averages of this self-serve process. The nursery had their summer college students take numerous samples, with each sample containing 9 bags, and weigh them. The average weight was 22 kg. The average range was .37kg.
What are the upper and lower control limits for the x bar chart and R chart?
The limits for the xbar chart are defined to be
xbar +/- A2*Rbar
where xbar is the mean of each bag, estimated from all the samples put together, where A2 is a sample-size-specific anti-biasing constant and where Rbar is the average range in the samples.
We are given that xbar = 22/m = 22/9 where m is the number of bags in each sample, and that Rbar = 0.37kg. Using look-up statistical tables we find that A2 = 0.337 when the sample size m =9 bags.
Therefore the control limits for the xbar chart are given by
22/9 +/- 0.337*0.37 = (2.320, 2.570) to 3dp.
The control limits of the R chart are defined to be
where D3 and D4 are sample-size-specific anti-biasing constants. Again using look-up statistical tables we find that D3 = 0.184 and D4 = 1.816 respectively. Rbar is the average of the ranges over all the samples as before.
Therefore the control limits for the R chart are given by
(0.184*0.37, 1.816*0.37) = (0.0681, 0.6719) to 4dp.
Limits for xbar chart (2.320, 2.570) to 3dp
Limits for R chart (0.0681, 0.6719) to 4dp