Question

QUESTION 25 (10 points) Process Capability Indexes A ball bearing design for wheel assemblies used in giant mining dump trucks requires the inner diameter to have a lower specification unit (LSL) of 1,097.50 mm and an upper specification unit (USL) of 1,102.50 mm. A simple random sample of 270 finished ball bearings, which were produced by an approximately normally distributed process, has the sample average of 1101.20 mm and the sample standard deviation of 0.60 mm. A too small inner diameter is especially not good because the ball bearing cannot be fitted into a regular axle. You would need to use 3 decimals in the following calculations. (a)[2] Draw a diagram of two scales, one for specifications, and the other for variations. (b)[1] Give an equation for the process capability index Cp, calculate and interpret it. (c)[2] Give an equation for the upper one-sided index Cpu, calculate and interpret it. (d)[2] Give an equation for the lower one-sided index Cpl, calculate and interpret it. (e)[1] Give an equation for the process capability index Cpk, calculate and interpret it (f)[1] Among the above 4 indexes, which one is the most closely watched? Explain your answer (g)[1] If you are the buyer of those ball bearings, what would you say to the manufacturer?

Answer 1

a.

The diagram is shown below. The blue line represents the specifications across the x axis. The y axis will represent the variation in number of a particular specification.

b.

The process capability equation is

Cp = (USL – LSL)/(6*standard deviation)

Cp = (1102.5-1097.5)/(6*0.6) = 1.38

c.

The upper sided index equation is

CpU = (USL – mean)/(3*standard deviation)

CpU = (1102-1101.2)/(3*0.6) = 0.44

d.

The lower sided index equation is

CpL = (mean – LSL)/(3*standard deviation)

CpL = (1101.2-1097.5)/(3*0.6) = 2.05

e.

The process capability index equation is

Minimum of (CpU, CpL)

The value is CpU and the value is

Cpk = 0.44

f.

Among the four indexes, the closely watched one is the Cpk. This is because Cp measures only the variation but not the possibility that the variation is not centered. On the other hand Cpk measures the centering of the process to check how many units will likely be out of specifications.

g.

If I was the buyer, I would tell the manufacturer that the process is skewed and will likely breach the lower specification. As a result, they need to improve the Cpk value to at least 1.