Question

18. i) Process Capability (See Page 317 - 319 of Goetsch and Davis Chap. 18)
    Formulas: (Do not use a d2 factor from fig. 18.22 ).
    Instead, use the formula Cp = (USL – LSL)/6 σ        Note that σ is the standard deviation. USL and LSL are the customer’s specification limits.
    Cpk is calculated in 2 ways:
    Cpk = (USL – Xdouble bar)/ 3 σ   Note that X double bar is the process mean
    Cpk = (X double bar – LSL)/ 3 σ
    
    Customers of Don Corleone Pizza have specified that family-sized pizza crusts they order should be 28 to 32 centimeters in diameter. Sample data recently collected indicate that the crusts average 30 centimeters in diameter, with an estimated standard deviation sigma (σ) of 0.5 centimeters.

(i)    Calculate the Cp index to determine if Don Corleone’s pizza crust production system is capable of meeting its customers’ requirements. Briefly explain your findings  

(ii)    Suppose that a new operator produces pizza crusts with a process average of 30.6 centimeters instead of 30 centimeters. Calculate the Cpk twice. How capable is the system capable of meeting its customers’ requirements? If not, what options does Don Corleone have to rectify the situation?

Answer 1

The formulas are already provided. The x double bar is the process mean.

Here x double bar = 30

USL = 32

LSL = 28

Standard deviation = 0.5

Using the formulas,

(i)

Cp = (USL-LSL) /6 standard deviation = (32-28)/(6*0.5) = 1.33

A 1.33 Cp is a reasonable value for process capability. This means that the spread of variation is well thin the tolerance width.

Cpk = min (CpU and CpL)

CpU = (USL – x double bar)/3 standard deviation = (32-30)/(3*0.5) = 1.33

CpL = (x double bar – LSL)/3 standard deviation = (30-28)/(3*0.5) = 1.33

This means the Cpk is also 1.33

Overall we can view this as the process being equally spread and the range of process is within the specification.

(ii)

If the new x double bar value is 30.6 then the value of Cp will remain 1.33 but the value of Cpk will change.

CpU = (32-30.6)/(3*0.5) = 0.933

CpL = (30.6-28)/(3*0.5) = 1.73

Cpk = 0.933

This shows that the process has shifted upward and the overall index indicates that the process is no longer capable.

In order to rectify this, we either need to bring back the mean to 30 instead of 30.6 or need to reduce the value of standard deviation.