Question

The Stryker Baseball Bat Company manufactures wooden and aluminum baseball bats at its plant in New England. Wooden bats produced for the mass market are turned on a lathe, where a piece of wood is shaped into a bat with a handle and barrel. The bat is cut to its specified length and then finished in subsequent processes. A specific style of wooden bat is supposed to have a mean barrel circumference of 10 inches with ±0.5 inches tolerance at its thickest point.
The company is trying to buy a new machine to cut the bats due to a projected increase in sales. There are three different candidate machines that the company is trying to buy at different costs. The following table summarizes the process average, process standard deviation (SD), and purchase cost of each machine. Which machine, if any, should the company purchase (they are trying to buy the cheapest machine that can get the job done)?

Machine....Average.......SD.........Purchase Cost
-------------------------------------------------------------------
...1..............10.14........0.12...........$20,000
...2..............9.92........0.13...........$18,000
...3..............10.00........0.15...........$15,000

1. Determine the Upper and Lower Specification Limit?

2. Calculate the process capability index of machine 1, machine 2 and machine 3?

3. Based on process capability index of machine 1, is machine 1 capable?

4.Based on process capability index of machine 2, is machine 2 capable?

5. Based on process capability index of machine 3, is machine 3 capable?

Answer 1

Solution:

(1) Upper Specification Limit (USL) and Lower Specification Limit (LSL) is given as,

USL = 10 inches + 0.5 inches

USL = 10.5 inches

LSL = 10 inches - 0.5 inches

LSL = 9.5 inches

(2) Process capability index (Cpk):

Cpk = Minimum of [(USL - Mean) / 3 $\sigma$ ] and [(Mean - LSL) / 3 $\sigma$ ]

where, $\sigma$ = Standard Deviation (SD)

Machine 1:

Cpk = Minimum of [(10.5 - 10.14) / (3 x 0.12)] and [(10.14 - 9.5) / (3 x 0.12)]

Cpk = Minimum of (1.00 and 1.78)

Cpk = 1.00

Machine 2:

Cpk = Minimum of [(10.5 - 9.92) / (3 x 0.13)] and [(9.92 - 9.5) / (3 x 0.13)]

Cpk = Minimum of (1.49 and 1.08)

Cpk = 1.08

Machine 3:

Cpk = Minimum of [(10.5 - 10.00) / (3 x 0.15)] and [(10.00 - 9.5) / (3 x 0.15)]

Cpk = Minimum of (1.11 and 1.11)

Cpk = 1.11

(3) For a process to be capable of meeting specification limits, Cpk should be equal or greater than 1.33. For Cpk values between 1.00 and 1.33, the process is capable but with tight control.

Based on process capability index of machine 1 (Cpk = 1.00) , machine 1 is capable but with tight control.

(4) Based on process capability index of machine 2 (Cpk = 1.08) , machine 2 is capable but with tight control.

(5) Based on process capability index of machine 3 (Cpk = 1.11) , machine 3 is capable but with tight control.