# Operations Management question (Specification Limits): Ecril Technologies makes car parts using vacuum formed moulds. The President wants the process of creating moulds to be reviewed to determine...

Operations Management question (Specification Limits):

Ecril Technologies makes car parts using vacuum formed moulds. The President wants the process of creating moulds to be reviewed to determine if the life span of the moulds, or the rate they have to replace moulds, is acceptable. Each mould produces on average 1,459 acceptable parts with a standard deviation of 81. The upper and lower control limits are 1,297 and 1,945 parts respectively.

**Calculate the Cp and Cpk to determine if the mould making process is capable of producing sufficient parts to specification? Explain your response.**

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### 1 Answer

1) Cp is the Process Capability Ratio. It is the number of times the spread of the process (6sigma) fits into the tolerance width (USL - LSL) (upper statistical limit or control limit - lower statistical limit or control limit) ie

Cp = (LSU-LSL)/6sigma = (1945-1297)/(6*81) = 4/3

This demonstrates that the spread of the process fits 1 and the 1/3rd times into the tolerance width for number of acceptable parts. This indicates that the production of moulds falls well within the tolerance range considering spread alone. The Cp measure, however, does not take *location* of the process into account. The Cpk measure on the other hand, does.

2) Cpk is the minimum of the 2-sided Process Capability Ratio (CpL,CpU). It is the minimum of CpL (lower Cp) and CpU (upper Cp) where these are the number of times half of the spread of the process, or the half-width, (3sigma) fits into each half of the tolerance interval (LSL,USL) where the centre is placed at the sample mean mu (=1459 here), which NB may not be halfway between LSL and USL. The Cpk measure then focuses on the worst side of the process spread with reference to the tolerance interval. If the process is not balanced equally over the tolerance interval, but is shifted upwards or downwards from the central target, Cpk will indicate this.

Cpk = min((mu-LSL)/3sigma,(USL-mu)/3sigma) = min((1459-1297)/(3*81),(1945-1459)/(3*81)) = min(2/3,2) = 2/3

The half-width of the process, 3sigma, only fits 2/3 times into the lower end of the tolerance interval indicating that the centre of the process spread is below that of the central target. In terms of process capability, a Cpk of 2/3 equates to 2.3% of moulds produced being below specification

(Phi(Cpk*3)*100 = percentage below (or above) specification, where Phi is the CDF of the standard Normal distribution).

Armed with this piece of information, the president can decide whether this percentage of defectives is acceptable in terms of costs of replacement versus costs of improving the manufacture of the moulds.

**Cp = 4/3 and Cpk = 2/3**

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