Operations and Supply Chain Management The Core

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320 OPERATIONS AND SUPPLY CHAIN MANAGEMENT
Excel:
Ch10_SPC
Example 10.2
The quality assurance manager is assessing the capability of a process that puts pressurized grease in
an aerosol can. The design specifications call for an average of 60 pounds per square inch (psi) of pressure
in each can, with an upper specification limit of 65 psi and a lower specification limit of 55 psi.
A sample is taken from production and it is found that the cans average 61 psi, with a standard
deviation of 2 psi. What is the capability of the process? What is the probability of producing a defect?
SOLUTION
Step 1—Interpret the data from the problem
​LSL = 55 USL = 65
X ​ = 61 σ = 2​
Step 2—Calculate the C pk
​C​ pk ​= min​ ​________
X ​ − LSL
​, USL − ________ X ​
[ 3σ 3σ
​] ​
​C​ pk ​= min​ _ 61 − 55 65 − 61 ​
​ ​, ​ _
[ 3(2) 3(2)
​] ​
​C​ pk ​= min[1,0.6667 ] = 0.6667
​
This is not a very good capability index. We see why this is true in step 3.
Step 3—Calculate the probability of producing a defective can:
Probability of a can with less than 55 psi
X − X ​
Z = ​ _____ ​= ​ ______ 55 − 61
​ = − 3
σ 2 ​
NORM.S.DIST(− 3) = 0.001349898
Probability of a can with more than 65 psi
X = ​ X ​
Z = ​ _____ ​= ______ 65 − 61
​ ​ = 2
σ 2 ​
1 − NORM.S.DIST(2) = 1 − 0.977249868 = 0.022750132
Probability of a can with less than 55 psi or more than 65 psi
​Probability = 0.001349898 + 0.022750132 = 0.024100030​
Or approximately 2.4 percent of the cans will be defective.
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The following table is a quick reference for the fraction of defective units for various
design specification limits (expressed in standard deviations). This table assumes that the
standard deviation is constant and that the process is centered exactly between the design
specification limits.