Operations Management
Homework 6
1. Kellogg’s is monitoring its process of making Frosted Mini Wheats. Every hour, they take a sample of 4 boxes of cereal coming off of the production line and weigh each box. The measurements taken from the first 10 samples are given below. All measurements are in ounces.
|
Sample
|
x1
|
x2
|
x3
|
x4
|
x
|
R
|
|
1
|
16.21
|
16.27
|
16.72
|
17.47
|
16.67
|
1.25
|
|
2
|
16.98
|
17.54
|
16.94
|
15.53
|
16.75
|
2.00
|
|
3
|
14.80
|
15.64
|
17.63
|
16.58
|
16.17
|
2.83
|
|
4
|
15.69
|
15.83
|
16.74
|
16.85
|
16.28
|
1.16
|
|
5
|
16.35
|
17.47
|
16.96
|
15.95
|
16.68
|
1.53
|
|
6
|
16.82
|
17.03
|
16.63
|
17.27
|
16.94
|
0.65
|
|
7
|
16.95
|
17.38
|
17.18
|
16.76
|
17.06
|
0.62
|
|
8
|
15.98
|
16.74
|
17.67
|
18.48
|
17.22
|
2.50
|
|
9
|
18.73
|
17.31
|
17.80
|
18.70
|
18.14
|
1.42
|
|
10
|
16.68
|
16.51
|
16.78
|
16.97
|
16.73
|
0.47
|
a. Calculate the upper and lower control limits for the x bar-chart and the R-chart associated with this data. See below for the table of the D2, D3, and D4 values. (3 points)
b. Plot the x bar-chart and R-chart for Kellogg’s cereal production process. You may use the blank charts below to help you. Also determine whether or not Kellogg’s process of making cereal is in control. If it is not in control, state everything that looks wrong on either the x bar chart or R-chart. You may use software like Excel for the plots. (4 points)
|
n
|
D2
|
D3
|
D4
|
|
2
|
1.128
|
0
|
3.267
|
|
3
|
1.693
|
0
|
2.575
|
|
4
|
2.059
|
0
|
2.282
|
|
5
|
2.326
|
0
|
2.116
|
|
6
|
2.534
|
0
|
2.004
|
|
10
|
3.078
|
0.233
|
1.777
|
|
15
|
3.473
|
0.348
|
1.652
|
|
20
|
3.727
|
0.414
|
1.586
|
|
25
|
3.922
|
0.459
|
1.541
|
2. Suppose that we are making an auto part and that the mean length of the part is 5 inches with a standard deviation of 0.25 inches, and the length is normally distributed. The upper specification limit is 5.5 inches and the lower specification limit is 4.2 inches. The z-score table is provided on the next page.
a. What is the process capability index, Cpk? (2 points)
b. Under this process, what is the rate of defects per unit (DPU) and the defects per million units (DPMU)? (2 points)
c. If you have five auto parts made under this process, what is the probability that all five of them are not defective? (2 points)
d. Suppose you can modify the process to change the mean, while the standard deviation will remain the same. What should the mean be changed to, to minimize the DPU? What will be the DPU with this change? (2 points)