ST5208 Analytics for Quality Control and Productivity Improvements
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Department of Statistics and Data Science
ST5208 Analytics for Quality Control and Productivity Improvements
1. Readings from an industrial process are given below (read down from left). They are taken in one hour intervals.
619 658 640
612 664 647
589 659 633
610 635 624
625 642 630
653 647 625
The target value is µ0 = 650. The sample average of these 18 readings is = 634, the sample standard deviation is 19.5, the range is R = 75 and the average moving range is MR = 11.4.
(a) Provide the formulas for the computation of the sample standard deviation,
range and average moving range. Show how each of these values can be used to estimate the process standard deviation.
(b) Find the control limits of a control chart using one of the three standard
deviation estimates in (a). Explain your choice of standard deviation estimate.
(c) You are provided with specification limits of 650 士 50. Is the use of modified control limits appropriate here? Explain.
(d) Construct a two-sided tabular Cusum chart corresponding to k = 0.5 with ARL0 = 370. There is no need to plot the control chart. Interpret the Cusum chart. Name one advantage of using the Cusum chart.
2. A process engineer ran a 22 experiment with two replicates. The factors are A and B .
Run |
1 |
2 |
(1) |
66 |
76 |
a |
73 |
81 |
b |
96 |
84 |
ab |
94 |
88 |
(a) Test if the main effect of A is significant at level α = 0.05.
(b) A colleague of yours questioned the need of having the design point ab. His view
is that to estimate the main effect of A, the information in (1) and a suffices while to estimate the main effect of B, the information in (1) and b suffices. What is your reply to your colleague?
(c) Your boss thinks that dealing with just two factors for eight runs is too wasteful. He would like to have as many factors as possible with the same number of runs. He thinks that the presence of two-factor interactions should not be ignored but the estimation of these interaction effects is unnecessary. Also all main effects should be estimable. Provide a design (with explicit design points) that you feel will satisfy all your boss’s requirements.
(d) Comment on the following statement: “In a robust design involving uncontrol- lable noise factors, it makes sense that at least one two-factor interaction effect involving the noise factors should be estimable”.
3. A soft-drink bottler has a lower specification limit of 180 psi on the bursting strength of the bottles. A sample of 8 bottles was chosen from a lot and tested. The bursting strengths are
230, 250, 220, 210, 190, 215, 210, 230.
The sample mean is = 219 while the process standard deviation is known to be σ = 20.
(a) Should the soft-drink bottler accept the lot assuming that he adopts MIL STD
105E under normal inspection with an acceptable quality of 1.5 nonconforming per 100 items?
(b) Using the scheme in (a), what is the probability of rejecting a random lot
assuming that there is indeed 1.5 nonconforming per 100 items?
(c) Consider instead acceptance sampling for variables with parameter k = 2. Would you accept the lot? The supplier argues that none of the bottles tested has bursting strength below the specification limit and hence you cannot reject the lot. What is your reply?
(d) What value of k should you adopt [in part (c)] such that the probability of rejecting a random lot when there is 1.5 nonconforming per 100 items is the same as that in (b)? You may assume normality.
4. Two quality characteristics are to be monitored simultaneously. The covariance matrix (in correlation form) and the in-control process mean are respectively
Σ = 土. and µ = 土 .0(0) .
(a) Find the control limit for the chi-square control chart based on an α-risk of
0.05. Suppose that a sample of size one results in the standardized observation
vector [1, 3]7 . Compute the test statistic to be plotted on the chi-square control chart. Is an out-of-control signal generated?
(b) A MEWMA control chart is to be designed to provide good protection against a shift to a new mean of [0.5, 1]7 . If an in-control average run length of 200 is satisfactory, what value of λ and what upper control limit H would you recommend? Explain why by using the table below.
5. The first five observations from a process with target value 50 are given below.
Observation, t |
yt |
1 |
52 |
2 |
54 |
3 |
60 |
4 |
64 |
5 |
57 |
Suppose this process has a manipulatable variable, whose value at time t is xt . An integral controller is to be set up for this process. Assume that the process gain g = 1.2 and λ = 0.2 in the EWMA forecasting procedure will provide adequate one-step-ahead predictions. Compute the adjustment made to the manipulatable variable xt - xt_1 and the adjusted output yt(|) for t = 1, . . . , 5.
2022-04-20