Operations Management Practice Slides with solution

Few practice problems which will help students to understand solution

1. Control Limits for Charts  

2. Control Limit for R Chart

3. A hard-bake process is used in conjunction with photolithography in semiconductor manufacturing. We wish to establish statistical control of the flow width of the resist in this process using and R charts. Twenty-five samples, each of size five wafers, have been taken when we think the process is in control. The interval of time between samples or subgroups is one hour. The flow width measurement data ( in microns ) from these samples are shown in table in next slide  

4. R Chart For samples of n = 5, we find from Appendix Table that D 3 = 0 and D 4 = 2.114. Therefore, the control limits for tthe R chart are

5. Chart   To find the control limits on the chart, we use A 2 = 0.577 from Appendix Table for samples of size n = 5

6. Estimating Process Capability The and R charts provide information about the performance or process capability of the process. From the chart, we may estimate the mean flow width of the resist in the hard-bake process as =1.5056 microns . The process standard deviation may be estimated using equation  

7. The specification limits on flow width are 1.50 0.50 micron, hence the flow width should be between   1.0 to 2.0 micron E stimate of the fraction of nonconforming wafers produced p= P{x<1.00} + P{x>2.00} = +1-   = +1-   ≈0.00015+1-0.99980 = 0.00035 That is, about 0.035% [350 parts per million (ppm)] of the wafers produced will be outside of the specifications

8. Another way to express process capability is in terms of the process capability ratio ( PCR) C p , which for a quality characteristic with both upper and lower specification limits ( USL and LSL, respectively) is This implies that the “natural” tolerance limits in the process (three-sigma above and below the mean ) are inside the lower and upper specification limits. Consequently, a moderately small number of nonconforming wafers will be produced. The PCR C p may be interpreted another way . The quantity is simply the percentage of the specification band that the process uses up.

9. For the hard-bake process an estimate of P is That is, the process uses up about 84% of the specification band

10. Construction of x-bar & s chart Setting up and operating control charts for x-bar and s requires about the same sequence of steps as those for x-bar and R charts, except that for each sample we must calculate the sample average and the sample standard deviation s . If σ 2 is the unknown variance of a probability distribution, then an unbiased estimator of σ 2 is the sample variance

11. Constructing x-bar and s chart

12. For x-bar chart For s chart

13. Frozen orange juice concentrate is packed in 6-oz cardboard cans . These cans are formed on a machine by spinning them from cardboard stock and attaching a metal bottom panel . By inspection of a can, we may determine whether, when filled , it could possibly leak either on the side seam or around the bottom joint. Such a nonconforming can has an improper seal on either the side seam or the bottom panel . Set up a control chart to improve the fraction of nonconforming cans produced by this machine

14. To establish the control chart, 30 samples of n = 50 cans each were selected at half-hour intervals over a three-shift period in which the machine was in continuous operation. The data are shown in Table next slide. We construct a phase I control chart using this preliminary data to determine if the process was in control when these data were collected. Since the 30 samples contain nonconforming cans, we find from equation

15. The sample fraction nonconforming from each preliminary sample is plotted on this chart. We note that two points , those from samples 15 and 23, plot above the upper control limit , so the process is not in control. These points must be investigated to see whether an assignable cause can be determined .

16. Analysis of the data from sample 15 indicates that a new batch of cardboard stock was put into production during that half-hour period. The introduction of new batches of raw material sometimes causes irregular production performance , and it is reasonable to believe that this has occurred here . Furthermore, during the half-hour period in which sample 23 was obtained, a relatively inexperienced operator had been temporarily assigned to the machine, and this could account for the high fraction nonconforming obtained from that sample. Consequently, samples 15 and 23 are eliminated , and the new centre line and revised control limits are calculated

17. Note that we have not dropped samples 15 and 23 from the chart, but they have been excluded from the control limit calculations, and we have noted this directly on the control chart. This annotation of the control chart to indicate unusual points, process adjustments, or the type of investigation made at a particular point in time forms a useful record for future process analysis and should become a standard practice in control chart usage.

18. Note also that the fraction nonconforming from sample 21 now exceeds the upper control limit. However, analysis of the data does not produce any reasonable or logical assignable cause for this, and we decide to retain the point . Therefore , we conclude that the new control limits in Figure can be used for future samples. Thus, we have concluded the control limit estimation phase (phase I) of control chart usage.

19. A process that produces titanium forgings for automobile turbocharger wheels is to be controlled through use of a fraction nonconforming chart . Initially , one sample of size 150 is taken each day for 20 days, and the results shown in Table are observed. (a) Establish a control chart to monitor future production. (b) What is the smallest sample size that could be used for this process and still give a positive lower control limit on the chart?

20. Variable Sample Size – Method 1

21. 2 nd Method – using average sample size The second approach is to base the control chart on an average sample size, resulting in an approximate set of control limits . This assumes that future sample sizes will not differ greatly from those previously observed . If this approach is used, the control limits will be constant, and the resulting control chart will not look as formidable to operating personnel as the control chart with variable limits .