MIE1727 Assignment 2

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Apr 3, 2024

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University of Toronto Department of Mechanical and Industrial Engineering MIE1727: Quality Assurance I (Winter 2023) Assignment 2 Due: Monday, March 20, 2023, at 5pm. Submit to Crowdmark. Please post questions to Piazza Question 1 (12 points) Provide answers to the follow questions. a) In some cases, the quality characteristic that you are measuring may be eligible for variables control chart or an attributes control chart. Discuss the pros and cons of choosing one or the other. (3 points) b) Explain the difference between control charts for nonconformity (c-chart) number of nonconforming (np-chart) and fraction of nonconforming (p-chart). (3 points) c) What is the difference between actual process capability and process potential capability? Why should we calculate the actual process capability ratio for some cases? Explain. (3 points) d) Estimating the process capability can be done using a histogram and probability plotting. Clearly state the advantages and disadvantages of both methods. (3 points) Question 2 (8 points) A process has an in ‐ control fraction nonconforming of p = 0.01. The sample size is n = 300. What is the probability of detecting a shift to an out ‐ of ‐ control fraction nonconforming of p = 0.05 on the first sample following the shift? Question 3 (10 points) A production line runs in batches of size 1000 where 64 units from each batch is sampled for quality assurance. The nominal fraction nonconforming is known to be ? = 0.1. a) Set up the control chart. You do not have to provide a graphical representation. (3 points) b) Suppose the customer risk is set to 0.5. What is the maximum fraction nonconforming that produces this risk level? You may estimate this answer using numerical tools. (3 points) c) Determine the sample size required to force a positive lower control limit for this chart. (2 points) d) Discuss why a positive LCL is preferable to LCL= 0. (2 points)

University of Toronto Department of Mechanical and Industrial Engineering MIE1727: Quality Assurance I (Winter 2023) Question 4 (10 points) A new scientific journal from Elsevier publisher has a new specification with managing the first decision of the new research paper submissions within 24 hours. Table 1 shows the number of decisions made each day for the last 20 days and the number of submissions that required more than 24 hours to be issued. Table 1: research papers’ submissions data . Day #Submissions Late decisions Day #Submissions Late decisions 1 200 3 11 219 0 2 250 4 12 238 10 3 240 2 13 250 4 4 300 5 14 302 6 5 200 2 15 219 20 6 250 4 16 246 3 7 246 3 17 251 6 8 258 5 18 273 7 9 275 2 19 245 3 10 274 1 20 260 1 a) Set up the fraction nonconforming control chart for this process, using the variable ‐ width control limit approach. Plot the preliminary data in Table 1 on the chart. Is the process in statistical control? (3 points) b) Assume that assignable causes can be found for any out ‐ of ‐ control points on this chart. What center line should be used for process monitoring in the next period, and how should the control limits be calculated? (3 points) c) Set up the fraction nonconforming control chart for this process, using the average sample size control limit approach. Plot the preliminary data in Table 1 on the chart. Is the process in statistical control? Compare this control chart to the one based on variable ‐ width control limits. (4 points)

University of Toronto Department of Mechanical and Industrial Engineering MIE1727: Quality Assurance I (Winter 2023) Question 5 (10 points) The IT department at the University of Toronto measures their efficacy by how many service tickets are completed within 48 hours of creation. Twenty months of data are shown in Table 2 . Table 2: Number of service tickets resolve outside of 48 hours. Month Total tickets >48 hours Month Total tickets >48 hours 1 150 2 11 100 1 2 150 1 12 100 0 3 250 9 13 100 1 4 250 7 14 200 4 5 200 5 15 200 3 6 200 3 16 200 4 7 200 6 17 200 10 8 250 8 18 200 4 9 250 7 19 150 0 10 250 6 20 150 2 a) Construct the trial control chart for this process. You are invited (but not required) to use Minitab, but still required to show the computations for your limits. (2 points) b) Discuss how you may use the results from part (a) to phase II control. (1 points) c) Repeat this exercise for an average sample size. Include necessary cautions. (3 points) d) Repeat this exercise using standard deviation units. (2 points) e) Since there are only a few sample sizes construct a special control chart that has four different sets of control limits all on one set of axes, one for each sample size. Discuss how it would be used, and how it compares to the chart from part (a). (2 points) Question 6 (15 points) The following fraction nonconforming control chart with ? = 400 is used to control a process: UCL = 0.0962 Center line = 0.0500 LCL = 0.0038 a) Find the width of the control limits in standard deviation units. (2 points)

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University of Toronto Department of Mechanical and Industrial Engineering MIE1727: Quality Assurance I (Winter 2023) b) What would be the corresponding parameters for an equivalent control chart based on the number nonconforming? (2 points) c) Use the Poisson approximation to the binomial to find the probability of a type I error. (2 points) d) Use the Poisson approximation to the binomial to find the probability of a type II error if the true process fraction nonconforming is 0.0600. (2 points) e) Draw the OC curve for this control chart. (2 points) f) Find the ARL when the process is in control and the ARL when the process fraction nonconforming is 0.0600. (2 points) g) Suppose the process fraction nonconforming shifts to 0.15. What is the probability of detecting the shift on the first subsequent sample? (3 points) Question 7 (4 points) A process is in statistical control with 𝑥̿ = 299 𝑅 ̅ = 7. The control chart uses a sample size of n = 8. Specifications are at 300 ± 9. The quality characteristic is normally distributed. a) Estimate the potential capability of the process. (1 points) b) Estimate the actual process capability. (1 points) c) How much improvement could be made in process performance if the mean could be centered at the nominal value? (2 points) Question 8 (5 points) A random sample of n = 40 of polymeric fibers resulted in a mean diameter of 0.1264 μm. and a standard deviation of 0.0003 μm. We assume that the diameters are normally distributed. a) With 95% confidence, what are the limits where 95% of the fiber diameters should fall? (2 points) b) Construct a 95% confidence interval on the true mean diameter. Explain the difference between this interval and the one obtained in question (a). (3 points) Question 9 (10 points) The repeatability and the reproducibility are important parameters in scientific experiments. Therefore, two batches of the same hydrophobic material have been used

University of Toronto Department of Mechanical and Industrial Engineering MIE1727: Quality Assurance I (Winter 2023) to measure their capacity in absorbing ten toxic solvents, three times each. The data are shown in Table 3 . Table 3: absorption capacity data. Batch1 measurements (g/g) Batch2 measurements (g/g) Toxic solvents Rep.1 Rep.2 Rep.3 Rep.1 Rep.2 Rep.3 1 50 49 50 50 48 51 2 42 42 41 41 41 41 3 63 60 60 64 62 61 4 49 51 50 48 50 51 5 38 39 38 38 39 38 6 52 50 50 52 50 50 7 61 61 61 61 60 60 8 42 40 39 43 38 40 9 50 51 50 51 48 49 10 57 56 59 56 57 58 c) Estimate repeatability and reproducibility of this material efficiency. (3 points) d) Estimate the standard deviation of measurement error. (3 points) e) If the specifications are at 50 ± 10, what can you say about their capability? (4 points) Question 10 (10 points) In order to test the measurement system, an operator measures 10 parts three times each. The results are given in Table 4. a) Using Minitab, graph the x-bar and R charts and describe any measurement error that may be indicated from these charts. (3 points) b) Compute total variability, product variability, and % variability due to the measurement system. (4 points) c) The specifications on the part are 100 ± 15. Compute the 𝑃 / 𝑇 ratio and discuss the results. (3 points)

University of Toronto Department of Mechanical and Industrial Engineering MIE1727: Quality Assurance I (Winter 2023) Table 4: Measurement data. Part number Measurement 1 2 3 1 99 98 98 2 98 98 96 3 100 99 98 4 100 100 97 5 95 93 97 6 96 95 97 7 100 101 100 8 95 97 98 9 100 98 99 10 101 103 100 Question 11 (12 points) To ensure chemical purity of a commercial organic chemical, measurements of the level of certain intermediate chemical material are taken every 30 minutes. Data from 22 samples appear below in Table 5 . Table 5: Measurements data. Sample number Level Sample number Level 1 15.3 12 15.9 2 15.7 13 14.7 3 14.4 14 15.2 4 14 15 14.6 5 15.2 16 13.7 6 15.8 17 12.9 7 16.7 18 13.2 8 16.6 19 14.1 9 15.9 20 14.2 10 17.4 21 13.8 11 19.7 22 14.8 a) Use (I, MR) charts with 3 sigma limits to get the estimates of the in-control process parameters. We did not directly discuss (I, MR) charts in class, but you can refer to section 6.4 for details. To control future production, design a one-sided CUSUM chart for detecting a shift in the process mean from 𝜇 0 to 𝜇 1 = 15.25, where 𝜇 0 is the in-control estimate of the process mean.

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University of Toronto Department of Mechanical and Industrial Engineering MIE1727: Quality Assurance I (Winter 2023) When the process is in control, we want to have the ARL= 370. When the process is out of control and the process mean shifts from 𝜇 0 to 𝜇 1 = 15.25, we want to have the ARL ≤ 10 . Determine the sample size and the control limits to satisfy both requirements. Calculate the ARL for 𝜇 1 = 15.2 for the designed CUSUM chart and compare with the ARL for 𝜇 1 = 15.2 for the 𝑥 ̄ chart with 3-sigma limits, and the sample size you have determined for the CUSUM chart. (6 points) b) Design a two-sided CUSUM chart for detecting shifts in the process mean from 𝜇 0 to 𝜇 1 = 15.5, and from 𝜇 0 𝑡? 𝜇 2 = 14 , sample size ? = 1. Use the in-control estimates of the process parameters obtained in part (a). When the process is in control, we want to have the ARL= 370. Analyze the data using the designed two- sided CUSUM chart. Calculate the ARL for the shifted mean value equal to 14 for the designed two-sided CUSUM chart and compare with the ARL of the I chart with 3-sigma limits. (6 points)