IE 691-Advanced Methods for Quality Improvement MIDTERM EXAMINATION (Take-Home) Total of 100 points I have neither given nor received aid on this examination, nor have I concealed any violation of the Honor Code NAME: ________________________________ ID#: ____________________ Signature, Date: ___________________________ DUE: Before 11:59:59 pm, March 12, 2004. Total number of pages = 12 OPEN BOOKS, OPEN NOTES. BEST OF LUCK! 1 Problem 1: In the semiconductor industry, the production of microcircuits (integrated circuits-IC) involves many steps. The wafer fabrication process typically builds these microcircuits on silicon wafers, and there are many microcircuits at each wafer. Each production lot consists of between 16 and 48 wafers. Some processing steps treat each wafer separately, so that the batch size for that step is one wafer. It is usually necessary to estimate several components of variation: within-wafer, between-wafer, between-lot, and the total variation. (a) Suppose that one wafer is randomly selected from each lot and that a single measurement on a critical dimension of interest is taken. Which components (within-wafer, between-wafer, between-lot) of variation could be estimated with these data? What type of control charts would you recommend? (b) Suppose that each wafer is tested at five fixed locations (say, the center and four points at the circumference). The average and range of these within-wafer of these within-wafer measurements are x ww and Rww , respectively. What components (within-wafer, between-wafer, between-lot) of variability are estimated using control charts based on these data? (c) Suppose that one measurement point on each wafer is selected and that this measurement is recorded for five consecutive wafers. The average and range of these between-wafer measures are x BW and RBW , respectively. What components (within-wafer, between-wafer, between-lot) of variability are estimated using control charts based on these data? Would it be necessary to run separate X and R charts for all five locations on the wafer? (d) Consider the question in part (c). How would your answer change if the test sites on each wafer were randomly selected and varied from wafer to wafer? (e) What type of control charts and rational subgroup scheme would you recommend to control the batch-to-batch variability? (f) A critical dimension (measured in m ) is of interest to the process engineer. Suppose that five fixed positions are used on each wafer (position 1 is the center) and that two consecutive wafers are selected from each batch. The data from several batches are shown as below (Table 1). Please (a) check the <a href="/keyword/normality-assumption/" >normality assumption</a> of the measured data. (b) Construct separate control charts (3 pairs) that allow (b.1) within-wafer, (b.2) between wafers, and (b.3) lot-lot variability to be evaluated. 2 Table 1 Measured wafer dimensional data 3 Problem 2: When we analyze the control charts, the X and R chart should be analyzed and interpreted separately as well as together. A discussed in during the lectures, a stable and in-control process can have 99.73% of points randomly distributed within the control limits on the charts. The X and R charts points should not be correlated to each other (follow each other patterns). However, in certain situations, these charts might show similar trends (they &quot;move&quot; together). Let say that we have two processes whose corresponding control charts are shown as Figs. 2-1 and 2-2. It was observed that the points have some level of correlation between X and R charts as it is marked with circles in Figs. 2-1 and 2.2. (a) Please identify types of the correlations in these two processes, and then draw sketches of the process data distributions and explain the consequence(s) of aforementioned correlation(s) between X and R charts. (b) Let say that Figs 2-1 (a) and (b) represent data taken at Stage 1 of the process in Fig. 2-3, and Figs 2-2 (a) and (b) represent data taken at Stage 2 of the process in Fig. 2-3. Also, let say that both Figs 2-1 and 2-1 measure the same point on the product (Fig. 2-4). However, measurements were taken in different stages (Stages 1 and 2 in Figs 2-3 and 2-4). Please provide analysis of the data presented in Figs. 2-1 and 2-2. Specifically identify all relations between: i. X in Fig. 2-1 X in Fig. 2-2 ii. R in Fig. 2-1 R in Fig. 2-2 iii. ( X , R) in Fig. 2-1 ( X , R) in Fig. 2-2 iv. X in Fig. 2-1 R in Fig. 2-1 v. X in Fig. 2-2 R in Fig. 2-2 4 (a) (a) (b) (b) Fig. 2-1 Control charts for measurement point at Stage 1 Fig. 2-2 Control charts for measurement point at Stage 2 Stage 2 Stage 1 Measurement station Fig. 2-3 Flowchart of Multi-stage Manufacturing Process 5 Same Measurement point ... Stage 1 (Underbody) Stage 2 (Full body) Fig. 2-4 Multi-stage Vehicle Body Assembly Process 6 Problem 3: A quality characteristic associated with circuit boards is measured to the nearest tenth of a mil (i.e., 0.0001 inch). The current specification is 0 ~ 10 mils, but the company is interested in the possibility of being able to meet a reduced specification of 0 ~ 6 mils. Data collected from current production are give in the table below. (a) Determine a reasonable interval width and compile a frequency distribution. (b) Plot the associated relative frequency histogram and interpret the results. (c) Plot the associated ogive and use it to estimate the proportion of circuit boards for which the measure is 6 mils or greater. (d) Construct X and R charts. Comment. (e) Check the <a href="/keyword/normality-assumption/" >normality assumption</a> of the process data. How about the process capability? (f) If we can want to setup X and R charts to monitor the process in the near future, how should we deal with the current data (25 samples) to calculate the control limits? Hint: for the concept of process capability, if 6 sigma &lt; specification, the process is capable, otherwise it is incapable. Sigma represents the estimated variance of measured process data. 7 Problem 4: It is desired to study an injection molding process with a p chart. However, because the number of parts produced per hour varies, an engineer who studies the process is uncertain as to how the data should be analyzed. The data shown below summarize the number ...