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appear to be operating in a state of statistical control at the desired target level 16-40. The concentration of a chemical product is measured by taking four samples from each batch of material. The average concentration of these measurements is shown for the last 20 batches in the following table: Batch 1 2 3 4 5 6 7 8 9 10 Concentration 104.5 99.9 106.7 105.2 94.8 94.6 104.4 99.4 100.3 100.3 Batch 11 12 13 14 15 16 17 18 19 20 Concentration 95.4 94.5 104.5 99.7 97.7 97.0 95.8 97.4 99.0 102.6
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(a) Suppose that the process standard deviation is 8 and that the target value of concentration for this process is 100. Design a CUSUM scheme for the process. Does the process appear to be in control at the target (b) How many batches would you expect to be produced with off-target concentration before it would be detected by the CUSUM control chart if the concentration shifted to 104 Use Table 16-9. 16-41. Consider a standardized CUSUM with H 5 and K 1 2. Samples are taken every two hours from the process. The target value for the process is 0 50 and 2. Use Table 16-9. (a) If the sample size is n 1, how many samples would be required to detect a shift in the process mean to 51 on average (b) If the sample size is increased to n 4, how does this affect the average run length to detect the shift to 51 that you determined in part (a) 16-42. A process has a target of 0 100 and a standard deviation of 4. Samples of size n 1 are taken every two hours. Use Table 16-9. (a) Suppose the process mean shifts to 102. How many hours of production will occur before the process shift is detected by a standardized CUSUM with H 5 and K 1 2 (b) It is important to detect the shift de ned in part (a) more quickly. A proposal is made to reduce the sampling frequency to 0.5 hour. How will this affect the CUSUM control procedure How much more quickly will the shift be detected (c) Suppose that the 0.5 hour sampling interval in part (b) is adopted. How often will false alarms occur with this new sampling interval How often did they occur with the old interval of two hours (d) A proposal is made to increase the sample size to n 4 and retain the two-hour sampling interval. How does this suggestion compare in terms of average detection time to the suggestion of decreasing the sampling interval to 0.5 hour
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While the control chart is a very powerful tool for investigating the causes of variation in a process, it is most effective when used with other SPC problem-solving tools. In this section we illustrate some of these tools, using the printed circuit board defect data in Example 16-4. Figure 16-17 shows a U chart for the number of defects in samples of ve printed circuit boards. The chart exhibits statistical control, but the number of defects must be reduced. The average number of defects per board is 8 5 1.6, and this level of defects would require extensive rework. The rst step in solving this problem is to construct a Pareto diagram of the individual defect types. The Pareto diagram, shown in Fig. 16-22, indicates that insuf cient solder and solder balls are the most frequently occurring defects, accounting for (109 160) 100 68% of the
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Figure 16-22 Pareto diagram for printed circuit board defects.
observed defects. Furthermore, the rst ve defect categories on the Pareto chart are all solderrelated defects. This points to the ow solder process as a potential opportunity for improvement. To improve the ow solder process, a team consisting of the ow solder operator, the shop supervisor, the manufacturing engineer responsible for the process, and a quality engineer meets to study potential causes of solder defects. They conduct a brainstorming session and produce the cause-and-effect diagram shown in Fig. 16-23. The cause-and-effect diagram is widely used to display the various potential causes of defects in products and their interrelationships. They are useful in summarizing knowledge about the process. As a result of the brainstorming session, the team tentatively identi es the following variables as potentially in uential in creating solder defects: 1. 2. Flux speci c gravity Solder temperature
Solder Wave turbulance Exhaust Conveyor speed Maintenance Conveyor angle Contact time Wave fluidity Type Solder defects Alignment of pallet Solderability Pallet loading Orientation Temperature Temperature Wave height Specific gravity