TQM & SPC Problem Solutions

13 a c p 150 100 6 10 56 083333 c pk min 150 100310

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13. a. c p = (150 - 100) / ( 6 * 10) = 5/6 = 0.83333 c pk = MIN ( (150-130)/(3 * 10), (130-100)/(3*10) ) = MIN ( 2/3, 3/3) = 0.66667 b. c pk is a better measure of process capability in this situation (and in most situations). The reason is that c p assumes that the specification limits are equally distant from the mean of the process. In many situations, including this problem, the mean is not centered in the specification limits and c p actually over-states the process capability. c. The percentage of product that can be expected to fall outside the specification limits is the sum of the percentage of those observations that are expected to be more extreme than the USL and LSL. Since the specification limits are not equidistant from the mean of the process, the expected percentages above the USL will not be equal to the expected percentage below the LSL. The answer can be found by utilizing the solution developed for c pk in part (a) above. Any observation farther than 2 standard deviations above the mean is beyond the upper specification limit. From appendix A (Areas under the standard normal probability distribution), the expected percentage of observations more extreme (greater) than the USL is 100*(1 - (.5 + .4772)) = 2.28 %. Likewise, the percentage of points expected to appear below the LSL is given by the expected percentage of observations more extreme (less) than 3 standard deviations from the mean, or 100*(1 - (.5 + .4987)) =0.13%. The solution, then, is 2.28% + .13% = 2.41% of the product can be expected to fall outside the specification limits. 14. a. If c p = (1000 - 800) / ( 6 * SD), then a little algebra yields SD = 200/ (6 *1.5)= 22.22 b. Since the process is centered in the USL and LSL, the mean value is: 9 - 9
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(1000 + 800) / 2 = 900. c. Several possibilities may exist for the company. A few are listed here: 1) The producer could attempt to negotiate wider specification limits. This may not be acceptable to the customer in the short or long term. 2) The producer could attempt to modify the product or production process so that the finished product would meet the customer's requirements. This may entail additional time, cost, and/or may or may not be feasible. However, there are many short and long term advantages to choosing this approach. 3) The producer could choose to turn down this order if it is not capable of producing according to customer specifications. While this choice may not be palatable to some, situations exist where it may be necessary. 4) The producer could produce to capability, and then perform 100% inspection to weed out all products, which fall outside the customer's specifications. However, this choice is contrary to the principles of continuous improvement of quality and should only be used in the short run while improvements are being made. 9 - 10
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