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I'm busy with a small project where a large amount of samples have been taken from a manufacturing process (2700 samples of 11 items). A specified Upper and Lower Specification Limit has been set, and items under the LSL are said to cost $3 to fix, while items above the USL are said to cost $5 to fix. The data is spread with a uniform distribution.

How would I go about deciding where to centre the process (given that the distribution would stay the same along the centre line) to minimize total cost? I know how to do it iteratively, but I'd like a more optimal way to solve this problem.

EDIT: Here is an example of the data I'm working with.

One sample would be, for instance

  • 45.62565379
  • 47.06496942
  • 46.39000538
  • 46.44387364
  • 45.81911053
  • 45.25935862
  • 48.75357907
  • 46.50918593
  • 46.87072887
  • 46.60195194
  • 48.09000017

There are 2701 more samples like the one above (albeit with different values) making up my population. The population mean is 47.66 and population standard deviation is 1.425. The UCL is 48.98 and the LCL is 46.34. The USL has been set to 50 and the LSL to 45.

Currently the process is centered around the population mean, but the amount of samples with means above 50 is proportionally larger than that of the amount of samples with means under 45, meaning that the process is more expensive, as it costs $5 to fix a batch above the USL and only $3 to fix it under the LSL. How do I decide where to centre the process if its distribution around the centre line will remain the same to minimize cost?

Martin
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    Please provide a numerical example. – PascalVKooten Oct 09 '14 at 08:54
  • You should put here you iterative code so that it can be optimized. Also see here for a reproducible example: http://stackoverflow.com/questions/5963269/how-to-make-a-great-r-reproducible-example – rnso Oct 09 '14 at 09:04
  • I haven't written the iterative code, as it would be a very slow process to run. It would involve calculating the total cost, shifting the centreline by a small amount, recalculating the total cost and comparing it to the previous cost to see if it improved. – Martin Oct 09 '14 at 09:10
  • i don't get it. Your process generates uniform data, right? so you know the min and max (a,b) of your distribution. Therefore an optimal solution is to set 'b' exactly at your USL limit (since cost(usl) > lsl). But maybe you did not mean uniform distribution (which is strange indeed for an industrial process) – agenis Oct 09 '14 at 09:31
  • Hm. I see your point. I'm not sure why I overlooked that... I'm sure there was a reason. Maybe the data is normally distributed after all. I'll check it out and get back here if it is. – Martin Oct 09 '14 at 09:39
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    Whether your code is slow or not is not the issue. You're expected to show some effort and present the state of your work here. The point of SE is to get help improving your skill/knowledge. – Carl Witthoft Oct 09 '14 at 11:37

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