Improving quality performance (x and R tables, Six Sigma)

The company Dairy Products produces milk powder and other dairy products, like milk and butter. The milk powder is packaged in large sacks of 80 pounds each. The packing and filling machines used to put milk powder into these large sacks have not been operating well lately.

Dairy Products’ management has decided to have a look at the x and R charts to see whether there are structural deficiencies that need correction. The samples are taken from the last fifteen days, one in the morning shift

(a) and the other in the afternoon shift (b).

Required:
1. Make an x and R chart. Can you detect from these charts whether there are structural problems with the packing and filling machines? Which structural elements are visible in these charts?
2. The management team decides to impose a lower and upper control level. The lower control level is 70 pounds and the upper is 90 pounds. What are the chances of the existing system surpassing the upper control level? What are the chances of falling below the lower control level?
3. What is the Six Sigma capacity index for this process, if we correct the index for the mean being off-centre? Comment on the outcome.
4. Suppose management decides to apply the 12@s distribution to the 70-90 control limits, reducing the risk of surpassing these limits. What are the new 3@s control limits (make sure you take into account the mean being off-centre)?

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Sample information (each sample contains 6 independent observations) Samples Batch 1 1 2 ABA B 3 A B 4 A d88 = 326 4 6 68 80 78 82 78 91 71 73 85 91 84 75 71 81 91 81 83 68 B 66 87 69 5 A 78 84 79 75 B 77 81 86 83 6 A 83 66 66 67 B 73 63 84 73 86 7 A 75 82 84 69 88 B 71 85 86 72 80 8 65 82 87 84 68 B 76 93 89 86 9 69 80 76 87 B 92 83 74 10 A 71 86 74 B 84 94 11 A 81 90 B 68 90 83 12 A 71 93 71 95 74 B 85 84 76 75 84 13 A 82 74 70 95 90 B 67 97 91 80 76 71 14 A 79 B 15 A B 82 86 71 77 85 28 81 8986