Pancake Jacks will begin to bottle syrup for distribution in local markets. The

Question

Pancake Jacks will begin to bottle syrup for distribution in local markets. The specification limits are set at 16.0 ounces on the lower end (as printed on the label) and 16.5 ounces on the higher end (no more would fit in the bottle). Bottles that are filled too low cost $0.80 each to go back and top off. Bottles that are filled too high cause leakage into the machines, averaging $10.00 per bottle to clean up. The cost of syrup is 18 cents per ounce.

The company uses a filling machine with a natural process variability of = 0.10 ounces. Pete would like to set the target at 16.10 ounces, and Shirley would like to set the target at 16.25 ounces. By computing expected filling and fixing costs per unit of each option, determine which of those two options appears to be most economical. Please round to four decimal places in your calculations, including monetary amounts, i.e., 1/100 of a penny.

Can you identify a target that is better than either Shirley’s suggestion or Pete’s suggestion? What would that target be? Support your decision.

Explanation / Answer

FIrst we will Evaluate Pete' suggestion

The mean filling of bottle = 16.10 ounce

Expected Total Cost = Lower Filled bottles go back and top off cost + HIgh filled bottles to clean up + Cost of syrup

Lower Filled bottles go back and top off cost = Pr (Lower filled Bottles) * 0.80 = Pr(X < 16.00; 16.10; 0.10) * 0.80

Z - value = (16.00 - 16.10)/ 0.10 = -1

so Pr(X < 16.00; 16.10; 0.10) = (Z < -1) = 0.1587

Lower Filled bottles go back and top off cost = 0.1587 * 0.80 = $ 0.1270

HIgh filled bottles to clean up = 10 * Pr(X > 16.50; 16.10; 0.10)

Z = (16.50 - 16.10)/ 0.10 = 4

so Pr(X > 16.50; 16.10; 0.10) = (Z > 4) = 0.0001

High Filled bottles = 0.0001 * 10 = $ 0.001

Cost of syrup = 0.18 * 16.10 = $2.898

Expected Total Cost per bottle = 2.898 + 0.1270 + 0.001 = $ 3.0260 per bottle

FIrst we will Evaluate Shirley' suggestion

The mean filling of bottle = 16.25 ounce

Expected Total Cost = Lower Filled bottles go back and top off cost + HIgh filled bottles to clean up + Cost of syrup

Lower Filled bottles go back and top off cost = Pr (Lower filled Bottles) * 0.80 = Pr(X < 16.00; 16.25; 0.10) * 0.80

Z - value = (16.00 - 16.25)/ 0.10 = -2.5

so Pr(X < 16.00; 16.25; 0.10) = (Z < -2.5) = 0.0062

Lower Filled bottles go back and top off cost = 0.0062 * 0.80 = $ 0.00496

HIgh filled bottles to clean up = 10 * Pr(X > 16.50; 16.10; 0.10)

Z = (16.50 - 16.25)/ 0.10 = 2.5

so Pr(X > 16.50; 16.10; 0.10) = (Z > 2.5) = 0.0062

High Filled bottles = 0.0062 * 10 = $ 0.062

Cost of syrup = 0.18 * 16.25 = $2.925

Expected Total Cost per bottle = 2.925 + 0.062 + 0.00496 = $ 2.9920 per bottle

So , Shirley's suggestion epected cost per bottle is less than the pets's suggessted expected cost per bottle. the cost per bottle for shirley is lesser than we shoudl go for it.

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