3. Calculate the critical fractile based on the underage and average costs given 1 and 2 above. This gives the optimal in-stock rate for those costs. The corresponding optimal order-up-to point is the smallest order-up-to point that has an in-stock rate equal to or higher than the critical fractile. Use your table of order-up-to points and in-stock rates to find the optimal order-up-to point based on your calculated critical fractile.

Critical Fractile = (cost of underage)/(cost of average + cost of underage)

= 1/(0.2 + 1)

Critical Fractile = 0.8333

Optimal Stock Rate = 83.33%

Using Excel sheet for different order up to point = 59 to 63 (Stock rate = 84% > 83.33%)

Decision Variable
Order up to Point	59
Total Results
Total Demand	2012
Total Sales	1905
Average Inventory	39.10
Total Lost Sales (units)	107
Total Gross Profit	$1,905
Total Inventory Holding Cost	$391.02
Net Profit	$1,514
In-Stock Rate	84%
Fill Rate	94.7%

Decision Variable
Order up to Point	60
Total Results
Total Demand	2012
Total Sales	1913
Average Inventory	40.07
Total Lost Sales (units)	99
Total Gross Profit	$1,913
Total Inventory Holding Cost	$400.74
Net Profit	$1,512
In-Stock Rate	84%
Fill Rate	95.1%

Decision Variable
Order up to Point	61
Total Results
Total Demand	2012
Total Sales	1921
Average Inventory	41.05
Total Lost Sales (units)	91
Total Gross Profit	$1,927
Total Inventory Holding Cost	$410.49
Net Profit	$1,511
In-Stock Rate	84%
Fill Rate	95.5%

Decision Variable
Order up to Point	62
Total Results
Total Demand	2012
Total Sales	1929
Average Inventory	42.03
Total Lost Sales (units)	83
Total Gross Profit	$1,929
Total Inventory Holding Cost	$420.26
Net Profit	$1,509
In-Stock Rate	84%
Fill Rate	95.9%

Decision Variable
Order up to Point	63
Total Results
Total Demand	2012
Total Sales	1937
Average Inventory	43
Total Lost Sales (units)	75
Total Gross Profit	$1,937
Total Inventory Holding Cost	$430.05
Net Profit	$1,507
In-Stock Rate	84%
Fill Rate	96.3%

Optimal order up to point (average) = (59+60+61+62+63)/5

Optimal order up to point = 61

Question

4. The number calculated from 3 does not match the order-up-to point determined in #1 (a-v). Brown wonders if his cost of underage depends on his order-up-to point. Evaluate Brown's thought that if his order-up-to point is high enough, the cost of underage is close to his margin minus the holding cost, but as his order-up-to point gets lower and lower the cost of underage should get closer to just the margin.