Question

1. Suppose you sell three types of donuts: plain, nut covered and chocolate covered. The daily demand of each type of donuts is normally distributed and independent of each other. The means and standard deviations are given in the following table:

	Mean	Standard deviation
Plain	16	4
Nut covered	20	10
Chocolate covered	25	5

Assume that the production lead time of making donuts is zero, and you replenish your inventory of donuts every morning (i.e. review period is one day). Besides, you want to achieve a service level of 80% for each type of donuts.

1. If you make three types of donuts at the beginning of the day, how much safety stock do you need to carry in total to achieve a service level of 80% for each type of donuts? (Please round it up when the safety stock for any type of donut is not an integer, such as 1.01≈ 2)

2. Instead of making three types of donuts at the beginning of the day, you can only make plain donuts and dip plain donuts in nuts or chocolate on demand (assume for no extra cost). If you follow this strategy, how much safety stock do you need to carry in total to achieve a service level of 80% for each type of donuts? (Please round it up when the safety stock is not an integer, such as 1.01≈ 2)

3. If you follow the strategy in part b, compared to part a, the total safety stock would go down by how much percentage? (Please round it to the closest integer percentage, e.g. 10.1%≈10%)

Answer 1

According to the periodic review system, the safety stock formula is given as follows:

Safety stock during lead time and review period = zσT+L = z*σd√(L + T)

T = review period = 1

L = lead time = 0

SS = z*σd√(0 + 1) = z*σd

For 80% of cycle service level, z = 0.8416 (from excel function “=Normsinv())

SS = z*σd = 0.8416*σd

Part a: Safety stock calculation of the products:

Product	Mean	σd	SS = zσT+L = 0.8416* σd	Adjusted SS
Plain	16	4	3.3664	4
Nut Covered	20	10	8.4162	9
Chocolate Covered	25	5	4.2081	5
			Total	18

Total safety stock = 18 units

Part b:

If the plain donuts are made, than the total of mean demand of donuts = mean demand of plain donuts = 16 + 20 + 25 = 61 units

To determine, the standard deviation of mean demand of the plain donuts, calculate total variance of three donuts-type as follows:

Product	Mean	σd	Variance = (σd)2
Plain	16	4	4*4 = 16
Nut Covered	20	10	100
Chocolete Covered	25	5	25
Total	61		141
		SD	11.87434

Standard deviation of the mean demand of plain donuts = √total variance = √141 = 11.8743

For CSL of 80%,

SS = zσT+L = 0.8416* σd = 0.8416*11.874 = 9.99

Adjusted SS = 10 units

Safety stock of Plain donut = 10 units

Part c:

Reduction in the safety stock due to aggregation = (Total SS of donuts - SS of pain donuts)/Total SS of donuts

Reduction = (18 – 10) / 18 x 100 = 44.44 %

The reduction in the safety stock = 44%