Question

A company serves two markets in North America and has to decide on optimal order quantity and safety stock. The demand in the first market is equal to 1, 2, 3, or 4 with equal probability, whereas the demand in the second market is equal to 3, 4, or 5 with equal probability. The company has a target level of product availability of CSL=90% on both markets.

a) (1.5 points) What is the optimal order quantity for market 1? What is the optimal safety

stock for market 1?

b) (1.5 points) What is the optimal order quantity for market 2? What is the optimal safety

stock for market 2?

c) (0.5 points) What are optimal order quantity and safety stock for the decentralized

distribution system?

d) (2.5 points) What are optimal order quantity and safety stock for the centralized distribution system?

Answer 1

a)

Demand	Probability	Cumulative probability
1	0.25	0.25 < 0.9
2	0.25	0.50 < 0.9
3	0.25	0.75 < 0.9
4	0.25	1.00 > 0.9

So, the optimum order size for covering the demand in 90% cases = 4.

Average demand is 2.5,

So, safety stock = 4 - 2.5 = 1.5

b)

Demand	Probability	Cumulative probability
3	0.333	0.333 < 0.9
4	0.333	0.667 < 0.9
5	0.333	1.000 > 0.9

So, the optimum order size for covering the demand in 90% cases = 5.

Average demand is 3,

So, safety stock = 5 - 3 = 2

c)

The safety stock will be just the sum of a) and b) i.e. 1.5+2 = 3.5

The order size has to be reported separately i.e. 4 and 5, we cannot add them. Adding them means nothing in real as they will be ordered separately.

d)

Market-1	Market-2	Total demand	Probability
1	3	4	0.25 * 0.333 = 0.0833
1	4	5	0.0833
2	3	5	0.0833
1	5	6	0.0833
2	4	6	0.0833
3	3	6	0.0833
2	5	7	0.0833
3	4	7	0.0833
4	3	7	0.0833
3	5	8	0.0833
4	4	8	0.0833
4	5	9	0.0833

Combine the probability of the same total demand numbers in the following table and compute the cumulative probability after that only.

Total demand	Probability	Cumulative probability	Demand * prob
4	0.0833	0.0833	0.333
5	0.1667	0.2500	0.833
6	0.2500	0.5000	1.500
7	0.2500	0.7500 < 0.9	1.750
8	0.1667	0.9167 > 0.9	1.333
9	0.0833	1.0000	0.750
Total	1.0000		6.500

So, the optimum order size for covering the demand in 90% cases = 8.

Average demand is = 6.5,

So, safety stock = 8 - 6.5 = 1.5