Question

4. During the past fifty lead times, demands for the AAA toothpaste have been as follows:

What are the reorder points to achieve the following indicated service level? What amount of safety stock should be kept?

a) 99%, b) 95%, c) 90%, and d) 50%

Demand	2	3	4	5	6	7	8
Frequency	1	5	15	17	8	3	1

Answer 1

Step 1 : We need to first find the probability of demand of each of the level and the cumulative probability

Probability of demand of 2 units during the lead time = Frequency/Total observations = 1/50 = 0.02 = 2%
Probability of demand of 3 units during the lead time = Frequency/Total observations = 5/50 = 0.1 = 10%
Probability of demand of 4 units during the lead time = Frequency/Total observations = 15/50 = 0.3 = 30%
Probability of demand of 5 units during the lead time = Frequency/Total observations = 17/50 = 0.34 = 34%
Probability of demand of 6 units during the lead time = Frequency/Total observations = 8/50 = 0.16 = 16%
Probability of demand of 7 units during the lead time = Frequency/Total observations = 3/50 = 0.06 = 6%
Probability of demand of 8 units during the lead time = Frequency/Total observations = 1/50 = 0.02 = 2%

Step 2 : Next we find the cumulative probability of the demand. Cumulative probability at each level is the sum of the probability at each level of demand and all the previous level of demand

Cumulative probability of demand of 2 units during the lead time = 2% = 2%
Cumulative probability of demand of 3 units during the lead time = 2% + 10% = 12%
Cumulative probability of demand of 4 units during the lead time = 12% + 30% = 42%
Cumulative probability of demand of 5 units during the lead time = 42% + 34% = 76%
Cumulative probability of demand of 6 units during the lead time = 76% + 16% = 92%
Cumulative probability of demand of 7 units during the lead time = 92% + 6% = 98%
Cumulative probability of demand of 8 units during the lead time = 98% + 2% = 100%

Step 3 : Now we find the average demand during the lead time which is the sum of product of the probability and the demand. Hence average lead time demand is as follows

Lead time demand = (2*2% + 3*10% + 4*30% + 5*34% + 6*16% + 7* 6% + 8*2%) = 4.78

Now Safety Stock is given by the formula, Safety Stock = Reorder Point - Average Lead Time demand

Now Reorder point at each service level is the demand level at which the cumulative probability calculated in step 2 is more than or equal to the required service level.

For service level of 99%, we see that cumulative probability of 99% or more occurs at demand level of 8.

Thus Safety Stock for 99% service level = 8-4.78 = 3.22

For service level of 95%, we see that cumulative probability of 95% or more occurs at demand level of 7.

Thus Safety Stock for 95% service level = 7-4.78 = 2.22

For service level of 90%, we see that cumulative probability of 90% or more occurs at demand level of 6.

Thus Safety Stock for 90% service level = 6-4.78 = 1.22

For service level of 50%, we see that cumulative probability of 50% or more occurs at demand level of 5.

Thus Safety Stock for 50% service level = 5-4.78 = 0.22