Question

Consider a manufacturer of personal computers that sells PCs through the 4 Best Buy stores in the Triangle area. The weekly demand at the Durham store is normally distributed with mean 30 and standard deviation 8. The weekly demand at each of the Cary and two Raleigh stores is normally distributed with mean 20 and standard deviation 5. Inventory is reviewed on a continuous basis and the lead time from the manufacturer’s plant to the retail stores is 5 days.

a. Please compute the safety stock levels at each of the four retail stores if the manufacturer wants to achieve a probability of no stock-out of 98%.

b. If each computer costs $1500 and the holding cost is based on an annual interest rate of 20%, what is the total annual cost of carrying safety inventory across the four retailers?

c. The PC manufacturer decides that they don’t need the retailers to sell their computers. They decide to withdraw their inventory from the retail stores and satisfy customer demand from a central warehouse in Cary. Assume that customers are indifferent between buying at a retail store versus waiting for their computer to be shipped from the warehouse, i.e. demand does not change. The warehouse will also follow a continuous inventory review policy and the lead time from the plant is still 5 days. Please compute the annual cost of carrying safety inventory under this supply chain design using the cost parameters from part (b).

Answer 1

a) At 98% confidence interval the z value is 2.33 (this is from the normal distribution table)

Whenever there is a variable demand, the safety stock is given by the equation

Safety stock = z*SD*sqrt(L)

Here SD is the standard deviation and L is the lead time.

Using these information we can calculate the safety stock level for each of the four retail stores.

Durham safety stock = 2.33*8*sqrt(5) = 41.68 units

Cary and two Raleigh stores safety stock = 2.33*5*sqrt(5) = 26.05 units

b) Holding cost (H) = 20% of 1500 = 300

Annual carrying cost for Durham is 41.68*300 = 12504

Annual carrying cost for Cary and two Raleigh stores = 26.05*300 = 7815

However the cost for 3 of them will be 7815*3 = 23445

Total annual carrying cost of the safety stock for the 4 stores is 35949

c) If the manufacturer wants to handle the demand then they will need to adjust the demand accordingly. Now we know the four independent normal distribution data. These needs to be combined. In probability theory the sum of two normal distribution results in a normal distribution where the new mean is the sum of the independent means and the new variance is the sum of the independent variances. Thus the new demand will be

Mean = 30 + 20 + 20 + 20 = 90

Variance = 8^2 + 5^2 + 5^2 + 5^2 = 139

This means the SD = sqrt(139) =11.78

This means the safety stock needed is 2.33*11.78*sqrt(5) = 61.37

The annual cost of this safety stock holding will be 61.37*300 = 18412