Question

You are hired to manage the inventory of product X. The annual demand for product X is normally distributed with mean 1000 and standard deviation 400. The warehouse you rent charges you $1 per item per month for holding product X. Each time you place an order, you pay $60 for customs fee. Please use the above information to answer the following three questions.

a. What is your optimal order quantity that minimizes the total annual ordering and holding costs?

b. Suppose you have a supply lead time of 0.25 years. How much safety stock should you hold for a 95% service level?

c. You recently bought a warehouse so there is no rental fee for holding cost. Each inventory of product X costs you $300. You also realize that saving money in the bank, instead of buying inventory, will generate an interest of 2% annually. What is the optimal order quantity that minimizes the total annual ordering and holding costs? (hint: holding cost now becomes an opportunity cost of saving the money at the bank to earn interest)

Answer 1

(a)

Annual demand, D = 1,000
Unit holding cost, h = $1
Ordering cost, K = $60

Optimal order quantity, Q* = (2.D.K / h)^1/2 = sqrt(2*1000*60/1) = 346 (rounded off)

Total annual ordering cost = (D/Q*) x K = (1000/346)*60 = $173.4
Total annual holding cost = (Q*/2) x h = (346/2)*1 = $173

(b)

Service level = 95%
Average lead time, L = 0.25 years

Z = normsinv(0.95) = 1.645

Safety stock = Z * σ * √L = 1.645*400*√0.25 = 329

(c)

Holding cost, h = opportunity lost by holding inventory = 2% * $300 = $6

Optimal order quantity, Q* = (2.D.K / h)^1/2 = sqrt(2*1000*60/6) = 141 (rounded off)

Total annual ordering cost = (D/Q*) x K = (1000/141)*60 = $425.5
Total annual holding cost = (Q*/2) x h = (141/2)*6 = $423