The Ratas Company is using mainly teak lumber in the manufacture of outdoor furniture. They purchase teak lumber from a far away lumberjack in medium-to-large amounts and store it in their lumber yard near the factory. Each order placed to the lumberjack costs K to them. They use up lumber in the manufacture of outdoor furniture at the rate of lambda tons/month. Ratas assumes that the inventory carrying costs for lumber are made up of two parts: inventory charge (ic/ton/month) proportional to the average inventory, and warehousing cost (w/ton/month) proportional to the maximum inventory. Ratas may allow for stock-outs of lumber, though manufacturing of furniture may stop for a while, until the replenishment order arrives. They assume a shortage cost of pi/ton/month, when they are out of stock. Ratas assumes 20 days in a month.

a) Formulate the total monthly cost function G(Q,B) for Ratas. (Let Q and B be the order quantity and maximum stock-out level for lumber, respectively).

b) Derive the optimum order quantity, Q* and the backorder level, B*. Calculate the values of Q* and B* if K=500 TL/order, c=120,000 TL/ton, w=5 TL/ton/month, pi=500 TL/ton/month, i=1%/month, lambda=5 tons/month.

c) What should the minimum capacity of lumber yard (in tons) if Ratas continues to order in the optimum order quantity?

d) What should the reorder point of Ratas be, if the lumberjack usually sends the order to Ratas in 5 days?

e) What is Ratas’s longest delay (in days) in manufacturing, when Ratas is out of lumber stock?