Exercise Set 7.6
Inventory Control

A. Problem

Holding inventory is costly for firms. They have to pay rent, utilities and insurance for warehouses, and salaries for warehouse employees. On the other hand, placing orders more often (in order to maintain low inventory levels) is also costly -- there is the cost of sending purchase orders, processing incoming inventory, and utilities and phone bills for the purchasing dept. Also, low inventory levels may result in dissatisfied customers. Hence the need for inventory control.

B. Assumptions

  1. Demand is known and constant
  2. The only variable costs are:
    1. Ordering cost: The cost of placing an order
    2. Carrying cost: The cost of holding inventory over time (also called holding cost)

C. Notation

Q = Number of pieces per order

Q* = Optimal number of pieces per order (also called EOQ, economic order quantity)

D = Annual demand in units, for the inventory item

Co = Ordering cost for each order

Ch = Carrying cost per unit per year

D. Cost components

Annual ordering cost
= (Annual demand / No. of units per order) x (Order cost per order)
= (D/Q)Co

Annual carrying cost
= (Average inventory level) x (Carrying cost per unit per year)
= (Q/2)Ch

E. Firm's optimization problem

To minimize total cost by choosing optimal value of Q

Total cost = (D/Q)Co + (Q/2)Ch

F. Questions

  1. Obtain the FOC for a minimum.
  2. Obtain the optimal value of Q:
    Q* = (2DCo/Ch)(1/2)
  3. Is the SOC for a minimum met?
  4. At Q*, what is the relationship between the Ordering cost and Carrying cost?

G. Graphs

  1. Plot costs versus Q.
  2. As Q rises, the ordering cost [ increases / decreases ].
  3. Carrying cost is linear. What is the slope of the line?
  4. What is the shape of the Total Cost curve?
  5. What is the significance of the intersection of the ordering cost curve and carrying cost curve?

H. Comparative Statics

What is the change in the EOQ due to:

  1. An increase in Co?
  2. A decrease in Ch?
  3. An increase in demand?

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