Model Assumptions

• Demand is deterministic and constant throughout the year.
• Inventory depletes at a constant rate.
• Replenishment is instantaneous when inventory reaches zero.
• No shortages or backorders are allowed.

These assumptions are illustrated in the inventory cycle diagram (Fig. 18.1), where inventory level drops linearly and is instantly restored to maximum \( Q \).

Inventory Cost Components

Let:

• \( Q \): Order quantity
• \( D \): Annual demand (units/year)
• \( A \): Ordering cost per order
• \( h \): Annual holding cost per unit

Ordering Cost

Number of orders per year: \( \frac{D}{Q} \)
Annual ordering cost: $$ O(Q) = \frac{D}{Q} \cdot A $$

Holding Cost

Average inventory level: \( \frac{Q}{2} \)
Annual holding cost: $$ H(Q) = \frac{Q}{2} \cdot h $$

Total Inventory Cost

$$ T(Q) = O(Q) + H(Q) = \frac{D}{Q} \cdot A + \frac{Q}{2} \cdot h \quad \text{(Eq. 18.1)} $$

Derivation of EOQ

To minimize total cost \( T(Q) \), set the derivative to zero: $$ \frac{dT(Q)}{dQ} = -\frac{DA}{Q^2} + \frac{h}{2} = 0 $$ Solving for \( Q \): $$ Q^* = \sqrt{\frac{2AD}{h}} \quad \text{(Eq. 18.2)} $$ This is known as the Wilson-Harris formula.

Optimal Metrics

• Total cost at EOQ: $$ T(Q^*) = \sqrt{2ADh} $$
• Optimal number of orders per year: $$ n^* = \frac{D}{Q^*} $$
• Optimal cycle time (in days): $$ t^* = \frac{Q^*}{D/365} = \frac{365}{n^*} $$
• EOQ is inversely proportional to the square root of holding cost: $$ Q^* \propto \frac{1}{\sqrt{h}} $$

Safety Stock Consideration

Let \( s \) be the stock-out cost per incidence. The adjusted carrying cost becomes \( h + s \). Safety stock is: $$ Q_s = Q^* \cdot \sqrt{\frac{s}{s + h}} $$