Inventory Models: EOQ Formula, Derivation, Examples

The Economic Order Quantity (EOQ) model answers a basic business question: how much stock should you order each time to keep total inventory costs as low as possible? Ordering too much wastes money on storage; ordering too little wastes money on repeated order fees.

It matters because inventory ties up cash. Every shop, warehouse and factory balances ordering costs against holding costs daily, and EOQ gives the exact order size that minimises their sum — one of the oldest and most used results in operations research.

Key Concepts

• Demand rate D: units needed per year (assumed constant).
• Ordering cost S: fixed cost per order placed (paperwork, delivery).
• Holding cost H: cost of storing one unit for one year.
• Order quantity Q: the batch size; the decision variable.
• Reorder point: the stock level that triggers a new order: R = d·L, where d is daily demand and L is lead time in days.
• Cycle stock: average inventory Q/2, since stock runs from Q down to zero.

Formulas

• Total annual cost: TC(Q) = (D/Q)·S + (Q/2)·H + P·D (P = unit purchase price).
• EOQ: Q* = √(2DS / H)
• Number of orders per year: N = D/Q*; time between orders: T = Q*/D years.

Assumptions: constant known demand, instant delivery, no stockouts, no quantity discounts, holding cost proportional to inventory. The square-root form means doubling demand only increases EOQ by about 41%.

Worked Example

A shop sells D = 1,200 notebooks per year. Ordering cost S = ₹50 per order; holding cost H = ₹2 per notebook per year.

1. Q* = √(2 · 1200 · 50 / 2) = √(120,000/2) = √60,000 ≈ 244.9 → order 245 notebooks.
2. Orders per year: N = 1200/245 ≈ 4.9 → about 5 orders.
3. Annual ordering cost: (1200/245)·50 ≈ 4.898·50 ≈ ₹244.90.
4. Annual holding cost: (245/2)·2 = 245·1 = ₹245.00.
5. Total (excluding purchase price): ≈ ₹489.90. Note ordering and holding costs are equal at EOQ — that balance is exactly what the formula achieves.

Common Misconceptions

• "Bigger orders are always cheaper per unit." They cut ordering cost but raise holding cost; EOQ finds the crossover point.
• "EOQ works with discounts." Quantity discounts need a modified model — the plain EOQ assumes constant unit price.
• "You must order fractional EOQ exactly." The cost curve is flat near the optimum; rounding to a practical number costs almost nothing.

Real-World Applications

• Retailers set standard reorder quantities for thousands of products.
• Manufacturers size raw-material purchase lots.
• Spare-parts managers decide stocking levels for maintenance supplies.

The EOQ model balances two opposing costs — ordering and holding — with a single square-root formula. Its assumptions are strict, but the insight that total cost is minimised where the two costs meet shapes nearly all inventory policy.