Every supply chain manager, warehouse operator, and retail merchant balances on a knife's edge between two costly blunders: ordering in tiny batches (generating excessive shipping, customs, and clerical costs) or ordering massive stockpiles (running out of warehouse space and tying up hundreds of thousands of dollars in stagnant inventory).
First derived by Ford W. Harris in 1913, the Economic Order Quantity (EOQ) provides the mathematical answer to this fundamental trade-off by identifying the precise order batch size that minimizes total inventory-related expenditure.
Calculate Your Optimal Order Size
Input your annual demand and carrying costs to generate immediate EOQ lot sizes:
1. The Two Opposing Cost Forces
EOQ balances two distinct cost functions that pull in opposite directions:
Ordering Costs (S)
Fixed expenses incurred each time a purchase order is placed:
- Purchase order administration & invoice processing
- Fixed freight shipment and delivery fees
- Receiving dock labor, inspection, and unloading
- Trend: Increases if you place many small orders.
Carrying / Holding Costs (H)
Expenses incurred to store and preserve one unit over a full year:
- Warehouse rent, climate control, and shelving security
- Inventory insurance, property taxes, and shrinkage
- Product obsolescence and perishable spoilage
- Cost of capital tied up in unsold inventory
- Trend: Increases as batch order size grows.
2. Mathematical Derivation of EOQ
Total Annual Inventory Cost (TC) is the sum of annual ordering costs and annual holding costs:
Total Inventory Cost Equation:
Total Cost (TC) = (D / Q) × S + (Q / 2) × H
By taking the first derivative of TC with respect to order quantity Q, setting it to zero, and solving for Q, we arrive at the classical Wilson EOQ formula:
The Optimal Order Quantity Formula:
Q* = √ [ (2 × D × S) / H ]
3. Step-by-Step Worked Supply Chain Scenario
Consider an industrial distributor distributing specialty hydraulic filters:
- Annual Demand (D): 12,000 units/year
- Order Setup Cost (S): $100 per shipment
- Holding Cost (H): $4.00 per unit per year (warehouse storage + 15% cost of capital)
| Candidate Order Quantity (Q) | Annual Orders (D / Q) | Annual Ordering Cost | Annual Holding Cost | Total Annual Cost |
|---|---|---|---|---|
| 200 Units (Too Small) | 60 orders | $6,000 | $400 | $6,400 |
| 400 Units | 30 orders | $3,000 | $800 | $3,800 |
| 775 Units (Exact EOQ) | 15.5 orders | $1,549 | $1,549 | $3,098 (Cost Minimum) |
| 1,200 Units | 10 orders | $1,000 | $2,400 | $3,400 |
| 3,000 Units (Too Large) | 4 orders | $400 | $6,000 | $6,400 |
Notice the elegant symmetry at exact EOQ: Annual Ordering Cost ($1,549) precisely equals Annual Holding Cost ($1,549). Any deviation away from 775 units—whether ordering 200 units or 3,000 units—causes the total inventory expenditure to more than double.