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Economic Order Quantity (EOQ) is the mathematically optimal order size that minimizes the total combined cost of ordering inventory and holding it in stock. First derived by Ford Whitman Harris in a 1913 paper in Factory, The Magazine of Management and later popularized by R.H. Wilson in the 1930s, the EOQ model has remained a foundational tool in inventory management for over a century. The core insight is elegant: ordering too frequently wastes money on per-order costs (purchasing, receiving, paperwork), while ordering too infrequently wastes money on holding costs (warehousing, capital tied up in inventory, obsolescence, insurance). The EOQ formula finds the exact quantity where these two cost curves cross — the minimum point of total inventory cost. At this quantity, annual ordering cost exactly equals annual holding cost. This mathematical elegance makes EOQ both useful and memorable, though its real-world application requires careful attention to its assumptions: constant and known demand, constant ordering cost, constant unit cost regardless of quantity, and instantaneous replenishment. In practice, EOQ is used as a planning baseline rather than a rigid prescription. Supply chain managers calculate EOQ to understand the cost-optimal order size, then adjust for real-world constraints: supplier minimum order quantities (MOQs), quantity discount breakpoints, storage capacity limitations, and lead time variability. When demand is seasonal or variable, EOQ is recalculated for each planning period or integrated with safety stock models that handle demand uncertainty. Modern ERP systems (SAP, Oracle, Dynamics) implement EOQ calculations automatically within their materials requirements planning (MRP) modules. Companies like Amazon have extended EOQ into sophisticated dynamic reorder systems that recalculate optimal order quantities in real time based on demand forecasts, freight rates, and storage costs — but the underlying Wilson-Harris formula remains the mathematical core.
EOQ Formula (Wilson Formula): EOQ = √(2DS / H) Where: D = Annual demand (units per year) S = Ordering cost per order ($/order) — cost of placing and receiving one purchase order H = Annual holding cost per unit ($/unit/year) — typically 20–30% of unit cost Related Calculations: Number of orders per year = D ÷ EOQ Time between orders = 365 ÷ (D ÷ EOQ) days Total annual ordering cost = (D ÷ EOQ) × S Total annual holding cost = (EOQ ÷ 2) × H [average inventory = EOQ/2] Total inventory cost = Total ordering cost + Total holding cost Worked Example: D = 12,000 units/year S = $60 per order Unit cost = $10, holding cost rate = 25% H = $10 × 25% = $2.50/unit/year EOQ = √(2 × 12,000 × 60 ÷ 2.50) = √576,000 = 759 units Orders per year = 12,000 ÷ 759 ≈ 15.8 orders Total annual cost = (15.8 × $60) + (759/2 × $2.50) = $948 + $949 = $1,897
- 1Determine annual demand (D) — use 12 months of historical sales data adjusted for known trends or seasonality; for new products, use forecasts from comparable items or market research.
- 2Calculate ordering cost per order (S) — include all costs associated with placing and receiving a single purchase order: procurement staff time, purchase order system costs, receiving and inspection labor, and any fixed supplier delivery charges per order.
- 3Calculate annual holding cost per unit (H) — this is typically expressed as a percentage of unit cost (20–30% is standard) covering: cost of capital (opportunity cost of money tied up in inventory), warehousing space cost, inventory insurance, shrinkage/damage/obsolescence risk, and handling costs.
- 4Apply the EOQ formula: √(2DS/H). The result is the quantity in units that minimizes total annual inventory cost.
- 5Calculate the implied order frequency: D ÷ EOQ gives orders per year; 365 ÷ (D/EOQ) gives days between orders. This determines your replenishment schedule.
- 6Verify total cost at EOQ and compare to your current order quantity — if your current ordering pattern differs significantly from EOQ, calculate the cost penalty to quantify the savings opportunity from adjusting.
- 7Check EOQ against real-world constraints: supplier MOQ, pallet quantities, storage capacity, and quantity discount breakpoints — adjust the order quantity and re-evaluate total cost at each feasible alternative.
Ordering 707 units ~14 times per year minimizes total inventory cost. Ordering 500 units would cost $1,900/year; ordering 1,000 units costs $1,900/year — symmetric cost penalty on both sides of EOQ.
High unit cost creates high holding cost, driving EOQ to just 46 units — essentially weekly ordering. For expensive inventory, frequent small orders minimize capital tied up in stock, outweighing the ordering cost penalty.
Very low unit cost and high ordering cost (truck freight) push EOQ to 100,000 units — effectively quarterly ordering. Holding cost is negligible relative to the cost of frequent shipments for bulk commodities.
When a 6.25% quantity discount is available at 1,000 units (vs. EOQ of 707), the product cost saving ($5,000/year) outweighs the extra inventory cost ($357/year). Always model total cost including product price at discount breakpoints.
Retail buyers use EOQ to set purchase order quantities for replenishment items, reducing the total cost of inventory management across thousands of SKUs and ensuring capital is not unnecessarily tied up in slow-moving stock.
Manufacturing procurement teams calculate EOQ for raw materials and components, balancing the cost of frequent supplier deliveries against the capital cost of holding raw material safety stocks at factory level.
E-commerce fulfillment companies use EOQ-based replenishment algorithms within their WMS to automate purchase order generation, triggering orders at the reorder point for the EOQ quantity without manual intervention.
Supply chain consultants use EOQ analysis to demonstrate the cost penalty of clients' current ordering patterns — typically finding that businesses with ad-hoc ordering habits spend 15–35% more on inventory-related costs than the EOQ-optimized baseline.
Economic Production Quantity (EPQ) is the EOQ variant for manufactured items where production is not instantaneous.
EPQ accounts for the production rate (p) and demand rate (d): EPQ = √(2DS/H × p/(p-d)). This is relevant for make-to-stock manufacturers balancing setup costs against in-process inventory holding costs during production runs.
Quantity discount analysis requires comparing total cost (product cost +
Quantity discount analysis requires comparing total cost (product cost + ordering cost + holding cost) at multiple order size breakpoints — not just minimizing ordering and holding costs in isolation. When a supplier offers a 5% discount for orders above a threshold, the product cost saving may outweigh the holding cost penalty of carrying more inventory, making the larger order size cheaper overall.
Probabilistic EOQ models (stochastic EOQ) incorporate demand variability by
Probabilistic EOQ models (stochastic EOQ) incorporate demand variability by treating demand as a probability distribution rather than a constant. These models — including the newsvendor model for perishables and the (Q, r) inventory policy for continuous review — extend the basic Wilson formula to handle real-world demand uncertainty while still using EOQ as the base order quantity.
| Order Qty as % of EOQ | Total Cost vs. Optimal | Practical Interpretation |
|---|---|---|
| 50% of EOQ | +25% total cost | Ordering twice as often as optimal — high transaction cost |
| 75% of EOQ | +6% total cost | Slightly under-ordering — minor penalty |
| 100% (EOQ) | Optimal — 0% penalty | Mathematically minimum total inventory cost |
| 125% of EOQ | +3% total cost | Slightly over-ordering — very minor penalty |
| 150% of EOQ | +8% total cost | Excess inventory — holding costs begin to dominate |
| 200% of EOQ | +25% total cost | Ordering half as often as optimal — mirrored penalty |
What is Economic Order Quantity and why does it matter?
Economic Order Quantity (EOQ) is the order size that minimizes total annual inventory costs — the sum of ordering costs (cost of placing each order) and holding costs (cost of storing inventory). It matters because businesses systematically over-order or under-order without an analytical framework. Over-ordering ties up capital and increases storage costs; under-ordering leads to frequent orders with high transaction costs and potential stockouts. EOQ finds the mathematically optimal balance point.
What costs are included in 'ordering cost'?
Ordering cost (S) includes every expense associated with placing and receiving a single purchase order: procurement staff time to source, negotiate, and process the PO; the cost of operating purchasing software systems; receiving dock labor to unload and check the delivery; quality inspection costs; and any fixed freight or delivery charges that apply per order regardless of quantity. Ordering cost does NOT include the unit product cost — only the transaction overhead per order.
How do I calculate the holding cost rate?
The holding cost rate is typically expressed as a percentage of unit cost per year, covering: cost of capital (your weighted average cost of capital, typically 8–15%), warehouse space cost (rent, utilities, racking allocated per SKU), inventory insurance (0.5–2% of value), shrinkage and obsolescence risk (1–5% depending on category), and handling costs. For most businesses, a holding rate of 20–30% of unit cost is a reasonable starting estimate. High-fashion and technology products may use 40–50% due to rapid obsolescence.
What are the limitations of the EOQ model?
EOQ assumes constant, known demand — it breaks down for seasonal or highly variable demand patterns. It assumes instantaneous replenishment — in practice, lead times require a separate reorder point calculation. It assumes constant unit cost — quantity discounts change the optimal order size. It assumes known, constant ordering and holding costs — these often vary in practice. Despite these limitations, EOQ provides a useful cost-minimizing baseline from which real-world adjustments are made.
How does EOQ relate to the reorder point?
EOQ answers 'how much to order'; the reorder point (ROP) answers 'when to order'. They work together: you order EOQ units whenever inventory falls to the reorder point (Average Demand × Lead Time + Safety Stock). EOQ determines your ordering cycle length and average inventory level; ROP ensures you trigger the order with enough lead time to avoid a stockout. Both calculations are required for a complete inventory replenishment policy.
Should I use EOQ if my supplier has a minimum order quantity?
If the supplier's MOQ exceeds your calculated EOQ, you must order at least the MOQ — you cannot order below it. In this case, calculate total annual cost at the MOQ and determine whether the supplier relationship is cost-competitive given the forced inventory level. If MOQ is below EOQ, simply use EOQ. If EOQ falls within a quantity discount tier, calculate total cost (product + ordering + holding) at the EOQ quantity, the discount threshold, and the next discount tier to identify the true cost-minimizing order size.
How is EOQ used in ERP systems?
ERP systems like SAP, Oracle NetSuite, and Microsoft Dynamics incorporate EOQ as a standard replenishment policy option in their MRP/inventory management modules. Users input demand forecasts, ordering costs, and holding cost rates; the system calculates EOQ and uses it to generate purchase order recommendations. Many systems implement variants like Economic Production Quantity (EPQ) for manufactured items, or dynamic EOQ that recalculates automatically as inputs change.
プロのヒント
The EOQ cost curve is surprisingly flat near the optimum — ordering 20% more or less than EOQ costs only 2–3% more than the optimal. This flatness means you have flexibility to round EOQ to practical quantities (pallet multiples, container minimums) without meaningful cost penalty. Use EOQ to set your ordering range, then choose the most operationally convenient quantity within ±25% of EOQ.
ご存知でしたか？
Ford Whitman Harris derived the EOQ formula in 1913 while working as a production engineer at Westinghouse. He published it in a single page in Factory magazine — one of the most cited papers in operations management history. Remarkably, the formula went largely uncredited for decades because Harris never received academic recognition; it was R.H. Wilson's 1934 application paper that made 'Wilson's formula' the popular name for Harris's discovery.