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Estamos preparando um guia educacional completo para o Economic Order Quantity (EOQ). Volte em breve para explicações passo a passo, fórmulas, exemplos reais e dicas de especialistas.
The Economic Order Quantity (EOQ) model is a fundamental inventory optimization technique that determines the ideal order quantity — the amount to order each time — that minimizes the total annual cost of ordering and holding inventory. Developed by Ford W. Harris in 1913 and popularized by R.H. Wilson, the EOQ model balances two competing cost drivers: ordering costs decrease as order size increases (fewer orders needed), while holding costs increase with order size (more inventory to store). The optimal order quantity sits at the minimum of the total cost curve, where ordering costs equal holding costs. Ordering costs include the administrative expense of placing and processing a purchase order, supplier setup charges, transportation costs, and receiving and inspection costs. These costs are incurred each time an order is placed, so larger order quantities mean fewer orders per year and lower total ordering costs. Holding costs (also called carrying costs) include storage space, insurance, obsolescence, spoilage, opportunity cost of capital tied up in inventory, and handling costs. These costs scale with the average inventory level — larger orders mean higher average inventory. The EOQ formula assumes constant and known demand, instantaneous replenishment, fixed ordering and holding costs, and no quantity discounts. While these assumptions simplify reality, the EOQ model provides a useful baseline and is robust: total cost is relatively flat near the EOQ, so small deviations from the theoretical optimum do not significantly increase total cost. This 'flat bottom' property means the EOQ framework is practically useful even when assumptions are not perfectly met. Extensions of the basic EOQ model include the Reorder Point (ROP) — the inventory level that triggers a new order, considering replenishment lead time and safety stock — and the Economic Production Quantity (EPQ) for manufacturing settings where production is continuous rather than instantaneous. Quantity discount models modify EOQ to account for price breaks at higher order volumes, requiring a comparison of total costs at each price break point. EOQ analysis is a cornerstone of inventory management, supply chain optimization, and working capital management. Even in an era of sophisticated enterprise resource planning (ERP) systems and machine learning demand forecasting, EOQ principles underpin the replenishment logic in most modern inventory systems.
EOQ = √(2 × D × S / H) Total Annual Cost = (D/EOQ) × S + (EOQ/2) × H Reorder Point = Lead Time (days) × Daily Demand + Safety Stock
- 1Estimate annual demand (D) in units from historical sales data or demand forecasts.
- 2Determine the ordering cost (S) per order: include administrative time, PO processing, transportation minimums, and receiving costs.
- 3Calculate the annual holding cost per unit (H): H = Item Cost × Holding Rate. Typical holding rates are 20–30% per year, reflecting storage, insurance, capital cost, and obsolescence.
- 4Apply the EOQ formula: EOQ = √(2DS/H). This is the quantity to order each time a replenishment is triggered.
- 5Calculate number of orders per year: N = D / EOQ. Calculate order cycle time: T = 365 / N days.
- 6Determine the reorder point: ROP = (Daily demand × Lead time in days) + Safety Stock. Safety stock buffers against demand or lead time variability.
- 7Calculate total annual cost to verify: TC = (D/EOQ) × S + (EOQ/2) × H. Confirm ordering cost equals holding cost at the EOQ.
At EOQ, ordering cost = holding cost = $707 each
EOQ = √(2 × 10,000 × $50 / $2) = √500,000 = 707 reams. Number of orders = 10,000 / 707 = 14.1 per year (about every 26 days). Annual ordering cost = 14.1 × $50 = $707. Annual holding cost = (707/2) × $2 = $707. Total = $1,414. This confirms the EOQ property: at the optimal quantity, ordering cost equals holding cost. Any deviation from 707 units increases total cost, though the total cost curve is fairly flat near the optimum.
High unit cost drives up holding cost and reduces EOQ
H = $80 × 25% = $20/unit/year. EOQ = √(2 × 5,000 × $200 / $20) = √100,000 = 316 units. Orders per year = 5,000 / 316 = 15.8 (every 23 days). Annual ordering cost = 15.8 × $200 = $3,162. Annual holding cost = (316/2) × $20 = $3,162. Total = $6,325. The high holding cost per unit ($20) is driven by the expensive $80 unit price — capital cost is the dominant component, which is why high-value inventory should be ordered more frequently in smaller quantities.
Quantity discounts do not always overcome higher holding costs
EOQ = √(2 × 2,000 × $30 / $4) = 173 units. TC at EOQ = $693. At 500 units: ordering cost = (2,000/500) × $30 = $120; holding cost = (500/2) × $4 = $1,000; total = $1,120. Purchase price saving = $500/year. Net total at 500 units = $1,120 − $500 = $620 — marginally better than EOQ's $693. In this case, the discount is worth taking. Always compare total costs including purchase savings, not just ordering and holding costs.
EOQ must be recalculated for each demand regime
Using monthly demand and holding cost: Peak EOQ = √(2 × 8,000 × $150 / $5) = √480,000 = 693 units. Off-season EOQ = √(2 × 1,000 × $150 / $5) = 245 units. A single EOQ using blended annual demand would significantly overstock in off-season and understock at peak. For seasonal businesses, the EOQ should be recalculated by season. In practice, companies build planned inventory ahead of peak season using a separate 'build plan' logic on top of the base EOQ replenishment system.
Retail and wholesale replenishment order sizing
Manufacturing production run length optimization
Hospital pharmacy inventory management (critical drugs)
Raw material purchasing for consumer goods manufacturers
E-commerce fulfillment center stocking optimization
In practice, this edge case requires careful consideration because standard assumptions may not hold. When encountering this scenario in economic order quantity (eoq) calculator calculations, practitioners should verify boundary conditions, check for division-by-zero risks, and consider whether the model's assumptions remain valid under these extreme conditions.
In practice, this edge case requires careful consideration because standard assumptions may not hold. When encountering this scenario in economic order quantity (eoq) calculator calculations, practitioners should verify boundary conditions, check for division-by-zero risks, and consider whether the model's assumptions remain valid under these extreme conditions.
In practice, this edge case requires careful consideration because standard assumptions may not hold. When encountering this scenario in economic order quantity (eoq) calculator calculations, practitioners should verify boundary conditions, check for division-by-zero risks, and consider whether the model's assumptions remain valid under these extreme conditions.
| Cost Component | Typical Range (% of Unit Value/Year) | Notes |
|---|---|---|
| Cost of Capital / Opportunity Cost | 6–12% | Tied to WACC or borrowing rate |
| Storage / Warehousing | 2–5% | Rent, utilities, racking, climate control |
| Insurance | 0.5–1% | Coverage on stored inventory value |
| Obsolescence / Technology Risk | 1–10% | High for electronics, fashion, perishables |
| Spoilage / Deterioration | 0.5–5% | High for food, chemicals, pharmaceuticals |
| Handling / Labor | 0.5–2% | Cycle counting, putaway, picking overhead |
| TOTAL TYPICAL RANGE | 15–30% | Use 20–25% as a general rule of thumb |
What are the key assumptions of the EOQ model?
The classic EOQ model assumes: (1) demand is constant, known, and continuous throughout the year; (2) lead time is known and constant; (3) each order arrives in one instantaneous batch; (4) no quantity discounts — unit cost is constant regardless of order size; (5) holding and ordering costs are constant; (6) no stockouts are permitted. These assumptions simplify the real world considerably. The model is most applicable to stable, commodity-type products with predictable demand. For products with lumpy, seasonal, or highly variable demand, more sophisticated methods (simulation, dynamic programming) may be needed.
What is included in ordering cost?
Ordering costs are all costs associated with placing and receiving a single purchase order, regardless of order size. Typical components include: purchasing department labor time to create and approve the PO, electronic transmission costs, supplier setup charges (in manufacturing: machine changeover time and materials), inbound freight and handling for the minimum shipment, receiving and inspection labor, and quality testing costs. Ordering costs can range from as low as $5–$10 for fully automated EDI orders to hundreds or thousands of dollars for complex custom procurement. Accurately estimating ordering cost is critical because errors here have significant impact on EOQ.
How do I calculate the holding cost rate?
The holding cost rate (expressed as a percentage of unit value per year) typically includes: cost of capital (6–12%, reflecting the interest rate on the money tied up in inventory or the company's WACC), storage space (2–5% of value, including rent, utilities, and racking), insurance (0.5–1%), obsolescence and spoilage risk (1–5%, depending on product type), deterioration and damage (0.5–2%), and inventory management overhead (0.5–1%). Total holding cost rates typically range from 15% to 35% of inventory value per year. For perishable or rapidly obsolescing products, rates can be 40% or higher.
What is the reorder point, and how does it relate to EOQ?
The reorder point (ROP) is the inventory level at which a new order should be triggered so that replenishment arrives before stockout. ROP = (Average daily demand × Replenishment lead time in days) + Safety stock. EOQ determines how much to order; ROP determines when to order. Together they form a complete inventory policy: order EOQ units whenever inventory falls to ROP. Safety stock protects against demand or lead time variability and is typically calculated as a multiple of the standard deviation of demand during lead time, based on the desired service level (e.g., 95% in-stock probability).
Does EOQ work for manufacturing (not just purchasing)?
Yes. The manufacturing equivalent is the Economic Production Quantity (EPQ), also called the production run quantity. EPQ = EOQ × √(D / (D − P)) where P is the daily production rate. The EPQ is always larger than EOQ because production replenishes inventory gradually (unlike an instantaneous purchase delivery), so the average inventory during a production run is lower than it would be for an equivalent purchase order. EPQ minimizes the total of setup costs (equivalent to ordering costs) and holding costs for a production environment. It is widely used in production scheduling for batch manufacturing.
How does EOQ change if holding costs double?
Since EOQ = √(2DS/H), doubling H reduces EOQ by a factor of √2 ≈ 1.414 — so EOQ decreases by about 29%. Higher holding costs make large inventory less economical, so the optimal strategy shifts to ordering smaller quantities more frequently. Conversely, doubling ordering cost S increases EOQ by the same factor — more expensive orders encourage fewer, larger orders. This square root relationship means EOQ is relatively insensitive to moderate errors in cost estimation: a 100% error in H or S produces only a 41% error in EOQ, and a much smaller percentage error in total cost.
What is the total cost at EOQ, and is it sensitive to deviations?
Total cost at EOQ = √(2DSH). At EOQ, ordering cost equals holding cost (each = TC/2). The total cost curve has a flat bottom near the EOQ — total cost is relatively insensitive to moderate deviations. For example, ordering 20% more or less than EOQ only increases total cost by about 2%. This 'robust' property is practically important: it means you don't need perfect precision in your estimates. However, for very high-value inventory, even small percentage errors in total cost represent significant dollar amounts, so precision matters more in high-stakes situations.
Dica Pro
Run an EOQ sensitivity table: calculate total cost at 50%, 75%, 100%, 125%, and 150% of EOQ. The relatively flat total cost near EOQ gives you practical wiggle room to adjust for supplier minimum quantities, truck-load economics, or purchasing convenience.
Você sabia?
Ford W. Harris developed the EOQ formula in 1913 while working as an engineer at Westinghouse Electric. He published it in an engineering trade journal under the title 'How Many Parts to Make at Once.' The formula was largely forgotten until R.H. Wilson independently rediscovered it in 1934 and popularized it in business literature — hence its occasional name 'Wilson's formula.'