Key takeaways
– Economic Order Quantity (EOQ) is a classic inventory model that gives the optimal order size Q* that minimizes the sum of annual ordering (or setup) costs and annual holding (carrying) costs.
– EOQ assumes steady (deterministic) demand, fixed ordering cost, fixed holding cost per unit, and constant lead time; these assumptions limit its direct applicability for highly variable or seasonal demand.
– EOQ is a building block for inventory policy: compute EOQ, then set a reorder point (R) that accounts for lead time and safety stock; adjust for quantity discounts, production processes (EPQ), and stochastic demand as needed.
What is EOQ?
Economic Order Quantity (EOQ) is the number of units a firm should order each time to minimize the combined annual costs of ordering and holding inventory. It was first described by Ford W. Harris in 1913 and remains widely used in inventory management and operations planning.
The basic EOQ formula and intuition
– Symbols:
– D = annual demand (units per year)
– S = fixed cost per order (ordering or setup cost, $ per order)
– H = annual holding cost per unit (carrying cost, $ per unit per year)
– EOQ formula:
Q* = sqrt( (2 × D × S) / H )
Why this works (intuition):
– Ordering cost is proportional to the number of orders per year = D / Q, so annual ordering cost = (D / Q) × S (decreases as Q increases).
– Holding cost is proportional to average inventory = Q / 2, so annual holding cost = (Q / 2) × H (increases as Q increases).
– EOQ balances these two opposing costs where their marginal rates are equal.
Other useful formulas
– Annual total relevant cost (excluding purchase cost) at order quantity Q:
Total cost = (D / Q) × S + (Q / 2) × H
– Minimum annual ordering + holding cost at Q*:
Minimum cost = sqrt(2 × D × S × H)
– Reorder point (R) when demand is constant and lead time is L (in same time units as demand):
R = d × L + Safety stock
where d = demand rate per unit time (D divided by number of time units in a year) and Safety stock depends on desired service level and demand/lead-time variability.
– Production EOQ (EPQ) for when items are produced internally at rate p > demand rate d:
Q*_EPQ = sqrt( (2 × D × S) / (H × (1 − d/p)) )
Practical numerical example (jeans)
– Given: D = 1,000 pairs/year; S = $2 per order; H = $5 per pair per year.
– EOQ: Q* = sqrt( (2 × 1,000 × 2) / 5 ) = sqrt(800) ≈ 28.3 → order about 28 pairs per order.
– Interpretation: Ordering 28 pairs each time minimizes the sum of ordering and holding costs under the model’s assumptions.
– Reorder point example: if daily demand d = 1,000/365 ≈ 2.74 pairs/day and lead time L = 7 days, then R ≈ 2.74 × 7 ≈ 19 pairs (plus safety stock if demand/lead time are variable).
When is EOQ high or low?
– EOQ increases (larger order size) when:
– Annual demand D increases.
– Ordering/setup cost S increases (expensive to place each order).
– Holding cost H decreases (cheaper to hold inventory).
– EOQ decreases (smaller order size) when:
– D decreases.
– S decreases (cheap/orders are inexpensive).
– H increases (high storage, insurance, obsolescence, or capital costs).
Practical steps to implement EOQ in your business
1. Gather accurate inputs
– Estimate annual demand (D) from sales history or forecasts.
– Calculate ordering cost (S): include all fixed costs tied to placing/receiving an order—purchase processing, transport setup, inspection, administrative time.
– Calculate holding cost per unit per year (H): often expressed as a percentage of unit cost (carrying rate) plus insurance, storage, obsolescence, shrinkage, and opportunity cost of capital. H = carrying rate × unit cost + other per-unit annual costs.
2. Choose consistent time units
– Make sure D, H, and S use the same time basis (usually per year).
3. Compute EOQ (Q*)
– Use Q* = sqrt( (2DS) / H ). Round to practical order sizes (e.g., packaging or pallet sizes).
4. Set the reorder point (R)
– Base R on average demand during lead time: R = d × L.
– Add safety stock if demand or lead time are variable: Safety stock = z × σL (z from desired service level; σL = standard deviation of demand during lead time).
5. Compute expected annual costs for management review
– Annual ordering cost = (D / Q*) × S
– Annual holding cost = (Q* / 2) × H
– Compare combined cost to current policy.
6. Adjust for practical constraints
– Round to supplier case/pallet sizes.
– Consider quantity discounts: run EOQ for discounted price ranges or use a quantity-discount EOQ variant.
– For made-to-order or production-run items, use EPQ formula.
7. Implement in systems and policies
– Configure ERP/WMS reorder points and order quantities.
– Document order rules and exceptions.
8. Monitor, measure, and revise
– Track actual demand, lead times, stockouts, and carrying costs.
– Recompute EOQ periodically (quarterly or annually) and whenever major cost drivers change.
Limitations and when EOQ may not be appropriate
– EOQ assumes deterministic and constant demand and constant lead time. Many real-world systems have seasonality, trends, or stochastic demand — in those cases you must add safety stock or use stochastic inventory models (r,Q) or (s,S) policies.
– EOQ ignores quantity discounts unless explicitly modeled (a larger order with a lower unit price changes H and purchase cost trade-offs).
– It assumes unlimited storage capacity and ignores supplier constraints, minimum order quantities, and transportation economies of scale.
– It treats holding and ordering costs as known and linear; in practice, carrying costs can include non-linear items like obsolescence and spoilage.
– Perishable goods, fashion items, or those with high obsolescence risk require different approaches.
Extensions and related models
– EOQ with quantity discounts: evaluate EOQ in each price break interval and compare total costs (including purchase cost).
– EPQ (economic production quantity): for manufacturing where items are produced at a finite rate p.
– Stochastic inventory models: include lead-time variability and demand uncertainty; determine safety stock using service-level targets.
– Multi-item and joint replenishment models: consider common ordering costs across items or shared shipments.
Quick checklist before applying EOQ
– Are demand and lead time stable or variable? If variable, compute safety stock.
– Have you included all components of holding cost (capital, storage, obsolescence, shrinkage)?
– Are there quantity discounts, minimum order quantities, or supplier constraints?
– Is it more appropriate to use production-run models (EPQ) or periodic review policies?
– Is your ERP/WMS set up to enforce calculated reorder points and order quantities?
The bottom line
EOQ is a simple, powerful starting point for minimizing ordering and carrying costs. It helps you choose an order size that balances ordering frequency with inventory levels. For best results, use EOQ as part of a broader inventory management program: collect good input data, add safety stock for variability, model discounts and production constraints, automate orders in your systems, and review parameters routinely.
Sources
– Investopedia, “Economic Order Quantity (EOQ)” (source URL provided by user).
– Ford W. Harris, original EOQ formulation (1913).