Economic order quantity: calculate the cost curve before choosing a batch

Ordering a small quantity frequently reduces average inventory but increases the number of orders. Ordering a large quantity less often does the reverse. The economic order quantity, or EOQ, identifies the lowest combined ordering and holding cost under a deliberately simple set of assumptions.

MIT's inventory lecture presents the model with uniform known demand and replenishment received as a complete quantity. That setting matters. The basic calculation is a useful cost benchmark, but it does not by itself solve uncertain demand, quantity discounts or a constrained warehouse.

Start by asking whether the decision is how much to replenish repeatedly. A one-time purchase for a short selling season has a different problem: unsold stock and missed sales can dominate. EOQ assumes a continuing cycle rather than a single uncertain selling opportunity.

Consider an invented item with annual demand of 12,000 units, a relevant fixed cost of $75 per replenishment order, and an annual holding cost of $6 per unit of average inventory. Purchase price is unchanged across order sizes in this illustration.

Annual ordering cost is demand divided by order quantity, multiplied by the $75 order cost. Average cycle inventory is half the order quantity because stock declines evenly from the replenishment quantity to zero. Annual holding cost is that average multiplied by $6.

The $6 must be a cost per unit per year, not a percentage entered without the unit's value. A percentage carrying charge can be converted into dollars using the relevant inventory value, with any other applicable holding costs included on a consistent basis.

Write annual relevant cost as 12,000 × $75 / Q + $6 × Q / 2, where Q is the order quantity. The first term falls as Q rises; the second rises. Their sum has a minimum at the square root of 2 × demand × order cost / holding cost.

For this example, EOQ is the square root of 300,000, or approximately 547.72 units. At that quantity, annual ordering cost is about $1,643.17 and annual holding cost is also $1,643.17. Combined relevant cost is approximately $3,286.34.

The equality of these two costs is a property of this basic model at its continuous optimum. It is a useful calculation check. It is not a rule that every real inventory policy should force ordering and holding costs to match despite constraints or additional cost terms.

A supplier may sell in packs or require particular lot sizes. The calculated 547.72 units is then a reference point for comparing feasible choices. Evaluate the full cost at each practical quantity rather than assuming that rounding upward always gives the better result.

At 300 units, annual ordering cost is $3,000 and holding cost is $900, totaling $3,900. At 600 units, the amounts are $1,500 and $1,800, totaling $3,300. The 600-unit policy costs only about $13.66 more per year than the unconstrained optimum under these assumptions.

That small difference can justify choosing 600 when it fits packaging or administration better. It does not justify ignoring a large omitted cost. The comparison is persuasive only if the $75 and $6 estimates represent the economics of the actual alternatives.

At the theoretical quantity, average cycle stock is about 273.86 units. Immediately after replenishment, physical stock is about 547.72 units under the simplified zero-lead-time model. A storage constraint applies to the relevant peak, not automatically to the average used for holding cost.

The order cycle is approximately 16.66 days if demand is spread evenly over a 365-day year. If demand occurs only on operating days, use that calendar consistently. The annual demand does not justify assuming sales happen on every calendar day.

Supplier payments may occur before, on or after delivery. The cost-minimizing average inventory calculation does not establish the cash needed on the payment date. Put the proposed purchasing pattern into the dated cash forecast before treating the EOQ result as affordable.

With 12,000 units bought each year at an unchanged unit price, annual purchase cost is the same for every order quantity. It can be omitted when locating the minimum, although it remains a real cost of running the business.

If purchase price changes at quantity thresholds, that simplification fails. Compare purchase cost, ordering cost and holding cost at the relevant feasible quantities. A discount can shift the best choice even when its larger stockholding requirement is expensive.

The order cost should also reflect what changes when another order is placed. A fixed purchasing-team salary does not necessarily create a $75 cash payment each time. Administrative effort can still have a capacity cost, but label that basis instead of presenting every allocation as an avoidable cash expense.

Uncertain demand and replenishment timing introduce a separate protection-stock question. EOQ chooses a recurring lot size in its simplified setting; it does not calculate the buffer needed to achieve a specified service level.

Perishable stock, limited shelf space, supplier capacity, finite production rates and minimum commitments can constrain the choice. A large quantity that is mathematically cheap to hold under the model may be unusable if units expire before they are needed.

Review the cost curve when an input changes materially. Because quantity appears both in the denominator of ordering cost and the numerator of holding cost, a different demand or cost estimate changes the balance. Preserve the old assumptions so the reason for a policy change remains understandable.

A useful recommendation might choose 600 units, show its $3,300 modeled annual relevant cost, and explain the packaging advantage and cash requirements. The decimal optimum alone is less useful than a feasible policy with a visible cost consequence.

Keep the demand estimate, order-cost basis, holding-cost basis and omitted constraints alongside the result. That lets the next reviewer update the decision when conditions change, instead of inheriting an order quantity whose original economic logic has been lost.