Inventory Control: EOQ Model with Worked Example

Inventory protects production and customers against delays, but it ties up cash, takes space and risks damage and obsolescence. Ordering too often is also costly, because each order or production setup has fixed administrative, transport or changeover costs. The EOQ model, one of the oldest results in operations management (Harris, 1913), balances these opposing costs.

Assumptions of the basic EOQ model

• Demand is known and constant over time.
• Lead time is known and constant.
• The whole order arrives at once.
• Ordering (or setup) cost per order is fixed regardless of quantity.
• Holding cost is proportional to average inventory.
• No quantity discounts; no stockouts allowed.

These assumptions are simplifications, but the model is robust: total cost changes little for order sizes near the EOQ.

Inventory pattern and costs

With order quantity Q, inventory drops steadily from Q to zero, then a new order arrives. Average inventory is Q/2.

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Annual costs as a function of Q:

$$\text{Ordering cost} = \frac{D}{Q}S \qquad \text{Holding cost} = \frac{Q}{2}H$$ $$TC(Q) = \frac{D}{Q}S + \frac{Q}{2}H$$

where D is annual demand (units per year), S the cost per order or setup, and H the holding cost per unit per year. H is often expressed as a percentage i of unit cost C, so H = iC; typical carrying rates are 15 to 30% per year, covering capital, storage, insurance, obsolescence and handling.

Deriving the EOQ

Setting the derivative of TC with respect to Q to zero:

$$-\frac{DS}{Q^2} + \frac{H}{2} = 0 \quad\Rightarrow\quad EOQ = \sqrt{\frac{2DS}{H}}$$

At this quantity, ordering cost equals holding cost, and total cost is TC* = √(2DSH).

Reorder point and safety stock

With daily demand d and lead time L days, the reorder point is:

$$ROP = d \times L + SS$$

Safety stock SS protects against variation in demand and lead time. A common formula for variable demand is SS = z σ_d √L, where z is the service level factor (1.65 for about 95%) and σ_d the standard deviation of daily demand.

Extensions of the EOQ model

• Economic production quantity (EPQ): when items are produced gradually at rate p rather than arriving all at once, EPQ = √(2DS ÷ (H(1 − d/p))). S is the machine setup cost.
• Quantity discounts: compare total cost including purchase price at each price break.
• Planned backorders: allow shortages at a cost when customers will wait.
• Periodic review systems: order up to a target level at fixed intervals.
• ABC analysis: apply tight control and EOQ calculations to high value A items, simpler rules to low value C items.

Where EOQ is used

EOQ calculations are common for purchased parts and materials with steady usage, such as fasteners, bearings, packaging, chemicals and spare parts for maintenance stores. In manufacturing, the EPQ version sets batch sizes for parts made on shared machines, where each changeover has a setup cost. Distribution centres use EOQ logic to decide how often to replenish fast moving items from suppliers. ERP systems often include EOQ as one of several lot sizing rules, alongside lot for lot and fixed period ordering.

Steps to apply the EOQ model

1. Estimate annual demand from sales forecasts or consumption history.
2. Estimate ordering or setup cost: purchasing staff time, transport, receiving, inspection or machine changeover.
3. Estimate holding cost: cost of capital plus storage, insurance, handling, obsolescence and damage, usually as a percentage of unit cost.
4. Calculate EOQ, then round to practical pack, pallet or container sizes.
5. Set safety stock and reorder point based on lead time and demand variation.
6. Review parameters regularly, at least yearly or when demand or costs change.

A note on units

Keep units consistent: if demand is annual, holding cost must be per unit per year. If demand is monthly, use a monthly holding cost. Mixing periods is the most common mistake in EOQ calculations and gives answers that are wrong by a factor of √12 or more.

Sensitivity and practical use

Because the total cost curve is flat near its minimum, ordering 20% more or less than the EOQ raises cost by only about 2%. This makes the model useful even with rough estimates. More important is the insight it gives: reducing ordering or setup cost reduces the optimal batch size, which is the logic behind quick changeover and small lot production in lean manufacturing, where excess stock is one of the seven wastes. In MRP systems, EOQ is one of the lot sizing options used in production planning and control.