2. How Much Should I Order?

Chapter 1 introduced several inventory policies. This chapter asks a narrower question: when a policy uses a fixed order quantity, how large should that batch be? We will build a simple cost model, find the order quantity that minimizes it, and then examine what the result can and cannot tell us.

The model in this chapter is a starting point, not a complete description of an international supply chain. It assumes predictable demand and immediate replenishment. Later sections show how production limits, backorders, and quantity discounts change the analysis.

Model a simple replenishment system

Begin with one item, one stocking location, and one supplier. Demand is constant and known. Every time the stock is replenished, the same quantity $Q$ arrives immediately. This is a simplified continuous-review $(s,Q)$ system with a reorder point of zero and zero lead time.

Let:

Symbol	Meaning	Units in this model
$D$	Demand over one year	units/year
$Q$	Quantity in each replenishment order	units/order
$K$	Fixed cost of placing and receiving one order	currency/order
$h$	Annual variable holding cost for one unit	currency/(unit-year)

Use one consistent time basis. Since demand is annual, holding costs must also be annual. If a supplier quotes order costs in another currency, convert before combining the values.

Cycle stock

With immediate replenishment and steady demand, inventory declines from $Q$ to zero between arrivals. Its average over the cycle is the midpoint:

$$\text{Average cycle stock} = \frac{Q + 0}{2} = \frac{Q}{2}$$

This average is called cycle stock. It is the inventory held to cover ordinary demand between replenishments; it does not include safety stock.

Identify the costs that change with $Q$

Holding cost is the annual cost of carrying variable inventory: capital tied up in stock, insurance that scales with stock value, deterioration, and other per-unit costs. If $h$ is the annual holding cost per unit, then:

$$\text{Annual cycle-stock holding cost} = h\frac{Q}{2}$$

Some warehouse expenses are fixed over the range being studied, such as rent for space already leased. Those expenses matter to the business, but they do not change when $Q$ changes, so they do not select the best $Q$ in this model.

The fixed order cost $K$ is incurred each time an order is placed. Ordering $D$ units per year in batches of $Q$ requires approximately $D/Q$ orders per year. Therefore:

$$\text{Annual fixed order cost} = K\frac{D}{Q}$$

Costs that apply to every unit regardless of batch size, such as a constant per-unit purchase price or per-unit freight charge, total the same amount for a given annual demand. They can be left out when finding the basic EOQ, but they must be included when comparing options that change the unit price or other per-unit charges.

The relevant annual cost is the sum of the two terms that depend on order quantity:

$$C(Q) = h\frac{Q}{2} + K\frac{D}{Q}$$

As $Q$ increases, average cycle stock and its holding cost increase. But the number of orders per year, and therefore fixed order cost, decreases. EOQ finds the balance between those two effects.

Find the economic order quantity

Differentiate the relevant cost with respect to $Q$ and set the derivative to zero:

$$\frac{dC}{dQ} = \frac{h}{2} - \frac{KD}{Q^2} = 0$$

Solving gives the economic order quantity (EOQ):

$$Q^* = \sqrt{\frac{2KD}{h}}$$

For positive $K$, $D$, and $h$, this point is a minimum. At the EOQ, annual cycle-stock holding cost equals annual fixed order cost:

$$h\frac{Q^*}{2} = K\frac{D}{Q^*}$$

The minimum relevant annual cost is:

$$C(Q^*) = \sqrt{2KDh}$$

The balance gives an immediate intuition. Higher fixed order costs favor larger batches and fewer orders. Higher per-unit holding costs favor smaller batches and more frequent orders. Demand growth increases the EOQ, but only with the square root of demand.

Worked example: estimate an order quantity

A business imports a component with annual demand of 1,000 units. A purchase order and its receiving work cost $50 per order. The unit purchase price is$10. Annual storage expense is estimated at $2 per unit, and other annual holding expenses (such as capital and insurance) are estimated at 15% of the purchase price.

First calculate the annual holding cost per unit:

$$h = 2 + 0.15(10) = 3.50\;\text{currency/(unit-year)}$$

Then use annual demand $D=1{,}000$, order cost $K=50$, and holding cost $h=3.50$:

$$Q^* = \sqrt{\frac{2(50)(1{,}000)}{3.50}} \approx 169\;\text{units}$$

At this quantity, the average cycle stock is about $169/2 = 84.5$ units. The annual holding cost is about $3.50(169)/2 = 295.75$, and the annual fixed order cost is about $50(1{,}000)/169 = 295.86$. They are approximately equal, with the tiny difference caused by rounding.

The model's minimum relevant annual cost is about $591.61$. It excludes the $10{,}000$ annual purchase spend and any constant per-unit freight or handling charge, because those amounts do not change with $Q$ in this example. Add them back for a broader budget total, but do not mistake them for costs optimized by the basic EOQ.

Use forecasts and check sensitivity

The $D$ in EOQ can be a well-supported demand forecast rather than a blind copy of historical sales. Use a forecast that covers the same time horizon as $h$ and reflects known changes such as a new customer, a market expansion, or a product phase-out. A poor forecast can make a precise-looking EOQ operationally wrong.

Cost estimates are uncertain too. Fortunately, the basic EOQ cost curve is relatively flat near its minimum. The ratio between the cost at a chosen quantity $Q$ and the minimum cost is:

$$\frac{C(Q)}{C(Q^*)} = \frac{1}{2}\left(\frac{Q}{Q^*} + \frac{Q^*}{Q}\right)$$

For the example, choosing a convenient batch of 200 instead of about 169 increases the modeled relevant cost by roughly 1.4%. That can be a worthwhile trade if 200 fits a supplier's case pack or simplifies receiving. Check the actual constraint and all applicable costs; a discount or freight break can change the comparison.

From model to operation: Model, Optimize, Apply, Learn

Treat the formula as one loop in a planning process:

1. Model: define the item, time horizon, demand, order cost, and holding cost; state what is excluded.
2. Optimize: calculate $Q^*$ under those assumptions.
3. Apply: translate the result into an order quantity that fits case packs, supplier minimums, transport, and internal processes.
4. Learn: compare actual ordering effort, inventory, and service with the model; revise assumptions when reality differs.

EOQ determines an order quantity, not a reorder point. With positive supplier lead time, the trigger must account for demand while replenishment is in transit; with uncertain demand or lead time, a buffer may also be needed. Those questions belong with reorder points and safety stock, not the zero-lead-time EOQ derivation above.

Extensions: what changes the simple model?

Replenishing through production

When a factory produces the batch while customers continue consuming it, inventory does not jump instantly to $Q$. Let $P$ be the production rate and $D$ the annual demand rate, with $P>D$. During production, inventory accumulates at the net rate $P-D$. The maximum cycle stock is $Q(1-D/P)$, so average cycle stock is half that amount.

The relevant annual cost becomes:

$$C_{prod}(Q) = K\frac{D}{Q} + h\frac{Q}{2}\left(1-\frac{D}{P}\right)$$

Optimizing gives:

$$Q^*_{prod} = \sqrt{\frac{2KD}{h(1-D/P)}}$$

As production capacity approaches demand ($P$ approaches $D$ from above), inventory builds more slowly during a run. This production-batch model is not a shortcut for ignoring capacity: the assumptions require a stable production rate greater than the demand rate.

Allowing planned backorders

Some business customers accept a delay if the supplier communicates clearly and the contract defines service expectations. A backorder model can trade some waiting against lower cycle stock, but only if the delay or shortage has an explicit and credible cost.

Let $b$ be the annual penalty for keeping one unit backordered for one year. In a simple deterministic model that permits planned backorders, the optimum order cycle has:

$$Q^*_{backorder} = \sqrt{\frac{2KD(h+b)}{hb}}, \qquad B^* = Q^*_{backorder}\frac{h}{h+b}$$

Here $B^*$ is the maximum planned backlog in the cycle. These results rely on the model's assumptions and a defensible penalty estimate; they are not a recommendation to promise customers a service level the operation cannot meet.

Evaluating an all-units quantity discount

An order discount changes the purchase cost with $Q$, so the purchase spend can no longer be ignored. For each price tier $i$, let $c_i$ be the unit price and calculate a feasible candidate using that tier's holding cost (often $h_i = r c_i$ when the holding rate $r$ is proportional to item value). Compare total annual costs:

$$C_i(Q) = K\frac{D}{Q} + h_i\frac{Q}{2} + c_iD$$

An EOQ calculated for a discounted price is useful only if it falls within that tier's quantity range. Otherwise, test the applicable tier boundary. Compare the feasible EOQ candidates and boundaries using the full cost, including purchase spend; do not accept a discount just because its unit price is lower.

Assumptions and practical limits

The basic EOQ is useful because it makes the order-cost versus holding-cost trade-off visible. It is not a universal buying instruction. Its simple form assumes:

• One item with stable, known demand over the planning horizon.
• A fixed order cost and a constant per-unit annual holding cost.
• Immediate, reliable replenishment, with no demand or lead-time uncertainty.
• No binding budget, storage, minimum-order, capacity, or quantity-discount constraint.

International procurement often violates several assumptions at once: ocean lead times vary, shipments are consolidated, currency changes, and orders face minimum quantities or container constraints. Start with EOQ as a benchmark, then test the constraints and uncertainty that matter to the actual lane.

Practice: calculate and apply EOQ

Concept check

Quiz

EOQ and cost drivers

1 / 4

In the basic EOQ model, what happens to annual fixed order cost as Q gets larger?

Pick an answer to see if you are right.

Rank the EOQ decision procedure

Put the steps in the order a planner should use to turn a cost model into an operational batch.

Quiz

From inputs to an order quantity

1 / 1

Apply the basic deterministic EOQ method, then translate it into a usable order rule.

OrderDrag the handle to rearrange — top should be first.

1. ⋮⋮1Compare feasible quantities and apply a practical batch; monitor results and revisit assumptions.
2. ⋮⋮2Align the demand horizon, order cost, and holding cost units; state the model assumptions.
3. ⋮⋮3Calculate Q* = sqrt(2KD/h) from the validated inputs.
4. ⋮⋮4Check case packs, minimum orders, freight breaks, and other operational constraints.

Drag items into the correct order (or use ↑↓), then check.

Takeaways

• Basic EOQ balances annual cycle-stock holding cost against annual fixed order cost.
• With $D$ in units/year, $K$ in currency/order, and $h$ in currency/(unit-year), $Q^*=\sqrt{2KD/h}$ is measured in units.
• At the EOQ, the two relevant annual costs are equal; purchase spend that is constant across quantities does not alter the basic answer.
• A nearby practical quantity can have nearly the same modeled cost, but constraints and price tiers must be checked explicitly.
• EOQ is a benchmark under assumptions, not a reorder point and not a substitute for handling variable demand or lead time.

Previous: Inventory Policies and Replenishment.

Next: When Should I Order? — account for supplier lead time and review schedules.