A business that orders too often pays heavy ordering costs. A business that orders too much ties up cash in the store and pays to house, insure and guard it. Somewhere in between is an order size that keeps the total of these costs as low as possible.
That order size is the economic order quantity, usually called the economic order quantity (EOQ), and finding it is the best-known calculation in inventory management.
The EOQ formula is simple, but the examiners ask more than “plug in the numbers”. They want the costs of holding stock, the assumptions behind the model, what happens when a supplier offers a discount for bulk orders, and, in older papers, the stock levels that trigger re-ordering. This page covers all of these, with two fully worked calculations and a bank of practice questions.
In 30 seconds
Stock costs money in four ways: the purchase price, ordering costs, holding costs and shortage costs. Bigger orders cut ordering costs but raise holding costs. The EOQ = √(2 × Co × D ÷ Ch), where Co is the cost per order, D is annual demand and Ch is the holding cost per unit per year. At the EOQ, annual ordering cost equals annual holding cost.
If the supplier gives a discount for large orders, calculate the total annual cost (purchases + ordering + holding) at the EOQ and at each discount threshold, and choose the lowest.
The costs of holding and ordering stock
Many businesses, especially manufacturers and retailers, keep large stocks so that they can meet demand at once. A stock-out can mean a lost sale and a lost customer, who may go to a competitor. But keeping stock is not free.
| Cost group | What it includes |
|---|---|
| Purchase price | The price paid to the supplier for the goods |
| Ordering (re-order) costs | The cost of making an order: delivery cost if the buyer pays it; the administrative cost of placing the order, such as telephone calls; the cost of checking goods when they arrive; and, if the item is made in house, the set-up cost of each production batch |
| Holding costs | Cost of the capital tied up in stock; insurance; warehousing; and losses from obsolescence, deterioration and theft |
| Shortage costs | Profit lost on sales that could not be made; later profit lost because customers leave; and the cost of stopping production when raw materials run out |
The cost of capital in stock
Money tied up in stock must be financed, either from the firm’s own funds, which could have earned a return elsewhere, or by borrowing, which carries interest. Either way, there is a cost.
Example. A firm holds between 0 and 6,000 units of an item costing ₦500 a unit. Demand is steady. The interest rate is 8% a year.
- Find the largest stock value.
6,000 × ₦500 = ₦3,000,000 - Find the average stock value. Demand is steady, so stock falls from the maximum to zero. The average is half the maximum.
3,000,000 ÷ 2 = ₦1,500,000 - Apply the interest rate. This is the cost of financing the stock.
1,500,000 × 8% = ₦120,000 a year
Add warehouse rent and storekeepers’ pay and the total cost of holding is higher.
Some holding costs vary with the amount of stock (interest, losses through deterioration and theft), and some are fixed (warehouse rent, storekeepers’ wages). Changing the order size alters only the variable holding costs, so those are the ones that matter in the EOQ model.
The trade-off
Take a firm that needs 12,000 units of a part a year, used at a steady rate. If it orders once, the order size is 12,000 and the average stock is 6,000. If it orders twice a year, each order is 6,000 and the average stock is 3,000. If it orders four times, each order is 3,000 and the average stock is 1,500.
| Order size (Q) | Average stock (Q ÷ 2) | Orders a year (D ÷ Q) | Effect |
|---|---|---|---|
| 12,000 | 6,000 | 1 | High holding cost, low ordering cost |
| 6,000 | 3,000 | 2 | Middle |
| 3,000 | 1,500 | 4 | Low holding cost, high ordering cost |
Smaller orders reduce stock and holding costs but need more orders, which increases ordering costs. The business wants the order size that minimises the total. Remember that only relevant costs matter in this decision: costs that change when the order size changes. Any cost the firm will pay whatever order size is chosen can be ignored.
The economic order quantity model
The economic order quantity (EOQ) is the order size that minimises the total of annual ordering costs and annual holding costs for an item. The model rests on simplifying assumptions.
| Assumption | What it means for the calculation |
|---|---|
| No bulk discounts: every unit costs the same whatever the order size | The annual purchase cost does not change with order size, so it can be left out of the decision |
| The lead time (time between ordering and delivery) is constant and known | A new order can be timed to arrive exactly when stock runs out, so maximum stock equals the order size Q |
| There is no risk of running out of stock | Shortage costs can be ignored, and no safety stock is kept |
| Annual demand is constant through the year | Stock falls steadily from Q to zero, so average stock is Q ÷ 2 |
Given these assumptions, the relevant costs are just the annual ordering cost and the annual holding cost.
The formula
EOQ = √( 2 × Co × D ÷ Ch )
Co = cost of placing one order. D = annual demand in units. Ch = cost of holding one unit in stock for a year. The answer Q is the number of units to buy in each order.
It comes from the following relationships:
- Stock falls from Q to 0 each cycle, so average stock = Q ÷ 2 and annual holding cost = (Q ÷ 2) × Ch.
- Orders per year = D ÷ Q, so annual ordering cost = (D ÷ Q) × Co.
- Total relevant cost = (D ÷ Q) × Co + (Q ÷ 2) × Ch. The EOQ is the Q that makes this smallest.
At the EOQ the annual ordering cost and annual holding cost are equal. This is a very useful check on your answer.
Worked calculation
Example. A firm uses 36,000 units of a component a year. Each order costs ₦4,500 to place. The unit price is ₦150, and the annual holding cost is 8% of the price, which is ₦12 a unit.
- Calculate the EOQ. D is annual demand (36,000), Co is the cost of one order (4,500) and Ch is the holding cost per unit (12). The unrounded EOQ is 5,196.15. Use it in the next steps so the two costs match exactly.
EOQ = √(2 × 4,500 × 36,000 ÷ 12) = √27,000,000 = 5,196 units - Count the orders. Annual demand divided by the order size.
36,000 ÷ 5,196.15 = 6.9282 orders a year - Work out the annual costs. Ordering cost is orders × cost per order. Holding cost is average stock (half the order size) × holding cost per unit. The table below sets them out.
| Cost | Working | ₦ |
|---|---|---|
| Ordering | 6.9282 × 4,500 | 31,177 |
| Holding | (5,196.15 ÷ 2) × 12 = 2,598.08 × 12 | 31,177 |
Check. At the EOQ the two costs are equal, and they are.
| Cost | Working | ₦ |
|---|---|---|
| Ordering and holding | 31,177 + 31,177 | 62,354 |
| Purchases | 36,000 × 150 | 5,400,000 |
| Total cost | 62,354 + 5,400,000 | 5,462,354 |
The costs the EOQ model controls (₦62,354) are small beside the purchase price (₦5,400,000). The purchase price is irrelevant to the order size here because it is the same whichever order size is chosen.
Be careful with the inputs
- If demand is given per quarter or per month, convert it to an annual figure first. For example, 12,000 units every three months is 48,000 a year.
- If the holding cost is given as a percentage of the purchase price, convert it into naira per unit: 6% of ₦80 is ₦4.80.
- Round the EOQ to the nearest whole unit only at the end, as the question may state.
Quantity discounts
The EOQ model assumes that the unit price is the same whatever the order size. If the supplier offers a lower price for orders above a certain size, the purchase price becomes a relevant cost, because it now depends on the order size. The best order size is then either:
- the EOQ itself, or
- the smallest order size that earns each available discount.
To decide, calculate the total annual cost for each candidate: annual purchase cost + annual ordering cost + annual holding cost. Choose the order size with the lowest total.
Worked calculation with two discount levels
Example. Annual demand is 24,000 units. The normal price is ₦50. Placing an order costs ₦400. Holding cost is 10% of the purchase price. The supplier offers a price of ₦49 for orders of 4,000 or more, and ₦48 for orders of 8,000 or more.
- Find the holding cost per unit. It is 10% of the purchase price.
10% × ₦50 = ₦5 - Calculate the EOQ at the normal price.
EOQ = √(2 × 400 × 24,000 ÷ 5) = √3,840,000 = 1,960 units - Test it against the thresholds. 1,960 is below both 4,000 and 8,000, so it earns no discount.
- Total the costs at each candidate order size. The candidates are the EOQ and the smallest order that earns each discount. For each one, add purchases, ordering and holding costs. The table below does this.
| Order 1,960 (price ₦50) | Order 4,000 (price ₦49) | Order 8,000 (price ₦48) | |
|---|---|---|---|
| Purchases (24,000 × price) | 1,200,000 | 1,176,000 | 1,152,000 |
| Ordering (24,000 ÷ Q × ₦400) | 4,899 | 2,400 | 1,200 |
| Holding (Q ÷ 2 × 10% of price) | 4,899 | 9,800 | 19,200 |
| Total | 1,209,798 | 1,188,200 | 1,172,400 |
The lowest total is at an order size of 8,000 units. Ordering and holding costs rise at larger orders, but the saving on the purchase price is far bigger.
Points to watch
- Holding cost and the discounted price. If holding cost is a percentage of price, it falls when the price falls. The calculation above uses 10% of the discounted price (₦4.90 and ₦4.80). In some textbook examples the holding cost is kept at the original naira figure. The answer to a discount question is almost always the same either way, but follow the instructions in the question.
- If the EOQ at a discounted price falls above its threshold. The EOQ itself is then a valid order size at that price, so use it rather than the threshold.
- The purchase cost must be included. Comparing only ordering plus holding cost would wrongly favour small orders.
- Non-cost factors. A large order needs storage space and cash, and the goods may deteriorate or become obsolete before use. These may limit the discount option.
Re-order level and stock levels
Background: the syllabus does not set out the stock level formulas below. They appeared in questions under the previous syllabus, and are standard stores control ideas, so they are included in case a question uses them.
The EOQ tells you how much to order. The re-order level tells you when. With a lead time, the firm must place the order while stock is still enough to cover usage until delivery.
| Level | Formula | Purpose |
|---|---|---|
| Re-order level | Maximum usage × maximum lead time | The stock level at which a new order is placed, so that even in the worst case stock does not run out |
| Minimum level | Re-order level − (average usage × average lead time) | A warning level. Stock should not normally fall below it. It acts as a buffer |
| Maximum level | Re-order level + re-order quantity − (minimum usage × minimum lead time) | A warning level. Stock should not normally rise above it, to avoid overstocking |
Example. Usage is between 80 and 120 units a day, averaging 100. Lead time is between 10 and 20 days, averaging 15. The re-order quantity is 3,000 units.
- Re-order level. Use the worst case: the highest usage for the longest lead time.
120 × 20 = 2,400 units - Minimum level. Take off the usage you expect in a normal lead time.
2,400 − (100 × 15) = 900 units - Maximum level. Add the re-order quantity, then take off the lowest usage in the shortest lead time.
2,400 + 3,000 − (80 × 10) = 4,600 units
Factors that affect the maximum level include storage space, the cost of holding stock, the risk of obsolescence or deterioration, the cash available, insurance cost, and any minimum order size or discount from suppliers.
The ideas in more depth
Why the EOQ is not very sensitive
The total cost curve is flat near the minimum. If you order a little more or a little less than the EOQ, the total changes only slightly. This means that rounding the EOQ to a convenient order size, such as a full pallet, costs very little.
Limitations of the model
- Demand is rarely perfectly constant. It may be seasonal or uncertain.
- Lead times vary, so safety stock is often needed.
- Ordering and holding costs are hard to measure accurately, because part of each is fixed.
- It ignores price changes, discounts and limited storage space, unless extended.
- It treats each item separately, while in practice orders for many items may be combined.
Linking EOQ to the buying cycle
The stores department uses stock records, the minimum level and the EOQ to decide when to raise a purchase requisition and for how many units. The EOQ therefore connects directly to the documents in the previous topic.
Worked example: choosing the order size
Scenario. Makurdi Grains Ltd uses 60,000 bags of an additive each year. Each bag costs ₦20. An order costs ₦300 to process, and holding cost is 10% of the purchase price. The supplier offers a 3% discount (price ₦19.40) on orders of 6,000 bags or more, and a 5% discount (price ₦19.00) on orders of 10,000 bags or more. Advise on the best order size.
- EOQ at ₦20. Holding cost is 10% of ₦20.
Holding cost = ₦2
EOQ = √(2 × 300 × 60,000 ÷ 2) = √18,000,000 = 4,243 bags - Total cost at each candidate order size. The candidates are the EOQ and the smallest order for each discount: 6,000 and 10,000 bags. For each, the holding cost is 10% of that price.
| Order 4,243 (EOQ) | Order 6,000 | Order 10,000 | |
|---|---|---|---|
| Price per bag ₦ | 20.00 | 19.40 | 19.00 |
| Holding cost per bag ₦ | 2.00 | 1.94 | 1.90 |
| Purchases (60,000 × price) ₦ | 1,200,000 | 1,164,000 | 1,140,000 |
| Ordering (60,000 ÷ Q × 300) ₦ | 4,243 | 3,000 | 1,800 |
| Holding (Q ÷ 2 × holding cost) ₦ | 4,243 | 5,820 | 9,500 |
| Total cost ₦ | 1,208,485 | 1,172,820 | 1,151,300 |
At the EOQ, ordering and holding cost are each 4,242.64 before rounding, which is why the total is 1,208,485.
Decision. The lowest cost is at 10,000 bags per order. It saves ₦1,208,485 − ₦1,151,300 = ₦57,185 a year compared with ordering the EOQ. Before accepting, the firm should check that it has space and cash for 10,000 bags and that the additive will not spoil before it is used.
Common mistakes in the exam
- Forgetting the 2 in the formula, or putting D and Co the wrong way round.
- Using quarterly or monthly demand without converting to a year.
- Using the holding cost as a percentage instead of converting it to naira per unit.
- Including the purchase price when finding the EOQ without discounts. It does not change the answer.
- Leaving the purchase cost out of the total in a discount question.
- Only calculating the total cost at the discount thresholds and forgetting the EOQ itself.
- Not checking that ordering cost equals holding cost at the EOQ.
- Writing the assumptions of the model without saying what each implies.
Quick revision
- Costs: purchase, ordering, holding, shortage. Bigger orders: lower ordering costs, higher holding costs.
- EOQ = √(2 Co D ÷ Ch). At EOQ, annual ordering cost = annual holding cost. Orders per year = D ÷ Q. Average stock = Q ÷ 2.
- Assumptions: no discounts, constant known lead time, no stock-outs, constant demand.
- With discounts: compare total annual cost (purchases + ordering + holding) at the EOQ and at each threshold.
- Background: re-order level = max usage × max lead time. Minimum level = ROL − average usage × average lead time. Maximum level = ROL + ROQ − minimum usage × minimum lead time.
Practice questions
These are original questions written in the style of the Institute of Chartered Accountants of Nigeria (ICAN) exam. Try each one before you open the answer.
Section A style: multiple choice
Question 1. Which cost is a holding cost?
- Cost of placing an order
- Insurance of stock in the warehouse
- Lost profit from a stock-out
- Delivery charge on each order
Show the answer
B. Holding costs include capital cost, insurance, warehousing and losses from obsolescence, deterioration and theft.
Question 2. Annual demand is 10,000 units, the cost per order is ₦200 and the holding cost is ₦4 per unit per year. The EOQ is:
- 500
- 707
- 1,000
- 2,000
Show the answer
C. √(2 × 200 × 10,000 ÷ 4) = √1,000,000 = 1,000.
Question 3. The number of orders a year in Question 2 is:
- 5
- 10
- 20
- 50
Show the answer
B. 10,000 ÷ 1,000 = 10.
Question 4. The annual holding cost at the EOQ in Question 2 is:
- ₦1,000
- ₦2,000
- ₦4,000
- ₦8,000
Show the answer
B. (1,000 ÷ 2) × ₦4 = ₦2,000, which equals the ordering cost 10 × ₦200.
Question 5. At the EOQ:
- Ordering cost is zero
- Annual ordering cost equals annual holding cost
- Holding cost is double ordering cost
- Purchase cost is minimised
Show the answer
B. This is a property of the model.
Question 6. Which is an assumption of the basic EOQ model?
- Bulk discounts are available
- Demand is constant and known
- Stock-outs are frequent
- Lead time is unknown
Show the answer
B. Demand is constant throughout the year.
Question 7. If orders are for Q units and demand is steady with no safety stock, the average stock held is:
- Q
- Q ÷ 2
- 2Q
- Q ÷ 4
Show the answer
B. Stock falls steadily from Q to zero.
Question 8. When a supplier offers a discount for large orders, the purchase price:
- Remains irrelevant
- Becomes relevant to the order size decision
- Is the only relevant cost
- Makes the EOQ formula invalid for all orders
Show the answer
B. It now varies with order size.
Question 9. Demand is 3,000 units every quarter. The annual demand to use in the EOQ formula is:
- 3,000
- 6,000
- 12,000
- 36,000
Show the answer
C. 3,000 × 4 = 12,000.
Question 10. A unit costs ₦250 and holding cost is 8% of price. The holding cost per unit per year is:
- ₦8
- ₦20
- ₦25
- ₦50
Show the answer
B. 8% × ₦250 = ₦20.
Section B style: written questions
Question 11 (10 marks). (a) Explain the four groups of costs associated with inventory, with examples. (8 marks) (b) State why holding costs and ordering costs conflict. (2 marks)
Question 12 (20 marks). Awka Beverages Ltd uses 48,000 kg of an ingredient a year. Placing an order costs ₦540. The normal price is ₦100 per kg and holding cost is 6% of purchase price.
(a) State four assumptions of the EOQ model. (4 marks)
(b) Calculate the EOQ, the number of orders a year, and the annual ordering and holding costs at the EOQ. (6 marks)
(c) The supplier offers a price of ₦98 per kg for orders of 4,000 kg or more and ₦96 per kg for orders of 8,000 kg or more. Advise the company on the best order size, taking holding cost as 6% of the purchase price. (10 marks)
A further written question
Question 13 (10 marks). Background topic. Daily usage of a component varies from 80 to 120 units and averages 100. Lead time varies from 10 to 20 days and averages 15. The re-order quantity is 3,000 units. (a) Calculate the re-order level, the minimum level and the maximum level. (6 marks) (b) Explain the purpose of the minimum and maximum levels. (2 marks) (c) State two limitations of the EOQ model. (2 marks)
Model answers
Show the model answer to Question 11
(a) Costs. Purchase price: what is paid to the supplier for the goods. Ordering costs: the cost of making an order, such as delivery paid by the buyer, the administrative cost of placing the order (for example, telephone calls), the cost of checking goods on arrival and, for items made in house, the set-up cost of each batch.
Holding costs: the cost of capital tied up in stock, insurance, warehousing and losses from obsolescence, deterioration and theft. Shortage costs: profit lost on sales that cannot be made, later loss of profit through loss of customer goodwill, and the cost of production stoppages caused by a shortage of materials.
(b) Large orders reduce the number of orders and so reduce ordering costs, but they raise average stock and so raise holding costs. Small orders do the reverse. The EOQ is the order size that balances the two.
Show the model answer to Question 12
(a) Assumptions. (1) No bulk discounts, so every unit costs the same whatever the order size. (2) The lead time is constant and known. (3) There is no risk of a stock-out, so shortage costs are ignored. (4) Demand is constant through the year, so average stock is half the order size.
(b) EOQ.
- Holding cost per kg.
6% × ₦100 = ₦6 - EOQ.
√(2 × 540 × 48,000 ÷ 6) = √8,640,000 = 2,939 kg - Orders a year.
48,000 ÷ 2,939 = 16.33 - Annual ordering cost.
16.33 × ₦540 = ₦8,818 - Annual holding cost. Average stock is half the order size.
(2,939 ÷ 2) × ₦6 = 1,469.5 × ₦6 = ₦8,817 - Check. The two costs are equal, apart from rounding.
(c) Discount decision. The EOQ of 2,939 kg is below both thresholds, so compare it with the two discount order sizes.
| Order 2,939 kg (EOQ) | Order 4,000 kg | Order 8,000 kg | |
|---|---|---|---|
| Price per kg ₦ | 100 | 98 | 96 |
| Holding cost per kg (6%) ₦ | 6.00 | 5.88 | 5.76 |
| Purchases (48,000 × price) ₦ | 4,800,000 | 4,704,000 | 4,608,000 |
| Ordering (48,000 ÷ Q × 540) ₦ | 8,818 | 6,480 | 3,240 |
| Holding (Q ÷ 2 × holding cost) ₦ | 8,817 | 11,760 | 23,040 |
| Total cost ₦ | 4,817,635 | 4,722,240 | 4,634,280 |
Advice: order 8,000 kg at a time, since the total annual cost of ₦4,634,280 is the lowest, saving ₦183,355 against ordering the EOQ. The company should first check that it has enough storage space and cash for 8,000 kg and that the ingredient will not deteriorate before use.
Show the model answer to Question 13
(a) Stock levels.
- Re-order level. Maximum usage × maximum lead time.
120 × 20 = 2,400 units - Minimum level. Re-order level − (average usage × average lead time).
2,400 − (100 × 15) = 900 units - Maximum level. Re-order level + re-order quantity − (minimum usage × minimum lead time).
2,400 + 3,000 − (80 × 10) = 4,600 units
(b) The minimum level is a warning level. If stock falls below it, usage is higher or delivery later than normal and there is a risk of a stock-out, so management should act. The maximum level is a warning level in the other direction. If stock rises above it, the firm is overstocked, tying up cash and increasing holding costs and the risk of deterioration.
(c) Any two of:
- demand is rarely perfectly constant
- lead times vary so safety stock is needed
- ordering and holding costs are hard to measure accurately
- it ignores discounts, price changes and storage limits unless extended
- it treats each item separately
Pathfinder on Economic order quantity and inventory control
These are the ICAN Pathfinder questions that test this topic. ICAN owns the questions and the suggested solutions, so we point you to them and do not copy them. Open your Pathfinder, attempt each question under exam conditions, then check your answer against the notes above.
From the previous syllabus. Asks for the re-order, minimum and maximum stock levels (2 marks each), five factors in setting the maximum level, and a 6-mark EOQ calculation.
Revise first: Re-order level and stock levels
From the previous syllabus. Asks which statement is not an assumption of the EOQ model.
Revise first: The economic order quantity model
From the previous syllabus. Asks which item is NOT among the costs balanced in finding the economic order quantity.
Revise first: The costs of holding and ordering stock
Multiple-choice items often ask for the EOQ, orders per year or annual holding cost from given data. Check Section A of every diet you practise.
Revise first: Quantity discounts
Source: ICAN Pathfinder, Management Accounting. Diets covered: May Diet, 2019 and May Diet, 2017 (previous syllabus). Stock levels are background, not in the syllabus. May Diet, 2025 and November Diet, 2024 are not covered here.
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