---
title: "EOQ Formula, Worked Example and When to Drop It"
description: "The economic order quantity formula with a worked example, why ordering 20% to 25% off the optimum costs only 2.5% more, and when a lot discount wins."
url: "https://staging.khoi.co.uk/en/blog/eoq-muoi-phut-va-gioi-han"
lang: en-GB
author: "Lê Quốc Khôi"
date: 2026-11-03
updated: 2026-11-03
topics: ["Supply chain & operations"]
tags: ["Economic order quantity", "Inventory management", "Reorder point", "Safety stock", "Quantity discounts"]
---

# EOQ in ten minutes: the formula, its limits and when to drop it

## Key takeaways

- EOQ is the quantity per order that makes ordering cost plus holding cost lowest when demand is steady. Q* = √(2DS/H), where D is annual demand, S the cost of placing one order and H the cost of holding one unit for a year.
- The total-cost curve is flat around Q*: ordering 25% more or 20% less than Q* raises ordering plus holding cost by only 2.5%. If the true ordering cost is double or half the figure you used, the total differs by about 6%. This is arithmetic, not a research finding.
- Hypothetical example, an electrical-supplies wholesaler: moving from 1,000 reels a month to EOQ saves about VND 2.7m a year, on annual purchases of VND 1.8bn. A 2% lot discount on orders of 1,200 reels saves about VND 31m, more than ten times as much.
- Drop EOQ when demand swings strongly or a lot discount applies, and use a different formula when the firm makes what it sells (Chopra, 2019, ch. 11; Slack et al., 2022, ch. 13). For seasonal goods, one fixed reorder point all year can cause stock-outs: Chopra's example is a point worth ten days of demand in the low season but only two in the peak (Chopra, 2019, ch. 12).

You run purchasing for a wholesaler of electrical and plumbing supplies. Cable sells steadily all year, and your team orders 1,000 reels at the start of every month because it is "easy to remember". A formula gives a better quantity in ten minutes. The real question is how much money that formula is worth, and when it goes wrong.

## What is EOQ, and how is the formula built?

The [economic order quantity (EOQ)](https://staging.khoi.co.uk/en/glossary#economic-order-quantity) is the quantity per order that makes the sum of ordering and holding cost smallest, provided demand is steady. The two textbooks cited here both cover EOQ (Chopra, 2019, ch. 11; Slack et al., 2022, ch. 13); the section at the end says where the formula is usually traced to and what I could not read.

There are three inputs: D, annual demand; S, the cost of each order; and H, the cost of holding one unit for a year. Ordering plus holding cost is (D ÷ Q) × S + (Q ÷ 2) × H. A large order means few orders but a lot of stock held; a small one is the reverse. Finding the minimum of that sum gives Q* = √(2DS ÷ H), and at Q* the ordering cost equals the holding cost. The derivation is my own. The library records the assumptions that come with it: steady demand, a fixed lead time and no quantity discount (Chopra, 2019, ch. 11; Slack et al., 2022, ch. 13).

## How much money separates ordering 1,000 reels a month from EOQ?

Less than you would think. Everything below is hypothetical: the setting, demand and costs are mine, made up to show the arithmetic.

**A hypothetical example.** An electrical-supplies wholesaler in Can Tho sells 12,000 reels of cable a year and buys them at VND 150,000 a reel. Each order costs VND 600,000 in paperwork, freight and receiving. I assume holding cost is 25% of the purchase price a year, so H = VND 37,500 per reel per year.

Q* = √(2 × 12,000 × 600,000 ÷ 37,500) = √384,000 ≈ 620 reels, about 19 orders a year, one every 2.7 weeks.

| Order quantity Q (reels) | Orders per year | Ordering cost (VND m) | Holding cost (VND m) | Total (VND m) | Above the minimum |
|--:|--:|--:|--:|--:|--:|
| 300 | 40 | 24.00 | 5.63 | 29.63 | 27.5% |
| 500 | 24 | 14.40 | 9.38 | 23.78 | 2.3% |
| 620 | 19.4 | 11.61 | 11.63 | 23.24 | 0% |
| 800 | 15 | 9.00 | 15.00 | 24.00 | 3.3% |
| 1,000 (current) | 12 | 7.20 | 18.75 | 25.95 | 11.7% |
| 1,200 | 10 | 6.00 | 22.50 | 28.50 | 22.6% |

Moving from 1,000 reels to 620 saves 25.95 minus 23.24, about VND 2.7m a year. Annual purchases are 12,000 × 150,000 = VND 1.8bn, so the saving is about 0.15% of what you buy. The curve is flat: ordering 25% more than Q* (775 reels) or 20% less (496) costs only about 2.5% extra. In general, if Q is r times Q*, ordering plus holding cost is (r + 1/r) ÷ 2 times the minimum; at r = 0.5 or r = 2 that is 1.25.

So EOQ is worth doing quickly once, not worth doing finely. Suppose the true ordering cost is double the VND 600,000 you used: the true Q* is about 41% larger than your figure, and using the old figure makes ordering plus holding cost about 6% higher, because (√2 + 1/√2) ÷ 2 ≈ 1.06. If the true cost is half, the result is the same. When the inputs are vague, round to the pack size instead of fussing over single reels.

## When does a lot discount beat EOQ?

When the price cut is worth more than the cost of holding extra stock, which basic EOQ never asks. The library says plainly to avoid EOQ when lot discounts apply (Chopra, 2019, ch. 11; Slack et al., 2022, ch. 13).

Suppose the supplier takes 2% off (VND 147,000 a reel) for orders of 1,200 or more. Compare total annual cost, purchase price included:

| Option | Purchases (VND m) | Ordering cost (VND m) | Holding cost (VND m) | Total (VND m) |
|:--|--:|--:|--:|--:|
| Order 620 reels at VND 150,000 | 1,800.00 | 11.61 | 11.63 | 1,823.24 |
| Order 1,200 reels at VND 147,000 | 1,764.00 | 6.00 | 22.05 | 1,792.05 |

The large discounted order is about VND 31m a year cheaper, more than ten times what EOQ saved. Holding cost in the second row uses the lower price: 25% × 147,000 = VND 36,750 a reel, times the 600 reels held on average. The result holds up reasonably well: with a 40% holding rate instead of 25%, the large order is still about VND 24m cheaper, and by my calculation the discount only loses if the holding rate exceeds roughly 80% a year (both calculations recompute Q* for the no-discount option). Perishable or expiring goods are different, because holding cost is then no longer a tidy percentage.

## When to order, not just how much?

When stock reaches the reorder point: expected demand during the wait for delivery plus a safety buffer (Chopra, 2019, ch. 12). In the example, average demand is 12,000 ÷ 52 ≈ 231 reels a week; suppose a two-week lead time and a weekly standard deviation of 60 reels (an assumption). Demand over the two-week wait is 462 reels. To avoid a stock-out in 95% of cycles, the safety buffer from the standard statistical formula is 1.645 × 60 × √2 ≈ 140 reels. The reorder point is about 600 reels: when stock falls to 600, order another 620.

One pitfall is leaving 600 fixed all year. Chopra's example is a reorder point that equals ten days of demand in the low season but only two in the peak, so stock-outs are certain (Chopra, 2019, ch. 12). In my example, if peak demand rises to 460 reels a week, demand over the two-week wait is 920, and a 600-reel reorder point covers only 1.3 weeks, less than the lead time. For seasonal goods, recalculate mean demand and its standard deviation for each time of year. See also [Reorder point and safety stock](https://staging.khoi.co.uk/en/framework/reorder-point-safety-stock) and [Economic order quantity (EOQ)](https://staging.khoi.co.uk/en/framework/economic-order-quantity) in the library.

## When should you drop EOQ?

| Situation | Problem with EOQ | What to do instead |
|:--|:--|:--|
| Demand swings strongly with the season | The steady-demand assumption fails; the library says to avoid EOQ when demand swings strongly (Chopra, 2019, ch. 11; Slack et al., 2022, ch. 13) | Recalculate by season, or order from a period-by-period demand plan |
| Quantity or truck-load discounts | The purchase price changes with Q, which the basic formula ignores | Compare total cost including purchases, as in the table above |
| The firm makes what it sells | A different variant of the formula is needed (Chopra, 2019, ch. 11; Slack et al., 2022, ch. 13) | Use the produce-in-house variant |
| No reliable figures for S and H | A precise-looking answer from guessed inputs | Order by pack size on a fixed cycle and review after a quarter |
| Cheap items tying up little capital | The effort costs more than the saving | Order on a fixed cycle |

The "What to do instead" column, the truck-load row and the last two rows are my own reading, not the books'. In this example, where EOQ saves only 0.15% of purchase value, a buyer's time is better spent negotiating price and discounts.

## Going deeper: what did I read, and what did I not?

The formula is usually traced to F. W. Harris's paper "How many parts to make at once", published in 1913 and reprinted in 1990 in *Operations Research*. The 1990 reprint is not open access and I could not read the full text, so I do not cite it for any claim here. The formula, the r-times result and every figure in the example are my own calculations, recomputed in code, and you can check them yourself.

I also did not read the chapters of Slack and colleagues (chapter 13) or Chopra (chapters 11 and 12) directly. What I attribute to the two books is what the framework summaries already published in the site's library record: EOQ's assumptions, when to avoid it, the produce-in-house variant, and the seasonal reorder-point example. The rest, including the 1.645 factor for the safety buffer, is standard statistics and I attribute it to neither book. One last limit: EOQ does not price stock-outs, so any shortage problem has to be handled through the safety buffer at the reorder point, and the quality of that buffer depends on the standard deviation of demand you estimate from your own data.

## Frequently asked questions

### What goes into the holding cost?

In this post's example I fold the interest on capital tied up in stock, warehouse space, shrinkage and insurance into one rate of 25% of the purchase price a year. That is my assumption, not a standard figure. For a real number, take your firm's cost of capital and add the storage and shrinkage costs in your own books. Because Q* depends only on the square root of the rate, an error of a few tens of per cent does not change the answer much.

### What counts as the cost of placing an order?

Everything that arises per order regardless of quantity: preparing and approving the order, the freight for one delivery, receiving and checking at the warehouse, and payment. The price of the goods is not part of it. In the example I assume VND 600,000 per order. If the freight per trip changes with the quantity, this simple model no longer holds.

### How does EOQ differ from the reorder point?

EOQ answers how much to order each time. The reorder point answers when: when stock falls to the demand expected during the wait for delivery plus a safety buffer. They work together: once you know Q, you order Q each time stock reaches the reorder point.

### What if the supplier only sells in cartons of 50?

Round Q* to the nearest multiple of the pack size. The total-cost curve is flat around Q*, so being off by 20% to 25% changes ordering plus holding cost by only about 2.5%, though being off by half raises it by 25%; the table in the middle of the post has the figures. If the supplier sets a minimum order well above Q*, work out the total cost at that minimum rather than at Q*.

## References

- Chopra, S. (2019) *Supply chain management: strategy, planning, and operation (7th global edn)*. Harlow: Pearson. Available at: https://www.pearson.com/se/Nordics-Higher-Education/subject-catalogue/decision-science/Chopra-supply-chain-management.html.
- Slack, N., Brandon-Jones, A. and Burgess, N. (2022) *Operations Management (10th edn)*. Harlow: Pearson. Available at: https://www.pearson.com/nl/en_NL/higher-education/subject-catalogue/decision-science/slack-operations-management-10thedition.html.

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