The all-units discount method worked on one part number, and why the deeper discount is not the cheaper one.
With quantity discounts, compute the EOQ at each price, raise it to the break quantity when it falls below, and compare total annual cost including the purchase cost. For part AF-7, buying 95,680 units a year at 385 EUR an order, a 0.5 per cent discount for orders of 8,000 saves 10,628 EUR a year. A 1 per cent discount for orders of 20,000 saves only 4,385: the deeper discount is the worse deal.
Worked in full in Supply Chain Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The plain economic order quantity assumes the unit price does not depend on the batch. Suppliers rarely agree. As soon as a price break appears, the purchase cost, which the EOQ formula leaves out because it is the same at every batch, becomes the largest line in the comparison, and the cost curve stops being smooth: it drops at each break. The method that handles this is old and short, and it fits on one sheet.
| Input | Value |
|---|---|
| Annual demand D (1,840 a week, 52 weeks) | 95,680 units |
| Cost of placing one order S | 385 |
| List price, delivered | 28.60 |
| Holding rate (capital, storage, obsolescence, insurance) | 13.4% |
| Break 1: any quantity | 28.600 |
| Break 2: 8,000 units or more, 0.5% off all units | 28.457 |
| Break 3: 20,000 units or more, 1.0% off all units | 28.314 |
These are all-units discounts: reach the break and every unit in the order is billed at the lower price. That is the common form in purchasing contracts and the one the method below solves. The holding cost is a rate on the price paid, so it falls slightly as the price falls.
=MAX(SQRT(2*D*S/(c*r)),Qmin) for Q, then =D*c+D/Q*S+Q/2*c*r for the total.At list price the EOQ is 4,384.5 units, as in the plain EOQ calculation for AF-7. At 28.457 it is 4,395.5 and at 28.314 it is 4,406.6. A one per cent price change moves the EOQ by half a per cent, because it sits under a square root. Neither discounted EOQ comes near its break, so in both bands the candidate quantity is the break: 8,000 and 20,000.
| Candidate | Order qty | Orders a year | Purchase | Ordering | Holding | Total |
|---|---|---|---|---|---|---|
| EOQ at list | 4,384.5 | 21.82 | 2,736,448 | 8,402 | 8,402 | 2,753,251 |
| Break 2 | 8,000 | 11.96 | 2,722,766 | 4,605 | 15,253 | 2,742,623 |
| Break 3 | 20,000 | 4.78 | 2,709,084 | 1,842 | 37,941 | 2,748,866 |
Break 2 wins. Against the EOQ at list it saves 13,682 of purchase cost and adds 3,054 of ordering and holding, a net 10,628 EUR a year. Break 3 saves twice as much on purchase, 27,364, but adds 22,979 of inventory cost, because the average cycle stock rises from 2,192 units worth 62,698 to 10,000 units worth 283,140. Its net saving is 4,385, and it is 6,243 a year worse than break 2.
The break quantities can be tested the other way round, by solving the discount that makes each break-even. Moving from 4,384.5 to 8,000 units costs so little in inventory that any discount above 0.114 per cent pays for it. Moving to 20,000 needs 1.225 per cent just to match break 2. The supplier offered 1.0, so the buyer should either decline or counter at 1.25 per cent or more.
The order interval tells the same story in plain language. At the EOQ AF-7 is reordered every 2.4 weeks, at 8,000 every 4.3 weeks, at 20,000 every 10.9 weeks. Ten weeks of stock of a filter element is a lot of shelf, and the obsolescence component of the holding rate exists precisely for it.
| Case | Plain EOQ (units) | Break 2 saves | Break 3 saves |
|---|---|---|---|
| Holding rate 7.4% (capital only) | 5,900 | 13,141 | 17,057 |
| Holding rate 13.4% (full) | 4,385 | 10,628 | 4,385 |
| Holding rate 25.0% | 3,210 | 3,572 | −22,311 |
The ranking flips with the holding rate. At the capital-only rate of 7.4 per cent, break 3 looks best by almost 4,000 a year. At 25.0 per cent, which is not unusual for parts with short lives, it loses 22,311 against simply ordering the EOQ. Break 2 is robust to the size of its discount: at 0.2 per cent it still saves 2,373, at 0.3 per cent 5,124, at 0.75 per cent 17,507.
Comparing inventory cost only. Ordering and holding at 8,000 cost 19,858 a year against 16,803 at the EOQ, so a model that leaves out the purchase line rejects the discount. The purchase cost is the only line the discount moves, and it is 2.7 million a year. It has to be in the comparison.
Valuing the stock at the cost of capital alone. As the table shows, a holding rate that ignores storage, obsolescence and insurance makes stock look cheap, and cheap stock makes the biggest break look best. The decision is only as good as the rate.
Forgetting what else the batch changes. In the simple model the safety stock does not depend on the batch, so it drops out of the comparison. In practice a bigger batch means fewer replenishment cycles a year, 4.78 instead of 21.82, and therefore fewer exposures to a stock-out; it also means warehouse space that may not exist. Check both before signing.
The full holding rate and AF-7's order quantity come from the free workbook for this case, where every input is a blue cell you can overwrite with your own part.
Yes, but only as a first step inside each price band. Compute the EOQ at each discounted price; if it falls below that band's minimum quantity, the best order in the band is the minimum itself. For AF-7 the EOQ barely moves, from 4,384.5 at list to 4,406.6 at 1 per cent off, so every discount band is reached only by ordering its minimum: 8,000 or 20,000 units.
The one at which purchase savings cover the extra holding and fall in ordering cost. For AF-7, ordering 8,000 instead of 4,384.5 breaks even at a 0.114 per cent discount, because the extra inventory cost is only 3,054 a year. Moving to 20,000 units needs 1.225 per cent, against the 1.0 per cent offered, so it loses 6,243 a year against the 8,000 break.
Because the large break mostly buys stock. At a 7.4 per cent capital-only rate the 20,000 break looks best, saving 17,057 against 13,141 for the 8,000 break. At the full 13.4 per cent it saves 4,385 against 10,628, and at 25.0 per cent it costs 22,311 more than the plain EOQ.
This article is one calculation from Supply Chain Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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