How much to order: EOQ

Every time you place an order, it costs something fixed: paperwork, calls, a delivery run, receiving and checking. Every unit you keep in stock costs something too: cash tied up, storage, spoilage. Order rarely in big batches and you save on ordering but pay to hold a mountain of stock. Order often in small batches and you hold little but pay the ordering cost again and again.

The economic order quantity (EOQ) is the order size where the total of those two costs is lowest:

where $D$ is annual demand, $S$ the cost per order and $H$ the cost of holding one unit for a year.

Take a café in Pune that uses 1,200 kg of coffee beans a year. Each order costs about ₹500 to place, and holding a kilogram for a year costs about ₹200. Then $EOQ = \sqrt{2 \times 1200 \times 500 / 200} = \sqrt{6000} \approx 77.5$ kg, about 15 orders a year. At that size the yearly ordering cost and holding cost come out equal, around ₹7,750 each.

The useful surprise is how forgiving it is. If the café orders 100 kg instead (29% more than the EOQ), its total cost rises by only about 3%. So you can round EOQ to a pack size, a truck load or a supplier’s minimum without losing much.

Two costs pulling in opposite directions

With annual demand $D$, order size $Q$, cost per order $S$ and holding cost $H$ per unit per year:

• You place $D/Q$ orders a year, so the yearly ordering cost is $\frac{D}{Q}S$.
• Stock falls from $Q$ to zero and back again, so on average you hold $Q/2$, and the yearly holding cost is $\frac{Q}{2}H$.

The first term falls as $Q$ grows; the second rises. Setting the slope of $TC$ to zero gives the minimum:

At $Q^{*}$ the two costs are exactly equal, a handy check on any calculation. The purchase price does not appear, because you buy $D$ units a year whatever the order size (until discounts enter, in the deep dive). Holding cost is usually set as a rate times the unit cost: $H = i \times C$.

Why it is so forgiving

How much you lose depends only on how far you are from $Q^{*}$:

Ordering half or double the EOQ raises cost by 25%, but ordering 20% too much or too little costs under 3% extra. So round to cases, pallets or truckloads freely. The square root also means EOQ responds slowly to its inputs: if demand quadruples, the EOQ only doubles, and you order twice as often.

How much, and when

EOQ answers how much. The reorder point, $ROP = d\,L + SS$, answers when. Together they form the classic (Q, R) policy: whenever the inventory position falls to $R$, order $Q$.

When the supplier offers a discount

Suppose the bean supplier charges ₹800 a kg, or ₹780 a kg for orders of 100 kg or more, and the café’s holding rate is 25% of the price a year. EOQ alone can’t settle this, because the purchase cost now depends on $Q$. The method:

1. Compute the EOQ at each price.
2. If that EOQ falls below the price’s minimum, raise it to the minimum.
3. Compare total cost including the purchase cost, $D \times C$, and pick the lowest.

The larger order costs ₹258 more to order and hold, but the discount saves ₹24,000 on purchases, so it wins by about ₹23,700 a year. Discounts often swamp the EOQ trade-off, which is why buyers must judge them on total cost, not unit price.

Making instead of buying

When a plant makes the batch itself, stock builds up gradually while production runs at rate $P$ and demand uses it at rate $D$. Peak stock is lower than $Q$, which gives the economic production quantity:

Here $S$ is the set-up cost of a production run. The faster production is relative to demand, the closer this gets to the plain EOQ.

Demand that changes from week to week

EOQ assumes steady demand. MRP systems face lumpy, time-phased requirements, such as 0, 120, 0, 40 and 300 units in the coming weeks. For these, planners use lot-sizing rules: lot-for-lot (order exactly each period’s need), fixed period (cover, say, two weeks at a time), or heuristics such as Silver–Meal and least unit cost that balance set-up and holding cost period by period. The Wagner–Whitin algorithm finds the true optimum by dynamic programming. ERP systems, including SAP, offer a menu of these procedures in the material master.

What lean does to the formula

EOQ takes the ordering or set-up cost $S$ as given. Lean manufacturing attacks $S$ itself, with quick changeovers (SMED), simpler ordering and standard containers. Because $Q^{*}$ grows with $\sqrt{S}$, cutting set-up cost by 75% halves the batch size, the cycle stock and the total ordering-plus-holding cost. The two schools agree more than they argue: lean changes the inputs, and the formula follows.