EOQ with Price Breaks¶

Definition¶

The course's EOQ model with price breaks gives an all-unit discount: every ordered unit has price $c_2<c_1$ when the batch meets a threshold $q$ . Following the “at least” convention of the worked examples,

\begin{gathered} c(y)=\begin{cases}c_1,&0<y<q,\\c_2,&y\ge q,\end{cases}\\ TCU(y)=Dc(y)+\frac{KD}{y}+\frac{hy}{2}. \end{gathered}

Holding rate $h$ is fixed across price levels; demand is constant, replenishment instantaneous, and shortages prohibited.

Interpretation¶

With an all-unit discount, ordering at least $q$ units lowers the price of every unit in the batch. Buying more than the EOQ can then pay off even though it raises the holding cost, so the purchasing cost $Dc(y)$ must be included in the comparison instead of being left out as in the classical EOQ model.

Two total-cost curves for two unit prices, each drawn as solid where its price applies; the minimum of the higher-price curve at 22 is compared with the cost at the price break 35, which is lower.

The blue curve is the total cost at price $170$ , which applies below $35$ units, and the red curve the total cost at price $160$ from $35$ on. The blue curve is lowest at $y_m=22$ $22$ , with cost $450$ per day. The red curve does not apply at $22$ , but at the threshold $35$ it costs $442.07$ , which is lower, so the optimal order is $35$ .

From the break $q$ on, every unit costs $c_2<c_1$ $c_{1}$ , so the total cost curve jumps down. Compare the best quantity on each side of $q$ .

Both curves have the same shape and differ only by the constant $D(c_1-c_2)$ $c_{2})$ , so each is lowest at $y_m=\sqrt{2KD/h}$ $2 K D / h$ . On the discounted curve, the cheapest quantity that qualifies is $y_m$ if $y_m\ge q$ $q$ , and otherwise $q$ itself, since that curve rises to the right of $y_m$ . The comparison is therefore between at most two candidates, $y_m$ at the higher price and $q$ at the lower price.

The price-break indifference quantity turns this comparison into a threshold rule. A common slip is to take the discounted price at $y_m=22$ $22$ , which gives $430$ per day, but $22$ units do not qualify for that price.

Key Results¶

Let $y_m=\sqrt{2KD/h}$ $2 K D / h$ . If $y_m\ge q$ $q$ , order $y_m$ at the discount price.
If $y_m<q$ $q$ , compare the full costs at $y_m$ and $q$ and choose the cheaper; both are optimal at equality.
This page treats $y=q$ as qualifying for the discount, as the worked examples do; check the convention when a problem states the threshold as $y>q$ $q$ .
The discount is all-unit, not a discount only on units above the threshold.

Examples¶

For phones, $y_m=22$ $22$ : cost at 22 is 450 per day, while cost at the qualifying 35 is about 442.07; order 35.
For the snack bar, the 100-kilogram threshold is not worthwhile; the optimal quantity remains about 48.30.

Prerequisites¶

Price-Break Indifference Quantity

Sources¶

OR Modelling · Week 5

Aliases: quantity discount model, all-unit quantity discount model Symbols: square root

Last updated: 20 October 2026 03:02