Shortage Cost¶

Definition¶

A shortage cost penalizes unmet demand. In the EOQ model with planned shortages, customers wait for replenishment and the parameter $p$ charges each outstanding unit per unit of waiting time. With batch size $y$ and stock $s$ on hand just after each delivery, so that the maximum backlog is $y-s$ $s$ , the average shortage cost rate is

p\frac{(y-s)^2}{2y}.

Interpretation¶

A shortage cost puts a price on making customers wait. A planned backlog lets the shop keep less stock, but every waiting unit is charged $p$ per unit of time, so the model can trade holding cost against waiting customers.

One cycle of a stock level that starts at s, falls below zero into a backlog of y - s, and is then refilled, with the stock part and the shaded backlog part marked.

In one cycle of length $t_0=y/D$ $y / D$ , the stock falls from $s$ to zero at time $s/D$ and then becomes negative: the shaded triangle below the axis is the backlog, with depth $y-s$ and width $(y-s)/D$ $s) / D$ , so its area $(y-s)^2/(2D)$ adds up units times waiting time.

The cost rate $p(y-s)^2/(2y)$ is $p$ times this area divided by the cycle length. In the EOQ model with planned shortages the optimal backlog shrinks as $p$ grows. This $p$ is not the selling price $p$ of the newsvendor model, which charges a lost margin once per unit of unmet demand, not per unit of waiting time.

Key Results¶

The red triangle has area $(y-s)^2/(2D)$ $s)^{2} / (2 D)$ . Multiplying by $p$ gives shortage cost $p(y-s)^2/(2D)$ per cycle; dividing by $t_0=y/D$ gives the cost rate $p(y-s)^2/(2y)$ $s)^{2} / (2 y)$ .
Demand is backordered and later filled: the arriving batch first clears the backlog $y-s$ and leaves $s$ units on hand.
Use the same time unit for $p$ , $h$ , and $D$ .
In the newsvendor notation, $p$ instead denotes selling price; symbols must be interpreted within their model.

Examples¶

The phone shop compensates waiting customers by 6 per phone per day.
The EOQ practice exercise permits planned shortages at 10 per unit per week.

Prerequisites¶

Sources¶

OR Modelling · Week 5

Aliases: stockout cost, backlog cost Keywords: backorder cost

Last updated: 27 October 2026 11:52