435. Lost Sales

435.1. EOQ with lost sales

Relax one dimension from basic EOQ: excess demand is no longer forbidden — but unlike planned backorders, customers facing a stockout do not wait. Each unit of unmet demand is permanently lost at penalty per unit (lost margin + goodwill).

435.1.1. Setup

New variables (beyond basic EOQ):

• = lost-sale cost per unit ($ per unmet unit)
• = stockout duration per cycle (decision variable; could be 0)

The inventory profile in each cycle is:

• Receive units, drain at rate until empty after time .
• Stay stocked-out for time — during this period units of demand walk away.
• Order again. Cycle length .

435.1.2. Cost model

Per cycle:

• Order:
• Holding: — average inventory over time .
• Lost-sales: — units lost at each.

Per unit time, divide by :

Two decision variables now: and .

435.1.3. When is it optimal to plan stockouts? (FOC in )

Take , quotient rule:

Set numerator to zero, expand :

This says: at the boundary , lost sales become attractive only if there, i.e., .

Plug in . Then , so the condition becomes:

So lost sales has a threshold behavior:

435.1.4. Threshold interpretation

The threshold is exactly the basic-EOQ cost per unit served:

So the rule is intuitive: operate iff the per-unit lost-sale cost exceeds the per-unit cost of running EOQ. Below that, you’d rather lose the customer than pay the inventory overhead.

435.1.5. Final formulas

Below the threshold, the model degenerates — there’s no interior planned-stockout solution under deterministic constant demand.

435.1.6. Why no interior solution?

Under deterministic constant demand with instantaneous replenishment, there is no benefit to planned stockouts — you can always order the right amount at the right time. Lost sales becomes interesting only when demand is stochastic (newsvendor / models with lost sales) or when there’s a minimum order quantity forcing you to order in lumps that don’t divide annual demand cleanly.