Lost Sales

451. Lost Sales

451.1. 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 $𝜋$ per unit (lost margin + goodwill).

451.1.1. Setup

New variables (beyond basic EOQ):

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

The inventory profile in each cycle is:

Receive $𝑄$ units, drain at rate $𝐷$ until empty after time $𝑄 / 𝐷$ .
Stay stocked-out for time $𝜏$ — during this period $𝐷 𝜏$ units of demand walk away.
Order again. Cycle length $𝑇 = 𝑄 / 𝐷 + 𝜏$ .

451.1.2. Cost model

Per cycle:

Order: $𝑆$
Holding: $ℎ \cdot (𝑄 / 2) \cdot (𝑄 / 𝐷)$ — average inventory $𝑄 / 2$ over time $𝑄 / 𝐷$ .
Lost-sales: $𝜋 𝐷 𝜏$ — $𝐷 𝜏$ units lost at $𝜋$ each.

Per unit time, divide by $𝑇 = 𝑄 / 𝐷 + 𝜏$ :

TC (𝑄, 𝜏) = \frac{𝑆 + ℎ 𝑄^{2} / (2 𝐷) + 𝜋 𝐷 𝜏}{𝑄 / 𝐷 + 𝜏}

Two decision variables now: $𝑄$ and $𝜏$ .

451.1.3. When is it optimal to plan stockouts? (FOC in $𝜏$ )

Take $𝜕 TC / 𝜕 𝜏$ , quotient rule:

\frac{𝜕}{𝜕 𝜏} TC = \frac{𝜋 𝐷 \cdot (𝑄 / 𝐷 + 𝜏) - (𝑆 + ℎ 𝑄^{2} / (2 𝐷) + 𝜋 𝐷 𝜏) \cdot 1}{{(𝑄 / 𝐷 + 𝜏)}^{2}}

Set numerator to zero, expand $𝜋 𝐷 (𝑄 / 𝐷 + 𝜏) = 𝜋 𝑄 + 𝜋 𝐷 𝜏$ :

𝜋 𝑄 + 𝜋 𝐷 𝜏 - 𝑆 - ℎ 𝑄^{2} / (2 𝐷) - 𝜋 𝐷 𝜏 = 0

𝜋 𝑄 - 𝑆 - ℎ 𝑄^{2} / (2 𝐷) = 0

This says: at the boundary $𝜏 = 0$ , lost sales become attractive only if $𝜕 TC / 𝜕 𝜏 \leq 0$ there, i.e., $𝜋 𝑄 \leq 𝑆 + ℎ 𝑄^{2} / (2 𝐷)$ .

Plug in $𝑄 = 𝑄_{basic}^{*} = \sqrt{2 𝑆 𝐷 / ℎ}$ . Then $ℎ 𝑄^{2} / (2 𝐷) = 𝑆$ , so the condition becomes:

𝜋 \cdot \sqrt{\frac{2 𝑆 𝐷}{ℎ}} \leq 2 𝑆 \Leftrightarrow 𝜋 \leq \sqrt{\frac{2 𝑆 ℎ}{𝐷}}

So lost sales has a threshold behavior:

Lost-sale cost $𝜋$	Optimal action
$𝜋 \geq \sqrt{2 𝑆 ℎ / 𝐷}$	Operate as basic EOQ ( $𝜏^{*} = 0$ , no stockouts).
$𝜋 < \sqrt{2 𝑆 ℎ / 𝐷}$	Don’t operate — lose all demand. Inventory not worth the trouble.

451.1.4. Threshold interpretation

The threshold $\sqrt{2 𝑆 ℎ / 𝐷}$ is exactly the basic-EOQ cost per unit served:

\frac{{TRC}_{basic}^{*}}{𝐷} = \frac{\sqrt{2 𝑆 𝐷 ℎ}}{𝐷} = \sqrt{\frac{2 𝑆 ℎ}{𝐷}}

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.

451.1.5. Final formulas

𝜋 \geq \sqrt{\frac{2 𝑆 ℎ}{𝐷}} \Rightarrow 𝑄^{*} = \sqrt{\frac{2 𝑆 𝐷}{ℎ}}, 𝜏^{*} = 0

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

451.1.6. 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 / $(𝑄, 𝑟)$ 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.

Example

Given (shared EOQ params + a lost-sale penalty):

Annual demand: $𝐷 = 12000$ units/year
Order cost: $𝑆$ = $50 / order
Holding cost: $ℎ$ = $2 / unit / year
Lost-sale penalty: vary $𝜋$ to see threshold

Step 1 — compute the threshold

𝜋_{threshold} = \sqrt{\frac{2 𝑆 ℎ}{𝐷}} = \sqrt{\frac{2 \cdot 50 \cdot 2}{12000}} = \sqrt{\frac{200}{12000}} \approx

$0.13 / unit

Step 2 — decide

$𝜋$ = $1 / unit (typical lost margin): $1 ≫ 0.13$ → operate as basic EOQ.
$𝜋$ = $0.10 / unit (very cheap to lose): $0.10 < 0.13$ → don’t operate.

Step 3 — compare to basic EOQ

Basic EOQ assumes you always serve demand: $𝑄^{*} = 775$ , ${TRC}^{*} \approx 1549$ $/year.
Lost sales adds a test: only operate if $𝜋$ exceeds the per-unit cost of EOQ. For any normal product where lost margin > a few cents per unit, basic EOQ wins and the lost-sales option is never used.

Under deterministic demand, lost sales is a go / no-go decision, not a planned-shortage strategy. The interesting variants live in stochastic demand models — see newsvendor and $(𝑄, 𝑟)$ policies.