Backorders

438. Backorders

438.1. EOQ with planned backorders

Relax one dimension from basic EOQ: excess demand is no longer forbidden — stockouts are allowed at penalty rate $𝑏$ per unit per year. Now there are two decisions: order quantity $𝑄$ and maximum backorder $𝐵$ .

438.1.1. Setup

New variables (beyond basic EOQ):

$𝐵$ = maximum backorder quantity (decision variable)
$𝑏$ = backorder penalty cost per unit per year

The inventory profile in each cycle:

Receive $𝑄$ units, immediately fill the backlog $𝐵$ — on-hand jumps to $𝑄 - 𝐵$ .
On-hand drains at rate $𝐷$ over time $\frac{𝑄 - 𝐵}{𝐷}$ , hits zero, then accumulates backlog up to $𝐵$ over time $\frac{𝐵}{𝐷}$ , total cycle $𝑇 = \frac{𝑄}{𝐷}$ .

Two area integrals over a cycle, divided by the cycle length $𝑇 = \frac{𝑄}{𝐷}$ :

Average on-hand: triangle of height $𝑄 - 𝐵$ , width $\frac{𝑄 - 𝐵}{𝐷}$ → $\frac{{(𝑄 - 𝐵)}^{2}}{2 𝐷} ∣ 𝑇 = \frac{{(𝑄 - 𝐵)}^{2}}{2 𝑄}$
Average backlog: triangle of height $𝐵$ , width $\frac{𝐵}{𝐷}$ → $\frac{𝐵^{2}}{2 𝐷} ∣ 𝑇 = \frac{𝐵^{2}}{2 𝑄}$

438.1.2. Cost model

Total relevant cost (drop the constant purchase-cost term):

TRC (𝑄, 𝐵) = 𝑆 (\frac{𝐷}{𝑄}) + ℎ \frac{{(𝑄 - 𝐵)}^{2}}{2 𝑄} + 𝑏 (\frac{𝐵^{2}}{2 𝑄})

Two changes from basic EOQ TRC, both highlighted in red: the holding term now sees $(𝑄 - 𝐵)$ instead of $𝑄$ (less average on-hand), plus a new shortage term.

438.1.3. Derive $𝐵^{*}$ first (FOC in $𝐵$ )

Holding $𝑄$ fixed, take $𝜕 TRC / 𝜕 𝐵$ :

\frac{𝜕}{𝜕 𝐵} TRC = ℎ \cdot 2 (𝑄 - 𝐵) \frac{(- 1)}{2 𝑄} + 𝑏 \cdot 2 \frac{𝐵}{2 𝑄} = - \frac{ℎ (𝑄 - 𝐵)}{𝑄} + 𝑏 \frac{𝐵}{𝑄}

Set to zero, multiply by $𝑄$ :

- ℎ (𝑄 - 𝐵) + 𝑏 𝐵 = 0

- ℎ 𝑄 + ℎ 𝐵 + 𝑏 𝐵 = 0

𝐵^{*} = \frac{ℎ 𝑄}{ℎ + 𝑏}

So the backlog is the share $ℎ / (ℎ + 𝑏)$ of the order quantity.

438.1.4. Derive $𝑄^{}$ (FOC in $𝑄$ , after substituting $𝐵^{}$ )

Take $𝜕 TRC / 𝜕 𝑄$ and set to zero (algebra for the holding term uses the product rule):

- \frac{𝑆 𝐷}{𝑄^{2}} + \frac{ℎ (𝑄^{2} - 𝐵^{2})}{2 𝑄^{2}} - 𝑏 \frac{𝐵^{2}}{2 𝑄^{2}} = 0

Multiply by $2 𝑄^{2}$ :

- 2 𝑆 𝐷 + ℎ (𝑄^{2} - 𝐵^{2}) - 𝑏 𝐵^{2} = 0

ℎ 𝑄^{2} - (ℎ + 𝑏) 𝐵^{2} = 2 𝑆 𝐷

Substitute $𝐵^{*} = ℎ \frac{𝑄}{ℎ + 𝑏}$ so $(ℎ + 𝑏) {(𝐵^{*})}^{2} = ℎ^{2} \frac{𝑄^{2}}{ℎ + 𝑏}$ :

ℎ 𝑄^{2} - ℎ^{2} \frac{𝑄^{2}}{ℎ + 𝑏} = 2 𝑆 𝐷

𝑄^{2} \cdot ℎ \frac{ℎ + 𝑏 - ℎ}{ℎ + 𝑏} = 2 𝑆 𝐷

𝑄^{2} \cdot \frac{ℎ 𝑏}{ℎ + 𝑏} = 2 𝑆 𝐷

{(𝑄^{*})}^{2} = \frac{2 𝑆 𝐷}{ℎ} \cdot \frac{ℎ + 𝑏}{𝑏}

Take the square root:

𝑄^{*} = \sqrt{\frac{2 𝑆 𝐷}{ℎ}} \cdot \sqrt{\frac{ℎ + 𝑏}{𝑏}}

The first factor is the basic EOQ; the second is a multiplier $> 1$ that grows the order when backorders are cheap (small $𝑏$ ).

438.1.5. Final formulas

𝑄^{*} = \sqrt{\frac{2 𝑆 𝐷}{ℎ}} \cdot \sqrt{\frac{ℎ + 𝑏}{𝑏}} 𝐵^{*} = \sqrt{\frac{2 𝑆 𝐷 ℎ}{𝑏 (ℎ + 𝑏)}} {TRC}^{*} = \sqrt{2 𝑆 𝐷 ℎ} \cdot \sqrt{\frac{𝑏}{ℎ + 𝑏}}

Sanity check: as $𝑏 \to \infty$ (backorders prohibitively expensive), the multiplier $\sqrt{\frac{ℎ + 𝑏}{𝑏}} \to 1$ and $𝐵^{*} \to 0$ — recovers basic EOQ. As $𝑏 \to 0$ (backorders are free), $𝑄^{*} \to \infty$ and ${TRC}^{*} \to 0$ .

Example

Given (shared EOQ params + a backorder penalty):

Annual demand: $𝐷 = 12000$ units/year
Order cost: $𝑆$ = $50 / order
Holding cost: $ℎ$ = $2 / unit / year
Backorder penalty: $𝑏$ = $8 / unit / year

Step 1 — backlog share

𝐵^{*} = \frac{ℎ 𝑄^{*}}{ℎ + 𝑏} = \frac{2}{2 + 8} 𝑄^{*} = 0.2 𝑄^{*}

So the optimal backlog is 20% of the order quantity.

Step 2 — order quantity

𝑄^{*} = \sqrt{\frac{2 \cdot 50 \cdot 12000}{2}} \cdot \sqrt{\frac{2 + 8}{8}} = \sqrt{600000} \cdot \sqrt{1.25} \approx 774.6 \cdot 1.118 \approx 866 units

Step 3 — backlog and cost

𝐵^{*} = 0.2 \cdot 866 \approx 173 units

{TRC}^{*} = \sqrt{2 \cdot 50 \cdot 12000 \cdot 2} \cdot \sqrt{\frac{8}{10}} \approx 1549 \cdot 0.894 \approx 1386 $/year

Compare to basic EOQ on the same inputs:

Basic EOQ: $𝑄^{*} = 775$ , ${TRC}^{*} \approx$ $1549
With backorders: $𝑄^{*} = 866$ (larger), $𝐵^{*} = 173$ , ${TRC}^{*} \approx$ $1386 (10% lower)

Allowing planned backorders is worthwhile here because $𝑏$ is only $4 \times ℎ$ — backorders cost a bit more than holding but not so much that we want zero backlog.