434. Backorders

434.1. EOQ with planned backorders

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

434.1.1. Setup

New variables (beyond basic EOQ):

• = maximum backorder quantity (decision variable)
• = backorder penalty cost per unit per year

The inventory profile in each cycle:

• Receive units, immediately fill the backlog — on-hand jumps to .
• On-hand drains at rate over time , hits zero, then accumulates backlog up to over time , total cycle .

Two area integrals over a cycle, divided by the cycle length :

• Average on-hand: triangle of height , width →
• Average backlog: triangle of height , width →

434.1.2. Cost model

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

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

434.1.3. Derive first (FOC in )

Holding fixed, take :

Set to zero, multiply by :

So the backlog is the share of the order quantity.

434.1.4. Derive (FOC in , after substituting )

Take and set to zero (algebra for the holding term uses the product rule):

Multiply by :

Substitute so :

Take the square root:

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

434.1.5. Final formulas

Sanity check: as (backorders prohibitively expensive), the multiplier and — recovers basic EOQ. As (backorders are free), and .