EOQ with planned shortages: the backorder-allowance model
When stockouts are tolerated at a known per-unit-per-time penalty b, the EOQ model extends to plan a positive shortage B*. The optimal order quantity Q* grows by a factor of sqrt((H + b) / b) and B* = Q* * H / (H + b)[1]. As b grows large the model collapses to classic Wilson with B* tending to zero.
When the model fits
Backorder policy is economically defensible when the per-unit shortage cost b can be measured (an explicit late fee, a contractual penalty, a documented marginal lost-margin rate) and when customers accept the wait. It is the right tool for B2B contract goods, spare parts with priced expedite agreements, and direct-to-distributor wholesale where the buyer expects allocations during peak.
Boundary behaviour
As b tends to infinity, (H + b)/b tends to 1 and Q* collapses to the Wilson Q*, B* tends to 0. As b tends to 0 (stockouts cost nothing), Q* tends to infinity and B* approaches Q*: order once, ship out of the lot, let the rest backorder. That degenerate case is why a finite, measured b is essential.
Backorder policy vs safety-stock-based service level
A safety-stock policy says "hold extra inventory so 99 percent of cycles avoid stockout." A backorder policy says "accept the stockout, charge the buyer (or pay the penalty), reorder to plan." The right choice depends on whether b is known and whether customers will wait. Most B2C e-commerce uses safety stock; most B2B contract supply uses a hybrid with both a service-level floor and a backorder allowance above it.