EOQ with planned shortages
When stockouts are tolerated and customers wait (backorders), the total cost function adds a third term: shortage cost b per unit per year carried as backorder. Closed form: Q* = sqrt(2DS/H) * sqrt((H+b)/b), with optimal planned shortage B* = Q* * H/(H+b)[1].
Intuition
The factor sqrt((H+b)/b) inflates Q* upward because spreading some shortage across the cycle is cheaper than carrying every unit on the shelf. As b approaches infinity (shortages catastrophic), the factor approaches 1 and Q* collapses to Wilson with B* = 0. As b approaches H, the model becomes indifferent and B* approaches Q/2.
When deliberate stockouts make business sense
Industrial MRO with patient internal customers. Long-tail spare parts where the replacement is "sometime this week." B2B contracts where backorder is explicitly permitted (and not penalised by SLA). In retail and D2C, b is effectively infinite because the lost-sale economics make stockouts uneconomic; the Wilson model with safety stock is the right tool there[2].
Backorder vs safety stock
Backorder allowance is a deterministic planned shortage. Safety stock is the buffer that absorbs stochastic demand variance during lead time. They solve different problems: you can run a (Q, r) policy with both, where Q comes from the backorder formula and r = d*L + Z*sigma_LT comes from the ROP model.