Economic production quantity (EPQ)
When you produce in-house at rate p and demand drains at rate d, demand reduces inventory while production builds it. Maximum on-hand is Q*(1 - d/p), not Q. Solving the total cost function with that adjustment gives Q* = sqrt(2DS / [H*(1 - d/p)])[1]. Requires p > d, otherwise demand outruns supply and the model is undefined.
How (1 - d/p) changes the inventory profile
Under Wilson EOQ the inventory profile is a sawtooth: instantaneous rise to Q, linear decline to zero, repeat. Under EPQ the rise is a slope of (p - d)/p, never reaching Q; the decline is the standard d slope; the cycle is longer than Wilson’s for the same Q*. Average on-hand is (Q*/2) * (1 - d/p), which is why holding cost is reduced relative to Wilson.
When EPQ collapses to Wilson
As p approaches infinity, (1 - d/p) approaches 1 and EPQ Q* approaches Wilson Q*. This is the precise mathematical meaning of "instantaneous resupply" in the original Harris paper: a supplier dropping a full batch at once is an EPQ with infinite production rate[2].
Setup cost vs ordering cost
S in the EPQ context is "setup cost" (changeover labour, scrap-on-ramp, first-article inspection) rather than "order cost." Numerically identical, accounting differs. The SMED (single-minute exchange of die) program that Toyota used to drive S toward zero forces Q* downward toward 1, which is the algebra of just-in-time.