ThirdOrderLossFunctionIfc

Represents the 3rd order loss function.

Denominated per the support, as SecondOrderLossFunctionIfc is:

G3(x) = (1/6)E[max(X-x,0)*max(X-x-1,0)*max(X-x-2,0)]    X discrete
G3(x) = (1/6)E[max(X-x,0)^3]                            X continuous

and in both cases the tail accumulation of the order below it — a sum of G2 over the integers above x on a discrete support, an integral of G2 from x on a continuous one.

What it is for. A second order loss function gives the expected shortage when the stock level is fixed. When the level is itself random — the position an (r, Q) policy occupies is uniform across the reorder band — the shortage has to be averaged over its distribution, and averaging a second order loss function across a range produces a difference of third order ones:

sum over j in (b, c] of G2(j) = G3(b) - G3(c)

That identity is what makes the variance of the backorder level computable in closed form rather than by summing across every level in the band.

This is deliberately not part of LossFunctionDistributionIfc. Not every distribution that has the first two orders has a closed form for the third, and a numerical default would be an approximation wearing the same signature as the exact implementations beside it. A caller that needs the third order accepts this type, or tests for it.

Inheritors