MMP.17:5.2 - Correct a cheap allocation model, then change its regime

A network model returns delivered quantities (q_1(d)) and (q_2(d)) through two channels for demand (2\leq d\leq4). In the normal regime all demand is served, so (q_1+q_2=d). A cheap approximation splits it equally: (L(d)=(d/2,d/2)).

The source supplies (q(2)=(1.2,0.8)) and (q(4)=(3,1)). The first-channel discrepancies from (L) are (0.2) and (1). Linear interpolation of that discrepancy gives

[ \widehat d_1(d)=0.4d-0.6,\qquad \widehat q_1(d)=0.9d-0.6,\qquad \widehat q_2(d)=d-\widehat q_1(d). ]

At demand (3), the replacement returns ((2.1,0.9)). It preserves total delivery and nonnegativity throughout the declared interval. Those properties follow from the construction; agreement between the cases has not yet been established.

The receiving question is whether the first delivery stays at or below (2.25) at demand (3). The provisional value (2.1) is only (0.15) below the limit, and there is no supported error bound for this interpolation. A source query at (3) returns ((2.4,0.6)), ruling out the proposed limit. Add its discrepancy (0.9) and interpolate separately over ([2,3]) and ([3,4]). The new surrogate matches all three cases and preserves the balance. Further-input accuracy remains an empirical question or requires a separate bound.

Now channel 2 becomes unavailable. The normal-regime fit does not describe this input. Suppose the changed source account states that channel 1 serves all demand up to (3) units and any excess remains unserved. The needed output now includes unserved demand (u):

[ q_1=\min(d,3),\qquad q_2=0,\qquad u=d-q_1. ]

At demand (4), the result is ((3,0,1)). The balance has become (q_1+q_2+u=d). This branch follows from the newly supplied operating rule, not from extrapolating the normal-regime cases. Its formula is already cheap enough to use; training another replacement adds no benefit here. The caller can retain the normal-regime surrogate for its supported questions and use this formula for the stated unavailable-channel regime.