C.29.1:5.1 - Transfer a reservation update to available stock

Situation and question. A reservation account stores on-hand quantity n and reserved quantity r. Both are integers, with 0 ≤ r ≤ n. One reservation is allowed when n − r ≥ 1. While this operation is performed, there are no deliveries, shipments or cancellations. The question is whether a smaller display can both decide that operation’s availability and calculate its effect.

The source reservation update is:

U(n,r) = (n,r+1), defined when n−r ≥ 1.

Start with the proposed display H(n,r) = n, labelled “available.” The allowed states (1,0) and (1,1) have the same display, one. In (1,0), another reservation is allowed; in (1,1), it is not. Thus availability is not a function of H alone. Moreover, a permitted reservation leaves H unchanged, whereas the amount still reservable decreases.

The first result is that the total on hand omits a distinction needed by the reservation question. H remains useful as the on-hand total.

Construct the repaired display F(n,r) = a = n − r. Define its receiving reservation operation by V(a) = a − 1 for integer a ≥ 1. Now compare both orders:

F(U(n,r)) = F(n,r+1) = n−(r+1)
          = (n−r)−1 = V(F(n,r)).

The operation conditions also agree: U is defined exactly when F(n,r) ≥ 1, which is the condition for V. After an allowed update, 0 ≤ r + 1 ≤ n and a − 1 ≥ 0, so the result remains inside each account’s domain.

For n = 5 and r = 2, the source becomes (5,3), and the display moves from 3 to 2. After k reservations, the same reasoning gives available stock a − k, for integer 0 ≤ k ≤ a. Each intermediate reservation is allowed; the next reservation at k = a is not.

Returned result. Available stock is sufficient to decide and update this reservation operation. It does not determine n and r separately: (5,2) and (7,4) both give a = 3. If the next question asks how many items are reserved, retain that distinction. If the next operation is shipment or cancellation, recover its definition and compare it separately. Correct transfer of the reservation equation does not establish that a physical stock count is accurate.