Theorem 5 — an optimal policy whose offer sets grow with remaining capacity and shrink with remaining time

Let $(\lambda,\rho)$ be a Markov chain choice model with the standing assumptions and $\sum_{j\in N}\lambda_j\le1$, let $r\in\mathbb R^n$ be the product revenues, $T$ the number of periods and $c$ the initial capacity. For $1\le t\le T$ and $1\le x\le c$, an optimal subset $\hat S_t(x)$ is an optimal solution of the problem on the right side of the (Single Resource) dynamic program,

Then there exists an optimal policy, that is, a choice of optimal subsets $\hat S_t(x)$ for all $1\le t\le T$ and $1\le x\le c$, such that

The first property says that the optimal offer set becomes smaller as remaining capacity decreases, so the optimal policy is a protection level policy: each product $j$ has a threshold $\bar x_{jt}$ and is offered in period $t$ exactly when the remaining capacity is at least $\bar x_{jt}$. The second says the optimal offer set becomes smaller when more periods remain.

Formalization Note The paper writes $\subset$ for non-strict inclusion. The first inclusion is stated for $x\ge2$: with zero units the dynamic program has no decision ($V_t(0)=0$ is a boundary condition), and the paper's later chain $\hat S_t(0)\subset\hat S_t(1)$ reads $\hat S_t(0)$ as $\emptyset$, for which the inclusion is trivial. The optimality clause is required of every $\hat S_t(x)$ in range, so the statement cannot be met by an arbitrary family of sets. One hypothesis is added, as in the marginal-value milestone: $\sum_j\lambda_j\le1$ (implied by the model of at most one arrival per period). No sign condition is placed on the revenues.