Proposition 1 — Lower certificate for the optimal cost

For every probability measure $P$ on the nonnegative real numbers with integrable identity function and strictly positive real mean $\mu=\int d\,P(\mathrm{d}d)$, every integer $L\ge1$, and every pair of real numbers $h>0$ and $p>0$, the certificate infimum defined below is at most the policy cost infimum defined below. For a sequence $d=(d_i)_{i\ge0}$ of nonnegative real numbers, real numbers $r,z$, and every integer $m\ge0$, define $F_m(d,r)=\max_{0\le k\le m}\sum_{i=0}^{k-1}(r-d_i)$, with the empty sum at $k=0$ equal to $0$, and define $E_m(d,r,z)=(F_m(d,r)+\sum_{i=0}^{m-1}(d_i-r)-z)^+$, where $y^+=\max(y,0)$ and the value is embedded in $[0,\infty]$. A real pair $(r,z)$ is feasible when $0\le r\le\mu$, $z\ge0$, and, for both $m=L$ and $m=L+1$, its nonnegative Lebesgue integral under the infinite independent product measure satisfies $\int E_m(d,r,z)\,P^{\otimes\mathbb N}(\mathrm dd)\le m(\mu-r)$ in $[0,\infty]$. The certificate infimum is $A=\inf_{(r,z)\text{ feasible}}[hz+p(\mu-r)]$, with each objective embedded as a finite nonnegative real in $[0,\infty]$. For the policy infimum, let $(D_t)_{t\ge0}$ have law $P^{\otimes\mathbb N}$ and independently let $U$ have the restriction of Lebesgue measure to $[0,1]$. A policy specifies, for every integer $t\ge0$, a measurable map $\pi_t:[0,\infty)^t\times\mathbb R\to[0,\infty)$ and places the finite nonnegative order $q_t=\pi_t((D_i)_{0\le i<t},U)$; no boundedness or integrability of these maps is required, and at $t=0$ there are no demand coordinates. Its state starts at $I_0=0$ and $x_{0,j}=0$ for $0\le j<L$ and evolves by $I_{t+1}=(I_t+x_{t,0}-D_t)^+$, $x_{t+1,j}=x_{t,j+1}$ for $0\le j<L-1$, and $x_{t+1,L-1}=q_t$. Its period cost in $[0,\infty]$ is $C_t=hI_{t+1}+p(D_t-I_t-x_{t,0})^+$, and its cost is $J(\pi)=\limsup_{T\to\infty}\frac{1}{T+1}\sum_{t=0}^{T}\int C_t\,\mathrm d(P^{\otimes\mathbb N}\otimes\mathrm{Leb}|_{[0,1]})$, where $T$ ranges over nonnegative integers and the nonnegative integrals and the limit superior may be infinite. Define $O=\inf_{\pi}J(\pi)$ over all these policies. The asserted inequality is $A\le O$. Both infima and the inequality are in the extended nonnegative reals, the infimum of an empty family is $+\infty$, and no feasible pair or policy attaining either infimum is asserted to exist. Zero demand values, $r=0$, $r=\mu$, $z=0$, and $L=1$ are included whenever the stated conditions hold; subtraction in the state and loss expressions is truncated at zero, as indicated by the positive part, while the sums defining $F_m$ and the expression inside $E_m$ use ordinary real subtraction.