Theorem 1 — Exact lead-time bound and the $7/3$ guarantee

For every probability measure $P$ on the nonnegative real numbers such that $d\mapsto d$ is integrable and its real mean $\mu=\int d\,P(\mathrm{d}d)$ is strictly positive, and every integer $L\ge 1$ and real numbers $h>0$ and $p>0$, the following three inequalities hold. To specify their quantities, let $(D_t)_{t\ge0}$ have the infinite product law $P^{\otimes\mathbb N}$ and let $U$ be independent of that sequence with law equal to Lebesgue measure restricted to the closed interval $[0,1]$. An admissible policy consists of a measurable map $\pi_t:[0,\infty)^t\times\mathbb R\to[0,\infty)$ for each integer $t\ge0$; its order at time $t$ is $q_t=\pi_t((D_i)_{0\le i<t},U)$, so the history at $t=0$ has no demand coordinates. All orders and state coordinates are finite nonnegative real numbers, with no boundedness or integrability condition on orders beyond measurability. Starting with $I_0=0$ and $x_{0,j}=0$ for $0\le j<L$, define $I_{t+1}=(I_t+x_{t,0}-D_t)^+$, $x_{t+1,j}=x_{t,j+1}$ for $0\le j<L-1$, and $x_{t+1,L-1}=q_t$, where $y^+=\max(y,0)$. The period cost is $C_t=hI_{t+1}+p(D_t-I_t-x_{t,0})^+$, viewed in the extended nonnegative reals $[0,\infty]$, and the cost of the policy is $J(\pi)=\limsup_{T\to\infty}\frac{1}{T+1}\sum_{t=0}^{T}\int C_t\,\mathrm d(P^{\otimes\mathbb N}\otimes\mathrm{Leb}|_{[0,1]})$, with $T$ ranging over nonnegative integers and the integrals being nonnegative Lebesgue integrals, possibly infinite. Put $O=\inf_{\pi}J(\pi)$ over all such admissible policies. For any finite $S,a\ge0$, let $J_{S,a}$ be the same average cost for the recursion with $q_t=\min\{(S-I_t-\sum_{j=0}^{L-1}x_{t,j})^+,a\}$, and put $B=\inf_{S\ge0}\inf_{a\ge0}J_{S,a}$; zero values of either parameter are included. For a demand sequence $d=(d_i)_{i\ge0}$, real numbers $r,z$, and an integer $m\ge0$, set $F_m(d,r)=\max_{0\le k\le m}\sum_{i=0}^{k-1}(r-d_i)$, including the empty sum $0$ at $k=0$, and $E_m(d,r,z)=(F_m(d,r)+\sum_{i=0}^{m-1}(d_i-r)-z)^+$, viewed in $[0,\infty]$. A pair $(r,z)\in\mathbb R^2$ is feasible precisely when $0\le r\le\mu$, $z\ge0$, and $\int E_m(d,r,z)\,P^{\otimes\mathbb N}(\mathrm dd)\le m(\mu-r)$ for each of $m=L$ and $m=L+1$, where each integral is a nonnegative Lebesgue integral and the finite nonnegative right side is embedded into $[0,\infty]$. Put $A=\inf_{(r,z)\text{ feasible}}[hz+p(\mu-r)]$, again as an infimum in $[0,\infty]$, and $K=1+\frac{4L^2}{(L+1)(3L-1)}$. The assertion is $B\le KA\ \land\ B\le KO\ \land\ KO\le\frac73O$. All costs, infima, inequalities and products in this assertion use the extended nonnegative reals; the displayed finite nonnegative coefficients are embedded in that space, an empty infimum is $+\infty$, and $0\cdot(+\infty)=0$. The infima defining $A$, $B$ and $O$ assert no attainment by any pair of parameters or policy. The assumption $L\ge1$ includes $L=1$ and keeps the displayed denominators nonzero; demands may equal zero even though their mean is strictly positive.