Proposition 3 — Ordinary base-stock cost bound

For every probability measure $P$ on the nonnegative real numbers whose identity function is integrable and whose real mean $\mu=\int d\,P(\mathrm{d}d)$ is strictly positive, every integer $L\ge1$, every pair of real numbers $h>0$ and $p>0$, and all real numbers $r,z$ satisfying the feasibility conditions below, the stated bound holds for a particular policy cost. For a nonnegative real demand sequence $d=(d_i)_{i\ge0}$ and integer $m\ge0$, let $F_m(d,r)=\max_{0\le k\le m}\sum_{i=0}^{k-1}(r-d_i)$, including the empty sum $0$ at $k=0$, and let $E_m(d,r,z)=(F_m(d,r)+\sum_{i=0}^{m-1}(d_i-r)-z)^+$, where $y^+=\max(y,0)$. The conditions are $0\le r\le\mu$, $z\ge0$, and $\int E_m(d,r,z)\,P^{\otimes\mathbb N}(\mathrm dd)\le m(\mu-r)$ for each of $m=L$ and $m=L+1$. These are nonnegative Lebesgue integrals under the infinite independent product demand law and inequalities in the extended nonnegative reals $[0,\infty]$, with each positive part and finite nonnegative right side embedded into that space. Set $S=((L+1)r+2z)^+$, a finite nonnegative real; the hypotheses make $S=(L+1)r+2z$. Let $(D_t)_{t\ge0}$ have law $P^{\otimes\mathbb N}$ and independently let $U$ have law equal to Lebesgue measure restricted to $[0,1]$. Starting from $I_0=0$ and $x_{0,j}=0$ for all $0\le j<L$, define $q_t=\min\{(S-I_t-\sum_{j=0}^{L-1}x_{t,j})^+,S\}$, $I_{t+1}=(I_t+x_{t,0}-D_t)^+$, $x_{t+1,j}=x_{t,j+1}$ for $0\le j<L-1$, and $x_{t+1,L-1}=q_t$ for every integer $t\ge0$. The recursion ignores $U$, but costs are integrated over the stated joint law. With $C_t=hI_{t+1}+p(D_t-I_t-x_{t,0})^+$ embedded in $[0,\infty]$, define $J_{S,S}=\limsup_{T\to\infty}\frac{1}{T+1}\sum_{t=0}^{T}\int C_t\,\mathrm d(P^{\otimes\mathbb N}\otimes\mathrm{Leb}|_{[0,1]})$, where $T$ ranges over nonnegative integers and each integral is a nonnegative Lebesgue integral. The assertion is $J_{S,S}\le\left[2hz+\left((2-\frac{1}{L+1})p+hL\right)(\mu-r)\right]^+$, with the finite right side embedded into $[0,\infty]$; this right side is already nonnegative under the hypotheses. All state, loss and order subtractions are truncated at zero as displayed, while the sums in $F_m$ and the expression inside $E_m$ use ordinary real subtraction. The average cost is defined in the extended nonnegative reals, allowing $+\infty$, and divides by the positive finite number $T+1$. This is a bound for the explicitly specified level and equal order cap $S$; it does not take an infimum or assert an optimum is attained. The cases $L=1$, zero demand values, $r=0$, $r=\mu$, and $z=0$ are included whenever feasible, and $L\ge1$ makes the displayed denominator nonzero.