There are two common approximations for the $(r,Q)$ inventory optimization problem that use the EOQ model. It is well known that one of them has a fixed worst-case error bound, but there is confusion in the literature about which one it is.
Approximation 1 (EOQ+SS Approximation): In this approximation, we set $Q$ using the EOQ formula and set $r$ so that the safety stock (SS) ensures a service level of $\alpha \equiv p/(p+h)$: $$\begin{align} Q & = \sqrt{\frac{2K\lambda}{h}} \\ r & = \mu + z_\alpha\sigma, \end{align}$$ where $K$ is the fixed cost, $\lambda$ is the mean demand per year, $h$ is the holding cost per item per year, $p$ is the stockout cost per item per year, $\mu$ and $\sigma$ are the mean and standard deviation of the lead-time demand, and $z_\alpha$ is the $\alpha$th quantile of the standard normal distribution.
We are treating the inventory level process as being decomposed into two parts, a top part that looks like the EOQ curve and a bottom part that is flat, at height $s = r - \mu = z_\alpha\sigma$ (the safety stock).
Approximation 2 (EOQB Approximation): Here, we use the economic order quantity with backorders (EOQB) to set $Q$: $$Q = \sqrt{\frac{2K\lambda(h+p)}{hp}}.$$ To set $r$ we use the fact that, for given $Q$, the optimal $r$ satisfies $$g(r) = g(r+Q),$$ where $g(\cdot)$ is the standard newsvendor cost function.
Which approximation has a fixed worst-case error bound?