Which EOQ-based $(r,Q)$ approximation has a fixed worst-case error bound?

Question

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?

Answer 1

Only the EOQB approximation (approximation 2) has a fixed worst-case error bound. Zheng (1992) proved an error bound of $\frac18$, and Axsäter (1996) proved a stronger bound of $(\sqrt{5}-2)/2 \approx 0.118$, which is tight.

The EOQ+SS approximation (approximation 1) does not have a fixed worst-case error bound; for any $m$, we can find a problem instance for which the ratio between the cost of the EOQ+SS solution and the cost of the optimal $(r,Q)$ solution is greater than $m$.

I've noticed that it's a common mistake to say that the EOQ+SS approximation has a fixed worst-case bound, even though it does not. My guess is that the confusion arises from the fact that lots of people know the EOQ+SS approximation (it is simpler and therefore taught in many OR/OM/IE courses and textbooks) while fewer people know the EOQB approximation; and from the fact that the papers by Zheng and Axsäter both use the term "EOQ" in the title, rather than EOQB.

Note: Worst-case bounds aside, the EOQ+SS approximation can result in much larger errors in practice than the EOQB tends to. However, these errors largely disappear if we set $Q$ according to the EOQ model and then set $r$ using $g(r)=g(r+Q)$, instead of using $r = \mu+z_\alpha\sigma$.