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

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?

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 $$m$$.
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 $$r = \mu+z_\alpha\sigma$$.