Expected marginal seat revenue

EMSRb

One of the most widely used RM heuristics is EMSRb. It is simple and produces under certain conditions close to optimal results. Belobaba reports studies in which both EMSRa and EMSRb were compared. He shows that EMSRb is consistently within 0.5 percent of the optimal solution, while EMSRa under certain conditions can deviate more than 1.5 percent from the optimal solution. However, with mixed order of arrival and frequent re-optimization both methods perform well. There is also study by Polt that shows mixed results.

EMSRb is also based on an approximation that compares two classes, but it does take the statistical averaging effect into account. Instead of aggregating protection levels, as EMSRa does, it aggregates demand. Suppose we are again in stage ${\displaystyle j+1}$ and we want to calculate the protection limit ${\displaystyle y}$_j. Then first all future demand for classes j, j-1,…, 1 is aggregated:

${\displaystyle S_j=\sum _{k=1}^j{D}_k}$

and the weighted revenues are calculated:

${\displaystyle {\overline{r}}_j=\frac{\sum _{k=1}^j{r}_k\cdot \overline{D_k}}{\sum _{k=1}^j\overline{D_k}}}$

Then, again with Littlewood’s rule, the protection limit for classes j and higher is calculated such that:

${\displaystyle P(S_j>y_j)=\frac{r_{j+1}}{{\overline{r}}_j}}$

Rearranging gives:

EMSR Littlewood's rule

${\displaystyle y_j^{\star }=F_j^{-1}(1-\frac{r_{j+1}}{{\overline{r}}_j})}$

${\displaystyle y_j^{\star }}$ is the optimal protection limit, ${\displaystyle F_j(x)}$ is a continuous distribution used to model the demand. Usually demand is considered to be independent and distributed normally with a mean and a variance . Using that the protection limits can be calculated as:

${\displaystyle y_j={\mu }_j+z_{\alpha }\cdot {\sigma }_j}$

with the mean and variance of the demand to come as ${\displaystyle {\mu }_j=\sum _{k=1}^j{\mu }_k}$ and ${\displaystyle {\sigma }_j^2=\sum _{k=1}^j{\sigma }_k^2}$ respectively. ${\displaystyle z_{\alpha }}$ is calculated with the inverse of the normal distribution ${\displaystyle z_{\alpha }={\varphi }^{-1}(1-\frac{r_{j+1}}{{\overline{r}}_j})}$. This is done for each j, giving the protection limit for every class.