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The earliest Revenue Management model is known as Littlewood’s rule.

The two class model

Littlewood proposed the first static single resource quantity based RM model. It was a solution method for the seat inventory problem for a single leg flight with two fare classes. Those two fare classes have a fare of $R_1$ and $R_2$, whereby $R_1>R_2$. The total capacity is $C$ and demand for class $j$ is indicated with $D_j$. The demand is distributed via a distribution that is indicated with $F_j()$. The demand for class 2 comes before demand for class 1. The question now is how much demand for class 2 should be accepted so that the optimal mix of passengers is achieved and the highest revenue is obtained. Littlewood suggests closing down class 2 when the certain revenue from selling another low fare seat is exceeded by the expected revenue of selling the same seat at the higher fare. In formula form this means: accept demand for class 2 as long as:

$R_2\ge R_1*\text{Prob}(D_1>x)$

where

$R_2$ is the value of the lower valued segment

$R_1$ is the value of the higher valued segment

$D_1$ is the demand for the higher valued segment and

$x$ is the capacity left

This suggests that there is an optimal protection limit $y_1^{\star }$. If the capacity left is less than this limit demand for class 2 is rejected. If a continuous distribution $F_j(x)$ is used to model the demand, then $y_1^{\star }$ can be calculated using what is called Littlewood’s rule:

$y_1^{\star }=F_1^{-1}(1-\frac{R_2}{R_1})$

This gives the optimal protection limit, in terms of the division of the marginal revenue of both classes.

Alternatively bid prices can be calculated via

$\pi (x)=R_1*\text{Prob}(D_1>x)$

Littlewood's model is limited to two classes. P. Belobaba developed a model based on this rule called Expected marginal seat revenue, abbreviated as EMSR, which is an $n$-class model

References

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See also

• Yield management
• Expected marginal seat revenue