Littlewood's rule

Littlewood proposed the first static single-resource quantity-based RM model.[1] 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 ${\displaystyle R_{1}}$ and ${\displaystyle R_{2}}$, whereby ${\displaystyle R_{1}>R_{2}}$. The total capacity is ${\displaystyle C}$ and demand for class ${\displaystyle j}$ is indicated with ${\displaystyle D_{j}}$. The demand has a probability distribution whose cumulative distribution function is denoted ${\displaystyle F_{j}}$. 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.[2] In formula form this means: accept demand for class 2 as long as:

${\displaystyle R_{2}\geq R_{1}\cdot \operatorname {Prob} (D_{1}>x)}$

where

${\displaystyle R_{2}}$ is the value of the lower valued segment

${\displaystyle R_{1}}$ is the value of the higher valued segment

${\displaystyle D_{1}}$ is the demand for the higher valued segment and

${\displaystyle x}$ is the capacity left

This suggests that there is an optimal protection limit ${\displaystyle y_{1}^{\star }}$. If the capacity left is less than this limit demand for class 2 is rejected. If a continuous distribution ${\displaystyle F_{j}(x)}$ is used to model the demand, then ${\displaystyle y_{1}^{\star }}$ can be calculated using what is called Littlewood’s rule:

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

${\displaystyle \pi (x)=R_{1}\cdot \operatorname {Prob} (D_{1}>x)}$

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 ${\displaystyle n}$-class model [3]

• Yield management
• Expected marginal seat revenue