Littlewood's rule

The earliest revenue management model is known as Littlewood’s rule, developed by Ken Littlewood while working at British Overseas Airways Corporation.

The two class model

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:

Littlewood's rule

${\displaystyle y_{1}^{\star }=F_{1}^{-1}\left(1-{\frac {R_{2}}{R_{1}}}\right)}$

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]

References

1. Pak, K. and N. Piersma (2002). Airline Revenue Management: An overview of OR Techniques 1982–2001. Rotterdam, Erasmus university
2. Littlewood, K. (1972). "Forecasting and Control of Passenger Bookings." Proc. 12th AGIFORS Symposium, reprinted in Journal of Revenue and Pricing Management, Vol. 4 (2005), http://www.palgrave-journals.com/rpm/journal/v4/n2/pdf/5170134a.pdf
3. Belobaba, P. P. (1987). Air Travel Demand and Airline Seat Inventory Management. Flight Transportation Laboratory. Cambridge, MIT. PhD

See also

• Yield management
• Expected marginal seat revenue