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 has a probability distribution whose cumulative distribution function is denoted 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 \cdot \operatorname{Prob}( D_1x )

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:

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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 \cdot \operatorname{Prob} ( D_1x )

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

References

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