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Revision as of 21:56, 27 January 2014

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.^[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 $R_{1}$ 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.^[2] In formula form this means: accept demand for class 2 as long as:

R_{2} \geq R_{1} * 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 $y_{1}^{⋆}$ . 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 $y_{1}^{⋆}$ can be calculated using what is called Littlewood’s rule:

y_{1}^{⋆} = 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

π (x) = R_{1} * 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 ^[3]

References

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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.<ref>Pak, K. and N. Piersma (2002). Airline Revenue Management: An overview of OR Techniques 1982-2001. Rotterdam, Erasmus university</ref> 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 <math>R_{1}</math> and <math>R_{2}</math>, whereby <math>R_{1}> R_{2}</math>. The total capacity is <math>C</math> and demand for class <math>j</math> is indicated with <math>D_{j}</math>.
+The demand is distributed via a [[Distribution (mathematics)|distribution]] that is indicated with <math>F_{j}( )</math>. 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.<ref>Littlewood, K. (1972). "Forecasting and Control of Passenger Bookins." AGIFORS Symposium Proc. 12</ref> In formula form this means: accept demand for class 2 as long as:
+:<math>R_{2}\ge R_{1} * \text{Prob}( D_{1}>x ) </math>
+where<br />
+:<math>R_{2}</math> is the value of the lower valued segment
+:<math>R_{1}</math> is the value of the higher valued segment
+:<math>D_{1}</math> is the demand for the higher valued segment and
+:<math>x</math> is the capacity left
+This suggests that there is an optimal protection limit <math>y_{1}^{\star}</math>. If the capacity left is less than this limit demand for class 2 is rejected. If a [[continuous distribution]] <math>F_{j}(x)</math> is used to model the demand, then <math>y_{1}^\star</math> can be calculated using what is called ''Littlewood’s rule'':
+:<math>y_{1}^{\star} = F_{1}^{-1}(1-\frac{R_{2}}{R_{1}})</math>
+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
+:<math>\pi(x) = R_{1} * \text{Prob}( D_{1}>x )</math>
+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 <math>n</math>-class model <ref>Belobaba, P. P. (1987). Air Travel Demand and Airline Seat Inventory Management. Flight Transportation Laboratory. Cambridge, MIT. PhD</ref>
+==References==
+<!--- See [[Wikipedia:Footnotes]] on how to create references using <ref></ref> tags which will then appear here automatically -->
+{{Reflist}}
+==See also==
+* [[Yield management]]
+* [[Expected marginal seat revenue]]
+{{DEFAULTSORT:Littlewood's Rule}}
+[[Category:Pricing]]

Office Open XML file formats: Difference between revisions

Revision as of 21:56, 27 January 2014

The two class model

References

See also

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