Yield Management: What Quantities of Airline Tickets to Sell at What Prices and What Times

Objective

The objective is to maximize the following quadratic function:

	`ExpectedYield`
	$= \left(\text {if }\Argument{current\_ period} \le 1, \text {then}\underset {\text {option} \in \text {OPTIONS}}{\underset {\text {class} \in \text {CLASSES},}{\sum _{i \in \text {SCENARIOS},}}}\Argument{prob[i]} \cdot \Variable{R1[i,class,option]}\right)$
	$+ \left(\text {if }\Argument{current\_ period} \le 2, \text {then} \underset {\text {option} \in \text {OPTIONS}} {\underset {\text {class} \in \text {CLASSES},}{\sum _{(i,j) \in \text {SCENARIOS2},}}}\Argument{prob[i]} \cdot \Argument{prob[j]} \cdot \Variable{R2[i,j,class,option]} \right)$
	$+ \left( \text {if }\Argument{current\_ period} \le 3, \text {then} \underset {\text {option} \in \text {OPTIONS}}{\underset {\text {class} \in \text {CLASSES},}{\sum _{(i,j,k) \in \text {SCENARIOS3},}}} \Argument{prob[i]} \cdot \Argument{prob[j]} \cdot \Argument{prob[k]} \cdot \Variable{R3[i,j,k,class,option]} \right)$
	$+ \underset {\text {class} \in \text {CLASSES}}{\sum _{\text {period} \in 1\dots \Argument{current\_ period}-1,}} \Argument{actual\_ revenue[period,class]}$
	$- \Argument{plane\_ cost} \cdot \Variable{NumPlanes}$

where

$\begin{align*} \Variable{R1[i,class,option]} & = \Argument{price[1,class,option]} \cdot \Variable{P1[class,option]} \cdot \Variable{S1[i,class,option]} \\ \Variable{R2[i,j,class,option]} & = \Argument{price[2,class,option]} \cdot \Variable{P2[class,option]} \cdot \Variable{S2[i,j,class,option]} \\ \Variable{R3[i,j,k,class,option]} & = \Argument{price[3,class,option]} \cdot \Variable{P3[class,option]} \cdot \Variable{S3[i,j,k,class,option]} \end{align*}$