We extend our two$-$class model to the setting where some of the rejected low$-$fare customers seek to purchase high$-$fare tickets. We have a total capacity C$.$ Let f$1$ and f$2$ be the price of the low$-$fare and high$-$fare tickets. Let the random variables D$1$ and D$2$ capture the low$-$fare and high$-$fare demands. Assume that D$1$ and D$2$ are independent. To model the opportunity to sell high$-$fare tickets to rejected low$-$fare customers, we assume that a fraction $\backslash$alpha in $[0,1]$ of the rejected low$-$fare demand will seek to book high$-$fare tickets.

Thus, given a booking limit b$,$ the total high$-$fare demand D$2$ is given by D$2=$ D$2+\backslash$alpha max$\{$D$1$ b$,0\}.$

Note that the booking limit decision will influence the total high$-$fare demand in the case

when low$-$fare demand exceeds the booking limit$.$

$($a$)$ Let E$\{$Z$($b$,$ D$1,$ D$2)\}$ denote the expected total revenue given that the booking limit

for low$-$fare customers is set at b$.$ Argue that

E$\{$Z$($b$,$ D$1,$ D$2)\}=$ f$1$ E

n

min$\{$b$,$ D$1\}$

o

$+$ f$2$ E

n

min $\{$C $$ min$\{$b$,$ D$1\},$ D$2+\backslash$alpha max$\{$D$1$ b$,0\}$