Chapter 11.8, Problem 31AYU

Airline Revenue An airline has two classes of service: first class and coach. Management's experience has been that each aircraft should have at least 8 but no more than 16 first- class seats and at least 80 but no more than 120 coach seats

(a) If management decides that the ratio of first-class seats to coach seats should never exceed $1:12$, with how many of each type of seat should an aircraft be configured to maximize revenue?

(b) If management decides that the ratio of first-class seats to coach seats should never exceed $1:8$, with how many of each type of seat should an aircraft be configured to maximize revenue?

(c) If you were management, what would you do?

[Hint: Assume that the airline charges $\$C$ for a coach seat and $\$F$ for a first-class seat; $C>0,F>C$.]

To determine

To solve: The given linear programming problem.

Answer to Problem 31AYU

Solution:

a. The revenue is maximized when the aircraft is configured with 10 First Class seats and 120 coach seats.

b. The revenue is maximized when the aircraft is configured with 15 First Class seats and 120 coach seats.

c. First Class Seats to Coach Seats does not exceed $1:8$ ratio.

Explanation of Solution

Given:

• Each aircraft should have at least 8 but not more than 16 first class seats.
• Each aircraft should have at least 80 by not more than 120 coach seats.
• $\$$ $C$ is the charges for a coach seat and $\$F$ is the charges for First class seat such that $C>0$ and $F>C$.

Calculation:

Begin by assigning symbols for the two variables.

$x$ be the number of First Class seats.

$y$ be the number of Coach seats.

If $p$ be the total revenue,

$P=Fx+Cy$

The goal is to maximize $P$ subject to certain constraints on $x\text{and}y$. Because $x\text{and}y$ represents number of seats, the only meaningful values of $x\text{and}y$ are non-negative.

Therefore, $x\ge 0,y\ge 0$.

Also from the given data we get,

$8\le x\le 16$

$80\le y\le 120$

a. If the ratio of first class to coach seats should not exceed $1:12$, the following can be written $\frac{x}{y}\le \frac{1}{12}$ or $12x\le y$.

Therefore, the linear programming problem may be stated as,

Maximize, $P=Fx+Cy$.

Subject to,

$x\ge 0$

$y\ge 0$

$x\ge 8$

$x\le 16$

$y\ge 80$

$y\le 120$

$y\ge 12x$

The graph of the constraints is illustrated in the figure below.

The corner points are as follows:

Corner points are	Value of objective function $P=Fx+Cy$
$\left(8,96\right)$	$8F+96C$
$\left(8,120\right)$	$8F+120C$
$\left(10,120\right)$	$10F+120C$

Since $F>C>0$, the revenue is maximized when the aircraft is configured with 10 First Class seats and 120 coach seats.

b. If the ratio of first class to coach seats should not exceed $1:8$, the following can be written $\frac{x}{y}\le \frac{1}{8}$ or $8x\le y$.

Therefore, the linear programming problem may be stated as,

Maximize, $P=Fx+Cy$.

Subject to,

$x\ge 0$

$y\ge 0$

$x\ge 8$

$x\le 16$

$y\ge 80$

$y\le 120$

$y\ge 8x$

The graph of the constraints is illustrated in the figure below.

The corner points are as follows:

Corner points are	Value of objective function $P=Fx+Cy$
$\left(8,120\right)$	$8F+120C$
$\left(8,80\right)$	$8F+80C$
$\left(10,80\right)$	$10F+80C$
$\left(15,120\right)$	$15F+120C$

Since $F>C>0$, the revenue is maximized when the aircraft is configured with 15 First Class seats and 120 coach seats.

c. If the ratio of first class seats to coach seats should not exceed $1:12$, the total revenue is $10F+120C$. If the ratio of first class seats to coach seats should not exceed $1:8$, the total revenue is $15F+120C$. Since $F>C>0$, the revenue is maximized when First Class Seats to Coach Seats ration does not exceed $1:8$ ratio.