Q. 31 - Precalculus Enhanced with Graphing Utilities

Question

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 not more than $120$ coach seats.

(a) If management decides that the ratio of first class 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 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; $C>0,F>0$.] 

Step-by-Step Solution

Answer

Part a. The revenue is maximum when number of first class seats is $10$ and number of coach seats is $120$.

Part b. The revenue is maximum when number of first class seats is $15$ and number of coach seats is $120$.

Part c. If I was management I would choose the condition of part b that the ratio of first class to coach seats should never exceed $1:8$.

Part a. Step 1. Given Information

It is given that each aircraft should have at least $8$ but no more than $16$ first class seats and at least $80$ but not more than $120$ coach seats.

Part (a) Step 2. Form the inequalities

Let $x$ be the number of first class seats and $y$ be the number of coach seats.

As first class seats varied from $8$ to $16$, so $8\le x\le 16$.

Similarly, coach seats varied from $80$ to $120$, so $80\le y\le 120$.

Also for this case the ratio of first class seats to coach seats should never exceed $1:12$, so

$$\begin{gathered} \frac{x}{y}\le \frac{1}{12} \\ 12x\le y \end{gathered}$$

If the cost of one coach seat is $$C$ and the cost of one first class seat is $$F$, then the revenue function for $x$ first class seat and $y$ coach seat is given as

$R=Fx+Cy$ and we need to maximize the function.

Part (a) Step 3. Graph the inequalities

The inequalities are graphed and the feasible region has been shaded as 

Part (a) Step 4. Find revenue for the corner points

The feasible region is bounded by three corner points $(8,120),(8,97),(10,120)$. And the revenue for each point is given as

$\left(x,y\right)$	$R=Fx+Cy$
$\left(8,120\right)$	$R=8F+120C$
$\left(8,97\right)$	$R=8F+97C$
$\left(10,120\right)$	$R=10F+120C$

So it is observed that in this case, the revenue is maximum when an aircraft has $10$ first class seats and $120$ coach seats.

Part (b) Step 1. Form the inequalities

If $x$ is the number of first class seats and $y$ is the number of coach seats than the first two inequalities are

$$\begin{gathered} 8\le x\le 16 \\ 80\le y\le 120 \end{gathered}$$

Also in this case the ratio of first class to coach seats should never exceed $1:8$.

$$\begin{gathered} \frac{x}{y}\le \frac{1}{8} \\ 8x\le y \end{gathered}$$

The revenue function that needs to be maximized remains the same $R=Fx+Cy$ where $F$ is the cost of one first class seat and $C$ is the cost of one coach seat.

Part (b) Step 2. Graph the inequalities

The inequalities are graphed and the feasible region has been shaded as 

Part (b) Step 3. Find revenue for the corner points

The feasible region is bounded by three corner points $\left(8,80\right),\left(10,80\right),\left(15,120\right),\left(8,120\right)$. And the revenue for each point is given as

$\left(x,y\right)$	$R=Fx+Cy$
$\left(8,80\right)$	$R=8F+80C$
$\left(10,80\right)$	$R=10F+80C$
$\left(15,120\right)$	$R=15F+120C$
$\left(8,120\right)$	$R=8F+120C$

So it is observed that in this case, the revenue is maximum when an aircraft has $15$ first class seats and $120$ coach seats. 

Part (c) Step 1. Which situation is better

In the first case when the ratio of first class to coach seats should never exceed $1:12$ the revenue is maximum when the aircraft has $10$ first class and $120$ coach sets.

In the second case when the ratio of first class to coach seats should never exceed $1:8$ the revenue is maximum when the aircraft has $15$ first class and $120$ coach sets.

So overall revenue will be maximum in the second case than compared to first case.

So if I were management I would choose case two which states that he ratio of first class to coach seats should never exceed $1:8$.