26. An aeroplane can carry a maximum of 200 passengers. A profit of ?. 1000 is made on each executive class ticket, and a profit of ?. 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by executive class. Determine how many tickets of each type must be sold in order to maximise the profit for the airline. What is the maximum profit?

Let the airline sell x tickets of executive class and y tickets of economy class, respectively.

The mathematical formulation of the given problem can be written as given below:

$Maximisez=1000x+600y\dots \dots \dots \dots \left(i\right)$

Subject to the constraints,

$\begin{array}{}\end{array}x+y \le 200\dots \dots \dots \dots \dots ..\left(ii\right)$

$x\ge 20\dots \dots \dots \dots \left(iii\right)$

$y–4x\ge 0\dots \dots \dots \dots \dots \left(iv\right)$

$x,y\ge 0\dots \dots \dots \dots \dots \left(v\right)$

The feasible region determined by the constraints is given below:

A (20, 80), B (40, 160) and C (20, 180) are the corner points of the feasible region.

The values of z at these corner points are given below:

Corner Point	z = 1000x + 600y
A (20, 80)	68000
B (40, 160)	136000	Maximum
C (20, 180)	128000

136000 at (40, 160) is the maximum value of z.

Therefore, 40 tickets of the executive class and 160 tickets of the economy class should be sold to maximise the profit, and the maximum profit is? 136000.

<p>Let the airline sell x tickets of executive class and y tickets of economy class, respectively.</p><p>The mathematical formulation of the given problem can be written as given below:</p><p><span title="Click to copy mathml"><math><mrow><mi>M</mi><mi>a</mi><mi>x</mi><mi>i</mi><mi>m</mi><mi>i</mi><mi>s</mi><mi>e</mi><mi>z</mi><mo>=</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mi>x</mi><mo>+</mo><mn>6</mn><mn>0</mn><mn>0</mn><mi>y</mi><mo>…</mo><mo>…</mo><mo>…</mo><mo>…</mo><mrow><mo> (</mo><mrow><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p><p>Subject to the constraints, </p><p><span title="Click to copy mathml"><math><mrow><mtable columnalign="left"></mtable><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>&nbsp;</mo><mo>≤</mo><mn>2</mn><mn>0</mn><mn>0</mn><mo>…</mo><mo>…</mo><mo>…</mo><mo>…</mo><mo>…</mo><mo>.</mo><mo>.</mo><mrow><mo> (</mo><mrow><mi>i</mi><mi>i</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mrow><mi>x</mi><mo>≥</mo><mn>2</mn><mn>0</mn><mo>…</mo><mo>…</mo><mo>…</mo><mo>…</mo><mrow><mo> (</mo><mrow><mi>i</mi><mi>i</mi><mi>i</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mrow><mi>y</mi><mo>–</mo><mn>4</mn><mi>x</mi><mo>≥</mo><mn>0</mn><mo>…</mo><mo>…</mo><mo>…</mo><mo>…</mo><mo>…</mo><mrow><mo> (</mo><mrow><mi>i</mi><mi>v</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mrow><mi>x</mi><mo>, </mo><mi>y</mi><mo>≥</mo><mn>0</mn><mo>…</mo><mo>…</mo><mo>…</mo><mo>…</mo><mo>…</mo><mrow><mo> (</mo><mrow><mi>v</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></span></p><p>The feasible region determined by the constraints is given below:</p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1732774132phpfOMbAQ_480x360.jpeg" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1732774132phpfOMbAQ.jpeg" alt="" width="226" height="218"></picture></div></div><p>A (20, 80), B (40, 160) and C (20, 180) are the corner points of the feasible region.</p><p>The values of z at these corner points are given below:</p><table><tbody><tr><td><p>Corner Point</p></td><td><p>z = 1000x + 600y</p></td><td>&nbsp;</td></tr><tr><td><p>A (20, 80)</p></td><td><p>68000</p></td><td>&nbsp;</td></tr><tr><td><p>B (40, 160)</p></td><td><p>136000</p></td><td><p>Maximum</p></td></tr><tr><td><p>C (20, 180)</p></td><td><p>128000</p></td><td>&nbsp;</td></tr></tbody></table><p>136000 at (40, 160) is the maximum value of z.</p><p>Therefore, 40 tickets of the executive class and 160 tickets of the economy class should be sold to maximise the profit, and the maximum profit is? 136000.</p>

0 Upvotes 0 Upvotes 0 Downvotes

• • Report Abuse
• Comment