31. A toy company manufactures two types of dolls, A and B. Market research and available resources have indicated that the combined production level should not exceed 1200 dolls per week, and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other types by at most 600 units. If the company makes a profit of ?. 12 and ?. 16 per doll on dolls A and B, respectively, how many of each should be produced weekly in order to maximise the profit?

Let x and y be the number of dolls of type A and B, respectively, that are produced in a week.

The given problem can be formulated as given below:

$Maximisez=12x+16y\dots \dots \dots ..\left(i\right)$

Subject to the constraints,

$\begin{array}{}\end{array}x+y \le 1200\dots \dots \dots \dots \left(ii\right)$

$y\le x/2orx\ge 2y\dots \dots \dots .(ii\mathrm{i)}$

$x–3y\le 600\dots \dots \dots \dots .\left(iv\right)$

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

The feasible region determined by the system of constraints is given below:

A (600, 0), B (1050, 150) and C (800, 400) are the corner points of the feasible region.

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

Corner Point	z = 12x + 16y
A (600, 0)	7200
B (1050, 150)	15000
C (800, 400)	16000	Maximum

The maximum value of z is 16000 at (800, 400).

Hence, 800 and 400 dolls of type A and type B should be produced, respectively, to get the maximum profit of? 16000.

<p>Let x and y be the number of dolls of type A and B, respectively, that are produced in a week.</p><p>The given problem can be formulated 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>2</mn><mi>x</mi><mo>+</mo><mn>1</mn><mn>6</mn><mi>y</mi><mo>…</mo><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>1</mn><mn>2</mn><mn>0</mn><mn>0</mn><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>y</mi><mo>≤</mo><mi>x</mi><mo>/</mo><mn>2</mn><mi>o</mi><mi>r</mi><mi>x</mi><mo>≥</mo><mn>2</mn><mi>y</mi><mo>…</mo><mo>…</mo><mo>…</mo><mo>.</mo><mrow><mo> (</mo><mrow><mi>i</mi><mi>i</mi><mi>i)</mi></mrow></mrow></mrow></mrow></math></span></p><p><span title="Click to copy mathml"><br><math><mrow><mrow><mrow><mo></mo></mrow></mrow><mrow><mo>x–</mo><mn>3</mn><mi>y</mi><mo>≤</mo><mn>6</mn><mn>0</mn><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 system of constraints is given below:</p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1732774272phptZvW7z_480x360.jpeg" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1732774272phptZvW7z.jpeg" alt="" width="228" height="207"></picture></div></div><p>A (600, 0), B (1050, 150) and C (800, 400) 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 = 12x + 16y</p></td><td>&nbsp;</td></tr><tr><td><p>A (600, 0)</p></td><td><p>7200</p></td><td>&nbsp;</td></tr><tr><td><p>B (1050, 150)</p></td><td><p>15000</p></td><td>&nbsp;</td></tr><tr><td><p>C (800, 400)</p></td><td><p>16000</p></td><td><p>Maximum</p></td></tr></tbody></table><p>The maximum value of z is 16000 at (800, 400).</p><p>Hence, 800 and 400 dolls of type A and type B should be produced, respectively, to get the maximum profit of? 16000.</p>

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