Name: Mathematical Formulation of LPP: Illustration 2
Uploaded: 2019-07-19
Duration: 8 min 40 s
Description: LPP can be used in many real-life scenarios. Here is one of them. Watch this video to get an idea of how a manufacturing unit plans production using LPP.

Question

Optimize the objective function $P = 50 x + 20 y$ , given the four constraints:

$x + 4 y \leq 21$

$- 4 x + 6 y \leq 0$

$0 \leq x \leq 15$

$y \geq 0$

Accepted Answer

Given LPP is

Optimize $P = 50 x + 20 y$

subject to

$x + 4 y \leq 21$

$- 4 x + 6 y \leq 0$

$0 \leq x \leq 15$

$y \geq 0$

Let us first find the feasible region of the LPP

Consider

$x + 4 y \leq 21$

We graph the line $x + 4 y = 21$

This straight line is obtained by joining the points $(0, 5.25)$ , and $(21, 0)$ .

Since the inequality is $\leq$ . So, we represent $x + 4 y \leq 21$ by the solid line in the $x y$ -plane.

This line divides the $x y$ -plane into two half planes. One plane lies above the line $x + 4 y = 21$ , and the other plane lies below the line $x + 4 y = 21$ .

Since $0 + 4 (0) \leq 21$

$\Rightarrow 0 \leq 21$ , which is true.

So, the lower half plane that contains the point $(0, 0)$ represent the solution region for $x + 4 y \leq 21$ .

Now, consider

$- 4 x + 6 y \leq 0$

$\Rightarrow 6 y \leq 4 x$

$\Rightarrow 3 y \leq 2 x$

We graph the line $2 x = 3 y$

This line passes through the origin. Also, we represent it by a solid line in the $x y$ -plane.

And, the region that lies below the line $2 x = 3 y$ represent the solution region of $3 y \leq 2 x$ including all the points on the line $2 x = 3 y$ .

Now consider

$0 \leq x \leq 15$

We know that the region that lies between the $y$ -axis, and the line $x = 15$ represent the solution region for $0 \leq x \leq 15$ .

And, $y \geq 0$ represent the region that lies in the first, and the second quadrants including the points on the line $y = 0$ ( $x$ -axis).

Thus, the feasible region of the given system of constraints is represented by the shaded region in the following graph:

The corner points of the feasible region are $A (5.727, 3.8118), B (15, 1.5), C (15, 0),$ and $D (0, 0)$ .

The values of $Z$ at these corner points are as follows:

$\begin{array}{l} Corner Point (x, y) & Value of z at (x, y) \\ A (5.727, 3.8118) & P = 50 (5.727) + 20 (3.812) = 362.59 \\ B (15, 1.5) & P = 50 (15) + 20 (1.5) = 780 \\ C (15, 0) & P = 50 (15) + 20 (0) = 750 \\ D (0, 0) & P = 0 \end{array}$

Since the feasible region is bounded.

Hence, the maximum value of $P$ is $780$ that occurs at the point $(15, 1.5)$ .

Size	Maximum number of people	Cost	Insurance
Small	$A (5.727, 3.8118), B (15, 1.5), C (15, 0),$	$$ 11000$	$$ 120$
Large	$16$	$$ 24000$	$$ 80$

Type of investment	Interest per annum
Municipal bond	$2 %$
Bank Mutual fund	$3.2 %$
Speculative money market fund	$5 %$

Rose Harrison and Clara Huizink Solutions for Exercise 15: Practice 4

Rose Harrison Mathematics Solutions for Exercise - Rose Harrison and Clara Huizink Solutions for Exercise 15: Practice 4

Questions from Rose Harrison and Clara Huizink Solutions for Exercise 15: Practice 4 with Hints & Solutions

Learn and Prepare for any exam you want !