Name: Linear Programming: Manufacturing Problem
Uploaded: 2019-07-19
Duration: 7 min 3 s
Description: In this video we are going to learn linear programming manufacturing problem comprehensively with the help of real life examples. Check it out!

Question

Apply corner point method to find the optimal solution of the following linear programming problem. Find also the optimum value of the objective function.

Maximize $z = x + 3 y$

subject to $5 x + 4 y \geq 20$ ,

$7 x + 8 y \leq 56$

where $x \geq 0, y \geq 0$

Accepted Answer

Given LPP is

Maximize $z = x + 3 y$

subject to $5 x + 4 y \geq 20$

$7 x + 8 y \leq 56$ , where $x \geq 0, y \geq 0$

Let us first find the feasible region of the LPP

We first draw the graph of the line $5 x + 4 y = 20$ on the $x y -$ plane by joining the points $(0, 5),$ and $(4, 0)$ .

Now, the line divides the plane into two half. Let us identify the solution region of the inequation $5 x + 4 y \geq 20$ .

Since the coordinates of the point $(0, 0)$ does not satisfy the inequation $5 x + 4 y \geq 20$ . So, the solution region for this inequation is that half-plane which does not contain the point $(0, 0)$ . Also, since the inequality is slack, so all the points on the line are also included in the solution region.

Similarly, we can find the solution region for $7 x + 8 y \leq 56$ .

Also, $x \geq 0, y \geq 0$ represents the region in the first quadrant.

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 $A (0, 5), B (0, 7), C (8, 0),$ and $D (4, 0)$ .

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

$\begin{array}{l} Corner Point (x, y) & Value of z at (x, y) \\ A (0, 5) & z = 0 + 3 (5) = 15 \\ B (0, 7) & z = 0 + 3 (7) = 21 \to maximum \\ C (8, 0) & z = 8 + 3 (0) = 8 \\ D (4, 0) & z = 4 + 3 (0) = 4 \end{array}$

Since the feasible region is bounded.

Hence, the maximum value of $z$ is $21$ that occurs at the point $(0, 7)$ .

Apply corner point method to find the optimal solution of the following linear programming problem. Find also the optimum value of the objective function.

Maximize $z = x + 3 y$

subject to $5 x + 4 y \geq 20$ ,

$7 x + 8 y \leq 56$

where $x \geq 0, y \geq 0$

Important Questions on Linear Programming Problems

Learn and Prepare for any exam you want !

Apply corner point method to find the optimal solution of the following linear programming problem. Find also the optimum value of the objective function. Maximize z=x+3y subject to 5x+4y≥20, 7x+8y≤56 where x≥0, y≥0

Important Questions on Linear Programming Problems

Learn and Prepare for any exam you want !

Apply corner point method to find the optimal solution of the following linear programming problem. Find also the optimum value of the objective function.

Maximize $z = x + 3 y$

subject to $5 x + 4 y \geq 20$ ,

$7 x + 8 y \leq 56$

where $x \geq 0, y \geq 0$