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

An LPP has only three non-redundant constraints, two of them are given to be

7 x + 5 y \leq 35

and

y \leq 4

. The objective function

z = 2 x - 6 y

of the LPP is minimised at the points

(4, 1)

, which is not a corner point of the feasible region. Determine the corner point of the feasible region. Determine the third non-redundant constraint of the LPP.

Accepted Answer

Given constraints of an LPP are $7 x + 5 y \leq 35$ and $y \leq 4$

It is given that the objective function $z = 2 x - 6 y$ is minimum at $(4, 1)$ , which is not a corner point of the feasible region.

Let $m$ be the minimum value of $z$

So, we have

$2 x - 6 y \leq m$

Now, $m = 2 (4) - 6 (1) = 2$

$\Rightarrow 2 x - 6 y \leq 2$

$\Rightarrow x - 3 y \leq 1$ , which is the required third constraint.

Now, we need to draw the graphs of all the constraints to find the corner points of the feasible region

We have

$7 x + 5 y \leq 35$

$y \leq 4$

$x - 3 y \leq 1$

$x \geq 0, y \geq 0$

Consider the inequation, $7 x + 5 y \leq 35$

Since $7 x + 5 y = 35$ is a line joining the points $(0, 7),$ and $(5, 0)$ . Also, the co-ordinates of the point $(0, 0)$ satisfies the inequation $7 x + 5 y \leq 35$ . So, the solution region of this inequation is left-side of the line $7 x + 5 y = 35$ , including all the points on the line.

Similarly, the solution region for inequality $x - 3 y \leq 1$ is that side of the line $x - 3 y = 1$ (joining the points $(0, - \frac{1}{3}),$ and $(1, 0)$ ) which contains the point $(0, 0)$ , including the points on the line.

And, solution region of $y \leq 4$ lies below the horizontal line $y = 4$ .

$x \geq 0, y \geq 0$ represent the solution region in the first-quadrant.

Hence, the common solution region of all the constraints is represented by the graph shown below:

Hence, corner points of the required feasible region are

$(0, 0), (1, 0), (0, 4), (2.143, 4), and (4.231, 1.077)$

Important Questions on Linear Programming Problems

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