Question

The objective function Z = 4x + 3y can be maximized subjected to the constraints 3x + 4y ≤ 24, 8x + 6y ≤ 48, x ≤ 5, y ≤ 6; x, y ≥ 0

A. At only one point

B. At two points only

C. At an infinite number of points

D. None of these

Answer 1

Correct answer is C.

Given the objective function is Z = 4x + 3y

Constraints are:

3x + 4y ≤ 24

8x + 6y ≤ 48

x ≤ 5

y ≤ 6

x ≥ 0

y ≥ 0

If we consider these inequalities as equalities for some time,

We will have

3x + 4y = 24

8x + 6y = 48

x = 5

y = 6

x = 0

y = 0

If we plot all these lines on a graph we will have optimal area formed by the vertices, OABCD.

Now, to find where the function Z has maximized, let us substitute all these points in the objective function Z.

Here, we can clearly see that, the function Z is maximized at two points B & C giving the value 24.

There will be infinite/multiple optimal solutions for a LPP if it has more than one set of optimal solutions that can maximize/ minimize a problem.

This will clear the fact that, the function Z will maximize at infinite number of points.