What is the minimum value of the objective function, C, with the given constraints? C=2x+3y

A) 2x+3y≤24

B) 3x+y≤15

C) x≥2

D) y≥3

Given:

The objective function, ,

Constraints,

To find:

The minimum value of the given objective function.

Solution:

We have,

Objective function,            ....(i)

The related equations of given constraints are

      ...(ii)

             ...(iii)

, it is a vertical line parallel to y-axis and 2 units left from y-axis.

, it is a horizontal line parallel to x-axis and 4 units above from y-axis.

Table of value for (i),

x :   0     12

y :   8     0

Table of value for (ii),

x :   0      5

y :   15     0

Plot these points on a coordinate plane and draw these 4 related lines.

The sign of inequality of and is ≤, it means the related lines are solid lines and the shaded region lie below the related line.

For , left side of the line is shaded.

For , shaded region is above the line .

From the below graph it is clear that the vertices of the feasible (common shaded region) are (2,3), (4,3), (3,6) and (2,6.667).

Points                        

(2,3)                            (Minimum)

(4,3)                            

(3,6)                            

(2,6.667)                

Therefore, the objective function is minimum at point (2,3).