Operation Research-Graphical method-minimum sum

Question

Solve the problem by using graphical method:

 

Min Z= 20x1+40x2, 

Sub to

36x1+6x2 >= 108,

3x1+12x2 >= 36,

20x1+10x2 >= 700,

 x1,x2>=0.

Answer 1

Aman,

 

This is a linear programming problem. To restate the problem:

 

Objective function: z = 20x₁ + 40x₂

Constraints: 36x₁ + 6x₂ ≥ 108,

                     3x₁ + 12x₂ ≥ 36,

                   20x₁ + 10x₂ ≥700,

                     x₁ ≥ 0, y₁ ≥ 0.

 

To use the graphical method, you must graph the five constraints on a single set of axes, locate the shaded portion (the feasible region), find the coordinates of the vertices, plug each of them into the objective function, then compare the values. the smallest value is the minimum and the coordinates for x₁ and x₂ are the values which produce it.

 

Are you able to graph the feasible region? I cannot show it here due to the limitations of WyzAnt, so here are the vertices:

 

(0,0),

(35,0),

(0,70).

 

(It turns out that the first two inequalities are not necessary because the graph of the third is contained in both of them in the first quadrant.) After you plug these three points into the objective function, you find the following values:

 

(0,0) → z = 0,

(35,0) → z = 700

(0,70) → z = 2800.

 

Since you want the minimum value for z, it is clear that this will occur at the origin and the value is 0 (zero).

 

Let me know if you have any more questions on the solution.