Question

Consider the following Linear Problem

Minimize 2x1 + 2x2 equation (1)

subject to:

x1 + x2 >= 6 equation (2)

x1 - 2x2 >= -18 equation (3)

x1>= 0 equation (4)

x2 >= 0 equation (5)

13. What is the feasible region for Constraint number 1, Please consider the Non-negativity constraints.

14. What is the feasible region for Constraint number 2, Please consider the Non-negativity constraints.

15. Illustrate (draw) contraint 1 and 2 in a same graph and find interception within feasible region. Hint: consider the non-negativity constraints.

Answer 1

Ans 13

As per the query, Constraint number 1 is x1 + x2 >= 6 and the non-negativity constraints are x1>= 0 and x2 >= 0 (meaning you need to focus on quadrant 1 only).

To determine the feasible region (i.e. to find which side of your plotted line(s) is(are) relevant), follow these rules-

• Rearrange the constraint's equation in a way that "X2" is on the left and everything else on the right (e.g. x1 + x2 >= 6 can be rearranged as x2 >= 6 - x1 )
• Plot this "X2=" line (red line shown in the graph below)
• Your feasible region will lie above the line for a "greater than" sign (X2 ≥) and below the line for a "less than" sign (X2 ≤)....

Ans 14

Follow the same steps as above.

The second constraint can be rearranged as follows-

x1 - 2x2 >= -18

-2x2 >= -18 - x1

2x2 <= 18 + x1 (multiplying by -1 on both sides reverses the inequality)

x2 <= 9 + 0.5(x1) (the inequality sign indicates that the feasible region will lie below the given constraint; see the graph below)

Ans 15

The final feasible region will be the one common to all your constraints. This is represented by the mesh in the graph below. This feasible region contains the solution to your linear problem.