Converting the given in-equations into equations, we obtain the following equations:
x + 2y = 40, 3x + y = 30, 4x + 3y = 60, x = 0 and y = 0.
Region represented by x + 2y ≤ 40: The line x + 2y = 40 meets the coordinates axes at A1 (40, 0) and B1 (0, 20) respectively. Join these points to obtain the line x + 2y = 40. Clearly, (0, 0) satisfies the in-equationx + 2y ≤ 40. So, the region in xy-plane that contains the origin represents the solution set of the given in-equation.
Region represented by 3x + y ≥ 30: The line 3x + y = 30 meets X and Yaxes at A2 (10, 0) and B2 (0, 30) respectively. Join these points to obtain this line. We find that O (0, 0) does not satisfy the inequation 3x + y ≥ 30. So that region in xy-plane which does no contain the origin is the solution set of this inequation.
Region represented by 4x + 3y ≥ 60: The line 4x + 3y = 60 meets X and Yaxes at A3 (15, 0) and B1 (0, 20) respectively. Join these points to obtain the line 4x + 3y = 60. We observe that O (0, 0) does not satisfy the inequation 4x + 3y ≥ 60. So, the region not containing the origin in xy-plane represents the solution set of the given inequation. Region represented by x≥ 0, y ≥ 0: Clearly, the region represented by x ≥ 0 and y ≥ 0 is the first quadrant in xy-plane.
The shaded region A3A1 QP in fig. represents the common region of the regions represented by the above inequations. This region expresents the feasible region of the given LPP.
The coordinates of the corner-points of the shaded feasible are A3 (15, 0)A1 (40, 0), Q (4, 18) and P (6, 12). These points have been obtained by solving the equations of the corresponding intersecting lines, simultaneously.
The values of the objective function at these points are given in the following table:
Point (x, y)
|
Value of the objective function
Z = 20x + 10y
|
A3 (15, 0)
|
Z = 20 × 15 + 10 × 0 = 300
|
A1 (40, 0)
|
Z = 20 × 40 + 10 × 0 = 800
|
Q (4, 18)
|
Z = 20 × 4 + 10 × 18 = 260
|
P (6, 12)
|
Z = 20 × 6 + 10 × 12 = 240
|
Clearly, Z is minimum at P (6, 12). Hence, x = 6, y = 12 is the optimal solution of the given LPP. The optimal value of Z is 240.