Question

. Use the graphical method to solve the problem Max Z 13x,+17x Subject to +3x, 20 Sx,+4x, S50 x, 20

Answer 1

To solve the linear programs graphically, we need to follow the below steps

Step 1 : Plot the given expressions as equations on the graph

Step 2 : Identify the feasible region considering the inequalities

Step 3 : Find the coordinates of the corner points of the feasible region

Step 4 : Find the value of objective function at each of the corner points. Wherever the value is maximum for maximization problem and minimum for minimization problem, that point is the optimal solution

For the given problem

Step 1 : The given equation are plotted as below and the feasible regions is identified with shaded area

O=ZX — O=TX OS=2x++[X5 — ST=ZX+[X — OZ=ZX&+[X-— 25 10 D Graph

Step 2 : Feasible regions is identified by the shaded region in the above graph

Step 3 : There are 4 corner points for the feasible regions as follows

A (0,0); B(0,6.67); C(3.68,7.89); D(10,0)

Step 4 : As objective function is 13x₁+17x₂, the value of the objective function at each of the corner points is as below

At A, the value of objective function is 13*0+17*0 = 0

At B, the value of objective function is 13*0+17*6.67 = 113.39

At C, the value of objective function is 13*3.68+17*7.89 = 181.97

At D, the value of objective function is 13*10+17*0 = 130

Thus the maximum value of objective function occurs at (3.68,7.89). Hence the optimal solution is x₁ = 3.68 and x₂ = 7.89