Question

Solve the following linear programming problem graphically: 

Maximize Z=4X₁+4X₂, 

Subject to: 

3X₁ + 5X₂ ≤ 150 

X₁ - 2X₂ ≤ 10 

5X₁ + 3X₂ ≤ 150 

X₁, X₂ ≥ 0 

1) Using the line drawing tool, plot the constraints by picking two endpoints for each line. Do not plot the nonnegativity constraints. 

2) Using the point drawing tool, plot the five corner points which define the feasible region. 

The optimal solution is X₁ = _______ and X₂ = _______ (round your responses to two decimal places). 

Maximum profit is $_______  

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

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

Graph A(0,30) B(18.75,18.75) C(25.38,7.69) E(0,0) 5 1 0 30 35 D(10,0) 3X1+5X2=150 15 - 20 - 25 5X1+3X2=150 X1-2X2=10

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

Step 3 : There are 5 corner points for the feasible regions as follows

A (0,30); B(18.75,18.75); C(25.38,7.69); D(10,0); E(0,0);

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

At A, the value of objective function is 4*0+4*30 = 120

At B, the value of objective function is 4*18.75+4*18.75 = 150

At C, the value of objective function is 4*25.38+4*7.69 = 132.28

At D, the value of objective function is 4*10+4*0 = 40

At E, the value of objective function is 4*0+4*0 = 0

Thus the maximum value of objective function occurs at (18.75,18.75). Hence the optimal solution is X₁ = 18.75 and X₂ = 18.75

Maximum profit which is the value of objective function at optimal solution is $150