In: Statistics and Probability
2. Find the maximum value of the objective function z = 4x + 6y where x ≥ 0 and y ≥0, subject to the constraints
a. −x + y ≤ 11
b. x+ y ≤ 27
c. 2x+ 5y ≤ 90
please do a detailed graph so i can understand better.
Solution :
Consider the given Linrear Programming Problem as follows;
Objective function : Maximize z = 4x + 6y
Subject to the constraints;
a. −x + y ≤ 11
b. x+ y ≤ 27
c. 2x+ 5y ≤ 90
where x ≥ 0 and y ≥0.
To solve the given LPP graphocally we first plot the constraints on the graph. For that we porceed as follows;
a. we solve the equation; −x + y = 11 that is, for x= 0 we get y = 11 and for y=0 we get x = -11
x | 0 | -11 |
y | 11 | 0 |
b. we solve the equation; x + y = 27 that is, for x= 0 we get y = 27 and for y=0 we get x = 27
x | 0 | 27 |
y | 27 | 0 |
c. we solve the equation; 2x + 5y = 90 that is, for x= 0 we get y = 18 and for y=0 we get x = 45
x | 0 | 45 |
y | 18 | 0 |
Also we have x ≥ 0 and y ≥ 0.
In the graph above the shaded region is the feasible region. And hence the critical points are as follows;
(0,0), (0,11), (15,12) and (27,0).
We have been given the objective function z = 4x + 6y. At these critical points we get the value of objective function as follows;
Critical Points | Objective Value |
(0 , 0) | 4*0 + 6*0 = 0 |
(0 , 11) | 4*0 + 6*11 = 66 |
(15 , 12) | 4*15 + 6*12 = 60 + 60 = 120 |
(27 , 0) | 4*27 + 6*0 = 108 |
This implies that we get maximum value of objective function at point (15, 12).
The maximum value of objective function is 120.