Question

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.

Answer 1

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.