Question

In: Statistics and Probability

2. Find the maximum value of the objective function z = 4x + 6y where x...

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.

Solutions

Expert Solution

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.


Related Solutions

Find the maximum and minimum of the function f(x, y, z) = (x^2)(y^2)z in the region...
Find the maximum and minimum of the function f(x, y, z) = (x^2)(y^2)z in the region D = {(x, y, z)|x^2 + 2y^2 + 3z^2 ≤ 1}.
Maximize z=x+4y Subject to 2x+6y<= 36 4x+2y<= 32 x>= 0 y>= 0 Maximum is ___________ at...
Maximize z=x+4y Subject to 2x+6y<= 36 4x+2y<= 32 x>= 0 y>= 0 Maximum is ___________ at x = ______ y = ______
The function ​f(x,y,z)=4x−9y+7z has an absolute maximum value and absolute minimum value subject to the constraint...
The function ​f(x,y,z)=4x−9y+7z has an absolute maximum value and absolute minimum value subject to the constraint x^2+y^2+z^2=146. Use Lagrange multipliers to find these values.
find the range of the function x^2 +4x -7
find the range of the function x^2 +4x -7
z=3x^2+2y^2-xy-4x-7y+12 Find the extreme values; determine whether the function is at maximum, minimum, then evaluate the...
z=3x^2+2y^2-xy-4x-7y+12 Find the extreme values; determine whether the function is at maximum, minimum, then evaluate the critical values for the following function:
1.  For the function P(x) = 4x^2 - 16 / x^2 -5x find the a) List X...
1.  For the function P(x) = 4x^2 - 16 / x^2 -5x find the a) List X Intercept(s) if any b) List Y Intercept(s) if any c) List Horizontal Asymptote(s) if any d) List Vertical Asymptote(s) if any e) Domain 2. precalculus
Find the maximum and minimum and draw the graph of f(x) = 4x 2 - 40x + 80, for x = [0,8].
Find the maximum and minimum and draw the graph of f(x) = 4x 2 - 40x + 80, for x = [0,8].
Find the absolute maximum value and the absolute minimum value of the function f(x,y) = (1+x^2)(1−y^2)...
Find the absolute maximum value and the absolute minimum value of the function f(x,y) = (1+x^2)(1−y^2) on the disk D = {(x,y) | x2+y2⩽1}?
Given function f(x,y,z)=x^(2)+2*y^(2)+z^(2), subject to two constraints x+y+z=6 and x-2*y+z=0. find the extreme value of f(x,y,z)...
Given function f(x,y,z)=x^(2)+2*y^(2)+z^(2), subject to two constraints x+y+z=6 and x-2*y+z=0. find the extreme value of f(x,y,z) and determine whether it is maximum of minimum.
Questions 1 relate to the following constrained optimization problem: maximize z(x,y)=4x^2-2xy+6y^2 subject to x+y=72. subject to...
Questions 1 relate to the following constrained optimization problem: maximize z(x,y)=4x^2-2xy+6y^2 subject to x+y=72. subject to . A) What is the optimal value of X? B) What is the optimal value of Y? C) What is the maximized value of Z?
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT