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Linear Programming

Question
CBSEENMA12033526
Wired Faculty App

Maximize z = 9 x + 3 y subject to the constraints
2x + 3y ≤ 13
2x + y ≤ 5
x, y ≥ 0

Solution

We have to maximize
z = 9x + 3 y
subject to the constraints
2x + 3 y ≤ 13
2x + y ≤ 5
x, y ≥ 0
Consider a set of rectangular cartesian axes OXY in the plane.
It is clear that any point which satisfies x ≥ 0,y ≥ 0 lies in the first quadrant.
Let us draw the graph of 2x + 3y = 13
For x = 0,  3y = 13  rightwards double arrow space straight y space equals space 13 over 3
For y = 0,  2x = 13    rightwards double arrow space straight x space equals space 13 over 2
therefore space space space space line space 2 straight x plus 3 straight y space equals space 13 space space meets space OX space in space straight A open parentheses 13 over 2 comma 0 close parentheses space and space OY space in space straight B open parentheses 0 comma space 13 over 3 close parentheses.
Again we draw the graph of 2x + y = 5
For x = 0,  y = 5
For y = 0,  2x = 5  rightwards double arrow space space straight x space equals space 5 over 2
therefore space space space line space 2 straight x plus straight y space equals space 5 space meets space OX space in space straight C open parentheses 5 over 2 comma space 0 close parentheses space and space OY space in space straight D left parenthesis 0 comma space 5 right parenthesis.

Since feasible region satisfies all the constraints.
therefore    OCEB in the feasibe region. The corner points are O(0, 0),  straight C open parentheses 5 over 2 comma space 0 close parentheses comma space straight E left parenthesis 0.5 comma space 4 right parenthesis comma space space straight B open parentheses 0 comma space 13 over 3 close parentheses 
       At O(0, 0),  z = 9(0) + 3(0) = 0+ 0 = 0
At   straight C open parentheses 5 over 2 comma space 0 close parentheses comma space straight z space equals space 9 space open parentheses 5 over 2 close parentheses space plus space 3 space left parenthesis 0 right parenthesis space equals space 45 over 2 plus 0 space space equals space 45 over 2 space equals space 22.5
At   straight E thin space left parenthesis 0.5 comma space 4 right parenthesis comma space space straight z space equals space 9 left parenthesis 0.5 right parenthesis space plus space 3 space left parenthesis 4 right parenthesis space equals space 4.5 space plus space 12 space equals space 16.5
At space straight B space open parentheses 0 comma space 13 over 3 close parentheses comma space space straight z space equals space 9 space left parenthesis 0 right parenthesis space plus space 3 space open parentheses 13 over 3 close parentheses space equals space 0 plus 13 space equals space 13
therefore space space space Maximum space value space of space straight z space is space 22.5 space at space open parentheses 5 over 2 comma 0 close parentheses space space straight i. straight e. comma space space space when space straight x space equals space 5 over 2 comma space space straight y space equals 0


Some More Questions From Linear Programming Chapter

Solve the following Linear Programming Problems graphically:
Maximise    Z = 3x + 2y
subject to the constraints: x + 2y ≤ 10, 3x + y ≤ 15, x, y ≥ 0, 

Maximize z = 4x + 1y such that x + 2y ≤ 20, x + y ≤ 15, x ≥ 0, y ≥ 0.

Minimize z = 2x + 3y, such that 1 ≤ x + 2y ≤ 10, x ≥ 0, y ≥ 0.

Solve the following linear programming problem graphically:
Minimise Z = 200x + 500y
subject to the constraints x + 2y ≥ 10, 3x + 4 y ≤ 24,  x ≥ 0, y ≥ 0

Solve the following problem graphically:
Minimise and Maximise Z = 3x + 9y
subject to the constraints:
x + 3y ≤ 60
x + y ≥ 10
x ≤ y
x ≥ 0, y ≥ 0

Show that the minimum of Z occurs at more than two points.
Minimise and Maximise Z = 5x + 10y subject to constraints x + 2y ≤ 120,  x + y ≥ 60, x - 2 y ≥ 0, x, y ≥ 0.

Minimize z = 5x + 7y such that 2x + y ≥ 8, x + 2y ≥ 10, x, y ≥ 0.

Show that the minimum of Z occurs at more than two points. 
Minimise and Maximise Z = x + 2y subject to constraints x + 2y ≥ 100, 2x - y ≤ 0, 2x + y ≤ 200, x, y ≥ 0

 

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