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Linear Programming

Short Answer Type

1. Solve the following Linear Programming Problems graphically:
Maximise Z = 3x + 4y
subject to the constraints: x + y ≤ 4, x ≥ 0, y ≥ 0

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Long Answer Type

2. Solve the following linear programming problem graphically:
Maximise Z = 4x + y
subject to the constraints: x + y ≤ 50, 3x + y ≤ 90, x ≥ 0, y ≥ 0

335 Views

3. Find the maximum value of f = x + 2 y subject to the constraints:
2x + 3 y ≤ 6
x + 4 y ≤ 4
x, y ≥ 0

We are to maximize
f = x + 2y
subject to the constraints
2x + 3 y ≤ 6
x + 4 y ≤ 4
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.
Now we draw the graph of the line 2 x + 3 y = 6.
For x = 0, 3 y = 6, or y = 2
For y = 0, 2 x = 6, or x = 3
∴ line meets OX in A (3, 0) and OY in L (0, 2)
Let us draw the graph of line x + 4 y = 4
For x = 0, 4 y = 4, or y = 1
For y = 0, x = 4
∴ line meets OX in B (4, 0) and OY in M (0, 1)

Since feasible region is the region which satisfies all the constraints
∴ OACM is the feasible region. The comer points are
$straight O left parenthesis 0 comma space 0 right parenthesis comma space space space straight A left parenthesis 3 comma space 0 right parenthesis comma space space space straight C open parentheses 12 over 5 comma space 2 over 5 close parentheses comma space space space straight M left parenthesis 0 comma space 1 right parenthesis$
At $straight O left parenthesis 0 comma space 0 right parenthesis space straight f space equals 0 plus 0 space equals space 0$

$At space straight A left parenthesis 3 comma space 0 right parenthesis comma space straight f space space equals space 3 plus 0 space equals space 3 At space straight C open parentheses 12 over 5 comma space 2 over 5 close parentheses comma space space space straight f space equals space 12 over 5 plus 4 over 5 space equals space 16 over 5 space equals space 3.2 At space straight M left parenthesis 0 comma space 1 right parenthesis comma space space straight f space equals space 0 plus 2 space equals space 2$
$therefore space space space space maximum space value space equals space 3.2 space at space open parentheses 12 over 5 comma space 2 over 5 close parentheses.$

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

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5. Solve the following Linear Programming Problems graphically:
Minimise Z = - 3x + 4 y
subject to the constraints: x + 2y ≤ 8, 3x + 2y ≤ 12, x ≥ 0, y ≥ 0

149 Views

6. Solve the following Linear Programming Problems graphically:
Minimise Z = 5x + 3y
subject to the constraints: 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0.

116 Views