Linear Programming

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A diet is to contain at least 80 units of Vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F1 cost Rs. 4 per unit and F2 costs Rs. 6 per unit. One unit of food F1 contains 3 units of Vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of Vitamin A and 3 units of minerals. Formulate this as a linear programming problem and find graphically the minimum cost for diet that consists of mixture of these two foods and also meets the minerals nutritional requirements.

Solution

Let x be the number of units of food F₁ and y be the number of units of food F₂.

LPP is,

Minimize Z = 4x + 6y such that,

3x + 6y $\geq$ 80

4x + 3y $\geq$ 100

x, y $\geq$ 0

Representing the LPP graphically

Corner points are $[0, \frac{100}{3}], [24, \frac{4}{3}], [\frac{80}{3}, 0]$

Point	cost = 4x + 6y
$(0, \frac{100}{3})$	4 x 0 + 6 x $\frac{100}{3}$ = 0 + 200 = 200
$(24, \frac{4}{3})$	4 x 24 + 6 x $\frac{4}{3}$ = 96 + 9 = 104
$(\frac{80}{3}, 0)$	4 x $\frac{80}{3}$ + 6 x 0 = $\frac{320}{3}$ + 0 =106.67

From the table it is clear that, minimum cost is 104 and occurs at the point $(24, \frac{4}{3})$ .

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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Linear Programming

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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NCERT Sample Papers

Entrance Exams Preparation

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Linear Programming

Some More Questions From Linear Programming Chapter

Solve the following Linear Programming Problems graphically:Maximise Z = 3x + 2ysubject 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 + 500ysubject to the constraints x + 2y ≥ 10, 3x + 4 y ≤ 24, x ≥ 0, y ≥ 0

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

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

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

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