The maximum value of the objective function of an LPP subject to the constraints 2 ≤ x ≤ 6 and 3 ≤ y ≤ 5 is 12 and it is attained at the point 6 , 4 . Determine the objective function.

Given that an LPP is subject to the constraints 2 ≤ x ≤ 6 , and 3 ≤ y ≤ 5 Let the objective function of an LPP be z = a x + b y Then, maximum value of z is 12. Let us first find the feasible region of the given constraints. The common shaded region in the first quadrant represents the solution set of the given constraints Since the maximum value of an objective function is attained at the point 6 , 4 that lies on the line segment joining two corner points 6 , 3 , and 6 , 5 . We know that if two corner points of the feasible region are optimal solutions of the same type, i.e., both produce the same maximum or minimum, then any point on the line segment joining these two points is also an optimal solution of the same type. Thus, the optimal solution of the given LPP occurs at infinitely many points that lies on the line segment joining the corner points 6 , 3 , and 6 , 5 . Now, the coordinates of these points will satisfy the objective function, z = a x + b y = 12 ( maximum value ) So, satisfying the coordinates of the point 6 , 4 , we have 12 = 6 a + 4 b ⇒ 3 a + 2 b = 6 → i Now, satisfying the coordinates of point 6 , 3 , we have 12 = 6 a + 3 b ⇒ 2 a + b = 4 → i i Applying, 2 i i - i , we get 4 a + 2 b - 3 a + 2 b = 8 - 6 ⇒ a = 2 Thus, 2 2 + b = 4 ⇒ b = 4 - 4 = 0 (using i i ) Hence, z = a x + b y ⇒ z = 2 x + 0 y Hence, the required objective function is z = 2 x .

Show that S = { ( x , y ) ∈ R × R : 2 x + 5 y ≤ 6 } is a convex set.

To Prove: The set S = { ( x , y ) ∈ R × R : 2 x + 5 y ≤ 6 } is a convex set. A convex set is a set of points such that, given any two points A , B in that set, the line A B joining them lies entirely within that set. Intuitively, this means that the set is connected (so that you can pass between any two points without leaving the set) Clearly, the shaded region in the above graph represents the set of points that satisfies the inequation 2 x + 5 y ≤ 6 , and also it satisfies the definition of convex set . Hence, the set S = { ( x , y ) ∈ R × R : 2 x + 5 y ≤ 6 } is a convex set.

The constraints of an LPP are given to be 5 x + 3 y ≤ 15 , x - 3 y ≤ 3 , 0 ≤ x ≤ 7 , 0 ≤ y ≤ 6 . Determine which of these constraints are redundant.

Given constraints of an LPP are: 5 x + 3 y ≤ 15 , x - 3 y ≤ 3 , 0 ≤ x ≤ 7 , and 0 ≤ y ≤ 6 We need to find the redundant constraints from the given constraints Let us first find the feasible region (or common solution) of these constraints. Consider 5 x + 3 y ≤ 15 The graph of the line 5 x + 3 y = 15 on the x y - plane can be drawn by joining the points 0 , 5 , and 3 , 0 . Now, the coordinates of the point 0 , 0 satisfies the inequation, 5 x + 3 y ≤ 15 . So, the solution region of this inequation is left-side of the line 5 x + 3 y = 15 that contains the point 0 , 0 , including all the points on the line 5 x + 3 y = 15 . Similarly, we can draw the solution region for x - 3 y ≤ 3 . Now, x = 7 is a vertical line, and y = 6 is a horizontal line, and x , y ≥ 0 implies the solution region is in first quadrant including the x - and y - axes. Also, y ≤ 6 indicates that the solution region is below the horizontal line y = 6 , and x ≤ 7 indicate that the solution region is on the left-side of the vertical line x = 7 . Hence, the graph of the feasible region of all the given constraints is given as below: Clearly, if we exclude the constraints x - 3 y ≤ 3 , x ≤ 7 , and y ≤ 6 then our feasible region will not get affected. Therefore, the constraints x - 3 y ≤ 3 , x ≤ 7 , and y ≤ 6 are redundant.

E M B I B E

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 26 - Linear Programming Problems>EXERCISE>Q 3

EASY

12th West Bengal Board

IMPORTANT

Earn 100

Which of the following is not a convex set?

66.67% studentsanswered this correctly

Important Questions on Linear Programming Problems

HARD

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 26 - Linear Programming Problems>EXERCISE>Q 3

Consider the following linear programming problem:

Maximize $z = 3 x + 5 y$

subject to $5 x + 2 y \leq 20$ ,

$x \leq 2$

where $x \geq 0, y \geq 0$ .

Then

HARD

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 26 - Linear Programming Problems>EXERCISE>Q 3

Let

X_{1}

and

X_{2}

be two optimal solutions of an LPP, then

EASY

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 26 - Linear Programming Problems>EXERCISE>Q 3

The objective function

z = a x + b y

of an LPP is minimized at two corner points

(4, 5)

and

(8, 3)

of the feasible region. Then a root of the equation

a x^{2} + b x + 2 a = 0

is

EASY

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 26 - Linear Programming Problems>EXERCISE>Q 3

Which of the following is a convex set in two dimensions?

HARD

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 26 - Linear Programming Problems>EXERCISE>Q 1

The maximum value of the objective function of an LPP subject to the constraints

2 \leq x \leq 6

and

3 \leq y \leq 5

is

12

and it is attained at the point

(6, 4)

. Determine the objective function.

EASY

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 26 - Linear Programming Problems>EXERCISE>Q 2

Show that

S = {(x, y) \in R \times R : 2 x + 5 y \leq 6}

is a convex set.

MEDIUM

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 26 - Linear Programming Problems>EXERCISE>Q 3

The constraints of an LPP are given to be

5 x + 3 y \leq 15, x - 3 y \leq 3, 0 \leq x \leq 7, 0 \leq y \leq 6

. Determine which of these constraints are redundant.

HARD

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 26 - Linear Programming Problems>EXERCISE>Q 4

An LPP has only three non-redundant constraints, two of them are given to be

7 x + 5 y \leq 35

and

y \leq 4

. The objective function

z = 2 x - 6 y

of the LPP is minimised at the points

(4, 1)

, which is not a corner point of the feasible region. Determine the corner point of the feasible region. Determine the third non-redundant constraint of the LPP.

Learn and Prepare for any exam you want !

|

|

|

|

|

|

|

|