Linear Programmig Problem MCQ Quiz in বাংলা - Objective Question with Answer for Linear Programmig Problem - বিনামূল্যে ডাউনলোড করুন [PDF]

Corner points of the feasible region for the given figure are (0, 0), (0, 4), and (12, 6).
F = 3x - 4y

So Minimum value of F = -16 at point (0,4).

The Correct Option is (2).

Calculation:

Converting the inequations to equations, we obtain:

2x + y = 3, x + 2y = 6, x = 0, y = 0.

For 2x + y = 3 

• Putting x = 0, 0 + y = 3 ⇒ y = 3
• Putting y = 0, 2x + 0 = 3 ⇒ x = \(\frac{3}{2}\)
• This line meets the x-axis at \(\left ( \frac{3}{2},0 \right )\) and the y-axis at (0, 3). Draw a line through these points.
• We see that the origin (0, 0) does not satisfy the inequation 2x + y > 3
• Therefore, the region that does not contain the origin is the solution of the inequality 2x + y > 3

For x + 2y = 6

• Putting x = 0, 0 + 2y = 6 ⇒ y = 3
• Putting y = 0, x + 0 = 6 ⇒ x = 6
• This line meets the x-axis at (6, 0) and the y-axis at (0, 3). Draw a line through these points.
• We see that the origin (0, 0) does not satisfy the inequation x + 2y > 6
• Therefore, the region that does not contain the origin is the solution of the inequality x + 2y > 6

The solution to the inequalities is the intersection of the 2x + y > 3, x + 2y > 6 and x, y > 0. Thus, the shaded region represents the solution set of the given set of inequalities.

Now solving 2x + y = 3 and x + 2y = 6, we get the point P(0, 3).

Now, Putting (3, 5) in 2x + y > 3,

2(3) + 5 > 3 ⇒ 11 > 3 which is true.

Again, putting (3, 5) in x + 2y > 6,

3 + 2(5) > 6 ⇒ 13 > 6 which is true.

∴ (3, 5) lies in the common region.

So, the correct option is (4)

Solution:

Given that, Z = 11x + 7y and the constraints 2x + y ≤ 6, x ≤ 2, x ≥ 0, y ≥ 0

Let 2x + y = 6

Now, plotting all the constrain equations we see that the shaded area OABC is the feasible region determined by the constraints.

The feasible region is bounded. So, the maximum value will occur at a corner point of the feasible region.

Corner points are (0, 0), (2, 0), (2, 2) and (0, 6).

On evaluating the value of Z, we get

CONCEPT:

• At the corner point of the feasible reason, we have to check the value of the objective function to get the maximum value.

CALCULATION:

Given:

So, the condition of p and q, so that the minimum of Z occurs at (3, 0) and (1, 1) can be extracted by equating the minimum values of Z for (3, 0) and (1, 1).

⇒ p + q = 3p

⇒ 2p = q

∴ \(\rm p=\frac{q}{2}\)

• So, the correct answer will be option 2.

CONCEPT:

At the corner point of the feasible reason, we have to check the value of the objective function to get the maximum value.

CALCULATION:

Given: Z = px + qy

So, the condition of p and q, so that the minimum of Z occurs at (3, 0) and (1, 1) can be extracted by equating the minimum values of Z for (3, 0) and (1, 1).

⇒ p + q = 3p

⇒ 2p = q

⇒ \(\rm p=\frac{q}{2}\)

∴  the correct answer will be option 3

Explanation:

 Z = px + qy, where p, q > 0

Corner points of the feasible region determined by the system of linear constraints are (0, 3), (1, 1), and (3, 0)

Let z₀​ be the minimum value of z in the feasible region.

The value z_0​ is attained at both (3, 0) and (1, 1).

z_0​ = p × 3 + q × 0     ---- (1)

z₀ ​= p × 1 + q × ​1     ---- (2)

From Equations (1) and (2) we get 

⟹ p × 3 + q × 0 = p × 1 + q × ​1 

⟹ 3 p = p + q

⟹ 2 p = q

⟹ p = q/2

The Correct Option is (2).

Explanation:

Corner points of the feasible region for an LPP are (0, 2), (3, 0), (6,0), (6, 8) and (0,5).
F = 4x + 6y

So the Minimum value of F occurs at any point on the line segment joining the points (0, 2) and (3, 0).

The Correct Option is (4).

Solution:

Given that, Z = 5x + 3y and the constraints 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0

Taking 3x + 5y = 15, we have

And, taking 5x + 2y = 10, we have

Now, plotting all the constrain equations we see that the shaded area OABC is the feasible region determined by the constraints.

The feasible region is bounded. So, the maximum value will occur at a corner point of the feasible region.

Corner points are (0, 0), (2, 0), (0, 3) and \(\left ( \frac{20}{19},\frac{45}{19} \right )\)

On evaluating the value of Z, we get

Concept: 

There are three main components of linear programming:

• Decision Variables: These variables are the activities that share the available resources while also competing with one another. In other words, these are interrelated in terms of resource utilization and need to be solved simultaneously. They are assumed to be non-negative and continuous.  
• The Objective Function: Each linear programming problem is aimed to have an objective to be measured in quantitative terms such as profits, sales, cost, time, or some other parameter, etc which needs to be maximized or minimized. The objective function will vary depending upon the business requirements or other factors.
• Constraints: These represent real-life limitations such as money, time, labor or other restrictions and the objective function needs to be maximized or minimized while satisfying the constraints. They are represented as linear equations or inequations in terms of the decision variables.

For example, suppose x and y are the decision variables. The objective function will be given by:

Z = ax + by ….(1)

Where a and b are constants and Z is the function to be maximized or minimized. There will be conditions x ≥ 0, and y ≥ 0, which indicates the non-negative constraints on the decision variable.

The equation looks very simple since there are various assumptions involved while forming a linear programming example. These are mentioned below:

• Parameters such as resources available, profit contribution of unit decision variable and resource used by unit decision variable needs to be known.
• Decision variables are continuous. Hence the outputs can be an integer or a fraction.
• The contribution of each decision variable in the objective function is directly proportional to the objective function.

Calculation:

Given:

x, y: Number of units of A and B being sold respectively.

It is very beneficial if the information is converted to a tabular form to make the data visualization much easier. As per the above info, the table is formed below:

• Since x and y cannot be negative, x ≥ 0      …(2) and y ≥ 0      ….(3)
• One unit of A takes 1 unit of electricity and 3 units of diesel.   
• Similarly, one unit of B takes 1 unit of electricity and 2 units of diesel.   

⇒ x + y ≤ 5       ….(4) and 3x + 2y ≤ 12       ….(5)

• The objective function will be the profit obtained by selling x units of A and y units of B.

⇒ Z = 60x + 50y ….(6)

• Hence, the problem is represented mathematically by (2), (3), (4), (5), and (6).
• So, the correct answer is option 1.

Concept

Objective Function: In a Linear Programming Problem (LPP), the objective function is a linear equation involving decision variables that we aim to either maximize or minimize. The general form of an objective function is:

\( Z = ax + by \)

Linear Programming: Linear programming is a method for finding the maximum or minimum value of a linear function, subject to certain constraints. The objective function and the constraints are linear functions of the decision variables.

Calculation

Given the objective function:

\( Z = ax + y \)

This is a valid and simplified form of a linear objective function in Linear Programming.​

Important Points\(Z=ax+by+c\) 

This is also a valid form for the objective function, but it is more general than Option 1. It involves two variables, x and y, with coefficients a and b, and an additional constant term  c" id="MathJax-Element-9-Frame" role="presentation" style="position: relative;" tabindex="0">$c$ $c$ .

Options 3 and 4 are constraint equations, not an objective function.