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Linear Programming Important Questions

Linear programming is a process for optimizing operations under certain constraints. 

  • The main objective of linear programming is to maximize or minimize numerical values. 
  • It is composed of linear functions that are constrained by constraints in the form of linear equations or inequalities. 
  • Linear programming is considered a significant tool for determining optimal resource use. 
  • The phrase “linear programming” is made up of two words: linear and programming. 
  • The term “linear” refers to the connection between one or more variables. 
  • The term “programming” refers to the process of selecting the best solution from a set of options.

The basic components of the LP are as follows:

  • Decision Variables
  • Constraints
  • Data
  • Objective Functions

Short Answers Questions

Ques. The point which does not lie in the half-plane 2x + 3y -12 < 0 is

  1. (2, 3)
  2. (1, 2)
  3. (2,1)
  4. (-2, 3)

Ans. The correct answer is a. (2, 3)

Explanation: On substituting the value of point (2,3) in 2x + 3y – 12, we get

2(2) + 3(3) – 12

= 4 + 9 – 12

= 13 – 12 = 1

Which is greater than zero.

Ques. Which of the following is a type of Linear programming problem?

  1. Transportation problems
  2. Manufacturing problem
  3. Diet problem
  4. All of the above

Ans. The correct answer is d. All of the above

Explanation: Transportation problems, Manufacturing problems, and Diet problems, all are the different types of linear programming problems.

Ques. The optimal value of the objective function is attained at the points

  1. on Y-axis
  2. on X-axis
  3. corner points of the feasible region
  4. None of the above

Ans. The correct answer is c. corner points of the feasible region

Explanation: An optimal value is any point in the feasible region that provides the optimal value (maximum or minimum) of the objective function.

Ques. A set of values of decision variables that satisfies the linear constraints and non-negativity conditions of an L.P.P. is called its

  1. Optimum solution
  2. Feasible solution
  3. Unbounded solution
  4. None of the above

Ans. The correct answer is b. Feasible solution

Explanation: A feasible solution is a set of values for the decision variables that satisfies all the constraints of the LPP.

Ques. The linear inequalities or equations or restrictions on the variables of a linear programming problem are called

  1. Objective function
  2. a constraint
  3. Decision variables
  4. None of the above

Ans. The correct answer is b. a constraint

Explanation: A constraint is a set of linear inequalities, equations, or limits on the variables of a linear programming problem.

Ques. The objective function of a linear programming problem is

  1. function to be optimized
  2. a constraint
  3. A relation between the variables
  4. None of the above

Ans. The correct answer is a. function to be optimized

Explanation: The objective function of a linear programming problem is a function to be optimized.

Ques.  Region represented by x ≥ 0, y ≥ 0 is

  1. second quadrant
  2. fourth quadrant
  3. third quadrant
  4. first quadrant

Ans. The correct answer is d. first quadrant

Explanation: All the positive values of x and y will lie in the first quadrant.

Ques. What is Linear Programming?

Ans. Linear programming is a method of optimizing problems that are constrained in some way. It is the process of maximizing or reducing linear functions according to linear inequality constraints. 

Ques. What are the different types of linear programming?

Ans. The different types of linear programming are

  • Solving linear programming by Simplex method
  • Solving linear programming by graphical method
  • Solving linear programming using R
  • Solving linear programming with the use of an open solver.

Ques. What are the requirements of linear programming?

Ans. The following are the requirements of linear programming

  • Objective function
  • Non-negativity
  • Constraints
  • Finiteness
  • Linearity

Ques. What is meant by linear programming problems?

Ans. The linear programming problems (LPP) help with determining the best optimal solution to a linear function (also known as the objective function) that is constrained by a set of linear inequality constraints.

Long Answers Questions

Ques. What are the applications of linear programming?

Ans. The following are the applications of linear programming

  • Manufacturing Sector: Many manufacturing industries employ linear programming functions to maximize their profits while lowering product manufacturing costs.
  • Engineering Sector: Many engineering sectors throughout the world employ linear programming to solve design and production problems. It will produce the maximum output under the given conditions.
  • Energy Sector: The energy sector uses the linear programming method to increase output and production efficiency.
  • Transportation Sector: Transportation sectors use the linear programming function to reduce transportation costs and increase efficiency.

Ques. What are the fundamental theorems of linear programming?

Ans. The following are the two essential linear programming theorems:

  • Theorem 1: For a linear programming problem, let R be the feasible region and z = ax + by be the objective function. When the variables x and y are constraints defined by the linear inequalities, Z has an optimum value. The optimal value must be located at the corner of the feasible region.
  • Theorem 2: Let R represent the feasible region of a linear programming problem and z = ax + by representing the objective function. If S is bounded, then z has a maximum and a minimum optimization value on R, and each value corresponds to a JR corner point.

Ques. What are the characteristics of Linear Programming?

Ans. The following are the characteristics of Linear Programming

  • Objective function: The objective function is a linear function (z = px + qy, where p and q are constants) whose value is to be maximized or decreased.
  • Constraints: Constraints are linear inequalities or inequalities on the variables of a linear programming problem (LPP). The conditions where x ≥ 0 and y ≥ 0 are called non-negative restrictions.
  • Optimal Value: Every objective function has a maximum and a minimum value. As a result, the value of the objective function is referred to as the optimal value.
  • Optimization Problem: An optimization problem is one that involves maximizing or minimizing a linear function subject to constraints given by a set of linear inequalities.

Feasible Region: A feasible region is a common region found by all constraints, including non-negative constraints of an LP. A feasible region is also known as a convex polygon. At the same time, a region other than the feasible region is referred to as an infeasible region.

Ques. Solve the following LPP graphically:

Maximise Z = 2x + 3y, subject to x + y ≤ 4, x ≥ 0, y ≥ 0

Ans. Let us draw the graph of x + y =4 as below

Linear Programing

In the following image, the shaded region (OAB) represents the feasible region given by the system of constraints x ≥ 0, y ≥ 0, and x + y ≤ 4.

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

Corner Points are O(0, 0), A (4, 0) and B (0, 4).

Evaluate Z at each of these corner points.

Corner point Value of Z
O (0, 0) 2 (0) + 3(0) = 0
A (4, 0) 2 (4) + 3(0) = 8
B (0, 4) 2 (0) + 3 (4)= 12 (maximum)

Therefore, the maximum value of Z is 12 at the point (0, 4).

Ques. A manufacturing company makes two types of television sets; one is black and white and the other is colour. The company has the resources to make at most 300 sets a week. It takes Rs 1800 to make a black and white set and Rs 2700 to make a coloured set. The company can spend not more than Rs 648000 a week to make television sets. If it makes a profit of Rs 510 per black and white set and Rs 675 per coloured set, how many sets of each type should be produced so that the company has a maximum profit? Formulate this problem as an LPP given that the objective is to maximize the profit.

Ans. Let x and y represent the number of black and white sets and coloured sets produced each week, accordingly.

Thus x ≥ 0, y ≥ 0

The company can make at most 300 sets a week, therefore, x + y ≤ 300.

The weekly cost (in Rs) of manufacturing the set is 1800x + 2700y and the company can spend up to Rs. 648000.

Therefore, 1800x + 2700y ≤ 648000

or

2x + 3y ≤ 720

The total profit on x black and white sets and y coloured sets are Rs (510x + 675y).

Let the objective function be Z = 510x + 675y.

Therefore, the mathematical formulation of the problem is as follows.

Maximise Z = 510x + 675y subject to the constraints :

x + y ≤ 300

2x + 3y ≤ 720

x ≥ 0, y ≥ 0

The graph of x + y = 30 and 2x + 3y = 720 is given below.

Linear Programming

Corner point Value of Z
A(300, 0) 153000
B(180, 120) 172800 (Maximum)
C(0, 240) 162000

Therefore, the maximum profit will be obtained when 180 black-and-white sets and 120 coloured sets are manufactured.

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