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**

**(2, 3)****(1, 2)****(2,1)****(-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?**

**Transportation problems****Manufacturing problem****Diet problem****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**

**on Y-axis****on X-axis****corner points of the feasible region****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**

**Optimum solution****Feasible solution****Unbounded solution****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**

**Objective function****a constraint****Decision variables****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**

**function to be optimized****a constraint****A relation between the variables****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**

**second quadrant****fourth quadrant****third quadrant****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

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.

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.