##### Chapter 5

## Linear Programming

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

#### Introduction

**inear programming**is a branch of mathematics that deals with the methods to maximize or minimize profits. This is called optimizing a problem, and hence

**linear programming**finds importance in the field of telecommunication, commerce, management, science, etc.

The **linear programming** of LP is made up of several components. They are as follows:

- Data
- Objective Functions
- Constraints
- Decision Variables

You might be wondering if** linear programming** only has components or characteristics as well? To answer your questions, **linear programming** has five characteristics that are used while solving the problem:

- Linearity: The degree of every variable in linear function must be one, i.e., the relationship between the variables in the function shall be linear.
- Constraints: when it comes to resources in the linear functions, then they must be conveyed in the mathematical form.
- Objective Function: In linear programming, it is necessary to specify the objective function qualitatively.
- Non- negativity: In linear programming, the variable cannot have a negative value and must either be 1 or 0.

You might have come across the situation where you wanted to maximize the potential in the best possible way. For example, consider a company trying to maximize the profit while having limitations of labor and materials. In situations like these, optimization plays an important role, and mathematics can help them reach their end goal. But not just in terms of business, **linear programming** finds its importance in many other fields.

- Engineering:
**Linear programming**helps solve the problems around designing and manufacturing as it can help optimize the shapes. - Manufacturing: Linear expressions are used to maximize profits in the manufacturing industry.
- Energy industry: Helps in finding ways to optimize the power systems.
- Transportation:
**Linear programming**helps in optimizing the cost and time to achieve the maximum result.

Since **linear programming** is helpful in many different fields, the functional problems can be defined as linear programming problems. It has been noted that **linear programming** problems such as queries related to multi-commodity flow and network flows have contributed to much research on functional algorithms to find the solutions.

Linear graphs can be used in our daily life to represent the relationship between the different quantities. For example, if a school hires more teachers, then the number of students taking admission increases as well and vice-versa.

This relationship between the quantities can be in direct or indirect proportions. The relationship can be represented in a graphical manner called linear graphs.

Linear equations can help solve real-world problems efficiently, and for that, it is necessary that the real-life problems are converted to mathematical expressions. The expression should represent the relationship between the variables and all the necessary information.

Certain procedures are involved when converting the situation to a mathematical statement. The steps are as follows:

- To convert the problem to a mathematical statement, the algebraic expression should portray the problem efficiently.
- The unknown values must be assigned variables.
- It is necessary to obtain the data, keywords, and phrases by reading the problem multiple times.
- The information obtained must be organized sequentially.
- With the information obtained, you can frame your equation and solve it using systematic techniques.
- To make sure that the solution is correct, retrace it to the problem statement.

Linear Equations | Non- Linear Equations |

The equation is represented by a straight line in the graph. | The equation is represented by a curve in the graph. |

The degree of variables is 1. | The variables can have degrees of 2 or more. |

The equation for linear equation is:
Where, x, y= variables m= slope c= constant | The equation for non- linear equation is Where, x, y= variables a, b, c= constant |

1. What is the difference between the linear and non-linear equation?

A linear equation is represented by a straight line in the graph, while a non-linear equation is represented by a curve.

2. What are the advantages of **linear programming**?

Linear programming has various advantages:

- Help get the insight into the real-world problems.
- Helps in solving multi-dimensional problems.
- Helps in finding the best possible solution to a problem with given available options.

3. What are the different types of **linear programming**?

There are many different types of **linear programming,** and depending on the situation, you can choose the suitable method:

- Simplex method
- R method
- Graphical method
- Open solver method

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