Linear Programming

Dhristi JEE 2022-24

What is Linear Programming, Explain with Examples?

Linear programming is a mathematical method for optimizing operations given restrictions. Linear programming's basic goal is to maximize or minimize a numerical value. It consists of linear functions that are limited by linear equations or inequalities. A prominent technique for discovering the most effective use of resources is linear programming. It also has linear functions that are subjected to the constraints in the form of linear equations or in the form of linear inequalities. 


Linear programming is a popular technique for determining the most efficient resource allocation. "Linear programming" is made up of two words: "linear" and "programming." The term "linear" refers to a one-dimensional relationship between many variables. The term "programming" refers to the process of choosing the optimal answer from a variety of options.


Linear Programming

The challenge of maximizing or minimizing a linear function under linear constraints is known as linear programming (LP) or linear optimization. Equalities or inequalities can be used as restraints. Profit and loss calculations are part of the optimization challenges. Linear programming problems are a type of optimization problem that aids in determining the feasible region and optimizing the solution to get the highest or lowest function value.


Linear programming is a technique for analyzing various inequalities in a scenario and estimating the best value that can be obtained under given constraints. The following are some of the assumptions made when dealing with linear programming:

  • The total number of constraints should be expressed numerically.

  • The constraints and the objective function should have a linear relationship.

  • It is necessary to optimize the linear function (i.e., the objective function).


To put it another way, linear programming is an optimization approach for maximizing or minimizing the objective function of a mathematical model given a set of conditions expressed by a linear relationship. The linear programming problem's main goal is to find the best solution.


What is Constraint?

A constraint is an inequality that defines how the values of the variables during a problem are limited. In order for applied mathematics techniques to figure, all constraints should be linear inequalities.


When to Use Linear Programming?

Linear programming is a technique for solving problems that are constrained in some way. It also refers to the process of maximizing or minimizing linear functions which is constrained by a linear inequality. The challenge of solving linear programming is thought to be the simplest.


Components of Linear Programming

The normal components of Linear Programming are pointed out below:

  • Decision Variables

  • Constraints

  • Data

  • Objective Functions


Characteristics of Linear Programming

Given below are the five characteristics of linear programming problem:

  • Constraints- The limitations should be put up in the mathematical form, regarding the given resource.

  • Objective Function- In a problem, the objective function should be mentioned in a quantitative way.

  • Linearity- The relationship between two or more variables in the function should be linear. 

  • Finiteness-  There always should be finite and infinite input and output numbers. If the function has infinite factors, the optimal solution will not be feasible. 

  • Non-Negativity- The value of the variable should be positive or zero. It should not be negative.


Linear Programming Method (Simplex)

To solve the linear programming models, the easiest linear programming method is used to find the optimal solution for a problem. It also involves slack variables, tableau, and pivot variables for the optimization of a particular problem. The algorithm used here is given below

  • Change of variables and normalize the sign of independent terms

  • Normalise restrictions

  • Match the objective functions to zero

  • Write the initial tableau of the simplex linear programming method

  • Stopping condition

  • Input and output variable choices

  • Again update tableau.

  • Iteration should be continued until the optimal solution is gotten


Different Types of Linear Programming

The different types of linear programming are as follows:

  1. Solving linear programming by Simplex method

  2. Solving linear programming using R

  3. Solving linear programming by graphical method

  4. Solving linear programming with the use of an open solver.


Advantages and Uses of Linear Programming

The advantages of linear programming are as follows:

  1. Linear programming provides insights into business problems.

  2. It helps to solve multi-dimensional problems.

  3. According to change of the conditions, linear programming helps us in adjustments.

  4. By calculating the profit and cost of different things, Linear programming also helps to take the best solution.

Linear Programming is very much used in the field of Mathematics and some other fields like economics, business, telecommunication, and the manufacturing fields. 


Linear Programming Applications

A real-time example would be considering the limitations of labors and materials and finding the best production levels for maximum profit in particular circumstances. It is part of important areas of mathematics also known as the optimization of linear programming techniques. 


In order to find the most calculated solution to a problem with specified constraints, linear programming is utilised. We use linear programming to turn real-world issues into mathematical models. There's also an objective function, as well as linear inequalities and constraints.


The Applications of the Linear Programming in Some Other Fields are Given Below:

  • Engineering – It solves design and manufacturing problems as it is helpful for doing shape optimization.

  • Efficient Manufacturing – To maximize profit, companies use linear expressions.

  • Energy Industry – It provides methods to optimize the electric power system.

  • Transportation Optimisation – For cost and time efficiency.


What is Linear Programming in Business?

When a linear function is subjected to multiple constraints, we use linear programming to minimize or maximize it. This technique has also proven to be quite beneficial in directing quantitative judgments in various business planning, as well as in industrial engineering and, to a lesser extent, in the social and physical sciences.


How do You Write a Linear Programming Problem?

Understanding the problem is the first step in linear programming.

  • Give an explanation of the goal.

  • Define the decision variables.

  • Create the goal function.

  • Describe the limitations.

  • Include the limitations on nonnegativity.

  • Make it look nice.

When a linear function is subjected to multiple constraints, we use this technique to reduce or maximize it. This technique has also proven to be quite beneficial in directing quantitative judgments in various business planning, as well as in industrial engineering and, to a lesser extent, in the social and physical sciences.

FAQs on Linear Programming

1. What is Linear Programming Used for?

Linear programming is used in getting the most calculated solution for a problem with given constraints. In linear programming, we create our real-life problems into a mathematical model. It also involves an objective function, linear inequalities with the subject to constraints.

2. How Do You Define Linear Programming?

Linear programming is a process of optimizing the problems which are subjected to certain constraints. It also means that it is the process for maximising or minimizing the linear functions which comes under linear inequality constraints. The problem of solving linear programs is considered the easiest one.

3. What is Linear Programming in Business?

Linear programming is the technique where we minimize or maximize a linear function when they are subjected to various constraints. This process also has been very useful for guiding the quantitative decisions in different business planning, also in industrial engineering, and—to a lesser extent— also in the social and the physical sciences.

4. How Do You Write a Linear Programming Problem?

Steps to Linear Programming are:

  1. Understand the problem.

  2. Describe the objective. 

  3. Define the decision variables. 

  4. Write the objective function. 

  5. Describe the constraints. 

  6. Add the nonnegativity constraints. 

  7. Write it up pretty.

5. What are the characteristics of Linear Programming?

The five properties of the linear programming issue are as follows:

  • The goal function of a problem should be described quantitatively.

  • Decision Variables- The outcome will be determined by the decision variable. It provides the final answer to the problem. The first step in solving any problem is to identify the decision variables.

  • Linearity- The function must have a linear relationship between two or more variables. It signifies that the variable's degree is one.

  • Finiteness- Input and output numbers should be finite and limitless. The optimal solution is not possible if the function contains infinite factors.

  • Non-negativity- The value of the variable must be either positive or zero. It can't be a negative number.

  • Constraints- In terms of the resource, the constraints should be defined mathematically.

6. What are linear programming problems?

Linear Programming Problems (LPP) are problems in which the goal is to determine the best value for a given linear function. The best value can be either the highest or the lowest. The specified linear function is regarded as an objective function in this case. The objective function might have several variables that are subject to conditions, and it must meet a set of linear inequalities known as linear constraints. The linear programming challenges can be utilized to find the best answer for a variety of circumstances, including manufacturing issues, diet issues, transportation issues, and allocation issues, among others.

Comment