Linear Programming

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What is Linear Programming, Explain with Examples?

In Mathematics, the linear programming method is for optimizing the operations with some constraints. The main point of linear programming is to minimize or maximize the numerical value. It also has the linear functions that are subjected to the constraints in the form of linear equations or in the form of linear inequalities.

Linear programming is considered an important technique which is used to find the optimum resource utilization. The terminology “linear programming” has two words “linear” and “programming”. The word “linear” says about the relationship between multiple variables with degree one. The word “programming” says about the process of selecting the best solution from different alternatives.

What is Linear Programming?

Let’s define what is linear programming and some linear programming problems. It can be defined as an optimization of the linear programming technique for a system of linear constraints and a linear objective function. An objective function defines the number to be optimized, and therefore the goal of applied mathematics is to seek out the values of the variables that maximize or minimize the target function. Linear Programming is also known as Linear Optimization.

Linear programming method is also for the consideration of the different inequalities relevant to a particular situation and calculating the best value that is required to be obtained in those conditions. We now know what a linear programming problem is. Let's discuss the importance.

Some of the idea taken while working with applied Mathematics are:

  • The total number of constraints should be written in quantitative terms

  • The relationship between the constraints and therefore the objective function should be linear

  • The linear function (i.e., objective function) is to be optimized.

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 also can be used to solve a problem when the aim of the problem is to maximize some of the value and there is a linear system of inequalities that express the constraints on the problem.

Components of Linear Programming

The normal components of the 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 the 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:

  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:

  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. 

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.

FAQ (Frequently Asked Questions)

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.