Linear Programming

BASIC CONCEPTS OF LINEAR PROGRAMMING:

 In Mathematics, linear programming is a method of optimising operations with some constraints. The main objective of linear programming is to maximize or minimize the numerical value. It consists of linear functions which are subjected to the constraints in the form of linear equations or in the form of inequalities.

Linear programming is considered an important technique that is used to find the optimum resource utilisation. The term “linear programming” consists of two words as linear and programming. The word “linear” defines the relationship between multiple variables with degree one. The word “programming” defines the process of selecting the best solution from various alternatives.

Linear Programming is widely used in Mathematics and some other field such as economics, business, telecommunication, and manufacturing fields. In this article, let us discuss the definition of linear programming, its components, a simplex method with linear programming problems.

What is Linear Programming?

Linear programming (LP)  or Linear Optimisation may be defined as the problem of maximizing or minimizing a linear function that is subjected to linear constraints. The constraints may be equalities or inequalities. The optimisation problems involve the calculation of profit and loss.  Linear programming problems are an important class of optimisation problems, that helps to find the feasible region and optimise the solution in order to have the highest or lowest value of the function.

Linear programming is the method of considering different inequalities relevant to a situation and calculating the best value that is required to be obtained in those conditions.

Some of the assumption taken while working with linear programming are:

• The number of constraints should be expressed in the quantitative terms
• The relationship between the constraints and the objective function should be linear
• The linear function (i.e., objective function) is to be optimised

Components of Linear Programming

The basic components of the LP are as follows:

• Decision Variables
• Constraints
• Data
• Objective Functions

Linear Programming Problems

The Linear Programming Problems (LPP) is a problem that is concerned with finding the optimal value of the given linear function. The optimal value can be either maximum value or minimum value. Here, the given linear function is considered an objective function. The objective function can contain several variables, which are subjected to the conditions and it has to satisfy the set of linear inequalities called linear constraints. In Linear programming, the term “linear” represents the mathematical relationship that is used in the given problem (Generally, linear relationship) and the term “programming” represents the method of determining the particular plan of action. The linear programming problems can be used to get the optimal solution for the following scenarios, such as manufacturing problems, diet problems, transportation problems, allocation problems and so on.

Characteristics of Linear Programming

The following are the five characteristics of the linear programming problem:

Constraints – The limitations should be expressed in the mathematical form, regarding the resource.

Objective Function – In a problem, the objective function should be specified in a quantitative way.

Linearity – The relationship between two or more variables in the function must be linear. It means that the degree of the variable is one.

Finiteness –  There should be finite and infinite input and output numbers. In case, if the function has infinite factors, the optimal solution is not feasible. 

Non-negativity – The variable value should be positive or zero. It should not be a negative value.

Decision Variables – The decision variable will decide the output. It gives the ultimate solution of the problem. For any problem, the first step is to identify the decision variables.