Explain Linear Programming Problem - Introduction & Definition of Linear Programming Problem (LPP):
In this article, we explain Linear Programming Problem (LPP) and learn about introduction and definition of Linear Programming Problem (LPP).Introduction of Linear Programming Problem (LPP)
In linear programming, we formulate our real-life problem into a mathematical model. It involves an objective function, linear inequalities with subject to constraints (s.t.).
The technique of linear programming was developed by Russian mathematician L.V. Kantorovich. In short form, a Linear programming problem is also denoted as LPP.
Definition of Linear Programming Problem
Linear programming or linear optimization is a technique for determining an optimal (maximum or minimum) solution of limited resources. The resources may be in the form of men, materials and machines etc.
According to the famous economist Robbins, resources (land, labour, capital, materials, machines, etc.) are always limited. But each resource has different alternative uses. The problem for any person is to select only those alternatives which can maximize profit or minimize the cost of production. So, Linear programming techniques are used to select the best possible strategy from a number of alternatives.
Linear programming is often used in business to find maximum profit or minimum cost.
A linear programming problem (LPP) includes a set of simultaneous linear equations which represent the conditions of the problem and a linear function which expresses the objective function of the problem.
Hence, linear programming is used for obtaining the most optimal solution for a problem with given constraints.
In other words,
Hence, it is an optimization method applicable for the solution of optimization problems where the objective function and the constraints are linear.
That is, "linear programming is a mathematical technique for allocating limited resources in an optimum manner."
Some Definitions in Linear Programming Problem (LPP):
What is Optimization:
The technique of achieving the best solutions under the given circumstances is called Optimization.
What is Programming:
It is the mathematical technique to determine the optimum use of the limited available resources.
What is Linear Programming:
It is a decision-making technique under the given constraints under the condition that the relationship among the variables involved is linear.
Meaning of Linear Programming Problem (LPP):
There are two words in Linear Programming: Linear and Programming.
Linear - The word “linear” means the relationship which can be represented by a straight line. In other words, it is used to describe the relationship between two or more variables that are proportional to each other.
Programming - The word “programming” is concerned with the optimal allocation of limited resources. Hence, linear programming deals with that class of programming problems for which all relations among the variables are linear.
Hence, Linear Programming is the technique of making decisions under the given constraints under the condition that the relationship among the variables involved is linear. That is, Linear programming is a mathematical technique to allocate limited resources in an optimal way in which the objective function is also linear and the conditions (identities/inequalities) are also linear. The result of linear programming is to maximize profit and minimize cost.
Aims and Objectives of Linear Programming Problem (LPP):
1. The main aim of using LPP is to provide an information base for the most efficient allocation of scarce resources. That is, the important reason for using LP is to optimise the allocation of some resources. For example,
To choose an investment portfolio from a variety of bond and stock investment alternatives that maximises the return on investment.
To allot available machines, time and labour hours in each department along with raw materials, for the activities of production of various products which have been scheduled, and to determine the number of units of each product to maximise profit.
To identify which warehouse should deliver how much product to which market in order to reduce total transportation expenses.
2. The objectives of using LPP is to improve logical thinking and provide better guidance for future challenges.
3. The purpose of using LPP by a manager is to select the optimal solution by evaluating the costs and benefits of various possibilities and alternatives.
4. The aim of using LPP is to help in making adjustments according to changing conditions.
5. The goal of using LPP is to help in the solving of multi-dimensional problems.
6. The objective function in LP is a linear equation that represents some quantity (such as profits earned, expense, etc.) that must be maximised or minimised given the restrictions. For an objective function, the objective of linear programming is to maximise or minimise.
7. For multiple reasons, linear programming is widely used in the field of optimization. Many operational research functional problems can be represented as linear programming problems. Some particular problems of linear programming, such as network flow questions and multi-commodity flow questions, are considered important enough to warrant extensive study on functional algorithms to solve them.
(Source – Various books of college library)
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