Chapter  12

Linear Programming Problems

Introduction, related terminology such as constraints, objective function, optimization, different types of linear programming (L.P.) problems, mathematical formulation of L.P. problems, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints), described by amar sir. Furthermore, it helps the student while facing other exams in the future.

Introduction

This section the discussion of linear equations, linear inequalities, applications of linear inequalities in previous grades. It introduces the concept of optimisation problems and a special case of optimisation problem called linear programming problem, using an example and described by amarsir. An ideal example of optimisation would be maximising the profit and minimising the cost of a production unit. 

Linear Programming Problem and its Mathematical Formulation

Formulation of an LPP refers to translating the real-world problem into the form of mathematical equations which could be solved. It usually requires a thorough understanding of the problem.

Mathematical formulation of the problem

In formulating a problem for linear Programming problem .study analysis must be made of the following major components: 

  1. The environment
  2. The objectives
  3. The decision maker
  4. The alternative courses of action and constraints.

It defines the non-negative constraints, objective function, decision variables.

A linear programming problem is finding the optimal value [maximum or minimum] of a linear function of variables, which are subjected to certain conditions and satisfying a set of linear constraints.

The variables involved in the objective function are called decision variables.

The constraints are the restrictions on the variables.

Important Definitions and Results:

Linear Programming

It is an important optimization (maximization or minimization) technique used in decision making is business and everyday life for obtaining the maximum or minimum values as required of a linear expression to satisfying certain number of given linear restrictions.

Linear Programming Problem (LPP)

The linear programming problem in general calls for optimizing a linear function of variables called the objective function subject to a set of linear equations and/or linear inequations called the constraints or restrictions.