7/31/2019 Solving LP

1/12

Solving Systems of Linear EquationsK. Goodoory

7/31/2019 Solving LP

2/12

Introduction

The central mathematical problem of linearprogramming is to find the solution of a system oflinear equations which maximizes or minimizes a

given linear objective function

7/31/2019 Solving LP

3/12

Equivalent systems

Two systems of equations are said to be equivalent ifboth systems have the same solution set.

A solution to one system is automatically a solution

to the other system and vice versa The method of solving a system of equations is to get

an equivalent system which is easy to solve

By solving the simple system, we simultaneously get

the solutions to the original system

7/31/2019 Solving LP

4/12

7/31/2019 Solving LP

5/12

Basic and Nonbasic Variable

A variable x1 is said to be a basic variable in a givenequation if it appears with a unit coefficient in thatequation and zeros in all other equations

Those variables which are not basic are callednonbasic variables

By applying the elementary row operations, a givenvariable can be made a basic variable.

7/31/2019 Solving LP

6/12

7/31/2019 Solving LP

7/12

7/31/2019 Solving LP

8/12

7/31/2019 Solving LP

9/12

Principles of the Simplex Method

The simplex method as developed by G.B. Dantzig isan iterative procedure for solving linearprogramming problems expressed in standard form

The simplex method requires that the constraintequations be expressed as a canonical system fromwhich a basic feasible solution can be readilyobtained.

7/31/2019 Solving LP

10/12

7/31/2019 Solving LP

11/12

7/31/2019 Solving LP

12/12

Adjacent basic feasible solution

An adjacent basic feasible solution differs from thebasic feasible solution in exactly one basic variable.

In order to obtain an adjacent basic feasible solution,

the simplex method makes one of the basic variable anonbasic variable, and in its place a nonbasic variableis brought in as a basic variable