Introduction Big - M Method: Examples Two phase Method: Examples

OptimizationBig-M and Two-Phase Methods for LPP

DeepikaBITS Pilani, Hyderabad Campusdeepika@hyderabad.bits-pilani.ac.in

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October 16, 2021

Introduction Big - M Method: Examples Two phase Method: Examples

Outline

Introduction

Big - M Method: Examples

Two phase Method: Examples

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Introduction Big - M Method: Examples Two phase Method: Examples

Introduction: LPP

To solve a linear programming problem, we present generalised versionsof the simplex method:

• Big - M method

• Two phase method

These methods can be used to solve both maximisation and minimisationproblems with ≥, ≤ and = type constraints

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Introduction Big - M Method: Examples Two phase Method: Examples

Big - M Method for the Solution of LPP

• In many LP problems, often we end up with cases where the initialbasic feasible solution (BFS) does not exist.

• In such a case, we can employ the big - M method

• Basic Idea: Introducing artificial variables so that we get initial basicfeasible solution (BFS)

• To prevent these variables from being part of the optimal solution,we penalise them in the objective function

• Then apply the usual simplex method to obtain the optimal solution

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Introduction Big - M Method: Examples Two phase Method: Examples

Examples: Big - M Method for the Solution of LPP

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Examples: Big - M Method for the Solution of LPP

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Comments on Big - M Method

Relationship between optimal solutions of the modified problem and theoriginal problem:

• If the modified problem has no optimal solution, then the originalproblem has no optimal solution

• if any of the artificial variables are non-zero in the optimal solutionof the modified problem, then the original problem has no optimalsolution

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Introduction Big - M Method: Examples Two phase Method: Examples

Two Phase method for the Solution of LPP

• In many LP problems, often we end up with cases where the initialbasic feasible solution (BFS) does not exist.

• In such a case, we can employ the two phase method

• In the first phase (phase - I), we formulate a different LPP and findan initial BFS.

• In phase - II, we use the initial BFS to find the optimal solution ofthe original LPP

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Examples: Two phase Method

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