Wyndor Model: Optimality Test and Solving an Example, Lecture notes of Linear Programming

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Chapter 4: The Simplex Method

Cathal Heavey

September 26, 2011

Introduction

Background SimplexMethod

Background Simplex Method

Background SimplexMethod

Developed by George Dantzig in 1947

Remarkably efficient solution method

Section 4.1 introduces geometric interpretation

Sections 4.2-4.4 presents a method to solve standard LP problem (

form)

Sections 4.5-4.6 adapt method for other forms

Section 4.7 discusses postoptimality analysis

Section 4.8 discusses computer implementation issues

Finally section 4.9 briefly describes a new solution method interior-pointapproach

Background Simplex Method

Background SimplexMethod

Developed by George Dantzig in 1947

Remarkably efficient solution method

Section 4.1 introduces geometric interpretation

Sections 4.2-4.4 presents a method to solve standard LP problem (

form)

Sections 4.5-4.6 adapt method for other forms

Section 4.7 discusses postoptimality analysis

Section 4.8 discusses computer implementation issues

Finally section 4.9 briefly describes a new solution method interior-pointapproach

Background Simplex Method

Background SimplexMethod

Developed by George Dantzig in 1947

Remarkably efficient solution method

Section 4.1 introduces geometric interpretation

Sections 4.2-4.4 presents a method to solve standard LP problem (

form)

Sections 4.5-4.6 adapt method for other forms

Section 4.7 discusses postoptimality analysis

Section 4.8 discusses computer implementation issues

Finally section 4.9 briefly describes a new solution method interior-pointapproach

Background Simplex Method

Background SimplexMethod

Developed by George Dantzig in 1947

Remarkably efficient solution method

Section 4.1 introduces geometric interpretation

Sections 4.2-4.4 presents a method to solve standard LP problem (

form)

Sections 4.5-4.6 adapt method for other forms

Section 4.7 discusses postoptimality analysis

Section 4.8 discusses computer implementation issues

Finally section 4.9 briefly describes a new solution method interior-pointapproach

Background Simplex Method

Background SimplexMethod

Developed by George Dantzig in 1947

Remarkably efficient solution method

Section 4.1 introduces geometric interpretation

Sections 4.2-4.4 presents a method to solve standard LP problem (

form)

Sections 4.5-4.6 adapt method for other forms

Section 4.7 discusses postoptimality analysis

Section 4.8 discusses computer implementation issues

Finally section 4.9 briefly describes a new solution method interior-pointapproach

Background Simplex Method

Background SimplexMethod

Developed by George Dantzig in 1947

Remarkably efficient solution method

Section 4.1 introduces geometric interpretation

Sections 4.2-4.4 presents a method to solve standard LP problem (

form)

Sections 4.5-4.6 adapt method for other forms

Section 4.7 discusses postoptimality analysis

Section 4.8 discusses computer implementation issues

Finally section 4.9 briefly describes a new solution method interior-pointapproach

Corner Point Feasible Solutions (CPF))

Corner Point FeasibleSolutions (CPF)) Wyndor ModelOptimality testSolving the ExampleKey Concepts — 1–2Key Concepts — 3–4Setting Up The SimplexMethodAugmented (Standard)ModelSome DefinitionsBasic SolutionBasic Solution ExampleObjective FunctionEquation

For any Linear programming problem with

n

decision variables,

two CPF solutions are

adjacent

to each other if they share

n

constraint boundaries. The two adjacent CPF solutions areconnected by a line segment that lies on these same sharedconstraint boundaries. Such a line segment is referred to as an edge

of the feasible region.

Wyndor Model

Corner Point FeasibleSolutions (CPF)) Wyndor Model Optimality testSolving the ExampleKey Concepts — 1–2Key Concepts — 3–4Setting Up The SimplexMethodAugmented (Standard)ModelSome DefinitionsBasic SolutionBasic Solution ExampleObjective FunctionEquation

Wyndor Glass

CPF

Adjacent CPF Solutions

(0, 6) and (4,0)

(2, 6) and (0,0)

(4,3) and (0, 6)

(4, 0) and (2,6)

(0,0) and (4, 3)

Optimality test

Corner Point FeasibleSolutions (CPF))Wyndor Model Optimality test Solving the ExampleKey Concepts — 1–2Key Concepts — 3–4Setting Up The SimplexMethodAugmented (Standard)ModelSome DefinitionsBasic SolutionBasic Solution ExampleObjective FunctionEquation

Optimality Test:

Consider any linear programming problem that processes at

least one optimal solution. If a CPF solution has no

adjacent

CPF solutions

that are

better

(as measured by

Z

), then it must be an

optima

l solution.

(2, 6) must be optimal with

Z

as (0, 6) gives

Z

and (4, 3) gives

Z

Solving the Example

Corner Point FeasibleSolutions (CPF))Wyndor ModelOptimality test Solving the Example Key Concepts — 1–2Key Concepts — 3–4Setting Up The SimplexMethodAugmented (Standard)ModelSome DefinitionsBasic SolutionBasic Solution ExampleObjective FunctionEquation

Key Concepts — 1–

Corner Point FeasibleSolutions (CPF))Wyndor ModelOptimality testSolving the Example Key Concepts — 1–2 Key Concepts — 3–4Setting Up The SimplexMethodAugmented (Standard)ModelSome DefinitionsBasic SolutionBasic Solution ExampleObjective FunctionEquation

Solution Concept 1:

For any problem with at least one optimal solution,

optimal equal best CPF.

Solution Concept 2:

Simplex method is iterative

Initialization:

Key Concepts — 1–

Corner Point FeasibleSolutions (CPF))Wyndor ModelOptimality testSolving the Example Key Concepts — 1–2 Key Concepts — 3–4Setting Up The SimplexMethodAugmented (Standard)ModelSome DefinitionsBasic SolutionBasic Solution ExampleObjective FunctionEquation

Solution Concept 1:

For any problem with at least one optimal solution,

optimal equal best CPF.

Solution Concept 2:

Simplex method is iterative

Initialization:

Optimality Test: