THE STEPS OF THE SIMPLEX ALGORITHM

1 Introduction Algorithm2e is an environment for writing algorithms in LATEX2e An algorithm is de ned as a oating object like gures It provides macros that allow you to create di erent sorts of key words,

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THE STEPS OF THE SIMPLEX ALGORITHM - HEC Montréal

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Ecrire un algorithme qui permet à l’utilisateur de saisir une suite caractère se terminant par ‘*’, et qui affiche à la fin le nombre d’apparition de la lettre ‘A’ Solution 1 : en utilisant une boucle Répéter Algorithme Appatition ;

THESTEPSOFTHESIMPLEXALGORITHM

Page2of8

1. Introduction

2. Slackandsurplusvariables

mustbewritteninstandardform. a. Constraintsoftype(൑):foreachconstraint݅ofthistype,weaddaslack variable݁ ,suchthat݁ isnonnegative.

Example

൑ʹtranslatesinto͵ݔ b. Constraintsoftype(൒):foreachconstraint݅ofthistype,weaddasurplus variable݁ ,suchthat݁ isnonnegative.

Example

൒ʹtranslatesinto͵ݔ abbreviatedas(PL=)resp.(PG=).

3. BasicandnonǦbasicvariables

a) Set݊െ݉variablesequaltozero.ThesevariablesarecallednonͲbasicvariables (N.B.V). b) Solvethesystemforthe݉remainingvariables.These variablesarecalledbasic variables(B.V.) c) Thevectorofvariablesobtainediscalledthebasicsolution(itcontainsboth basicandnonͲbasicvariables).

Abasicsolutionisadmissible

Page3of8

4. Admissiblesolutions

point(cornersolution).

5. Solutionofalinearprogram(LP)

(LP)

Ex:ܽܯݔܼ

(LP)

Ex:ܽܯݔܼ

NonͲbasicvariables

݂݅x

1 ൌx 2 thenBasicvariables:

Page4of8

StepA:initialtable

Coef.inZ100012000000

BaseX 1 X 2 E 1 E 2 E 3 E 4 b i

Coef.ZBasicVar.

0E 1

1051000200

0E 2

23010060

0E 3

10001034

0E 4

01000114

z j

0000000

C j -z j

100012000000

Thegreenframecorrespondstoݖ

:thecoefficientsinൈܽ

Exampleforthecolumnofܺ

called(ܽ )ofthevariablesinthe objectivefunction(ܼ function,calculatedasfollows: StepB

Maximumoftheܥ

formaximumproblems.

Minimumoftheܥ

fortheminimumproblems.

Inourexample:ݔ

hasthegreatestܥ ;henceitentersinthesetofbasic variables.

Page5of8

StepC :selectionoftheleavingvariable

InaproblemofeitherminOR

max,theleavingvariableistheminimumof Pr

Inourexample,weneedtoevaluate:

Enteringvariable

Coef.inZ100012000000

BaseX 1 X 2 E 1 E 2 E 3 E 4 b i

Coef.ZBasic

Var. 0E 1

1051000200

0E 2

23010060

0E 3

10001034

0E 4

01000114

z j

0000000

C j -z j

100012000000

200/5=40

60/3=20

14/1=14՜istheminimum,hence݁

isthevariablethatleavesthesetofbasic variables. StepD :pivot

Coef.inZ100012000000

BaseX 1 X 2 E 1 E 2 E 3 E 4 b i

Coef.ZBasic

var. 0E 1

1051000200

0E 2

23010060

0E 3

10001034

0E 4

01000114

zquotesdbs_dbs7.pdfusesText_5

[PDF] THE STEPS OF THE SIMPLEX ALGORITHM - HEC Montréal

THESTEPSOFTHESIMPLEXALGORITHM

Contents

Page2of8

1. Introduction

2. Slackandsurplusvariables

Example

Example

3. BasicandnonǦbasicvariables

Abasicsolutionisadmissible

Page3of8

4. Admissiblesolutions

5. Solutionofalinearprogram(LP)

Ex:ܽܯݔܼ

Ex:ܽܯݔܼ

NonͲbasicvariables

݂݅x

Page4of8

StepA:initialtable

Coef.inZ100012000000

Coef.ZBasicVar.

1051000200

23010060

10001034

01000114

0000000

100012000000

Thegreenframecorrespondstoݖ

Exampleforthecolumnofܺ

Maximumoftheܥ

Minimumoftheܥ

Inourexample:ݔ

Page5of8

InaproblemofeitherminOR

Inourexample,weneedtoevaluate:

Enteringvariable

Coef.inZ100012000000

Coef.ZBasic

1051000200

23010060

10001034

01000114

0000000

100012000000

200/5=40

60/3=20

14/1=14՜istheminimum,hence݁

Coef.inZ100012000000

Coef.ZBasic

1051000200

23010060

10001034

01000114