chapter 7 linear programming

Chapter 7Linear Programming

I. Simplex Method

Ding-Zhu Du

LP examples

• A post office requires different numbers of full-time employees on different days of the week. The number of full-time employees required on each day is given in the table. Union rules state that each full-time employee must work five consecutive days and then receive two days off. The post office wants to meet its daily requirements using only full-time employees. Formulate an LP that the post office can use to minimize the number of full-time employees that must be hired.

6032 yx

yxz 54

Feasible domain

Optimal occurs at a vertex!!!

.0,9,0

6032 s.t.

54 min

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yxzax

Slack Form

.)(rank

.0

s.t.

min

mA

x

bAx

cxzx

What’s a vertex?

. ,),(2

1

if

vertexa called is polyhadren ain point A

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x

. of sin vertice found becan it then

solution, optimalan has over min If

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xcx

Ax = b, xx =

Fundamental Theorem

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does However, solutions. optimal are *)( line

on points feasible all that followsIt solutions. optimal

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havemust we2,)( and ,

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solutions. optimal all among components zero of

number maximum with *solution optimalan Consider

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x

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x

x

Proof.

ion.contradict a

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Proof (cont’s).

t.independenlinearly

are 0 with 1for all if

only and if vertex a is point feasible a

Then . ofcolumn th thedenote Let

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Characterization of Vertex

Proof

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and ,

0, smallly sufficientfor and 0Then

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0 if

settingby )( Define zero. are

allnot and 0such that 0 with for

exists e then thert,independenlinearly not are 0for If

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then t,independenlinearly are 0for If

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Basic Feasible Solution

0. and

|| = m = )(rank ifonly and if basis feasible

a is subset index an Then .)( Denote

0}.|{such that solution

feasible basic a exists thereif feasible is basisA

basis. a called torscolumn vec of

subsett independen maximum a ofset index The

solution. feasible basic a called also isA vertex

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Optimality Condition

condition.acy nondegenerunder

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ifonly and if optimal is Moreover,

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basic a with associated is basis feasibleEach

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III

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x

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I

Sufficiency

.0at minimum the

reaches ,0 and 0 Since

)(1

11

11

I

IIIII

IIIIIIIIIII

IIIII

IIII

x

cxxAAcc

xAAccbAcxcxccx

xAAbAx

bxAxA

bAx

Nondegeneracy Assumption

.0 , basis feasibleevery For 1 bAI I

Remark

solution. basic feasible oneexactly with associated is

basis feasibleevery condition,acy nondegenerUnder

Necessary

.,...,2,1 allfor 0 1. Case

.' and )( Denote

optimal.not is 0

thatshow We.0' ,* somefor Assume

.' Denote

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1

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mia

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solution. optimal

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. as togoes aluefunction v

object theHence, solution. feasible a is

1 and if '

* if

* and if 0

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1

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aluefunction vobject ofsolution

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such that * Choose

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(pivoting)

Simplex Method

method.

simplex called g,propromminlinear thesolve

tomethod a givesy necessarit of proof The

Simplex Table

1 0 0 2 1 4 36

0 1 0 5 2 2 24

0 0 1 3 1 1 30

0 0 0 2 1 3

0,,,,,

3624

24522

303 s.t.

23 min

654321

6321

5321

4321

321

z

xxxxxx

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1 2 1 4 36

1 5 2 2 24

1 3 1 1 30

2 1 3

}6,5,4{ 0

z

I

1 2 1 4 36

1/2 5/2 1 1 12

1 3 1 1 30

2 1 3

}2{})5{\}6,5,4({ 1

z

I

1 1/2 21/ 0 3 24

1/2 5/2 1 1 12

2/1 1 1/2 0 0 18

1/2 1/2 0 2 12

}2{})5{\}6,5,4({ 1

z

I

1 1/2 21/ 0 3 24

1/2 5/2 1 1 12

2/1 1 1/2 0 0 18

1/2 1/2 0 2 12

}6,2,4{ 1

z

I

1/3 1/6 61/ 0 1 8

1/2 5/2 1 1 12

2/1 1 1/2 0 0 18

1/2 1/2 0 2 12

}1{})6{\}6,2,4({ 2

z

I

1/3 1/6 61/ 0 1 8

3/1 2/3 8/3 1 0 4

2/1 1 1/2 0 0 18

2/3 1/6 1/6 0 0 28

}1,2,4{ 2

z

I

Puzzle 1

? basis feasible

1st or thesolution feasible basic1st thefind wedo How

Artificial Feasible Basis

m

m

z

z

z

z

zx

bIzAx

zzz

2

1

21

where

0 ,0

s.t.

min

Puzzle 2

LP? solve

tohow hold,not does assumptionacy nondegenerWhen

lexicographical ordering

.0 if positivehocally lexicograp is

A vector .1 somefor ...

if , as written ,n larger tha

hicallylexicograp be tosaid is .)...(

and )...,( vectorswoConsider t

1111

21

21

L

iiii

L

n

n

x >x

n i y, x=y, x, =yx

yx >y

x,y,,yyy=

x,,xxx=

Method

positive.hically lexicograp is row topthe

except blesimplex ta initial in the rowevery that makes This

columns. first at the placed is basis feasible initial the

such that columns of ordering therearrange Initially,

m

n

Method(cont’)

pivoting.

under preserved be willrow top theother than rows all of

spositveneshically lexicograp that theguarantees choice This

.0'for )'

' ... ,

'

' ,

'

'( among one

smallesthocally lexicograp theis )'

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'

' ,

'

'(such that

choose we, of choiceFor

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a

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b

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a

a

bi'

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Method (cont’)

.0

such that basis feasibleith solution w basic feasible optimalan

findor solution optimal of cenonexisten findseither algorithm theTherefore,

ing.nonincreas is aluefunction v objective theand oncemost at basis feasible

each visit algorithm that theguarantee rules additional above theTherefore,

ordering. hicallexicograpin strictly decreasse top themake pivot will

each positive,hically lexicograp are row top theother than rows all Since

1 AAc - c

I

II

Theorem

optimal. is with associatedsolution feasible

basic then the0, satisfies basis feasible a if Moreove,

0. such that basis

feasible with associatedsolution feasible basic optimalan hasit then

solution, optimalan has gprogramminlinear theIf

1

1

I

AAc - cI

AAc - cI

II

II

Puzzle 3Method?Simplex of timerunning theisWhat

Complexity of LP

time.-polynomialin computed becn

point extreme optimalan solution, optimalan Given

time.)(in runs MethodPoint Interior

time.)(in runs Method Ellepsoid

time.lexponentiain runs MethodSimplex

5.3

6

nO

nO

Thanks, End