chapter 7 linear programming
DESCRIPTION
Chapter 7 Linear Programming. Ding-Zhu Du. I. Simplex Method. LP examples. - PowerPoint PPT PresentationTRANSCRIPT
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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.
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6032 yx
yxz 54
Feasible domain
Optimal occurs at a vertex!!!
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.0,9,0
6032 s.t.
54 min
wyx
wyx
yxzax
Slack Form
.)(rank
.0
s.t.
min
mA
x
bAx
cxzx
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What’s a vertex?
. ,),(2
1
if
vertexa called is polyhadren ain point A
zyxzyzyx
x
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. of sin vertice found becan it then
solution, optimalan has over min If
.}0|{Let
xcx
Ax = b, xx =
Fundamental Theorem
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.constraint oneleast at violates' is, that ,in not
'point a havemust line theThus, line.any contain not
does However, solutions. optimal are *)( line
on points feasible all that followsIt solutions. optimal
also are and that means This .
havemust we2,)( and ,
,* Since distinct. are ,*, and 2/)(*
such that ,exist thereis, that not, is * suppose
on,contraditiBy . of vertex a is * that show willWe
solutions. optimal all among components zero of
number maximum with *solution optimalan Consider
x
x
y-xx*+
zy czcx* = cy =
/cy+czcx* = czcx*
cycxzyxzyx
zyx
x
x
Proof.
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ion.contradict a
,*than component -zero more one has which ,* and
'between solution optimalan findeasily can weNow,
.0 with somefor 0 constraint a violate
must ' Hence, 0. constraint latecannot vio '
that means This .any for 0 toequal is *)( of
component th theTherefore, .0* havemust
we,0 and 2/)( since ,0for
Moreover, . constraint latecannot vio ' Thus,
.*))(( ,any for that Note
xx
x
*>xjx
xxx
y-xx*+
i==x=yz
,zy+zy*=x* =x
Ax = bx
= by-xx*+A
jj
i
iii
iiiiii
Proof (cont’s).
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t.independenlinearly
are 0 with 1for all if
only and if vertex a is point feasible a
Then . ofcolumn th thedenote Let
jj
j
xnja
x
Aja
Characterization of Vertex
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Proof
.2
and ,
0, smallly sufficientfor and 0Then
.0 if 0
0 if
settingby )( Define zero. are
allnot and 0such that 0 with for
exists e then thert,independenlinearly not are 0for If
}.0|{set index by determineduniquely are
then t,independenlinearly are 0for If
0
zyxdxzdxy
d
x
xd
dd
axj
xa
xjx
xa
j
jjj
j
x jjjjj
jj
jj
jj
j
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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
1
bA
IA
II, ja=A
xjIx
I
I
jI
j
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Optimality Condition
condition.acy nondegenerunder
0
ifonly and if optimal is Moreover,
}.|{ where
0 and with solution feasible
basic a with associated is basis feasibleEach
1
1
IIII
III
AAcc
x
IjjI
xbAxx
I
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Sufficiency
.0at minimum the
reaches ,0 and 0 Since
)(1
11
11
I
IIIII
IIIIIIIIIII
IIIII
IIII
x
cxxAAcc
xAAccbAcxcxccx
xAAbAx
bxAxA
bAx
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Nondegeneracy Assumption
.0 , basis feasibleevery For 1 bAI I
Remark
solution. basic feasible oneexactly with associated is
basis feasibleevery condition,acy nondegenerUnder
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Necessary
.,...,2,1 allfor 0 1. Case
.' and )( Denote
optimal.not is 0
thatshow We.0' ,* somefor Assume
.' Denote
*
11
1
*
1
mia
bAbAAa
bA
x
xx
cIj
AAccc
ij
IIij
I
I
I
j
IIIII
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solution. optimal
an givenot do and 0 So,
. as togoes aluefunction v
object theHence, solution. feasible a is
1 and if '
* if
* and if 0
0,any for Then
.,...,2,1 allfor 0 1. Case
1
*
)(
*
bAxx
aIjab
jj
jjIj
x
mia
III
ijiji
j
ij
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.assumptionacy nondegenerby 0' since
''
aluefunction vobject ofsolution
feasible basic with basis feasible new a is 'Then
*}.{})'{\('Set
.1 with 'Let
. 0 |min
such that * Choose
.,...,2,1 somefor 0 2. Case
*
**
**
'*
***
*
*
i
Iji
ijI
ji
ijij
i
ji
i
ij
b
bca
bcbc
I
jjII
aIj
aa
b
a
b
i
mia
(pivoting)
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Simplex Method
method.
simplex called g,propromminlinear thesolve
tomethod a givesy necessarit of proof The
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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
xxxx
xxxx
xxxx
xxxz
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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
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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
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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
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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
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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
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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
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Puzzle 1
? basis feasible
1st or thesolution feasible basic1st thefind wedo How
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Artificial Feasible Basis
m
m
z
z
z
z
zx
bIzAx
zzz
2
1
21
where
0 ,0
s.t.
min
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Puzzle 2
LP? solve
tohow hold,not does assumptionacy nondegenerWhen
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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=
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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
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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 )'
' ... ,
'
' ,
'
'(such that
choose we, of choiceFor
'''
1
'
''
'
''
1'
''
'
ijij
in
ij
i
ij
i
ji
ni
ji
i
ji
i
aa
a
a
a
a
b
a
a
a
a
a
bi'
i'
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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
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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
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Puzzle 3Method?Simplex of timerunning theisWhat
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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
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Thanks, End