econ lecture

8/10/2019 Econ lecture

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Contents college 3 en 4

Book: Appendix A.1, A.3, A.4, 3.4, 3.5, 4.1, 4.2, 4.4, 4.6 (not: 3.6 -

3.8, 4.2 - 4.3

!xt"# lite"#t$"e on "eso$"ce const"#ined

p"o%ect sc&ed$ling ('ill e nded o$t


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)l#nning #nd sc&ed$lingopti*i+#tion tec&ni$es

isp#tc&ing $les

Co*posite isp#tc&ing $les

Ad#pti/e se#"c& 0n#*ic )"og"#**ing

(ntege" ine#" )"og"#**ing

C$tting pl#ne *et&ods

B"#nc& #nd Bo$nd

Be#* e#"c&


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ine#" p"og"#**ing ()*odel

):

#t"ix o"*:

n,,1%(x

x#...x#x#

x#...x#x#

x#...x#x#

:tos$%ect

xc...xcxc*in

%

*n*n22*11*

2nn2222121

1nn1212111

nn2211

=

+++

+++

+++

+++

6x

Ax

xc*in 7

o%ecti/e $nction

const"#ints

/#"i#le

"est"ictions

'&e"e:x, c: n-/ecto"A: *,n-*#t"ix: *-/ecto"


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ine#" p"og"#**ing

ex#*ple

2,1%(6x

12x4x3

4xx2:tos$%ect

xx*#x

%

21

21

21

=

+

+

+

x

x

12

4

x

x

43

12

:tos$%ectx

x

1

1*#x

2

1

2

1

2

1

7

o":

i i


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ine#" p"og"#**ingex#*ple: g"#p&ic#l

sol$tion (2

1 2 3 4 5 6

x1

1

2

3

4

5

6x2

o%ecti/e(

6xx 21 =+

4xx2 21 +

12x4x3 21 +

2,1%(6x

12x4x3

4xx2

:tos$%ect

xx*#x

%

21

21

21

=

+

+

+

sol$tion

sp#ce


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ine#" p"og"#**ing(cont.

ol$tion tec&ni$es: (d$#l si*plex *et&od

inte"io" point *et&ods (e.g. 9#"*#"k#"#lgo"it&*

Co**e"ci#l sol/e"s, o" ex#*ple:

C)! (;


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ntege" p"og"#**ing ()

*odels ntege" /#"i#le "est"iction

): intege" /#"i#les onl0

): p#"t intege", p#"t non-intege" /#"i#les

B): in#"0 (-1 /#"i#les


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ntege" p"og"#**ingex#*ple: g"#p&ic#l sol$tion

(2

1 2 3 4 5 6

x1

1

2

3

4

5

6x2

o%ecti/e(

6xx 21 =+

2,1%(=x

12x4x3

4xx2:tos$%ect

xx*#x

%

21

21

21

=

+

+

+

+

2 opti*#l sol$tions>


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o" intege" p"og"#**ing

*odels$ppose tt #ll coe?cients #"e intege" in t&e *odel:

i.e.

!x#*ple: t"#nspo"t#tion p"ole*

i A s t&e tot#l $ni*od$l#"it0 p"ope"t0

(i.e. e/e"0 s$#"e s$*#t"ix s dete"*in#nt ,1,-1

t&e"e is #n opti*#l intege" sol$tion x@

t&e si*plex *et&od 'ill nd s$c& # sol$tion

6x

Ax

xc*in 7

%,i,,# ii%


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ntege" p"og"#**ingt"icks

);B!: x o" x k

$se in#"0 indic#to" /#"i#le 0

"est"ictions:

k5o" x,

65o" x,

1

6

=

{ }6,10

0kxxon$ppe"o$nd#nis(0x


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);B!: xed costs: i xiD t&en costs C(xi

$se indic#to" /#"i#le 0i

"est"ictions :

(2

6x

Ax

x(C*ini*i+e

=

6.o" x

6,o" x

xck

6x(C

:'&e"e

i

i

iii

i

>

=

+

=

o" x,

o" x,

1

i

i

>

=

{ }6,10

xc0kx(C

0x

i

iiiii

ii

+=

i(


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E#"d /s. sot "est"ictions

"d "est"iction: *$st &old, ot&e"'ise$ne#siilit0

o" ex#*ple:

sot "est"iction: *#0 e /iol#ted, 'it& #pen#lt0

o" ex#*ple:

(3

6F,6x

F5xx

166Fxc*ini*i+e

21

7

+++

5xx 21 +


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Asol$te /#l$es:

sol$tion:

(4

+

+

+=

=

ttt

ttt

000

000

"ee0,6x

0x#

0*in

tt,%

ttt,%%

%

% t

+=

( )

60,60,6x

00x#

00*in

ttt,%

tttt,%%

%

t

tt

+=

+

+

+

+

go#l

/#"i#tion


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Con%$ncti/eGdis%$ncti/e p"og"#**ing

- con%$ncti/e set o const"#ints: *$st #ll e s#tised

- dis%$ncti/e set o const"#ints: #t le#st one *$st es#tised

ex#*ple (Appendix A.4:

(5

%k%

k%k

%%%

pxx

o"

pxx

x'*in

H1,6I0

01(pxx

0pxx

x'*in

2%k%

1k%k

%%%


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) ex#*plenonp"ee*pti/e single *#c&ine, tot#l

'eig&ted co*pletion ti*e (App. A.3

5$nction(o%ecti/ext'i*i+e*in

%%oo5ti*eco*pletionxt

%%oo5ti*eco*pletionxt

n

1%

1-C*#x

6t

%t%

1-C*#x

6t%t

%t

= =

=

=

== 1xif jt

ot&e"'is6#ndt,ti*e#t%%oi5,1x%t completes=

o%ecti/e $nction: *ini*i+e 'eig&ted co*pletion ti*e:

*odel denition:


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) ex#*ple (cont.

est"iction: #ll %os *$st e co*pleted once:

1x1-C*#x

6t%t =

=

est"iction: onl0 one %o pe" ti*e t:

tti*epe"%ooneex#ctl0:on("est"icti1x

td$"ingp"ocessinis%%o(i51x

n

1%

pt

ts%s

pt

ts%s

%

%

=

=

=

+

=

+

=

i %o % is in p"ocess d$"ing t, it *$st eco*pleted so*e'&e"e d$"ing Jt,tKp%L


17/35


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) ex#*ple (cont.Addition#l "est"iction: p"ecedence const"#ints

odel denition: MCC(% s$ccesso"s o %o %

%o % *$st e co*pleted eo"e #ll %os in MCC(%:

n,1,%MCC(%,k(o"txtx

k%ooti*est#"tptx

%%ooti*eco*pletiontx

1*#xC

6t

kt

1*#xC

6t

%t

k

1*#xC

6tkt

1*#xC

6t%t

=

=

=

=

=

=

=


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ntege" p"og"#**ingsol$tion tec&ni$es

Ee$"istic /s. explicit #pp"o#c&: t"#de-oN et'een sol$tion $#lit0 #nd co*p$t#tion ti*e

t"#de-oN et'een i*ple*ent#tion eNo"tGcosts #nd 0ield(i.e. p"ots g#ined "o* sol$tion $#lit0 i*p"o/e*ent

Ee$"istic *et&odsO o" ex#*ple: loc#l se#"c& (e.g. si*$l#ted #nne#ling, t#$ se#"c&, k-opt

(co*posite disp#tc&ing "$les (e.g. !, )7, #d#pti/e se#"c&

"o$nding "#ction#l sol$tions

e#* se#"c&


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!xplicit *et&odsO 3 c#tego"ies:

1. d0n#*ic p"og"#**ing

2. c$tting pl#ne (pol0&ed"#l *et&ods3. "#nc& #nd o$nd

o": &0"id *et&ods (co*in#tion o t&e#o/e

Co**e"ci#l ) sol/e"s $s$#ll0 $se #co*in#tion o &e$"istics #nd 2, 3

ntege" p"og"#**ingsol$tion tec&ni$es (cont.


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0n#*ic p"og"#**ing

)"ole* di/ided into st#ges

!#c& st#ge c#n /e /#"io$s st#tes

A "ec$"si/e o%ecti/e $nction is $sed toite"#te t&"o$g& #ll st#tes #nd #ll st#ges(o"'#"ds o" #ck'#"ds

Hx,i(i(Px,i(cI*ini(P tt1t1ttttx

ttt

+=

7,6,(txt =ti

(const#ntci(P 666 =


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C$tting pl#ne *et&ods

7!) : C"e#te # "el#x#tion o t&e p"ole*0

o*itting "est"ictions

(e.g. t&e integ"#lit0 "est"ictions7!) 1: ol/e t&e c$""ent p"ole*

7!) 2: sol$tion is ine#sile t&en gene"#te

# "est"iction tt c$ts o t&e sol$tion,#nd #dd it to t&e p"ole* 7!) 1

;t&e"'ise: ;Q!


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B"#nc& #nd o$nd

!n$*e"#tion in # se#"c& t"ee

e#c& node is # p#"ti#l sol$tion, i.e. # p#"t

o t&e sol$tion sp#ce

...

...

"oot node

c&ild nodes

c&ild nodes

e/el

e/el 1

e/el 2


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B"#nc& #nd o$ndex#*ple 1

is%$ncti/e p"og"#**ing (#ppendix A.4:dis%$ncti/e set o const"#ints: #t le#st one *$st es#tised

x% co*pletion ti*e o %o %

"est"iction:k,%(pxxo"pxxeit&e" %k%k%k

sol/e ) 'it&o$tdis%$ncti/e

"est"ictions( ) "el#x#tion

i dis%$nct."est". /iol#tedo" % k

e/el

e/el 1

...

k%k pxx %k% pxx


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B"#nc& #nd o$nd (cont.

Mppe" o$nd: e.g. # e#sile sol$tion

o'e" o$nd:

e.g. # sol$tion to #n Re#sie"S

p"ole*

Qode eli*in#tion (#t&o*Gdisc#"dnodes:

'&en lo'e" o$nd D $ppe" o$nd


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B"#nc& #nd o$nd (cont.

B"#nc&ing st"#teg0:

&o' to p#"tition sol$tion sp#ce

Qode selection st"#teg0:

se$ence o explo"ing nodes:

dept& "st (t"ies to ot#in # sol$tion #st

"e#dt&Gest o$nd "st (t"ies to nd t&e est sol$tion

'&ic& nodes to explo"e (lte" #nd e#* 'idt&

lte" 'idt&: Tnodes selected o" t&o"o$g& e/#l$#tion

e#* 'idt&: Tnodes tt #"e "#nc&ed on ( lte" 'idt&

Be#* se#"c&

B"#nc& #nd o$nd


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ingle *#c&ine, *#xi*$* l#teness,

"ele#se #nd d$e d#tes

lo'e" o$nd: ! K p"ee*ption

Jobs 1 2 3 4

p(j) 4 2 6 5

r(j) 0 1 3 5

d(j) 8 12 11 10

(U,U,U,U

(1,U,U,U (4,U,U,U(2,U,U,U

e/el

e/el 1(3,U,U,U

B"#nc& #nd o$nd


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o'e" o$nd o": (1,U,U,U

o'e" o$nd: *#x *#x(,1V-12,15-

11,5

Jobs 1 2 3 4

p(j) 4 2 6 5r(j) 0 1 3 5

d(j) 8 12 11 10

t

"(2 "(3 "(4

d(4Wd(3

d(3Wd(2

1 2 3 4 5 6 V 8 X 1 1112131415161V3 3 21 4

x


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(U,U,U,U

(1,U,U,U (4,U,U,U(2,U,U,U

e/el

e/el 1(3,U,U,U

x(cont.

B5 BV@

MB

(1,2,U,U (1,3,U,U Jobs 1 2 3 4

p(j) 4 2 6 5r(j) 0 1 3 5

d(j) 8 12 11 10

ine#sil

e:(1,3,4,3,

2B6@MB

(1,2,4,3

B5@MB(1,3,4,2

;Q!

(1,2,4,3 (1,3,4,2

B"#nc& #nd o$nd


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1 2 3 4 5 6

x1

1

2

3

4

5

6x2

o%ecti/e(

6xx 21 =+

4.2x

,8.6x

2

1

=

=

6x1 1x1

3x

,6x

2

1

==

2x

,1x

2

1

==

2x2 3x2

2x

,1x

2

1

==

3x

,6x

2

1

==

o%: 3 o%: 3

o%: 3 o%: 3

) sol$tion:

e#* se#"c ex#*p e


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e#* se#"c ex#*p esingle-*#c&ine, tot#l 'eig&ted

t#"diness

Mppe" o$nd: A7C "$le (#pp#"ent t#"dinesscost:sc&ed$le 1 %o #t # ti*ee/e"0 ti*e # *#c&ine co*es #/#il#le,

dete"*ine "#nking o %os:

= p96,tpd*#x(

%

%

%

%%

ep

't(

"$le

Y)7"$le

look-#&e#d p#"#*ete":

9 4.5 K ( .5

9 6 - 2 ( .5

d$e d#te "#nge

#cto"

( ) *#x*in*#x CGdd =

e#* se#"c ex#*p e


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e#* se#"c ex#*p e(cont.

single-*#c&ine, tot#l 'eig&ted t#"diness

(U,U,U,U

(1,U,U,U (4,U,U,U(2,U,U,U (3,U,U,U

1,2,3(%

p

'e

p

't(

%

%p9

6,tpd*#x(

%

%

%

%%

===

Mppe" o$nd 0 A7C "$le: 3,2,1%(66,tpd*#x( %% ==

Jobs 1 2 3 4

p(j) 10 10 13 4

d(j) 4 2 1 12w(j) 14 12 1 12

w(j)/p(j) 1.4 1.2 0.1 3

e#* se#"c ex#*p e


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(U,U,U,U

(1,U,U,U (4,U,U,U(2,U,U,U (3,U,U,U

Jobs 1 2 3 4

p(j) 10 10 13 4

d(j) 4 2 1 12

w(j) 14 12 1 12

w(j)/p(j) 1.4 1.2 0.1 3

Jobs C(j) d(j) T(j) w(j)*T(j)

1 10 4 6 84

2 24 2 22 264

3 37 1 36 36

4 14 12 2 24

Mppe" o$nd0 A7C "$le:

7ot#l 48

e#* se#"c ex#*p e


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(U,U,U,U

(1,U,U,U (4,U,U,U(2,U,U,U (3,U,U,U

MB48MB436MB814MB44

disc#"dedexplo"ed$"t&e"

(e#* 'idt& 2

4 nodes#n#l0+ed(lte" 'idt&4

x


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x(cont.

(U,U,U,U

(1,U,U,U (4,U,U,U(2,U,U,U (3,U,U,U

MB48 436 814 44

(1,2,U,U (1,3,U,U (1,4,U,U

MB48 V6 48

(1,4,2,3 (1,4,3,2

MB48 554

t

(2,1,U,U (2,3,U,U (2,4,U,U

436

(2,4,1,3 (2,4,3,1

436 68

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