econ lecture
TRANSCRIPT
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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
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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$nd
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B"#nc& #nd o$ndex#*ple 2
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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B"#nc& #nd o$ndex#*ple 2
o'e" o$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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B"#nc& #nd o$ndex#*ple 3
1 2 3 4 5 6
x1
1
2
3
4
5
6x2
o%ecti/e(
6xx 21 =+
4.2x
,8.6x
2
1
=
=
6x1 1x1
3x
,6x
2
1
==
2x
,1x
2
1
==
2x2 3x2
2x
,1x
2
1
==
3x
,6x
2
1
==
o%: 3 o%: 3
o%: 3 o%: 3
) sol$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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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
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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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
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