scheduling+concerns+opbmal+allocabon+or+assignment+ of ... ·...
TRANSCRIPT
Scheduling+concerns+opBmal+allocaBon+or+assignment+of+resources,+over+Bme,+to+a+set+of+tasks+or+acBviBes.+++Resourses:+machines,+people,+space+Tasks:+ + ++producBon,+jobs,+classes,+flights+
+– schedule+may+be+represented+by+GanS+charts+– m+machines+M1,...,Mm++++and+n+jobs+J1,...,Jn++++
+
Henry0Gan30(VS)01861::19190
Examples++
Gate+assignment+ Containers+
Processor+scheduling+Graph+drawing+
General+NotaBon+• m+machines+1,...,m+• n+jobs.+++J++is+set+of+all+jobs:+++J={1,...,n}+.+++Data+for+jobs:+• pj++:+processing+Bme+of+job+j+• pij+:+processing+Bme+of+job+j+on+machine+i++
+ +(if+processing+Bme+of+job+j+depends+on+the+machine)+• rj+:+release+date+of+job+j +(earliest+starBng+Bme)++• dj+:+due+date+of+job+j+ + +(commiSed+compleBon+Bme)+• wj+:+weight+of+job+j+ + +(importance)+++
ClassificaBon+of+Scheduling+Problems++
+
(Many)+Scheduling+problems+can+be+described+by+a+three+field+
notaBon+α|β|γ,+where++
+α+describes+the+machine+environment+
+β+describes+the+job+characterisBcs,+and+
+γ+describes+the+objecBve+criterion+to+be+minimized++
+
Remark:++A+field+may+contain+more+than+one+entry+but+may+also+be+empty+
+
EXAMPLE:++
+++1+|+rj+|++Σ
j+C
j+++++++++++Single+machine.++
+ + + +Jobs+have+release+Bmes.++
+ + + +ObjecBve+is+minimizing+the+sum+of+the+compleBon+Bmes.+++
+ α|β|γ,+
ClassificaBon+d+Machine+environment++(α+)+++
• Single+machine+(α+=+1)+• IdenBcal+parallel+machines+(α+=+P+or+Pm)++
– m+idenBcal+machines;+if+α+=+P,+the+value+m+is+part+of+the+input+and+if+α+=+Pm,+the+value+is+considered+as+a+constant+(complexity+theory)+
– each+job+consist+of+a+single+operaBon+and+this+may+be+processed+by+any+of+the+machines+for+pj+Bme+units++
• Unrelated+parallel+machines+(α+=+R+or+Rm)++– m+different+machines+in+parallel+
pij+is+the+process+Bme+of+job+j+if+scheduled+completely+on+machine+i+
+
ClassificaBon+d+Job+characterisBcs+(β+)+++
• release+dates+(rj+is+contained+in+β+field)+– if+rj+is+not+in+β+field,+jobs+may+start+at+any+Bme+– +if+rj+is+in+β+field,+jobs+may+not+start+processing+before+their+release+
date++
• preempBon+(pmtn+is+contained+in+β+field)++– processing+of+a+job+on+a+machine+may+be+interrupted+and+resumed+at+a+
later+Bme+even+on+a+different+machine+
• unit+processing+Bmes+(pj+=+1+or+pij+=+1+in+β+field)+– +each+job+(operaBon)+has+unit+processing+Bmes++
• other+’obvious’+specificaBons+(i.e.+dj+=+d)+++
ClassificaBon+d+Job+characterisBcs+(β+)+++
• precedence+constraints+(prec+in+β+field)+– between+jobs+precedence+relaBons+are+given:+a+job+may+not+start+its+
processing+before+another+job+has+been+finished+– may+be+represented+by+an+acyclic+graph+(verBces+=+jobs,+arcs+=+
precedence+relaBons)++
+
ClassificaBon+d+ObjecBve+criterion+(γ++)+++
NotaBons+– Cj+:+compleBon+Bme+of+job+j++
– Lj+=+Cj+−+dj+:+lateness+of+job+j+
ObjecBves:+
– Makespan+ + + + +(γ+=+Cmax+)+++ +Cmax+=+max+{C1,...,Cn}++
– Maximum+lateness+ + +(γ+=+Lmax+)+ +Lmax+=+max{L1,...,Ln}++
– Total+compleBon+Bme+ + +(+γ+=++Σj+Cj+)+
– Total+weighted+compleBon+Bme+(+γ+=++Σj+wjCj+)+
+
Many+more+models+in+literarture+!+
!""#$!"
!"
##
$%
!"&'(
)"*#*%+
)"*%*#+
)#*"*%+
)#*%*"+
)%*"*#+
)%*#*"+! % &
! %&
!%&
!% &
!% &
& ! %
7+9+14=30
2+9+14=25
7+12+14=33
2+7+14=23
5+12+14=31
5+7+14=26
'()*#)+,-*.+/0#1+2#*+#3./4#.*5
, %-%"%#%%%,./$0!-"#%
!""#$!"
!"
##
$%
!"&'(
678(#*9(#:+;)#./#7/<#+=4(= %*1#*2*1#0#
69-)>#:+;#!#?+/*=.;-*()##!%>#
:+;#%#?+/*=.;-*()#)#3%+!#>#
7/4#)+#+/0#
@3#2(#27/*#*+#A./.A.B(#4$">
2(#27/*#!% *+#;(#*9(#)A7,,()*>#
!# *9(#)(?+/4#)A7,,()*>#(*?0
!% !# !#3% !#0##0##0
!! C##!
!% C##!# D #%
!
!$E! C##!# D# #% D# F D #$E!
!$ C##!# D #%# D# F# D #$E! D #$
$!" C#$#! D#G$%!H#% D##F#D#%#$E!D# #$
!""#$!"
!"
##
$%
!"&'(
I 69(#J=+;,(A#.)#)+,K(4#;<#+=4(=./L#:+;)#./#567 +=4(=
I M)#7/<#+*9(=#)+=*./L>#*9.)#*78()# F 8 %)#&9': #+
!"#$
%"&'
!"#$%&'&(")'*+!"#$*,-(&.
!"#$%&!
&'()*'+,"(()-.'(/"0(1
!"#$%&'&(")'*+!"#$*,-(&.
/ ,00(1(,)&+2$"*.)"#('*+&(#,3+'*4"5(&6#7 ,-(7&
/ 8()&5'1&'9*,:;+/ %)*(<,*.+&6'&+$"*.)"#('*=&(#,+'*4"5(&6#7 ,-(7&
>%))()4+&(#,+(7+9"%)?,?+9.+'+$"*.)"#('*+()+()$%&+7(@,A+!2"34 = &6,+)%#9,5+"0+7&,$7+45"B7+'7+ !"3
!2"#4-= &6,+)%#9,5+"0+7&,$7+45"B7+'7+ !"#!2"$('5"4 = &6,+)%#9,5+"0+7&,$7+45"B7+'7+!"('5"
!"#$%&!
&'()*'+,"(()-.'(/"0(1
!"#$%&'&(")'*+!"#$*,-(&.
2-!!"#+"3
&4-!!"#+"3
!"#$%&!
&'()*'+,"(()-.'(/"0(1
!"#$%&'&(")'*+!"#$*,-(&.
2
!"#"$%&'"()&*+,)-./&+.012345
!"#$%&'()*+'*,-(.#/0
123".+'*,-(.#/0
4$&-(0.("+'*,-(.#/0
+55-,2(/3.(,)+'*,-(.#/0
630.780',9:75$-;,-/3)"$7(07<$-(;($%7=>7",/5&.3.(,)3'7$25$-(/$).0
?(053."#()*7-&'$07;,-7@$30>A75-,='$/0
B-3)"#C3)%C=,&)%7;,-7@#3-%A75-,='$/0
630.:75$-;,-/3)"$7(07<$-(;($%73)3'>.("3''>
D3>7.3E$7',)*F7,;.$)70&(.3='$7;,-70/3''75-,='$/07,)'>7
6()%7,5.(/3'70,'&.(,)0
630.F7'(E$7!GHC-&'$7.,70,'<$7IJJ7!!"
!"#"$$%$&'"()*+%,-&'*+*.*,*+/&!!"#
"0
"1
"2
345%#&647+8,-
$%&'&"()*+&%,-*
!"#$%&''()$%(*+,#)#$&,, -(./$#&0,1#0$)2&'$"#$%(*+,#)#$('#$-(.3
."/01-)'&"()*+&%,-*
9*&4#/+&'$%&''()$.#$/*&,,#0$)2&'$)2#$&5#0&6#$*&%21'#$,(&7:
{ }npppppC ,...,,max where, 21max =≥
∑ =≥
n
j j /mpC1max
{ }∑ =≥
n
j j /mpp, C1max max
! "#
!"#"!$%&"#"'$()
*#
*"
*$
#%&&'()*+)(,-&,.-&/0,12()&2(3-40(5&6()+-&&& { }∑ ==
n
j jOPT mpp, C
1max / max
"%&&'/54,7+*,&(&4158)-92(*.15-&5/507--20,16-&4*.-:+)-;(44185&& </=4&,/&(&4158)-&2(*.15-&15&(5&(7=1,7(7>&/7:-7&4,(7,158&?1,.&,.-&)/58-4,&</=&@
*# ! " # $ % & '( )
! " #
# $ % &
& '( )
$%&&'+,&,.14&4158)-92(*.15-&4*.-:+)-&15,/&$ 0(7,4&/A&)-58,.OPTCmax
#BC
!! ! "# $ %& '
"#$#"%&'#$#(%)*
{ }∑ ==
'
+ ++ %,-#,(1
"#$%&'$%&
!!
!(
!)
*+,-../,0##1
!! ! "# $ %& '
"#$#"%&'#$#(%)*
{ }∑ ==
'
+ ++ %,-#,(1
"#$%&'$%&
% '
!!
!(
!)
!
"# $ %&
!"#$%&'()*
!"#$%&'
()!!"#*%+
",)!!"#*%+
"-./0-*1%../)2-.3%4.5
")!!"#*%+
$%&'(&%)*+,"&-.)%/0123++,
"!420+!#*%+
-%."/()0#12
3454
36&7/(1)(81
3-$$/"'(#9)(":1
!"#"$"%"#&'!!"#
$%&'"()
*+""#!!"#
%,+""#!!"#
%-./0-!1"../+2-.3"4.5
%+""#!!"#
$%&'#()*+,-.%/0#1,2344567%83'%6/9:
( )*+,-'./0%'"#1/'+'*"%1'
( %,2-34*-'12-'5"6%1'+7+"*+0*-'./0'/#'12-'*"%1'/5'4#%,2-34*-3'./0%'82-#-7-6'+'$+,2"#-'0-,/$-%'"3*-
26
A D
F
B C E
Machine 3
Machine 2
Time 0
Machine 1
I H J
G
List+Scheduling+
27
Machine 3
Machine 2
Machine 1 A
D
F
B C E
Time 0
I H J
G
List+Scheduling+
28
Machine 3
Machine 2
Machine 1 A
D
F
B
C E
Time 0
I H J
G
List+Scheduling+
29
Machine 3
Machine 2
Machine 1 A
D
F
B
C
E
Time 0
I H J
G
List+Scheduling+
30
Machine 3
Machine 2
Machine 1 A
F
B
C
E
Time 0
I H J
G
D
List+Scheduling+
31
Machine 3
Machine 2
Machine 1 A
F
B
C
E
Time 0
I H J
G
D
List+Scheduling+
32
G
Machine 3
Machine 2
Machine 1 A
F
B
C
E
Time 0
I H J
D
List+Scheduling+
33
Machine 3
Machine 2
Machine 1 A
F
B
C
E
Time 0
I H J
G D
List+Scheduling+
34
Machine 3
Machine 2
Machine 1 A
F
B
C
E
Time 0
I
H
J
G D
List+Scheduling+
35
Machine 3
Machine 2
Machine 1 A
F
B
C
G
E
Time 0
I H
J
G D
List+Scheduling+
36
Machine 3
Machine 2
Machine 1 A
F
B
C
G
E
Time 0
I H
J
D
List+Scheduling+
37
Machine 3
Machine 2
Machine 1 A
D
F
B
C
G
E
0
I H
J
Machine 3
Machine 2
Machine 1
A
D
F
B C
G
E
0
I
H
J
Optimal Schedule
List schedule
List+Scheduling+