Seite - 172 - in Algorithms for Scheduling Problems
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Text der Seite - 172 -
Algorithms 2018,11, 35
FromFormulas (20)and(21),weobtain that
Hâ(Cs)=Hâ(Csâ1)+Hâ(Ls)âHâ(Lsâ1)+Hâ(Lsâ1|Ls) (22)
HereHâ(Lsâ1|Ls)denotes theconditionalentropyof the layer sâ1under theconditionthat the
probabilitiesandentropyof layer sare found.
Denote, forconvenience,H(Cs)=H(s)andHâ(Cs)=Hâ(s).
UsingthedeïŹnitionof theweightedentropyandFormula (21),weobtain
H(s)= c(s)Hâ(s)= c(s)(Hâ(Csâ1)+Hâ(Ls)âHâ(Lsâ1)+Hâ(Lsâ1|Ls)).
UsingtheFormulaof theconditionalentropy, thedeïŹnitionsofeventsAi(s),probabilities pi(s)
andmatricesM(2)(s),wecanwrite that
Hâ(Lsâ1|Ls)=â N
â
i=1 (
P(Ai(s)) · N
â
j=1 P ( Aj(sâ1)|Ai(s) ) · logP(Aj(sâ1)|Ai(s)) )
=âNâ
i=1 (
pi(s) · N
â
j=1 pË(1)ij (sâ1) · log (
pË(1)ij (sâ1) ))
Hâ(Lsâ1|Ls)=â N
â
i=1 (
pi(s) · N
â
j=1 pË(1)ij (sâ1) · log (
pË(1)(sâ1) ))
(23)
UsingFormula (22)wecanwrite
H(s)= c(s) (
Hâ(Csâ1)+Hâ(Ls)âHâ(Lsâ1)â N
â
i=1 (
pi(s) · N
â
j=1 pË(1)ij (sâ1) · log (
pË(1)ij (sâ1) )))
(24)
s=1,2, . . .
H(sâ1)âH(s)=
=H(sâ1)âc(s) (
Hâ(Csâ1)+Hâ(Ls)âHâ(Lsâ1)â N
â
i=1 (
pi(s) · N
â
j=1 pË(1)ij (sâ1) · log (
pË(1)ij (sâ1) )))
=
= (
1â c(s)c(sâ1) )
H(sâ1)âc(s)(Hâ(Ls)âHâ(Lsâ1))+c(s) N
â
i=1 (
pi(s) · N
â
j=1 pË(1)ij (sâ1) · log (
pË(1)ij (sâ1) )) .
Weobtain that
H(sâ1)âH(s)= (
1â c(s)c(sâ1) )
H(sâ1)âc(s)(Hâ(Ls)âHâ(Lsâ1))+c(s) · N
â
i=1 (
pi(s) · N
â
j=1 pË(1)ij (sâ1) · log (
pË(1)ij (sâ1) ))
(25)
s=1,2, . . .
Since the followingrelationsarevalid,
0<1â c(s)c(sâ1)<1,
0<âNâ
i=1 (
pi(s) · N
â
j=1 pË(1)ij (sâ1) · log (
pË(1)ij (sâ1) ))
< logN
0<Hâ(Ls)< logN, 0<Hâ(Lsâ1)< logN
lim
sââH(sâ1)=0, limsââ c(s)=0
weobtain that
lim
sââ(H(sâ1)âH(s))=0
172
zurĂŒck zum
Buch Algorithms for Scheduling Problems"
Algorithms for Scheduling Problems
- Titel
- Algorithms for Scheduling Problems
- Autoren
- Frank Werner
- Larysa Burtseva
- Yuri Sotskov
- Herausgeber
- MDPI
- Ort
- Basel
- Datum
- 2018
- Sprache
- englisch
- Lizenz
- CC BY 4.0
- ISBN
- 978-3-03897-120-7
- Abmessungen
- 17.0 x 24.4 cm
- Seiten
- 212
- Schlagwörter
- Scheduling Problems in Logistics, Transport, Timetabling, Sports, Healthcare, Engineering, Energy Management
- Kategorien
- Informatik
- Technik