Page - 172 - in Algorithms for Scheduling Problems
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Algorithms 2018,11, 35
FromFormulas (20)and(21),weobtain that
H∗(Cs)=H∗(Cs−1)+H∗(Ls)−H∗(Ls−1)+H∗(Ls−1|Ls) (22)
HereH∗(Ls−1|Ls)denotes theconditionalentropyof the layer s−1under theconditionthat the
probabilitiesandentropyof layer sare found.
Denote, forconvenience,H(Cs)=H(s)andH∗(Cs)=H∗(s).
Usingthedefinitionof theweightedentropyandFormula (21),weobtain
H(s)= c(s)H∗(s)= c(s)(H∗(Cs−1)+H∗(Ls)−H∗(Ls−1)+H∗(Ls−1|Ls)).
UsingtheFormulaof theconditionalentropy, thedefinitionsofeventsAi(s),probabilities pi(s)
andmatricesM(2)(s),wecanwrite that
H∗(Ls−1|Ls)=− N
∑
i=1 (
P(Ai(s)) · N
∑
j=1 P ( Aj(s−1)|Ai(s) ) · logP(Aj(s−1)|Ai(s)) )
=−N∑
i=1 (
pi(s) · N
∑
j=1 pˆ(1)ij (s−1) · log (
pˆ(1)ij (s−1) ))
H∗(Ls−1|Ls)=− N
∑
i=1 (
pi(s) · N
∑
j=1 pˆ(1)ij (s−1) · log (
pˆ(1)(s−1) ))
(23)
UsingFormula (22)wecanwrite
H(s)= c(s) (
H∗(Cs−1)+H∗(Ls)−H∗(Ls−1)− N
∑
i=1 (
pi(s) · N
∑
j=1 pˆ(1)ij (s−1) · log (
pˆ(1)ij (s−1) )))
(24)
s=1,2, . . .
H(s−1)−H(s)=
=H(s−1)−c(s) (
H∗(Cs−1)+H∗(Ls)−H∗(Ls−1)− N
∑
i=1 (
pi(s) · N
∑
j=1 pˆ(1)ij (s−1) · log (
pˆ(1)ij (s−1) )))
=
= (
1− c(s)c(s−1) )
H(s−1)−c(s)(H∗(Ls)−H∗(Ls−1))+c(s) N
∑
i=1 (
pi(s) · N
∑
j=1 pˆ(1)ij (s−1) · log (
pˆ(1)ij (s−1) )) .
Weobtain that
H(s−1)−H(s)= (
1− c(s)c(s−1) )
H(s−1)−c(s)(H∗(Ls)−H∗(Ls−1))+c(s) · N
∑
i=1 (
pi(s) · N
∑
j=1 pˆ(1)ij (s−1) · log (
pˆ(1)ij (s−1) ))
(25)
s=1,2, . . .
Since the followingrelationsarevalid,
0<1− c(s)c(s−1)<1,
0<−N∑
i=1 (
pi(s) · N
∑
j=1 pˆ(1)ij (s−1) · log (
pˆ(1)ij (s−1) ))
< logN
0<H∗(Ls)< logN, 0<H∗(Ls−1)< logN
lim
s→∞H(s−1)=0, lims→∞ c(s)=0
weobtain that
lim
s→∞(H(s−1)−H(s))=0
172
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book Algorithms for Scheduling Problems"
Algorithms for Scheduling Problems
- Title
- Algorithms for Scheduling Problems
- Authors
- Frank Werner
- Larysa Burtseva
- Yuri Sotskov
- Editor
- MDPI
- Location
- Basel
- Date
- 2018
- Language
- English
- License
- CC BY 4.0
- ISBN
- 978-3-03897-120-7
- Size
- 17.0 x 24.4 cm
- Pages
- 212
- Keywords
- Scheduling Problems in Logistics, Transport, Timetabling, Sports, Healthcare, Engineering, Energy Management
- Categories
- Informatik
- Technik