j=1
εjPr{γef f,j≥γth}, UjX8V
r?2`2 Pr{γef f,j≥γth} Bb i?2 HBMF `2HB#BHBiv Q7 i?2 j@i? bm#+?MM2HX h?2 aAL_ i?`2b?QH/
γth K2Mb i?2 KBMBKmK aAL_ `2[mB`2K2Mi 7Q` /i i`MbKBbbBQMX
jX9Xk 62KiQ+2HH *T+Biv
h?2 72KiQ+2HH +T+Biv C Bb /2}M2/ b i?2 p2`;2 i?`Qm;?Tmi Q7 72KiQ+2HH-r?B+? /2T2M/b QM i?2 +?MM2H b2H2+iBQM b+?2K2- i?2 MmK#2` Q7 mb2/ bm#+?MM2Hb- M/
i?2 /QTi2/ J*a Q7 2+? bm#+?MM2HX ++Q`/BM; iQ i?2 2[mBpH2Mi aAL_ γef f,j 7`QK i?2 11aJ K2i?Q/- r2 +M /2i2`KBM2 i?2 mb2/ J*a M/ i?2M i?2 bT2+i`mK 2{+B2M+v σjX bbmK2 i?i Bm Bb i?2 #M/rB/i? Q7 bm#@+``B2` M/ i?2`2 `2 M bm#@+``B2`b BM bm#+?MM2HX h?2 bm#+?MM2H #M/rB/i? Bb Bj = MBmX h?2M- i?2 72KiQ+2HH +T+Biv Bb 2[mH iQ
C = 1
1 + G
J j=1
εjBjσj, UjXeV
r?2`2 G Bb i?2 ;m`/ 7`+iBQMX
k9
jX9Xj 1z2+iBp2 aT2+i`mK 1{+B2M+v
h?2 2z2+iBp2 bT2+i`mK 2{+B2M+v Q7 i?2 2t+HmbBp2 M/ b?`2/ bT2+i`mK HHQ@
+iBQM b+?2K2b 7Q` i?2 72KiQ+2HHb `2 +H+mHi2/ BM i?2 7QHHQrBM;X AM i?2 2t+HmbBp2@
bT2+i`mK b+?2K2- i?2 iQiH HHQ+i2/ #M/rB/i? Bb2B bBM+2 i?2 72KiQ+2HHb M/ K+`Q@
+2HHb `2 HHQ+i2/ rBi? /Bz2`2Mi 7`2[m2M+v #M/bX bbmK2 i?i i?2 +?B2p2/ 72KiQ@
+2HH +T+Biv mM/2` i?2 HBMF `2HB#BHBiv `2[mB`2K2Mi Bb CexX h?2 2z2+iBp2 bT2+i`mK 2{+B2M+v Q7 QM2 72KiQ+2HH BM i?2 2t+HmbBp2@bT2+i`mK b+?2K2 Bb /2}M2/ b
SEex = Cex
2B. UjXdV
AM i?2 b?`2/ bT2+i`mK b+?2K2- i?2 72KiQ+2HHb mb2 i?2 bK2 7`2[m2M+v #M/ Q7 #M/@
rB/i? B b i?2 K+`Q+2HHbX G2i Csh #2 i?2 +?B2p2/ +T+Biv Q7 QM2 72KiQ+2HH rBi?
i?2 HBMF `2HB#BHBiv `2[mB`2K2MiX h?2`27Q`2- i?2 2z2+iBp2 bT2+i`mK 2{+B2M+v 7Q` i?2 b?`2/@ bT2+i`mK b+?2K2 Bb /2
SEsh = Csh
B . UjX3V
AM/22/- SEex M/ SEsh /2}M2/ BM UjXdV M/ UjX3V `2 i?2 K2bm`2K2Mib Q7 bT2+i`mK 2{+B2M+v BKT`Qp2K2Mi #v QM2 72KiQ+2HHX Hi?Qm;? i?2 2z2+iBp2 bT2+i`mK 2{+B2M+v SEex M/ SEsh `2 MQi i?2 +imH bT2+i`mK 2{+B2M+v Q7 ?2i2`Q;2M2Qmb K+`Q+2HHf72KiQ+2HH bvbi2K- i?2v +M #2 mb2/ iQ [mMiBiiBp2Hv +QKT`2 i?2 bT2+i`mK 2{+B2M+v Q7 i?2 2t+HmbBp2 M/ b?`2/ HHQ+iBQM b+?2K2bX AMimBiBp2Hv- rBi? i?2 /Qm#H2
#M/rB/i?- i?2 2t+HmbBp2@bT2+i`mK HHQ+iBQM b+?2K2 Kv ?p2 H2bb bT2+i`mK 2{@
+B2M+v i?M i?2 b?`2/ b+?2K2X >Qr2p2`- BM i?2 b?`2/@bT2+i`mK b+?2K2- 72KiQ+2HHb Kmbi `2/m+2 i?2 mb2/ bm#@+``B2`b 7Q` 2Mbm`BM; i?2 HBMF `2HB#BHBiv Q7 HH i?2 mb2`bX h?Bb bB;MB BM; i?2 HQ+iBQM r`2M2bb +T#BHBiv BM i?2 72KiQ+2HH bvbi2K iQ BKT`Qp2 i?2 2z2+iBp2 bT2+i`mK 2{+B2M+v Q7 i?2 b?`2/ bT2+i`mK b+?2K2X
k8
h?MFb iQ i?2 HQ+iBQM r`2M2bb +T#BHBiv b /Bb+mbb2/ BM a2+iBQM jXRXR- i?2 72KiQ+2HHb `2 r`2 Q7 i?2 2tBbi2M+2 Q7 M Qmi/QQ` mb2`X h?2`27Q`2- QMHv i?2 72K@
iQ+2HHb +HQb2 iQ i?2 Qmi/QQ` mb2` b?QmH/ /2+`2b2 i?2 mb2/ bm#@+``B2`b iQ `2/m+2 i?2 BMi2`72`2M+2X AM i?Bb bBimiBQM- i?2 +?B2p2/ 72KiQ+2HH +T+Biv Bb Csh1X PM i?2 +QM@
i``v- i?2 72KiQ+2HHb 7` rv 7`QK i?2 Qmi/QQ` mb2` +M biBHH mb2 KQ`2 bm#@+``B2`b rBi? ?B;?2` +T+Biv Csh0X amTTQb2 i?i 7Q` i?2 +QMbB/2`2/ 72KiQ+2HH- i?2 T`Q##BHBiv i?i i?2`2 Bb M Qmi/QQ` mb2` `QmM/ Bi Bb pX h?2`27Q`2- rBi? i?2 HQ+iBQM r`2M2bb-i?2 p2`;2 bT2+i`mK 2{+B2M+v 7Q` r`2M2bb-i?2 b?`2/ b+?2K2 +M BM+`2b2 iQ
SELA = pCsh1+ (1 − p)Csh0
B . UjXNV
h?2 2tBbiBM; T`Q##BHBiv p Q7 Qmi/QQ` mb2` `QmM/ 72KiQ+2HH /2T2M/b QM i?2 TQT@
mHiBQM /2MbBiv- 72KiQ+2HH /2MbBiv- 72KiQ+2HH i`MbKBbbBQM TQr2` M/ bQ QMX h?Bb T`Q##BHBiv +M #2 2biBKi2/ 7`QK i?2 HQM;@i2`K biiBbiB+bX
ke
kd
*>Sh1_ 9
.Bbi`B#mi2/ *?MM2H a2H2+iBQM 7Q`
62KiQ+2HH rBi? hrQ@iB2` AMi2`72`2M+2
9XR 62KiQ+2HH *T+Biv JtBKBxiBQM
*T+Biv M/ HBMF `2HB#BHBiv `2 #Qi? 2bb2MiBH 7+iQ`b BM i?2 /Bbi`B#mi2/ +?M@
M2H b2H2+iBQM T`BM+BTH2 7Q` i?2 P6.J@#b2/ 72KiQ+2HH bvbi2KbX 6`QK HBMF `2HB#BHBiv T2`bT2+iBp2- /2+`2bBM; i?2 MmK#2` Q7 bm#+?MM2Hb HHQ+i2/ iQ 72KiQ+2HH +M /2@
+`2b2 i?2 BMi2`72`2M+2 7Q` i?2 mb2`b `QmM/ i?2 72KiQ+2HHbX >Qr2p2`- 7`QK i?2 HBMF +T+Biv biM/TQBMi- BM+`2bBM; i?2 MmK#2` Q7 bm#+?MM2Hb HHQ+i2/ iQ 72KiQ+2HH +M T`QpB/2 ?B;?2` /i `i2X
hQ BKT`Qp2 i?2 i`/2Qz #2ir22M +T+Biv M/ HBMF `2HB#BHBiv- r2 7Q`KmHi2
M QTiBKBxiBQM T`Q#H2K iQ /2i2`KBM2 i?2 QTiBKH MmK#2` Q7 bm#+?MM2Hb HHQ+i2/
iQ 72KiQ+2HH- BKBM; iQ KtBKBx2 72KiQ+2HH +T+Biv bm#D2+i iQ i?2 HBMF `2HB#BHBiv
`2[mB`2K2Mi Q7 BM/QQ` M/ Qmi/QQ` mb2`bX
h?2 /2+BbBQM p`B#H2 BM i?2 QTiBKBxiBQM T`Q#H2K Bb i?2 +?MM2H mb;2 `iBQ ρX "b2/ QM i?2b2 +QMbB/2`iBQMb- r2 7Q`KmHi2 i?2 +T+Biv KtBKBxiBQM Bbbm2 b MQMHBM2` T`Q;`KKBM; T`Q#H2K BM i?2 7QHHQrBM;,
0≤ρ≤1Kt C = 1 1 + G
J j=1
εjBjσj U9XRV
bm#D2+i iQ,
ρ = 1 J
J j=1
εj U9XkV
Prel≥ Relth U9XjV
r?2`2 εj ∈ {0, 1}- M/ Relth Bb i?2 HBMF `2HB#BHBiv `2[mB`2K2Mi 7Q` HH i?2 mb2`bX
9Xk .Bbi`B#mi2/ *?MM2H a2H2+iBQM a+?2K2
q2 /2p2HQT irQ /Bbi`B#mi2/ +?MM2H b2H2+iBQM T`BM+BTH2b 7Q` i?2 72KiQ+2HH bvb@
i2Kb (kj), i?2 ;BM@Q`B2Mi2/ M/ BMi2`72`2M+2@pQB/M+2@Q`B2Mi2/ +?MM2H b2H2+iBQM b+?2K2bX h?2 }`bi b+?2K2 BKb iQ i`MbKBi /i BM i?2 bm#+?MM2H rBi? ?B;?2`
HBMF ;BMX "v +QMi``v- i?2 b2+QM/ QM2 i`MbKBib /i BM i?2 bm#+?MM2H rBi? HQr2`
BMi2`72`2M+2X "Qi? b+?2K2b QT2`i2 BM /Bbi`B#mi2/ KMM2`X 1+? 72KiQ+2HH +M b2@
H2+i i?2 bm#+?MM2Hb 7Q` i`MbKBbbBQM #v Bib2H7X h?2 F2v T`K2i2` Bb i?2 bm#+?MM2H mb;2 `iBQ ρ b b?QrM BM UdXRVX q2 /2iBH i?2 /2p2HQT2/ +?MM2H b2H2+iBQM b+?2K2b
b 7QHHQrb (k9)X
9XkXR Jt@JBM :BM@P`B2Mi2/ am#+?MM2H a2H2+iBQM a+?2K2
q2 2tTHBM i?2 T`Q+2/m`2b BM i?2 7QHHQrBM;X
UaRV *QKT`2 i?2 HBMF ;BM Q7 BM/BpB/mH bm#@+``B2` BM bm#+?MM2HX h?2 KBMBKmK bm#@+``B2` HBMF ;BM BM bm#+?MM2H Bb
hj = min
m=1,2,...,M{h(i)j,m}, 7Q` j = 1, . . . , J U9X9V r?2`2 M Bb i?2 iQiH MmK#2` Q7 bm#@+``B2`b BM bm#+?MM2H- M/ J Bb i?2 i?2 iQiH MmK#2` Q7 bm#+?MM2HbX
k3
UakV LmK#2` HH i?2 bm#+?MM2Hb ++Q`/BM; iQ i?2 KBMBKmK bm#@+``B2` HBMF ;BM- #v i?2 Q`/2` h1 ≥ h2 ≥ ... ≥ hρJ ≥ ... ≥ hJX
UajV a2H2+i i?2 }`bi ρJ bm#+?MM2Hb rBi? ?B;?2` HBMF ;BM 7Q` i`MbKBbbBQMX h?i Bb-i@i? bm#+?MM2H rBHH #2 b2H2+i2/- B7 hi ≥ hρJX
6Q` 2tKTH2- i?2`2 `2 9 bm#+?MM2Hb BM i?2 P6.J@#b2/ 72KiQ+2HH
bvbi2K-M/ 2+? bm#+?MM2H ?b j bm#@+``B2`bX h?2 HBMF ;BMb 7Q` i?2 bm#@+``B2`b BM 2+?
bm#+?MM2H `2 SCH1 4 {yXk- yX8- yX9}- SCH2 4 {@RX9- @yXj- @yXe}- SCH3 4 {@yXR-yXd- RXR} M/ SCH4 4 {@yX9- @yX3- yXj} U/"V- `2bT2+iBp2HvX ++Q`/BM; iQ UaRV- i?2 KBMBKmK HBMF ;BM hj 7Q` 7Qm` bm#+?MM2Hb `2 {yXk- @RX9- @yXR- @yX3} U/"VX bbmK2 i?i r2 M22/ iQ +?QQb2 irQ bm#+?MM2Hb 7Q` i`MbKBbbBQMX h?2M- ++Q`/BM; iQ UakV
M/ UajV- r2 b2H2+i SCH1 M/ SCH3 bBM+2 i?2b2 bm#+?MM2Hb ?p2 ?B;?2` KBMBKmK HBMF ;BMbX
9XkXk Jt@p; :BM@P`B2Mi2/ am#+?MM2H a2H2+iBQM a+?2K2
h?Bb b+?2K2 b2H2+ib i?2 bm#+?MM2Hb ++Q`/BM; iQ i?2 p2`;2 bm#@+``B2` HBMF
;BMX
UaRV *QKTmi2 i?2 p2`;2 bm#@+``B2` HBMF ;BM Q7 2+? bm#+?MM2HX
hj = 1 M
M m=1
h(i)j,m, 7Q` j = 1, . . . , J. U9X8V
UakV LmK#2` HH i?2 bm#+?MM2Hb ++Q`/BM; iQ i?2 p2`;2 bm#@+``B2` HBMF ;BM- #v i?2 Q`/2` h1 ≥ h2 ≥ ... ≥ hρJ ≥ ... ≥ hJX
UajV 6QHHQr i?2 bi2T UajV BM a2+iBQM 9XkXRX
*QMbB/2` i?2 2tKTH2 BM a2+iBQM 9XkXRX h?2 p2`;2 HBMF ;BM hj 7Q` i?2 7Qm`
bm#+?MM2Hb `2{yXjd- @yXd9- yX8N- @yXk3} U/"V- `2bT2+iBp2HvX 6QHHQrBM; UakV M/ UajV BM a2+iBQM 9XkXR- r2 rBHH b2H2+i SCH1 M/ SCH3- bBM+2 i?2b2 bm#+?MM2Hb ?p2 ?B;?2`
p2`;2 HBMF ;BMX
kN
9XkXj JBM@Jt AMi2`72`2M+2 pQB/M+2 am#+?MM2H a2H2+iBQM a+?2K2
h?2 T`Q+2/m`2b `2 /2b+`B#2/ BM i?2 7QHHQrBM;X q2 +QMbB/2` i?2 BMi2`72`2M+2 7`QK K+`Q+2HH M/ Qi?2` 72KiQ+2HHbX
UaRV *QKT`2 i?2 BMi2`72`2M+2 Q7 2+? bm#@+``B2` BM bm#+?MM2HX h?2 KtBKmK BMi2`72`2M+2 7Q` i?2 bm#@+``B2`b BM bm#+?MM2H Bb
Ij = max
UakV LmK#2` HH i?2 bm#+?MM2Hb ++Q`/BM; iQ i?2 KtBKmK bm#@+``B2`
BMi2`72`2M+2-#v i?2 Q`/2` I1 ≤ I2 ≤ ... ≤ IρJ ≤ ... ≤ IJX
UajV a2H2+i i?2 }`bi ρJ bm#+?MM2Hb rBi? HQr2` BMi2`72`2M+2 7Q` i`MbKBbbBQMX h?i Bb- i@i? bm#+?MM2H rBHH #2 b2H2+i2/- B7 Ii ≤ IρJX
bbmK2 i?i i?2 iQiH BMi2`72`2M+2 TQr2` 7Q` i?2 bm#@+``B2`b BM 2+? bm#+?M@
M2H `2 SCH1 4{@dy- @NR- @3j}- SCH2 4 {@3R- @dk- @Ryk}- SCH3 4{@3R- @RRj- @Ryk}
M/ SCH4 4 {@3N- @Ry3- @d8} U/"KV- `2bT2+iBp2HvX ++Q`/BM; iQ UaRV- i?2 KtBKmK bm#@+``B2` BMi2`72`2M+2 Ij 7Q` 7Qm` bm#+?MM2Hb `2 {@dy- @dk- @3R- @d8} U/"KVX +@
+Q`/BM; iQ UakV M/ UajV- r2 rBHH b2H2+i SCH3 M/ SCH4 /m2 iQ HQr2` KtBKmK BMi2`72`2M+2X
9XkX9 JBM@p; AMi2`72`2M+2 pQB/M+2 am#+?MM2H a2H2+iBQM a+?2K2
h?Bb b+?2K2 b2H2+ib i?2 bm#+?MM2Hb ++Q`/BM; iQ i?2 p2`;2 BMi2`72`2M+2 7Q`
i?2 bm#@+``B2`b Q7 bm#+?MM2HX
UaRV *QKTmi2 i?2 p2`;2 BMi2`72`2M+2 Q7 bm#+?MM2HX
Ij = 1
UakV LmK#2` HH i?2 bm#+?MM2Hb ++Q`/BM; iQ i?2 p2`;2 BMi2`72`2M+2- #v i?2 Q`/2`
I1 ≤ I2 ≤ ... ≤ IρJ ≤ ... ≤ IJX
UajV 6QHHQr i?2 bi2T UajV BM a2+iBQM 9XkXjX
*QMbB/2`BM; i?2 2tKTH2 BM a2+iBQM 9XkXj- i?2 p2`;2 BMi2`72`2M+2 Ij 7Q` 7Qm`
bm#+?MM2Hb `2{@d9X8- @deXj- @38Xd- @dNXe} U/"KVX h?2M- 7QHHQrBM; UakV M/ UajV- r2 b2H2+i SCH3 M/ SCH4 /m2 iQ HQr2` p2`;2 BMi2`72`2M+2 TQr2`X