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� � ��� -,"��)�aN^L-PQcONSU�z�RTL4PQI�?7P/`�I{|�LOU�|9y�RTU_UvN^wO\SI�z�RT` wONSLM|iP_N^RTL R���` ]O\SP_N^yO\SN^z{|iPQNO2TI/z cO|�L54TI�UN^LVNSLOdONO2gNSdO]M|�\MLERgdOI�U{D � �)-,� ��� � $d� -,� $�"*� P_cM|iP z}|�]OUvI PQcEI I�?7P_I{t_LO|�\�]ELOz{I�tvP |iN^L4P��Vz{|�yEP_]Ot_I�dw$��Uvz{I�LM|�t_NSRTU*|�t_I LER�P*N^dOI�L$P_NSKMI�d�N^LaPQcONSU I�?E|�`byO\^IiDI c7]EU76�NS��8 I0cM|&2TI

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St = {st ∈ RN×t|stij ∈ {0, 1}, i = 1, . . . , N, j = 1 . . . , t}

�)R�t*I#?E|�`�yE\^I,6MNG�?P_cOI�yE\�|�LOLEN^L54�z{R�LOU_NSUvPQU*R���P 8 Rb�%]EPQ]OtvI0P_N^`bI y�I�t_NSRgdOU*|�LEd�8 I/cM|&2�I�P 8 RLORgdEI{U76MP_cOIWN^L4PQI�t_yOtvI�P |oPQN^R�LaR��?Uvz{I�LM|�t_NSR

s2 =

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1 10 1

)

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b(st)R��*U_z�I{LM|it_N^R

st ∈ St, t > 1UvR=P_cM|iP

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ij = stij, i = 1, . . . , K, j =

1, . . . , t − 1D ��L RiPQcOI�t 8 R�t_dOUS6G8 Iz{|�L dOI�KOLOI PQcOI/yOt_I�2gNSRT]OU Uvz{I{LO|�t_NSRa|iP

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yOt_RTwO|�wON^\SNSP���R���U_z{I�LM|�tvN^Rst ∈ St

z}|�L�w�I0z{|�\^z�]O\�|oPQI{da�%tvRT`

p(st) =N∏

i=1

[(p+i )qt+

i(st)(p−i )qt−

i(st)],

8cOI{tvIqt+i (st)

|�LOdqt−i (st)

tvI{yOtvI{U_I�L4P/PQcEI�L$]O` w�I{t0R�� PQN^`bI{U P_cOI�dEI{`�|�LOd 4TRgI�U�]Ey3RTtdOR=8L ]OL4P_N^\ y�I{tvN^R7d

tD�[gN^LEz{I PQcEII7g$]M|iP_N^RTL

qt+i (st) + qt−

i (st) = tcORT\SdOUM PQcOI�dOI�`V|�LEd

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Di(st) =

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i(st)(di)

qt+

i(st)Di

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st ∈ St |�\^\SRgz{|iPQNSRTL R���2TI{cEN^z{\SI{U xi(st)z}|�LAw�IM8t_NSPvPQI�L3|�U

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|�\^\SRgz{|iPQNSRTL z}|iLOLOR�P�w�I�z}|�\Sz{]O\^|iPQI�d0NSL PQcONSUG8�|}� � 8 IpLOI�I{dWPQR*z}|�\Sz{]O\^|iPQI?P_cOIp|�\^\SRgz{|iPQNSRTL �%tvRT`PQcOI*dOI{z�N^UvN^RTLOU P_cM|iPT8 I{t_I*`�|�dOI N^LP_cOIyOt_I�2gNSRT]OU�y�I{t_NSRgd Rxi(s

t $&%(' ) =∑N

j=1 xji(b(st $&%(' ))

� R=8a6MPQcOIWz{R�LOUvP_tQ|�NSL$P_U R���RT]OtRTyEP_N^`bN^�}|oPQN^R�LVyOtvRTwO\^I�`xw�I{z�RT`�I

FTD ��LONSP_N�|�\�z{RTLEUvPQt_|�N^L4P_U7R :|X� ∑N

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xi(s0) ≤ 1, i = 1, . . . , N

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:|X� ∑N

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�|iP `bRTUvP R�LOIF2TI�cON^z�\^I dONSU_yM|oPQz cOI�dPQR�LOR7dOI

j�

� LM|�\SR�4TRT]EU_\S� PQR ��s�s �%RTtWyOtvR�3vI{z P y�RTtvPv�%RT\^NSRaU_I{\SI{z PQN^R�LG6��%R�t0P_cOI�dEI�PQI�t_`bN^LM|oPQN^R�L=Ri� RTw9A3vI{z�P_NB2�I9�%]OLEz�PQNSRTLG6T8 I9LOI�I{d c t_I�U_RT]Et_z{InU_]OtvyO\^]EU_I{USc�R�� I{|�z c P_I{t_`bN^LO|�\ Uvz{I{LO|�t_NSROD ��L RT]OtyOt_RTwE\^I{` PQcOI�U_I-t_I�U_RT]Et_z{I-Uv]Ot_yO\S]OU_I�U9|�t_I P_cOI=|�`�R�]OL4PQU�]OLEdOI{\SNB2TI�t_I�d NGPQI{`bU 8NGPQc�z cOR>AU_I{L dOI�\^NO2TI{t �=UvPQt_|iPQIS4��TD �)R�t�I{|�z c UvP_tQ|iP_I74��� %�%I{|�U_NSwO\^Ibz{RT`/wON^LM|oPQN^R�L R��

xij(st)c U*�WP_cOI{UvI

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1),

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RS(st) = RS(b(st)) −

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t).

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EV =`bN^L ∑

st∈St

p(st)RS(st), F[�

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∆+(st)t_I�yOt_I�U_I{L4P_UWP_cOI dOI�27N�|iP_N^RTL

�%t_RT` I�?gy�I�z�PQ|iPQNSRTLC2i|�\^]OI ]OyX8�|�tvdOU*|�LOdaPQcEI 2i|�\^]OI R��∆−(st)

dOR=8LX8�|�tvdOU76MtvI{Uvy�I�z�PQNO2TI�\S�TD�)R�t I}|�z c9PQI{tv`�NSLM|�\�U_z{I�LM|�tvN^R56OR�LO\S�VRTLOIWR���PQcOI�U_I0dOIS27N�|oPQN^R�LM|�\ 2i|�tvN�|�wO\SI{U�z{|�L9dONBi�I{t��%tvRT`�ED� L9RTw�3vI{z PQNO2TIW�%]OLOz�P_N^RTLaP_cM|iP*|�\SU_RP |,HiI{Ut_NSU.H�NSL4PQRb|�z{z�RT]OL4P&6Mz{|�L�PQc$]OUw�I 8tvNSP_P_I{L9|�U

`bN^L ∑

st∈St

p(st)[RS(st) + ρ(∆+ + ∆−)] Y��

��L1 Y,��6MPQcOI0z{R7I (bz�N^I{L4Pρ > 0

658cONSz c dOI�y�I�LOdOU R�LndOI�z{NSU_N^R�Ln`�|,H�I�t7c U tvN^U.H�|iP_P_NSPQ]EdOI0|�LOdN^U dOI PQI{tv`�NSLOI{d9N^L�UvRT`�I dEI{z{NSU_NSRTL9AuP_cOI{RTtvI�P_cON^z{|�\�`�I PQcOR7dG6Oy�I{LO|�\SP_N^�{I�U��%t_RT` 2i|�t_N^|�wON^\SNSP���DH I{y�I{LOdEN^L54aRTL z{|�U_I>6�RTLOI/z{RT]O\Sd |�\SU_R�z�RTLOU_NSdOI{t U_I{yO|�tQ|iP_IK2i|�\^]EI{UJ�%R�tJPQcEI z{R7I (bz{NSI{L4PQU Ri�dOIS27N�|oPQN^R�LM|�\ 2i|�tvN�|�wO\SI{U�D ��L � ]EUvP |o�%U_U_R�LGc UJ|�LOd [E|�\SR5c U :Y��T�4f,� yOt_I�\^N^`bN^LO|�tv��I�?E|�`byO\^I>6OPQcOIt_I{UvRT]Otvz{I Uv]Ot_yO\S]OU_I�Uf8 I{tvI `bRTLOI���|iLOd�RTLE\S� dOI�2gN^|iPQNSRTLOU�dOR=8LX8�|it_dOU 8 I{tvInPQ|,H�I�L N^L4PQR|�z{z�RT]OL4P}Dn��L3PQcOI�dOI{\SNB2�I{tv�=yOtvRTwO\SI{`�6 4TI�LOI{t_|�\^\G� U_y�R�H�I�LG6uRTLOI�z{RT]E\^dAUv]OyOy�RTU_IVP_cM|iP0PQcOIdOIS27N�|oPQN^R�L dOR=8L 8 |�tvdOUWN^U z{R�LOU_NSdOI{tvI{d�8 R�t_U_IP_cM|�L PQcEIdOIS27N�|oPQN^R�L ]OyX8�|�tvdOU{DKI c7]EU76�PQcOIz{R7I (bz{NSI{L4P*R��

∆+ 8 R�]O\^d9w�IWUv`V|�\S\^I�t PQcO|�L9P_cOIWz{R7I (bz{NSI{L4P*R��∆−

D

� � � � � S� � � ��� �� ����

��LAPQcONSU/z cM|�ygPQI{tS6J8 I�wE]ON^\Sd | L$]O`bI{t_NSz}|�\pz}|iU_I�I#?O|i`�yO\SIVR��*|n`�R7dOI{\pdOI�U_z{tvN^w�I{d NSL3PQcOIyOt_I�2gNSRT]OUJz cM|�yEP_I{t{D k I��%R�t_` ]O\^|iPQI P_cOI0yOt_RTwE\^I{` 8NGPQc9KL?gI{d yM|itQ|�`bI�P_I{t_UJ|�LOdnU_RT\O2TIWPQcOI

Page 11: B D0D0D0DD0DD0D0DD0DWDD0DD0D0DD0DD0DWDD0D0D F}Xsalserver.org.aalto.fi/vanhat_sivut/Opinnot/Mat-2.4108/... · 2005-02-21 · " #$ " %#$&' )(* + #-,/.0.213#4 + 5 5 60#7+ 48%#9 : ;=

RTyEPQNS`V|i\�dOI{\SNB2�I{tv�9U PQtQ|oPQI74i�a�%RTt*P_cON^U*yM|�t PQNSz{]O\^|�t yOt_R�wO\^I�` 8NSPQc9z{R�`�yO]gPQI{t A�wO|�U_I�d PQR7RT\^U�Dk3I0|�NS` PQRMHiI{I{y9PQcOIWI#?O|i`�yO\SIW|�UU_NS`�yO\SI |�LOd9N^\S\^]OU PQt_|iPQNO2TI |iU y�RTUvU_NSwO\^IiD

� ��� ��� ��� ��� � ��� �� � ���� ����� �

V I P]OU�|�UvU_]O`bIVP_cM|iPF8 IacM|&2TI�P 8 RD2TI�cON^z�\^I{U� K = 2

��|�LOdAPQcEt_I{IV\^R7z}|iP_N^RTLOU �L$]O`/w�I�tR�� LOR7dOI{U

N = 3� D � dON^Uvz{tvI�PQI9PQNS`�I�y�I{t_Uvy�I�z�P_NB2TI z{RTLOUvN^U PQUVRi�

t ∈ {0, 1, 2}6 UvR3PQcOI

dOI{z�N^U_NSRTLOU0|�tvI`V|�dEI�|iPt = 0

|iLOdt = 1

D0I cEIbdEI{z{NSU_NSRTL-`V|,H$NSL54ayOt_R7z{I�U_U0|�U0|C8cORT\SIz}|�L9w�IWdOI�U_z�t_N^w�I{d9yOcM|iU_I0w$��yOcM|�UvI/|�U

FTDH I{z�N^dOI!8cOI�t_IJP_R/|i\^\^R7z}|oPQI�PQcEI!2�I{cONSz{\SI{UpN^LONGPQN�|i\^\S��D�I cOI NSLONSP_N�|�\g|�\S\^R7z}|iP_N^RTL NSU�dOI�P_I{tNA`�NSLOI{d-w4� P_cOI02�|i\^]OI�U0Ri� wONSLM|�t � dOI�z{N^UvN^RTL�2i|�tvN�|�wO\SI{U

xij(s0)Uv]Oz c P_cM|iP

xi(s0) =

∑3j=1 xij(s

0)

YgD I cOI0N^LONGPQN�|i\�dEI{`�|�LOd9N^U t_I{|�\^NS�{I�dn|iLOd�yM|�tvP_\S�VUQ|iP_N^UvKOI{d t = 0

�CEDH I{z�N^dOIM8cOI�t_IPQRn|�\^\SRgz{|iPQI0PQcOI02�I{cONSz{\^I�UWLOI�?7P}D�I cONSU N^U0|�\^UvRadOI�P_I{tv`�NSLOI{d-w4� PQcOI2i|�\^]OI�UR��

xij(s0), i, j = 1, 2, 3

U_]Oz c9PQcM|oPxi(s

1) =∑3

j=1 xji(b(s1))

XODH I{`�|�LOd9N^LaP_cOIWLOI�?7P*y�I�t_NSRgd�N^U tvI}|�\SN^��I{d�|�LOd9yM|�t PQ\G��UQ|oPQN^U KMI{d t = 1

�fgDH I{z�N^dOIM8cOI�t_I�PQRn|�\^\SRgz{|iPQI/P_cOI02TI�cON^z�\^I{U0N^L P_cOI�UvI{z�RTLOd3y�I{tvN^R7d�DMI cOI�|�\^\SRgz{|iPQNSRTLR���PQcEI U_I�z{RTLOd�y�I{tvN^R7dVN^U dOI�P_I{tv`�NSLOI{d�w4�bPQcOId2�|i\^]OI�U R���wONSLM|�t ��dEI{z{NSU_NSRTL 2i|�t_N^|�wO\SI{Uxij(s

1)U_]Oz c9PQcM|oP

xi(s2) =

∑3j=1 xji(b(s

2))

�ED I cOIWKMLM|�\�dEI{`�|�LOd9N^U t_I{|�\^NS�{I�dn|iLOd�yM|�tvP_\S�VUQ|iP_N^UvKOI{d t = 2

� D

[gN^LOz�I 8 I cM|&2TIWC/\SRgz{|iPQNSRTLOU76$NSL�y�I�t_NSRgdnF 8 I cO|=2�I PQR�PQ|�\�R��23 = 8

Uvz{I{LO|�t_NSRTU{D k I N^LEdOI�?PQcOIWU_z�I{LM|it_N^R�U*NSL9|�%R�\^\^R=8NSL54K8 |���D

s1(1) s1(2) s1(3) s1(4) s1(5) s1(6) s1(7) s1(8)1 1 1 1 0 0 0 01 1 0 0 1 1 0 01 0 1 0 1 0 1 0

< N�|�N^LEdOI�?gN^LL4bP_cOI�Uvz{I�LM|�t_NSRTUJ|iPt = 1

|�UJ|�w�R[2�I,6�8 I/z}|iL I}|iU_N^\G�C8t_NGPQIWPQcOI0U_z�I{LM|it_N^R�U |iPt = 2

N^L |�L3NS\^\S]OUvP_tQ|iP_NB2TI 8�|}�=w4�-z{RT` wEN^LONSL54aPQcOIbz{RT\S]O`bL 2�I{z�P_RTt_U�N^L4PQtvRgdE]Oz{I�dA|�w�R=2TI�D�)R�t I�?E|�`byO\^I>6

s2(1, 8) =

1 01 01 0

F��

Page 12: B D0D0D0DD0DD0D0DD0DWDD0DD0D0DD0DD0DWDD0D0D F}Xsalserver.org.aalto.fi/vanhat_sivut/Opinnot/Mat-2.4108/... · 2005-02-21 · " #$ " %#$&' )(* + #-,/.0.213#4 + 5 5 60#7+ 48%#9 : ;=

t_I{yEt_I{UvI{L4PQU |bUvz{I{LO|�t_NSR�8cOI�t_I�|�\^\�PQcOI0dOI{`�|�LOdEUZ4�R�]Oyn|iPKMtvUvPJy�I�t_N^R7d9|�LOd�4TRbdOR=8L |iPU_I{z�RTLOd�y�I{tvN^R7d�D � |iP_]OtQ|�\S\S�,6T�%RTt

t = 264PQcOI*L$]O` w�I{t�R���U_z�I{LM|�tvN^RTUpN^U

82 = 6464U_NSLOz{I*IS2�I{t �

z{RT` wEN^LM|iP_N^RTLaRi�?dOI KMLOI�d U_NSL54T\SI�A�y�I{tvN^R7d�z cM|�LL4TI{Ut_I�yOt_I�U_I�L$P_U |�U_z�I{LM|�tvN^RODk3I�z cOR7RTU_I�P_cOIayM|�t_|�`bI�PQI�t_UR��*PQcEIa`�R7dOI{\ |�U �%RT\S\^R=8U{D I cOI�N^LONGPQN^|�\�dOI{`�|�LOdOUNSLAPQcOI\^R7z}|iP_N^RTLOU�|�t_I

D1 = 58, D2 = 20, D3 = 30.�)R�t�PQcOIWyM|�t_|�`bI�PQI�t_U R�� |wON^LOR�`�N^|�\�\^|iP_P_N^z�I,698 IWz cOR7RTU_I

(u1 u2 u3) = (2 1 2),

|�LOdG6)|�U]OUv]M|�\ 6

(d1 d2 d3) = (1/u1 1/u2 1/u3) = (0.5 1 0.5).

I cOI/\SR�4TN^z w�I�cON^LOd9PQcONSUz cORTN^z�I�NSU76MLM|oPQ]Ot_|�\^\G��6EP_cM|iP*NS��PQcOI0dOI{`�|�LOd�KMtvUvPd4TRgI�UJ]Ey |�LOd9N^LPQcOIWLOI#?gPJy�I{t_NSRgdadER[8L 6�NGP t_I PQ]OtvLOUPQRPQcOIW\SIS2TI�\G8cOI{tvI�NGP RTtvN^4TNSLM|�\^\G�08�|iU{DI cOI 2i|�\S]OI{U Ri��P_cOIWyOt_RTwO|�wON^\SNSP_N^I{Up�%RTt�PQcEI0dEI{`�|�LOdOU PQRM4TR�]Oyn|�tvI dOI�P_I{t_`bN^LEI{dn|�U

(p+1 p+

2 p+3 ) = (0.1 0.5 0.5),

|�LOdaPQcOI z�RTt_tvI{Uvy�RTLEdON^L54yEt_RTwM|iwON^\SNSPQI�U �%RTt PQcOI dOI�`V|iLOdOU�PQR04�RdOR=8Lnz{|�L�w�I z{|�\^z�]O\�|oPQI{d�%t_RT`

p−i = 1 − p+i , i = 1, 2, 3

D� ?gy�I{z P |iP_N^RTL 2i|�\^]EI R��$PQcOI�dOI�`V|iLOd �%RTt�\SRgz{|iPQNSRTL

i|iP

tz}|iL0w�Iuz}|i\^z{]E\�|iP_I{d �%tvRT`

EV (Di(t)) =∑

st∈St p(st)Di(st),P_cOI�2i|�\S]OI R��)I�2TI�tv� yO|�tQ|�`bI�P_I{t�N^L/P_cOI �%RTtv` ]O\^|*cM|iU?w�I{I{Lz{|�\^z�]O\�|oPQI{d

w�I �%RTt_IiD [gN^LEz{IaPQcOI9z cM|�LL4TI{U�R�� dOI�`V|iLOdOU�|iPbLORgdEI{U�|it_I�N^LOdEI{y�I{LOdOI�L4P�R�� I}|�z c R�PQcOI�t76|�L I}|�UvN^I�t 8�|}� P_Raz}|�\Sz{]O\^|iPQI0PQcOI�U_IF2i|�\S]OI{U NSU dOI PQI{tv`�NSLON^LL4�|�L I#?Ey�I{z PQI�d ` ]E\SPQNSyO\^NSz}|iP_NB2�Iz cM|�L54TI �%RTtaLERgdOI

iNSL | wEN^LORT`bN�|i\*\^|iP_P_N^z�I w4�

(p+i ui + p−i di)

6 |iLOd 8 I-z}|iL]4TI PaPQcOII�?gy�I�z�PQ|iPQNSRTLf2i|�\S]OI|iP

t27N�| ` ]O\GPQNSyO\S�7N^LL4 PQcOI*I�?gy�I{z P |iP_N^RTLM2i|�\^]OI*|iP

t−1w4�PQcONSUu��|�z�P_RTt{D

� U 8 I z}|iL�LOR�P_N^z{I*�%t_RT` yEN^z�P_]Ot_I�F�6$PQcOI dEI{`�|�LOda|iP \^R7z}|oPQN^R�LnFJcM|�U |0dOI{z�t_I}|iU_N^LL4 PQt_I�LOd�DH I{`�|�LOd�|iPu\SRgz{|iPQNSRTL�Y t_I�`V|�NSLOU z�RTLOUvPQ|�L4PpR[2�I{tuPQNS`�I>6,8cOI�t_I}|iUpP_cOI dOI{`�|�LOdb|iP�\SRgz{|iPQNSRTLC/PQI{LEdOU PQRN^LEz{t_I{|�U_IiDJI c$]OU76M�%t_R�`x| U P |iP_N^z 27N^I�8y�R�N^L4P&67PQcOI R�yEPQNS`V|�\�dOI�\^NB2�I{t �VU PQtQ|oPQI74�N^I{U8 RT]E\^d w�I/PQcEI/RTLOI�U PQcO|iP |�\S\^R7z}|iP_I PQcOIK2�I{cONSz{\SI{U UvR�PQcM|oP \^R7z}|oPQN^R�LOUWF|�LOd CV|�t_I 2gNSU_NGPQI{d|iP*IS2TI�tv�

t ∈ {0, 1, 2}D

I cOI{tvI��%RTtvI,6 w4� PQcEI�z cOR�U_I{L3z{RT` wEN^LM|iP_N^RTL-R�� yM|�t_|�`�I PQI�t_U76�8 I�z{|�L3U PQ]OdE�-P_cOI�]EUQ|�wONS\^NGP��R���PQcOIRTygPQN^`bN^�{|iPQNSRTL9`�R7dOI{\:DW��L-|�dg�gLO|�`�NSz/`bRgdEI{\^\SN^L54bI{LX27N^tvRTLO`bI{L4P&6�|i\^U_Ra\SRgz{|iPQNSRTL Yz}|�L w�IC8 RTt PQc9Aa27NSU_NSP_N^L54 w$� |j2TI{cEN^z{\SI,R�NG� 6 �%RTt I#?O|i`�yO\SI,6�U_z�I{LM|it_N^R

s1(2)tvI}|�\SN^�{I�U76uPQcEI

- ��� $L-,'udOI{`�|�LOd9N^L9LOR7dOI�CV|iPt = 1

NSU30 · 0.5 = 15

]OLONSP_U{D���L9PQcONSUJUvNSP_]M|iPQNSRTLG6Oz�\^I{|�t_\G��6

FTF

Page 13: B D0D0D0DD0DD0D0DD0DWDD0DD0D0DD0DD0DWDD0D0D F}Xsalserver.org.aalto.fi/vanhat_sivut/Opinnot/Mat-2.4108/... · 2005-02-21 · " #$ " %#$&' )(* + #-,/.0.213#4 + 5 5 60#7+ 48%#9 : ;=

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

30

40

50

60

Time

Exp

ecte

d de

man

dExpected demands over time periods

Location 1Location 2Location 3

��N^4T]OtvI F�R � ?gy�I�z�P_I{d dOI�`V|iLOdOU*|iP PQNS`�I y�I�t_N^R7dOU

\^R7z}|iP_N^RTLnYw�I�z{RT`bI{U*` ]Oz c `bRTtvI�|oP_PQt_|�z�P_NB2�I�|i\SPQI�t_LM|oPQNB2�I0|�UJNGP!8 RT]E\^d cO|=2�I/w�I{I�L NS��PQcOIRTyEPQNS`�NS�}|iP_N^RTL�cM|�d9w�I�I{Ln|�z{z�RT`byO\^NSU_cOI�d9R�L�P_cOIWwM|�UvN^UR���I�?gy�I�z�PQ|iPQNSRTL\2�|i\^]OI�U{D��L=R�t_dOI�t PQR\4TI P PQcEI U_RT\S]EPQNSRTL w4�-�n|iPQ\^|�w U_Ri� P 8�|�tvI,6G8 ILOI{I�d PQRaI�?gyOtvI{UvUWP_cOIyOtvRTwO\^I�`�%RTt_` ]E\�|iP_I{d NSL z cM|�yEP_I{tWXa|�LEd=Uvy�I�z{NGKMI{d-N^L z cM|�yEP_I{t0f96�NSL UvPQ|�LOdM|it_d �%RTt_` R��uP_cOIRTyEP_NOA`�NS�}|iP_N^RTL yOtvRTwO\SI{`9D�I cEI�RTw 3vI{z�P_NB2�Ib�%]OLEz�PQNSRTL=cM|�U PQR9w�I�R�� �%RTt_`

cTx6 8cEI{t_I

cTNSU�|

1 × n2�I{z PQRTt*|�LOd

xN^U|�L

n × 12TI�z�P_RTtJz�RTL4P |�NSLON^LL4dOI{z�N^U_NSRTL\2i|�t_N^|�wO\^I�U{D

��L3RT]Ot I#?E|�`�yE\^Iz}|iU_I,6�PQcOI�dOI�z{NSU_N^R�L=`�|,HiI{tWN^U tvN^U.H�A*LOI�]EPQt_|�\d ρ = 0

� |�LOdW8 I�z{|�L ]EU_IPQcOI0RTw�3vI�z�PQNO2TIW�%]OLOz PQN^R�L F[� DTI cOI�t_I �%RTt_I>6�z�\^I}|it_\S�,6L8 I�dOR�LORiPJcO|=2�I�P_RbNSLOz{\S]OdOI P_cOI�dEIS27NOA|iPQNSRTLM|�\ 2�|it_N�|iwO\^I�U N^L-RT]Ot `bR7dOI{\:D su]Ot_I�wONSLM|�t � N^L4P_I74TI�t yOt_R�4�tQ|�`b`�NSL54Vz{|�L-w�I�z{|�t_tvN^I{dw$��wEN^L4PQyOtvR�4FA��%]OLOz PQN^R�L9Ri�u� |oPQ\�|iw�E ygPQN^`bN^�{|iPQNSRTL I�R7RT\^w�R&? Dk3I9cM|&2TI

32 = 9wEN^LM|�t �AdOI�z{NSU_N^R�Le2i|�tvN�|�wO\SI{U �%RTt P_cOI9z{]Otvt_I{L4P�`�R�`�I�L$PU

t = 0� |�LOd

|�dOdONGPQN^R�LM|�\T� �%RTt�IS2�I{t ��Uvz{I�LM|�t_NSR |oPt = 1

D _*I{LOz�I,6iP_cOI�PQR�PQ|�\TL$]O` w�I{t�R��gwON^LM|itv� 2i|�tvN�|�wO\SI{UN^U

9 + 8 · 9 = 81D

��L |�\S\�U_z�I{LM|it_N^R�U76OPQcEI0wEN^LM|�t � 2i|�tvN�|�wO\SI{Uxij(s

t)|it_I z{R�\^\^I�z�P_I{d�N^L9|z{RT\S]O`bL2�I{z PQRTtw4�

F�Y

Page 14: B D0D0D0DD0DD0D0DD0DWDD0DD0D0DD0DD0DWDD0D0D F}Xsalserver.org.aalto.fi/vanhat_sivut/Opinnot/Mat-2.4108/... · 2005-02-21 · " #$ " %#$&' )(* + #-,/.0.213#4 + 5 5 60#7+ 48%#9 : ;=

x(st) =(

x11(st) x12(s

t) x13(st) x21(s

t) x22(st) . . . x33(s

t))T

.

k3IWz{R�\^\^I�z�P P_cOI 2�|it_N�|iwO\^I�U�N^L9|z{RT\S]O`bLC2TI{z PQRTtJ|�z�z{R7dON^L54P_R�| �%RT\S\^R=8N^LL4 U_ISg7]EI{LOz�I,R

x =

x(s0)x(s1)(1)x(s1)(2)DDDx(s1)(8)

=

x11(s0)

x12(s0)DDD

x33(s0)

x11(s1(1))

x12(s1(1))

x13(s1(1))

x21(s1(1))DDD

x33(s1(1))

x11(s1(2))DDD

x33(s1(8))

.

� U_NSL54PQcONSULOR�P |oPQN^R�LG6gP_cOIWRTw�3vI{z PQNO2TI �%]OLOz PQN^R�L F[��6MI#?gyO\^NSz{NSP_\S�bz}|i\^z{]E\�|iP_I{dG6Ow�I{z�RT`bI{U

EV = M +(

cT

0cT

1

)

x, :CX�

8cOI{tvIM2�I{z PQRTtcT

0∈ R1×9 NSU PQcOIz�RgI)(�z�N^I�L$Pa2�I{z�P_RTt �%RTt wONSLM|�t � dOI�z{NSU_N^R�LOU 2i|�t_N^|�wO\^I�U |iP

t = 0|�LEd�2�I{z�P_RTt

cT

1∈ R1×72 �%RTt wON^LO|�tv�VdOI{z�N^U_NSRTLOU 2i|�t_N^|�wO\SI{U |iP t = 1

6Et_I{Uvy�I�z�P_NB2TI�\S��DI cOI 2�|i\^]OI�U R���PQcOI z�RT`�y�RTLOI�L4PR��

c0

P_cM|iP` ]E\SPQNSyO\^NSI{Uxij(s

0)NSL F[� z{|�Law�I z{|�\^z�]O\�|oPQI{d

�%t_RT`

−∑

s2∈S2

p(s2)(Di + Dj(b(s2))),

|�LOdG6MtvI{U_y�I{z PQNO2TI{\G��6O�%RTt�P_cOIWz{RT`by�RTLOI{L4P*R��c1

P_cM|iP` ]E\SPQNSyO\^NSI{Uxij(s

1(k)), k = 1 . . . 8N^L

F[�#658 IWRTwEPQ|�N^L

−∑

s2∈S2

p(s2)Dj(s2).

k3I/|�\^UvRVz}|iLDg$]ONSP_I/I}|iU_N^\G��I�?gyOtvI{U_UJPQcOI z�RTLOU PQtQ|iN^L4PQUJN^L��%RTt_`Ax ≤ b

RTtS6��%RTt*I7g$]M|oPQN^R�Lz{RTLOU PQt_|�N^L4PQUS6

Ax = bD���L=|�\S\)6)PQcOIUvN^��I R��p`V|oPQt_N ?

ANSU

(3 + 27) × 81z�RTt_tvI{U_y�RTLOdEN^L54

N^LOISg7]O|�\^NGP���z�RTLOU PQtQ|iN^L4PQU F}w � |�LOdW :Yiw ��6�8cOI{tvI}|�U�P_cOI{tvI |�tvI9 + 24

I7g$]M|�\SNSP��bz{R�LOUvP_tQ|�NSL$P_U�%t_RT`; F�|X��|�LEd :Y�|X� NSL�P_cOIW\^NSUvP RTL�yM|>4TI0�ED

F�C

Page 15: B D0D0D0DD0DD0D0DD0DWDD0DD0D0DD0DD0DWDD0D0D F}Xsalserver.org.aalto.fi/vanhat_sivut/Opinnot/Mat-2.4108/... · 2005-02-21 · " #$ " %#$&' )(* + #-,/.0.213#4 + 5 5 60#7+ 48%#9 : ;=

� ��� � � � � ����� � � � ����� � ��� ����� �

� NO2TI{LAP_cOIayOtvRTwO\^I�`�NSL3PQcEN^U/�%RTtv`U6:8 Iaz{|�L U_RT\O2TI�P_cOI�`bN^LONS`�NS�}|iP_N^RTL yOtvRTwO\SI{` w$� PQcOIwON^L4PQyEt_R�4W�%]OLOz PQN^R�LbRi��� |oPQ\�|iw�D � U@8 IJz{|�LbU_I{I�%t_R�` I7g$]M|iP_N^RTLU �CX��64PQcEI!2i|�\S]OIR���PQcOI*N^LON APQN�|i\EUvP_Rgz H

M >> 0 %z{RTLEUvP |iL$P � z}|�Lbw�I UvI{yM|it_N^��I{d��%t_RT`�RTw�3vI�z�PQNO2TI �%]OLOz PQNSRTL�D I cEI{t_I �%RTt_I>6

PQcOIaRTw 3vI{z�P_NB2�I��%]OLOz�P_N^RTL 8 Iaz cOR7RTU_I�P_R w�IV`�NSLON^`bN^��I{dG6?|�z�P_]M|�\^\G�-N^U/P_cOIxAWdEI{y�I{LOdOI�L4P

yM|�tvP*NSL :CX�#6OPQcOIWz cM|�LL4TIWR���P_cOIWUvP_Rgz H)D�?N^tvUvP&6o\^I P?]EU�U PQ]OdE�WP_cOI RTygPQN^`bN^��I{d0|�\S\^R7z}|iP_N^RTL R��92TI�cON^z�\^I{U?|iP

t = 06>8cONSz c/NSU�dOI PQI�t_`bN^LOI�d

w$�=PQcEI�RTyEP_N^`�|�\ 2i|�\^]OI�U/R��wEN^LM|�t �=dOI{z�N^U_NSRTLQ2i|�tvN�|�wE\^I{Uxij(s

0)D � z{z�RTt_dONSL54 P_R PQcOI�dOI�A

U_z{tvN^w�I{d9dOI{z�N^UvN^RTLa`�|,H$N^L54yEt_R7z{I{UvU76MwM|iU_I{d9RTLaPQcEI0NSLE�%RTtv`V|iP_N^RTL�R���PQcOIWNSLONSP_N�|�\)dOI�`V|�LEdOU76|�\^UvR�P_cOI�|�\S\^R7z}|iP_N^RTLnR���2�I{cONSz{\SI{U0|iP

t = 1NSU dOI�P_I{tv`�NSLOI{d-w4� dOI�z{N^UvN^RTLEUW|iP

t = 0DFI cOI

t_I{Uv]O\SP_U 4TNO2TI{L w$� � |iP_\�|�w |�t_I/U_cOR=8L N^L9P |�wE\^IVF/U_RVP_cM|iP NG��|Vz�I{\^\�N^UJ`V|it.H�I�dj8NSP_c �����6PQcOIWz{R�t_t_I�U_y�RTLOdONSL54bLORgdEI N^U 27N^UvNSPQI�dnw4��|F2�I{cONSz{\^I0|iPz�RTt_tvI{Uvy�RTLEdON^L54

tD

t = 0 t = 1LOR7dOI�F � �

LOR7dOI�YLOR7dOI0C � �

I?|�wE\^I�F�R�E yEPQNS`V|�\�LOR7dOI{U �%RTt 2TI�cON^z�\^I�UP_R�27NSU_NSP*|oPt = 0

|�LOdt = 1

wO|�U_I�d RTLn|&2�|iN^\�|iwO\^IN^LE�%RTtv`V|oPQN^R�La|iP

t = 0� P

t = 0PQcOI |&2i|�N^\^|�wO\^I?N^LE�%RTtv`V|oPQN^R�L |�t_IuP_cOI NSLONSP_N�|�\odOI�`V|�LEdOU7R

D1 = 58, D2 = 20, D3 =30DTP �VI�?gy�I{z�P_N^RTL 6)|iP

t = 16EV (D1) = 37.7, EV (D2) = 20, EV (D3) = 37.5

DJI c$]OUS6NSPN^U R�w 27NSRT]OU PQcM|oPRTwEP |iN^LOI�d9tvI{Uv]O\SP_UK �%RTt t_NSU.H�LOI{]EP_tQ|�\�dEI{z{NSU_NSRTL�`V|,HiI{t*�|it_I z{R�t_t_I�z�P{D� I�?7P76L8 I U PQ]Odg�VPQcEI RTyEP_N^`�|�\)|�\^\SRgz{|iPQNSRTL�|iP

t = 2DJI cEI t_I�U_]O\GPQU P_cM|iP dOI�y�I�LOdaRTLC8cONSz c

U_z{I�LM|�tvN^RbRgz�z{]Otvt_I{dn|iPt = 1

6M|it_IWyOt_I�U_I�L$P_I{dnN^LaP |�wE\^I0YgD��`�`bI{dON^|iPQI z{RTLOz�\^]OUvN^RTL�NSU�PQcOI UvN^`bN^\�|it_NSP���Ri��P_cOIWRTyEPQNS`V|i\�UvRT\^]EP_N^RTLaNSLaU_I�2TI{t_|�\�U_z�I{LM|�tvN^RTU|iP

t = 1D V I P/]OU�dOI{LER�PQI�PQcOI�RTyEP_N^` ]O` U_RT\S]EPQNSRTL-N^L

s1(k)w4�

x∗(s1(k)) � 8 I�LOR�P_N^z{IPQcM|iP

s1(1) s1(2) s1(3) s1(4) s1(5) s1(6) s1(7) s1(8)1 1 1 1 0 0 0 01 1 0 0 1 1 0 01 0 1 0 1 0 1 0LORgdEIF∗ ∗ ∗ ∗ ∗ ∗LORgdEI0Y

∗ ∗ ∗ ∗ ∗ ∗LORgdEIWC∗ ∗ ∗ ∗

I?|�wE\^I Y�RTE yEPQNS`V|�\MLERgdOI�U P_R 27N^UvNSP |iPt = 2

z�RTLOdONGPQNSRTLM|�\OP_R 8cONSz caU_z{I�LM|�tvN^R/R7z{z�]Ot_tvI{d�|iPt = 1

F}X

Page 16: B D0D0D0DD0DD0D0DD0DWDD0DD0D0DD0DD0DWDD0D0D F}Xsalserver.org.aalto.fi/vanhat_sivut/Opinnot/Mat-2.4108/... · 2005-02-21 · " #$ " %#$&' )(* + #-,/.0.213#4 + 5 5 60#7+ 48%#9 : ;=

0 0.5 1 1.5 21

1.5

2

2.5

3Scenarios 1 and 3

0 0.5 1 1.5 21

1.5

2

2.5

3Scenarios 2 and 4

0 0.5 1 1.5 21

1.5

2

2.5

3Scenarios 5 and 7

0 0.5 1 1.5 21

1.5

2

2.5

3Scenarios 6 and 8

�?N^4T]Et_I0Y9R�E yEPQNS`�NS�{I�d�PQR�PQ|�\�t_R�]EPQI�U�%R�t!2�I{cONSz{\SI{U 8NSP_c9tvI{Uvy�I�z�P*PQRbRgz�z{]Otvt_I{d9U_z�I{LM|it_N^R�|iPt = 1

x∗(s1(1)) = x

∗(s1(3))

x∗(s1(2)) = x

∗(s1(4))

x∗(s1(5)) = x

∗(s1(7))

x∗(s1(6)) = x

∗(s1(8)), �X �

IS2TI�L NS�:8 I0z cM|�L54�I�PQcEI 2i|�\S]OI{U*R�� UvRT`bI�yM|itQ|�`bI�P_I{t_UK �dEI{U_yENSPQIu2� D I c7]EU76Y8 I/z�RTLOz�\^]OdOI

PQcM|iPUvN^LOz�I PQcOIWdOI�`V|�LEdn|oP\^R7z}|iP_N^RTL�YNSU z{RTLEUvP |iL$P76MRTLO\G�VPQcEIa2i|�tvN�|iP_N^RTLEU N^La\SRgz{|iPQNSRTLOU F|�LOd�C0|>i�I{z P P_cOIJUvRT\^]gPQN^R�L�|oP

t = 1DJP ��t_I#A�z cOI{z H$N^LL4/PQcOI*N^LEdOI�?gN^LL40Ri� PQcEIJUvz{I{LO|�t_NSRTU

s1 6NSP NSU z{\SI}|�tJPQcM|iPJPQcOIyEt_IS27NSRT]OU ISg7]O|iPQNSRTLOU cORT\Sd NS�

D2(st)N^UJz{RTLOU P |�L4P{D � U 8 Iz�RTL4PQN^L$]OI

PQRaUvP_]OdE�9PQcONSU I#?O|i`�yO\SI,6�PQcOI L$]O`/w�I�t R��T4TI�L7]EN^LOI�\S�ndONBi�I{tvI{L4P Uvz{I{LO|�t_NSRTU PQcO|iPW|�tvI/dOI{|�\SP8NSPQc9N^US6OwM|�UvI{d9RTL �X �#6E�%RT]Ot NSLOUvP_I}|�d�R��?I{N^4�c$P{DI cOI�`bRTt_I U_y�I{z{NGKMz�UvRT\^]EP_N^RTL 2i|�\S]OI{U�R��

xij(st)c U NSLOUvP_I}|�d/R��

xi(st)c U*� 4TNO2TI�Lw$� � |iP_\�|�w�N^U

4TtQ|�yEcON^z{|�\^\G�dOI{`bRTLOU PQt_|iPQI�d�N^LbK�4T]OtvI YgD ��L�PQcOI*K�4T]OtvIJPQcEI |�z PQ]M|�\�t_R�]EPQI�U �%RTtpN^LOdENB27N^dO]O|�\2TI{cEN^z{\SI{U�|�t_IbdOI�`�RTLEUvPQt_|iPQI�d�D �MRTt I#?O|i`�yO\SI,6�N^L-PQcOIb]OyOy�I{t0\^I � PWz{RTtvLOI{t0R�� PQcOIbyON^z PQ]OtvI,6NSP�z}|�LAw�IbU_I{I�LAPQcM|oP NS� U_z�I{LM|�tvN^R

s1(1)RTt

s1(3)Rgz�z{]Ot/|iP

t = 16�P_cOIVRTygPQN^`�|�\utvRT]EPQI�U

RTwEP |iN^LOI�d �%t_RT` � |iP_\�|�wn�%RTt RT]EtJP 8 R2�I{cONSz{\SI{U |�tvI(1 → 3 → 1)

�dORiP_PQI�d wOt_R,H�I{L-\SN^LOI&�|�LOd

(3 → 1 → 3) %U_RT\SN^d�wOt_R�HiI{L \SN^LOI&� DlI cOI z�RTt_tvI{U_y�RTLOdEN^L54=U_RT\S]EPQNSRTL�N^L PQI{tv`�UbR��

wON^LM|itv� 2i|�tvN�|�wE\^I{U NSU �NG�s1(1)

R7z{z{]Et_U �

F�f

Page 17: B D0D0D0DD0DD0D0DD0DWDD0DD0D0DD0DD0DWDD0D0D F}Xsalserver.org.aalto.fi/vanhat_sivut/Opinnot/Mat-2.4108/... · 2005-02-21 · " #$ " %#$&' )(* + #-,/.0.213#4 + 5 5 60#7+ 48%#9 : ;=

x13(s0) = x31(s

0) = x31(s1(1)) = x13(s

1(1)) = 1,

8cON^\SI R�PQcOI�txij(s

0)c U*|�LOd

xij(s1(1))

c UI7g$]M|�\��gDI cOI�c \SI}|�yEU7c�N^LARTygPQN^`�|�\�t_R�]EPQI�U `V|}�=w�I�]OLONSL$P_]ONSP_NB2�I,R PQcEI�t_RT]gPQI{U

(1 → 1 → 1)|�LOd

3 → 3 → 3)UvI{I{`�`/]Oz c `bRTtvI�|iP_P_tQ|�z PQNB2�I�PQcM|iL�P_cOIaRTwEPQ|�N^LOI�d U_RT\S]EPQNSRTL�D � z PQ]M|�\S\S�,6

PQcOI�U_IVtvRT]EP_I{U0|�\^UvR |it_I�RTyEP_N^`�|�\ DFI cEI{t_I�|�tvIbUvIS2�I{tQ|i\ RTygPQN^`�|LR 4Tt_|�yOcONSz}|�\S\S�,6�IS2�I{tv�=Uv]Oz cU_I�P R���P 8 R wOt_R,H�I{L \SN^LOI�U PQcM|iP/z{RTLELOI{z P/PQcEIVy�RTN^L4PQU0`�|�t.HiI{d 8NGPQc �>� N^L K54T]Ot_I�Yn|�LEdPQcOI0wOt_R,H�I{L \^N^LEI{U dORbLOR�P*cM|&2TI 2TI�tvPQNSz}|�\�z�RT`�y�RTLOI�L4PQU76)tvI{yOtvI{U_I�L4PQU |iL RTyEP_N^` ]E`xN^L9I}|iz cU_z{I�LM|�tvN^RODFI cEI�|iw�R=2TI��|�z P z}|iL=w�I�z{RTLEz{\^]EdOI{d-U_NSLOz{IP_cOIRTw�3vI�z�PQNO2TI�%]OLOz PQNSRTL dOR7I{UWLOR�PdOI{y�I{LOd�RTLPQt_|=2�I{\Mz�RTUvP_U

cij

D I c$]OUS67P_cOI|�\^\SRgz{|iPQNSRTL/Ri� 2�I{cONSz{\SI{Up|iPt ∈ {0, 1, 2}

NSUuz{t_]Ez{N�|i\N^LaRT]Ot `bR7dOI{\ 6gLORiP PQcOI NSLOdONB27NSdO]M|�\�t_RT]gPQI{U��%R�t 2TI�cON^z�\^I{U�DJk3I 8NS\^\)�%]OtvP_cOI{tv\S��dEN^U_z�]OU_U P_cON^UN^U_Uv]OIWN^L�z cM|�ygPQI{tJ�ED�h� PQcOI0z{R�LOU_NSUvPQI�LOz��nR���PQcOI0U_RT\S]EPQNSRTL�RTwEPQ|�N^LOI�dnNSUJz�RTLOUvN^dOI�t_I{d 6 |�U_NB4TLONSKOz}|�L4P*RTwOUvI{t`2i|iP_N^RTLN^U�PQcO|iP*NSP dOI�y�I�LOdOU*RTL�PQcOIWU_z�I{LM|it_N^RP_cM|iPR7z{z{]Et_UJ|iP

t = 1DJk3IW|i\^U_RKMLEd9P_cOIWRTwEP |iN^LOI�d

U_RT\S]EPQNSRTLOU��%I}|�UvN^wO\SI�D�[gNSLOz{I�PQcOIpU_z�I{LM|it_N^R�A�z{R�L$P_N^L54�I{L4P�U_RT\S]EPQNSRTLOU�|�tvIpNSdOI{L4P_N^z}|i\�8NSPQc0t_I�U_y�I{z PPQR�P_cOI |i\^\^R7z}|oPQN^R�LaR�� 2TI�cON^z�\^I�U |iP

t = 0|�LOd

t = 16 8 I/U PQ]OdE�9PQcEI�tvI}|�UvRTLM|�wONS\^NGP��aR�� P_cOI

U_RT\S]EPQNSRTLOU w4��PQcEI�|i\^\^R7z}|oPQN^R�L�|iPt = 2

Dk3I0U_I�I0�%tvRT`xPQcEI�UvRT\^]EP_N^RTLEU N^L9U_z{I�LM|�tvN^RTU

s1(1), s1(3), s1(5), s1(7)P_cM|iP*NS��PQcOI0dOI�`V|�LEd

|iP \^R7z}|oPQN^R�L CC4TR7I{U ]Oy-|iP KOt_UvP y�I�t_N^R7dG6�NSP NSU |�\O8�|}�7U 27N^UvNSP_I{d-|iPt = 2

D �h�uP_cOIdOI�`V|�LEd|�z�P_]M|�\^\G�aN^LEz{t_I{|�U_I�U76�NSP*I7g$]M|i\^U

30 · 2 = 60|iP*PQcOI0dOI�z{N^UvN^RTL9`�|,H$N^L54�`bRT`�I�L4P&6�|�LOd�PQc$]OU

U_]OtvI{\S��I�?gz{I�I{dOU*PQcOIWdOI�`V|�LEdnR���\^R7z}|oPQN^R�L�Y Y��X��|oPt = 2

D_JR=8 I�2TI�t76 NG�RTLOIaR�� P_cOIaR�P_cOI{t Uvz{I�LM|�t_NSRTU

s1(2), s1(4), s1(6), s1(8)Rgz�z{]OtS6�dOI{`�|�LOd |iP

\^R7z}|iP_N^RTLCW|iPt = 1

w�I{z{R�`�I�U30 ·0.5 = 15

6$|�LOd�N^LP_cOI{UvI*Uvz{I�LM|�t_NSRTU76$\SRgz{|iPQNSRTLC N^UuLOI�2TI�t2gNSU_NGPQI{d |iP

t = 2D I cEI z{RTLEdONSP_N^RTLM|i\�I#?gy�I�z�P |oPQN^R�Lj2i|�\^]OI0�%RTtJP_cOI dOI�`V|iLOd-|iP \SRgz{|iPQNSRTLnC

|iPt = 2

NSU0NSL P_cOI{UvIVU_z�I{LM|it_N^R�U15 · 1.25 = 18.75

DV��L PQcEI{U_IVU_z{I�LM|�tvN^RTUS6�PQcEN^U 2i|�\^]OIbN^Uw�I�\^R=8 P_cOI�z{RTLOdENSPQNSRTLM|�\�I�?gy�I{z�PQ|iPQNSRTLW2�|i\^]OI�U0Ri� R�P_cOI{tW\SRgz{|iPQNSRTLOU76�|�LOd=N^L PQcOIbz}|iU_I�R�� |t_N^U`HfApLOI�]EPQt_|�\�dOI{z�N^UvN^RTL�`�|,H�I�t76O\SRgz{|iPQNSRTL�CN^U�PQcEI{t_I �%RTt_IWLOI�2TI�tZ27N^UvNSP_I{d9|iP

t = 2D

� \^UvRJRiPQcOI�t�c z cOI{z H$NSL54TU7coR��OP_cOI |�w�R=2�I�H$N^LEd�NSLOdON^z{|iPQIuP_cOI `bR7dOI{\X4TNO2TI�U�tvI}|�UvRTLM|�wO\SI�t_I{Uv]O\SP_U{DE LEI�R�� PQcOIbU_I�2TI�tQ|�\pUvPQ|iPQNSzU_RT\S]EPQNSRTLOU76�wM|iU_I{d RTL3I#?gy�I�z�P |oPQN^R�L 2i|�\S]OI{Uf %K54T]Ot_I9F[� R�� dOI�A`V|�LEdOUF8 R�]O\^d w�I

(1 → 1 → 1)|�LOd

(3 → 3 → 3)�%RTtIS2�I{t � U_z{I�LM|�tvN^ROD I cEI9z�RTtNA

t_I{Uvy�R�LOdON^LL4Q2i|�\^]EI{UbR�� wONSLM|�t � 2i|�t_N^|�wO\^I�UPQcM|iPbNS`�yO\SI{`bI{L4PPQcEN^U�dOI{\SNB2�I{tv� U PQt_|iPQIS4���|�t_IdOI{LORiPQI{d=w4�

x∗��� �����

D I cOI02i|�\S]OI R���PQcEIRTw�3vI{z PQNO2TI�%]OLOz PQN^R�L z}|�L-LOR=8�w�Iz{|�\^z�]O\�|iP_I{d w4�M + c

Tx∗��� ����� = M − 234.58

DWI c7]EU76�PQcOIaI#?Ey�I{z PQI�d dOI�z{tvI}|�UvInNSL3PQcOIaU PQR7z H 6 RTt�PQcOII�?gy�I�z�P_I{d |�`bRT]OL4PR���dOI�\^NB2�I{tvN^I{US6ON^U*YiC�XODZf�� ]OLONGPQU{Dk NSPQc P_cOI�RTyEPQNS`V|i\?U_R�\^]EP_N^RTL�4TNO2TI�L w4�=� |oPQ\�|iwG6�8 IM4TI P

M + cTx∗ = M − 235.7225

DI cOI{tvI��%RTtvI,6TUvz{I{LO|�t_NSR>A�wM|�U_I�d/RTygPQN^`bN^�{|iPQNSRTL NSLOz{tvI}|�UvI{U�PQcOIpI�?gy�I{z�P_I{d|i`�RT]EL$P�R��EdOI{\SNB2�I{t_NSI{Uw$�-F�DSF}X4YTf/]OLONGPQURTt�ED X4��b z�RT`�yO|�t_I�d9P_R�|U P |iP_N^z U_R�\^]EP_N^RTL�D

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I cOI�I�?gy�I�z�P_I{d�NS`�yEt_R=2TI�`�I�L$P N^U tvI{\�|oPQNB2�I{\G�3Uv`V|�\S\ NSLAPQcONSU I�?E|�`byO\SI � P_cOIayOtvRTwM|�wONS\^NGPQN^I�UR�� PQcEI/U_z�I{LM|it_N^R�Ud8cOI�t_I/P_cOI/dE�7LM|i`�NSz0UvRT\^]gPQN^R�LndONOi I�t_U�%t_RT`2P_cOI/U P |iP_N^zWRTLOI/|�t_I�U_`�|�\^\:D�)]EtvPQcEI{t_`bRTtvI,6gP_cOI dE�7LM|�`bN^z dOI{z�N^U_NSRTLa`�|,H$N^L54 dERgI�U LOR�P NSLE�M]OI�LOz{I PQcEI dOI{\SNB2TI�t_NSI{U�`�|�dOI|iP

t = 0|iLOd

t = 1|iLOdAU_RnPQcOI�`�|[3vRTtWyM|itvP Ri��PQcOIVdOI{\SNB2TI�t_NSI{U�|�t_I�U_N^`bN^\^|�t P_R UvPQ|iPQNSz

|�LOd�dE�7LM|i`�NSz U_R�\^]EP_N^RTLOU�D H I�y�I�LOdONSL54RTLVP_cOI Uv]O` Ri��PQcOIJyOt_R7dO]Oz PQU�R�� PQcOI�U_I P_I{t_`bU767PQcOIR[2�I{t_|�\^\ NSLE�M]OI�LOz{I9R�� P_cOI�dE�7LM|�`bN^z�dOI�z{N^UvN^RTL `�|,H$N^L54-tvI{`�|�N^LOUNSLOU_NB4TLONGKMz}|�L4PF8NGPQc PQcOIyM|�tQ|i`�I PQI{t 2i|�\S]OI{Uz cOR�U_I{LnN^LaPQcEN^U I�?E|�`byO\SI�D_JR=8 I�2TI�t76)PQcOI `�|�N^L��%Rgz�]OU R���RT]Ot I#?O|i`�yO\SI 8 |�U PQR�dOI{`bRTLOU PQt_|iPQI cER[8 PQcOI UvRT\^]gPQN^R�L�dOI�Ay�I�LOdOU�RTLbPQcOIJz{]Otvt_I�L$P�U_z{I�LM|�tvN^R\ PQcOIJdOI{y�I{LEdOI{LOz�I RTLDc 8cM|iP cM|�yOy�I{LOI�d�N^LbPQcOIJyOt_I�27N^RT]OUPQN^`bI NSL$P_I{tN2�|i\)c � D k IWz{RT]O\Sd9|�\^UvRwO]ON^\SdaI�?E|�`byO\^I�U 8cOI{tvI PQcOIWI�?gy�I{z PQI{dnN^`byOt_R=2�I{`bI{L4PN^U\�|�t`4TI{t � NS� 6��%RTtWI#?O|i`�yO\SI,6�PQcOI�c dOI�P_I{t_`bN^LEN^UvP_N^z>c dOI�`V|�LEd D2 = 24

6�UvPQ|iPQNSz�U_RT\S]EPQNSRTL-dORgI�ULOR�P z cO|�L54TIK8cEI{t_I{|�U PQcEI/I�?gy�I{z PQI{d=|�`bRT]OL4P R��udOI�\^NB2�I{tvN^I{UJN^LnRTyEPQNS`�NS�{I�d U_R�\^]EP_N^RTL9t_NSU_I{UPQR Y�C��ED�f�YTYTf=]ELONSP_U� Y7DSFWb cONB4TcOI�t�PQcO|�L�z{RTtvt_I�U_y�RTLOdONSL54 2i|�\^]EInRTwgP |�NSLOI{d 8NGPQc U P |iP_N^zUvPQt_|iPQIS4��L� D�hP U_cERT]O\^d w�I�LOR�PQNSz{I�d PQcM|oPd8cOI�L z{RT`byM|�tvN^L54bPQcOI�UvPQ|iPQNSz/|�LEd dE�7LM|�`bN^z0UvP_tQ|iP_I74TNSI{U76�PQcOI- ��� $L-,'p|�wOU_R�\^]EP_I�dONOi I�t_I�LOz{I w�I�P 8 I{I�L-dOI{\SNB2TI�tv�n|�`bRT]OL4PQUJN^L9PQI{tv`�NSLM|�\�U_z�I{LM|it_N^R�U z}|iL w�II{NSP_cOI{ty�RTUvNSPQNO2TIaRTtLEI744|iP_NB2�I�D �MRTt I#?O|i`�yO\SI,6�NS�JU_z�I{LM|it_N^R

s1(5)R7z{z{]Et_Ub|iP

t = 16pP_cOI

|�z�P_]M|�\�dOI�`V|�LEdOU*|iPLOR7dOI{U*|�tvI(D1(s

1(5)) D2(s1(5)) D3(s

1(5))) = (29 20 60)D

� R=8a6)|�z{z�RTt_dEN^L54 P_R�P_cOI dE�7LM|�`bN^z RTyEPQNS` ]O` U_RT\S]EPQNSRTLG6E|iPt = 1

PQcOI 2�I{cONSz{\^I�U U_cOR�]O\^d�w�IdOt_NO2TI{LAP_R \SRgz{|iPQNSRTLOU�Yn|�LEd C N^L-PQcEI�LOI�?7P/y�I�t_N^R7d�D��h�U_z�I{LM|it_N^R F�R7z{z{]Et_U/NSL P_cOI�LOI#?gPy�I�t_N^R7dU

t = 2�#6$dOI{`�|�LOd

D1(s2(5, 1)) = 58

6$|�LOd�PQcOIdE�7LM|�`bN^z U PQt_|iPQIS4��M4TNB2�I{Up|�z PQ]M|�\S\S�8 RTtvU_Ibt_I�U_]O\GP PQcM|�L P_cOI�UvP |oPQN^zU PQt_|iPQIS4��TDM_*R[8 IS2�I{tS6�UvN^LOz�I�P_cOI�yOt_R�wM|�wONS\^NSP��

p+1 = 0.1

NSUU_`�|�\^\ 6iPQcEI0� � ������� � KMLM|�\My�I{tvN^R7dF2�|i\^]OI�Ri�

D1N^U

0.1 ·58+0.9 ·14.5 = 18.85NSL�U_z{I�LM|�tvN^R

s1(5)6�8cEI{t_I{|�UWPQcOIbdOI�P_I{tv`�NSLON^U PQN^z 2�|i\^]OI�R��

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� � � �a��� �0�� � ������S�9�� �/���

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� ��� � ����� � � ��� �

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|�LOdG6pN^L 4TI�LOI{t_|�\�z}|iU_I,6@8cOI{tvI�t_N^U`H3LOI�]EPQt_|�\^NGP��3NSULOR�Pb|�UvU_]O`bI{d 62 · 2Nt $&%(' |idOdONSP_N^RTLO|�\

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P � P_cOI{UvI|�U_Uv]O`byEPQNSRTLOU76 RTLO\S��P_cOI|�\S\^R7z}|iP_N^RTL�N^Lg�M]OI{LEz{I{U P_cOI RTw 3vI{z�P_NB2�IF2�|i\^]OIiD �MRTt I#?�A|�`�yE\^I,6�NG� PQcEI�RTygPQN^`�|�\?|�\^\SRgz{|iPQNSRTLOU |iP

t = 0|iLOd

t = 1|�t_I�U_]Ez c3PQcM|oP�P_cOI�yEt_R7dO]Oz�P_U

|�t_I dEI{\^NO2TI�t_I{d P_R�\^R7z}|oPQN^R�LOU�F |�LOd-CL6�IS2TI�tv�-|it_z/UvI�P P_cM|iP z�RTLOLOI�z�P_U PQcOI LERgdOI�UF|�LOd-CN^U0RTyEPQNS`V|i\ D�I c7]EU76uN^L=RT]Ot0U_y�I{z�N�|�\�z{|�U_I>6u|�\^UvRnUvN^`byO\^I�t `�R7dOI�\T8NGPQc

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t ≥ 1|iU

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RS(st) = RS(b(st)) +

N∑

i=1

xi(st)Di(s

t)Pi −

N∑

i=1

N∑

j=1

cijxij(b(st)),

8cOI{tvIcij

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tvI{yOtvI{UvI{L4PQU | ]OLONGPuyOt_NSz{IR�� PQcOIVyOt_R7dO]Oz P}DDI c$]OUS6udOI�KMLEN^L54

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