, . ( ). , . .
, .
2.8.4. .
, : (1), (2) (3). k=1 2 : . , N . ij(k) i j k.
0.2 0.5 0.3 0.3 0.6 0.1
{ij(1)}= 0 0.5 0.5, {ij(2)}= 0.1 0.6 0.3
0 0 1 0.05 0.4 0.55
:
(k=1), ( 1) 0.2, 0.5 0.3. ( 3) 1 .
(k=2), ( 1) 0.3, 0.6 0.1. ( 3) 0.05 , 0.4 0.55 .
rij(k) ( ), , i j k.
7 6 3 6 5 1
{rij(1)}= 0 5 1, {rij(2)}= 7 4 0.
0 0 1 6 3 2
:
(k=1), ( 1) 7 , 6 3. ( 3, , 1 ) 1 ().
(k=2), ( 1) 6, 5 1 ( ). ( 3) 6, 3 2.
vi(k) , i k,
vi(k)=∑jpij(k)rij(k).
(k=1),
v1(1)=0.2´7+0.5´6+0.3´3=5.3,
v2(1)=0´0+0.5´5+0.5´1=3,
v3(1)=0´0+0´0+1´ (1)= 1.
|
|
(k=2)
v1(2)=0.3´6+0.6´5+0.1´ (1)=4.7,
v2(2)=0.1´7+0.6´4+0.3´0=3.1,
v3(2)=0.05´6+0.4´3+0.55´ (2)=0.4.
, fn(i) n , :
f1(i)=maxk{vi(k)},
fn(i)=maxk{vi(k)+∑jpij(k)fn-1(j)}, n=2,3,,N. (2.8.4)
N=4. 2.8.4 2.8.7.
n=1 . 2.8.4
i | vi(k) | |||
k=1 | k=2 | f1(i) | k* | |
5.3 | 4.7 | 5.3 | ||
3.1 | 3.1 | |||
1 | 0.4 | 0.4 |
n=2 . 2.8.5
i | vi(k)+pi1(k)f1(1)+pi2(k)f1(2)+pi3(k)f1(3) | |||
k=1 | k=2 | f2(i) | k* | |
5.3+.2´5.3+.5´3.1+.3´.4= =8.03 | 4.7+.3´5.3+.6´3.1+.1´.4= =8.19 | 8.19 | ||
3+0´5.3+.5´3.1+.5´.4= =4.75 | 3.1+.1´5.3+.6´3.1+.3´.4= =5.61 | 5.61 | ||
1+0´5.3+0´3.1+1´0.4= = 0.6 | .4+.05´5.3+.4´3.1+.55´.4= =2.13 | 2.13 |
n=3 . 2.8.6
i | vi(k)+pi1(k)f2(1)+pi2(k)f2(2)+pi3(k)f2(3) | |||
k=1 | k=2 | f3(i) | k* | |
5.3+.2´8.19+.5´5.6+.3´2.13= =10.38 | 4.7+.3´8.19+.6´5.61+.1´2.13= =10.74 | 10.74 | ||
3+0´8.19+.5´5.61+.5´2.13= =6.87 | 3.1+.1´8.19+.6´5.61+.3´2.13= =7.92 | 7.92 | ||
1+0´8.19+0´5.61+1´2.13= = 1.13 | .4+.05´8.19+.4´5.6+.55´2.13= =4.23 | 4.23 |
n=4 . 2.8.7
i | vi(k)+pi1(k)f3(1)+pi2(k)f3(2)+pi3(k)f3(3) | . | ||
k=1 | k=2 | f4(i) | k* | |
5.3+.2´10.74+.5´7.92+.3´4.23= =12.68 | 4.7+.3´10.74+.6´7.92+.1´4.23= =13.097 | 13.10 | ||
3+0´10.74+.5´7.92+.5´4.23= =9.075 | 3.1+.1´10.74+.6´7.92+.3´4.23= =10.195 | 10.19 | ||
1+0´10.74+0´7.92+1´4.23= = 3.23 | .4+.05´10.74+.4´7.92+.55´4.23 =6.4315 | 6.43 |
, 1-,2- 3- (k*=2) , 4- (n=1) , . f4(1)=13.10 , f4(2)= 10.19 f4(3)=6.43 .
.
. -, ; n- : pij(k,n) rij(k,n). -, , fN(i) . α , α=1/(1+t), t , (4.9.4) :
|
|
fn(i)=maxk{ vi(k)+α∑jpij(k)fn-1(j)}, n=2,3,,N. (2.8.5)
. α=0.6. ( 2.8.8).
. 2.8.8
i | n=1 | n=2 | n=3 | n=4 | ||||
f1(i) | k* | f2(i) | k* | f3(i) | k* | f4(i) | k* | |
5.3 | 6.94 | 7.77 | 8.26 | |||||
3.1 | 4.61 | 5.43 | 5.92 | |||||
0.4 | 1.44 | 2.19 | 2.66 |
, . .
. . ( ) , . (2.8.4) n → ∞.
, i tn-1 j tn. . ( ) (. 2.2), . N (N+1) N .
2.8.6.
, , . . , , , , ( , , , ..). , , .. . , . : , .
(, , ). , , . , , , , . () , .
i ( 1) i. , . n, , :
|
|
1. , .
2. , .
..
n <(1+ 2 ++n-1)/(n 1), (2.8.6)
n+1 >(1 + 2++n)/n. (2.8.7)
2.8.5. , , . 2.8.9.
2.8.9
26,7** | ||
27,5 | ||
33,3 |
, , ..
3 =20 <(1+ 2)/2=30, 4 =30 >(1 + 2+3)/3=26,7.
, , . . r , d=1/(1+r/100). (2.8.6) (2.8.7) :
n <(1+ 2 d ++n-1 d n-2)/(1+ d++d n-2), (2.8.8)
n+1 >(1 + 2 d ++n d n-1)/(1+ d++ d n-1). (2.8.9)
2.8.6. , , r =5%. 3 . 2.8.10 (d=0,952):
2.8.10
i | ci | - |
30.49 | ||
27.16* | ||
28.82 | ||
30.02 | ||
32.96 |
, .. (2.8.8) (2.8.9) n=3. , .
.
cij , i, j.
:
fi , , i, i+1,, n, , i.
f1( ), :
fn+1 =0,
fi =minj>i{cij + fj}, i=n, n1, , 1. (2.8.10)
, , , :
ik k i ;
rik k i.
fi(k) , i, i+1,, n, , i k .
i,
fi(k) =rik+1 +fi+1(k+1),
,
fi(k) =ik +ri1 +fi+1(1).
,
fi(k)=min{rik+1+fi+1(k+1),ik+ri1+fi+1(1)}, i=1,2,,n, (2.8.11)
fn+1(k)=0 k. .
|
|
n , (n+1)- .
f1(k0), k0 . , i=1, .
2.8.7. , 2.8.11 2.8.12.
2.8.11. rik 2.8.12. ik
, , , , , 3 4, , 5 . 2.8.12 .
(2.8.11):
f6(k) =0 k.
i=5 ( 5 4):
f5(4) =min{r55 +f6(5), 54 +r51 +f6(1)}=min{200+0,115+10+0}=125,
f5(3) =min{r54 +f6(4), 53 +r51 +f6(1)}=min{85+0,110+10+0}=85,
f5(2) =min{r53 +f6(3), 52 +r51 +f6(1)}=min{40+0,90+10+0}=40,
f5(1) =min{r52 +f6(2), 51 +r51 +f6(1)}=min{20+0,70+10+0}=20.
i=4 ( 4 3):
f4(3) =min{r44 +f5(4), 43 +r41 +f5(1)}=min{120+125,105+14+20}=139,
f4(2) =min{r43 +f5(3), 42 +r41 +f5(1)}=min{52+85,85+14+20}=119,
f4(1) =min{r42 +f5(2), 41 +r41 +f5(1)}=min{28+40,65+14+20}=68.
i=3 ( 3 2):
f3(2) =min{r33 +f4(3), 32 +r31 +f4(1)}=min{68+139,80+16+68}=164,
f3(1) =min{r32 +f4(2), 31 +r31 +f4(1)}=min{32+119,60+16+68}=144.
i=2 ( 2 1):
f2(1) =min{r22 +f3(2), 21 +r21 +f3(1)}=min{36+164,55+18+144}=200.
.. ,
f1(0) = 11 +r11 +f2(1)=100+20+200=320.
, :
, , .
, k . , ( ).
pj , j (j<k);
rj j ;
sj , j.
r1 (, , ..).
, , :
, k;
, k;
.
, (2.8.10) (2.8.11)
k-1 k-1
fi =mink=1,2,,K{åfi+j j+fi+k(1 - åj)+Rk}, i=1,2,,n, fn+1=0, (2.8.12)
j=1 j=1
, , i+k , , . , , :
|
|
k-1 k-1
Rk = r1 + r2(1 - p1)+ r3(1 - p1 - p2)++ rk(1 - åpj) + åsjpj. (2.8.13)
j=1 j=1
, , , , , , k- , k- .
(2.8.12),(2.8.13) , .
.
, ( k- ). :
f =mink=1,2,,K{Rk/k}, (2.8.14)
k ,
k-1 k-1
k =åjj+k(1 - åj). (2.8.15)
j=1 j=1
, , (2.8.14), , , ( 2.8.5).
2.8.8. .2.8.13 ( 2 4).
2.8.13
k | k | rk | sk | Rk | k | Rk/k |
¼ | ||||||
1.75 | 65.14 | |||||
¼ | 2.5 | 51.6 | ||||
61.33 | ||||||
1/2 | 3.5 | 60.57 |
5 2.8.13:
R1 = r1 =100,
R2 = r1 + r2(1 - p1)+ s1p1=100+12´3/4+20´1/4=114,
R3 = r1 + r2(1 - p1)+ r3(1 - p1 - p2) + s1p1+ s2p2=114+20´3/4=129,
R4 = 129+20´1/2+180´1/4=184,
R5 = 184+56´1/2=212.
6 2.8.15:
1 = 1,
2 = 1´1 +2(1 - p1)=1/4+2´3/4=1.75,
3 = 1´1 +2´p2+3(1 - p1 - p2)=1/4+2´0+3´3/4=2.5,
4 = 1´1+2´2+3´3+4´ (1 - 1/4 - 1/4)=1/4+3/4+4´1/2=3,
5 = 1´1+2´2+3´3+4´4+5´ (1/2)= 1/4+3/4+5´1/2=3.5.
7 , k=3, .
, . , , Rk/k, k=5, åj(rj/k), k=4.
20%.
(: ).
(, r d=1/(1+r/100)) (2.8.12) (2.8.13) :
k-1 k-1
fi=mink=1,2,,K{ådjfi+jj+dkfi+k(1 - åj)+Rk}, i=1,2,,n, fn+1=0, (2.8.16)
j=1 j=1
k-1 k-1
Rk=r1+d1r2(1 - p1)+d2r3(1 - p1 - p2)++dk-1rk (1 - åpj) + ådj-1sjpj. (2.8.17)
j=1 j=1
d (2.8.14) (2.8.15) :
f =mink=1,2,,K{Rk/(1 - kd)}, (2.8.18)
kd
k-1 k-1
kd=ådjj+dk(1 - åj). (2.8.19)
j=1 j=1
f , .