308 downloads
179 Views
457KB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
-
! "# !$ %& ' %(# ! ) ! '
2003
4
5
1.
5
2.
5
3. 3.1.
! * # !$#
7 9
4.
"
10
5.
#
11
6.
$ !% 6.1. ! ! ! 6.2. ! 6.3. !
12 12 15 16 17
1.
17
2.
& ! %
17
3.
' ( &$
18
4.
– 4.1. +! 4.2. * !- " 4.3. , ! !-
19 19 20 22
5.
)
% 5.1. - !
5.2. .
" 5.3. ,
23 23 24 25
6.
% ! ( 6.1. /
6.2. . 6.3. 0 ! 1 ! 6.4. 0 ! # 6.5. 2 ! !3 3 6.6. 2 ! #
!
26 26 27 27 28 31 31
32
1.
32
2.
$ *
32 2
3.
+
33
4.
, - *
35
5.
.( (
36
6.
$ %
37
7.
$
39
! "
42
1.
42
2.
$&
43
3.
& "
44
4.
$ * !
45
5.
/ % "
5.1. !$
5.2. 4 ! !$# / 5.3. # ! !$ ! 5.4. / !
5.5. / $/
5.6. / ! $/ !
# $%
46 46 46 49 49 50 51 52
1.
52
2.
$ " *
52
3.
$
54
4.
,
0 $!
55
5.
$ *%
58
6.
% 6.1. 2 !$ 6.2. !$ 5 -! 6
59 59 60
7.
$ 7.1. 2 !$ 7.2. !$
62 62 62
$
64
3
&'(
" !$ 787977 – %& ' 787:77 – %(# ! ) ! ') # "# ; " < ) ) ; " ! # ! # #
) /! ! ; ) 3 ) = ! ) ; ) ) ! ) ! ! * ! ; 3" > !$ / !$ ;
! !$ 787977 787:77 ! ; # / ) # " ) "# ! 1 # ! 5 ? ! ) ) ; ) ! ) 6) 3
) !$/ # "1 # !; #
4
'('&()*'
1.
( ! ! !3 ! -! ! ; ! !$ ) !
! -!
) 3 ! !# 5 )
) 6 # ! ! 5 !)
) ! ) 1 6 > 3 !
! ) 3 ! # !
# # @! # / 3 # ) ! $
) ! # / 3 # ) ) ; !$ $ ! .! ! 1 ! ) !3
# !3 $3 n ; ! A ! !) !# ! ) $ !
; ! ) ) $ !$ != /!; $) !$ # ) !$ < !$# 1 # ! # !; ! !$ !$
( ! !! !$/ # < ! < !$ ; $ 5:7BCD78–B7ECDDD6) ! !!
# 1 # ! $ ! ; CDE9) CDE: # ! > 3 ! ! ) CDFC
! ! *G0 4 $ 1 !; ! $ ! ! !) 9:7 < << $ ! !
1 *G0 B ! !$3 .! 1# B != 1. 4 ! ? 2. , ) ) !$? 3. 2 /? 4. , # ? 5. * !$ ? 6. !!? 7. -? 8. H A !$) 3 3) !$ ! ; 3 # ! < 3 $) ) !$3) ; ! ! !$ !$/ ) ; ! !
2. *# ! ! !
!C A !$ ! " !3"
5
< !
X1
4 "
!
1 2 n 0 2 ! + #
1 x11 x21 xn1 c1 v1 M1
4 !3" ! 2 … … x12 … x22 I … xn2 … c2 III v2 … m2 …
X2
n x1n x2n xnn cn vn mn Xn
A Y1 Y2 II Yn
< ! X1 X2 Xn
IV
∑Z
i
∑Y
i
<!3 ! ! ! 52(6)
3 ! 1 I 4 – 1 ! !# <! xij I
i- !) !$ !$# j- !$3
n
* 1! i- ∑ xij ! i-
\ j =1
!) !$/ !$ * 1!; n
j- ! ∑ xij ! j- ! i =1
II < ! 4 ; $ ) # " ! " ; ! 2 !$ ! " !) ! )
!) " $) 1
< # ; !3 . III , # !$ #
* # 5 !6 ! ! ! * ! ;
Zj - Zj = cj + vj + mj, j=1, 2, …,n IV + # !
$ 4 !; !$ # # !)
! !$ ! ! < ! * ; ! ! ) ! + + + + + + = ) + + + + + + = ) + + + + + + = ) + + + + + + = ) ! 6
(1)
n
∑x j =1
ij
+ Yi = X i , i = 1, n .
(2)
* ) ! !3 !$ ! 3 3) 1 ) ! ! 2() ! n
∑x i =1
ij
+ Z i = X i , j = 1, n .
(3)
*! n 5E6= n
n
n
n
i =1
i =1
∑∑ xij + ∑ Yi = ∑ X i . i =1 j =1
*! n 5F6= n
n
n
n
j =1
j =1
∑∑ xij + ∑ Z j = ∑ X j . j =1 i =1
4 !# # )
) !) 3; n
n
i =1
j =1
∑ Yi = ∑ Z j ,
(4)
- .
3. , ! 2( !3 # # ; #
! , CDE: J* *** !
! # ***
CDEFKE9) !3 3" / # ; ! !# # C9 ! /! ) 9 !
!$ # 9 ! # 0 ! ! !
1 !!$ << $ CDF7-# #
# (! ! 9C×9C ! CDCD
CDED 0 ! ! !) # ! xij 3 ; ) !$ ! aij =
xij Xj
, i , j = 1, n
(5)
3 !$ ) !3 A 1 aij 3 !"" <! aij ) ! ; i- ! # ) j- !$ ! ! $ 1 # !$#
$) ; ! 1 -! 1 aij, !$ $ # ! ! ! ! ! 1
" A 1 aij !3 ! / # ! ) 1 #
3 # ! 1 2 3
7
a11 a21 A= a n1
a12 a1n a 22 a 2 n ` , a n 2 a nn
# ! * !$ 1 # !$# 5:6 ;
$ ! !# = xij = aij X j , i , j = 1, n .
(6)
* !$ 586 5C6
/= a11 X 1 + a12 X 2 + ... + a1 j X j + ... + a1n X n + Y1 = X 1 , a X + a X + ... + a X + ... + a X + Y = X , 22 2 2j 2n 2 2 j n 21 1 ai1 X 1 + ai 2 X 2 + ... + aij X j + ... + ain X n + Yi = X i , a n1 X 1 + a n 2 X 2 + ... + a nj X j + ... + a nn X n + Yn = X n .
(7)
< - ! ! = X1 Y1 X2 Y2 X= , Y = , Xn Yn 5L6
$ AX + Y = X .
(8)
* 5L6 ! 5B6 1 - ; !$3 ! ! ! !$3 % – ' 5 !$3 $ 6 !$ !$ % – ' / $
= C 4$ ! !) X ; 3 3) Y A ! # 5B6= Y = X − AX .
(9)
E M !) Y) ! 3 ; 3 X)
$ .
! ! ! # # !) ; # I # .! / 1
/ 5B6
= X − AX = Y , !
( E − A) X = Y ,
8
(10)
1 0 0 0 1 0 E= ` 0 0 1 5C76 # = X = (E − A ) Y = BY . −1
(10)
M$ ! B = (E − A)
−1
(11)
1 ! B 4$ 3 ; 3 !3"# I #= Y1 = 1, Y2 = Y3 = … = Yn = 0. 4 Y 5C76
/ = X 1 b11 b12 X 2 = b21 b22 X n bn1 bn 2
b1n 1 b2 n 0 . ` bnn 0
23 X1 = b11, X2 = b21, … Xn = bn1. , ) 1! ! B ! !)
3 # ) !$ !
0 ! ! $ !$ # ! B. -! B 3 !"" )
– .
3.1.
4 1 ! A 1! !$= aij ≥ 0, i , j = 1, n . , 1 aij ! !3 ! i- !) aij ≤ 1, i , j = 1, n . 5B6 Y = X − AX ≥ 0 ,
!$ 23 X ≥ AX
(12)
A) ! 3" 5CE6) . .! ! ) ) 1! ; ! ! $/
9
4. ( ! ! $ ! 1 ! <
! ! !
"! !# 2 j- ! Lj ,
! 5!"" 6 tj =
Lj Xj
,
j = 1, n .
(13)
< #
; # ) # #
2 Ti ! ! ; i- 4 aijTi !
"! )
!3; i- !) !$ ! !
j- ! , ! ! j-
! 5!"" 6 = n
T j = t j + ∑ aij Ti ,
j = 1, n
(14)
i =1
! = T1 = t1 + a11T1 + a 21T2 + ... + ai1Ti + ... + a n1Tn , T = t + a T + a T + ... + a T + ... + a T , 12 1 22 2 i2 i n2 n 2 2 T j = t j + a1 j t1 + a 2 j t 2 + ... + aij ti + ... + a nj t n , Tn = t n + a1n t1 + a 2 n t 2 + ... + ain ti + ... + a nn t n .
(15)
< - t = (t1, t2, …, tn) 1 - T = (T1, T2, …, Tn) 1 ! < # # 5C:6
/= T = t + TA .
(16)
A 1 $ ! # # ; ! 3"# !) $ t $ < 1; ! 1 ; / 5C86= T − TA = t , T (E − A) = t , T = t (E − A) , −1
5CC6 T = tB 2 L "
# !# n
n
j =1
j =1
L = ∑ L j = ∑ t j X j = tX . * ! 5C76 X = BY ) 1
10
(17)
tX = tBY , 5CL6 ! = tX = TY .
(18)
5CB6 ) $ ) !
)
2 $ 3 1 ) ; $ /!
!33) 1 ! ! )
$ " 1 $ / $ $) ; ! ) 1 * 1 ! $) ! $ )
1 !$/3 !3
3 !$) ) $ )
#
) ) 3 *G0 1 ; ) $ , / 1 ! *G0)
!$/ 1 3 !
5. 4 !)
!# & !3 ; 5 ) ) 6 5$) !) ; 6 M$ $ ! 3 3 ! 4$ j- !
Φj , !
# ! fj =
j
,
Xj
j = 1, n .
(18)
<! fj 3 !"" ". %!"" " Fi I # !#) ; # ! ! i- ! j-
! ! ! ! ! ) ! 1 ; ! C , # !# # Fj) ! 3 ; fj)
# j- ! # !#) # # !
! # aij , ' = C) &
$ n
F j = f j + ∑ aij Fi ,
j = 1, n .
(19)
i =1
M$ aijFi ! # i- !) # ) !
! j- ! < 5CD6 ; ! ! 5C:6 < - f = (f1, f2, …, fn) 1 - F = (F1, F2, …, Fn) 1 ! < # # 5CD6
/= F = f + FA .
(20)
, I )
# !) $ !$ ; ) 1 f $ ; ) 1 ! # /
(20): 11
F = f (E − A) = fB . −1
(21)
!$ 1 ! $ ! I
3" "
6. 4$
= # !$#
0.1 0.2 0.3 A = 0.3 0.4 0.2 , 0.5 0.1 0.3 100 Y = 200 , 300 1 t = (1.2,1.4 ,1.9 ) 1 #
f = (1.5, 2.0,1.2 ) . , $ ! ! ! ) ; ! !
6.1.
!
.! ! ! !
3 3 ! ! X = (E − A) Y . −1
4 !$ ) ! ! A . ! ; ! ) 1! ! !
$/ 1 ! ) ! 1 ; ! <! B = (E − A )
−1
!
(E − A)−1 =
(E − A) E−A
.
M$ E − A – !!$ ) ( − ) – ) ; ! ! 1! (E − A)T . & ! ! ! !! $
12
( ( ( 1 = ( ( ( = ( ( ( + ( ( ( + ( ( ( − ( ( ( − ( ( ( − ( ( ( ( ( ( 0! ! 1!
!3 ! !!) !
! ) # ! 1! 2 !!$ ;
!3) ! !
)
! 1 ) ! ! 1! d12 D $ D12 = (− 1)
1+ 2
d 21 d 23 d 31 d 33
= −(d 21d 33 − d 23d 31 );
! ! 1! d31 $ D31 = (− 1)
3+1
d12 d13 d 22 d 23
= d 12 d 23 - d13d 22 .
<! B # E–A: 1 0 0 0.1 0.2 0.3 0.9 − 0.2 − 0.3 E − A = 0 1 0 − 0.3 0.4 0.2 = − 0.3 0.6 − 0.2 . 0 0 1 0.5 0.1 0.3 − 0.5 − 0.1 0.7 . ! # !!$ E − A : 0.9 − 0.2 − 0.3 E − A = − 0.3 0.6 − 0.2 = − 0.5 − 0.1 0.7 0.9 ⋅ 0.6 ⋅ 0.7 + ((− 0.2) ⋅ (− 0.2) ⋅ (− 0.5)) + ((− 0.3) ⋅ (− 0.3) ⋅ (− 0.1)) − ((− 0.3) ⋅ 0.6(− 0.5)) − ((− 0.2) ⋅ (− 0.3) ⋅ 0.7) − (0.9 ⋅ (− 0.2) ⋅ (− 0.1)) = 0.378 − 0.02 − 0.009 − 0.09 − 0.042 − 0.018 = 0.199. , (E − A) ) $ ! != 0.9 − 0.3 − 0.5 (E − A) = − 0.2 0.6 − 0.1 . − 0.3 − 0.2 0.7 T
# ! ! 1! (E − A) : T
13
0.6 − 0.1 = 0.4 , − 0.2 0.7 1+ 3 − 0.2 0.6 A13 = (− 1) = 0.22 , − 0.3 − 0.2 2 + 2 0.9 − 0.5 A22 = (− 1) = 0.48, − 0.3 0.7 3+1 − 0.3 − 0.5 A31 = (− 1) = 0.33, 0.6 − 0.1
− 0.2 − 0.1 = 0.17 , − 0.3 0.7 2 +1 − 0.3 − 0.5 A21 = (− 1) = 0.31, − 0.2 0.7 2 + 3 0.9 − 0.3 A23 = (− 1) = 0.27 , − 0.3 − 0.2 3+ 2 0.9 − 0.5 A32 = (− 1) = 0.19 , − 0.2 − 0.1 3+ 3 0.9 − 0.3 A33 = (− 1) = 0.48. − 0.2 0.6
A11 = (− 1)
1+1
A12 = (− 1)
1+ 2
4 ! !3" = 0.40 0.17 0.22 (E − A) = 0.31 0.48 0.27 . 0.33 0.19 0.48 .! 1! 1 !!$ ! 1; !# !$#
0.40 B = (E − A) = 0.31 0.33 −1
0.17 0.22 2.01 1 0.48 0.27 ⋅ = 1.56 0.199 0.19 0.48 1.66
0.854 2.41 0.955
1.11 1.36 . 2.41
<! $3 " ) 3" ; ! # $ ! 2.01 0.854 1.11 100 704 X = B ⋅ Y = 1.56 2.41 1.36 ⋅ 200 = 1045 . 1.66 0.955 2.41 300 1080 ! A X1, X2, X3, !
; !# = x11 = a11 X 1 = 0.1 ⋅ 704 = 70.4; x12 = a12 X 2 = 0.2 ⋅ 1045 = 209; x13 = a13 X 3 = 0.3 ⋅ 1080 = 324; x21 = a 21 X 1 = 0.3 ⋅ 704 = 211; x22 = a22 X 2 = 0.4 ⋅ 1045 = 418; x23 = a 23 X 3 = 0.2 ⋅ 1080 = 216; x31 = a31 X 1 = 0.5 ⋅ 704 = 352; x32 = a32 X 2 = 0.1 ⋅ 1045 = 105; x33 = a33 X 3 = 0.3 ⋅ 1080 = 324. ! # ! ! ! ! 5 !E6
14
" # ! 4 " ; ! 1 2 3 !$ ; ! - ; Zj, j=1,2,3 < !
4 !3" ; ! 1 2 3 70.4 209 324 211 418 216 352 105 324
A ; Yi, i=1,2,3
< ! Xi, i=1,2,3
100 200 300
704 1045 1080
633
732
864
600
71.0
313
216
600
704
1045
1080
!E )
Z1 +Z2 +Z3 = Y1 +Y2 +Y3 = 600, $ !3 !$ !$ ; #
6.2.
M ) # ! j- ! ;
3 1 2 3 - t = (t1 , t2 ,t3 ) . A 1 !
"! ; ) !$ -! ! < 1 ! T = (T1 ,T2 ,T3 ) 1 ! T = tB .
<! ! = 2.01 0.854 1.11 T = (1.2,1.4 ,1.9 )1.56 2.41 1.36 ; 1.66 0.955 2.41 1 2 2 01 1 4 1 56 1 9 1 66 T1 = . ⋅ . + . ⋅ . + . ⋅ . = 7.74 , T2 = 1.2 ⋅ 0.854 + 1.4 ⋅ 2.41 + 1.9 ⋅ 0.955 = 6.22 , T3 = 1.2 ⋅ 1.11 + 1.4 ⋅ 1.36 + 1.9 ⋅ 2.41 = 7.81. 4 !
# # ! L = tX = t1 X 1 + t2 X 2 + t3 X 3 = 1.2 ⋅ 704 + 1.4 ⋅ 1045 + 1.9 ⋅ 1080 = 4360. ) 1 ! ) !;
= TY = T1Y1 + T2Y2 + T3Y3 = 7.74 ⋅ 100 + 6.22 ⋅ 200 + 7.81 ⋅ 300 = 4361. * !$ ! !$/ !
15
/ ! !# !# 1, 2, 3 ! ! 5 !E6 1
t1, t2, t3 ) ! ! ! 5 #
!#6 $ #
4 ; " !
6.3.
1 2 3
4 !3" ! !
"! 1 2 3 84.5 251 389 295 585 302 669 200 616
M M
3 ; !#
3 5 ; 6 120 845 280 1463 570 2052
%
A 1 3 ! # ; )
# !) ! ; 2 3
= ( ) . A 1 ! = ( ) 3 I ) # # # !# !
! * ! 5EC6 F = fB . <! 1 ! 1 ! = 2.01 0.854 1.11 F = (1.5, 2.0, 1.2 ) 1.56 2.41 1.36 ; 1.66 0.955 2.41 F1 = 1.5 ⋅ 2.01 + 2.0 ⋅ 1.56 + 1.2 ⋅ 1.66 = 8.12 , F2 = 1.5 ⋅ 0.854 + 2.0 ⋅ 2.41 + 1.2 ⋅ 0.955 = 7.25, F3 = 1.5 ⋅ 1.11 + 2.0 ⋅ 1.36 + 1.2 ⋅ 2.41 = 7.27. 1, 2, 3 ! ! 5 !E6 1
f1, f2, f3 ) ! ! ! & # % 4 ; " !
1 2 3
4 !3" ! 2"! ; !# # 1 2 3 106 314 486 422 836 432 422 126 389
16
& ) ; & ) "; "!
!
; ! ; 150 1056 400 2090 360 1296
'( +,-** (
1.
< # 1 #
# ! #-!
!$3 ! !/ ,
! ! ) " ) ! $ ! ; I ) 3" !$3 !$) ! ) ; !$ 3 3" < ! ) !$ 5 "6
3" ! 56 !
$# ) ; !$ ! / 3" ;
! * ! !$ !
linear programming) ! ! !
!$ ! ) ; !$ # ! < / ! $3 < ! ! /! CDFD ) !; ) < !$ A 5CDC2CDB86) !$ <
CDL: N ! 3 !$ ! N A ! ; ! 5 1 ,+ A 6 ; < A ! ! $ ; 3 $ O:P) ! < CD9L A ! ! ; . ( . ! ; . ! ! " /
! ) !; / - < . !
$ O8P)
2. ! 4$ ! 3 P1 P2 4 1 !$;
3 $ S1, S2, S3 5 ) ) $ 6 +!
!E) !$) ! !
! ' *
) P1 P2. 7 6 4 1 1 4
)
140 S1 (+) 64 S2 (,) 64 S3 (- ) .
5
6
, $ I # x1 x2) "
!$ ! !$ f(x1, x2) = 5x1 + 6x2 ⇒ max, !!$ = 17
(1)
7 x1 + 6 x 2 ≤ 140 , 4 x1 + x 2 ≤ 64 ,
(2)
x1 + 4 x 2 ≤ 64 , I ! !$= x1 ≥ 0, x2≥ 0.
(3)
M 5C6) 5E6) 5F6 !
! 5M46 , ; 3
) # ! ! ) ; # !# !
3. " # !$ M !
$ !3 3 3 5C6 A 5E6 !
! $ x1, x2 ! ! < ! ! # ; /3 3 ) ! ! – > ; !3 ) # ! 3 5E6) ! #
$
C 1 ) ! 3
3 /3 3 ! ! $ 4 " $ # ) !
7x1 + 6x2 = 140, ! ! x1Q7) !
8x2 = 140, x2QC97K8QEFL?
! x2Q7) !; Lx1 = 140, x1QC97KLQE7 * 57) EFL6 5E7) 76 .! !
! ! 3-$ !$3 ) 5C)C6) ! ; ! 7 ⋅ 1 = 6 ⋅ 1 = 13 < 140 ,
) ! ! ! $) " 5C) C6 0 !
2! $3 # ! !$ OABCD < 1 ; !$ ! ) ! !$
; ) ! /
1 .! / ! ! ∂f ∂f , ∇f = = (5,6) . ∂x1 ∂x 2 ! ) ! !
.! ! ! $ !3" = :– !$ x1, 6 – !$ x2. .! # ! ! ; , 3 !) # ! < )
) f(x1, x26Q87) 5C6 ! 5x1 + 6x2= 60. ) ! ! ) 1 .
18
C > / M4 R )
! f = 87 ! !$) ; ! $ ) ! # ! !$/
4; " !3 f = 87 !!!$ !3 ) ! )
!$ ! B -
$ 1 ) ! ; L
+8 +9
= C97) = 89)
/ 3) # : x1 = 8, x2 = C9 4 1# # # # !
f ( x1 , x1 ) = 5 ⋅ 8 + 6 ⋅ 14 = 124 . A ) M4 / !$ ! #
# # 2 ) ! C) ! $ = ; ! $3 ! ! !$ OABCD? !$ /
! ! !
4. – 4.1.
(
> ) ! , ! 3 2 –
!!$# # , ) ; ! 5E6 x3) $= 140 – 7x1 – 6x2 = x3 ≥ 0. 23 19
7x1 + 6x2 + x3 =140. < !!$ x4, x5) ! 5E6 7 x1 + 6 x 2 + x 3
= 140,
4 x1 + x 2 + x 4 = 64 , x1 + 4 x 2 + x5 = 64 ,
(4)
A ) !!$ !$= x j ≥ 0,
j = 1,5 .
(5)
2 596 !3 # $3 ; 3
#) ! 3" 596) 5:6) ; . # 5:6 ) ! $
!$
) ! 3 !$
) 3; 2 !$ ) # ) 3
4 " $ ! @! 1
# # !$) ! ; # / M4 2 ! #
! < # #)
/ ) ! !
# $ !$ ! -
!
- !$/ ! ) ; $ , ! ! #
# # " # ! ) )
, = : ⋅ 9 K E = C7 , $ /$ C7 # * !– # # #
5!) ) 6 ! + !/ $ ! ! )
/ ; /
# #
4.2.
!- )
, ! f=c1x1+c2x2+c3x3+c4x4+c5x5
(6)
# a11 x1 + a12 x 2 + x 3 a 21 x1 + a 22 x 2 + a 31 x1 + a 32 x 2 +
= b1 , x4
= b2 ,
(7)
x5 = b3 ,
! # !$ 5:6 5L6 ) !
$ x1, x2 ! $
# !3) x3, x4, x5 <
!$ ! ! x1=0, x2=0, x3=b1, x4=b2, x5=b3,
(8)
! !
f = c3b1+c4b2+c5b3. 20
(9)
< 5L6 x3, x4, x5 x1, x2: x 3 = b1 − a11 x1 − a12 x 2 , x 4 = b2 − a 21 x1 − a 22 x 2 ,
(10)
x5 = b3 − a 31 x1 − a 32 x 2 . $
4 5C76 !3 3 586 ! # !
f = (c1–c3a11–c4a21–c5a31)x1 + (c2–c3a12–c4a22–c5a32)x2 + c3b1 + c4b2 + c5b3.
(11)
@! 1 x1, x2 5CC6 !$) !3 ; 3 !$ !$) 1 ) $ ; - ! !$
2 !$ $ 1 5CC6
$ ) !$ / A 1 ; ! ! $ < ; : Z1 = c3a11 + c4a21 + c5a31, Z2 = c3a12 + c4a22 + c5a32.
(12)
< 1# # ! !$
/= c1 – Z1 < 0, c2 – Z2 < 0,
(13)
4$ ! !$ 5CF6 !) $) ! ! ; ) 1 x1 5CC6 ! !) ) ! x1) ! !3
3 3
5D6 4 ) !$ $ ! ; x1 3 !) ! ) x2 ) $
$ x2Q7 5C76 ) ! x1 $/; 3 x3, x4, x5) ! 1 x1 5C76 !$ ! $ x1 !/$
) # x3, x4, x5 !$ 4 ! !3
! 5C76) ! = 0 = b1 – a11x1, 0 = b2 – a21x1, 0 = b3 – a31x1. @! ! $ b b b x1 = min 1 , 2 , 3 , a11 a 21 a 31
(14)
!$ x3, x4, x5 !$) $ ) !$; ! !$ 4 ! ) 1 x3) $ ; # # x2, x3 # x1, x4, x5. .! !$/# ! ! !$ $ ; 5L6) # ) ! 1 # ; # 3 3 , x3
x1) 1 x1 ! $
C - ! a11 . ! ! !3$ ; 3 x1 $ .! 1
! a21 )
a31 $ < !$
1# 5L6 1 1 x1 + a12 x 2 + a13 x3
a 122 x 2 + a 123 x3 + x 4 1 1 a 32 x 2 + a33 x3 +
21
= b11 , = b1 , 2
x 5 = b1 . 3
(15)
M$ a ij1 1 M )
!3 > / !# 2 / 5L6 ! $
! 5C:6 ) # ) ! ! !$ .! ! !- 3
! ! 5 !E6
4.3.
% ! -
< ! %x1' %x5' !
1 596)
! %bi'
3 # -! ! %bi'
;
# # " ! < ! %/' 3 ; ) ! %cj » – 1 ! ; < # ! 3 1 ! <
" 3 " * !-
c
j
1
2
3
1 x3 2 x4 3 x5
0 0 0
4 cj–Zj 1 x3 0 2 x4 0 3 4 1 2 3 4
x2 6 cj–Zj x1 5 x4 0 x2 6 cj–Zj
c1=5 c2=6 c3=0 c4=0 c5=0 x1 x2 x3 x4 x5 7 4 1
6 1
5 5.5 3.75 0.25 3.5 1 0 0 0
bi
bi aik
4
1 0 0
0 1 0
0 0 1
140 23.3 64 64 64 16
6 0 0
0 1 0
0 0 1
0 -1.5 -0.25
0 44 8 48 12.8
1 0 0 0 1 0
0 0 0.182 -0.682 -0.045 -0.636
0 0 0 1 0 0
0.25 -1.5 -0.273 0.773 0.318 -0.545
16 96 8 18 14 124
64
3 ! ! 5CE6) 5CF6 .! 1 ; 3 1! ! «cj » 1! ! %x1»,…, «x5')
3) ! 3 3"# cj 4 !
«cj » ! %bi'
! ! 5D6 2
! %bi». ) ! ! = x1=0, x2=0, x3=140, x4=64. x5=64, f=0. < ! ! $ ! !$#
! # %x1' %x2' 33 1# # )
! .! ! $ 3 !$; / < / 1 x2)
8 5 6 * ! %x2' #. + $) 3 3 ! !3$ ) ! ; b / i ! ai2S7)
# ! ! ! 4 !$; ai 2
1 / 16) x5)
) 1 3 ! ; !3$ ! $ 22
*
%x5' #. -!) ! !3" !3"
! ) 0#. < 3 !3 ! !$
1 ! # # .! 1! $ !3" ! / 3"
1!
3 !3 ! $3 . ! ; 1! !3" ! 3"# ; ! !$
! A ) 3 !
! 6 1 !)
1
2 . ! !3 1! ! ! = x1=0, x2=16, x3=44, x4=48, x5=0, f=96. < ! $ ! !$
! x1. b - < ! ! " / i !
ai 1 ai1S7) # ) 1 / 8 x3 - ; # !3" ! ) !3"
! ! ! %x1') / 3" 1!
5.5. 4 !
/ !
! $3 !3 ! < $ ! ! !$)
! /
x1=8, x2=14, x3=0, x4=18, x5=0, f=124 ! !$ > / ) !$/ !$ ! 124 $ ! ; 8 14
5. % 5.1.
+! !% !
4$ $ $ $ y1, y2, y3
# - ) ! ) "# ; ! ) !$/) !$ ! #
! < ! )
# !
!) ! ! # ) # !E
* 1 !$3 ! # !) !)
! ) ! 7 y1 + 4 y 2 + y 3 ≥ 5, 6 y1 + 1 y 2 + 4 y 3 ≥ 6.
(16)
y1 ≥ 0, y2 ≥ 0, y3 ≥ 0.
(17)
J !$= 4 !$ !$ $ y1, y2, y3, "
# ! !$) $ g = 140y1 + 64y2 + 64y3⇒ min.
(18)
M 5CB6) 5C86) 5CL6 /3 5C6) 5E6) 5F6
*
5C6) 5E6) 5F6 1 # !
23
5.2.
, )
4$ M4 ! ! n #
f = c1x1+c2x2+…+cnxn
(19)
! m a11 x1 + a12 x 2 + + a1n x n ≤ b1 , a 21 x1 + a 22 x 2 + + a 2 n x n ≤ b2 , a m1 x1 + a m 2 x 2 + + a mn x n ≤ bm ,
(20)
# !$ x1 ≥ 0, x2 ≥ 0, …, xn ≥ 0.
(21)
.
! g = b1y1 + b2y2 +…+ bmym
(22)
# a11 y1 + a 21 y 2 + + a m1 y m ≥ c1 , a12 y1 + a 22 y 2 + + a m 2 y m ≥ c 2 , a1n y1 + a 2 n y 2 + + a mn y m ≥ c n ,
(23)
!$ # y1 ≥ 0, y2 ≥ 0, …, ym ≥ 0.
(24)
< 5EE6) 5EF6) 5E96 ! # !
) 1 !
5 6) 1; !
@! 1
a11 a 21 A= a m1
a12 a 22 am2
a1n a2 n , a mn
(25)
) 5E76 5EF6)
) 1
! 1
#
24
y1
a11
a12
…
a1n
b1
y2
…
a21
a22
…
a2n
b2
…
…
…
…
…
…
ym
am1
am2
…
amn
bm
g=b1y1+ b2y2+…+bmyn
2 J! ;
; ;
… x1 x2 xn
2
…
… cn c1 c2
J! f=c1x1+c2x2+…cnxn
max E *#
!
5.3.
* $
1 02 2 " 3 fmax=gmin. 1 " 2 04 . X * = (x1* , x*2 ,… , x*n ) – 0
(19)–(21)2 Y * = (y1* , y*2 ,… , y*m ) – 0 (22)–(24)4 5 2 X*, Y* 0 2 2 03 m (26) x*j ∑ aij yi* − c j = 0 , j = 1,n , i =1 n y *i ∑ aij x*j − bi = 0 , i = 1,m. j =1
(27)
4 ! $ $3 ) ) ) !
5CD6–5EC6 $ !$ /)
# ; bi , i = 1,m 5
6) $
$ fmax = fmax(b1, b2, …, bm). % 0 " # 2 44 25
∂f max = y *i , i = 1,m. ∂bi
(28)
5EB6 ) –! ∆bi ; 3 !$
! ∆f m a x
=
∆b i y i * .
(29)
*! $ ) 5ED6 ! !/$ !$/# # ∆bi) ; 3 /
6. & # ! !3 !$ /
* ! < A # $ 6 , !
! .
6.1.
-
.
! 3
! ; $ / ! 2 ) ! !$ /;
) !$ /
! )
) !$ 3 * !
! !$ /
* * x1 = 8, x 2 = 14 , f max = 124 * ! 5E86 ! –! !$ /
! !$ ) 3"
!; < x1* > 0, x *2 > 0 ) 1 ; 5C86 !3 7 y1 + 4 y 2 + y 3 = 5, 6 y1 + 1 y 2 + 4 y 3 = 6.
(30)
* # 5F76 5EL6 ) ! ; –! ! ) ; 3" !$ / !3 4
!$ /
x1* = 8, x*2 = 14 5E6) ! 7 x1* + 6 x*2 = 7 ⋅ 8 + 6 ⋅14 = 140 = 140 , 4 x1* + 1x*2 = 4 ⋅ 8 + 1 ⋅ 14 = 46 < 64 , 1x1* + 4 x*2 = 1 ⋅ 8 + 4 ⋅14 = 64 = 64. <
! !$ / ! ; 598T896) 1 !$ /
; !3= y *2 = 0 1 ) ! 5F76 7 y1 + y 3 = 5, 6 y1 + 4 y 3 = 6, / 3) y1* = 7 11 = 0.636 ,
y* = 6 11 = 0.545 . 3
@! / ! $ "$3 ! ) / ; !$ ! !) !$ !; 3" ! = 26
5 ! 0 2 2 # 4 .! / ! ! ! ! # ! !; !$# # x3, x4, x5 !!$ –0.636, 0 –0.545)
y1* = 0.636 , y * = 0 , y * = 0.545 ) /) ! "$3
2
3
6.2.
, !!! %
) ! $3 !$ ) 3 " 4;
! ! !$ $ I ! !
< / !3 .! 1# ; ! !$= y1* = 0.636 > 0 , y *3 = 0.545 > 0 *$ !; ) ! = y *2 = 0 < ) 1 !
# ) 1 ) $)
! ) y1* > y *3 .
6.3.
./%% !
) 5E86 ) ! j- ; /! !$ ! ) x *j > 0 )
m
∑a i =1
ij
m
∑a i =1
ij
y *i = c j @!
y *i > c j ) 3" x*j = 0 , ) ! m
∆ j = ∑ aij y*i − c j
(31)
i =1
!$ ! ! j-
@! ∆ j > 0 ) ) "# !
!$/) !$ ! !) ! !$ ) ; @! ∆ j ≤ 0 ) ; / 3 !$ $ , ! * S1 (+) S2 (,) S3 (- )
0.636 0 0.545
P3 4 6 3
P4 7 4 1
P5 6 2 4
4
5
5
4$ /$ # P3, P4, M #
! !F <! 5FC6= P5 ∆ 3 = 4 ⋅ 0.636 + 6 ⋅ 0 + 3 ⋅ 0.545 − 4 = 4.18 − 4 = 0.18 > 0, ∆ 4 = 7 ⋅ 0.636 + 4 ⋅ 0 + 1 ⋅ 0.545 − 6 = 5.0 − 6 = −1.0 < 0, ∆ 5 = 6 ⋅ 0.636 + 2 ⋅ 0 + 4 ⋅ 0.545 − 5 = 6.0 − 5 = 1.0 > 0. 27
, ∆ 3 >0, ∆ 5 >0) $ 3 P3 P5) 3 P4 $ ) ∆ 4 < 0 .
6.4.
. !
< ! !3
@! !$ !!$ ) !$ ! <
) !$ !$
1 ) /! ; #U 4 M4 5CD6–5EC6 m !!$#
# xn+1, xn+2,…,xn+m , 5E76
/ AX = b ,
a11 a 21 A= a m1
a12 a22 am 2
a1n a2n a mn
1 0 0 1 0 0
x1 x2 0 b1 b2 0 , X = x n , b = . x n +1 1 bm x n+m
4$ !$ ! $ X X* > 7
X0 = 7 < !3 ) // !$ ! )
– !$ * A $
< A* !3 ! A) 3" ; !$ ! ) A0 !$ ! ,
$ A* X * + A 0 X 0 = b . , A0 X 0 = 0, A* X * = b .
(32)
5FE6 3 A* −1 ) ! X * = A*−1b
(33)
A* −1 = D ,
(34)
2
5FF6
/= X * = Db .
(35)
D # !
b ! !$ ; X* 4$
!$ ∆b ! b+∆b 2 !$; / X*+∆X* * ! 5F:6 X * + ∆X * = D(b + ∆b ) , 3 ) ! $ 5F:6) ∆ X* = D ∆ b .
28
# $ /
) 1 $) b 3
# y. 2 ∆b (j − ) $/ bj) " 3 / ; ) ∆ ( + ) 3" !$ ! -
" # ! x* ∆b (j − ) = min i i d ij
! dij > 0,
(36)
x* ∆b (j + ) = max i ! dij<0. i d ij
(37)
M$ dij – 1! D. ) i– ! # b j − ∆b (j − ) b j + ∆b (j + ) ; D ! / M4 !– * !; D !3 ! ! / ! !) 3" # ; 4 1 ! $ ) # #
/! < ! <! !- !F
# 3 x3. x4, x5 < !: ! / ; ! ! 0 1 - !-
x4 x1 x2
x1 0 1 0
x2 x3 0 -0.682 0 0.182 1 -0.045
x4 1 0 0
x5 0.773 -0.273 0.318
D 3 !# ! ) ! $ ) ; ! %/7 /! x1, x2, x4 < !$ !
0.182 0 − 0.273 0.318 D = − 0.045 0 − 0.682 1 0.773
(38)
A ! 596 7 6 1 0 0 A = 4 1 0 1 0 1 4 0 0 1
(39)
, !$ / $ x1*=8, x2*=14, x3*=0, x4*=18. x5*=0,
X =(x1, x2, x4)T X0 = (x3, x5)T) A* 3 ! 1, 2, 4 # ; 5FD6= 7 6 0 * A = 4 1 1 . 1 4 0 *
) D=A*-1 .! 1 D A*: 29
0.182 0 − 0.273 7 6 0 1 0 0 0.318 4 1 1 = 0 1 0 . DA = − 0.045 0 − 0.682 1 0.773 1 4 0 0 0 1 *
4 DA* )
) D !$ * "$3 1! ! D # ! !
b1) "$3 ! – ! b2 * 3 ; !$ ! .! " ! ! 5F86) 5FL6) ; / ! # b1
b2
b3
* 0.182 0 − 0.273 x1 = 8 0.318 x*2 = 14 − 0.045 0 − 0.682 1 0.773 x*4 = 18
< ! D !$ ! !$ 1!) 1 8 x1* (− ) = 44 . ∆b1 = min = d 11 0.182 < ! D !$# 1! ) 1 x*2 x*4 14 , 18 = (+) b max , ∆ 1 = = max − 0.045 − 0.682 d 21 d 31 max (− 308, − 26.4 ) = − 26.4 = 26.4 . 0 ! ! x* ∆b2(− ) = min 4 d 32
18 = min = 18 . 1
, !$# 1! ! ) ∆b2(+ ) = +∞ , !$ !$/ !
3 /;
) ! !3 !
1 ; 3 # !
$ 14 18 x * x* , ∆b3(− ) = min 2 , 4 = min = 0.318 0.773 d 23 d 33 min(44, 23.3) = 23.3 , 8 x* ∆b3(+ ) = max 1 = max = max( −29.3 ) = 29.3 . − 0.273 d 13 ) ! # !3"= !
(b1 − ∆b1(− ) , b1 + ∆b1(+ ) ) = (140 − 44, 140 + 26.4) = (96, 166.4) , ! (64 − 18, 64 + ∞ ) = (46, + ∞ ) , ! $ – (64 − 23.3, 64 + 29.3) = (40.7 , 93.3) .
30
6.5.
2 !!! 3% !3
<! I ! ! ! ) ; !$ ! !$
! ! I ; - ! $ 4$ ! $ !$ I 11 ! # )
; 140+11=151. - I #
! ; (96 , 166.4 ) ) ) ! # ; ! !
∆ f=y1*⋅∆ b 1=0.636⋅11=7. !$ ! ! 5F:6= x1 b1 + ∆b1 0.182 0 − 0.273 151 10.0 0.318 64 = 13.5 x 2 = D b2 = − 0.045 0 x b − 0.682 1 0.773 64 10.5 4 3 , )
# ! !$ !$ /; # I # /
6.6.
2 !!!
!$
! 5CB6
g min = g (Y * ) = 140 y1* + 64 y*2 + 64 y*3 = 140 ⋅ 7 / 11 + 64 ⋅ 0 + 64 ⋅ 6 / 11 = 124. - !$ ! fmax=124,
*!
fmax =g min $ = 3 ! ) $ ! 3 ; !$ ! !$ !$3 !$ fmax ! $ ; !$ $ # ! !$ ; gmin.
31
'( . -
1.
3 ) # $ / ; ! # ! !$# , 3)
) # #) / # # < 1 ! ; 3 ! ) ! ) " !
2 $3 # ! ! $)
! ) $ !$ ; ! # ! !# @ !$ – 3 ! A ! $ ! + ! $
! ! ) 3 " ; !$) 3 3 * ) 3" !) ; 3 ) #
0 2 !$ ! ! ; ! !$# 4 ! ! 3
# ) $ / ! / ! ; / ( $) / 3 ! ; ) ! 3 ) !
! ! # ) ! / ; / !$#
! 3 #
# – 1 !$ *!
# ! ! ) ! ) ) )
# ! # ) 3 . , ; $ !) !3"# !# #
! / ) ! !
) ! ) ) / # # ! # ) # $
! ) 1!$) !$ !3
) ! ! # - ) 3" !$ ; / @! !# ! # )
/ 4 # !$# ! ; ) $ / /$
– 1
$ 3 ) ! 3"3 ) $ ! !$ !$ /)
! $ !$ /
2. ' 3 3 ) 3 A B 4$ A m !# A1, A2, …, Am) B $ n !# B1, B2, …, Bn. 2 aij / A) ! ! 3 Ai) B ; ! 3 Bj , ! / ! 3
32
1 * B A1 * A2 A … Am
B1 a11 a21 … am1
B2 a12 a22 … am2
… … … … …
Bn a1n a2n … amn
4 / A < ; B)
) 4 !
! ! ) A ! B C) ! !) ! C
! < ! E ! / " ' * B B1 – %2!' B2 – %/ ' * A1 – %2!' -1 1 A 1 -1 A2 – %/ ' . ! $3 ! !3" ! − 1 1 P = . 1 − 1
3. ( m × n
P = (aij ), i = 1, m; j = 1, n . 2 αi $/ / A Ai αi = min aij . j =1,n
- ! ! !$ 1! i- ! ; / ! A) ! $ Ai)
B $ / 3"
A * # αi !$/
α = max αi = max min aij . i =1,m
i =1,m j =1,n
@! A $ $ ) ; #/# ! # !$ !$ /) 1 /
α - ! ! !$ !$#
/ ! . B $ / 2 β j !$/
/ B Bj β j = max aij . i =1,m
33
<! β j $ !$ 1! j- ! ! # ; 5 / 6 B ! $ 3)
!$ / != β = min β j = min max aij . i =1,n
i =1,n j =1,m
<! β !
4 # ; B $ !$/ β . @! # 3) # "
α=β=ν ) 3" ) 3
) # $ – !$ / 4
3 ) 0 . < ! F ! ) !!$# ! #
αi β j . $ 1 ! A1 A2 A3 βj
B1 2 7 5
B2 4 6 3
B3 B4 7 5 8 7 4 1
7
6
8
αi 2 6 1
7
@! A $ A2) $ ; / α = 6 < 1# ! # B ) $
B2) $/ / β = 6 < 3 $) B, $ $ /) 3 B2)
A) 1# ! # /) 3 A2) )
$
A2, B2 ! / ) 8 -! ! a 22 = 6 !$
!$
! , 1! ; ) ! ! ! 3 ) /
# $ ! ! 9
& 1 ! A1 A2 A3 A4
B1 B2 B3 B4 B5 3 4 5 2 3 1 8 4 3 4 10 3 1 7 6 4 5 3 4 8
βj
10
8
5
7
αi 2 1 1 3
8
A) $ $ /) 3 A4 1# ! # B) ) 3 B3 </ A / B F A
) B B3)
$ 3 A1 !$ / : < 3 $ B 1
34
$ 3 B4 $ / E ! , )
" / # #
4. ) * ' 4 ) / $ m×n $ $ - !
)
* Ai A # Ak) ! ; / i- $/ 3"# / k- ) )
# !$ !$/
aij ≥ akj , j = 1,n . * Ai # 3 Ak) ! / i-
3" / k- aij = a kj , j = 1, n . * Bj B # Bp) ! ; / j- ! !$/ 3"# / p- ! ) ; ) # !$ $/
a ij ≤ aip ,i = 1, m. * Bj # 3 Bp) ! / j- !
3" / p- ! a ij = aip ,i = 1, m. @! - $ 3" ! !3" ) 1 3
$ 4$ :×: ) ! : 0 A1 A2 A3 A4
B1 B2 B3 B4 B5 4 7 2 3 4 3 5 6 8 9 4 4 2 2 8 3 6 1 2 4
A5
3
5
6
8
9
M ) A5 ! 3 A2) 1 !33 # ; $ * ) !) A1 A4 ) A1 A4 A4 $ 2; A5 A4) ! F×:) ! 8 4 A1 A2 A5
B1 B2 B3 B4 B5 4 7 2 3 4 3 5 6 8 9 3 5 6 8 9
! 8) ! ) B1 B2)
/ B) 3" B1 $/ /) ; 35
3"# B2 A ) B3 B4 B5 2 ; ) # $ 3") $ B2, B4 B5) ! F×E) ! L 5 A1 A2 A5
B1 B3 4 2 3 6 3 6
, $ ) A2 A5 !3 ) 3 A5, !$ ! E×E) ! B 5 A1 A2
B1 B3 4 2 3 6
5. +# # @! ! ) # ; !$ / < 1 ! !$ !$ / !
*/ SA A # A1, A2, … ,Am p1, p2, … , pm)
m
∑p i =1
i
= 1.
2
SA=(p1, p2, … , pm). */ 3 3 B
SB=(q1, q2, … , qn) ! !$ / -
/ # SA*, SA* ) ! ; !$ ) $ $ </) ; 3" !$ /3) J !
α ≤ ν ≤β. 4 !$ 3 2
3 % 2 2 02 0 4 4$ SA*=(p1, p2, … , pm), SB*=(q1, q2, … , qn) – !$# @! ; # / 3 ! $3) .
36
0 ! 3 ) !3"
1 0 2
0 4
6.
4$
! 5aij6 !$3 3 A SA*=(p1, p2, … , pm), p1 + p2 + … + pm=1.
(1)
2 !$ SA A / $/ ; / ! B !$
( $) / aij ! !$ - $)
! ! !$/ ! M? 1 !; M) / SA*, SB* @! aij ! !$) )
$ / !$ ! != > 0. 4$ B !$ 3 3 B1) A !$ !$3
3 SA*) $ A1, A2, …, Am p1, p2, …, pm. * / A 5 / 6 *
a11p1 + a21p2 +…+ am1pm $ $/ 1 ! ! # B1, B2,…, Bn B a11 p1 + a21 p2 + + am1 pm ≥ ν , a12 p1 + a 22 p2 + + am 2 pm ≥ ν , a1n p1 + a 2 n p2 + + amn pm ≥ ν.
(2)
A ! y1 = p1 ν , y 2 = p2 ν , , y m = pm ν .
(3)
< 1# # 5E6
/ a11 y1 + a 21 y 2 + + a m1 y m ≥ 1, a12 y1 + a 22 y 2 + + a m 2 y m ≥ 1, a1n y1 + a 2 n y 2 + + a mn y m ≥ 1.
(4)
4 ! 5C6 ! y1 + y 2 + + y m = 1 ν = g .
(5)
A $ !) ) ; $
g = , ) # !3"
= y1, y2,…, ym, (4) " 5:6 ,
) ) !
; ! 37
2 !$ B " ! 4$ B
!$ SB*=(q1, q2,…, qn6) A )
/ B $ !$/ a11q1 + a12 q2 + + a1n qn ≤ ν , a 21 q1 + a 22 q2 + + a 2 n qn ≤ ν , a m1q1 + a m 2 q2 + + a mn qn ≤ ν.
(6)
< q1, q2, … , qn ! 3 ! 3 q1 + q2 + … + qn=1.
(7)
4 ! 586 5L6
x1 = q1 ν , x2 = q2 ν , , x n = qn ν .
(8)
! a11 x1 + a12 x2 + + a1n xn ≤ 1, a 21 x1 + a22 x 2 + + a 2 n xn ≤ 1, a m1 x1 + a m 2 x 2 + + a mn x n ≤ 1,
(9)
x1 + x 2 + + x n = 1 ν = f .
(10)
J!$ B / ! !
f= , ) # !$# # x1, x2,…, xn) 3"# !$
5C76 ! 5D6 * ! 5D6) 5C76 596) 5:6 )
!
!$ a11 a 21 P= ... a m1
a12 a1n a22 a 2 n , ... ... ... a m 2 a mn
!$ 5D6 !$ 3 P) 596 ! *
#
! ! A B) C
38
4
! B … x1 x2 xn
2 J! ;
; ; B A
a11
a12
…
a1n
1
4 y2
!
A …
a21
a22
…
a2n
1
…
…
…
…
…
…
ym
am1
am2
…
amn
1
…
1
1
…
1
2
! ; B J!
B
g=y1+ y2+…+yn
y1
f=x1+x2+…+xn
C * $
! ! # ) ! ) ! ! #
) !3 / #) !$ $ ; #
) / .! / ! $ ) "
7. 4 $ A1, A2, A3 * 3
# # $ B1, B2, B3, B4 !$) 3 ! ; !# # ) !B 6 1 * <
;
A1 A2 A3
B1 3 9 7
B2 3 10 7
B3 6 4 5
B4 8 2 4
- 1
5 A6) ! ; !$ 5 B) 0 ! )
) B1
B2) 1! ! B1 $/ 3"# 1! ! B2) 1
! B2 !3$ ) !$ ! ! 3
3×3 3 6 8 P = 9 4 2 7 5 4 2 ! 33 #33 ) !D 39
7 8 A1 A2 A3 βj
B1 B2 B3 3 6 8 9 4 2 7 5 4 9 6 8
αi 3 2 4
# 3= = 3 < = 6, 1 ! ) / # #
/
SA*=(p1, p2, p3)6 SB*=(q1, q2, q3,). < yi= pi/, i=1, 2, 3; xj= qj/, jQC) E) F ) !$ $ # C) ;
! A 5 6 g=y1+ y2+y3min, 3 y1 + 9 y 2 + 7 y 3 ≥ 1, 6 y1 + 4 y 2 + 5 y 3 ≥ 1, 8 y1 + 2 y 2 + 4 y 3 ≥ 1, yi 0, i = 1, 2, 3. B 5 6 f=x1+x2+x3 3x1 + 6 x2 + 8 x3 ≤ 1, 9 x1 + 4 x2 + 2 x3 ≤ 1, 7 x1 + 5 x2 + 4 x3 ≤ 1, xj 0, j = 1, 2, 3. =
4 1# ) !!$ ; 3 y1 + 9 y 2 + 7 y 3 − y 4 6 y1 + 4 y 2 + 5 y3 8 y1 + 2 y 2 + 4 y 3 3x1 + 6 x2 + 8 x3 + x4 9 x1 + 4 x 2 + 2 x3 7 x1 + 5 x2 + 4 x3
= 1, − y5 = 1, , yi ≥ 0, i = 4 , 5, 6; − y 6 = 1,
(11)
= 1, + x5 = 1, + x6 = 1,
(12)
x j ≥ 0,
j = 4 , 5 , 6.
4 ! /3
! ; / - /) ) # ; "$3
! # #
4 ! 5CC6 y1 = y2 = y3 = 0, ! y4 = -1, y5 = -1, y6 = -1, 40
$ / ! M ! 5CE6 ; ) ) ! / x1 = x2 = x3 = 0, x4 = 1, x5 = 1, x6 = 1, 1 ! B / " / !
!C7 9 !!B
1
2
3
c c =1 - x j
1
1
c2=1
c3=1
c4=0 c5=0 c6=0
x2
x3
x4
x5
x6
0 1 0 0 0 1
0 0 1 0 0 0
1 1 1 0 1/6 1/3
1 0 -1/9 -14/9 2/9 -1/9 -y3
1/6 1/6 4/27 2/27 1/27 5/27
1 2 3 4 1 2
x4 0 x5 0 x6 0 cj–Zj x2 1 x5 0
3 9 7 1 1/2 7
6 4 5 1 1 0
8 2 4 1 4/3 -10/3
1 0 0 0 1/6 -2/3
3 4 1 2 3 4
x6 0 cj–Zj x2 1 x5 0 x1 1 cj–Zj
9/2 1/2 0 0 1 0
0 0 1 0 0 0
-8/3 -4/3 44/27 22/27 -16/27 -18/27
-5/6 0 -1/6 0 7/27 0 7/27 1 -5/27 0 -2/27 0 -y1 -y2
bi
bi aik 1/6 1/4 1/5 1/3 1/7 1/27 1/7 3/10 0
) !3" / ! ) !3; " !$3 / 3 B: f = 1/ = 5/27, x1 = 1/27, x2 = 4/27, x3 = 0. 23
= 1/f = 27/5, q1 = x1 = 1/5, q2 = x2 = 4/5, q3 = x3 = 0. / !
/ 3" / ! , ) ! A / !; 3"= y1 = 2/27, y2 = 0, y1 = 1/9. 23
p1 = y1 = 2/5, p2 = y2 = 0, p3 = y3 = 3/5. < ! p1 + p2 +p3 = 2/5 + 0 + 3/5 = 1. ) 3 A2 $ !) ! A1 ! !$
EK:) A3 FK: " I ) !$
27/5 = 5.4
41
'( * /&0,-** (
1.
. $ ) !
! # !$# ! 0 5! 1 #6 #
) !$ $ ! !
1 < # ! #) ) !
I ! ! !$
1 4$ ! m / - $ # ;
! W – 0 4$ " / ! /
!$# / # m
W = ∑ wi ,
(1)
i =1
wi – / i– / H # /
xi 2
x = (x1 , x 2 , , x m )
(2)
$ ! , ! x) / ! ; !$ ) ! ! / 1 ! m ! < !
$ # /= 1) 4 $ /
$ ? 2) 2 $ !$ 1 ! 3? 3) 4 !$ 1 ! 3 M$ / –
1 ! 3 m
W = ∑ wi , i =1
wi –
i– <! W ! $ /
" / $) # !)
/
/ ) # !$ / ! / 4! / ; $ ! / # "# !
" "# / # ! i– / ) /
1 / ! !$) ) ! !$ / #
/# / # !3 / * # / ! ! $ ) ! ; !$ / ) # 4 ! ! / $ ; !$ ! / ! # ! ! m– / 4 ! 1 ) # m-2–
/ ) ! !$ ! ! m-1– / , ) $ ; ! !) $ $ ! ; # !$# ! M $ !$ S0, ! ! !$ ! , ) ! 1 – ; ! ! # ! !$
/
42
2. ,! < ) $ $
)
! # ; ! 4$ !) # " H0 3" $ V0 ! $ Hk $ Vk # 3
I !3 H1 !33 H2 ) #
3 ! V1 V2 Hk 8
8
10 10 11 11 H0
A
8
7
10
9
10
V0
10 10
7 12
11 9
13 9
B 11
9
11
10
7
8
7
9
11
10 9
9
9 12
12
13
12
14 13 Vk
C # 3 * !
1 !/ $ !$/ ) ! $
# ) ! !$ ) ! $/ $ !/ !$/
) I ! ! 1 !$/ , ) $
– ! 4$ $ ! $ $3 / ) – 5C6 ; !$# !#
3 ! ) ; !$# –
3 B 7 7 4# !$ A 3 B ! $ $ $)
B1 11 !$ # !
4 # B / = : ! F
!
!) B– / < ! B ; 11 B2 # ! B1 B2 5E6 @! B- / B1)
L) ! B2) CC -
; C B1 1 / # ! ; B 7 17 10 7 ! ! 4 !) L– / !$ #
9 11 ! C1, C2, C3 5F6 < # !$
; # ! *" ; 16 10 11 B2 # ! C1 C3 ! B .! ! C2 C2 $ ) 1 L– / ) ; 11 B– / !! !$ # !
4$ C2 B1 ! DVLQC8)
22 C3 B2 = C7VCCQEC 2 !$ # C2
B $ B1.
43
, ! ! !$ ! 9 A ; ! A) ) !$ # ! L7 ; !$ $ A B 9 !$ $ @! $ A B) / !$
54
12 42
8
7
51
10 41
10
10
60
11 49
11
Hk
13 29
12 17
8 8
33
10 7
9 9
24
8
B
7
9
11
16
10 11
9
11
10
11
40
7 33
7 26
9 22
9
10
12
13
14
12 58
11 50
10 45
9 39
13 36
9
A H0
70 V0
9 2 !$ /
Vk
# ) !! CCVC7VBVCEVCFVCEVC7VLQBF ) / ; # 1 1 !)
! !$ ! 1
! " ! !$ !
3. ! 4$
K) !$ m ; .1, .2, …, .m A .i ! x #
ϕ i (x6*! !$ K ) !$ ; !$/ # M$
! ! !$ !
/ ) 1 ! ( $ / ! .1) – .2 ! ! – ) !3 * ; / # ! S – !
"
! # G ! !3 x1, x2, …, xm, !; , !$ !) $ ! x1, x2, …, xm, # ! m
W = ∑ ϕ i (xi ) ⇒ max .
(3)
i =1
! i– / ! !$ / 5 1 /
6) ! ! i − / ! S 2 ! !$
/ Wi(S6) 3" ! !$ !) $ )
! i– ) x i (S). ! m– /
44
x m (S)= S, / ! ! 3 !
! !$ / !) ! ! ;
Wm(S) = ϕm(S). 4 !) 5m–1)– / 4$ /!
S 2 W m −1 (S ) ! !$ / # !# / #=
(m–1)– m– @! 5m–1)– / (m − 1) – 3 ! x, ! / S–x. </ # !# / # ϕm-1(x)+Wm(S–x). x) 1 / ! W m −1 (S ) = max{ϕ m −1 ( x ) + W m (S − x )}. x≤S
(4)
. ! 5m–2)–) 5m–3)– / .! !3 i– / # $
! !$ /
/ 1 / ! W i (S ) = max{ϕ i ( x ) + W i +1 (S − x )} x≤S
(5)
3" !$ ! x i (S)– x) 1
4 ! ! / ) .1 M$
$ $ S) 1 ! K 2 !$ ; / W * =W 1 (K ) = max{ϕ1 ( x ) + W 2 (K − x )} x≤K
(6)
,
x) 586) $ !$ ! x1*
/ 4 ! ! x1* # K − x1* ; !$ 3 $ ! !$ ! ! /
x2(S6) # ! !$ ! ! / ) x*2 = x2 (K − x1* ) ,
) ! ) ! $ / x *3 = x 3 (K − x1* − x*2 ),
4. ' 2" ) ; ! 0 ) ! –
A ! !$# / !; 3" ) !
(!! 5CDE7-CDB:6) = % S 02 ! 0 2 0 0 0 # 0 4 5:6 5:6 ; (!! 45
5. - 4$ # K=7 !$ $
5m=96 A ! ) ! $
∆x = C) $ ! $ !$ ! ! ) I
) ! # ! / $ d = : &
#
! C ϕ 1 (x) ϕ 2 (x) ϕ 3 (x) ϕ 4 (x) 0.1 0.6 0.3 1.0 0.5 1.1 0.6 1.2 1.2 1.2 1.3 1.3 1.8 1.4 1.4 1.3 2.5 1.6 1.5 1.3
x 1 2 3 4 5
5.1.
# !
, I ) ! x1, x2, x3, x4)
$
! != x1 + x2 + x3 + x4=K, " / ! ! W*=max{ϕ 1 (x1) + ϕ 2 (x2) +ϕ 3 (x3) +ϕ 4 (x4)}.
5.2.
1 -
4 ! 3 3) ! 9– / $
S 3" x4(S6 W4(S6 A ! /
! S ≤ d = 5 ! / !3
3) 1 x 4 (S)=S, W4 (S)= ϕ 4 (S), $ ! !$ / 9– / # 9–
! # / 4$ I !# $ / S=1. A ! x) ! # $ ) $
7 C * ; ! 596 W3 (1) = max{ϕ 3 (x ) + W4 (1 − x )}. x ≤1
(7)
<!
# #= x=0, ϕ3(0)+W4(1) = 0+1 = 1; x=1, ϕ3(1)+W4(0) = 0.3+0 = 0.3. 23 ) $ 5L6 !$ x=7) M W3(1)=1, x3(1)=7 .! ; ! !$ / $ / ! $ ;
! S) $ 7) C) W) L * 3" !E
46
" : - - S
x
S–x
ϕ3(x)
W4(S–x)
ϕ3(x)+ W4(S–x)
1 1
2 0 1 0 1 2 0 1 2 3 0 1 2 3 4 0 1 2 3 4 5 1 2 3 4 5 2 3 4 5
3 1 0 2 1 0 3 2 1 0 4 3 2 1 0 5 4 3 2 1 0 5 4 3 2 1 5 4 3 2
4 0 0.3 0 0.3 0.6 0 0.3 0.6 1.3 0 0.3 0.6 1.3 1.4 0 0.3 0.6 1.3 1.4 1.5 0.3 0.6 1.3 1.4 1.5 0.6 1.3 1.4 1.5
5 1.0 0 1.2 1.0 0 1.3 1.2 1.0 0 1.3 1.3 1.2 1.0 0 1.3 1.3 1.3 1.2 1.0 0 1.3 1.3 1.3 1.2 1.0 1.3 1.3 1.3 1.2
6 1.0 0.3 1.2 1.3 0.6 1.3 1.5 1.6 1.3 1.3 1.6 1.8 2.3 1.4 1.3 1.6 1.9 2.5 2.4 1.5 1.6 1.9 2.6 2.6 2.5 1.9 2.6 2.7 2.7
2 3
4
5
6
7
x3(S) 7 0
W3(S) 8 1.0
1
1.3
2
1.6
3
2.3
3
2.5
2.6 3, 4
4, 5
2.7
4 ! !$# / # !$
! 8 !E ! S - !$
!) !
3" x ! !$ /
! # L B
2) # ! # !$#
#
x) ! 1 ! !$/ ! )
!/$
4 ! !$# / ! S = 8 L )
I ) !# 3) / d = : -
) x ≤ 5
S – x ≤ 5. <! # 3 ! !$ / / ; !F M / / ϕ2(x6 # !C) !
!$ / $ / W3(S–x) – ! E
47
$ : - - S x S–x ϕ2(x) W3(S–x) ϕ2(x)+ W3(S–x) 1 1
2 0 1 0 1 2 0 1 2 3 0 1 2 3 4 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5
2 3
4
5
6
7
3 1 0 2 1 0 3 2 1 0 4 3 2 1 0 5 4 3 2 1 0 6 5 4 3 2 1 7 6 5 4 3 2
4 0 0.6 0 0.6 1.1 0 0.6 1.1 1.4 0 0.6 1.1 1.2 1.4 0 0.6 1.1 1.2 1.4 1.6 0 0.6 1.1 1.2 1.4 1.6 0 0.6 1.1 1.2 1.4 1.6
5 1.0 0 1.3 1.0 0 1.6 1.3 1.0 0 2.3 1.6 1.3 1.0 0 2.5 2.3 1.6 1.3 1.0 0 2.6 2.5 2.3 1.6 1.3 1.0 2.7 2.6 2.5 2.3 1.6 1.3
6 1.0 0.6 1.3 1.6 1.1 1.6 1.9 2.1 1.4 2.3 2.2 2.4 2.2 1.4 2.5 2.9 2.7 2.5 2.4 1.6 2.6 3.1 3.4 2.8 2.7 2.6 2.7 3.2 3.6 3.5 3.0 2.9
x2(S) 7 0
W2(S) 8 1.0 1.6
1
2
2.1
2
2.4
1
2.9
2
3.4
2
3.6
4 ! !$ ! /
!$ ! S = K = 7. & : - - x 0 1 2 3 4 5
K-x
ϕ1(x)
W2(K–x)
ϕ1(x)+W2(K–x)
7 6 5 4 3 2
0 0.1 0.5 1.2 1.8 2.5
3.6 3.4 2.9 2.4 2.1 1.6
3.6 3.5 3.4 3.6 3.9 4.1
48
x1* 5
W* 4.1
;
5.3.
!$ / !: 0 K=7 S 1 2 3 4 5 6 7
x4(S) 1 2 3 4 5
i=4 W4(S) 1.0 1.2 1.3 1.3 1.3
x3(S) 0 1 2 3 3 3, 4 4, 5
i=3 W3(S)
x2(S)
i=2 W2(S)
0 1 2 2 1 2 2
1.0 1.3 1.6 2.3 2.5 2.6 2.7
x1(S)
1.0 1.6 2.1 2.4 2.9 3.4 3.6
5
i=1 W1(S)
4.1
< !: S ) # /
! ) ! !$ / $ W*=4.1, 1 3 ! !$ : L M ) !
/ $ L – 5 = E ! ! S = E ; # ) 3 ! !$ C A ! $ / ; E – 1 = C 4 S = C $ ! A ; ! / C ) ! ; 3 ) !$ ! $ x1=5, x2=1, x3=0, x4=1. < !: 1 ! !
-
5.4.
4$ #
! = ∆ K=E 2" #
; K + ∆K = 7 + 2 = D < 1 ! ! : ! $ !!$
) 3" ! !$ ! ! S = B S = D + ; !$ 1 !!$ ! EC) FC) 9C " : - - S 8
9
x 3 4 5 4 5
S–x 5 4 3 5 4
ϕ3(x) 1.3 1.4 1.5 1.4 1.5
W4(S–x)
ϕ3(x)+ W4(S–x)
1.3 1.3 1.3 1.3 1.3
2.6 2.7 2.8 2.7 2.8
49
! !$
/ x3(S)
W3(S)
5
2.8
5
2.8
$ : - - S 8
9
x
S–x
0 1 2 3 4 5 0 1 2 3 4 5
8 7 6 5 4 3 9 8 7 6 5 4
ϕ2(x)
W3(S–x)
ϕ2(x)+ W3(S–x) 2.8 3.3 3.7 3.7 3.7 3.2 2.8 3.4 3.8 3.8 3.9 3.9
2.8 2.7 2.6 2.5 2.3 1.6 2.8 2.8 2.7 2.6 2.5 2.3
0 0.6 1.1 1.2 1.4 1.6 0 0.6 1.1 1.2 1.4 1.6
- x2(S)
W2(S)
2, 3, 4
3.7
4, 5
3.9
4 ! !9C) ! !9 !$
! S = K = 9) / $ 3"
& : - - K=9 x
K–x
ϕ1(x)
W2(K–x)
ϕ1(x)+ W2(K–x)
0 1 2 3 4 5
9 8 7 6 5 4
0 0.1 0.5 1.2 1.8 2.5
3.9 3.7 3.6 3.4 2.9 2.4
3.9 3.8 4.1 4.6 4.7 4.9
x1*
W*
5
4.9
4 ! !: !$ !EC) FC) 9C) ! !:C 0 K=9 S 1 2 3 4 5 6 7 8 9
x4(S) 1 2 3 4 5
i=4 W4(S) 1.0 1.2 1.3 1.3 1.3
x3(S) 0 1 2 3 3 3, 4 4, 5 5 5
i=3 W3(S) 1.0 1.3 1.6 2.3 2.5 2.6 2.7 2.8 2.8
x2(S)
i=2 W2(S)
0 1 2 2 1 2 2 2, 3, 4 4, 5
x1(S)
1.0 1.6 2.1 2.4 2.9 3.4 3.6 3.7 3.9
5
i=1 W1(S)
4.9
!:C ) !$ / ! 9D) !$ ; ! x1=5, x2=2, x3=1, x4=1.
5.5.
- -
4$ "
! L) : ) !$ !
; 3 .! ! / $ ! !$
! / ! K = : !$ !9E !$ ; W2(0)=0. 50
& " : - - K=5 x 0 1 2 3 4 5
K-x
ϕ1(x)
5 4 3 2 1 0
0 0.1 0.5 1.2 1.8 2.5
W2(K–x)
ϕ1(x)+ W2(K–x)
2.9 2.4 2.1 1.6 1.0 0.0
x1* 0
2.9 2.5 2.6 2.8 2.8 2.5
W* 2.9
0 " K=5 S 1 2 3 4 5
i=4 x4(s) W4(S) 1 2 3 4 5
1.0 1.2 1.3 1.3 1.3
i=3 x3(S) W3(S) 0 1 2 3 3
i=2 x2(S) W2(S) 0 1 2 2 1
1.0 1.3 1.6 2.3 2.5
i=1 x1(S) W1(S)
1.0 1.6 2.1 2.4 2.9
0
2.9
!:E ) !$ / $ W*=ED) !$ ; ! = x1=0, x2=1, x3=3, x4=1.
5.6.
- -
4$ !$ ! K = L ) $
-
) /
S = 7. !: # !3" !$ / x2 = 2, x3 = 3, x4 = 2. !$ / W* = 3.6.
51
'(# .*&&(-1&'2 ( .
1.
, ! ! !
!$# 5*26 4 *2 !$ !; ) ! ) ) #
A *2 ! ! ! 3"# !
! ) ) ! ! ! *2
$ . M ! 3 *2 ! ) ! ; ) 2! ! !; *! # *2 ) ; - $
! ) ; *2 $ -
*
! !$ *2 ; ! ! $ ) !$ ! *2 !$; ! $ ! , ! ! *2) 3"
! ; 5! ! ) # !$ $) #
6 !
1 *2 , ! $ ! ) ! ; # ) ! # ! ) !
! *2 ! # = *2 *2 < *2
) / ) ! ) *2 !; ) )
! ! ) ! ! !;
< *2 $3 # !
" ; $ *2 $3 !$ $ !3 . ; ! 8 0 – 0 7 !
; # ! . ! 8 0 – 0 7 $)
) !) !$ 3"# ! ) " ; !$ ! / ! $ ,
3 *2 ) ! #
; *2) ) !
# ! < *2
; *2) ) ! ! ) ; 3"# ! ) $ % '
) !$ ; !
2. ' 4 ; ) ! /
*2 ! ! 4 ) ! ; S1, S2, S3)W
!$) #
# 4 ) !
# #
) ! 4 *2 ! ! ; ) *2 !; ) ) !
*! ) 3" ) ) ! ! !3 ; t0 # "
!$
52
t0
) /! 1 ; t < t0 5 /! 6
0
t > t0 5"6
t0 (S0) C A 3
t
) S -! t
!$; S0 !$ t0 )
! 1 ) ! ! ! ) ) ; !$ t0. 4 ! !$) ; # $ # #= % ' % ' @! ; !$ # % ') "
!$ !$ ) /! 1 ) ) ; !$ ! ! !$ ) ! $)
" !!$ ) ! !$ ! !
) ! $ ! ) ! ! < ! ; $ ! $ # 4 ! ! # ; !$ $ # – " . * 3 !$ ! ) #
– ! ) 3" 4 ! ; # S) " # ! ) # $
5 $6 ! 4 ! ! ) ; ! $ ) ! <
!$= S0 – ! ) S1 – ! ) ) S2 – ! ) ) S3 – ! 3 > E S0
10 01 S1
S2 13
31
20
02
23 S3
32
E > # ! ! ) 3"# S0 S3) $3 ; # # ! ) $3 # ; ! 53
3. 4 ! !$ $ # ) !3"#
! ) )
" ) ! M ) 3" *2) ; 3 9 ! # "# ; 4 ) ! !3
4 ) ! #
< ) $ ! $ ; ) ! ! ) ; $ !$/) $3) ! )
! $
4 ) ! ! !3# # ; 3"#
1
2 ! ) 3"# #)
! ) 3"# ) ) # "# )
!) !) # "# ! ; ) !) ! 5 !$
/! ! ) ! 6) !3" !$; ) !$ !$ ) # !
!$ !) )
$ !
$ !3; " " 4 ) ! !3 )
) ) # "# )
@! ) $3 ! t # ! ! $ 4 0) 5! 6) !
) ! 4 / 3 ; ! !$/ ! #) # #
! ! /) #
! ) ! !$ ) $ ! <! Ot
F) ! 3 !
m – ! ) #
1
T O
t F 2$ ! .! / $ ) ;
m ) # ! Pm (τ ) =
(λτ )m m!
e −λτ .
(1)
> ) ! ! m ! 4 .! ! ; !) !
4 M (m ) = mm = σ 2m = λτ. M$ M(m6 ! mm ! ! m, σ 2m – < $ )
54
P0 (τ ) = e − λτ .
(2)
4$ T ! ! )
! 1 ! ! F(t) = p(T < t). F(t6 $ $ ) ! T $/ t) 1
) ; t # 4 ! )
" ) t
5E6 e − λt )
F (t ) = p(T < t ) = 1 − e − λt .
(3)
4! $ ! ! ! ; !) $ ϕ(t ) = F ′(t ) = λe − λt .
(4)
!) ! 5F6 ! ! $3 ! 596
!$ ! 1 !$ , ) !
!$ !$ ! ; ! ! !$ ! )
$ M (T ) = mT = σT2 =
1 . λ
(5)
.! / $ 1! 5 !6
t 5F6 (λ∆t ) p∆t = p (T < ∆t ) = 1 − e −λ∆t = 1 − 1 − λ∆t + + ≈ λ∆t 2 2
(6)
4. ) ., ) 3 ) C ( $)
) " Si Sj) !3 ; / ij, i, j Q 7) C) E) F ) 01 1 $
! ) 10 – $ ! >
! ! . < $3 i- $ pi(t6 )
t # i- Si .! !3 t ; = 3
∑ p (t ) = 1 . i
(7)
i =0
t $ )
t t # $ S0 - $
1. , t S0 t 0 $ 1 * ) "
S0) $ 01 + 02)
$ )
t
S0) 5 01 + 02 t) $ ) = C - 01 + 02 t. 55
< $ ) ! ! $ S0) # ; = p0(t)[1 - 01 + 02 t] 2. , t S1 S2 t 0 S0 $ 1 < t # ; S1 $3 p1(t6 M t S1 ; S0 $3 10t)
$ ) ! S1
t S0 p1(t 10t 0 ! $ # S2 S0 p2(t 20t 4 ! $ = p1(t 10t + p2(t 20t 4 ! p0(t+ t) = p1(t 10t + p2(t 20t + p0(t)[1 - 01 + 02 t], 3
p0 (t + ∆t ) − p0 (t ) = p1 (t )λ10 + p2 (t )λ 20 − p0 (t )(λ 01 + λ 02 ) . ∆t 4# !) ! ! 3 p0(t6 !3"
!$ p0′ (t ) = p1 (t )λ10 + p2 (t )λ 20 − p0 (t )(λ 01 + λ 02 ) .
(8)
! )
/ ! !$# " !$# ; 4 5B6 ! !$#
! = p0′ = λ 10 p1 + λ 20 p2 − (λ 01 + λ 02 ) p0 , p1′ = λ 01 p0 + λ 31 p3 − (λ10 + λ13 ) p1 , p2′ = λ 02 p0 + λ 32 p3 − (λ 20 + λ 23 ) p2 , p3′ = λ 13 p1 + λ 23 p 2 − (λ 31 + λ 32 ) p3 ,
(9)
A ! * A ! ! ) ) ; 4 ! p(t6
t @! "3 ! ) !$#
) 3 ! " < !; # ) 0 2 # . ( $ !$ = lim pi (t ) = pi , i = 0 , 1, 2 , 3. . t→∞
, !$ ) # !3 5D6 ; ! !3"3 ! !$# =
(λ 01 + λ 02 ) p0 = λ10 p1 + λ 20 p2 , (λ10 + λ13 ) p1 = λ 01 p0 + λ 31 p3 , (λ 20 + λ 23 ) p2 = λ 02 p0 + λ 32 p3 , (λ 31 + λ 32 ) p3 = λ13 p1 + λ 23 p2 .
56
(10)
< 5C76 ! ! $ !3" ! !$# = pi2
2 # 2 – 2 # i- 2 2 ! . H 5C76 # 9 p1, p2, p3, p4) ; ! / !$ $3 !$ !) ! ; 2 / ! "$3 ! p1 + p2 + p3 + p4 = 1
(11)
4!$ $ pi Si ! )
# 1 4$ 3 !3" !
= 01 = 02 = 13 = 23 = 10 = 31 = 20 = 32 = 2, 5C76
/=
3 p0 = 2 p1 + 4 p2 , 4 p1 = p0 + 4 p3 , 5 p2 = 2 p0 + 2 p3 ,
(12)
6 p3 = 2 p1 + p2 . 2
$) ) ) 5CE6 5CC6
; / ) ! !3 > = p0 − 4 p1 2 p0
+ 4 p3 = 0 , − 5 p 2 + 4 p3 = 0 ,
2 p1 + p2 − 6 p3 = 0 ,
(13)
p0 + p1 + p2 + p3 = 1. !$ 5CF6) !3 p0 !$# = p0 − 4 p1
+ 4 p3 = 0 ,
8 p1 − 5 p2 − 6 p3 = 0, 2 p1 + p2 − 6 p3 = 0, 5 p1 + p2 − 3 p3 = 1. 4 ! $ E) !$ ! !3 p1: p0 − 4 p1
+ 4 p3 = 0 , 1 p2 − 3 p3 = 0, 2 − 9 p 2 + 18 p3 = 0,
p1 +
−
3 p2 + 12 p3 = 1. 2
!$ $ $ ! !3 p2 =
57
p0 − 4 p1
+ 4 p3 = 0 ,
p1 +
1 p 2 − 3 p3 = 0 , 2 p 2 − 2 p3 = 0 ,
(14)
9 p3 = 1. 5C96 # p3 =
1 , 9
$ p2 = 2 p3 =
2 , 9
p1 = −
1 2 p 2 + 3 p3 = 2 9
p0 = 4 p1 − 4 p3 =
4 . 9
) ! ) ) ! p0 = !$ $
4 = 0.44 ! 99X)
9
5. ' > # ! ; ! # ) # # ! -
9 01 S0
12 S1
10
21
23 S2
32
… …
k-1,k k,k-1
k,k+1 Sk
k+1,k
… …
n-1,n Sn n,n-1
9 > ! 4 ! ) ) " ! ) ; / ! 4 / ! !$# ; ) " # ) !3
!3 ! .! S0 = 01p0 = 10 p1. .! S1: 12 + 10) p 1 = 01 p0 + 21 p2. < ! 5C:6 ! 12p1 = 21 p2; . !) ! ) 58
(15)
23p2= 32 p3 ! < !$ ! !3" = λ 01 p0 = λ10 p1 , λ 12 p1 = λ 21 p2 , . . . . . . . . λ k −1,k pk −1 = λ k ,k −1 pk , . . . . . . . . λ n −1,n pn −1 = λ n ,n −1 pn . A 5C86 $ " ! =
(16)
p0 + p1 + … + pn = 1.
(17)
/ 1 5C86 p1 p0: λ p1 = 01 p0 . λ 10 ) 5CB6) != λ λ λ p2 = 12 p1 = 12 01 p0 . λ 21 λ 21λ10 .! !3 k C n: pk =
λ k −1,k ⋅ ⋅ ⋅ λ 12 λ 01 p0 . λ k ,k −1 ⋅ ⋅ ⋅ λ 21λ 10
(18)
(19)
(20)
4 !$ ) p0) 5CL6= λ ⋅ ⋅ ⋅ λ 12 λ 01 λ λ λ = 1, p0 1 + 01 + 12 01 + ... + n −1,n λ10 λ 21λ10 λ n ,n −1 ⋅ ⋅ ⋅ λ 21λ 10 3 ! ! p0: −1
λ ⋅ ⋅ ⋅ λ12 λ 01 λ λ λ . p0 = 1 + 01 + 12 01 + ... + n −1,n λ 10 λ 21λ 10 λ n ,n −1 ⋅ ⋅ ⋅ λ 21λ10
(21)
2 ! ! 5EC6 p0) !$ !$ ! !
(20).
6. < ! 1 *2 $= A – ,:-) ! ) !; # ? Q – ) !3 ) ! ; # ? P – ) )
! ? k – !
# ! 5! !$ 6
6.1.
2! !
! ! )
; $3 4$ ! $3 ! !
- ! $ $ ! ) !$
! 59
1 . (22) µ * # $ # #= S0 – ! S1 – !
: t . =
S0
S1
: > !$ * ) ! 5C76) ! !$# = λp0 = µp1 , µp1 = λp0 . < ) ! $ ) ! ! p0 + p1 = 1 , # !$ p0 =
µ , λ+µ
p1 =
λ , λ+µ
(23)
3 # # S0 5 ! 6
S1 5 !
6 @! !
)
) $ 1 p1) λ (24) . = p1 = λ+µ M ! ) ! ! ! ) ) ! ; !#
p0) µ Q = p0 = . (25) λ+µ 0 !33 3 $ !) $
; $ )
! ) $ !$3 3
$= λµ . (26) = p0 = λ+µ
#! ! <+=
6.2.
4$ n ! ! ; $3 $ ! ! * # $
#= S0 – ! 5 7 ! 6) Si , = 1, – i ! >
8
S0
S2
S1
… …
Sk
k
… … (k+1
Sn n
8 > !$ *2 S0 S1
$3 !$
#
) S0 S16 , !3; ! 60
! @! ! 5 S1), ! $ @! 3 ! 5 S26)
S1) ! ! ! !
$ # ! E* ! ) ; ! ) F, k ! – k > 8 # ! ) 1 ; ! 5EC6) ! −1
2 λ3 λn λ λ p0 = 1 + + 2 + ... + + . 2 ⋅ 3µ 3 n! µ n µ 2µ
(27)
* ! ! 5CB6 – 5E76 !$ !$# = p1 =
λ λ2 λ3 λk λn p 0 , p 2 = 2 p0 , p3 = p ,..., p ,..., p = = 0 k n 2µ 2 ⋅ 3µ 3 k! µk n! µ n µ
(28)
<!
λ (29) µ @ ! – ! )
# "# !
!$ 5ED6 / 5EL6)
5EB6 = ρ=
−1
ρ2 ρ3 ρn p0 = 1 + ρ + + + ... + , 2 2⋅3 n! p1 = ρp0 , p 2 =
ρ2 ρ3 ρk ρn . p 0 , p3 = p0 ,..., pk = ,..., pn = 2 2⋅3 k! n!
(30)
(31)
& ! 5F76) 5FC6 3 ! -! 50 A -! )
) ! C7CCBLB) F7ECDED) !
! # ! 6 < $ *2 $ $ ) !
)
ρn (32) P . = pn = p0 . n! 23 # !$3 3 $ – $ )
; ! = ρn (32) Q = 1 − P . = 1 − p0 . n! 0 !3 $ ρ A = λQ = λ 1 − p0 . ! n n
(33)
* !
# ! k ) !$ !3"
!3 $ A) !
) ! #
A ! !
M ! 3"# !
# ! k= 5FF6) 61
A , µ
(34)
ρ k = ρ 1 − p0 . n! n
(35)
7. 7.1.
2! !
A ) !3 ! / /!) /! $ ; / /! ! ! ! 4 ! !
! ! $ D7
* !!$ $ ; ! F , !$ $ ) ! !
; ) !3 ) $ ! ; ) #
< ! !$3 ! 5! 6 ; $
! = 90 1/, !
t . = 3 = 1 20 . $ !
µ=
1 t .
= 20
1 .
< $ ) !) ) ! – 1
$ ) # ! 5E96= . =
90 9 λ = = = 0.818 . λ + µ 90 + 20 11
M ) BCBX !$# ! ) ) ! ; * ! ! ) /# – 1 !$ ; $ *2) # ! 5E:6= Q=
20 2 µ = = = 0.181 , λ + µ 90 + 20 11
) !$ CBCX ! * ! ) # – 1 !3 $
*2) # ! 5E86= =
90 ⋅ 20 180 λµ = = = 16.3 . λ + µ 90 + 20 11
4 ! ) ! 5! 6 !
!/ ! ) !/ !$/ ! ! ! ) ; !$ !
7.2.
#! !
4$ ! ) /
# 3) !
!
! F ! # , $ 1
*2 62
A ! ! ! !
n = 3. # 3 $
! 5ED6= ρ=
λ 90 = = 4.5 . µ 20
< $ p0 ) ! ) # ! 5F76= −1
−1
4.52 4.53 ρ2 ρ3 p0 = 1 + ρ + + + = 1 + 4.5 + = 0.0325 . 2 2⋅3 2 2⋅3 < $ # ! 5FE6= P . =
4.53 ρn 0.0325 = 0.493 . p0 = 3! n!
2 !$ $) ! 5FF6 Q = 1 − P . = 1 −
ρn p0 = 1 − 0.493 = 0.507 . n!
0 !33 3 $ # ! 5FF6= ρn A = λ Q = λ 1 − p0 = 90 ⋅ 0.507 = 45.6 . n! * !
# ! 5! 6 ) 5F96 k=
A 45.6 = = 2.28 . 20 µ
! ! ! #) *2
1 ) !$ ! ! ! !
63
332 &(. 1. - - ! != ! ;
K << & ) 0 > /) . . ? 4 << & – = Y,) 1999.– FDC 2. <!$ @* ! = ) ) ! – = )
>! - !) 1988.– E7B 3. ! 1 = ! K G A) (0
( ) & ? 4 G A – = ( ) Y,) CDDL– 97L 4. & >4 ! !$ = ; – = & ) E77C– :99
,' 3'4. 5. </ 0 Z < A ! – http://www.mathsoc.spb.ru/pantheon/kantorov/vershik.html 6. Dantzig G.B. Reminiscences About the Origins of Linear Programming //Mathematical Programming: The State of the Art / Ed. by A. Bachem, M. Groetschel and B. Korte.- Berlin: Springer Verlag, 1983.– P. 79-B8 @$ = . . ( < ; ! # ! – http://www.webcenter.ru/~zwb/origins.htm
64