Экономико-математические методы и модели: Учебное пособие

...

Author: Тарасов В.Л.

308 downloads 171 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

DOWNLOAD PDF

-

! "# !$ %& ' %(# ! ) ! '

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

Our partners will collect data and use cookies for ad personalization and measurement. Learn how we and our ad partner Google, collect and use data. Agree & close