«  »

Математические методы исследования экономики

1

. .
, ,
, .
: ?
, .
( "kybernetike" -
) - , ,
.

- ,
.

, -
.

. ,
, ,
,
, . ,
- .
, .
?
. ,
: .
, , , .
.
, - ,
- .
.
: , , ,
. ,
.
, , ,
, .
, ?
, ,
.

,
. , ,
, ,
. , , ,
, .
,
- . ,
, , - ,
, , , ,
, ,
.
.
,
. . ,
, .
" "
. ,
. ,
,
, .

.
.
,
, -
.
. ,
-
. ,
.
-,
.
,
.
. ,
.
,
, ,
.

.
1)
2) Ŀ
3)
4)
5)
6)

ͻĿ
Ŀ ĿĿ Ŀ Ŀ Ŀ
ĴĴĴĴ
ٺ ٺ

ͼ Ŀ




, .
. , ,
. .
,
. :

1) .
, -
, . ,

.
?
. ,
, .
. ,
, ,
.
.
,
, ,
.


, ,
.
, .
,

(, , ...). ,
, ,
, .
,
,
.

2)
-
,

.


:
1.
.
.
,
.
2. , .
, ,
,
, , .
,
- , ,
,
,
.
, ,
.
,
, ,
.

.

.

ͻ

ͼ
Ŀ Ŀ Ŀ


Ŀ Ŀ Ŀ

Ŀ Ŀ Ŀ Ŀ Ŀ Ŀ







:
.



.


:
1) :
,
. ,
,
, , ,
- .
, , "
" ,

""
, .
2) :
.
,
.
, ,
.
3) :
. ,
: ,
, ,
.
() ,

(). ,

,
.
- ,
, , ,
. ,

, .
,
,
,
.

3) .
- ,

. :
. - :
{ }
. - (
) : { }
. - ,
: {
}


,
, ,
.
.
,
.
, .

. ,
, .
, , ,
.

4) .
.
,
. - ,
.
" ".

,
.
, ,
.

.
- ( ).
- , .
, ,
, ,
.
, .

, ,
, .
.
, ,
.

5) .
, , ,
.
, , ,
. ,
,
, .
, .

6)

,
.
,
.

,
. -
,
, .
,
.


-
-----------------------

-. - -,
.
:
1) ,
.

- ;
, , ,
.
,
.
2)
, , , ,
.
,
, .
3) ,
. , -,
,
-, - ,
,
. ,
,
.



------------------------
.
,
, , ,
.
,
,
, .
,
( ) .
-
-, ;
,
- . , ,
, , ,
, (
, , , ).
()
: - + = 0
() :
- + + = 0,
.

K - , Xk, k=1...K,
, .
k-
Aik - i- .
: E Aik*Xk .

,
- Xi .
( i
: - E Aik*Xk + 1.0Xj = 0 (2.2)
, ,
.
:
1. ,
i.
2. -
, +1.0.
Xj,
, .
, 2.2. +1.0Xj,

.
3. ( , , Xk),
,
, -Aik. ,
,
( )
.
:
- E Aik*Xk + 1.0Xj = 0 ,
+1.0Xj, .
, ,
,
. ,
Xj,
.
,
, ,
, ,
- .



.
- ,
,
- ,
.
: Ai1X1 + ... + AijXj + ... + AinXn <= Bi,
Aij - i- Xj, j = 1 ... n, Bi -
.
, Xn+1,
, :
Ai1X1 + ... + AijXj + ... + AinXn - Xn+1 = 0,
Xn+1 <= Bi,

Aij i- Xj, j = 1 ... n,
,
, Bi
i- . ,

, .
(
.

,

.

.
(
)
( )
.
, ,
. - i-
Pi, Pb
,
: P1X1 + ... + PiXi => PbXb

, Xb .

.



:
1) .
2) .
3) .
4) .
1) .
,
.
,
,
.
,
,
.
:
Xi , Ci
,
CiXi.
.

, :
P, /

5.50 Ŀ
5.20 Ŀ
5.00

>Q, ./


2) .

, -
, ( ).
() .

( - CRF).
, CRF 365
(
) ,
.

.

3) .

(.1): Xi -
, Ci - ,
, - CiXi.
,
:


3 Ŀ
2 Ĵ 1 Ĵ

>
X1 X2 X3

4) .
,
.
: ,
(, , ,
...), ,
.




.
:
1) - .
2) .
3)
.
4) ,
1.
./ ,
Xi MBSD, Ci
/.
, ,
, ,
- .



(),

,
.

:

.
,
, ,
,
.
.
,
, , .
BOLNP COLNP LNB LNC
+1.0
,
,
. ,
, ,
90BBG, 9BBG, HYDBBG.

, ,
:
1) j ,
( Xj - j- ). 2 - 6
.
2) +1.0 ,
.
3) ,

, -Aij
, Aij - i,
j.
4)
, +1.0
. ,
, .
5) , ,
+Aij, Aij - i
j, (
).
6) j Cj
.

,
,
,
,
( ).
.
,
:
1) j ,
(Xj - j- ).
2 - 6 .
2) +1.0 ,
.
3) -1.0
( EVOLPROD).
4) -
-Pi ,
.
5) , -
+Pi.
6) 2 - 5 j,
(), ( B, Xb -
).
:
) (EVOLPROD)
+1.0,
) , -
, +Pb,
) , -
, -Pb,
) ,
+1.0 , (
Xb),
)
Cb.



-


, ,

.
.
.
.
,
, .
- , ,
,
.

, ,
:
1.
2.
3.
--------------------------------------------------------------------------
1. :
1.1
1.2
1.3

.

. ,
,
.


.
. .
:
1.2. .
1.2. .
1.2.

,
.
. ,
. ,

.
1.3.
1.3. .

.
:
1
2
3 ,

1.1 ,

1.2 , ,
, ...
.

2.1 .
,
.
.
2.2 ,
. 0.7
1.45 .

31
.
,
.
3.2
, .
-------------------------------------------------------------------

.
.


.
:
1) ,

2) ,

3)




,
- .
,
, .
, ,
. -
,
,
,
.
,

.

.
,
.
.

.



,
, , .
,
. :
,
.
,
, (MIN,MAX)
:
= F(X1,X2,...,Xn) => MIN(MAX)
,
,
.

.
,
. : ,
?
- . , -
, ,
.
:
-
.
:
,
, ,
.
,
,
,
: ,
.
,

.
,
. ,
.
,
.
.
- ,
, ...
,
, - .

,
,
.

,
,
,
,
,
.
.
,
, ,
, .
, ,
-
.
,

.
, ,
.


.



. ,

.
, ,
.

.

.
,
,
.
, .

,
.



,
- .
,
.

,
,
.
, , ,
.
,

.

.

.


.
,
.

,
. ,


,
.
,
, ,

.
, -
.
1)
.
,
.
2) ,
.

.
3)
-
.



.
, ,
, .
,
, ,
,
.
,
,
.
,
,
,
.
,
.

, ,
, , ,
.

.
.

.
,
.
:
B1...n
X1...Xm Q1...Qr
max(min).
, ,
, .
"
".

. , ,

.
,
.
,
, "
".

. .
1) ,
,
.
2) ,
.
:
.
.
.
.
,
.

,
.


-


(
) -
(
),
:
1. .
2. .
3.
.


.

$ 1 -

.
()

.
) ij i-

:
E Aij*Xj + Xs = Bi
Xs
E Aij*Xj <= Bi
_
, .. E Aij*Xj = Bi

-
.
()


i-
,

_
E Aij*Xj = Bi + 1
_
E Aij*Xj +(-1) = Bi
Xs
-1 i-
.
i-

Bi . Bi
i-
Yi ( i- )

.

Bi .
)
.

_
E Aij*Xj < Bi
.
:
_ _
E Aij*Xj = Bi Yi = 0 ( )
_ _
E Aij*Yi = Cj Xj = 0 ( )
Y
-
.


.

, .
.
Xj
12.65 / (j = -12.65)


50 ./.
: Xj + Xs = 50
Xs - .
1.04 /
- ?
1.04 /, ,
1.04
.
1.04 /
,
12.65 /:
1.04 ,
j .
, 12.65 +
1.04 = 13.69 /
.
, 13.69 / -
,
:
:
13.69
/.
= -
, = 13.69
- 12.65 = 1.04 /.
Xj - ,
Xj
,


, ,

( ).


. ()


( ).

(),
.
Aj ( j - )

() , Aij
- i
j.



, -

.
,
.
,
, .
:
Cj
Bi
Aij


++++++++++++++++++++++++++++++++++++++++++

)
-----------------------

.
q
_ _
Cj = Cj + q Dj = Dj - q


Xj
, Cj = 20 /
(=0)
= 1.40 /
21.40 /

= 0
.
.
Xj 21.40 /
Xj ,

( Xj )
Xj .
Cj
_
_ Cj + q, - < q <= Dj
Dj - Xj
. q
.
Xj
( _)
. q = Dj = 0
Xj
. Xj
MIN (B/Aj)i

, -1 -1
B=B * b Aj=B *aj :

X5 3.2 0.6
Xb = X1 B = 1.5 Aj = 0.3
X6 5.6 -1.2

MIN (Bi/Aij) = 1.5/0.3 = 5.0
21.40
/
Xj Xj ;
Xj X5 X1 X6
0.6 0.3 -1.2 . 5.0
Xj X1
Xj . ,
,

.

)
-------------------


i- .

_
Cb = Cb + q*Ei, Ei - i-
= 1 .
0
E3 = 0
1
0
j-
_
Dj = Dj + q*Aij

_
Dj => 0 Dj^ + q*Aij => 0, Dj^ -
.

Ci
Ci + q,

MAX {Dj^/-Aij} <= q <= MIN {Dj^/-Aij}
i/Aij>0 i/Aij<0

Aij < 0 q < +oo
Aij > 0 q > -oo
:
= 31.5 -3.5X4 -0.1X3 -0.25X5

X1 = 3.2 -1.0X4 -0.5X3 -0.60X5
X2 = 1.5 +0.5X4 +1.0X3 -1.00X5
X6 = 5.6 -2.0X4 -0.5X3 -1.00X5

X2 2 + q

:

_
_ D4 = 3.5 + q*(-0.5)
_ D3 = 0.1 + q*(-1.0)
D5 = 0.25 + q*(+1.0)
Aij ,
.
q :
(0.25/-1.0) <= q <= MIN (3.5/0.5 , 0.1/1)
-0.25 <= q <= 0.1
q




q = 0.1 X3
X2
0.1 X3
MIN {3.2/0.5 , 5.6/0.5} = 6.4 X3 X1
.

:
=============
1. q

.
q ;
.
,

,
.

2.
.

3.
:
;
,

( )
.



+++++++++++++++++++++++++++++++++++++++++++
_
Bi = Bi + q 1 <= i <= m
, Bi
-
.
Bi .
-

(
)

)
----------------------------------
i-
. :
,
Bi (
=>).
Bi + q,
-Xs <= q <= +oo <=
-oo <= q <= Xs =>
Xs - .
-:
3X1 + 4X2 + 7X3 <= 100

3X1 + 4X2 + 7X3 + X4 = 100
X4 = 26
:
3X1 + 4X2 + 7X3 <= 74

74.



)
------------------------------------

,
- .


, Bi (
<=).
Xb
Xb ,
, . (
Xb ).

:Ak1X1+Ak2X2 +...+Xs = Bk
Xs - .
Bk + q, :
1.1) Ak1X1+Ak2X2 +...+(Xs-q) = Bk
(Xs - q) Xs
, Xs
Xb = B - As*(-q)
As - Xs.
Xs
: B - As*(-q) => 0
q:
MAX {Bi/-Ais} <= q <= MIN {Bi/-Ais}
i/Ais>0 i/Ais<0
Ais > 0 q > -oo,
Ais < 0 q < +oo
=> q ,
E AijXj => Bi
-E AijXj <= -Bi
1.1) +(Xs-q) -(Xs+q).
:
= 31.5 -3.5X4 -0.1X3 -0.25X

X1 = 3.2 -1.0X4 -0.5X3 -0.60X5
X2 = 1.5 +0.5X4 +1.0X3 -1.00X5
X6 = 5.6 -2.0X4 -0.5X3 -1.00X5

X4 - i
( <=). Bi q, :
X1 = 3.2 - 1.0*(-q)
X2 = 1.5 + 0.5*(-q)
X6 = 5.6 - 2.0*(-q)
3.2 1.0
B = 1.5 As = -0.5
5.6 2.0
,
X1 => 0 3.2 - 1.0*(-q) => 0, q => 3.2/-1.0,
X2 => 0 1.5 + 0.5*(-q) => 0, q <= 1.5/0.5,
X6 => 0 5.6 - 2.0*(-q) => 0, q => 5.6/-2.0
q :
MAX {3.2/-1.0; 5.6/-2.0} <= q <= 1.5/0.5,
-2.8 <= q <= 3.0


+++++++++++++++++
1.
----------------------------
,
.
()
.
,
Aq.
,
.
,
.

2.
---------------------------------
,
, ,
.
, ,
.
,

).
, ,

,
.




,
,
.
,
.
.

Ŀ Ŀ
Ŀ
Ŀ Ŀ Ŀ
"" Ĵ 2

Ŀ Ŀ
Ŀ
"" Ĵ 1



: .
100 . 3.25 /
30 .
3.40 /.
100 . 0.35 ./.

0.15 1 0.85 2,
0.25 1 0.75 2.
1 , , ,
2 0.5 1
0.5 , 2,
.
50 . 0.25
/ . . 2
3.8 /, 1
5.5 / 45
./. ,
40 ./ 1. 1
4 ./
5.20 /. 1
5.0 /.
:
1 5.75 /; 2 4.0 /.

:
X1 -
X2 -
X3 - 1
X4 - 2
X5 - 2
X6 - 1
X7 - 1
X8 -
(<=30)
X9 - ,
1
(<=45)
X10 - ,
1
(<=40)
X11 - ,
1 (<=4)
X12 - ,
(<=100)
X13 - ,
(<=50)
X14 - 2,



0.15X1 + 0.25X2 + X3 + 0.5X5 - X6 - X7 + X9 = 45 [ 1 ]
0.15X1 + 0.25X2 + X3 + 0.5X5 + X6 - X7 - X10 = 40 [ 2 ]
X2 + X8 = 30 [ 3 ]
X6 + X11 = 4 [ 4 ]
X1 + X2 + X12 = 100 [ 5 ]
X5 + X13 = 50 [ 6 ]
- 0.85X1 - 0.75X2 - X4 + X5 + X4 = 0 [ 7 ]


5.50*(0.15X1 + 0.25X2 + 0.5X5) + 3.80*(0.85X1 + 0.75X2 - 0.5X5)
- 0.35*(X1 + X2) - 3.25X1 - 3.40X2 - 0.25X5 - 0.1*(0.15X1 +
0.25X2) - 0.25X3 - 0.20X4 - 0.30X6 - 0.5X7 --- MAX

0.825X1 + 1.375X2 + 2.750X5 + 3.230X1 + 2.85X2 - 1.9X5 - 0.35X1
- 0.35X2 - 3.25X1 - 3.40X2 - 0.25X5 - 0.015X1 - 0.025X2 - 0.25X3
- 0.20X4 - 0.30X6 - 0.5X7 ---> MAX


0.44X1 + 0.45X2 + 0.6X5 - 0.25X3 - 0.2X4 - 0.3X6 - 0.5X7 ---> MAX
-----------------------------------------------------------------


---------------

0.44 / ( ,
X12 0.44).
100 + q,
MAX { 3/-0.15; 70/-1; 32/-0.85 } <= q <= MIN { 2/0.15 }
-20 <= q <= 13.33
, 0.44 .
,
113,33 ./.

0.6 /
( , X13 0.6)
50 + q,
MAX { 3/-0.5; 50/-1 } <= q <= MIN { 2/0.5 }
-6 <= q <= 4
, 0.6 .
,
54 ./.


-------------------

0.01 /
30 + q,
MAX { 3/-0.1; 30/-1 } <= q <= MIN { 2/0.1; 70/1; 32/0.1 }

-30 <= q <= 20
q = -30, X2=0, .
q = 20, X2=50, 50 .
.
, 0.01 .
, 30 ./,
,
50 ./,
- . 0.01
. ,
30 ./. , ,

30 / ( 20 ./)
3.40 + 0.01 = 3.41 /.


---------------------------------------
1. ( )

X4: = 0.2
2 4.00 - 0.2 = .80 /
,
.

X6: = 0.3
1 5.20 + 0.30 =
5.50 / ,
MIN {3/1, 4/1} = 3 ./ (
).

X7: = 0.5
1 5.00 + 0.5 = 5.50
/ , ,
3 ., .

X3: = 0.25
1 5.75 - 0.25 = 5.50 /
, , 2 ./
.

2. ( )
X2: = -3.40
2 + q,
0.01/-1 <= q <=
3.41 / ( 2 =
-3.40 - 0.01 ) X8, ..
; X8
MAX {2/-0.1; 70/-1; 32/-0.1} <= X8 <= MIN {3/0.1; 30/1}
.
X8 30 ./,
( X2 ).
,
, .

,
( ( X1 ) )

X1: = - 3.25
1 + q,
- 0.44 <= q <= 0.01
3.24 / ( 1
= -3.25 + 0.01 ), X8 ( ..
) X8 = 30, X1 = 100,
X2 = 0.
50-70 ./ 0.44 /.

, ,
, , i
( X1 = 70 )
,
, , .
0.01 /,
;
0.44 /,
, 20 ./,
.


---------------------------------------
1. ( )
:
X9 = 2 X10 = 3 , 1,
43 ( 2
3 ).
X11 , 1
4.

2. ( )

,
.

.

, .

X12:
100 + q,
MAX {3/-0.15; 70/-1; 32/-0.85} <= q <= MIN{2/0.15}
.. -20 <= q <= 13.33

X8: 30 + q,
MAX {3/-0.1; 30/-1} <= q <= MIN {2/0.1; 70/1; 32/0.1}
.. -30 <= q <= 20

X13: 50 + q,
MAX {3/-0.5; 50/-1} <= q <= MIN {2/0.5}
.. -6 <= q <= 4

, ,
,
.



===============================

:
50 ./ 2.80 /,
75 ./
2.50 /.
.
,
:
-------------------------------------------------------------------

($/) ./
-------------------------------------------------------------------
0.20 100
0.15 25
0.10 40
0.15 40
--------------------------------------------------------------------

:
--------------------------------------------------------------------
(/) ()
--------------------------------------------------------------------
1 6.00
2 5.00 60 ./
3 4.00
--------------------------------------------------------------------

( / )
--------------------------- ---------------------------------

--------------------------- ---------------------------------
proc1 proc2 gr1
proc1 0.15 0.12 tret1 0.15 0.20 0.18
proc2 0.10 0.10 tret2 0.35 0.38 0.40
proc3 0.20 0.15 tret3 0.50 0.42 0.42
proc4 0.23 0.25 ---------------------------------
proc5 0.32 0.33
---------------------------

--------------------------- -------------------------------

--------------------------- -------------------------------
proc4 proc5 proc3 gr2
gr1 0.15 0.10 ref1 0.45 0.30
gr2 0.20 0.20 ref2 0.55 0.70
gr3 0.25 0.35 ------------------------------
gr4 0.40 0.35
---------------------------

, - .

:

1 3 .
1 tret1 ref1.
3 tret3, ref2 gr4.
2: % => 55
2:
tret2 tret3 ref1 ref2 gr3 gr4
% 65 60 53 50 45 40

!
----------------------------------------------



x1 + x21 = 50 ....ruda1
x2 + x22 = 75 ....ruda2
x1 + x2 + x23 = 100 ....blok osn

0.15x1 + 0.12x2 = x3 ....pr1
0.10x1 + 0.10x2 = x4 ....pr2
0.20x1 + 0.15x2 = x5 ....pr3
0.23x1 + 0.25x2 = x6 ....pr4
0.32x1 + 0.33x2 = x7 ....pr5

0.15x6 + 0.10x7 = x9 ....gr1
0.20x6 + 0.20x7 = x10 ....gr2
0.25x6 + 0.35x7 = x11 ....gr3
0.40x6 + 0.35x7 = x12 ....gr4

0.15x3 + 0.20x4 + 0.18x9 = x13 ....tr1
0.35x3 + 0.38x4 + 0.40x9 = x14 ....tr2
0.50x3 + 0.42x4 + 0.42x9 = x15 ....tr3

0.45x5 + 0.30x10 = x16 ....ref1
0.55x5 + 0.70x10 = x17 ....ref2

0.5x13 + 0.5x16 = x18 .... Q1
0.3x15 + 0.3x17 + 0.4x12 = x19 .... Q3
65x14 + 60x15 + 53x16 + 50x17 + 45x11 + 40x12 = 55x20 .... Q2





-
,
,
.
IBM - IBM PC.
IBM PC ,
, ,
.
15*40*50 13 .
,
: ,
( ) ,
.
C IBM PC :
--------------
1. ,
:

) ,
,
;
) - .
.
IBM PC/AT Intel-80286
Intel-80287
.

2. , ,
. :
) () -
,
.
640 (- ).
. ,
,
.
) () -
.

. DOS,
,
,
-.

3. - ,
(, ...).

4. -
. ,

, ,
(, ...).
:
( LPT1,LPT2...)
( COM1,COM2...).
,
.

5. - ,
.
5.25 (133)
360 1.2 .
1.2
, IBM PC/AT.
360 .
3.5
(89) 0.7 1.4 .

-
.
-
, ..
.

. 5.25

. ,
.
.
.

6. - ,
,
: , ,
, ...

.
. IBM PC/AT
40 .

IBM PC - 6*20*51 ., ,
- --------- .
102
,(
), 128 ASCII
( ),
.
IBM PC,
, -
.
,
- (
, Home, End, Page Up, Page Down).
.
12 .

. .

: Enter, Control,
Altenate, Tab, Insert, Delete .
, .
CTRL, ALT DEL
( " DOS").
.
IBM PC
,
.

-
.
() .
: .

- , 25 80 .
.
,
...
.
.

. 640*350 ,
640 350
. IBM PC
EGA VGA.
, VGA
,
, .

.
,
. IBM PC . IBM

EPSON. :

(). ,
.

10 60 .
: , ,
...
.

IBM PC :
- .
:
,
.
.
,
.

.

-
. ,

.

-
.
.

: , ,
, ...



- ,
. ,
, .

. () .
,
-
, ,
.


, ,
...

.
IBM PC
MS DOS Microsoft Corp. MS DOS

,
.
MS DOS :
1) -
.

- ,
.
- .
2) - ,
.
.
3) IO.SYS MSDOS.SYS.
. IO.SYS
- .
MSDOS.SYS MSDOS.
4) DOS -
. COMMAND.COM
. ,
DIR COPY
. ()

.

DOS.
5) DOS -
.
, ,
...
6) - ,
- DOS
.
CONFIG.SYS.

DOS.

DOS
:
) .
) "Reset"
) CTRL, ALT, DEL

.

- .
.
.
.
- ,
- IO.SYS MSDOS.SYS
.
CONFIG.SYS
,
.
COMMAND.COM
. COMMAND.COM
AUTOEXEC.BAT
, .
, ,
- NORTON COMMANDER.
AUTOEXEC.BAT
DOS ,
: C:\>


.
(Critical Path Method) CPM
CPM
. -
, ,
, .
.,
, .
(.1)


------------> ---------------->

.1.
-.
- ,

. -
. - ,
,
.
. ,
. ,
,
.
,
. (,)
,
. .
: ) -,
p ,
...) )--,
( , ..)
) - ,

. .
- .
,
,
. ;
. ,
.
.

.
()-
. '' ,
, ,
,,, ,
. i-j ,
i j. ,
.
.
(1,2,3...) 10,20,30,40... .
.
, 11-12,14-18 ...
, ,
:
) , ?
) ?
) ?
,
.
, - . .
, .

, ,
.
.
,
. PM
:
.
a) ,
, .
,
, .

,
.

.
,
.
)
, .
.

.

, .,
, -
.
.
, -
( )
, .,
,
.
.
CPM
, .

.
.
.
+ .
.
.
,

6 :
1. 1 ( -
i, i j).
2. 2, 3
3. 4.
, i .
4.
5.
5. 7.
, j .
6.
6.
------------------------------------------------------------------------------
1 | 2 | 3 | 4 | 5 | 6 | 7
------------------------------------------------------------------------------
| |- |
i-j | - | | -
------------------------------------------------------------------------------
: .

.
, .
, ,
, .
.
,
, :
= -
= -
.
:.
.
,
.
, , ,
.
- - .
.
FF , j
Ei D
:
FF=Ej-Ei-D
,
.
IF
D, :
IF=Ej-Li-D
He ,
, . (
)
. ,
.,
,
.
.
,
.
.
,
.
, .
.
.
:
1) i :
Ei=Lj
2) j :
Ei=Lj
3)
j i:

Lj-Ei-D=0
T , .

.
. ,
.
, .
, .
,,
, .
,
.
, ,10-15%,
.

, , .








.
:
*****
.

. ,
, -
, , -
.
-
.
.
) ,
.
) ,
.
-

. , -
- -
, ,
.

, -

.
.
:
1) ,
.
2) , , -
.

.
.

:- ( ) -

, - ( ) -
, m- . -
.

, -
-
: a+4m+b
ij - Fij= 6
b-a
***= 6
-
, ,
.
,.e. *****
(F)= F(ij).
(i,j)

,
:
d()= dij(Fij)
(i,j)
-
-
. Tij
Tij .
-
( )

, -
.
Tij , -.

.
.
0.01. -
, .
,
0.01
, -
.

.
:
: a=4
: m=6
: b=7
;
. ,
,
,
, .
4 (
7) , ,0.01 . -
, 6 ., -
,
, , -
. -
b-
1. 4.0 6.0 7.0 -
,
.
, 6.0
,
6 ( ) 0.5.
-
. , , -
, ;
+4m+b
=------
6

4+4*6+7
------- = 5.8


, 6 ,
, 5.8 6.
2 - 2 .
, 5.8
0.5.
;
,
, .

, ;
=4
m=5
b=18
4+4*5+18
=-------- = 7.0
6

3.
, b-
. . .,
7.0 0.5. -
, ,
5.



1 2 ;
1 2
=4 =3
m=6 m=5
b=8 b=13

4+24+8 3+20+13
=------ =6 =------- =6
6 6


=6, , ,
.
D.
b-a 2
D( )= (-----)
6

8-4 2
D(1)=(-----) =0.444
6

13-3 2
D(2)=(------) =2.777
6
,
. ,

,
.
.

****,
.

t(ij) =0.5., Ts =
t(ij), 0.5.
B-,
. , ,
, -
; ,
.
, Ts
, Tc,
, Ts. Tc Tc
,
:
g(t)= .
:
A B C D
1-------------2-------------3---------------4----------5
a = 4 a = 3 a = 2 a = 4
m = 6 m = 8 m = 4 m = 5
b = 8 b = 9 b = 7 b = 6
******=6+7.33 + 4.17 + 5 = 22.5

g(t)=***********=1.5



.
-
Tc.
Ts
21 24 (22.5+-1.5) 0.68.
( +-g)

Z
-
Z =-----------------------------------------------------

-
, Z .
,
21.5 .
21.5 - 22.5
Z =-------------- = - 0.67.
1.5
Z 0.25.
**********************************************************************
, .
,
,
.
, = 80 .
=2 , 74
86 . 0.9987.
= 78 ,
=5, 0.9987
63 93. ,
.

?
?
?