519.852

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. .. ї,
, e-mail: kafedra_vm@mail.ru;
2 ї,
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INEQUALITIES METHOD IN THE LINEAR PROGRAMMING PROBLEM WITH A PARAMETER 1Kravchuk S.P., 2Kravchuk I.S., 1Tatarnikov O.V., 1Shved E.V.
]PlekhanovRussian University of Economics. Moscow, e-mail: kafedra_vm@mail.ru;
2Moscow State University of Railway Transport, Moscow
In the analysis of engineering and economic problems which can be transformed to problems of linear programming often it is necessary to carry out research of mathematical model sensitivity to variation in a certain interval of coefficients of a task. The coefficients variation can be presented in the form of their dependence on some parameter. When using a simplex method this analysis is being a labor-intensive process as demands receiving of optimal solutions for each set of variable coefficients. In this paper the method of the solution of a problem of linear programming with the variable coefficients which is reduced to the problem of system of the same sign inequalities which is in turn reduced to one inequality is proposed. This method allows receiving the optimal solution in the form of function of the parameter defining variation of task coefficients.
Keywords: inequality, linear programming, the simplex method, Gauss-Jordan method, the objective function,
extremum, matrix
[2, 3] , . , .. . , ( ) . .
[2] :
Z = x + 3x2 ^ min;
\ + % < 4;
2xj + 2>2\ (1)
Xj > 0,2 > 0.
[4, 5]:
. 1, Zmin = 1 x1 = 1,
2 = 0.
:
= (1 + 1)1 + (3 + 2)2 ^ ;
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; > 0,2 > 0.
-
(1). (2)
(+^^-^+^);^ <2\ (+11)+(+12):2 -4+^;
-(2+-(1+22)2 < -(2+2); (3) -* ^ 0; -2 < 0.
:
1
V
1 + , 3 + 2
1 + 11 1 + 12
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-1 0
0 -1
(3 + 2)
(1+^22) (1 + 12)
(3 + 2)
(1+%)
V
(1+ )(1+ 22) - (3+ 2)(2 + 21) 1+ ,
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1 + -1
2(1+22)-(2 +62)(3 + 2)

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, t = 1/2. , t - 1/2. , , , 10 % (2).
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2=0,4[-2),

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2
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{11} -5 {12} 1 -1 1 -1
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z
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D
{1} 0 0 0 0 0 0
6Z-3(2 + 62)
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z
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Z >b2 -bx -2;
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4 >~W,
Z> 0.

Z . e[0,9; 1,1]; Z = Z . (1/2) = 1;
min L 7 7 jy Cp minv y 3
Xj e[0,9; 1,1]; *lcp=*l(X) = l-
V
-5 + Cj - 2C2 1 + Q -1 1 -1

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:
3
(1+ Q)
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(5-1+22)
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z
Z-(2 + b2)(3 + C2) + (4+b1)(5-Cl+2C2) 4 + bx
: _(2+62)(3+2)(1 + 1)_
z=-
(6+2C2)
'=(i+0,542)(i+c1)=z-, ;
x1 = 1 + 0,5b2; x2 = 0.
X

1=0,2(-)/);
2=0,(-^),
1 2 (4),
) = (0,2 + 0,9)2;
1 = 0,2 + 0,9; 2 = 0.
(6)
(7)

7 . [0,81; 1,21]; 7 = 7 . (1/2) = 1;
- 7 7 7 7 4 3
1 [0,9; 1,1];
^1 =^1 (^2)=1*

(4), (6)
=0,2(-^);
012=-∞,2 21=-∞4^-
22 2 ^ 2
(8)
, |11| < 0,1; |12| < 0,1; |21| < 0,2; |22\ < 0,1.
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2 (2+^+,) (0.2+ 0$ ; (9)
2+.
21
1,1-0,2
, 2 0.
1+] 2 +21 1,10,2
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^ ^ ^ -1 '- ' 5
1 [0,82; 1,22]; ^(1/2) = 1.
(5), (6), (9) , (), , 2().
2().
. 2 ,
VI
= 1 .
- [1]. . . [1].
Z = -,
22 - 33 - 4 ^ MAX;
2 +2 -3 +4 < 1-2;
Xj +3 2 + 43 +24 < 20+ ; -41 +52 -23 + 4 < 5+;
x1jc2,x3,x4 > 0.
[1],
Z =
2, Lie 14
2~ii, 77-31
13

*7
17.64 6 5
(10)
85 238
7 33
ї:
33
(11)
, 23-
389
21
64
^55+7.
Z< 2-;
77-31
;
(12)
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7() , 7. . 7() . 3.
5 (12):
Zmax(^)=
7 . (10) . , . 1, -
55+7
16 2-,
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13

<

389
21

17.64 6 5
;-1
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(10) , -.

1. ., .. . - .: , 1966.
2. .., .., .. . -.: , 2010. - 6 (154).
3. .., .., .., .. . - .: , 2014. - 3, 1.
4. : . . .. . - .: -, 2000.
5. .., .. . - .: , 1963.
References
1. Holstein E.G., Yudin D.B. New directions in linear programming. Moscow: Soviet Radio, 1966.
2. Kravchuk S.P., Kravchuk I.S., Swed E.V. Extremum in the system of linear inequalities with two variables. St. Petersburg.: Modern aspects of the economy, no. 6 (154), 2010.
3. Kravchuk S. P., Kravchuk I. S., Tatarnikov O.V., Swed E.V Inequalities method in the linear programming problem. M.: Fundamental research, no. 3, pp. 1, 2014.
4. General course of higher mathematics for economists. Textbook ed. prof. VI. Ermakov. Moscow: INFRA-M, 2000.
5. Yudin D.B., Goldstein E.G. Linear programming. Moscow: Fizmatgiz, 1963.
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23.09.2014.