Ñîâðåìåííûå èíôîðìàöèîííûå
òåõíîëîãèè/1. Êîìïüþòåðíàÿ èíæåíåðèÿ
cand. tech. sci. Semakhin A.M.
Kurgan State University,
Russia
DUAL MODEL
OF INFORMATION SYSTEM
The theory of a
duality of problems of linear programming has great value in the theoretical plan
and represents the big practical interest. On the basis of the theory of a
duality the algorithm of the decision of problems of linear programming - a
dual simplex a method and effective methods of the analysis of models is
developed /1/.
Let's develop mathematical
model of linear programming and we shall carry out symmetric structural
transformation of conditions of a direct problem to a dual problem. We shall
define the optimum decision and we shall lead the analysis of dual model.
The direct
mathematical model is formulated as follows: from among the firms, rendering
services satellite Internet in territory of the Russian Federation, preliminary
selected by the average expert estimations, it is required to choose the
provider satellite Internet with the maximal size of the net present value
(NPV) and satisfying to financial restrictions.
The mathematical
model of a choice of the optimum investment project satellite Internet in a
general view is represented as follows:
under restrictions (1)
where 
is a target parameter, unit of
measurement;
is investment
expenses of i project in j period of time, million. roubles;
is available means of
financing in j period of time, million. roubles;
is a share of financing of the
investment project;
is a number of the
investment project;
is a number of the
period of time, year.
After calculation of
parameters the mathematical model of a choice of the optimum investment project
satellite Internet looks like:
under restrictions (2)
The dual model is
developed by rules.
1. To each
restriction of a direct problem there corresponds a variable of a dual problem.
2. Each variable
direct problem there corresponds restriction of a dual problem.
3. The matrix of
factors of system of restrictions of a dual problem turns out from a matrix of
factors of system of restrictions of a direct problem transposing.
4. The system of
restrictions of a dual problem enters the name in the form of inequalities of
opposite sense to inequalities of system of restrictions of a direct problem.
5. Free members of
system of restrictions of a dual problem are factors of function of the purpose
of a direct problem.
6. The dual problem
is solved on a minimum if criterion function of a direct problem is set on a
maximum and on the contrary.
7. As factors of
criterion function of a dual problem free members of system of restrictions of
a direct problem serve.
8. If a variable of
direct problem 
, that i condition of system of restrictions of a dual problem is an
inequality, if 
- any number i the condition of
a dual problem represents the equation.
9. If j the parity
of a direct problem is an inequality, a corresponding estimation j a resource -
variable 
, if j the parity represents the equation a variable of dual problem 
- any number /1/.
Let's lead
symmetric structural transformation of a direct problem to return according to
rules. In a general view the dual model will enter the name as follows.
under restrictions (3)
Where 
-pure current cost of monetary
streams, million roubles.;
-Investment expenses
of i-th project in j-th period of time, million roubles.;
- available means of
financing in j-th period of time, million roubles.;
A-estimation of money resources
of financing in j the period of time;
-Number of the
investment project;
- number of the
period of time, year.
The dual
mathematical model with numerical parameters of factors looks like.
under restrictions (4)
The optimum decision of a dual problem is
presented in table 1.
Table 1
The optimum decision of a dual problem
|
Variable |
Size of variables |
Dual estimation |
Extremum of criterion function |
|
|
0,1768 |
6,0000 |
2,00526 |
|
|
0,2853 |
0,0000 |
|
|
|
0,0000 |
2,3881 |
|
|
|
0,0000 |
1,5000 |
|
|
|
0,0000 |
1,0376 |
|
|
|
0,2526 |
0,0000 |
|
|
|
0,0000 |
0,3060 |
|
|
|
1,8225 |
0,0000 |
|
|
|
1,4259 |
0,0000 |
The bottom and top borders of intervals of
stability of the optimum decision to change of factors of criterion function
are presented in table 2.
The bottom and top borders of intervals of stability of the optimum
decision are resulted in change of the right parts of restrictions in table 3.
The bottom and top borders of intervals of
stability of the optimum decision are resulted in change of the right parts of
restrictions.
Results of the lead researches have allowed
to draw following conclusions.
1. The optimum decision of a direct problem
of a choice of the project satellite Internet defines the list of financed
projects and shares of financing.
2. The decision of a dual problem defines
optimum system of estimations of the resources used for realization of
projects.
Table 2
Factors of criterion function of dual model
|
Number of the subitem |
Variable |
The minimal value
of factor |
Reference value of factor |
The maximal value factor |
|
1 |
|
2,9305 |
6,5000 |
8,0740 |
|
2 |
|
2,4152 |
3,0000 |
5,2824 |
|
3 |
|
0,6119 |
3,0000 |
+ |
|
4 |
|
0,0000 |
1,5000 |
+ |
Table 3
Values of the right parts of system of restrictions of dual model
|
Number of the subitem |
Free member of
the right part of restrictions |
The minimal value
of the right part of restrictions |
Reference value
of the right part of restrictions |
The maximal value
of the right part of restrictions |
|
1 |
|
1,0265 |
1,5273 |
2,5321 |
|
2 |
|
- |
0,7412 |
0,9938 |
|
3 |
|
0,8290 |
1,3744 |
2,2840 |
|
4 |
|
- |
0,1451 |
1,9676 |
|
5 |
|
- |
0,5303 |
1,9562 |
References:
1. G. P. Formin.
Mathematical Methods And Models In Commercial Activities.: Textbook – M.:
Finansy I Statistika, 2001. – 544 p.