Symetric Structural Transformation of Direct Mathematical 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.

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, p. 123].

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.;

- 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 is defined dual a simplex-method. The algorithm dual a simplex-method includes stages:

1. Drawing up of the pseudo-plan.

2. Check of the plan for an optimality.

3. A choice of leaders of a line and a column.

4. Calculation of the new basic plan [1, p. 135].

The optimum decision of a dual problem is presented in table 1.

Table 1

The optimum decision of a dual problem

Table 2

Factors of criterion function of dual model

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.

Table 3

Values of the right parts of system of restrictions of dual model

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.

References:

1. G. P. Formin. Mathematical Methods And Models In Commercial Activities.: Textbook – M.: Finansy I Statistika, 2001. – 544 p.