ALGORITHM FOR THE SIMPLIFICATION OF SOLUTION TO DISCRETE OPTIMIZATION PROBLEMS

Discrete optimization, which has already been defined as a separate section of the optimization theory, in most cases operates the combinatorial methods of solution [1]. The problems of allocation of raw materials among different products, formation of the nutrition ration, materials cutting, etc. are solved using the methods of linear programming [2]. The typical tasks of the combinatorial methods are to obtain the original reference plan, optimality assessment, improvement of the plan and boundary estimation of objective function. Since most discrete optimization problems belong to the class of NP problems, the use of algorithms for a problem simplification without losing control of the solution accuracy is relevant [3]. For a simplification procedure, the known relationship of the systems of linear algebraic equations with the system of linear algebraic inequalities and the classic apparatus of linear algebra are used [4]. Direct computing simplification technique was implemented with the use of the Jordan-Gauss method [5]. ALGORITHM FOR THE SIMPLIFICATION OF SOLUTION TO DISCRETE OPTIMIZATION PROBLEMS


Introduction
Discrete optimization, which has already been defined as a separate section of the optimization theory, in most cases operates the combinatorial methods of solution [1].The problems of allocation of raw materials among different products, formation of the nutrition ration, materials cutting, etc. are solved using the methods of linear programming [2].The typical tasks of the combinatorial methods are to obtain the original reference plan, optimality assessment, improvement of the plan and boundary estimation of objective function.Since most discrete optimization problems belong to the class of NP problems, the use of algorithms for a problem simplification without losing control of the solution accuracy is relevant [3].For a simplification procedure, the known relationship of the systems of linear algebraic equations with the system of linear algebraic inequalities and the classic apparatus of linear algebra are used [4].Direct computing simplification technique was implemented with the use of the Jordan-Gauss method [5].

Literature review and problem statement
In many cases, mathematical models of project management are interpreted as problems of discrete optimization [6].Solution of problems of discrete optimization is associated with fundamental difficulties.Modern methods of exact and approximated solution of such problems are examined with respect to their belonging to the so-called problems of P and NP class (algorithms for polynomial and exponential implementation of solution) [7].
Combinatorial methods for accurate and practical solution of discrete optimization problems take one of the important places in obtaining optimal values of such tasks [8].To realize solution algorithms, it is necessary to obtain the original reference plan, optimality estimates and improvements in case it is not optimal.Modern combinatorial methods for practical solution of discrete optimization problems require development of algorithms that allow obtaining an approximated solution with the guaranteed estimation of deviations from an optimum [9].
Simplification algorithms in discrete optimization problems are an effective technique to search for a solution of an optimization problem [10].If we perform projecting of a multidimensional process on a two-dimensional plane, such technique will make it possible to clearly observe the permissible set (lattice) of the parameters of a problem [11].An assessment of values of the objective function of a problem can be performed from the bottom and from the top and dynamically assess the possibility of diversifying the basis optimal variable with guaranteed accuracy [12].In paper [13], the method of thermoeconomic optimization of power consuming systems with the linear structure on the graphs was developed.Analysis of solutions' stability in the problems of detecting duplicates in electronic documents was presented in research [14].Complexity of displaying linear relationships in projects was shown with the use of Markov chains in paper [15].
It is possible to resolve the contradictions between requirements regarding the completeness of the model representations of technical and social systems and the methods of obtaining solutions by rational simplification of algorithms for solution of complex systems of equations [16].The essence of not sufficiently resolved problem regarding the search for solutions in problems of linear programming is the need to create a method of simplification of combinatoric solution of a discrete optimization problem.
In the studies, published by different authors, it is emphasized that one of the unresolved components of the general problem of development of effective models is the stage of obtaining a solution to systems of equations of mathematical description of complex systems [8].In this case, the methods for solution simplification through introduction of specific constraints [9], transition to iterative search for a solution [11], the use of the decomposition of objects with representation of systems using graphs [13] have become widely used.These solution simplification options are based on the additional transformation of the mathematical description of systems with construction of unique algorithms for obtaining the solution to a problem [15].The lack of effective in the chosen aspect methods of simplification of combinatorial solution to discrete optimization problems is probably associated with the computational complexity and diversity of displayed systems [16].The combinatorial methods for solving such problems allow separation of an array of subsets of solutions from a set of admissible values.The algorithm for separation of such subsets is the basis for combinatorial methods, but the usual sorting out solutions can be whatever large and complicated for calculations [3].Given this, all possible simplifications of the original problem should improve the combinatorial algorithm.Any simplification algorithms decrease the number of combinatorial sorting out from a set of admissible values.This is due to a known fact of the algorithmic complexity of discrete optimization problems.

The aim and objectives of the study
The aim of present research is to develop the algorithm of simplification of the solution of multidimensional discrete optimization problems using the standard computational procedures of linear algebra and some techniques of linear optimization.
To accomplish the set goal, the following tasks were to be solved: -to separate the class of problems, which are subject to simplification; -to present calculations of the model example.

Development of the simplified solution to discrete optimization problems
Assume we have a general linear optimization problem in the form: where W I is the objective function.
It is known that this problem can always be reduced to the canonical form of recording: We will note that the forms of linear optimization problems are equivalent -they preserve a set of solutions.It is possible to achieve this by using techniques for transformation for a transition from one form of problems to another.
Thus, the equation of the system of constraints to a linear optimization problem is equivalent to the system of two inequalities: The variables that are arbitrary by sign can be represented as the difference of two non-negative variables: , The transition from constraints-inequalities to constraints-equations is performed by adding a non-negative (balance) variable: , To simplify the transformation of linear optimization problems, the transition from maximization to minimization of the objective function and vice versa is also used: Given this, without loss of generality, let us assume that we have a linear discrete optimization problem, given in the canonical form: W max, where the range of matrix of coefficients of the constraints system is equal to rang .A m = Then solving the system using the Jordan-Gauss method [5] by arbitrary basis combination of variables, we will obtain the projection of the n-dimensional original problem on (n-m)-dimensional space.In case n-m=2, we have projecting on a two-dimensional plane.
Let us consider a model example of the solution of a five-dimensional linear optimization problem, which is based on projecting multidimensional space to two-dimensional space.
To solve the linear optimization problem Solution.The constraints system (1) consists of three independent equations.Let us proceed from the canonical forms of the problem representation to the standard one.Such transition is performed by solving the system (1) by the Jordan-Gauss method (Table 1).We select the three x 3 , x 4 , x 5 as basis variables.From the last chain of Table 1, we have the resolved system .

1. Projection onto Ox 1 x 2
Rejecting the basis variables, we ensure a transition to two-dimensional inequalities.Projection of the five-dimensional original problem onto coordinate plane Ox 1 x 2 takes the form: , x ≥ Graphical solution will be shown in Fig. 1.
Other coordinates will be obtained from the resolved system (2).Thus, the optimal solution to the original problem is equal to

C =
Let us consider all possible reductions of the problem.

2. Projection onto Ox 1 x 5
We select x 2 , x 3 , x 4 as basis variables.We solve the original system of constraints relative to variables x 2 , x 3 , x 4 by the Jordan-Gauss method (Table 2), but we use system (2) that is equivalent to (1).
Neglecting non-negative basis variables x 2 , x 3 , x 4 , we carry out a transition to the constraints-inequalities.The projection of the five-dimensional polyhedral of the original problem (1) is onto coordinate plane Ox 1 x 5 , otherwise the equivalent transition from the canonical form of the LO problem to the standard one takes the form of: The graphical solution to the problem is shown in Fig. 2 where The optimal vertex has coordinates 1 11, x = 5 0. x = Coordinates of x 2 , x 3 , x 4 will be calculated from system (3).Finally, [ ] opt max 11, 5, 19, 0, 0 .

X =
The obtained optimal solution coincides with the previous ones, which proves correctness of the performed calculations.

3. Projection onto Ox 2 x 5
We select x 1 , x 3 , x 4 as basis variables.We solve the original system of constraints with respect to variables x 1 , x 3 , x 4 , using the table of the solution of the previous calculation (Table 2).
We have the resolved system with basis variables x 1 , x 3 , x 4 Neglecting the non-negative basis variables, we ensure a transition to constraints-inequalities.Projection of a five-dimensional polyhedral of the original LO problem onto coordinate plane Ox 2 x 5 takes the form: Graphical solution to the problem is shown in Fig. 3 where The optimal vertex has coordinates 2 5, x = 5 0. x = Coordinates x 1 , x 3 , x 4 will be calculated from system (4) We select x 1 , x 2 , x 3 as basis variables.We solve the original constraints system with respect to variables x 1 , x 2 , x 3 (Table 4), using the previous table of the solution (Table 3).
Projection of the five-dimensional polyhedral of the original LO problem onto coordinate plane Ox 4 x 5 takes the form: The graphical solution to the problem is shown in Fig. 4 where The optimal vertex has coordinates 4 0, x = 5 0. x = Coordinates x 1 , x 2 , x 3 will be calculated from system (5) The obtained optimal solution coincides with the previous calculations.

5. Projection onto Оx 2 x 3
We select x 1 , x 4 , x 5 as basis variables.We solve the original constraints system with respect to variables x 1 , x 4 , x 5 from Table 5, using the previous table of the solution (Table 1).We have the resolved system with basis variables x 1 , x 4 , x 5 .
From the last chain, we obtain the solved system Rejecting the basis variables, we provide a transition to two-dimensional inequalities.Projection of the five-dimensional original problem onto coordinate plane Оx 2 x 3 takes the form: The graphical solution is shown in Fig. 5 where The coordinates of the extreme vertex are the solution to the system Other coordinates will be obtained from the solved systems (6).Thus, the optimal solution to the original problem is equal to We select the combination of x 1 , x 3 , x 5 as basis variables.We solve the original constraints system with respect to variables x 1 , x 3 , x 5 from Table 6, using the previous table of solution (Table 5).

Table 6
Calculations by basis x 1 , x 3 , x 5 .We have the resolved system with basis variables x 1 , x 3 , x 5 We proceed to constraints-inequalities.This transition ensures projecting of the original five-dimensional problem (1) onto coordinate plane Оx 2 x 4 and takes the form: The graphical solution is shown in Fig. 6 Other coordinates will be obtained from the solved system (7).The optimal solution to the original problem is equal to: Let x 2 , x 4 , x 5 be the basis variables.We solve the original constraints system regarding variables x 2 , x 4 , x 5 (Table 7), but, to reduce the amount of calculations, we use the previous table of solution (Table 5).We have the resolved system with basis variables x 2 , x 4 , x 5 .
It takes the algebraic form of: We proceed to constraints-inequalities.The projection of the original five-dimensional problem (1) onto coordinate plane Оx 1 x 3 takes the form: The graphical solution is shown in Fig. 7 where Coordinates of the extreme vertex are the solution to the system Other coordinates will be obtained from the solved systems (8).Thus, the optimal solution to the original problem is equal to

8. Projection onto Оx 3 x 4
Let x 1 , x 2 , x 5 be the basis variables.We solve the original constraints system with respect to variables x 1 , x 2 , x 5 (Table 8).To reduce the amount of calculations, we use the previous table of solution (Table 7).We have the resolved system with basis variables x 1 , x 2 , x 5 .
It takes the algebraic form of: We proceed to constraints-inequalities.Projection of the five-dimensional problem (1) onto coordinate plane Оx 3 x 4 takes the form: Coordinates of the optimal vertex are the solution to the system Other coordinates will be obtained from the solved system (9).The optimal solution to the original problem is equal to Let x 1 , x 2 , x 5 be the basis variables.We solve the original system of constraints regarding variables x 1 , x 2 , x 5 (Table 9), but, to reduce the amount of calculation, we will use the previous table of solution (Table 8).We have the resolved system with basis variables x 1 , x 2 , x 5 .
It takes the algebraic form: We proceed to constraints-inequalities.The projection of the five-dimensional problem (1) onto coordinate plane Оx 1 x 4 takes the form: The graphical solution is shown in Fig. 9    Coordinates of the optimal vertex are the solution to the system Other coordinates will be obtained from the solved system (10).The optimal solution to the original problem is equal to We accept x 1 , x 2 , x 4 as basis variables.We solve the original system of constraints with respect to variables x 1 , x 2 , x 4 (Table 10), but, to reduce the amount of calculation, we use the table of solution (Table 8).
We proceed to constraints-inequalities.The projection of the original five-dimensional problem (1) onto coordinate plane Оx 3 x 5 takes the form: The graphical solution is shown in Fig. 10 where Coordinates of the optimal vertex are the solution to the system Other coordinates will be obtained from the solved system (10).The optimal solution of the original problem is equal to

Discussion of results of studying the optimization calculations
The proposed approach to simplification of the combinatorial solution to a discrete optimization problem has significant benefits unlike the known methods of determining the optimal solution, such as the simplex method or the method of artificial basis [3].In fact, the performed decomposition of the system reduces the dimensionality of the system of equations to be solved.The projection of the multi-dimensional system of an original problem onto a two-dimensional coordinate plane makes it possible to obtain a simple system  The applied value of the proposed approach is the use of obtained results to provide a possibility to improve complex systems that are described by systems of linear equations with existence of the systems of linear constraints.Multivaluedness of combinatorial projections determines the possibility to change the set of parameters of the problem.Projecting a multi-dimensional optimization process on a two-dimensional plane was proposed.
This simplification method can be applied only to the prepared classes of problems.The rank of m matrix of coefficients of the system of constraints for a linear discrete optimization must problem must satisfy the condition of n-m=2, where n is the dimensionality of the problem.Generalization of such projecting on a three-dimensional space is expedient.

Conclusions
1.It is shown that the solution to a linear optimization problem is possible by simplification with the use of decomposition of a system due to construction of projections of a multi-dimensional system of the original problem onto two-dimensional coordinate planes.
2. It was proved by the example of solving a typical model problem that the proposed approach allows obtaining a simple system of graphic solutions of a complex problem of linear discrete optimization.The obtained result makes it possible to conclude that in the general case, there is no need to search for solutions by all projections.It is enough to determine the solution only by one projection.

Fig. 1 .
Fig. 1.Projection onto Ox 1 x 2The coordinates of the extreme vertex are the solution to the system Selection of three variables out of five is possible with the use of ten methods3   5   10.

Fig. 6 .
Fig. 6.Projection onto Оx 2 x 4Coordinates of extreme vertex are the solution to the system

5 x=
of graphic solution to a complex problem of linear discrete optimization.From the practical point of view, the proposed method makes it possible to simplify the calculation complexity of optimization problems of such a class.It was shown that the optimal solution to the original problem by all 10 projections is equal to: The obtained scientific result allows us to conclude that in the general case there is no need to search for a solution by all projections.It will suffice to determine the solution only for one projection.

Table 3
Calculations by basis x 1 , x 3 , x 4

Table 4
Calculations by basis x 1 , x 2 , x 3We have the resolved system with basis variables x 1 , x 2 , x 3

Table 7
Calculations by basis x 2 , x 4 , x 5

Table 8
Calculations by basis x 1 , x 2 , x 5

Table 9
Calculations by basis x 1 , x 2 , x 5

Table 10
Calculations by basis x 1 , x 2 , x 4