Volumetric modelling of economic index structures

Preface. One of the most important principles of organizing economic index system is minimization of primary data and maximization of outcome data: as much as possible necessary secondary (derivative) indexes must be obtained from the least quantity of primary indexes, which must be as less duplicated as possible. While implementing this principle, it is very important to choose rational structure of economic index files. Rational structure also allows to process data faster, to manage economy objects and processes more operatively by making timely decisions. Tools of volumetric modelling can be used for rationalization of structures, because analysis of economic indexes shows major usage of indexes that have three requisite-attributes and correspond to volumetric 3-coordinate system in economy and business area. The objective of this article is to show mechanism of volumetric modelling and to show possibilities of model adaptation and usage in practice by choosing and evaluating the most rational economic index structure. 1. Development of model. The framework of volumetric model is shown in pic. As we can see, all file of particular economic index can be imagined as hexahedron, filled in with such values of this index: { } ijk k j i A , R , R , R 3 2 1 ;


Volumetric modelling of economic index structures
Preface.One of the most important principles of organizing economic index system is minimization of primary data and maximization of outcome data: as much as possible necessary secondary (derivative) indexes must be obtained from the least quantity of primary indexes, which must be as less duplicated as possible.While implementing this principle, it is very important to choose rational structure of economic index files.Rational structure also allows to process data faster, to manage economy objects and processes more operatively by making timely decisions.Tools of volumetric modelling can be used for rationalization of structures, because analysis of economic indexes shows major usage of indexes that have three requisite-attributes and correspond to volumetric 3-coordinate system in economy and business area.The objective of this article is to show mechanism of volumetric modelling and to show possibilities of model adaptation and usage in practice by choosing and evaluating the most rational economic index structure.
1. Development of model.The framework of volumetric model is shown in pic.As we can see, all file of particular economic index can be imagined as hexahedron, filled in with such values of this index: where indexes R and basis of particular index A (some part of elements in the space of hexahedron can be blank).
If number of symbols of given indexes is known, then all of its file D size Q can be expressed as sum of its separate parts ( ) if it is assumed, that ( ) According to formula (1) calculated size Q is random quantity as all ijk A are. ijk A values are not equal to zero with probability ijk p and are equal to zero with probability presence in file D and ijk q is probability of its absence.Values of probability ijk p can be considered as given, because we can determine them using analytical method or, for example, with the help of statistical analysis.Also we consider that quantities do not differ much by their sequence, and it means that there is quite strong probability that there will be a lot of non-zero ijk A in file D.
Then we can calculate mathematical hope Q′ and dispersion Q ∆ of index file part size ( ) t D Q according to these formulas (Âåíåöêèé, 1974):

;
(3) We can apply such conditions to mathematical hope and dispersion of whole index file size (independently from the way of organizing structure of file): Now there is a question, how exactly mathematical hope of economic index file size matches its true size.We can express relational size deviation ε from its mathematical hope in such way (Misevièius, 2001): If average square deviation is , then ratio between Q and Q′ can be expressed like: Then we can determine deviation ε without major relational bias according to this formula: Therefore, formula (3), when Q′ is changed to Q , can be named as model of economic index file, and formula (9) -relational bias of this model.
2. Adaptation of model.Economic indexes can be decomposed into separate parts of hexahedron according to one or two semantic attributes in these ten ways: 1) "zero" decomposition, when all three attributes are written next to basis of each index (it means that file is not decomposed according to attributes); 2) the file is decomposed into parts according to the first attribute 1 R , value of which is general to all indexes of one part of file, and is written in the heading of this part only once; 3) the file is decomposed similarly according to the second attribute 2 R ; 4) the same decomposition according to the third attribute 3 R ; 5) indexes are grouped by two attributes at the same time: 1 R (senior attribute) ir 2 R (junior attribute); general value of elder attribute is written in the heading of file part only once, and junior attribute is written in subheading only once too; 6) the file is decomposed similarly according to attributes 1 R and 3 R ; 7) the same decomposition according to attributes 2 R and 3 R ; 8) the same decomposition according to attributes 2 R and 1 R ; 9) the same decomposition according to attributes 3 R and 1 R ; 10) the same decomposition according to attributes 3 R and 2 R .Now we can educe such calculations for all possible ways of file decompositions from general formulas (2) and (3): where After calculations according to these formulas, it is possible to determine s Q with the least value of file size, i.e. the most rational way of organizing index file.
Similarly we educe calculations of dispersion (Kubilius, 1996) from general formula (4): where relational bias of particular model of economic index file according to formula (9): Now we calculate relational economy, which is obtained choosing the most rational structure: Q can be considered as mathematical hope of index file size, when the way of organizing it is chosen randomly.
3. Model in practice and conclusions.It is always possible to determine level of filling hexahedron with indexes by analytical method, and all ijk p values, as a rule, are the same.Therefore, while changing ijk p to p, and ijk q to q , we can express formulas (10) like that: ( ) We change formulas (11) similarly: We will illustrate specific example of selection of one rational economic index file structure.We have to find the best variant of investment distribution file according to projects, managers and periods.Its index structure is showed in table 1.

Suppose that ( )
. Suppose that attribute i R1 has 30 values (there are 30 projects running; 30 = l ), there are 50 managers ( 50 = m ), and there are 36 periods, for example, number of 3 year period months ).Although practically there are only 12-20 managers in one project and finances are assigned to 12-24 periods.We will use average values in calculations, i.e. respectively 16 and 18.We determine from attributes that probability of filling hexahedron by sums of finances is equal 0,16.All data required for calculations is showed in table 2 Consequently the least value is 5 Q , and it means that the most rational method of file organization is the fifth one, when values of index basis are grouped by two attributes: project (elder) and manager (junior) and the period is written by every value of assigned finances sum.
When we define data required for further calculations according to formulas ( 15 .As we can see, volumetric modelling of index file produces fair benefits.
It is necessary to point out that all calculations that are related with development and evaluation of economic index file model can be done automatically by using specific computer programme.

Table 1 .
Structure of index