Calculation of the Generalizing Indicator of Productivity of the Enterprises Activity Based on the Matrix-Rank Approach

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The level of achievement of the objectives of the enterprise activity, which corresponds to the calculated indicator, has been proposed for better understanding by the authors. This level was presented with the help of quantitative and quality indicators, which characterize a compliance of some results of economic, ecological, social, legal components (number of these components, KS = 4) productivity of activity of the enterprise in relation to its purposes. In addition, this level can be saved up potential for further conducting and covering socially responsible business processes as the productivity of the industrial enterprise activity.

In this regard, the indicator of productivity of the industrial enterprise activity need to estimate for a satisfaction of multilateral interest in this problem. The corresponding technique (Figure 1) was proposed by the authors for solution of this question of an assessment of the productivity indicator of the industrial enterprise activity.

As a basis of this technique of an assessment of the productivity indicator of the industrial enterprise’s activity there were used the following two approaches: the balanced scorecard for an assessment of the enterprise activity ^{[2, 13]} and the matrix-rank approach ^{[14, 15, 16, 17]}.

The indicator of productivity of the enterprise activity is defined based on the general indicators of the following four components: economic, ecological, social and legal. It allows considering most fully all fields of the industrial enterprise activity and its purposes ^[2].

The proposed technique of an assessment of productivity indicator of the industrial enterprise activity consists of the following steps.

Step 1. Calculation of the basic indicators of the enterprise activity, which are a basis for the next calculation of some intermediate indicators of the following components of productivity of the enterprise activity – economic, ecological, social and legal. Number of these indicators based on the purposes groups of the industrial enterprise should be selected for their further calculation.

Step 2. Number of the estimated indicators according to the chosen purposes groups of the industrial enterprises (Table 1) should be selected for further calculation of the intermediate indicators of the already chosen economic, ecological, social and legal components. The method of an expert assessment, which will allow allocating the five most important estimated indicators for each component of the productivity of the enterprise activity (Table 2), was applied to calculation of the most powerful chosen earlier indicators.

Table 1. Enterprise purposes for the chosen components of the productivity of the enterprise activity

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Table 2. List of the estimated indicators for a calculation of the generalizing indicator of the productivity of the enterprise activity

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Step 3. Calculation of the estimated indicators of productivity of the enterprise activity, which are a basis for the further calculation of the intermediate indicators of economic, ecological, social and legal components.

Step 4. Calculation of the intermediate indicators of economic, ecological, social and legal components of the productivity of the enterprise activity.

The matrix rank approach was used for a calculation of the intermediate indicators of some chosen earlier components of the productivity of an activity of the following enterprises: Enterprise 1; Enterprise 2; Enterprise 3; Enterprise 4; Enterprise 5; Enterprise 6 for 2009-2013. These enterprises were been depersonalized that those calculations will not influence to the real enterprises.

Step 4.1. We have carried out the rationing/correlation of the obtained values of the estimated indicators on some components of productivity of the enterprise activity to make them into uniform measurement system (for obtaining a possibility of the comparative analysis) as the dimensionless coefficients (taking into account the enterprise purposes).

Step 4.2-4.3. Formation of a matrix of the ranks for the chosen components of a productivity of the enterprise activity.

Considering the estimated indicators for a definition of the intermediate indicators for each component, the productivity of the enterprise activity was chosen proceeding from groups of the enterprise purposes. These purposes are equivalent within each component. Thus, the matrix of ranks can be proposed for calculation of the intermediate indicators of the introduced components.

The previous step 4.1 has allowed proposing the unified general range of values for each evaluated indicator in a range from 0 to 1. Thus, the matrix of ranks can be built with a certain differentiation step depending on a quantity of the purposes of the enterprise activity, beginning from two purposes (in the simplest case). The matrix of ranks has been offered to present as the two-dimensional matrix with certain quantity of the lines and columns. In that case, to each indicator can be appropriated respectively two possible values (S = 2) – 0,5 and 1. Then a dimension (N) of the matrix of ranks (Sample), if a quantity of the purposes will be divided (in our case, the quantity of those purposes is K_p = 5) will equal to quantity of lines. For example, S_KPnumber =2, and the quantity of columns is equal S_KPcol=3, then N = S_KPnumber × S_KPcol= S(_{KPnumber+ KPcol)} = 2₂× 2₃ = 32. However, in this case, the accuracy of the calculation of the generalizing indicator of a productivity of the enterprise activity will be rather low, considering wide range of values of estimated indicators.

The matrix of ranks Sample (Table 3) with the determined earlier parameters (S = 2, S_KPnumber =2, S_KPcol=3) has been presented for bigger convenience of an explanation of the matrix-rank approach using. However, the matrix of ranks with the differentiation step 0,1 (S = 10) and the obtained dimension N = 105 = 10 000 has been used for a calculation of the generalizing indicator of productivity of the real enterprises activity.

At first, the filling direction (growth or fall) of the ranks (i.e. their values) in this matrix is necessary to define for a formation of ranks in this matrix. This step is possible after titles formation (by vertical at S_KPcol=3 (in our case for an example), and by horizontal at S_KPnumber=2). Then, it can be obtained in that case, when this value (minimum) corresponds by the left lower cell of a matrix having taken the minimum value (in our case it is 0,5) for all estimated indicators. We will be able to obtain the following case, when the maximum value (in our case it is 1) corresponds to the right top cell of a matrix by analogy. Thus, the direction of setting of ranks in a matrix can be defined. The matrix of ranks, which could be obtained in this case, is called the rectangular Toeplitz matrix or a diagonal-constant matrix ^{[3, 11]}. In this matrix, the identical elements are located on all diagonals, which are parallel to the main diagonal (this is shown by gray color in Table 3).

Table 3. Matrix of ranks for calculation of the intermediate indicators of some components for a productivity of the enterprise activity (for S = 2, N = 32)

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Thus, a formation of ranks in this matrix can be initiated. Let us establish the minimum rank is 1 (but for a convenience the minimum rank can be equal to any initial value). Then the step can be equal five (a difference of the adjacent ranks), the ranks can be calculated based on this step (according to Table 1) and proceeding from a quantity of the estimated indicators by each component of a productivity of the enterprise activity. The determination order of the ranks, which are placed in the matrix’s cells and which can be determined by the importance coefficient of the estimated indicators according to the enterprise purposes (according to Table 1). However, because in our case, the importance coefficients of the purposes Kp = 5 by each component (their quantity equals 4, Table 1) are equivalent (i.e. they do not ranked). Therefore, the weights of the estimated indicators also are equivalent, i.e. the ranks in the matrix cells are distributed evenly by the direction of filling for the ranks (it will be a growth or fall of their values) in a matrix of ranks. In our case (Table 3), it will be from the left bottom corner (cell) to the right top corner (cell), i.e. by the directions of the main diagonal of a matrix (Table 3).

All values of the matrix of ranks should be normalized for bigger convenience of using of the intermediate indicators (from a point of view of an assessment of some economic indicators), as it is shown in Table 4.

Table 4. The normalized matrix of ranks calculation of the intermediate indicators of components of the productivity of the enterprise activity (it is also known as Toeplitz's matrix, according to property of a matrix; S = 2)

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For example, let us a value of estimated indicators will be as the vector EI = { 0,5; 0,5; 1; 1; 0,5 }. Then the intermediate indicator of the component will equal 0,3137.

Step 4.4. Therefore, the matrix of ranks can be built by the principle described above with the differentiation step 0,1 (S = 10) for increasing of an determination accuracy of the intermediate indicators of some components of the productivity of the enterprise activity. The following intermediate indicators such as economic, ecological, social and legal components can be determined by the estimated indicators and with the help of the normalized matrix of ranks for the following enterprises: Enterprise 1, Enterprise 2, Enterprise 3, Enterprise 4, Enterprise 5 and Enterprise 6 for the time period 2009-2013 (Table 5). In this case, the real enterprises were been depersonalized that some damage has not been inflicted to these enterprises, for example, their image or reputation.

Table 5. Economic, ecological, social, legal intermediate indicators of the components of the productivity activity for the considered above enterprises

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Therefore, the following conclusion can be made according to the Table 5, Enterprise 1 respectively became the best enterprise behind the level of the following economic (corresponding indicator is equal 0,5474) and legal (corresponding indicator is equal 0,2861) components of the productivity of the enterprise activity in 2013. Enterprise 3 has the best intermediate indicator of the following ecological (corresponding indicator is equal 0,1022) and social components. Enterprise 2 has the worst results on intermediate indicators of the ecological (corresponding indicator is equal 0,0202) and legal (corresponding indicator is equal 0,0011) components. The following Enterprise 3, Enterprise 5, Enterprise 6 have the lowest indicator of the economic component (corresponding indicator is equal 0,0002).

Step 5. The orthogonal reflection of some variants of the intermediate indicator’s combinations of the economic, ecological, social, legal components of the productivity of the considered above enterprises activity can be executed in the plane in the set Rectangular System of Coordinates (RSC) for a calculation of the generalized indicator of the productivity of the enterprise activity. That is the geometrical interpretation of the obtained earlier results can be executed as some segments in the plane, where the coordinates of vertices can be determined by all pairs of the possible combinations of the productivity components (because RSC has been used). Because the quantity of the productivity components is equal K^C = 4, then a quantity of all possible pair combinations taking into account transpositions will equal 24 (that is 12^thsegments). Then, a distance between the determined earlier points (paired combinations of the productivity components) in RSC can be calculated by the following formula ^[6]:

(1)

where D is the distance between the determined earlier points of all variants of the combinations for the intermediate indicators of the productivity components;

х₁, х₂, у₁, у₂ are the coordinates of some points (in our case, there are a value of the intermediate indicators of the productivity components).

For example, a calculation of those indicators for Enterprise 1 in 2013 is presented in Table 6.

Rule of a calculation of a median (because the geometrical interpretation has been used) can be used or applied to the obtained earlier set of segments (in our case, it is 12th segments) based on their length that will allow obtaining the generalized indicator of the productivity of the enterprise activity.

Table 6. Calculation of the parameters of all variants of combinations for the intermediate indicators of some components for Enterprise 1 in 2013 (an example)

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3. Results of the Use of the Proposed Method

Graphical mapping of the parameters of all options of the combinations of the intermediate indicators of some components for Enterprise 1 in 2013 is presented in Figure 2.

Figure 2. Graphical mapping of the parameters of all options of the combinations of the intermediate indicators of some components for Enterprise 1 in 2013

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Median in the statistic is a set of values, which can be calculated as a mean of the ordered variation row. Thus, the median divides of this row on two equal parts: one part has a value variation attribute less, than an average value, and the second part has bigger value, than an average value. The median indicates to a value of a variation attribute, where a half of some units of a population has reached of this value of a variation attribute ^[5].

The median (unlike exponential average values) is the concrete characteristic of the variation row and it has certain values therefore it is still called as the descriptive characteristic. Median’s such property is connected with that fact the individual deviations are repaid at the descriptive characteristic, as it can be for the averages values. Descriptive characteristics always match a certain variant ^[5]. This fact can be used for a median choice at definition of the generalizing indicator of the productivity of the enterprise activity. Dynamic changes of which is shown in Figure 3. The forecast for 2014 is shown in Fig. 4. The equation of some trend functions and value of a reliability of the approximation is shown in Table 7. The generalizing indicators of the productivity of the enterprise activity are shown in Table 8.

Table 7. Equation of the trend functions and value of a reliability of the approximation of the enterprises

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Table 8. Generalizing indicators of the productivity of the enterprise activity

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Figure 3. Dynamics of some changes (by years) for the generalizing indicator of the productivity of the enterprise activity

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Figure 4. Forecast of the generalizing indicator of the productivity of the enterprise activity for 2014

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