.


International Journal of Economics and Financial 
Issues

ISSN: 2146-4138

available at http: www.econjournals.com

International Journal of Economics and Financial Issues, 2016, 6(4), 1956-1964.

International Journal of Economics and Financial Issues | Vol 6 • Issue 4 • 20161956

Management of the Formation of Rating Preferences of Economic 
Entities upon Collective Choice

Yury D. Motorygin1*, Vladimir S. Artamonov2, Alexander V. Maximov3, Elena N. Trofimets4, 
Valeriy Y. Trofimets5

1Saint Petersburg University of State Fire Service of EMERCOM of Russia, 149, Moskovsky Avenue, Saint Petersburg, 196105, 
Russia, 2Russian Ministry for Civil Defense, Emergencies and Elimination of Consequences of Natural Disasters, Teatralny prospect, 
3., Moscow, 109012, Russia, 3Saint Petersburg University of State Fire Service of EMERCOM of Russia, 149, Moskovsky Avenue, 
Saint Petersburg, 196105, Russia, 4Saint Petersburg University of State Fire Service of EMERCOM of Russia, 149, Moskovsky 
Avenue, Saint Petersburg, 196105, Russia, 5Saint Petersburg University of State Fire Service of EMERCOM of Russia, 149, 
Moskovsky Avenue, Saint Petersburg, 196105, Russia. *Email: malygin_com@mail.ru

ABSTRACT

The applicability of a set of methods in the case of obtaining mixed results of the final ranking of economic entities in the process of their rating 
assessment has been substantiated. A new method of the formation of rating preferences upon collective choice has been proposed. The methodology 
is based on the following: An information base in the form of a summary table of ratings of compared entities; a procedure for the formation of rating 
preferences based on a rank-sum (or arithmetic means of ranks); a procedure for the formation of rating preferences with the joint application of 
rating arithmetic means and position indicators; a procedure for the formation of rating preferences using the Kemeny median. The proposals for the 
calculation of weighting factors of the voting procedures and modification of the procedure for profiling the preferences using the Kemeny median 
have been made. An algorithm for the use of the proposed method has been developed. Methodology testing has been conducted upon the rating 
assessment of the financial and economic condition of industrial enterprises.

Keywords: System Analysis, Management of Economic Systems, Economic-mathematical Modeling, Rating Assessment, Financial and Economic 
Condition 
JEL Classifications: P40, P49

1. INTRODUCTION

The lack of information on business entities that complicates 
the process of taking managerial decisions by the business 
management regarding the selection of partners in the course 
of their activities is one of the urgent problems of the modern 
economy. The lack of information makes analysts look for new 
ways of getting the information on specific business entities.

The formation of ratings, which is a comparison of economic 
entities by a range of qualitative and quantitative characteristics 
and their ranking in order to identify the best and worst ones, is 
one of the ways to disseminate information on economic entities 
available to all interested users. Thus, currently much attention 
is paid to the calculation of various ratings of different economic 

entities, starting with the countries and ending with individual 
enterprises.

A number of major specialized rating agencies operate in the world 
economy. They provide sufficiently high quality services for the 
calculation of ratings. The services of these agencies are very 
popular because of high utility of the information provided. The 
features of the national economy often complicate the calculation 
of ratings with the use of foreign methods. This resulted in the 
emergence of a large number of original methods, the authors of 
which tried to take into account the peculiarities of the activities 
of various economic entities. Nevertheless, the problem of 
constructing adequate methods for their ranking is still relevant, 
including in terms of methodology: Data bases of methods, weight 
coefficients of local criteria, methods of obtaining the integrated 



Motorygin, et al.: Management of the Formation of Rating Preferences of Economic Entities upon Collective Choice

International Journal of Economics and Financial Issues | Vol 6 • Issue 4 • 2016 1957

rating score and a number of other elements differ. Therefore, the 
use of different methods often leads to controversial conclusions 
about the final ranking of economic entities.

The aggregation of results obtained under different procedures 
is one of the approaches to improve the adequacy of the final 
ranking. Certainly, this approach leads to a significant increase in 
calculation costs, but, taking into account the modern development 
of computer technology, this aspect is of secondary importance 
for the problem under review.

2. LITERATURE REVIEW

The problem of the formation of ratings of economic entities has 
different aspects-practical, methodical, financial and economic, 
mathematical aspects-which was reflected in numerous publications 
on this theme. An overview of approaches to the formation of rating 
systems by Russian and Western agencies can be found in the 
works (Karminskiy et al., 2005; Karminskiy and Polozov, 2016). 
The banking sphere was an initial direction of ratings. A review 
of approaches to the construction of bank rating models, including 
as a basis for the systems of early warning of defaults is presented 
in the works (Altman and Saunders, 1998; Amato and Furfine, 
2004; Trueck and Rachev, 2008; Yuksel, 2010; Rognoni, 2011; 
Karminskiy, 2015). The article (Altman and Rijken, 2004) is one 
of the fundamental works in the sphere of rating models, which 
examines an issue of the stability of such models.

Currently, corporate ratings are gaining a growing importance. 
With regard to industrial enterprises, they play an important 
role in the organization of syndicated lending and public tenders 
and serve as a basis for the formation of corporate bond ratings 
(Carling et al., 2007). A large number of works are devoted to the 
issues of the rating assessment of the financial status of enterprises 
(Kotlyar, 1999; Schiborsch, 2000; Postyushkov, 2001; Kovalev 
and Volkova, 2002; Ginzburg, 2004, etc.).

The mathematical aspects of rating procedures are reflected in 
the numerous works on econometrics, mathematical statistics, 
quantitative methods of financial analysis, expert evaluations 
(Podinovskiy, 2007; Goryunov, 2012; Saaty and Vargas, 2013; 
Eiselt and Sandblom, 2013; Doumpos and Zopounidis, 2014; 
Batkovskiy, 2015, etc.).

The mathematical aspects of the rating procedures are reflected 
in numerous works on econometrics, mathematical statistics, 
quantitative financial analysis methods, expert evaluations 
(Podinovskiy, 2007; Goryunov, 2012; Saaty and Vargas, 2013; 
Eiselt and Sandblom, 2013; Doumpos and Zopounidis, 2014; 
Batkovskiy, 2015, etc.).

The integral use of economic-mathematical methods and models is 
one of the principles of their application in the solution of complex 
management problems. For this purpose, two directions - the 
parallel use of methods (simultaneous use of different methods 
at the same stage of the problem solution) and sequential use of 
methods (the use of different methods at different stages of the 
problem solution) - can be distinguished.

The need for the parallel use of different methods (models) is due 
to the need to improve (enhance) the solution, which is obtained 
by applying a single method (model). A set of models jointly used 
to solve the same problem, is called an ensemble (committee) of 
models. In the last decade, ensembles of models have become an 
area of active research in data mining (Nisbet et al., 2009; Perner, 
2010, Alvo and Yu, 2014; Larose and Larose, 2015, etc.).

3. METHODOLOGY

3.1. Methodology Information Base
A concept of a rank of an entity underlies the proposed 
methodology of forming the rating preferences of economic 
entities under collective choice.

A rank (r) of an entity shall mean a serial number of the entity in 
the list of analyzed entities after their ordering upon descending 
ranking values. In this case, the rule of rank assignment to the 
entities shall be as follows:
1. If in a set of rating values all numbers are different, each entity 

is given a unique rank ri;
2. If in a set of rating values there is a group of k equal ranking 

values xi = xi+1 = xi+2 =… = xi+k, then the rank of the respective 
entities will be the same and equal to the arithmetic means 
of their serial numbers. The entity following this group gets 
a rank equal to ri + k;

3. The entity with the maximum rating value shall have a rank 
equal to 1;

4. The entity with the minimum rating value has the highest rank 
value equal to n - kmin + 1, where n is a number of entities in 
the list, kmin means a number of repeating minimum rating 
values in the list of entities.

Using the introduced concept of a rank of an entity, a profile of 
entities preferences (entities ranking) is developed for each of the 
aggregated methods of the rating assessment. Preference profiles 
are grouped in a summary table of entities ranks (Table 1), which 
is an information base for the methodology of the formation of 
rating preferences under collective choice.

3.2. Formation of the Rating Preferences Based upon 
the Rank Sum or Ranking Arithmetic Means
Final ranking of entities can be constructed based on the data of 
Table 1 by calculating the sum of ranks for each of the entities 

analyzed (
m

j ij
i=1

R = r ,j=1...n∑ ) and their subsequent ordering in 
ascending order Rj. This method is called a rank sum method. It 
is obvious that ranking will be similar if we divide each of the 

Table 1: A summary table of entities ranks
Methods Compared entities

Entity 1 Entity 2 …… Entity j …… Entity n
Method 1 r11 r12 …… r1j …… r1n
Method 2 r21 r22 …… r2j …… r2n
…… …… …… …… …… …… ……
Method i ri1 ri2 …… rij …… rin
…… …… …… …… …… …… ……
Method m rm1 rm2 …… rmj …… rmn



Motorygin, et al.: Management of the Formation of Rating Preferences of Economic Entities upon Collective Choice

International Journal of Economics and Financial Issues | Vol 6 • Issue 4 • 20161958

values Rj by the number of the methods used m. This method is 
called a method of ranking arithmetic means.

The main advantage of the rank sum method and the method of 
ranking arithmetic means is in their simplicity. However, it should 
be noted that they have a significant disadvantage. This disadvantage 
is caused by the fact that the ranks are measured on an ordinal scale. 
The works (Schrader, 1971; Kemeny and Snell, 1972; Orlov, 1996) 
show that only members of the variation series, in particular the 
median, can be reasonably used as an arithmetic mean of all the mean 
values in an ordinal scale. However, in practice, it is impractical to 
completely ignore the arithmetic mean value, due to the familiarity, 
clarity and, as a consequence, its high prevalence. However, when 
using the arithmetic mean there occur well-known situations known 
as “the average temperature in the hospital” or “the central part of the 
donut.” Such contradiction can be resolved if we take into account 
the fact that the arithmetic mean is a sufficiently stable score at a 
small scatter of data being processed, otherwise its use is a purely 
formal procedure. Thus, it can be concluded that the final ranking 
upon the rank sum method or the method of ranking arithmetic 
means should be carried out only with adequate consistency of 
preference profiles obtained by different methods.

A concordance coefficient can be used to assess the degree of 
consistency of preference profiles. This coefficient is defined by 
the follwoing formula:

m
2 2

i
i=1

12S
W=

m n(n -1)-m T∑
 (1)

Where S - the sum of squares of the differences between the sum 
of ranks and their average value; m - the number of the methods 
used; n - the number of the entities analyzed; Ti - an indicator of 
linked ranks in ith preference profile.

If a summary table of entities is used as the information base, a 
value S can be determined by the following formula:

2n m

ij
j=1 i=1

1
S= r - m(n+1)

2
 
 
 

∑ ∑  (2)

Where rij - a rank assigned to the j
th entity upon the ith method.

An indicator of linked ranks Ti is calculated by the formula:

ip
3

i ki ki
k=1

T = (h -h )∑ , i = 1…m (3)

Where pi - the number of groups of equal ranks in the i
th preference 

profile; hki the a number of equal ranks in the kth group of linked 
ranks of the ith preference profile.

If there are no matching ranks in the preference profile, then hki = 0 
and pi = 0, consequently Ti = 0.

The analysis of the national and foreign literature allows to set 
the limit value for the concordance coefficient W = 0.5 (Schrader, 

1971; Kemeny and Snell, 1972). When W < 0.5 preference profiles 
are not consistent, when 0.5 ≤ W <0.7 - not sufficiently consistent, 
when 0.7 ≤ W < 0.9 - consistent, when W ≥ 0.9 - extremely 
consistent.

When W ≥ 0.7 preference profiles are considered consistent and 
in order to develop the final ranking of enterprises we can use 
the rank sum method or the method of ranking arithmetic means. 
When W < 0.7 preferences profiles are not sufficiently consistent 
and to develop the final ranking the use of the mentioned methods 
is not entirely correct, which is particularly manifested when 
W < 0.5. Therefore, there is a need to develop other approaches 
and methods that make it possible to overcome the disadvantages 
of the rank sum method or the method of ranking arithmetic 
means.

3.3. Formation of Rating Preferences upon the Joint 
Application of the Method of Ranking Arithmetic 
Means and the Method of Ranking Position Indicators
The proposed approach is designed to compensate in some way 
for the shortcomings of the rank sum method and the method of 
ranking arithmetic means and provides more correct development 
of the final ranking of the entities at a value of concordance 
coefficient W < 0.7.

The approach implies the joint application of the method of ranking 
arithmetic means and the method of ranking position indicators. 
This approach is consistent with the concept of sustainability, 
considering the use of a variety of methods for processing the 
same data in order to form conclusions, obtained simultaneously 
in all methods used.

To adjust the rankings obtained by the method of ranking 
arithmetic means, it is proposed to use the ranking position 
indicators. This approach is due to the fact that when measured 
on an ordinal scale (the rank scale is their representative) 
it is correct to use the members of the variation series (in 
particular, a median should be used as a mid-scale, which can 
be mathematically proved). Quartiles, deciles, percentiles, 
etc. can be the members of the variation series, describing the 
position of the data.

It is proposed to use three position indicators: A median, a lower 
quartile and an upper quartile.

A preference profile median shall mean the importance of the 
rank located in the middle of the profile. For a profile with an 
odd number of ranks, a rank located in the center of the profile 
will be a median. For a profile with an even number of ranks, an 
arithmetic mean of two central ranks will be a median.

Preference profile quartiles shall mean the significance of the 
ranks, dividing the preference profile into four equal parts. 
The lower quartile separates the one-fourth of the profile with 
the lowest values of ranks, while the upper quartile separates 
the one-fourth of the profile with the highest values of ranks. 
Respectively, a median is a medium quartile.



Motorygin, et al.: Management of the Formation of Rating Preferences of Economic Entities upon Collective Choice

International Journal of Economics and Financial Issues | Vol 6 • Issue 4 • 2016 1959

The approach implies the element-wise summation of the final 
ranks obtained by the method of arithmetic means and the method 
of ranking position indicators:

R =R +Rj
AP

j
A

j
P  (4)

Where Rj
A  - a final rank of the jth entity obtained by the method 

of ranking arithmetic means; Rj
P  - a final rank of the jth entity 

obtained by the method of ranking position indicators; Rj
AP  - a 

final rank of the jth entity obtained upon the joint application of 
the method of arithmetic means and the method of ranking position 
indicators.

The final ranking is the result of the ordering of final ranks Rj
AP  

in ascending order.

3.4. Formation of Rating Preferences Based on the 
Kemeny Median
With the formation of rating preferences, the natural presumption 
is that the final ranking should be as close as possible to the 
preference profiles (rankings) based on different methods. Such 
ranking will correspond to the Kemeny median:

n

A B
j=1P

P*= d(P ,P )argmin∑  (5)

Where d(PA,PB) - the distance between rankings constructed by 
using the methods A and B.

The construction of the final ranking with the use of the Kemeny 
median requires the introduction of the distance between any 
two rankings PA and PB. In the basic procedure for finding the 
Kemeny median, such distance is introduced on the basis of the 
matrix of relations and the matrix of losses, the construction 
and processing of which require a sufficiently large number of 
additional processing operations.

The specifics of the problem considered allows to avoid the 
construction of these matrices, since the ranks of entities derived 
from the calculated rating scores by the methods described above, 
not the expert opinions located in the space of non-numeric objects, 
serve as the processed data. Taking into account the specifics of 
the problem, a concept of distance between the rankings should 
be introduced.

Definition: The distance between the rankings PA and PB, described 
by the vectors rj

A  and rj
B  respectively, is a value d(PA,PB) equal 

to the sum of modules of differences of the elements located on 
the same places in their respective vectors, i.e., the distance 
between the rankings PA and PB and is calculated as follows:

n
A B

A B j j
j=1

d(P ,P )= r -r∑  (6)

Based on the introduced concept of distance between the rankings, 
the problem of finding the final ranking can be expressed in the 
following optimization formula:

m

i
i=1

d(P ,P*) min∑  (7)

Where P* - the final ranking.

The final ranking P* can be considered as a set element (Pi). In 
this case, the solution of the optimization problem will be reduced 
to one of the rankings that are based on the above methods. In our 
opinion, it is more reasonable to find a new ranking Pm+1 = P*, the 
sum of distances from which to all other rankings is minimal. At 
the same time, a danger of a situation of “getting into the centre 
of the donut” is smoothed over by the non-Euclidean nature 
of the introduced metrics and optimization nature of the task. 
Consequently, the final ranking P* obtained based on the modified 
Kemeny median is more objective in comparison with the ranking 
obtained based on the rank sum method.

3.5. The Algorithm of the Formation of Rating 
Preferences upon Collective Choice
Having considered the specific issues associated with the 
development of the method of the formation of rating preferences 
of economic entities upon collective choice, we propose an 
algorithm for the use of the developed method as a summary 
logical conclusion (Figure 1).

4. RESULTS

4.1. A Brief Description of the Task of the Rating 
Assessment of the Financial Status of Enterprises
Testing of the proposed methodology has been carried out on the 
task of the rating assessment of the financial status of industrial 
enterprises.

The financial status of an enterprise is a complex characteristic of 
its position, which is determined by a system of indicators reflecting 
the availability, distribution and use of financial resources of an 
enterprise. The rating assessment of the financial status is a tool 
for comparing the indicators of financial and economic activities 
of competing companies. As a rule, it is conducted in the interests 
of external users: Investors, shareholders, creditors, customers, tax 
authorities, auditors, etc.

The rating assessment is based on the official financial statements 
containing the limited information on the activities of the 
enterprise. It does not imply an in-depth study of various aspects of 
financial-economic activities of the enterprise and its main purpose 
is to get an integrated rating numeric score for the subsequent 
ranking of compared enterprises.

Today in Russia, there are different methods of the rating 
assessment of the financial status of enterprises. The conducted 
analysis of the subject area and literary sources of educational, 
scientific and applied nature has allowed to identify a number of 
methods, which are, in our opinion, the most widely used. Such 
methods are as follows: A method of Bakanov and Sheremet 
(Bakanov and Sheremet, 2001) a method of Ginzburg (Ginzburg, 
2004), a method of Grafova (Grafova, 2003), a method of the 



Motorygin, et al.: Management of the Formation of Rating Preferences of Economic Entities upon Collective Choice

International Journal of Economics and Financial Issues | Vol 6 • Issue 4 • 20161960

company INEK (Kotlyar, 1999), a method of Kovalev (Kovalev 
and Volkova, 2002), a method of Postyushkov (Postyushkov, 
2001), a method of Savitskaya (Savitskaya, 2000), a method of 
Selezneva and Ionova (Selezneva and Ionova, 2006), a method of 
Shadrina (Shadrina, 2008), a method of Schiborsch (Schiborsch, 
2000).

However, it should be noted that in a number of methods there 
is a clearly defined system of indicators, based on which the 
rating score is calculated. There are only general guidelines 
for its construction. This circumstance does not allow for the 
correct comparative analysis of these methods. Therefore, for 

further consideration we have selected methods, the authors of 
which, in addition to the method of the rating score calculation, 
provide a system of indicators, which is an information base 
for the rating calculation. These methods include the methods 
of Kovalev, the company INEC, Schiborsch, Ginzburg, 
Postyushkov.

To test the developed method, six industrial companies were 
selected (with the code names “Quantum,” “Prometheus,” 
“Emerald,” “Pulsar,” “Alpha” and “Spectrum”), which provided 
the necessary reports for the calculation of indicators of their 
financial status.

Figure 1: The algorithm of the formation of rating preferences upon collective choice



Motorygin, et al.: Management of the Formation of Rating Preferences of Economic Entities upon Collective Choice

International Journal of Economics and Financial Issues | Vol 6 • Issue 4 • 2016 1961

4.2. Rating Assessment of the Financial Status of 
Enterprises
In accordance with the algorithm of the application of the 
developed method, the rating assessment of enterprises by using 
the methods developed by Kovalev, INEK company, Schiborsch, 
Ginzburg and Postyushkov has been conducted at the first stage. 
The calculated final rating values are given in Table 2.

A summary table of ranks of enterprises is formed on the basis of 
Table 2 and the rules of rank assignment (Table 3).

At the next stage, the assessment of the degree of consistency of 
preference profiles is conducted. For this purpose the concordance 
coefficient is calculated by the formula (1):

5 2 2
2 2

i
i=1

12S 12×256.5
W= = =0.593

5 ×6×(6 -1)-5×12
m n(n -1)-m T∑

Since 0.5 ≤ W < 0.7, the preference profiles are deemed not 
sufficiently consistent. Thus, to construct the final ranking it 
makes most sense to use either a method of the collective rating 
formation based on the joint use of arithmetic means and ranking 
position indicators, or a method of the collective rating formation 
based on the modified Kemeny median.

Let us conduct the rating assessment of the selected enterprises, 
using the above methods.

Using the first method, we calculate the final ranks of the 
enterprises by using arithmetic means of ranks, based on 

ranking position indicators and on their joint application 
(Table 4).

The final ranking is the result of the final rank ordering Rj
AP in 

ascending order (Table 5).

The second method that has been proposed for the formation 
of rating preferences is based on the application of the Kemeny 
median.

The first phase of this method involves the construction of the 
matrix of distances between the rankings obtained by using 
methods of Kovalev, INEC company, Schiborsch, Ginzburg and 
Postyushkov upon the formula (6) (Table 6).

After the construction of the matrix of distances between the final 
rankings, we obtain the final ranking P* based on the solution of 
the optimization task (7).

The task (7) has a significant computational complexity, so for its 
solution it is advisable to use the appropriate software. The Excel 
add-in “Search for Solution” is recommended as such software. 
The Kemeny median found is given in Table 7.

The final ranking of enterprises P* is shown in Table 8.

5. DISCUSSION

Analyzing the results obtained (Tables 5 and 8), it is easy to note 
that the final rankings obtained on the basis of the two proposed 
methods are slightly different from one another. This applies to 

Table 2: A summary table of rating scores of enterprises
Method Enterprises

Quantum Prometheus Emerald Pulsar Alpha Spectrum
Kovalev’s 15.0 49.5 42.6 68.0 23.3 94.4
INEK 22.0 22.0 27.0 8.0 14.0 26.0
Schiborsch’s −0.5 7.6 10.9 11.6 −3.3 12.8
Ginsburg’s 288.0 192.0 167.0 167.0 266.0 152.0
Postushkov’s 1.456 1.219 1.046 −0.285 −0.788 2.649

Table 3: A summary table of ranks of enterprises
Method Enterprises

Quantum Prometheus Emerald Pulsar Alpha Spectrum
Kovalev’s 6.0 3.0 4.0 2.0 5.0 1.0
INEK 3.5 3.5 1.0 6.0 5.0 2.0
Schiborsch’s 5.0 4.0 3.0 2.0 6.0 1.0
Ginsburg’s 6.0 4.0 2.5 2.5 5.0 1.0
Postushkov’s 2.0 3.0 4.0 5.0 6.0 1.0

Table 4: The rating assessment of enterprises on the basis of joint application of arithmetic means and ranking position 
indicators
Calculated values Enterprises

Quantum Prometheus Emerald Pulsar Alpha Spectrum
Final ranking upon arithmetic means 5.0 3.5 2.0 3.5 6.0 1.0
Final ranking upon position indicators 5.0 4.0 2.5 2.5 6.0 1.0
Summary ranking upon arithmetic means and position indicators 10.0 7.5 4.5 6.0 12.0 2.0
Final ranking upon arithmetic means and position indicators 5.0 4.0 2.0 3.0 6.0 1.0



Motorygin, et al.: Management of the Formation of Rating Preferences of Economic Entities upon Collective Choice

International Journal of Economics and Financial Issues | Vol 6 • Issue 4 • 20161962

the enterprises “Emerald” and “Pulsar,” as well as enterprises 
“Quantum” and “Alpha,” which had the same position in the 
second ranking. If we take into account the results of the two 
final rankings, it can be assumed that the financial-economic 
condition of the enterprise “Emerald” slightly exceeds the 
financial-economic status of the enterprise “Pulsar,” while the 
financial-economic condition of the enterprise “Quantum” 
significantly exceeds the financial-economic condition of the 
enterprise “Alpha.”

Despite some differences in the final rankings, their concordance 
coefficient W = 0.985, indicating a high degree of consistency of 
the estimates, so the resulting ranking can be constructed by using 
the rank sum method (Table 9).

The obtained resulting ranking confirmed our assumptions about 
certain superiority of the enterprise “Emerald” to the enterprise 
“Pulsar,” and the enterprise “Quantum” - to the enterprise 
“Alpha.”

With the accumulation of statistical information on the rating 
assessment of enterprises, weight coefficients for each of the 
discussed methods can be introduced into the proposed procedure 
for finding the Kemeny median, which allows to increase even 
more the objectivity of the resulting rating score.

Obviously, if the ranking PA is closer to the final ranking P* than 
the ranking PB, the weight of the method A must be higher than 
the weight of the method B, and vice versa. Since the best ranking 
corresponds to a smaller distance, the weight coefficients can be 
found by the following formula:

i
i m

i=1 i

1
d(P ,P*)

v =
1

d(P ,P*)∑
 (8)

Where vi - a weight coefficient of the i
th method.

Thus, for the above example the calculated weights of methods 
are shown in Table 10.

It should be noted that the weight coefficients calculated upon the 
results of the current procedure may be used in the subsequent 
procedures. It is obvious that if the other enterprises are involved in the 
rating assessment, then the weights calculated by the formula (8) will 
differ from the weights calculated last time. In order to stabilize the 
weights, the current and previously calculated weights are averaged, 
thus a new weight takes account of the value of “old” weights. This 
procedure is recurrent in nature and corresponds to the procedure for 
exponential smoothing of weights with a coefficient equal to a = 0.5.

Thus, with regard to the weights of methods, the formula of finding 
the final ranking can be written as follows:

m
®

i i
i=1

(v ×d(P ,P*)) min∑  (9)

Note that, despite its advantages, the method of the formation of 
collective preferences based on the modified Kemeny median has 
one significant drawback - it is much more complicated in terms of 
the applied calculation procedures. In particular, in order to find the 
final ranking P* the algorithms based on the method of branches and 
borders or gradient search methods should be used, so the practical 
application of the method is associated with its implementation at the 
software level. In the absence of specific software the task (9) can be 
efficiently solved by using the Excel add-in “Search for Solution,” 
in which a method of generalized gradient and the evolutionary 
method are implemented to solve nonlinear tasks.

6. CONCLUSION

Currently, there are various methods of the formation of ratings 
of economic entities. Each of the methods is usually a set of 

Table 5: The final ranking of enterprises based on the 
joint application of arithmetic means and ranking position 
indicators
Rank Place Enterprise
1 1st Spectrum
2 2nd Emerald
3 3rd Pulsar
4 4th Prometheus
5 5th Quantum
6 6th Alpha

Table 6: Matrix of distances between the rankings
Method Method Sum of distances

Kovalev’s INEK Schiborsch’s Ginsburg’s Postushkov’s
Kovalev’s 0.0 11.0 4.0 3.0 8.0 26.0
INEK 11.0 0.0 10.0 9.0 8.0 38.0
Schiborsch’s 4.0 10.0 0.0 3.0 8.0 25.0
Ginsburg’s 3.0 9.0 3.0 0.0 10.0 25.0
Postushkov’s 8.0 8.0 8.0 10.0 0.0 34.0

Table 7: The Kemeny median of rankings of the enterprises
Calculated values Enterprises

Kovalev’s INEK Schiborsch’s Ginsburg’s Postushkov’s Kovalev’s
Kemeny median 5.0 4.0 3.0 3.0 5.0 1.0
Final ranking upon the Kemeny median 5.5 4.0 2.5 2.5 5.5 1.0



Motorygin, et al.: Management of the Formation of Rating Preferences of Economic Entities upon Collective Choice

International Journal of Economics and Financial Issues | Vol 6 • Issue 4 • 2016 1963

indicators of the financial-economic condition of a business entity, 
the distribution of weight coefficients on the local criteria and the 
procedure for their integration into the integral rating indicator. 
Therefore, there is a problem of choosing any one method and 
conducting on its basis the rating assessment of economic entities 
or the simultaneous use of several proven methods. In our opinion, 
the latter approach is more reliable, but in this case there is a 
need to form the collective rating by combining some rankings 
obtained with the use of each of the methods. This integration leads 
to a significant increase in calculation costs. Thus, the practical 
application of the proposed approach implies the development 
of the automation system of the formation of collective rating 
preferences.

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Table 8: The final ranking of enterprises upon the 
Kemeny median
Rank Place Enterprises
1 1st Spectrum
2.5 2nd-3rd Emerald, pulsar
4 4th Prometheus
5.5 5th-6th Quantum, alpha

Table 9: Resulting ranking of enterprises
Rank Place Enterprises
1 1st Spectrum
2 2nd Emerald
3 3rd Pulsar
4 4th Prometheus
5 5th Quantum
6 6th Alpha

Table 10: Weight coefficients of methods calculated upon 
the data of the working example
Methods Distance to the Kemeny 

median
Weight coefficients of 

methods
Kovalev’s 4 0.17
INEK 8 0.08
Schiborsch’s 2 0.33
Ginsburg’s 2 0.33
Postushkov’s 8 0.08



Motorygin, et al.: Management of the Formation of Rating Preferences of Economic Entities upon Collective Choice

International Journal of Economics and Financial Issues | Vol 6 • Issue 4 • 20161964

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