=Paper=
{{Paper
|id=Vol-2917/paper21
|storemode=property
|title=Analysis of the Effectiveness of the Successive Concessions Method to Solve the Problem of Diversification
|pdfUrl=https://ceur-ws.org/Vol-2917/paper21.pdf
|volume=Vol-2917
|authors=Anna Bakurova,Hanna Ropalo,Elina Tereschenko
|dblpUrl=https://dblp.org/rec/conf/momlet/BakurovaRT21
}}
==Analysis of the Effectiveness of the Successive Concessions Method to Solve the Problem of Diversification==
https://ceur-ws.org/Vol-2917/paper21.pdf
Analysis of the Effectiveness of the Successive Concessions
Method to Solve the Problem of Diversification
Anna V. Bakurova, Hanna M. Ropalo and Elina V. Tereschenko
National University “Zaporizhzhia Polytechnic”, 64 Zhukovsky str., Zaporizhzhia, 69063, Ukraine
Abstract
The subject of the paper is multicriteria problems that arise when modeling the complex
diversification of a centralized pharmacy network. The purpose of the work is to analyze the
peculiarities of solving the three-criteria problem of pharmacy network diversification by the
method of successive concessions in the MATLAB package. The paper solves the following
problems: research of the advantages of the proposed three-criteria model of pharmacy
network diversification in relation to the classical two-criteria model of portfolio theory;
construction of the relation of dominance on a set of criteria; determination of the area of
stability in the space of the parameters of the concessions method; evaluating the
effectiveness of the method for problems of different sizes. The following methods are used:
classical portfolio theory, multicriteria optimization, the successive concessions method,
computer modeling of the Pareto set. The results obtained: a study of the processes of
complex diversification of the pharmacy network by building portfolio models and solving
the relevant multicriteria problems by the successive concessions method. Acceptable sets
and sets of pareto-optimal portfolios for the risk management are graphically found, taking
into account the activity of the network itself and the client portfolio. Conclusions: The
results of computer modeling and numerical analysis of solutions by sequential concessions
will be useful for automating the business processes of pharmacy networks, risk
management, analysis of market data to improve their efficiency.
Keywords 1
multicriteria problem, entropy, optimal portfolio problem, successive concessions method,
Pareto set, pharmacy network
1. Introduction
The successive concessions method is used to solve multicriteria problems, in which all partial
criteria are arranged in the order of importance. It is considered that each criterion is much more
important than the next that it is possible to be limited to consideration of only pairwise connection of
criteria and to choose the acceptable concession for the next criterion taking into account behavior of
only one following criterion. The disadvantages of the method include the complexity of the
appointment and coordination of the size of concessions, the need for a priori ranking of criteria. The
complexity increases with the dimension of the problem and the number of criteria. The result does
not necessarily belong to a subset of Pareto-efficient solutions. Therefore, the work of the method of
concessions is combined with dialogue - by multiple solutions of the optimization problem, graphs of
the dependences of the solution at the next stage from the concession of the previous stage obtained.
The expert selects the assignments based on the analysis of such diagrams and the assessment of the
gains or losses appropriate to the assignment. This paper considers the problem of the effectiveness of
the concessions method to solve the problems of portfolio optimization, which are known due to the
MoMLeT+DS 2021: 3rd International Workshop on Modern Machine Learning Technologies and Data Science, June 5, 2021, Lviv-Shatsk,
Ukraine
EMAIL: abaka111060@gmail.com (A. V. Bakurova); annropalo@gmail.com (H. M. Ropalo); elina_vt@ukr.net (E. V. Tereschenko)
ORCID: 0000-0001-6986-3769 (A. V. Bakurova); 0000-0001-5241-6911 (H. M. Ropalo); 0000-0001-6207-8071 (E. V. Tereschenko)
2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
research of H.M. Markowitz [1] and his modern followers [2]. Such tasks are characterized by
multicriteria and prevalence, both in the financial and non-financial spheres.
2. Literature review
When applying the methods of multicriteria optimization, traditional definitions of the dominance
relation, effective boundary, Pareto set, etc. were used. In particular, in the work of M.Ehrgott [3] the
basic definitions, different approaches and methods of multicriteria optimization are systematically
stated.
The group of publications of the short literature review devoted to portfolio optimization unites the
use of evolutionary methods in research, which is a promising area because the tasks of portfolio
optimization belong to the class of complex NP tasks. In [4], a genetic algorithm was constructed to
solve the problem of the optimal loan portfolio, where a specific entropy criterion was used as a
measure of the degree of portfolio diversification. Paper [5] presents several two-criteria formulations
for the optimization of the molecules portfolio, taking into account the limited budget and fixed size
of the portfolio. The computational diversity of the set associated with the covariance matrix is
presented in this paper by the Solow-Polaski measure. The proposed approach is tested in
experimental conditions using accurate and evolutionary approaches. The authors [6] consider the
problem of optimizing the product range in conditions of uncertainty.
Another group of publications of the literature review consists of works devoted to the study of the
peculiarities of the methods of multicriteria optimization, especially the method of successive
concessions. [8] considered various issues of solving multicriteria problems of discrete optimization
and proposed the method of equivalence sets; investigated the advantages of this method in
comparison with other popular methods of solving multicriteria problems, in particular with the
successive concessions method, for the class of discrete optimization problems. In [9] the existing
achievements in the application of multicriteria optimization methods are considered to solve
problems of evaluating the activities of enterprises, a comparative analysis of modern methods is
made, which includes optimization methods both without the participation of decision maker and
with the participation of decision maker . In the analysis of the successive concessions method, a
class of tasks was indicated for which this method is recommended - optimization of the values of
indicators of the enterprise with the definition of sustainability. The advantages of the method also
include the implementation of the concept of restrictions that are imposed on the value of the criteria.
The disadvantage of this method is that the solution requires verification of its belonging to the area
of compromises. Improving the successive concessions method in solving problems of multicriteria
optimization in logistics is proposed in [10]. The author's approach is based on the introduction of
special formats for the task of choosing a city for the warehouse and the form of ownership. The
possibility of analytical determination of the order of certain criteria at the request of decision maker
is discussed, the influence of different variants of their ordering on the ranking of alternatives is
analyzed.
The variety of subject areas in which researchers solved multi-criteria problems by classical and
modern methods indicates the relevance of scientific research in this direction. The authors of this
paper in [11, 12] built multicriteria models of complex diversification of the centralized pharmacy
network, where the successive concessions method was used to solve the corresponding multicriteria
problems. Giving the advantages and disadvantages of this method, identified by other authors [8, 9]
for their problems, it is necessary to analyze the effectiveness of the successive concessions method to
solve diversification problems. The purpose of the work is to analyze the effectiveness of solving the
three-criteria problem of pharmacy network diversification by the successive concessions method in
the MATLAB package [13]. To achieve this goal, the following tasks are solved in the paper.
• Research of the advantages of the proposed three-criteria model of pharmacy network
diversification over the classical two-criteria model of portfolio theory.
• Construction of the dominance relation on a set of criteria.
• Determination the area of stability in the space of the parameters of the method of concessions.
• Evaluation the effectiveness of the method for tasks (networks) of different sizes.
3. Problem statement and results
The paper [11] proposes a mathematical model of diversification in the form of a three-criteria
optimization problem. Solving multicriteria optimization problems is not trivial, and requires special
methods and definition of decision maker of certain parameters. One of the classic methods is the
method of concessions.
We present one of the tasks of complex diversification, which is formalized as a portfolio model
for optimizing the distribution of finances between the outlets of the centralized pharmacy network.
Optimally distributing goods among outlets, the network gets the maximum profit in time to respond
to the circulation of medications in a certain period, because the demand for medications in a
particular pharmacy is not constant and goods not sold in one outlet can be sold on time in another
and bring profit without increasing the length of the composition.
The turnover of the pharmacy network is the sum of the turnover of its pharmacy establishments.
The turnover of an individual pharmacy is estimated to be sufficient at the level of UAH 2 million,
which defines additional limitations in Model 1.
Let, - the share of the -th pharmacy in the turnover of the pharmacy network, which is equal to
the share of distributed financial resources for the -th pharmacy;
- the expected turnover of the pharmacy (UAH);
- number of pharmacies in the network.
The model of the optimal portfolio considered in the paper has a composition of the vector
objective function, consisting of three criteria. The first two criteria correspond to the classical
portfolio theory: risk criterion - Risk, which decreases; Sum criterion - the profitability on the
portfolio, which is desirable to increase.
The third criterion - entropy - Entropy, is introduced by us as a value that characterizes the level of
diversification of the portfolio and the assessment of the diversity of its components, which the
pharmacy network is trying to increase in its activities. In addition, the model has its own specific
system of constraints, which is determined by the business process of the pharmacy network in the
structure of its activities.
In this model, the Risk criterion - structural risk - is the risk of irrational distribution of financial
resources of the centralized pharmacy network between outlets. Structural risk is defined as the
covariance of the turnover of -th and -th pharmacies.
So, we have such multi-criteria problem of quadratic programming.
∑( ̄) ( ̄)
∑
∑ ( ) (1)
∑
∑ [ ]
{
The solution of problem (1) is a vector X * x1 , x2 ..xn - the optimal plan for the distribution of
financial resources of the centralized pharmacy network between outlets.
Solving multicriteria problems of complex diversification will be considered in detail on the
example of the successive concessions method [14, 15].
The successive concessions method for solving multicriteria problems is applied when partial
criteria can be ordered in ecrease of their importance. To choose a diversification strategy, we first
choose the following order: Entropy-Risk-Sum.
In the first stage, we determine the optimal value of the first most important Entropy criterion in
the field of acceptable solutions.
∑ ( )
∑ ̄
(2)
∑
{
The optimal solution for the first partial criterion is .
In the second stage, we solve the conditional optimization problem by the next most important
Risk criterion, adding to the conditions that determine the acceptable solutions, the conditions for the
deviation of the first Entropy criterion from the found optimal value by no more than the
acceptable concession . So we have the formalization of the second stage:
∑ ∑( ̄ ) ( ̄ )
∑ ( )
(3)
∑ ̄
∑
{
The optimal solution according to the second criterion is obtained.
The procedure is repeated for the next most important criterion adding to the conditions that
determine the acceptable solutions, the conditions for the deviation of the first criterion and
the second criterion from the found optimal values , ot more than the values of
the acceptable concessions and .
∑ ̄
∑ ( )
∑ ̄ (4)
( )
∑
{
The solution obtained in the third stage is the solution of the three-criteria conditional optimization
problem (1).
4. Experiments
Experiments with the models were conducted on real data of one of the pharmacy chains operating
in Zaporizhzhia. All calculations were performed in the MATLAB package [13].
Here are some useful definitions for analyzing the effectiveness of solving the three-criteria
problem of diversification the pharmacy network by the successive concessions method in the
MATLAB package, which are taken from sources [14, 15].
Definition 1. Stability of the algorithm - the ability to perform calculations and obtain the final
result with a given accuracy when changing the parameters of the algorithm and input data in some
area, which is called the area of stability.
Definition 2. Convergence is a property of an algorithm by changing its parameters to perform
calculations with an arbitrarily small error for a given class of input data (i.e. when increasing the
number of iterations for matching algorithms, the error will tend to zero). Moreover, the increase in
accuracy is achieved by changing the internal parameters of the algorithm (for example, the maximum
acceptable difference between the previous and next approximation).
Definition 3. The correctness of the computational method is a property of the indisputable
existence of the solution the problem and ensuring stability of the computational algorithm that
implements this method.
Let’s consider the parameters that affect the results of solving the three-criteria problem of
diversification by the successive concessions method.
The method of concessions requires the decision maker to determine the relation of dominance on
multiple criteria.
The criteria are: total income of the pharmacy network for a certain period , the risk
of income loss, which is defined as covariance on the set of income of pharmacies and
the degree of diversification, which is estimated by the entropy of the investment portfolio of the
pharmacy network .
Therefore, the first parameter of the method is the order of solving optimization problems
according to one of the three criteria.
The vector of deviation values is the second parameter of the method. After solving the first one-
criteria problem, the system of constraints is supplemented by a new criterion, which limits the
deviation from the found of the optimal solution at the previous stages.
The number of pharmacies in the network, which determines the size of the problem, is the third
parameter of the method.
The fourth parameter concerns the initial approximation and accuracy of the method. In this study,
its characteristics are the accuracy of algorithms implemented by standard functions
OPTIMIZATION TOOLBOX MATLAB, as the authors have developed a software product for
solving research problems in the MATLAB package. Such function is the exitflag, which describes
the conditions of the exit. If the value of exitflag is greater than zero, it means that the function
matches the desired solution by X. If it is zero, then the maximum value of the function estimate or
iteration has been exceeded. But if the value of exitflag is less than zero, then the function does not
match to some solution.
Therefore, to analyze the effectiveness of the method of concessions on certain parameters, it is
necessary to perform four steps that meet the objectives of this paper.
Step 1. Research of advantages of the offered three-criterion model of diversification of a
pharmacy network concerning classical two-criterion model of the portfolio theory.
The initial approximation of the first stage is fixed as 0.1 from the lower limit of the admissibility
interval. For each subsequent stage, we take the optimal value of the previous stage as the initial
approximation.
Let's compare the solutions obtained using the classical two-criteria approach and the proposed
three-criteria approach.
Consider the five modifications of problem (1), which are formed due to the different sequences of
consideration of the criteria of profitability (SUM), risk (Risk) and the degree of diversification
(Entropy) by the successive concessions method.
So we have two two-criteria problems, which we denote according to the sequence of criteria:
and The results of solving these problems for a small network (the number
of pharmacies in the network n = 5) with estimates of the main parameters are given in Table 1.
Table 1
The results of solving two-criteria problems for a small network
RISK_SUM n=5 SOLUTION SUM_ RISK n=5 SOLUTION
Stage of the method 1 level 2 level 1 level 2 level
SUM 25.271 30.038 141.489 25.271
Risk 15.724 17.296 641.734 15.724
ENTR 1.499 1.515 0.0944 1.499
exitflag 1 5 1 5
X= 0.394 0.196 0.001 0.394
0.001 0.001 0.007 0.001
0.001 0.001 0.990 0.001
0.164 0.321 0.001 0.164
0.439 0.481 0.001 0.439
According to the data in Table 1, we can see that the best result among the two-criteria
modifications is achieved for the model RISK_SUM, n = 5.
When adding the third criterion the vector objective function, you can get a total of 3! =
6 combinations to consider the sequence of criteria by the method of concessions. But the purpose of
the third criterion is to increase the degree of diversification of the portfolio, so we consider it the
most important and put in the first place (in addition, a number of experiments have confirmed that
such arrangement gives better results than others).
Therefore, we will consider two three-criteria problems, which we will also denote according to the
sequence of consideration of criteria by the method of concessions: and
. The results of solving three-criteria problems (n = 5) with estimates of the main
parameters are given in Table 2, where an inefficient modification is also
given for comparison.
Table 2
Results of solving three-criteria problems for a small network
ENTROPY_RISK_SUM ENTROPY_ SUM_ RISK RISK_SUM_ENTROPY
n=5 n=5 n=5
SOLUTION SOLUTION SOLUTION
1 level 2 level 3 level 1 level 2 level 3 level 1 level 2 level 3 level
SUM 60.149 38.320 42.049 60.149 87.512 78.761 25.271 30.038 28.172
Risk 92.970 31.926 35.119 92.970 214.102 154.047 15.724 17.296 17.296
ENTR 2.322 2.089 2.0897 2.322 2.089 2.089 1.499 1.515 1.690
exitflag 1 5 5 1 4 4 1 1 5
X= 0.200 0.289 0.219 0.200 0.105 0.062 0.394 0.196 0.355
0.200 0.092 0.067 0.200 0.198 0.106 0.001 0.001 0.017
0.200 0.056 0.086 0.200 0.437 0.377 0.001 0.001 0.005
0.200 0.245 0.285 0.200 0.143 0.259 0.164 0.321 0.240
0.200 0.318 0.344 0.200 0.118 0.196 0.439 0.481 0.384
According to the estimates given in Table 2, it can be stated that the ENTROPY_RISK_SUM
model n = 5 has the best result. Comparing the results of calculations from tables 1 and 2, we obtain
experimental confirmation that the introduction of the entropy criterion allows to increase the income
of the pharmacy network, but also increases the risk in different proportions.
Step 2. Construction of the dominance relation on the set of criteria.
The dominance ratio on the set of criteria, based on the significance of a particular criterion for the
optimal decision-maker solution and the results of the experiments in step 1, is most successful in the
ENTROPY_RISK_SUM model, where the criteria are ranked in descending order of importance. So
this is the best result recommended for decision-maker, so Figures 1, 2, and 3 below show the
computer simulation results for the the results of computer modeling of the domain of acceptable
solutions, Pareto set and effective solution (red triangle) ENTROPY_RISK_SUM model.
First Step
600
400
Risk
200
0
3
2 150
100
1 50
Entropy 0 0 SUM
Figure 1: The result of computer simulation of the first step of the concessions method for the
model ENTROPY_RISK_SUM
Second Step
600
400
Risk
200
0
3
2 150
100
1 50
Entropy 0 0 SUM
Figure 2: The result of computer simulation of the second step of the concessions method for the
model ENTROPY_RISK_SUM
Third Step
600
Risk 400
200
0
3
2 150
100
1 50
Entropy 0 0 SUM
Figure 3. The result of computer simulation of the third step of the concessions method for the
model ENTROPY_RISK_SUM
As can be seen from Figures 1, 2 and 3, the visualization of solutions using computer modeling
eliminates the previously mentioned disadvantage of the concessions method that the result does not
necessarily belong to a subset of Pareto-efficient solutions, and therefore requires verification of its
membership compromises. Thanks to the graphical interpretation, the check is performed
automatically.
Step 3. Determination the area of stability in the space of the parameters of the concessions
method
Construct n area of stability in the space of parameters of the concessions method and
. The color will indicate the end conditions of the algorithm. Red color indicates the successful
completion of the algorithm with the value exitflag=1 (First order optimality conditions satisfied).
Marked in green Exitflag = 4 (Computed search direction too small). Marked in blue exitflag = 5
(Predicted change in objective function too small).
Figure 4: The area of stability in the space of parameters of the method of concessions and for
the model ENTROPY_RISK_SUM, n=5
Figure 5: The area of stability in the space of parameters of the method of concessions and for
the model ENTROPY_SUM_RISK, n=5
For the ENTROPY_SUM_RISK model, experiments have shown that the concessions method is
resistant to parameter changes for both small and medium networks (n <33) and large networks with n
= 65 objects.
Step 4. Evaluation the effectiveness of the method for tasks (networks) of different sizes.
Let us evaluate the efficiency of the method application for networks of different size n = 5, 33, 65
and problems with different composition of the vector objective function.
Previously, Table 1 and 2 presented the results of solving two-criteria and three-criteria problems
for small networks n = 5. Similar experiments were performed with medium networks, the results of
which are presented in Table 3 and 4.
Table 3
Results of solving two-criteria problems for a medium-sized network, n=33
RISK_SUM n=33 SOLUTION SUM_RISK n=33 SOLUTION
1 level 2 level 1 level 2 level
SUM 33.064 42.227 146.461 47.646
Risk 11.112 12.223 866.358 13.449
ENTR 2.843 3.233 0.364 3.411
exitflag 5 5 1 5
Table 4
Results of solving three-criteria problems for a medium-sized network, n=33
ENTROPY_RISK_SUM n=33 SOLUTION ENTROPY_SUM_RISK n=33 SOLUTION
1 level 2 level 3 level 1 level 2 level 3 level
SUM 86.535 60.892 66.475 86.535 127.774 114.996
Risk 104.181 26.347 28.981 104.181 308.114 123.268
ENTR 5.044 4.486 4.539 5.0444 3.985 3.985
exitflag 1 0 0 1 5 5
The results of experiments with large networks (n = 65) are shown in Table 5 and 6.
Table 5
The results of solving two-criteria problems for a large network, n=65
RISK_SUM n=65 SOLUTION SUM_RISK n=65 SOLUTION
1 level 2 level 1 level 2 level
SUM 40.425 50.334 200.644 85.064
Risk 10.659 11.725 2302.353 27.226
ENTR 2.943 3.208 0.727 3.487
exitflag 5 5 1 5
Table 6
The results of solving three-criteria problems for a large network, n=65
ENTROPY_RISK_SUM n=65 SOLUTION ENTROPY_SUM_RISK n=65 SOLUTION
1 level 2 level 3 level 1 level 2 level 3 level
SUM 60.598 66.851 93.322 180.035 162.032 60.598
Risk 26.015 28.969 124.682 749.963 210.482 26.015
ENTR 5.245 5.342 6.022 3.312 3.312 5.245
exitflag 1 0 0 1 5 4
Evaluation of the effectiveness of the method of concessions for networks of different sizes is
performed by the value of the exitflag function of the MATLAB package, which describes the exit
conditions. Therefore, pay attention to the resulting column of each SOLUTION table, which
corresponds to the last stage of the method of concessions.
For all problems of dimension n = 5 the value of exitflag is greater than zero, ie this function
coincides with the desired solution by X. For the problem with the objective function of the model
ENTROPY_RISK_SUM at n = 33 and 65 the value of exitflag is zero. This indicates that the
maximum value of the function or iteration estimate was exceeded. But in no case was the exitflag
value less than zero when the function did not match some solution.
Therefore, there is always a solution for the considered problems, and its belonging to the
acceptable area is checked graphically thanks to the developed software in the MATLAB package.
5. Discussion and Conclusion
This work continues the study diversification models of the centralized pharmacy network, which
was started in [11] and [12]. In the article [12], the authors developed four main models of complex
diversification, which focus on various aspects of risk management: financial allocation, customer
relations, supply and individual outlets. The model of pharmacy portfolio management is considered
in more detail in [11]. Recommendations for different initial conditions are developed on the basis of
[11], which are determined by variations in the ratio of customers of three types: loyal, random and
online customers.
In contrast to [11] and [12], this paper is devoted to the study of effectiveness tool used by the
authors to solve the relevant multicriteria tasks - the successive concessions method. Evaluation of the
effectiveness of the method was performed on four parameters. For the first the parameter of the
method is the sequence of solving optimization problems for one of the three criteria. The
experiments were conducted on different arrangements of risk, income and entropy, as well as a
comparison of the results of solving classical two-criteria tasks (risk-income) with three-criteria. The
second parameter of the method is the vector of deviation values, which is determined by the size of
the assignment. There are a number of pharmacies in the network that determine the size of the
problem the third parameter of the method. The fourth parameter concerns the initial approximation
and accuracy of the method. The acceptable sets and sets of pareto-optimal portfolios were found
graphically for all cases of parameter values.
Thanks to the developed software in the MATLAB package it is solved the task of studying the
advantages of the proposed three-criteria model of pharmacy diversification networks relative to the
classical two-criteria model of portfolio theory, built the relation of dominance on a set of criteria;
determining the area of stability in the space of the parameters of the concessions method - which is
the scientific novelty of this paper.
The results of computer simulation and numerical analysis performed in this paper successive
concessions will provide investors with an appropriate tool decision support will be useful for
automating pharmacy business processes networks, risk management, analysis of market data to
improve their efficiency functioning.
Among the areas of further research - a comparative analysis of the effectiveness of other methods
solving diversification problems and taking into account the dynamic factors that affect the degree of
the risk.
6. Acknowledgements
The work was carried out as the part of the research work “Mathematical modeling of socio-
economic processes and systems”, the registration number DB05038, at the Department of System
Analysis and Computational Mathematics of National University “Zaporizhzhia Polytechnic”.
7. References
[1] H. M. Markowitz, K. Blay, Risk–Income Analysis: The Theory and Practice of Rational
Investing (a four-volume series), McGraw-Hill, 2014.
[2] J. B. Guerard (ed.), Handbook of Portfolio Construction. Contemporary Applications of
Markowitz Techniques, Springer, New York, Dordrecht, Heidelberg, London, 2010, p.808. URL:
http://doi: 10.1007/978-0-387-77439-8.
[3] M. Ehrgott, Multicriteria Optimization, Springer, Heidelberg, 2005. doi:10.1007/3-540-27659-9
[4] B. Y. Kishakevich, Formation of Pareto optimal loan portfolios with the help of genetic
algorithm, Visnyk KhNU, Khmelnyck, 2010. [Publication in Ukrainian]
[5] Alireza Golia, Hasan Khademi Zarea, RezaTavakkoli-Moghaddamb, Ahmad Sadeghieha, Hybrid
artificial intelligence and robust optimization for a multi-objective product portfolio problem
Case study: The dairy products industry, volume 137 of Computers & Industrial Engineering,
November 2019, https://doi.org/10.1016/j.cie.2019.106090.
[6] I. Yevseyeva, E. B. Lenselink, A. de Vries, A. P. IJzerman, A. H. Deutz, and M. T. M.
Emmerich, Application of portfolio optimization to drug discovery, volume 475 of Information
Sciences,, pp. 29–43, 2019. URL: http://doi.org/10.1016/j.ins.2018.09.049
[7] Jian Xiong, Chao Zhang, Gang Kou, Rui Wang, Hisao Ishibuchi and Fawaz E. Alsaadi,
Optimizing Long-Term Bank Financial Products Portfolio Problems with a Multiobjective
Evolutionary Approach, volume of Hindawi Complexity, 2020, Paper ID 3106097.
https://doi.org/10.1155/2020/3106097
[8] R. V. Khachaturov, Generalized Equivalence Set Method for Solving Multiobjective
Optimization Problems, voume 6 of Journal of Computer and Systems Sciences International,
2019
[9] G. L. Brodetskiy, D. A. Gusev, О. А Mazunina, A.V. Fel, Possibilities of the Method of
Successive Concessions When Selecting for Many Criteria. Magazine, volume 3 (80) of
Logistika i upravleniye tsepochkami postavok, June 2017
[10] L. M Malyarets'; O. V. Mínênkova, Research of the problems of multicriteria medicine in
professional activity based on the methods of multicriteria optimization, volume 1 of Economic
problems, 2017. URL: http://search.ebscohost.com/login.aspx?direct=true&site=eds-
live&db=bth&AN=123664557 &custid=s6224580. Acesso em: 13 mar. 2020. [Publication in
Russian]
[11] A. V. Bakurova, H. M. Ropalo, E. V. Tereschenko, Modeling of optimal portfolio of clients of
centralized Pharmaceutical network, volume 6 of Technology audit and production reserves,
2019. doi:10.15587/2312-8372.2019.186789
[12] A. V. Bakurova, Modeling of complex diversification for centralized pharmacy network in R. N.:
A. V. Bakurova, H. M. Ropalo , Е. V. Tereschenko (Eds.), volume 166 of The International
Conference on Sustainable Futures: Environmental, Technological, Social and Economic Matters
(ICSF 2020), Kryvyi Rih, Ukraine, 2020 URL: http://doi.org/10.1051/e3sconf/202016609003
[13] Mathworks, 2019. MATLAB*, 2019. URL: https://www.mathworks.com/campaigns/products.
[14] R. N. Kvêtniy, I. V. Bogach, O. R. Boyko, O. Y. Sofina, O. M Shushura, Computer modeling of
systems and processes. Calculation methods, VNTU, Vínnitsya: 2012, p. 193. [Publication in
Ukrainian]
[15] G. M Gnatíênko, V. E. Snityuk, Expert decision-making technologies, Kiív, . p. 444.
[Publication in Ukrainian]