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1. A hybrid approach based on multi-criteria decision
making and data-based optimization in solving
portfolio selection problem
Meysam Doaei (  me.doaei@iau.ac.ir )
Islamic Azad University, Esfarayen Branch
Kazem Dehnad
Islamic Azad University, Esfarayen Branch
Mahdi Dehnad
Research Article
Keywords: Multi-criteria decision making, Data-based optimization, Investment risk, Stock return
Posted Date: February 15th, 2023
DOI: https://doi.org/10.21203/rs.3.rs-2576724/v1
License:   This work is licensed under a Creative Commons Attribution 4.0 International License.
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Additional Declarations: No competing interests reported.

2. A hybrid approach based on multi-criteria decision making and data-
based optimization in solving portfolio selection problem
In this paper, a two-phase approach based on multi-criteria decision making and
multi-objective optimization is developed to solve the problem of optimal portfolio
selection. In the first phase, the initial selection of suitable companies for
investment is done by considering the criteria extracted from the literature review.
In the second phase, a multi-objective mathematical optimization model is
developed to determine the optimal investment in each company according to risk
and return criteria. In order to deal with uncertainty conditions, a data-based
approach is used, which is one of the newest applied methods in this field.
According to the obtained results, it is observed that cash adequacy ratio with score
0.1604 is the most important criterion and operating profit with score 0.004 is the
least important one. In the alternative prioritization section, it is concluded that
Shraz, Shavan, Shenft and Vanft companies have a high priority for investment. In
solving the mathematical model under certain conditions, it is observed that the
Pareto members 152, 154 and 193 have the smallest distance from the ideal
solution (0.0121) and therefore each of them can be used as the final solution. In
solving the problem under uncertain conditions, numerical scenarios resulting from
changes in the prioritization of companies based on the coefficient v is used in the
VIKOR model. After solving the model, it is observed that the impact of different
scenarios on corporate investment is not negligible and consequently investors
need to pay attention to this fact.
Keywords: Multi-criteria decision making; Data-based optimization; Investment
risk; Stock return
Introduction
The economic and financial conditions of global markets, intense and close competition
and uncertainty of the business environment, which is often due to the outbreak of
Coronavirus in 2020, has caused private sector facing many problems to invest in
financial markets (Ferneini, 2020). In fact, the decision-making criteria that investors
have considered in previous years for the optimal selection of portfolios cannot now lead
to highly reliable answers since the investment environment is affected by new factors

3. such as economic fluctuations, the spread of diseases, widespread closure of
manufacturing and service companies, and investors need to increase their horizons to
more accurately predict the activities of target companies (Talan & Sharma, 2019). In
general, the corporate sustainability criteria can include attention to economic trends,
employment infrastructure and the ones related to the social dimension, as well as the
level of attention of the company to environmental criteria (Ruan, 2018). In Iran,
environmental criteria are less considered and it is practically impossible to extract
comprehensive information regarding sustainable development from companies
operating in the financial markets. Therefore, the dimensions of sustainable development
in Iran only include attention to the criteria of economic development and job creation.
The question that needs to be answered in the first place is how to limit the scope of
decision-making on choosing the right companies to invest in so that the optimal
composition of the stock portfolio with respect to sustainability criteria can be more
focused. In fact, it is important to be able to conduct initial screening to eliminate weaker
companies before a thorough analysis of companies operating in the financial markets to
direct capital to them. This is precisely a decision based on a set of management criteria
and sub-criteria, the output of which leads to a limited number of potential companies to
invest (Ho, Tsai, Tzeng, & Fang, 2011). One good way to answer this question is to use
the concept of corporate financial distress and bankruptcy. In general, corporate financial
distress, which sometimes leads to bankruptcy, refers to situations in which companies
are unable to meet their obligations to creditors, or have difficulty in fully complying with
those obligations (Kisman & Krisandi, 2019). Therefore, financial distress analysis can
play an increasingly important role in the optimal selection of the investment portfolio.
This is because financial distress and non-fulfillment of obligations imposes high costs
on shareholders, companies, creditors, and on the entire economy (Mselmi, Lahiani, &

4. Hamza, 2017). Therefore, considering the criteria and sub-criteria related to financial
distress can play an important role in the initial screening of potential companies for
investment.
In this research, a multi-criteria decision-making model has been developed for
the initial selection of suitable companies for investment by considering criteria related
to financial distress, economic trends and employment conditions in order to perform
appropriate screening. Then a data-based mathematical optimization model is developed
to determine the amount of investment in each company according to risk and return
criteria so that the final outputs can be used as investment management decisions for
users. The most important unknown part of this research is determining the appropriate
method for screening potential companies for investment. According to the conducted
studies in the research literature, this issue has not been discussed so far and it is therefore
considered as a significant innovation in this research. In this study, the answer to this
unknown question will be discussed through the development of a multi-criteria decision-
making model based on indicators of financial distress and sustainable development.
There are also ambiguities about how the criteria related to sustainable development,
including job creation and economic trends, affect the final decisions of investors, the
answers to which are examined in the mathematical modeling section and the definition
of new objective functions.
The most important variables in this study include determining the importance of
each criterion related to the financial distress of companies, financial trends and also job
creation. These variables are mainly analyzed in the development of multi-criteria
decision-making model. There are also other variables including determining the share of
investment in each company according to the objective functions of risk, return and job
creation, which is used in mathematical modeling. In this way, all the main variables of

5. the research can be optimally quantified. Finally, the most important purpose of this
research includes the development of an integrated model to deal with the new conditions
that have arisen in recent years, including high price fluctuations, conditions of
uncertainty in the future activity of companies, ambiguous markets affected by the
outbreak of coronavirus and so on. In this research, this issue will be examined through
the use of two powerful tools in the field of decision making, including multi-criteria
decision-making methods and data-based mathematical programming models.
Literature review
The constant growth of the world population, lack of resources and environmental
pressures are important factors in determining the transition to a greener and more stable
planet (Mansley, 2000). Over the past decade, governments around the world have
committed to addressing climate change issues by revitalizing the national economy
through sources of sustainable economic, social, and environmental growth (Kisman &
Krisandi, 2019). In the Paris Agreement, adopted in December 2015 by the 21st
Conference of the Parties to the United Nations Framework Convention on Climate
Change, the nations agreed to strengthen the global response to climate change threats by
maintaining global warming in this century (Arif et al., 2020). To move towards low-
carbon economies, poverty reduction and sustainable livelihoods, investment in green
employment, biodiversity conservation, renewable energy, sustainable water
management and waste management must be implemented nationally. However,
advanced economies have recently suffered from a lack of investment in public
infrastructure, while developing economies do not have access to modern services for
their growing populations (Caplan, Griswold, & Jarvis, 2013). Accordingly, the ability to
gather the right type of investment for the infrastructure sector is of fundamental
importance. Therefore, climate policymakers have a responsibility to create incentives to

6. promote green growth and encourage private sector investment in sustainable projects
(Shabbir & Wisdom, 2020). The growing importance of sustainable and environmental
investments in financial markets also has implications. Financial markets are responding
to the growing demand for low-carbon projects around the world to meet the challenges
of climate change. In fact, new financial tools have been developed with the aim of
directing capital to green projects, and are expanding with the gradual recognition of their
benefits. Mathematical optimization models can be used to optimize low carbon and
healthy climate resistant infrastructures (Arif et al., 2020). The following are some of the
most recent studies in the field of sustainable investment management.
Cesarone, Scozzari, and Tardella (2020) examined portfolio selection by
considering risk management criteria. In order to solve the problem, they presented a
hybrid approach based on simulation and optimization methods. In this approach, a
greedy classical single-discipline innovative algorithm is used that can produce
appropriate solutions. According to the numerical results, it has been observed that the
criteria related to risk management have a much greater impact on the final output than
the economic criteria. Castilho, Gama, Mundim, and de Carvalho (2019) proposed a
method based on classical mean-variance analysis using machine learning in order to
optimize the portfolio selection problem in stock exchange networks. Uncertain future
returns and PER ratios of each asset are approximated using fuzzy L-R numbers, as well
as budget, scope, and cardinality constraints. Galankashi, Rafiei, and Ghezelbash (2020)
used the fuzzy analytical network process method to evaluate and select a stock portfolio
on the Tehran Stock Exchange. First, a literature review was performed to determine the
main criteria for selecting the portfolio, and then a Likert questionnaire was used to
finalize the list of criteria. Final criteria were applied in the fuzzy analytical network
process to rank 10 portfolios. The results showed that profitability, growth, market and

7. risk are the most important criteria for portfolio selection. Vuković, Pivac, and Babić
(2020) compared portfolio selection using a combination of multi-criteria decision
making and modern portfolio theory that includes only stock market indices. The sample
under analysis includes 18 shares collected from the Croatian capital market from January
2017 to January 2019. The results show that there is a significant difference in stock
rankings. However, stocks that did not enter the portfolio in the selection of modern
portfolio theory were ranked lowest due to the MCDM hybrid approach, which confirms
that these stocks are the worst for investment. S. Rezaei and Vaez-Ghasemi (2020)
presented a hybrid model for selecting and planning the optimal stock combination
according to the goals and priorities to achieve the highest compatibility between the final
selection and the initial ranking of each share. The proposed model consists of three steps:
coverage analysis method (for initial stock revision), multi-criteria decision making
(TOPSIS) in conditions of uncertainty to evaluate and rank stocks in separate stages, and
categorized linear programming to choose the best portfolio by increasing the score
according to the priorities and limitations of the organization. Xu, Ren, Dong, and Yang
(2020) selected a portfolio of renewable energy desalination systems with a sustainable
perspective within a multi-criteria decision-making framework under data uncertainty. A
mathematical framework was proposed to deal with data uncertainty. A fuzzy network
analysis method was used to assign weight to related criteria. Finally, the rational ranking
of the alternatives was done. Stanković, Petrović, and Denčić-Mihajlov (2020) stated that
despite the widespread use of modern stock portfolio theory and Markowitz's approach
to optimization, based on quadratic planning and the distribution of return probabilities
as the main parameters, these approaches were criticized. The standard method of mean
variance has been modified using more appropriate risk criteria in the optimization
algorithm. The purpose of this paper is to show the efficiency of these models and also to

8. justify their application in stock portfolio management in the Belgrade Stock Exchange.
Yoshino, Taghizadeh-Hesary, and Otsuka (2021) examined the impact of the Covid virus
19 and the achievement of sustainability goals in the portfolio selection problem. This
paper theoretically shows that the current investors allocation by considering
sustainability goals based on different consulting firms will lead to a change in the
investment portfolio. The allocation of stocks can be done globally by taxing pollution
and waste such as CO2, NOx and plastics at the same tax rate, and the global pollution
tax will lead to the allocation of stocks. Guo and Ching (2021) examined the issue of
selecting a high-order Markov switching portfolio with a capital gains tax. In this paper,
capital gains-losses compensation is studied according to the effect of loss transportation.
Markov switching mechanism is offered in the market of different countries. The average
variance model of Markov switching order is made randomly. A particle swarm
optimization algorithm based on Monte Carlo simulation was proposed to solve the
problem. The table 1 investigates the most important studies in the field of research.
Table 1. Some of the most important related papers
Case
stud
y
Tool
Dimensions of sustainable
development
Mathematica
l
optimization
MCD
M
Other
s
Economica
l
Socia
l
Environmenta
l
✓
✓
✓
✓
✓
(
Peylo, 2012
)
✓
✓
✓
✓
(
Garcia-
Bernabeu,
Pla, Bravo, &
Perez-
Gladish,
2015
)
✓
✓
✓
✓
✓
✓
(
Bilbao-
Terol,
Jiménez-
López,
Arenas-Parra,
& Rodríguez-
Uría, 2018
)
✓
✓
(
Cui, Gao, &
Shi, 2019
✓
✓
✓
(
Cesarone,
Scozzari, &

9. Case
stud
y
Tool
Dimensions of sustainable
development
Mathematica
l
optimization
MCD
M
Other
s
Economica
l
Socia
l
Environmenta
l
Tardella,
2019
)
✓
✓
(
Castilho et
al., 2019
)
✓
✓
(
Cao, 2019
)
✓
✓
(
García,
González-
Bueno,
Oliver, &
Tamošiūnienė
, 2019
)
✓
✓
(
Wang, He,
& Shi, 2019
)
✓
✓
✓
(
Galankashi
et al., 2020
)
✓
✓
✓
✓
(
Vuković et
al., 2020
)
✓
✓
✓
✓
✓
✓
(
S. Rezaei &
Vaez-
Ghasemi,
2020
)
✓
✓
✓
✓
✓
(
Xu et al.,
2020
)
✓
✓
(
Guo &
Ching, 2021
)
✓
✓
✓
✓
(
Yoshino et
al., 2021
)
✓
✓
✓
✓
✓
✓
The current
study
Based on the conducted literature review, it can be seen that an integrated model
based on multi-criteria decision-making methods and mathematical optimization to use
sustainable development criteria and also criteria related to high financial distress to
provide the optimal composition of the portfolio has not been presented so far. However,
using the results of this research can lead to highly reliable solutions. The most important
innovations of this research are described below.
1- Providing a hybrid approach based on multi-criteria decision making and data-
based mathematical optimization
2- Considering the criteria of financial distress for the initial screening of companies
3- Using sustainable development criteria to provide final results

10. Materials and methods
The level of need to examine the problem of this research can be found in the turmoil in
the Iranian financial markets. At present, the use of the former analysis methods does not
meet the needs of investors to provide reliable answers. In other words, some numerical
analyses may now show a kind of grow in the future for a company, but what actually
happens is the opposite, and the directed investment in that company is virtually
eliminated. One of the main reasons is the consideration of some criteria such as the
dimensions of sustainable development in the future activity of companies and also the
non-consideration of new conditions in their field of work. Therefore, providing a suitable
approach to consider a wider range of information and criteria in order to obtain final
solutions can lead to high-reliability solutions. Some of the benefits of conducting this
research can be considered in providing highly reliable solutions to determine the share
of investment in different companies by considering the criteria of sustainable
development and paying attention to the financial distress of companies. In fact,
conducting this research will provide a broader view of decision-making criteria in this
area, as well as the use of new tools. In addition, the high flexibility of the proposed
approach can pave the way for its improvement and the introduction of more criteria and
sub-criteria.
The proposed framework of this research consists of two phases. In the first phase,
using a multi-criteria decision-making model, companies suitable for investment are
evaluated by considering the criteria related to financial distress, economic trends and,
employment conditions. In order to be able to do a proper screening, a mathematical
optimization model is developed to determine the volume of investment in each company
according to risk and return criteria so that the final outputs can be used as investment
management decisions by decision makers. In order to ensure the obtained answers, the
necessary sensitivities are analyzed so that in addition to examining the behavior of the

11. proposed framework in different situations, the managers can observe the final results in
different situations and make the best decision.
Phase 1: Multi-criteria decision model
In this phase, the criteria and sub-criteria are determined, as well as the evaluation of
companies listed on the stock exchange. The steps of this phase are as follows: first, the
criteria and sub-criteria related to the evaluation of companies are extracted from the
research literature and validated according to the opinion of experts. Then, using a
questionnaire completed by experts, pairwise comparisons were performed to perform
the best-worst method calculations to determine the weight of each criterion. After
determining the global weights of the criteria, using the option-criteria matrix provided
by the experts, the final prioritization of the companies is determined using the VIKOR
method.
The reason for choosing the proposed decision-making methods
In general, multi-criteria decision-making methods include two categories of criteria
weighting methods and prioritization methods. Weighting methods often include
Analytic hierarchy process (AHP), Analytic network process (ANP) and Best-Worst
Method (BWM) and of course the methods presented based on these three methods. AHP
and ANP methods, despite being widely used in various fields, require a lot of
calculations, but the number of calculations of ANP method is much more than AHP
method. However, the best-worst method has fewer calculations. If n criteria exist, the
number of AHP calculations is equal to n (n-1) / 2 and the number of best-worst method
calculations is equal to 2n-3 (J. Rezaei, 2015, 2016). Another important issue that
guarantees the superiority of the BWM method over other methods is the existence of a
mathematical model to determine the final weight of the criteria. In AHP and ANP

12. methods, all calculations are based on the preferences of experts, which can cause
disturbances in determining the final weight of the criteria and subconsciously affect the
final weighting. However, in the best-worst method using a mathematical model, there is
a guarantee that the final weights of the criteria are global optimal and far from the
personal preferences of the experts. Therefore, using the best-worst method can produce
more reliable solutions (J. Rezaei, 2016).
The best-worst method
The best-worst method is one of the powerful methods in solving MCDM problems that
is used to obtain the weights of alternatives and criteria (J. Rezaei, 2016). The structure
of the BWM method consists of the following steps:
Step 1. Create a decision benchmark system: The decision benchmark system
consists of a set of criteria identified by reviewing the literature and opinions of experts
and are considered as {𝑐1, 𝑐2, … , 𝑐𝑛}. Decision criteria values can reflect the performance
of different options.
Step 2. Determine the best and worst of the main criteria as well as the sub-criteria:
Based on the decision criteria system, the best and worst criteria should be identified by
decision makers. The best criterion is indicated by the symbol 𝑐𝐵 and the worst criterion
is represented by the symbol 𝑤𝐵.
Step 3. Perform reference comparisons for the best criterion: In this step, the
priority of the best criterion over other criteria is determined by using numbers between
1 and 9. The results of this vector are shown as follows:
1
𝐴𝐵 = (𝑎𝐵1, 𝑎𝐵2, … , 𝑎𝐵𝑛)
Where 𝑎𝐵𝑗 indicates the priority of the best selected criterion B over each criterion
j. It is clear that 𝑎𝐵𝐵 = 1.

13. Step 4. Perform reference comparisons for the worst criteria: Similarly, using
numbers between 1 and 9, the priority of all criteria over the worst selected criterion is
calculated. The results of this vector are shown as follows:
2
𝐴𝑤 = (𝑎1𝑊, 𝑎2𝑊, … , 𝑎𝑛𝑊)𝑇
Where 𝑎𝑗𝑊 indicates the priority of each criterion j over the worst chosen criterion
W. It is clear that 𝑎𝑊𝑊 = 1.
Step 5. Determine the optimal weights (𝑊1
∗
, 𝑊2
∗
, … , 𝑊
𝑛
∗
): In this step, in order to
achieve the optimal weights of the criteria, the absolute difference {|𝑤𝐵 − 𝑎𝐵𝑗𝑤𝑗|, |𝑤𝑗 −
𝑎𝑗𝑊𝑤𝑊|} must be minimized for all js, which is formulated as the following optimization
problem.
3
𝑚𝑖𝑛 𝑚𝑎𝑥
𝑗
{|𝑤𝐵 − 𝑎𝐵𝑗𝑤𝑗|, |𝑤𝑗 − 𝑎𝑗𝑊𝑤𝑊|}
𝑆. 𝑡.
∑ 𝑤𝑗 = 1
𝑗
𝑤𝑗 ≥ 0, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗
Problem (3) can be converted to the following model:
4
𝑚𝑖𝑛 𝜉𝐿
𝑆. 𝑡.
|𝑤𝐵 − 𝑎𝐵𝑗𝑤𝑗| ≤ 𝜉𝐿
, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗
|𝑤𝑗 − 𝑎𝑗𝑊𝑤𝑊| ≤ 𝜉𝐿
, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗
∑ 𝑤𝑗 = 1
𝑗
𝑤𝑗 ≥ 0, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗
Model (4) is linear and has a unique solution. Therefore, by solving this model,
the optimal weights (𝑤1
∗
, 𝑤2
∗
, … , 𝑤𝑛
∗
) and the optimal value 𝜉𝐿∗
are obtained. For the
above model, values close to zero 𝜉𝐿∗
indicate a high level of compatibility (J. Rezaei,
2016).
VIKOR method
The VIKOR technique is an adaptive ranking method and is often used in situations with
different conflicting criteria (Opricovic, 1998). This method creates a compromise

14. solution based on "proximity to the ideal solution and mutual agreement through
concessions". This method has been widely used by many researchers to rank options
(Gupta, 2018). The steps of the VIKOR method are as follows.
Step 1: Obtain a pairwise matrix for each alternative so that each criterion is
evaluated using a verbal scale.
Step 2: Calculate the mean decision matrix using Equation 5.
5
𝑓𝑖𝑗 =
1
𝑘
∑ 𝑥𝑖𝑗
𝑡
𝑘
𝑡=1
𝑖 = 1,2, … , 𝑚; 𝑗 = 1,2, … , 𝑛
Where 𝑥𝑖𝑗
𝑡
is the value of the i-th alternative relative to the j-th criterion by the t-
th expert.
Step 3: Calculate the values of the best 𝑓
𝑗
∗
and the worst 𝑓
𝑗
−
for all criteria using
Equations (6) and (7).
6
𝑓𝑗
∗
= 𝑚𝑎𝑥𝑓𝑖𝑗, 𝑖 = 1,2, … , 𝑚; 𝑗 = 1,2, … , 𝑛
7
𝑓𝑗
−
= 𝑚𝑖𝑛𝑓𝑖𝑗, 𝑖 = 1,2, … , 𝑚; 𝑗 = 1,2, … , 𝑛
Where 𝑓
𝑗
∗
represents the positive ideal solution and 𝑓
𝑗
−
represents the negative
ideal solution for the j-the criterion.
Step 4: Calculate the values of 𝑆𝑖 and 𝑅𝑖 for i = 1,2,…, m using equations (8) and
(9).
8
𝑆𝑖 = ∑ 𝑤𝑗
(𝑓𝑗
∗
− 𝑓𝑖𝑗)
(𝑓𝑗
∗
− 𝑓𝑗
−
)
𝑛
𝑗=1
9
𝑅𝑖 = 𝑚𝑎𝑥 [𝑤𝑗
(𝑓𝑗
∗
− 𝑓𝑖𝑗)
(𝑓𝑗
∗
− 𝑓𝑗
−
)
]
Where 𝑆𝑖 represents the distance of alternative i from the positive ideal solution
and 𝑅𝑖 represents the distance of alternative i from the solution of the negative ideal and
𝑤𝑗 represents the weights of the factors obtained through fuzzy BWM analysis.
Step 5: Calculate the value of 𝑄𝑖 based on Equation (10).
10
𝑄𝑖 = 𝑣 [
𝑆𝑖 − 𝑆∗
𝑆− − 𝑆∗
] + (1 − 𝑣) [
𝑅𝑖 − 𝑅∗
𝑅− − 𝑅∗
]

15. Where 𝑆−
= 𝑚𝑎𝑥
𝑖
𝑆𝑖, 𝑆∗
= 𝑚𝑖𝑛
𝑖
𝑆𝑖 , 𝑅−
= 𝑚𝑎𝑥
𝑖
𝑅𝑖, 𝑅∗
= 𝑚𝑖𝑛
𝑖
𝑅𝑖 . Parameter v
are also introduced as weight for the group utility maximum strategy, which in this study
is considered equal to 0.5.
Step 6: Rank alternatives using 𝑄𝑖 Values.
Step 7: The options are sorted based on the minimum values obtained 𝑄𝑖 so that
the following two conditions are met at the same time:
First condition (acceptance property): alternative 𝐴1
is selected 𝑄(𝐴2) −
𝑄(𝐴1
) ≥ 1/𝑚 − 1, such that 𝐴2
is the alternative that is in the second rank, and m is
equal to the total number of alternative.
Second condition (consistency of acceptance in decision making): Also 𝐴1
must
get the first rank based on the values of 𝑆𝑖 and / or 𝑅𝑖.
Step 8: The alternative with the lowest value in 𝑄𝑖 comes first.
Phase 2: Designing a multi-objective data-based optimization model
The data required for the present study include the risk and return on investment of
companies under study. To calculate these parameters, the historical data available on the
website of the Center for Research, Development and Islamic Studies of the Stock
Exchange and Securities Organization is used, in which information about the price are
adjusted by increasing capital for the surveyed companies daily. To calculate the return,
the daily return of the company is first measured and then it is assumed that the annual
return of the companies is equal to the average daily return, which is calculated using the
geometric average. The amount of risk is also calculated through the standard deviation
of returns in daily time intervals. The process of data analysis methods is data selection,
data cleaning and preparation, determining the objective function, selecting the portfolio
based on the mathematical model and examining the uncertainty conditions, respectively.
In general, the general structure of the research model can be presented as follows.

16. 𝑥𝑖 The amount of investment in the i-th company
𝑃𝑖 investment priority in each company
𝛽𝑖 Risk in the i-th company
𝑟𝑖 Return on i-th company
𝑈𝑖 The maximum amount of budget available for the i-th company
11
𝑀𝑖𝑛 ∑ 𝑃𝑖𝑥𝑖
𝑖=1
12
𝑀𝑖𝑛 ∑ 𝛽𝑖𝑥𝑖
𝑖=1
13
𝑀𝑎𝑥 ∑ 𝑟𝑖𝑥𝑖
𝑖=1
𝑠. 𝑡.
14
∑ 𝑥𝑖
𝑖=1
= 1
15
0 ≤ 𝑥𝑖 ≤ 𝑈𝑖
The first objective function minimizes VIKOR scores in selected companies for
investment. In other words, minimizing VIKOR scores is equivalent to maximizing the
priority of choice for investment. The second objective function minimizes the amount
of risk and the third objective function maximizes the rate of return. Constraint (14)
ensures that the total shares purchased from all companies in the market is equal to 1.
Constraint (15) also ensures that the percentage of stock purchases from each company
is limited to a predetermined amount. In order to deal with the uncertainty conditions of
risk and return parameters, the problem model in uncertainty conditions is formulated
using a data-based approach.
In the data-based approach, the probability of G distribution is unknown and a set
of G-related historical data is available. It is assumed that G is observed only by a finite
set of samples and historical data. It is defined as {𝜉1
, 𝜉2
, … , 𝜉|𝑁|
}, in which N represents
the number of samples. In the proposed framework, historical examples are considered

17. as discrete scenarios of the problem, but the probability of occurrence of scenarios is
unknown.
In order to estimate the probability of occurrence of each of the scenarios denoted
by 𝜋𝑛 (n = 1,2,…, | N |), an ambiguous set D is presented to ensure that the correct
distribution of G is within this set as follows.
𝐷:
{
∑ 𝜋𝑛
𝑛=𝑁
= 1,
𝜉 ≤ ∑ 𝜋𝑛𝜉𝑛
𝑛=𝑁
≤ 𝜉
∑ (𝜋𝑛 (∑ 𝜋𝑛𝜉𝑛 − 𝜉𝑛
𝑛=𝑁
)
2
) ≤ 𝜔
𝑛=𝑁
𝜋𝑛 ≥ 0 ∀𝑛 ∈ 𝑁 }
In set 𝐷, the first constraint ensures that the probability set of the observed
samples is 1. The second constraint guarantees that the mean value of the uncertain
parameter 𝜉 is between predefined lower and upper bounds. The third constraint ensures
that the variance of the observed samples is limited to a predetermined upper bound. The
fourth constraint guarantees that value of samples’ probability is nonnegative.
In order to reduce the calculations and obtain control parameters, 𝜇̂ =
∑ 𝜉𝑛
𝑛=𝑁
|𝑁|
is
considered as an estimate of the mean value of the samples and the ambiguous set 𝐷 is
approximated as equation (4) (Fattahi, 2020).
𝐷
̅:
{
∑ 𝜋𝑛
𝑛=𝑁
= 1,
|∑ 𝜋𝑛𝜉𝑛
𝑛=𝑁
− 𝜇̂| ≤ 𝜌
∑(𝜋𝑛(𝜉𝑛 − 𝜇̂)2)
𝑛=𝑁
≤ 𝜔
𝜋𝑛 ≥ 0 ∀𝑛 ∈ 𝑁 }
,
Assuming that the stochastic parameters follow normal distribution function, the
sample mean follows the 𝑡 distribution with | 𝑁 | − 1 degrees of freedom, the mean 𝜇̂,

18. and the standard deviation
𝑠
√𝑁
where 𝑠2
=
∑ (𝜉𝑛−𝜇
̂)2
𝑛=𝑁
|𝑁|−1
. By setting 𝜌 as 𝑡𝛼
2
,|𝑁|−1, the value
of ∑ 𝜋𝑛𝜉𝑛
𝑛=𝑁 is in the confidence interval (1 − 𝛼)% for the true value of 𝐺 mean based
on the second constraint of 𝐷
̅. In addition, by setting 𝜔 as
(|𝑁|−1)𝑠2
𝜒1−𝛽,|𝑁|−1
2 where 𝜒1−𝛽,|𝑁|−1
2
is
Chi-square distribution with | 𝑁 | − 1 degrees of freedom. The true variance of 𝐺 with
confidence (1 − 𝛽)% will be less than 𝜔. Finally, based on the third constraint of 𝐷
̅, the
approximated variance of samples, ∑ (𝜋𝑛(𝜉𝑛 − 𝜇̂)2)
𝑛=𝑁 , will be limited to 𝜔. By learning
from historical data, a predetermined set of 𝐷
̅ is made for the true distribution, and final
decisions are made based on the fact that the true distribution can vary in the confidence
set. In the equation (5), the worst case of distribution is minimized in 𝐷
̅ so that robust
results are obtained.
min
𝑥∈𝑋
{𝑐𝑇
𝑥 + max
𝐺∈𝐷
̅
𝐸𝐺[𝑍(𝑥, 𝜉)]} .
Solution method
There is a set of solutions, named optimal pareto front solutions, instead of a single
optimal solution in multi-objective problems. To achieve optimal pareto front, different
methods have been proposed. The augmented epsilon constrained method is proved as an
efficient method among other existing methods. Let assume the general structure of a
multi objective model is as follows.
max (𝑓1(𝑥), 𝑓2(𝑥), … , 𝑓
𝑝(𝑥))
𝑠. 𝑡
𝑥𝜖𝑆
where 𝑥 is decision variable vector, f1(x), f2(x), … , fp(x) are objective functions,
and 𝑆 is the feasible space. According to the proposed structure by Mavrotas and Florios
(2013), model (38) is converted to model (39) to achieve optimal pareto front solutions.

19. max ( 𝑓1(𝑥) + 𝑒𝑝𝑠 × (
𝑠2
𝑟2
+ 10−1
×
𝑠3
𝑟3
+ ⋯ + 10−(𝑝−2)
×
𝑠𝑝
𝑟𝑝
))
𝑠. 𝑡
𝑓2(𝑥) − 𝑠2 = 𝑒2
𝑓3(𝑥) − 𝑠3 = 𝑒3
⋮
𝑓𝑝(𝑥) − 𝑠𝑝 = 𝑒𝑝
𝑥𝜖𝑆
𝑠𝑖𝜖𝑅
where 𝑒2, 𝑒3, … , 𝑒𝑝 are right hand side values, 𝑟2, 𝑟3, … , 𝑟𝑝 are the objective
functions domain, 𝑠2, 𝑠3, … , 𝑠𝑝 are auxiliary variables of constraints. 𝑒𝑝𝑠 is within
[10−6
, 10−3]. Figure 1 shows steps of implementing augmented epsilon constraint
method based on (Mavrotas & Florios, 2013).
Figure 1: flow chart of augmented eps-constraint method
Where:

20. 𝑒𝑘 = 𝑙𝑏𝑘 + 𝑖𝑘 × 𝑠𝑡𝑒𝑝𝑘
𝑙𝑏𝑘: lower bound for objective function k
𝑠𝑡𝑒𝑝𝑘 =
𝑟𝑘
𝑔𝑘
: step for the objective function k
𝑔𝑘: number of intervals for objective function k
𝑛𝑝: number of Pareto optimal solutions
Computational results
In this section, the numerical results obtained from solving the research problem are
presented, which is a combination of decision-making levels in order to evaluate the
companies under study and also to optimize the selected portfolio for investment. For this
purpose, first the numerical results obtained from solving the best-worst method and
VIKOR method as a decision-making phase are presented and then the computational
results in the optimization phase are examined. In this phase, there are two parts, which
first include solving the mathematical model in certain terms and then solving the
problem in terms of uncertainty using a data-based approach.
Introduction of the studied companies
Due to the nature of the research, this chapter uses the data provided by a limited group
of companies active in Iran capital market, which are mainly active in the field of energy
(industries related to petrochemical and oil and gas). After conducting the necessary
studies, Shepna, Shatran, Shavan, Shabriz, Shebandar, Shabhran, Shabas, Shiraz, Shanrel,
Shespa, Shenft and Oil companies were selected as companies to invest. It is noted that
in order to solve the proposed research approach, the financial information of these
companies, which is available in Excel file as an attachment to this research, will be used.

21. Numerical results of multi-criteria decision making
Due to the structure of the best-worst method, it is necessary to perform reduced pairwise
comparisons for each of the main criteria and sub-criteria. In fact, it is enough for each of
the criteria and their sub-criteria, the most important and least important criteria are
selected separately and then pairwise comparisons between other criteria with the best
and worst criteria are done. But the important issue is to introduce the criteria and their
sub-criteria, which are described in the table 2.
Table 2- Evaluation criteria and sub-criteria of the studied companies
Sub criteria
Criterion
Current ratio
Liquidity
Instant ratio
Liquidity ratio
Cash adequacy ratio
Cash flow ratio
Net working capital
Net profit to sell
Profitability
Gross profit to sell
Operating profit margin
Special profit to sell
Profit to special benefit
Return on assets ROA
Percentage of return on capital
ROE return on investment
Return on working capital
Constant returns
Return on sales (operating profit / net sales)
Operating profit or profit before interest and taxes
Net sales (operating income)
P / E
Market P / B
Stock prices on the last day of the year
Export
Export
Since small quantities are available for all criteria and sub-criteria, there will be
no need to use the expert community and form a decision board. In fact, by quantitative
analysis of the values associated with each of the criteria and sub-criteria, the scoring
pattern can be applied accurately. But as mentioned, small values related to each of the
criteria and indicators need to be available for each of the companies under study. This

22. set of criteria is available through the analysis of financial information of companies as
well as data available on the Cadal site. Appendix A provides the relevant data. Finally,
after performing the necessary calculations, the final weight of the criteria and sub-criteria
is presented as table 3.
Table 3- Final weight of criteria and sub-criteria using the best-worst method
Criterion
Local
weight
Indicator
Local
weight
Overall
weight
Score
Liquidity 0.4
Current ratio 0.149 0.0596 3
Instant ratio 0.112 0.0448 4
Liquidity ratio 0.149 0.0596 3
Cash adequacy ratio 0.401 0.1604 1
Cash flow ratio 0.149 0.0596 3
Net working capital 0.039 0.0156 8
Profitability 0.2
Net profit to sell 0.07 0.014 9
Gross profit to sell 0.07 0.014 9
Operating profit margin 0.07 0.014 9
Special profit to sell 0.07 0.014 9
Profit to special benefit 0.106 0.0212 7
Return on assets ROA 0.106 0.0212 7
Percentage of return on capital 0.035 0.007 10
ROE return on investment 0.176 0.0352 6
Return on working capital 0.07 0.014 9
Constant returns 0.106 0.0212 7
Return on sales (operating
profit / net sales)
0.07 0.014 9
Operating profit or profit before
interest and taxes
0.02 0.004 12
Net sales (operating income) 0.03 0.006 11
Market 0.2
P / E 0.5 0.1 2
P / B 0.25 0.1 2
Stock prices on the last day of
the year
0.25 0.1 2
Export 0.2 Export 0.2 0.04 5
As can be seen, the criterion of cash adequacy ratio with a score (0.1604) is the
most important. Also, P / E, P / B and stock prices are ranked second on the last day of
the year with a value of (0.1). But the criterion of operating profit or profit before interest
and tax with a score (0.004) is the least important. Using these values, the final ranking
of companies can be obtained using the VIKOR method.

23. Prioritization of companies
In this research, VIKOR method is used in order to prioritize companies. The input data
of VIKOR method include determining the weight of criteria and sub-criteria obtained in
the previous section. There is also a need to use an alternative-criteria matrix, which
includes the importance of each criterion for each company in the numerical range of 1
to 5. It is clear that number 1 is the least important and number 5 is the most important.
Similar to scoring the best-worst method, in the VIKOR method, numerical values of all
criteria are available to all companies, so accurate scores can be made. For this purpose,
the quantification interval of each criterion for each company is examined based on 5
divisions, so that numbers 1 to 5 for each criterion can be assigned to the companies. The
table 4 provides the final prioritization of options.
Table 4- Prioritization of companies based on different values of parameter v
V=0 V=0.1 V=0.2 V=0.3 V=0.4 V=0.5 V=0.6 V=0.7 V=0.8 V=0.9 Variance
Shapna 0.4 0.43 0.45 0.48 0.51 0.54 0.56 0.59 0.62 0.64 0.07719503
Shetran 1 1 1 1 1 1 1 1 1 1 0
Shavan 0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.05 0.01500445
Shabriz 0.4 0.4 0.39 0.39 0.39 0.39 0.38 0.38 0.38 0.37 0.00897341
Shabendar 0.4 0.43 0.46 0.49 0.53 0.56 0.59 0.62 0.65 0.68 0.08971303
Shabehran 0.4 0.42 0.44 0.46 0.48 0.5 0.51 0.53 0.55 0.57 0.05421678
Shebas 0.4 0.42 0.44 0.46 0.49 0.51 0.53 0.55 0.57 0.59 0.06081874
Sheraz 0 0 0 0 0 0 0 0 0 0 0
Sheranel 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.05781785
Shaspa 0.4 0.45 0.51 0.56 0.61 0.66 0.71 0.77 0.82 0.87 0.14973085
Shenaft 0.15 0.18 0.21 0.24 0.27 0.3 0.33 0.36 0.39 0.42 0.08447032
Venaft 0.4 0.38 0.35 0.33 0.3 0.28 0.25 0.23 0.2 0.18 0.07096325
As can be seen, Shiraz has a zero number in all values of v, which indicates the
highest priority for investment. Shatran Company also has the lowest priority for
investment. But the diagram of changes in companies' priorities based on different values
of v is as figure 2.

24. Figure 2- Analysis of changes in companies' preferences for different values v
The trend of change of priority or the same variance of VIKOR values for different
numbers v is also presented in the figure 3.
Figure 3 - The trend of changes in variance of VIKOR for each company
In a general conclusion, it can be said that Shiraz, Shavan, Shenaft and Venaft
companies have a high priority for investment. However, due to the multi-objective nature
of the mathematical model, it is not possible to be satisfied only with the priority provided
by the best-worst method. Therefore, in the following, the optimal amount of investment
in all companies will be determined.
0.00
0.20
0.40
0.60
0.80
1.00
1.20
V=0 V=0.1 V=0.2 V=0.3 V=0.4 V=0.5 V=0.6 V=0.7 V=0.8 V=0.9
VIKOR
score
Shapna Shetran Shavan Shabriz Shabendar Shaberhan
Shebas Sheraz Sheranel Shesta Shenaft Venaft
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Variance

25. Numerical results of multi-objective optimization
In this section, the numerical results obtained by solving the multi-objective optimization
model in both certain and uncertain modes are presented. It is noteworthy that in the case
of certainty, the priority selection of each company based on the value of v = 0.5 is
considered in the VIKOR method. But in the case of uncertainty, numerical scenarios are
considered for different values of v. For example, if the priority of companies for
investing is calculated based on 10 different values of v, we will have 10 scenarios in the
mathematical model.
Numerical results in certainty mode
As mentioned earlier, first the mathematical model is solved in the condition of certainty
and without numerical scenarios to determine the investment priority and the necessary
analysis is performed. For this purpose, risk and return values should be presented for
each company as an input parameter. Based on field findings, the values of risk and return
for each of the twelve companies are as table 5.
Table 5 - Risk and return of the studied companies
Shapna
Shetran
Shavan
Shabriz
Shabendar
Shabehran
Return
3.18725
1.3862
0.93049
1.20123
2.17486
1.40096
Risk
1.79923
0.06281
0.66794
0.24162
0.9404
0.1764
Shebas
Sheraz
Sheranel
Shaspa
Shenaft
Venaft
Return
0.44227
1.38391
1.7352
1.18769
1.4217
1.02604
Risk
1.8533
0.30309
0.44753
0.1609
0.70002
0.24535
The priority value for investment is also available in the table 6, which is in fact
the same as VIKOR coefficient (v = 0.5).
Table 6- Investment priority in the studied companies (smaller amounts have a
higher priority)
Shapna
Shetran
Shavan
Shabriz
Shabendar
Shabehran
Return
0.535172945
1
0.026119403
0.385172945
0.55696399
0.495172945
Shebas
Sheraz
Sheranel
Shaspa
Shenaft
Venaft
Return
0.506665482
0
0.501441602
0.661441602
0.299821725
0.277262497

26. According to previous research in this field, the maximum amount of investment
in each company is equal to 15% of the total available capital. According to this
information, the problem model can be solved optimally and the Pareto front can be
presented as figure 4.
Figure 4- Pareto front resulting from the certain solution
It is clear that the Pareto solutions have a non-dominated structure and they are
distributed throughout the justified space. In fact, the performance of the Epsilon
constraint method in solving the problem is acceptable. But the important point in
providing the final solution is the selection of one of the members of the Pareto front,
which of course is always considered as a management problem. In a clearer explanation,
unlike single-objective optimization problems, where only one global optimal solution is
generated, in multi-objective problems, several optimal solutions are generated at the
Pareto front, each of which can be used as the final solution. But which one to choose as
the solution that can be provided to the users is a fundamental question. In the research
literature, this choice is often made randomly, which is certainly not the best way to make
a decision. Of course, there are other methods based on Monte Carlo simulation, each of
which requires a long solution time and a certain level of computational error. In this

27. research, a new approach based on distance from the ideal solution is presented in order
to provide the best possible solution. Given that the ideal point is a hypothetical solution
in which all the objective functions are at their best at the same time. It is clear that such
a solution does not really exist and it is only used for measuring the quality of each Pareto
member.
Selection of the best performing Pareto member
In this method, first the mathematical model is solved for each of the objective functions
and the optimal value of the objective functions is calculated separately. Then the
Euclidean distance of each Pareto member to the ideal point is calculated and the Pareto
member with the shortest distance to the idea point is selected as the final solution. The
steps of this method include the following.
Step 1: Solve the mathematical model for each of the objective functions separately and
store the optimal values in 𝑍1
∗
،𝑍2
∗
‫و‬ 𝑍3
∗
Step 2: Solve the mathematical model using the Epsilon constraint method and store the
solutions in the optimal set 𝑃𝑆∗
.
Step 3: Calculate the Euclidean distance of the members of the set 𝑃𝑆∗
with (𝑍1
∗
, 𝑍2
∗
, 𝑍3
∗)
and produce the MID set.
Step 4: Select the Pareto member with the lowest MID value as the final solution.
The following equation for calculating MID is given below.
𝑖 ∈ 𝑃𝑆∗
𝑀𝐼𝐷𝑖 = √∑(𝑍𝑗
∗
− 𝑍𝑖𝑗)
2
𝑛𝑜𝑏𝑗
𝑗=1

28. Where 𝑛𝑜𝑏𝑗 equals the number of objective functions and 𝑍𝑖𝑗 equals the value of
the j function for the i parto member. Based on this relationship, a member of the party
with the highest efficiency can be selected.
After performing the calculations and determining the Euclidean distance of each
Pareto solution from the ideal point, the resulting values are as table 7.
Table 7. Distance of each Pareto member from the ideal point
Pareto
member
Ideal
distance
Pareto
member
Ideal
distance
Pareto
member
Ideal
distance
Pareto
member
Ideal
distance
Pareto
member
Ideal
distance
1 0.0265 41 0.0222 81 0.0165 121 0.0142 161 0.0173
2 0.0242 42 0.027 82 0.018 122 0.0161 162 0.0145
3 0.0307 43 0.0283 83 0.0146 123 0.0131 163 0.0166
4 0.0326 44 0.0298 84 0.0161 124 0.015 164 0.0189
5 0.0278 45 0.0241 85 0.0144 125 0.0171 165 0.0161
6 0.0297 46 0.0254 86 0.0161 126 0.0194 166 0.0184
7 0.0251 47 0.0269 87 0.0129 127 0.0162 167 0.0209
8 0.027 48 0.0227 88 0.0146 128 0.0185 168 0.0181
9 0.0291 49 0.0242 89 0.0165 129 0.021 169 0.0206
10 0.0245 50 0.0202 90 0.0152 130 0.0178 170 0.0233
11 0.0266 51 0.0217 91 0.0173 131 0.0203 171 0.0232
12 0.0222 52 0.0179 92 0.0196 132 0.023 172 0.0274
13 0.0243 53 0.0194 93 0.0162 133 0.0198 173 0.0275
14 0.0266 54 0.0173 94 0.0185 134 0.0225 174 0.0278
15 0.0201 55 0.019 95 0.021 135 0.0265 175 0.0245
16 0.0222 56 0.0154 96 0.0176 136 0.027 176 0.0246
17 0.0245 57 0.0171 97 0.0201 137 0.0236 177 0.0249
18 0.0203 58 0.019 98 0.0228 138 0.0241 178 0.0219
19 0.0226 59 0.0154 99 0.0259 139 0.0209 179 0.0222
20 0.0293 60 0.0173 100 0.0266 140 0.0214 180 0.0197
21 0.031 61 0.0194 101 0.0237 141 0.0189 181 0.0202
22 0.0264 62 0.0158 102 0.0246 142 0.0196 182 0.0174
23 0.0281 63 0.0179 103 0.021 143 0.0166 183 0.0179
24 0.0237 64 0.0202 104 0.0219 144 0.0173 184 0.0186
25 0.0254 65 0.0166 105 0.0185 145 0.0182 185 0.0158
26 0.0212 66 0.0189 106 0.0194 146 0.0152 186 0.0165
27 0.0229 67 0.0214 107 0.0162 147 0.0161 187 0.0146
28 0.0189 68 0.0203 108 0.0171 148 0.0142 188 0.0155
29 0.0206 69 0.0266 109 0.0182 149 0.0153 189 0.0129
30 0.0225 70 0.0277 110 0.015 150 0.0136 190 0.0138
31 0.0185 71 0.0237 111 0.0161 151 0.0149 191 0.0123
32 0.0204 72 0.0248 112 0.0174 152 0.0121 192 0.0134
33 0.0225 73 0.021 113 0.0142 153 0.0134 193 0.0121
34 0.0185 74 0.0221 114 0.0155 154 0.0121 194 0.0134
35 0.0206 75 0.0185 115 0.0138 155 0.0136 195 0.0123
36 0.0229 76 0.0196 116 0.0153 156 0.0125 196 0.0138
37 0.0189 77 0.0209 117 0.0123 157 0.0142 197 0.0129
38 0.0212 78 0.0173 118 0.0138 158 0.0161 198 0.0146
39 0.0237 79 0.0186 119 0.0155 159 0.0133 199 0.0139

29. Pareto
member
Ideal
distance
Pareto
member
Ideal
distance
Pareto
member
Ideal
distance
Pareto
member
Ideal
distance
Pareto
member
Ideal
distance
40 0.0197 80 0.0152 120 0.0125 160 0.0152 200 0.0158
According to the calculations, Pareto members 152, 154 and 193 have the shortest
distance from the ideal point (0.0121) and therefore each of them can be provided to the
decision makers as a final solution. The information of each of these selected Parthian
members is described in table 8.
Table 8. Values of the objective functions of each of the selected Pareto members
Priority
return
risk
Member 152
0.28
1.35
0.43
Member 154
0.28
1.37
0.43
Member 193
0.29
1.37
0.42
It can be seen that the values of the first to third objective functions for each of
the selected members are very similar to each other. In the end, the optimal investment
structure in each company according to the values of the objective functions is as table 9.
Table 9 - Investment amount of each Pareto member for numbering
Member 152 Member 154 Member 193
Shapna 0.06 0 0.04
Shetran 0.04 0 0
Shavan 0 0 0.15
Shabriz 0.15 0.15 0.15
Shabendar 0.15 0.021 0
Shabehran 0 0.15 0.062
Shebas 0 0
Sheraz 0.15 0.15 0.15
Sheranel 0.15 0.15 0.15
Shaspa 0.0003 0.117 0
Shenaft 0 0.112 0.148
Venaft 0.15 0.15 0.15
As can be seen, the composition of company investments in each Pareto member
are different. But choosing any of them will have similar implications for investors. The
figure 5 provides a graphical comparison of the investment portion of each Pareto
member.

30. Figure 5 - Comparison of the share of capital in each company
Numerical results under uncertainty
In this section, the numerical results of the proposed model under uncertainty are
examined. It should be noted that the uncertainty structure presented in this study is based
on scenarios and the previous data. It is clear that risk and return data is calculated based
on recorded financial information and therefore have no uncertainty. However,
determining the amount of investment priority in each company is based on the
calculations of the VIKOR method, which is taken from the historical data of the
companies. This issue was previously examined in the computational phase of the multi-
criteria decision method. Therefore, it can be said that numerical definition scenarios are
indirectly based on historical data and therefore have a good level of functionality in
solving mathematical models. In order to define the scenarios, the changes made in the
prioritization of companies based on the coefficient v in the VIKOR model are used. The
reason for this uncertainty is that determining the exact value for v has always been a
managerial issue and no precise scientific structure has been defined for it in previous
research. Therefore, in this study, different scenarios are defined based on the levels of
change v. In order to analyze the results, first 10 different scenarios including values of v
= 0 to v = 0.9 with an incremental step of 0.1 are considered and then the mathematical
0
0.05
0.1
0.15
0.2
Member 152 Member 154 Member 193
investment
percentage
Shapna Shetran Shavan Shabriz Shabendar Shabehran
Shebas Sheraz Sheranel Shaspa Shenaft Venaft

31. model is solved based on it. It should be noted that the occurrence probability of each
scenario is presented in Table 10.
Table 10 - occurrence probability of each scenario
Scenari
o 1
Scenari
o 2
Scenari
o 3
Scenari
o 4
Scenari
o 5
Scenari
o 6
Scenari
o 7
Scenari
o 8
Scenari
o 9
Scenari
o 10
0.15 0.05 0.16 0.04 0.12 0.08 0.09 0.11 0.15 0.05
The numerical value of the investment priority based on each of the scenarios is
as table 11.
Table 11 - The amount of investment priority in companies based on 10 scenarios
V=0 V=0.1 V=0.2 V=0.3 V=0.4 V=0.5 V=0.6 V=0.7 V=0.8 V=0.9
Shapna 0.4 0.43 0.45 0.48 0.51 0.54 0.56 0.59 0.62 0.64
Shetran 1 1 1 1 1 1 1 1 1 1
Shavan 0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.05
Shabriz 0.4 0.4 0.39 0.39 0.39 0.39 0.38 0.38 0.38 0.37
Shabendar 0.4 0.43 0.46 0.49 0.53 0.56 0.59 0.62 0.65 0.68
Shabehran 0.4 0.42 0.44 0.46 0.48 0.5 0.51 0.53 0.55 0.57
Shebas 0.4 0.42 0.44 0.46 0.49 0.51 0.53 0.55 0.57 0.59
Sheraz 0 0 0 0 0 0 0 0 0 0
Sheranel 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58
Shaspa 0.4 0.45 0.51 0.56 0.61 0.66 0.71 0.77 0.82 0.87
Shenaft 0.15 0.18 0.21 0.24 0.27 0.3 0.33 0.36 0.39 0.42
Venaft 0.4 0.38 0.35 0.33 0.3 0.28 0.25 0.23 0.2 0.18
After solving the model using numerical data, the Pareto front is presented as
figure 6.
Figure 6 - Pareto front in the presence of ten numerical scenarios

32. The order on the Pareto front reflects a non-dominated structure. The boundaries
of the Pareto front are also shown in the table 12.
Table 12 - Border points of the Pareto front
Risk Return Priority
Max Min Max Min Max Min
0.29 0.22 1.43 1.27 0.49 0.35
Similar to the analysis performed in the certain case to find the top member in
order to provide a solution to investors, the table 13 is presented.
Table 13 - Distance from the ideal point for Pareto members in case of ten scenarios
Pareto
member
number
Distance
from
the
ideal
point
Pareto
member
number
Distance
from
the
ideal
point
Pareto
member
number
Distance
from
the
ideal
point
Pareto
member
number
Distance
from
the
ideal
point
1 0.0265 51 0.0283 101 0.0133 151 0.0301
2 0.0288 52 0.0298 102 0.0152 152 0.0265
3 0.0265 53 0.0241 103 0.0173 153 0.027
4 0.0338 54 0.0254 104 0.0141 154 0.0236
5 0.0307 55 0.0269 105 0.0162 155 0.0241
6 0.0326 56 0.0227 106 0.0185 156 0.0209
7 0.0297 57 0.0242 107 0.021 157 0.0214
8 0.027 58 0.0202 108 0.0176 158 0.0221
9 0.0291 59 0.0217 109 0.0201 159 0.0189
10 0.0245 60 0.0179 110 0.0169 160 0.0196
11 0.0266 61 0.0194 111 0.0194 161 0.0166
12 0.0222 62 0.0173 112 0.029 162 0.0173
13 0.0243 63 0.019 113 0.0297 163 0.0152
14 0.0266 64 0.0154 114 0.0259 164 0.0161
15 0.0201 65 0.0171 115 0.0266 165 0.0142
16 0.0222 66 0.019 116 0.0275 166 0.0153
17 0.0245 67 0.0154 117 0.023 167 0.0125
18 0.0203 68 0.0173 118 0.0237 168 0.0136
19 0.0226 69 0.0158 119 0.0246 169 0.0121
20 0.0251 70 0.0179 120 0.021 170 0.0134
21 0.0324 71 0.0202 121 0.0219 171 0.0121
22 0.0341 72 0.0166 122 0.0185 172 0.0136
23 0.0293 73 0.0189 123 0.0194 173 0.011
24 0.031 74 0.0214 124 0.0162 174 0.0125
25 0.0264 75 0.0178 125 0.0171 175 0.0142
26 0.0281 76 0.0203 126 0.0182 176 0.0116
27 0.0237 77 0.0297 127 0.015 177 0.0133
28 0.0254 78 0.0308 128 0.0161 178 0.0152
29 0.0273 79 0.0266 129 0.0142 179 0.0126
30 0.0212 80 0.0277 130 0.0155 180 0.0145
31 0.0229 81 0.0237 131 0.0125 181 0.0166
32 0.0248 82 0.0248 132 0.0138 182 0.014
33 0.0206 83 0.021 133 0.0153 183 0.0161
34 0.0225 84 0.0221 134 0.0123 184 0.0184
35 0.0185 85 0.0234 135 0.0138 185 0.0158

33. Pareto
member
number
Distance
from
the
ideal
point
Pareto
member
number
Distance
from
the
ideal
point
Pareto
member
number
Distance
from
the
ideal
point
Pareto
member
number
Distance
from
the
ideal
point
36 0.0204 86 0.0185 136 0.0125 186 0.0181
37 0.0225 87 0.0196 137 0.0142 187 0.0206
38 0.0185 88 0.0209 138 0.0161 188 0.0233
39 0.0206 89 0.0173 139 0.0131 189 0.0205
40 0.0229 90 0.0186 140 0.015 190 0.0232
41 0.0189 91 0.0152 141 0.0171 191 0.0305
42 0.0212 92 0.0165 142 0.0141 192 0.0306
43 0.0237 93 0.018 143 0.0162 193 0.0274
44 0.0174 94 0.0146 144 0.0134 194 0.0275
45 0.0197 95 0.0161 145 0.0155 195 0.0278
46 0.0222 96 0.0144 146 0.0178 196 0.0246
47 0.0301 97 0.0161 147 0.0203 197 0.0249
48 0.0314 98 0.0129 148 0.0173 198 0.0219
49 0.0329 99 0.0146 149 0.0198 199 0.0222
50 0.027 100 0.0165 150 0.0296 200 0.0197
It is clear that only member 173 has the least distance from the ideal point andit
is chosen as the final solution. The optimal amount of investment of this Pareto member
is as table 14.
Table 14 - The optimal amount of investment in each of the ten scenarios
Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5
Shapna 0.047 0.1
Shavan 0.109 0.15 0.15 0.15 0.15
Shabriz 0.15 0.15 0.15
Shabendar 0.15 0.15 0.1
Shabehran 0.15 0.15 0.15 0.15 0.15
Sheraz 0.15 0.15 0.15 0.15 0.15
Shernal 0.15 0.15 0.15 0.15 0.15
Shasta 0.094
Shenaft 0.15 0.15 0.15 0.15 0.1
Venaft 0.1 0.15
Scenario 6 Scenario 7 Scenario 8 Scenario 9 Scenario 10
Shavan 0.15 0.15 0.15 0.15 0.15
Shabriz 0.15 0.15 0.15 0.15 0.15
Shabehran 0.15 0.15 0.15 0.15 0.15
Sheraz 0.15 0.15 0.15 0.15 0.15
Shernal 0.15 0.15 0.15 0.15 0.15
Shenaft 0.1 0.1 0.1 0.1 0.1
Venaft 0.15 0.15 0.15 0.15 0.15
It can be seen that in some scenarios, the amount of investment in companies is
equal to each other. The reason for this can be attributed to the proximity of the level of
prioritization for investing in any company. However, the impact of different scenarios
on corporate investment is not negligible and creates the need for investors to pay

34. attention to this fact. In order to more accurately investigate the effect of increasing
scenarios on the quality of solutions, the value of MID based on changes in the number
of scenarios is presented as table 15.
Table 15- Comparison of the results of the sensitivity analysis of the mathematical
model
MID
Number of
scenarios
MID
Number of
scenarios
MID
Number of
scenarios
MID
Number of
scenarios
1.6596
160
1.9102
110
2.0176
60
2.034
10
1.5574
170
1.83
120
2.0046
70
2.0251
20
1.5457
180
1.8282
130
2.0037
80
2.0215
30
1.5247
190
1.76
140
2.0024
90
2.0207
40
1.5155
200
1.7512
150
1.9379
100
2.0195
50
It is clear that as the number of scenarios increases, the MID value decreases,
which indicates that the numerical results are better. In fact, as the number of scenarios
increases, the model is able to provide solutions closer to the ideal point and thus the best
results are provided. The figure 7 shows the process of change.
Figure 7- The trend of MID changes based on the number of scenarios
Conclusion and future suggestions
In this research, a multi-criteria decision-making model is developed for the initial
evaluation of companies suitable for investment by considering criteria related to
financial distress, economic trends and employment conditions in order to perform
1.5
1.6
1.7
1.8
1.9
2
2.1
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
MID
number of senarios

35. appropriate screening. Then a mathematical optimization model is developed to
determine the volume of investment in each company according to risk and return criteria
so that the final outputs can be provided to users as investment management decisions.
The most important unknown part of this research is determining the appropriate method
for screening potential companies for investment. According to studies in the research
literature, this issue has not been discussed so far and it is therefore considered as a
significant innovation in this research. In this study, the answer to this question is
discussed through the development of a multi-criteria decision model based on indicators
of financial distress and sustainable development.
According to the numerical results, it can be seen that the criterion of cash
adequacy ratio with a score (0.1604) is the most important. Also, P / E, P / B and stock
prices are ranked second with a value of (0.1) on the last day of the year. But the criterion
of operating profit or profit before interest and tax with a score (0.004) is the least
important. Using these values, the final ranking of companies can be obtained using the
VIKOR method. In the alternative prioritization section, it is concluded that Shraz,
Shavan, Shenft and Vanft companies have a high priority for investment. However, due
to the multi-objective nature of the mathematical model, it is not possible to be satisfied
only with the priority provided by the best-worst method. In solving the mathematical
model in uncertain conditions, it is observed that the Pareto members 152, 154 and 193
have the smallest distance from the ideal point (0.0121) and therefore each of them can
be used as the final solution to the decision.
The corporate investment combination is different for each Pareto member. But
choosing any of them will have similar implications for investors. In solving the problem
under uncertainty, it is clear that the data related to risk and return is calculated based on
recorded financial information with no uncertainty. However, determining the amount of

36. investment priority in each company is based on the calculations of the VIKOR method,
which is taken from the historical data of the companies. This issue was previously
examined in the computational phase of the multi-criteria decision method. Therefore, it
can be said that numerical definition scenarios are indirectly based on historical data and
therefore have a good level of functionality in solving mathematical models.
In order to define the scenarios, changes in the prioritization of companies based
on the coefficient v in the VIKOR model have been used. The reason for this uncertainty
is that determining the exact value for v has always been a managerial issue and no precise
scientific structure has been defined for it in previous research. Therefore, in this study,
different scenarios have been defined based on the levels of changes v. In order to analyze
the results, first 10 different scenarios including values of v = 0 to v = 0.9 with incremental
step of 0.1 are considered and then the mathematical model is solved based on it. In
addition, it can be seen that in some scenarios, the amount of investment in companies is
equal to each other. The reason for this can be attributed to the proximity of the level of
prioritization for investing in any company. However, the impact of different scenarios
on corporate investment is not negligible and creates the need for investors to pay
attention to this fact.
In order to develop the theoretical and practical dimensions of the present study,
it is suggested that the proposed set of criteria and sub-criteria be developed in accordance
with the new political conditions of the region, Iran. Because these changes can directly
affect the final results. Moreover, conducting a study to investigate the causes and barriers
to investment in the Iranian capital market can expand the scope of research application.
In addition, the development of a mathematical model and the consideration of new
parameters and variables are presented as a final proposal.

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