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doi: 10.3934/naco.2021019
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A novel methodology for portfolio selection in fuzzy multi criteria environment using risk-benefit analysis and fractional stochastic

Department of Industrial Engineering, Yazd University, Yazd Iran

Received  November 2020 Revised  April 2021 Early access June 2021

This article proposes an efficient approach for solving portfolio type problems. It is highly suitable to help fund allocators and decision makers to set up appropriate portfolios for investors. Stock selection is based upon the risk benefits analysis using MADM approach in fuzzy environment. This sort of analysis allows decision makers to identify the list of acceptable portfolios where they can assign some portions of their asset to them. The purpose of this article is two folds; first, to introduce a methodology to select the list of stocks for investment purpose, and second, to employ a stochastic fractional programming model to assign money into selected stocks. This article proposes a hybrid methodology for finding an optimal or new optimal solution of the problem. This hybrid approach considers risks and benefits at the time of stocks prioritization. This is followed by solving a fractional programming to determine the percentages of the budget to be allocated to stocks while dealing with two sets of suitable and non-suitable stocks. For clarification purposes, a sample example problem is solved.

Citation: Yahia Zare Mehrjerdi. A novel methodology for portfolio selection in fuzzy multi criteria environment using risk-benefit analysis and fractional stochastic. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021019
References:
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References:
[1]

M. Abdel-Baset and I. M. Hezam, An improved flower pollination algorithm for ratio optimization problems, Applied Mathematics and Information Sciences Letters, 3 (2015). 83-91. Google Scholar

[2]

L. AdamM. BrandaH. Heitsch and R. Henrion, Solving joint chance constrained problems using regulation and benders' decomposition, Annals of Operations Research, 292 (2020), 683-709.  doi: 10.1007/s10479-018-3091-9.  Google Scholar

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A. Alinedjad and Y. Zare Mehrjerdi, A new approach for portfolio performance evaluation in MVS modeling using data envelopment analysis: (Case study: Iran stock market), Sharif Industrial Journal of Management and Industrial Engineering, 2013. Google Scholar

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M. Amiri, An integrated eigenvector-DEA-TOPSIS methodology for portfolio risk evaluation in the FOREX spot market, Expert Systems with Applications, 37 (2010), 509-516.   Google Scholar

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M. AmiriM. Shariatpanah and M. Benekar, Optimal portfolio selection using multi criterion decision making, Journal of Securities Exchange, 11 (2010), 5-24.   Google Scholar

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S. Amirian and M. Amiri, The effects of using fuzzy multi attributes approaches on selective portfolio returns in Tehran securities exchange market, in The 10th International Industrial Engineering Conference, Tehran University, Tehran, Iran, 2013. Google Scholar

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H. Arsham and A. B. Kahn, A complete algorithm for linear fractional programs, Computers & Mathematics with Applications, 20 (1990), 11-23.  doi: 10.1016/0898-1221(90)90344-J.  Google Scholar

[8]

E. Babaee Tirkolaee, A. Mardani, Z. Dashtian, M. Soltani and G. W. Weber, A novel hybrid method using fuzzy decision making and multi-objective programming for sustainable-reliable supplier selection in two-echelon supply chain design, Journal of Cleaner Production, 250 (2019). Google Scholar

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A. Baykasoglu and I. Golcuk, Development of a novel multiple-attribute decision making model via fuzzy cognitive maps and hierarchical fuzzy TOPSIS, Information Sciences, 301 (2015), 75-98.   Google Scholar

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[11]

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[13]

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A. Bilbao-TerolB. Pérez-GladishM. Arenas-ParraM. Victoria and R. Ura, Fuzzy compromise programming for portfolio selection, Applied Mathematics and Computations, 173 (2006), 251-264.  doi: 10.1016/j.amc.2005.04.003.  Google Scholar

[15]

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[16]

P. Brockett, William W. Abraham Charnes, Cooper Kuyuk Kwon and W. Timothy Ruefli, Chance Constrained Programming Approach to Empirical Analyses of Mutual Fund Investment Strategies, National Science Foundation under Grant SES 8722504 and by the IC., 1992. Google Scholar

[17]

G. F. Can and S. Demirok, Universal usability evaluation by using an integrated fuzzy multi criteria decision making approach, International Journal of Intelligent Computing and Cybernetics, 12 (2019), 194-223.   Google Scholar

[18]

A. Charnes and W. W. Cooper, Programming with linear fractional functions, Naval Research Logistics Quarterly, 9 (1962), 181-186.  doi: 10.1002/nav.3800090303.  Google Scholar

[19]

A. Charnes and W. W. Cooper, Chance-constrained Programming, Management Science, 6 (1962), 73-79.  doi: 10.1287/mnsc.6.1.73.  Google Scholar

[20]

Z. ChenS. Peng and A. Lisser, A sparse chance constrained portfolio selection model with multiple constraints, Journal of Global Optimization, 77 (2020), 825-852.  doi: 10.1007/s10898-020-00901-3.  Google Scholar

[21]

Z. ChenS. Peng and J. Liu, Data-driven robust chance constrained problems: a mixture model approach, J. Optim. Theory Appl., 179 (2018), 1065-1085.  doi: 10.1007/s10957-018-1376-4.  Google Scholar

[22]

P. ChunhachindaK. DandapaniS. Hamid and A. J. Prakash, Portfolio selection and skewness: Evidence from international stock markets, Journal of Banking and Finance, 21 (1997), 143-167.   Google Scholar

[23]

H. Dalman, An interactive fuzzy goal programming algorithm to solve decentralized bi-level multi-objective fractional programming problem, Available at http://sciencewise.info/media/pdf/1606.00927v1.pdf. Google Scholar

[24]

M. L. De PradoR. Vince and Q. J. Zhu, Optimal risk budgeting under a finite investment horizon, Risks, 7 (2019), 1-15.   Google Scholar

[25]

W. Dinkelbach, On non-linear fractional programming, Management Science, 13 (1967), 492-498.  doi: 10.1287/mnsc.13.7.492.  Google Scholar

[26]

M. Doumpos and Zo pounidis, Multi-objective optimization models in finance and investment, Journal of Golobal optimization, 76 (2020), 243-244.  doi: 10.1007/s10898-019-00873-z.  Google Scholar

[27]

M. D{ü}rC. Khompatraporn and Z. B. Zabinsky, Solving fractional problems with dynamic multistart improving hit-and-run, Ann. Oper. Res., 156 (2007), 25-44.  doi: 10.1007/s10479-007-0232-y.  Google Scholar

[28]

D. DuttaR. N. Tiwari and J. R. Rao, Multiple objectives linear fractional programming, A Fuzzy Set Theoretic Approach, Fuzzy Sets and Systems, 52 (1992), 39-45.  doi: 10.1016/0165-0114(92)90034-2.  Google Scholar

[29]

M. ElahiM. Yousefi and Y. Zare Mehrjerdi, Portfolio optimization with mean-variance approach using hunting search meta-heuristic algorithm, Financial Research Journal, 16 (2011), 37-56.   Google Scholar

[30]

S. FallahpourH. Safari and N. Omrani, Portfolio selection using fuzzy logarithm modeling and PROMETE approach, Financial Strategic Management Journal, 2 (2013), 103-120.   Google Scholar

[31]

T. B. Farag, A Parametric Analysis on Multicriteria Integer Fractional Decision-Making Problems, PhD Thesis, Faculty of Science, Helwan University, Helwan, Egypt, 2012. Google Scholar

[32]

G. GuastarobaR. Mansini and Sp eranza, On the effectiveness of scenario generation techniques in single-period portfolio optimization, European Journal of Operational Research, 192 (2009), 500-511.  doi: 10.1016/j.ejor.2007.09.042.  Google Scholar

[33]

P. Guo, X. Chen, M. Li and J. Li, Fuzzy chance constrained linear fractional programming approach for optimal water allocation, Stoch. Environ. Res. Risk Assess, (2014), 1601-1612. Google Scholar

[34]

S. N. Gupta, A chance constrained approach to fractional programming with random numerator, Journal of Math. Model Algorithm, 24 (2009), 1-5.  doi: 10.1007/s10852-009-9110-8.  Google Scholar

[35]

G. A. HanasusantoV. RoitchD. Kuhn and W. Wiesemann, Ambiguous joint chance constraints under mean and dispersion information, Oper. Res., 65 (2017), 751-767.  doi: 10.1287/opre.2016.1583.  Google Scholar

[36]

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Figure 1.  Steps to develop a hierarchical fuzzy TOPSIS-SAW-GP-Fractional Programming approach for portfolio risk-benefit analysis
Figure 2.  The hierarchy for the selection of the most suitable Portfolio
Table 1.  Sample of variants of MADM approaches
Variants of Fuzzy Authors and years
TOPSIS and MADM of publications
1 Fuzzy TOPSIS Mirabi et al. (2012),
Baykasoglu and Golcuk(2015)
2 Hierarchical fuzzy TOPSIS Zare Mehrjerdi (2020),
Baykasoglu (2013)
3 Type-2 fuzzy TOPSIS Baykasoglu and
Golcuk (2017)
4 Interval value fuzzy TOPSIS Zare Mehrjerdi (2013)
5 Group DM Fuzzy TOPSIS Wange (2008),
Zare Mehrjerdi (2013)
6 AHP and Fuzzy AHP Aydein Celen (2014)
7 ANP and Fuzzy ANP Pahlavan et al. (2012),
Amiri, M. (2010)
8 ELECTRE Chen-Tung-Chen (2009),
Thien Phue Ho Quang (2014)
9 Compromise Programming Bilbao-Terol et al. (2006)
10 PROMEETHE Fallahpour et al. (2014)
Variants of Fuzzy Authors and years
TOPSIS and MADM of publications
1 Fuzzy TOPSIS Mirabi et al. (2012),
Baykasoglu and Golcuk(2015)
2 Hierarchical fuzzy TOPSIS Zare Mehrjerdi (2020),
Baykasoglu (2013)
3 Type-2 fuzzy TOPSIS Baykasoglu and
Golcuk (2017)
4 Interval value fuzzy TOPSIS Zare Mehrjerdi (2013)
5 Group DM Fuzzy TOPSIS Wange (2008),
Zare Mehrjerdi (2013)
6 AHP and Fuzzy AHP Aydein Celen (2014)
7 ANP and Fuzzy ANP Pahlavan et al. (2012),
Amiri, M. (2010)
8 ELECTRE Chen-Tung-Chen (2009),
Thien Phue Ho Quang (2014)
9 Compromise Programming Bilbao-Terol et al. (2006)
10 PROMEETHE Fallahpour et al. (2014)
Table 2.  Sample of chance constrained programming applications in Portfolio selection
CCP and its Variants Authors and Years
rough variables of publications
1 CCP with random Tavana, M., Khanjani, R.,
and Di Caprio, D. (2019)
2 CCP with multi period Hassanlou, K. (2017)
portfolio selection
3 Joint CCP and Adam, L., Branda, M., Heitsch,
Portfolio selection H., and Henrion, R. (2018)
4 CCP with Multi objective modeling Miryekemani, S.A., Sadeh,
of Portfolio selection and GA E., Amini Sabegh, Z. (2017)
5 Sparse CCP Chen, Z., Peng, S.,
Portfolio selection and Lisser, A. (2020).
6 CCP and Robust and Sengupta, R.N.,
reliable portfolio optimization and Kumar, R. (2017).
7 CCP and Data Chen, Z., Peng,
driven robust S., Liu, J.(2018)
8 Ambiguous joint CCP Hanasusanto, G.A., Roitch, V.,
Kuhn, D., Wiesemann, W. (2017)
9 CCP and Type-2 fuzzy fractional Zhou, C., Huang, G.,
integrated modeling and Chen, J. (2019).
10 CCP with a sparse model Xu, F., Wang, M., Dai,
Y.H., Xu, D. (2018)
11 CCP and Data Alinedjad and Zare
envelop analysis Mehrjerdi (2013)
12 CCP and Fuzzy computer Liu (2009),
simulation Zare Mehrjerdi et al. (2010)
CCP and its Variants Authors and Years
rough variables of publications
1 CCP with random Tavana, M., Khanjani, R.,
and Di Caprio, D. (2019)
2 CCP with multi period Hassanlou, K. (2017)
portfolio selection
3 Joint CCP and Adam, L., Branda, M., Heitsch,
Portfolio selection H., and Henrion, R. (2018)
4 CCP with Multi objective modeling Miryekemani, S.A., Sadeh,
of Portfolio selection and GA E., Amini Sabegh, Z. (2017)
5 Sparse CCP Chen, Z., Peng, S.,
Portfolio selection and Lisser, A. (2020).
6 CCP and Robust and Sengupta, R.N.,
reliable portfolio optimization and Kumar, R. (2017).
7 CCP and Data Chen, Z., Peng,
driven robust S., Liu, J.(2018)
8 Ambiguous joint CCP Hanasusanto, G.A., Roitch, V.,
Kuhn, D., Wiesemann, W. (2017)
9 CCP and Type-2 fuzzy fractional Zhou, C., Huang, G.,
integrated modeling and Chen, J. (2019).
10 CCP with a sparse model Xu, F., Wang, M., Dai,
Y.H., Xu, D. (2018)
11 CCP and Data Alinedjad and Zare
envelop analysis Mehrjerdi (2013)
12 CCP and Fuzzy computer Liu (2009),
simulation Zare Mehrjerdi et al. (2010)
Table 3.  Sample of portfolio applications
Portfolio Application areas Authors and Years of publications
1 Multi-objective capital allocation Mizgier, Kamil, J., Joseph M.
for supplier development under risk Pasia, Srinivas Talluri (2017)
2 Multi-objective Optimization of Credit Mizgier, Kamil J., Pasia, Joseph (2016),
Capital Allocation in Financial Institutions Doumpos, M., and Zopounidis, (2020
3 Large Scale Portfolio optimization Qu, B.Y., Zhoi, Q., Xiao, J.M.,
using Multi-objective Programming Liang, J.J., Suganthan, P.N (2017)
4 Financial portfolio optimization, Özceylan, Eren, Kabak, Mehmet,
Bank portfolio management Dağdeviren, Metin (2016), Soleymani, F.,
monetary policy, economic uncertainty and Paquet, E. (2020), Amirian, S., Amiri,
M. (2013), Udomrachtavanich, W. (2005)
5 Application of Markowitz portfolio Ivanova, M., Dospatleiv, L. (2016)
optimization on Bulgarian stock market
6 Behavioral portfolio selection Simo-Kengne, B.D., Ababio,
and optimization K.A., Ur Koumba, J.M (2018)
7 Scenario-based portfolio selection Liesiö, J., Salo A. (2012)
8 Nonlinear bi-level programming approach Ma, S. (2016)
for product portfolio management
9 Effectiveness of scenario generation Guastaroba, G., Mansini,
techniques in single-period R., Speranza, (2009)
portfolio optimization
10 Optimal risk budgeting under De Prado, M.L., Vince,
a finite investment horizon R., and Zhu, Q.J. (2019)
Portfolio Application areas Authors and Years of publications
1 Multi-objective capital allocation Mizgier, Kamil, J., Joseph M.
for supplier development under risk Pasia, Srinivas Talluri (2017)
2 Multi-objective Optimization of Credit Mizgier, Kamil J., Pasia, Joseph (2016),
Capital Allocation in Financial Institutions Doumpos, M., and Zopounidis, (2020
3 Large Scale Portfolio optimization Qu, B.Y., Zhoi, Q., Xiao, J.M.,
using Multi-objective Programming Liang, J.J., Suganthan, P.N (2017)
4 Financial portfolio optimization, Özceylan, Eren, Kabak, Mehmet,
Bank portfolio management Dağdeviren, Metin (2016), Soleymani, F.,
monetary policy, economic uncertainty and Paquet, E. (2020), Amirian, S., Amiri,
M. (2013), Udomrachtavanich, W. (2005)
5 Application of Markowitz portfolio Ivanova, M., Dospatleiv, L. (2016)
optimization on Bulgarian stock market
6 Behavioral portfolio selection Simo-Kengne, B.D., Ababio,
and optimization K.A., Ur Koumba, J.M (2018)
7 Scenario-based portfolio selection Liesiö, J., Salo A. (2012)
8 Nonlinear bi-level programming approach Ma, S. (2016)
for product portfolio management
9 Effectiveness of scenario generation Guastaroba, G., Mansini,
techniques in single-period R., Speranza, (2009)
portfolio optimization
10 Optimal risk budgeting under De Prado, M.L., Vince,
a finite investment horizon R., and Zhu, Q.J. (2019)
Table 4.  Sample of research articles associated with fractional programming
Authors Purpose of research Solution methodology
1 Dalman, H. (2016) bi-level multi-objective fractional Interactive fuzzy goal
programming problem programming algorithm
2 Xiao, L. (2010) Solving linear fractional Neural network method
programming
3 Farag, T.B. (2012) integer fractional Parametric analysis
decision-making problems
4 Hezam, I.M., solving fractional programming Meta-heuristic algorithms,
Raouf MMH (2013) problems and complex variable Particle swarm optimization
Osama Abdel, Hezam fractional programming
IM, Raouf OA (2013)
5 Dür M, Solving fractional problems Dynamic multi-start
Khompatraporn C, improving hit-and-run
Zabinsky ZB (2007)
6 Udhayakumar, solving chance constrained Simulation based
et al. (2010) fractional programming genetic algorithm
7 Sameeullah Linear fractional Genetic Algorithm
et al. (2008) programming.
8 Gupta (2009) chance constrained approach Convex programming
to fractional programming
with random numerator
9 Wang C-F, linear fractional Global optimization
Shen P-P (2008) programming algorithm
10 Biswas and quadratic fractional Goal programming
Bose (2011) bi-level programming. approach
11 Charnes and converted the fractional Linear programming
Cooper (1962, 1973) programming (FP) into
equivalent linear
programming
12 Pal (2003, 2013), Fractional chance Fuzzy modeling and goal
Zare Mehrjerdi (2011) constraint programming programming approach
13 Zhang, Li, Two stage stochastic Decision support system
and Guo (2017) chance constrained
fractional programming
14 Zare Mehrjerdi Linear fractional Optimization
and Faregh (2017), programming Arsham
(1990), Dutta (1992)
15 Amiri, M., The effects of using fuzzy multi Multi attribute decision
Shariatpanah, attributes approaches on selective making
M., Benekar, portfolio returns in Tehran
M. (2010) securities exchange market
16 Phuc Ho Quang, Applications in Portfolio Multiple criteria
T. (2014) Selection Problems decision making
17 Y. Simaan. (1997 Estimation risk in mean variance model
portfolio selection and the mean-absolute
deviation model
18 Zhu, H., Hung, Stochastic linear fractional linear fractional
G. H. (2011) programming approach for programming
sustainable waste management
Authors Purpose of research Solution methodology
1 Dalman, H. (2016) bi-level multi-objective fractional Interactive fuzzy goal
programming problem programming algorithm
2 Xiao, L. (2010) Solving linear fractional Neural network method
programming
3 Farag, T.B. (2012) integer fractional Parametric analysis
decision-making problems
4 Hezam, I.M., solving fractional programming Meta-heuristic algorithms,
Raouf MMH (2013) problems and complex variable Particle swarm optimization
Osama Abdel, Hezam fractional programming
IM, Raouf OA (2013)
5 Dür M, Solving fractional problems Dynamic multi-start
Khompatraporn C, improving hit-and-run
Zabinsky ZB (2007)
6 Udhayakumar, solving chance constrained Simulation based
et al. (2010) fractional programming genetic algorithm
7 Sameeullah Linear fractional Genetic Algorithm
et al. (2008) programming.
8 Gupta (2009) chance constrained approach Convex programming
to fractional programming
with random numerator
9 Wang C-F, linear fractional Global optimization
Shen P-P (2008) programming algorithm
10 Biswas and quadratic fractional Goal programming
Bose (2011) bi-level programming. approach
11 Charnes and converted the fractional Linear programming
Cooper (1962, 1973) programming (FP) into
equivalent linear
programming
12 Pal (2003, 2013), Fractional chance Fuzzy modeling and goal
Zare Mehrjerdi (2011) constraint programming programming approach
13 Zhang, Li, Two stage stochastic Decision support system
and Guo (2017) chance constrained
fractional programming
14 Zare Mehrjerdi Linear fractional Optimization
and Faregh (2017), programming Arsham
(1990), Dutta (1992)
15 Amiri, M., The effects of using fuzzy multi Multi attribute decision
Shariatpanah, attributes approaches on selective making
M., Benekar, portfolio returns in Tehran
M. (2010) securities exchange market
16 Phuc Ho Quang, Applications in Portfolio Multiple criteria
T. (2014) Selection Problems decision making
17 Y. Simaan. (1997 Estimation risk in mean variance model
portfolio selection and the mean-absolute
deviation model
18 Zhu, H., Hung, Stochastic linear fractional linear fractional
G. H. (2011) programming approach for programming
sustainable waste management
Table 5.  Reported use of decision making approaches with risks and benefits analysis and strategies prioritization associated with Joyful Organization studies
MADM MODM Integrating
approaches approaches approaches
(1) risks and benefits hierarchical gp (this study) this research
analysis for portfolio fuzzy topsis
analysis (hftopsis) (this study)
(2) assessment of this study x this study
portfolio alternatives
and prioritization
of them
portfolio analysis optimization madm and
approaches modm integration
approach
MADM MODM Integrating
approaches approaches approaches
(1) risks and benefits hierarchical gp (this study) this research
analysis for portfolio fuzzy topsis
analysis (hftopsis) (this study)
(2) assessment of this study x this study
portfolio alternatives
and prioritization
of them
portfolio analysis optimization madm and
approaches modm integration
approach
Table 6.  Portfolio risks
Portfolio Risks Descriptions
Broker (C1) Since investors in the developing countries have
little access to brokers on the international markets so
the risk can be relatively high for investors.
Technical Analysis (C2) Access to professional technical analysts
is not a simple task. This because (1) there are not too
many of such analysts and (2) they are far less
experienced in this task in general.
Capital management (C3) Capital management techniques are related
to three categories of: Tools, Objectives and Costs.
More on these terminologies can be studied in
the work of Amiri, et al. (2010).
Trading System (C4) A good on-line trading system with fast access to
internet may help capital management to
access the main trading board and then buy or
sell as required. Most likely in developing countries the
internet access is not fast and always available for
political and social reasons as it is in many
middle eastern countries.
Technology (C5) Due to the fact that foreign exchange is rapidly
growing and will continue to do so and
reaching over 3 trillion dollars (Amiri et al. 2010),
hence we there is a need to spend more on
technology and expect a good level of risk at
any level of trading.
Trading Psychology (C6) The most effective factors in trading
psychology can be identified as: trading commitment,
personal trading style, personal discipline,
trading coach, courage, and familiar with the
secrets of successful traders.
Portfolio Risks Descriptions
Broker (C1) Since investors in the developing countries have
little access to brokers on the international markets so
the risk can be relatively high for investors.
Technical Analysis (C2) Access to professional technical analysts
is not a simple task. This because (1) there are not too
many of such analysts and (2) they are far less
experienced in this task in general.
Capital management (C3) Capital management techniques are related
to three categories of: Tools, Objectives and Costs.
More on these terminologies can be studied in
the work of Amiri, et al. (2010).
Trading System (C4) A good on-line trading system with fast access to
internet may help capital management to
access the main trading board and then buy or
sell as required. Most likely in developing countries the
internet access is not fast and always available for
political and social reasons as it is in many
middle eastern countries.
Technology (C5) Due to the fact that foreign exchange is rapidly
growing and will continue to do so and
reaching over 3 trillion dollars (Amiri et al. 2010),
hence we there is a need to spend more on
technology and expect a good level of risk at
any level of trading.
Trading Psychology (C6) The most effective factors in trading
psychology can be identified as: trading commitment,
personal trading style, personal discipline,
trading coach, courage, and familiar with the
secrets of successful traders.
Table 7.  Portfolio Benefits
Portfolio's Benefits Descriptions
Dividend (C7) Dividend can be defined according to following formula
where dai is the nominal annual revenue and Ph, i is the
highest price of asset i in the year before.
Dividend of a portfolio is defined as the weighted sum
of the dividends of individual stocks in the portfolio.
$ d_{i}=\frac{d_{i}^{a}}{p_{i}^{h}} $
Short term and long Some researchers considered short term and long term
term returns (C8) returns for 12 months performance and 36 months
Related formulas for these performances are shown below where
$ P_{T, i}, P_{T-1, i}, P_{T-3, i} $ are defined as
price of asset i at periods $ T, T-1, T-3, $ respectively.
The short term performance for asset $ i $ is as shown below:
$ r_{i}^{12}=\frac{P_{T, i}-P_{T-1, i}}{P_{T-1, i}} $
The long term performance for asset i is as shown below
$ r_{i}^{36}=\frac{P_{T, i}-P_{T-3, i}}{P_{T-3, i}} $
Liquidity (C9) For a specific asset, liquidity is set to be a proportion
of that asset which is called turnover rate.
Often investors prefer to deal with greater liquidity.
Standard and Poor's Based upon the Standard & Poor fund services,
Star Ranking (C10) the performance ranking which is based on an annual
basis is shown by star ranking. Taking $ SR_{i} $ as the number
of stars assigned to investment fund $ i $, we can define following
objective function for the problem under study.
$ f_{4}(x)=\sum_{i=1}^{M}SR_{i}x_{i} $
where $ SR_{i} $ denotes the number of stars
assigned to investment fund $ i $.
Financial System reporting This means that accounting system is in good
such as ERP system (C11) health and organization deals with low risk to
do stock trading. ERP financial systems are able to
report the status of the company in the right time
and at the right amount of information needed for
making right decisions.
Green Vision of This vision of company indicates that
the Company (C12) risks are low and benefits are high.
Portfolio's Benefits Descriptions
Dividend (C7) Dividend can be defined according to following formula
where dai is the nominal annual revenue and Ph, i is the
highest price of asset i in the year before.
Dividend of a portfolio is defined as the weighted sum
of the dividends of individual stocks in the portfolio.
$ d_{i}=\frac{d_{i}^{a}}{p_{i}^{h}} $
Short term and long Some researchers considered short term and long term
term returns (C8) returns for 12 months performance and 36 months
Related formulas for these performances are shown below where
$ P_{T, i}, P_{T-1, i}, P_{T-3, i} $ are defined as
price of asset i at periods $ T, T-1, T-3, $ respectively.
The short term performance for asset $ i $ is as shown below:
$ r_{i}^{12}=\frac{P_{T, i}-P_{T-1, i}}{P_{T-1, i}} $
The long term performance for asset i is as shown below
$ r_{i}^{36}=\frac{P_{T, i}-P_{T-3, i}}{P_{T-3, i}} $
Liquidity (C9) For a specific asset, liquidity is set to be a proportion
of that asset which is called turnover rate.
Often investors prefer to deal with greater liquidity.
Standard and Poor's Based upon the Standard & Poor fund services,
Star Ranking (C10) the performance ranking which is based on an annual
basis is shown by star ranking. Taking $ SR_{i} $ as the number
of stars assigned to investment fund $ i $, we can define following
objective function for the problem under study.
$ f_{4}(x)=\sum_{i=1}^{M}SR_{i}x_{i} $
where $ SR_{i} $ denotes the number of stars
assigned to investment fund $ i $.
Financial System reporting This means that accounting system is in good
such as ERP system (C11) health and organization deals with low risk to
do stock trading. ERP financial systems are able to
report the status of the company in the right time
and at the right amount of information needed for
making right decisions.
Green Vision of This vision of company indicates that
the Company (C12) risks are low and benefits are high.
Table 8.  weights by main objectives
Goal
Risks $ (0.39, 0.58, 0.74) $
Benefits $ (0.38, 0.56, 0.71) $
Goal
Risks $ (0.39, 0.58, 0.74) $
Benefits $ (0.38, 0.56, 0.71) $
Table 9.  weights by risks and benefits components
Risks Benefits
Risk1 (C1) $ (0.30, 0.48, 0.67) $ $ (0, 0) $
Risk1 (C2) $ (0.32, 0.49, 0.67) $ $ (0, 0) $
Risk1 (C3) $ (0.30, 0.48, 0.65) $ $ (0, 0) $
Risk1 (C4) $ (0.32, 0.52, 0.69) $ $ (0, 0) $
Risk1 (C5) $ (0.31, 0.50, 0.67) $ $ (0, 0) $
Risk1 (C6) $ (0.34, 0.52, 0.70) $ $ (0, 0) $
Ben1 (C7) $ (0, 0) $ $ (0.34, 0.53, 0.71) $
Ben2 (C8) $ (0, 0) $ $ (0.31, 0.49, 0.68) $
Ben3 (C9) $ (0, 0) $ $ (0.33, 0.53, 0.71) $
Ben4 (C10) $ (0, 0) $ $ (0.34, 0.53, 0.71) $
Ben5 (C11) $ (0, 0) $ $ (0.43, 0.63, 0.78) $
Ben6 (C12) $ (0, 0) $ $ (0.41, 0.60, 0.76) $
Weights $ (0.39, 0.58, 0.74) $ $ (0.38, 0.56, 0.71) $
Risks Benefits
Risk1 (C1) $ (0.30, 0.48, 0.67) $ $ (0, 0) $
Risk1 (C2) $ (0.32, 0.49, 0.67) $ $ (0, 0) $
Risk1 (C3) $ (0.30, 0.48, 0.65) $ $ (0, 0) $
Risk1 (C4) $ (0.32, 0.52, 0.69) $ $ (0, 0) $
Risk1 (C5) $ (0.31, 0.50, 0.67) $ $ (0, 0) $
Risk1 (C6) $ (0.34, 0.52, 0.70) $ $ (0, 0) $
Ben1 (C7) $ (0, 0) $ $ (0.34, 0.53, 0.71) $
Ben2 (C8) $ (0, 0) $ $ (0.31, 0.49, 0.68) $
Ben3 (C9) $ (0, 0) $ $ (0.33, 0.53, 0.71) $
Ben4 (C10) $ (0, 0) $ $ (0.34, 0.53, 0.71) $
Ben5 (C11) $ (0, 0) $ $ (0.43, 0.63, 0.78) $
Ben6 (C12) $ (0, 0) $ $ (0.41, 0.60, 0.76) $
Weights $ (0.39, 0.58, 0.74) $ $ (0.38, 0.56, 0.71) $
Table 10.  the decision Matrix
Risk1(C1) Risk2(C2) Risk3(C3) Risk4(C4) Risk5(C5) Risk6(C6)
Index funds (48, 68, 86) (50, 70, 87) (49, 69, 87) (43, 63, 80) (43, 62, 78) (48, 68, 84)
Computer (51, 71, 87) (50, 70, 87) (43, 63, 82) (43, 63, 80) (31, 49, 68) (46, 66, 83)
Durable goods (50, 70, 87) (50, 70, 87) (49, 69, 87) (43, 63, 80) (24, 39, 58) (27, 45, 63)
Pharmaceutical (53, 73, 89) (50, 70, 87) (49, 69, 87) (43, 63, 80) (31, 48, 66) (38, 57, 74)
Chip Industry (49, 69, 86) (50, 70, 87) (34, 53, 71) (43, 63, 80) (30, 48, 67) (28, 46, 65)
Real States (46, 66, 83) (50, 70, 87) (49, 69, 87) (43, 63, 80) (37, 55, 72) (38, 56, 74)
Life Insurance (44, 64, 82) (50, 70, 87) (49, 69, 87) (43, 63, 80) (29, 47, 66) (33, 51, 69)
Health (43, 63, 82) (50, 70, 87) (49, 69, 87) (43, 63, 80) (35, 53, 71) (41, 59, 76)
Insurance
Tourism (50, 70, 88) (50, 70, 87) (49, 69, 87) (43, 63, 80) (34, 52, 70) (32, 49, 68)
industry
Auto industry (50, 70, 87) (50, 70, 87) (36, 54, 72) (43, 63, 80) (33, 50, 68) (37, 53, 70)
Benefit1 Benefit2 Benefit3 Benefit4 Benefit5 Benefit6
(C7) (C8) (C9) (C10) (C11) (C12)
Index funds (58, 78, 94) (50, 70, 87) (55, 75, 93) (63, 83, 96) (63, 83, 97) ((25, 41, 61)
Computer (48, 68, 85) (47, 66, 84) (51, 71, 87) (45, 65, 82) (48, 68, 85) (40, 60, 78)
Durable goods (25, 42, 61) (32, 49, 68) (44, 63, 80) (32, 49, 67) (53, 73, 87) (19, 35, 54)
Pharmaceutical (39, 57, 74) (43, 61, 78) (42, 62, 78) (44, 63, 78) (52, 72, 86) (19, 33, 52)
Chip Industry (35, 53, 69) (41, 59, 76) (44, 63, 78) (52, 71, 85) (53, 73, 87) (12, 26, 46)
Real States (35, 54, 72) (37, 57, 74) (20, 37, 56) (21, 37, 56) (53, 73, 88) (16, 30, 49)
Life Insurance (46, 65, 81) (47, 67, 82) (21, 37, 57) (33, 50, 68) (53, 73, 87) (42, 61, 78)
Health (43, 63, 79) (46, 66, 82) (30, 47, 64) (26, 43, 61) (54, 74, 88) (22, 38, 57)
Insurance
Tourism (47, 67, 82) 47, 67, 82) (22, 40, 59) (40, 58, 73) (53, 73, 87) (27, 43, 62)
industry
Auto industry (49, 68, 83) (50, 69, 84) (38, 56, 71) (31, 48, 64) (53, 73, 87) (22, 38, 56)
$ X^{-} $ (43, 63, 82) (50, 70, 87) (34, 53, 71) (43, 63, 80) (24, 39, 58) (27, 45, 63)
$ X^{+} $ (58, 78, 94) (50, 70, 87) (55, 75, 93) (63, 83, 96) (63, 83, 97) (42, 61, 78)
Risk1(C1) Risk2(C2) Risk3(C3) Risk4(C4) Risk5(C5) Risk6(C6)
Index funds (48, 68, 86) (50, 70, 87) (49, 69, 87) (43, 63, 80) (43, 62, 78) (48, 68, 84)
Computer (51, 71, 87) (50, 70, 87) (43, 63, 82) (43, 63, 80) (31, 49, 68) (46, 66, 83)
Durable goods (50, 70, 87) (50, 70, 87) (49, 69, 87) (43, 63, 80) (24, 39, 58) (27, 45, 63)
Pharmaceutical (53, 73, 89) (50, 70, 87) (49, 69, 87) (43, 63, 80) (31, 48, 66) (38, 57, 74)
Chip Industry (49, 69, 86) (50, 70, 87) (34, 53, 71) (43, 63, 80) (30, 48, 67) (28, 46, 65)
Real States (46, 66, 83) (50, 70, 87) (49, 69, 87) (43, 63, 80) (37, 55, 72) (38, 56, 74)
Life Insurance (44, 64, 82) (50, 70, 87) (49, 69, 87) (43, 63, 80) (29, 47, 66) (33, 51, 69)
Health (43, 63, 82) (50, 70, 87) (49, 69, 87) (43, 63, 80) (35, 53, 71) (41, 59, 76)
Insurance
Tourism (50, 70, 88) (50, 70, 87) (49, 69, 87) (43, 63, 80) (34, 52, 70) (32, 49, 68)
industry
Auto industry (50, 70, 87) (50, 70, 87) (36, 54, 72) (43, 63, 80) (33, 50, 68) (37, 53, 70)
Benefit1 Benefit2 Benefit3 Benefit4 Benefit5 Benefit6
(C7) (C8) (C9) (C10) (C11) (C12)
Index funds (58, 78, 94) (50, 70, 87) (55, 75, 93) (63, 83, 96) (63, 83, 97) ((25, 41, 61)
Computer (48, 68, 85) (47, 66, 84) (51, 71, 87) (45, 65, 82) (48, 68, 85) (40, 60, 78)
Durable goods (25, 42, 61) (32, 49, 68) (44, 63, 80) (32, 49, 67) (53, 73, 87) (19, 35, 54)
Pharmaceutical (39, 57, 74) (43, 61, 78) (42, 62, 78) (44, 63, 78) (52, 72, 86) (19, 33, 52)
Chip Industry (35, 53, 69) (41, 59, 76) (44, 63, 78) (52, 71, 85) (53, 73, 87) (12, 26, 46)
Real States (35, 54, 72) (37, 57, 74) (20, 37, 56) (21, 37, 56) (53, 73, 88) (16, 30, 49)
Life Insurance (46, 65, 81) (47, 67, 82) (21, 37, 57) (33, 50, 68) (53, 73, 87) (42, 61, 78)
Health (43, 63, 79) (46, 66, 82) (30, 47, 64) (26, 43, 61) (54, 74, 88) (22, 38, 57)
Insurance
Tourism (47, 67, 82) 47, 67, 82) (22, 40, 59) (40, 58, 73) (53, 73, 87) (27, 43, 62)
industry
Auto industry (49, 68, 83) (50, 69, 84) (38, 56, 71) (31, 48, 64) (53, 73, 87) (22, 38, 56)
$ X^{-} $ (43, 63, 82) (50, 70, 87) (34, 53, 71) (43, 63, 80) (24, 39, 58) (27, 45, 63)
$ X^{+} $ (58, 78, 94) (50, 70, 87) (55, 75, 93) (63, 83, 96) (63, 83, 97) (42, 61, 78)
Table 11.  Positive distance for Risks
Risk1 Risk2 Risk3 Risk4 Risk5 Risk6 Total
Index funds 0.0258 0.0000 0.0915 0.0000 0.1420 0.1363 0.3956
Computer 0.0405 0.0000 0.0573 0.0000 0.613 0.1249 0.2840
Durable goods 0.0339 0.0000 0.0915 0.0000 0.0000 0.0000 0.1254
Pharmaceutical 0.0471 0.0000 0.0915 0.0000 0.0533 0.0723 0.2642
Chip Industry 0.0290 0.0000 0.0000 0.0000 0.0523 0.0072 0.0885
Real States 0.0112 0.0000 0.0915 0.0000 0.0985 0.0656 0.2668
Life Insurance 0.0032 0.0000 0.0915 0.0000 0.0461 0.0367 0.1775
Health
Insurance 0.0000 0.0000 0.0915 0.0000 0.0834 0.0891 0.2640
Tourism
industry 0.0356 0.0000 0.0915 0.0000 0.0778 0.0278 0.2326
Auto industry 0.0339 0.0000 0.0070 0.0000 0.0667 0.0539 0.1616
Risk1 Risk2 Risk3 Risk4 Risk5 Risk6 Total
Index funds 0.0258 0.0000 0.0915 0.0000 0.1420 0.1363 0.3956
Computer 0.0405 0.0000 0.0573 0.0000 0.613 0.1249 0.2840
Durable goods 0.0339 0.0000 0.0915 0.0000 0.0000 0.0000 0.1254
Pharmaceutical 0.0471 0.0000 0.0915 0.0000 0.0533 0.0723 0.2642
Chip Industry 0.0290 0.0000 0.0000 0.0000 0.0523 0.0072 0.0885
Real States 0.0112 0.0000 0.0915 0.0000 0.0985 0.0656 0.2668
Life Insurance 0.0032 0.0000 0.0915 0.0000 0.0461 0.0367 0.1775
Health
Insurance 0.0000 0.0000 0.0915 0.0000 0.0834 0.0891 0.2640
Tourism
industry 0.0356 0.0000 0.0915 0.0000 0.0778 0.0278 0.2326
Auto industry 0.0339 0.0000 0.0070 0.0000 0.0667 0.0539 0.1616
Table 12.  Positive Distance for Benefits
Benefit 1 Benefit 2 Benefit 3 Benefit 4 Benefit 5 Benefit 6 Total
Index 0.0000 0.0000 0.0000 0.0000 0.0000 0.1343 0.1343
funds
Computer 0.0559 0.0196 0.234 0.1051 0.878 0.0037 0.2954
Durable 0.2326 0.1246 0.0664 0.2157 0.0583 0.1876 0.8852
goods
Pharmac- 0.1257 0.0521 0.0767 0.1213 0.0678 0.2041 0.6477
eutical
Chip 0.1568 0.0635 .0714 0.0686 0.0618 0.2742 0.6963
Industry
Real 0.1442 0.0765 0.2570 0.3141 0.0558 0.2351 1.0827
States
Life 0.0743 0.0185 0.2518 0.2083 0.0618 0.0000 0.6148
Insurance
Health
Insurance 0.0897 0.0218 0.1796 0.2660 0.0523 0.1628 0.7720
Tourism
industry 0.0647 0.0185 0.2298 0.1562 0.0583 0.1182 0.6458
Auto 0.0540 0.0033 0.1167 0.2304 0.0618 0.1672 0.6334
industry
Benefit 1 Benefit 2 Benefit 3 Benefit 4 Benefit 5 Benefit 6 Total
Index 0.0000 0.0000 0.0000 0.0000 0.0000 0.1343 0.1343
funds
Computer 0.0559 0.0196 0.234 0.1051 0.878 0.0037 0.2954
Durable 0.2326 0.1246 0.0664 0.2157 0.0583 0.1876 0.8852
goods
Pharmac- 0.1257 0.0521 0.0767 0.1213 0.0678 0.2041 0.6477
eutical
Chip 0.1568 0.0635 .0714 0.0686 0.0618 0.2742 0.6963
Industry
Real 0.1442 0.0765 0.2570 0.3141 0.0558 0.2351 1.0827
States
Life 0.0743 0.0185 0.2518 0.2083 0.0618 0.0000 0.6148
Insurance
Health
Insurance 0.0897 0.0218 0.1796 0.2660 0.0523 0.1628 0.7720
Tourism
industry 0.0647 0.0185 0.2298 0.1562 0.0583 0.1182 0.6458
Auto 0.0540 0.0033 0.1167 0.2304 0.0618 0.1672 0.6334
industry
Table 13.  : Negative distance for Risks
Risk1 Risk2 Risk3 Risk4 Risk5 Risk6 Total
Index funds 0.0200 0.0000 0.0000 0.0000 0.0000 0.0000 0.0200
Computer 0.0060 0.0000 0.0295 0.0000 0.0683 0.0086 0.1124
Durable goods 0.0121 0.0000 0.0000 0.0000 0.1420 0.1363 0.2904
Pharmaceutical 0.0000 0.0000 0.0000 0.0000 0.0785 0.0540 0.1324
Chip Industry 0.0169 0.0000 0.0915 0.0000 0.0784 0.1265 0.3133
Real States 0.0349 0.0000 0.0000 0.0000 0.0355 0.0612 0.1316
Life Insurance 0.0436 0.0000 0.0000 0.0000 0.0847 0.0908 0.2190
Health
Insurance 0.0471 0.0000 0.0000 0.0000 0.0494 0.0401 0.1366
Tourism
industry 0.0106 0.0000 0.0000 0.0000 0.0531 0.1014 0.1651
Auto industry 0.0121 0.0000 0.0831 0.0000 0.645 0.0747 0.2344
Risk1 Risk2 Risk3 Risk4 Risk5 Risk6 Total
Index funds 0.0200 0.0000 0.0000 0.0000 0.0000 0.0000 0.0200
Computer 0.0060 0.0000 0.0295 0.0000 0.0683 0.0086 0.1124
Durable goods 0.0121 0.0000 0.0000 0.0000 0.1420 0.1363 0.2904
Pharmaceutical 0.0000 0.0000 0.0000 0.0000 0.0785 0.0540 0.1324
Chip Industry 0.0169 0.0000 0.0915 0.0000 0.0784 0.1265 0.3133
Real States 0.0349 0.0000 0.0000 0.0000 0.0355 0.0612 0.1316
Life Insurance 0.0436 0.0000 0.0000 0.0000 0.0847 0.0908 0.2190
Health
Insurance 0.0471 0.0000 0.0000 0.0000 0.0494 0.0401 0.1366
Tourism
industry 0.0106 0.0000 0.0000 0.0000 0.0531 0.1014 0.1651
Auto industry 0.0121 0.0000 0.0831 0.0000 0.645 0.0747 0.2344
Table 14.  Negative Distance for Benefits
Benefit 1 Benefit 2 Benefit 3 Benefit 4 Benefit 5 Benefit 6 Total
Index 0.2326 0.1246 0.2570 0.3141 0.0878 0.1346 1.1506
funds
Computer 0.1736 0.1039 0.2323 0.2053 0.0000 0.2697 0.9848
Durable 0.0000 0.0000 0.1881 0.0975 0.0310 0.0827 0.3993
goods
Pharmac- 0.1049 0.0717 0.1780 0.1914 0.0214 0.0694 0.6369
eutical
Chip 0.0753 0.0605 0.1842 0.2444 0.272 0.0000 0.5915
Industry
Real 0.0858 0.0472 0.0000 0.0000 0.0329 0.0384 0.2042
States
Life 0.1565 0.1055 0.0025 0.1045 0.0272 0.2742 0.6705
Insurance
Health
Insurance 0.1409 0.1020 0.0771 0.0466 0.0367 0.1067 0.5100
Tourism
industry 0.1658 0.1055 0.0249 0.1583 0.0310 0.1516 0.6372
Auto 0.1774 0.1213 0.1403 0.0860 0.0272 0.1040 0.6562
industry
Benefit 1 Benefit 2 Benefit 3 Benefit 4 Benefit 5 Benefit 6 Total
Index 0.2326 0.1246 0.2570 0.3141 0.0878 0.1346 1.1506
funds
Computer 0.1736 0.1039 0.2323 0.2053 0.0000 0.2697 0.9848
Durable 0.0000 0.0000 0.1881 0.0975 0.0310 0.0827 0.3993
goods
Pharmac- 0.1049 0.0717 0.1780 0.1914 0.0214 0.0694 0.6369
eutical
Chip 0.0753 0.0605 0.1842 0.2444 0.272 0.0000 0.5915
Industry
Real 0.0858 0.0472 0.0000 0.0000 0.0329 0.0384 0.2042
States
Life 0.1565 0.1055 0.0025 0.1045 0.0272 0.2742 0.6705
Insurance
Health
Insurance 0.1409 0.1020 0.0771 0.0466 0.0367 0.1067 0.5100
Tourism
industry 0.1658 0.1055 0.0249 0.1583 0.0310 0.1516 0.6372
Auto 0.1774 0.1213 0.1403 0.0860 0.0272 0.1040 0.6562
industry
Table 15.  Ranking of alternatives
Rows Stocks Names Si* Si- Si*+Si- Ci Rank
1 Index funds 0.5298 1.1707 1.7005 0.6884 1
2 Computer 0.5795 1.0972 1.6767 0.6544 2
3 Durable goods 1.0106 0.6897 1.7003 0.4056 8
4 Pharmaceutical 0.9119 0.7693 1.6812 0.4576 7
5 Chip Industry 0.7848 0.9048 1.6897 0.5355 3
6 Real States 1.3496 0.3359 1.6854 0.1993 10
7 Life Insurance 0.7922 0.8895 1.6817 0.5289 4
8 Health Insurance 1.0359 0.6466 1.6825 0.3843 9
9 Tourism industry 0.8784 0.8023 1.6808 0.4774 6
10 Auto industry 0.7950 0.8906 1.6856 0.5284 5
Rows Stocks Names Si* Si- Si*+Si- Ci Rank
1 Index funds 0.5298 1.1707 1.7005 0.6884 1
2 Computer 0.5795 1.0972 1.6767 0.6544 2
3 Durable goods 1.0106 0.6897 1.7003 0.4056 8
4 Pharmaceutical 0.9119 0.7693 1.6812 0.4576 7
5 Chip Industry 0.7848 0.9048 1.6897 0.5355 3
6 Real States 1.3496 0.3359 1.6854 0.1993 10
7 Life Insurance 0.7922 0.8895 1.6817 0.5289 4
8 Health Insurance 1.0359 0.6466 1.6825 0.3843 9
9 Tourism industry 0.8784 0.8023 1.6808 0.4774 6
10 Auto industry 0.7950 0.8906 1.6856 0.5284 5
Table 16.  Final ranking by HFTOPSIS and SAW
Rows Stocks Names Rank by Hierarchical TOPSIS Rank by SWA
1 Index funds 1 1
2 Computer 2 2
3 Durable goods 8 7
4 Pharmaceutical 7 8
5 Chip Industry 3 4
6 Real States 10 10
7 Life Insurance 4 3
8 Health Insurance 9 9
9 Tourism industry 6 6
10 Auto industry 5 5
Rows Stocks Names Rank by Hierarchical TOPSIS Rank by SWA
1 Index funds 1 1
2 Computer 2 2
3 Durable goods 8 7
4 Pharmaceutical 7 8
5 Chip Industry 3 4
6 Real States 10 10
7 Life Insurance 4 3
8 Health Insurance 9 9
9 Tourism industry 6 6
10 Auto industry 5 5
Table 17.  Fuzzy returns of 10 securities (units per stock
Security 1 2 3 4 5
$ i $
Security 6 7 8 9 10
$ i $
Fuzzy $ (0.2, 2.6, 3.7) $ $ (0.4, 2.5, 3.6) $ $ (0.2, 3.4, 4.6) $ $ (0.8, 1.2, 2.8) $ $ (0.3, 2.3, 3.9) $
$ \xi_{i} $ return
Security 1 2 3 4 5
$ i $
Security 6 7 8 9 10
$ i $
Fuzzy $ (0.2, 2.6, 3.7) $ $ (0.4, 2.5, 3.6) $ $ (0.2, 3.4, 4.6) $ $ (0.8, 1.2, 2.8) $ $ (0.3, 2.3, 3.9) $
$ \xi_{i} $ return
Table 18.  Expected Values
$ \eta_{1} $ $ \eta_{2} $ $ \eta_{3} $ $ \eta_{4} $ $ \eta_{5} $ $ \eta_{6} $ $ \eta_{7} $ $ \eta_{8} $ $ \eta_{9} $ $ \eta_{10} $
Values 2.30 1.70 2.60 1.95 2.025 2.275 2.25 2.90 1.50 2.2
$ \eta_{1} $ $ \eta_{2} $ $ \eta_{3} $ $ \eta_{4} $ $ \eta_{5} $ $ \eta_{6} $ $ \eta_{7} $ $ \eta_{8} $ $ \eta_{9} $ $ \eta_{10} $
Values 2.30 1.70 2.60 1.95 2.025 2.275 2.25 2.90 1.50 2.2
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