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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:
Steps to develop a hierarchical fuzzy TOPSIS-SAW-GP-Fractional Programming approach for portfolio risk-benefit analysis
The hierarchy for the selection of the most suitable Portfolio
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)
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)
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)
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
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
Portfolio risks
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.
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 $(0.38, 0.56, 0.71)$
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 $(0.39, 0.58, 0.74)$ $(0.38, 0.56, 0.71)$
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)
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
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
: 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
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
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
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
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
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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