A novel methodology for portfolio selection in fuzzy multi criteria environment using risk-benefit analysis and fractional stochastic

Yahia Zare Mehrjerdi

doi:10.3934/naco.2021019

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doi: 10.3934/naco.2021019

A novel methodology for portfolio selection in fuzzy multi criteria environment using risk-benefit analysis and fractional stochastic

Yahia Zare Mehrjerdi

Department of Industrial Engineering, Yazd University, Yazd Iran

Received November 2020 Revised April 2021 Published 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.

Keywords: Portfolio Selection, Fuzzy Hierarchical TOPSIS, Fractional Programming, Chance Constrained programming, goal programing.

Mathematics Subject Classification: 90C32, 97D30.

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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Figure 1. Steps to develop a hierarchical fuzzy TOPSIS-SAW-GP-Fractional Programming approach for portfolio risk-benefit analysis

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Figure 2. The hierarchy for the selection of the most suitable Portfolio

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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)

	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)

	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)

	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

	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	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'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.

	Goal
Risks	$ (0.39, 0.58, 0.74) $
Benefits	$ (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) $

	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	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

	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

	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

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	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

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

	$ \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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American Institute of Mathematical Sciences

A novel methodology for portfolio selection in fuzzy multi criteria environment using risk-benefit analysis and fractional stochastic

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