# Optimization of Risk and Return Using Fuzzy Multiobjective Linear Programming

Authors: Darsha Panwar; Manoj Jha; Namita Srivastava
Journal: Advances in Fuzzy Systems
 (2018)

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## Abstract

Stock selection poses a challenge for both the investor and the finance researcher. In this paper, a hybrid approach is proposed for asset allocation, offering a combination of several methodologies for portfolio selection, such as investor topology, cluster analysis, and the analytical hierarchy process (AHP) to facilitate ranking the assets and fuzzy multiobjective linear programming (FMOLP). This paper considers some important factors of stock, like relative strength index (RSI), coefficient of variation (CV), earnings yield (EY), and price to earnings growth ratio (PEG ratio), apart from the risk and return and stocks which are included within these same factors. Employing fuzzy multiobjective linear programming, optimization is performed using seven objective functions viz., return, risk, relative strength index (RSI), coefficient of variation (CV), earnings yield (EY), price to earnings growth ratio (PEG ratio), and AHP weighted score. The FMOLP transforms the multiobjective problem to a single objective problem using the “weighted adaptive approach” in which the weights are calculated by AHP or choices by the investors. The FMOLP model permits choices in solution.


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## Body

## 1. Introduction

Due to the uncertainty of return it is not easy to select the stocks. The main aim of portfolio selection is to obtain an accurate ratio of the assets to ensure that the investor gets the maximum return with minimum risk.Professor Markowitz initially presented the problem of portfolio selection [1]. He proposed the Markowitz model or mean-variance model (MV) for portfolio selection reiterating the fact that investing in more than one stock is less risky than investing in a single stock. Konno and Yamazaki [2] introduced an improved version of the Markowitz model in which the risk is calculated by the mean absolute deviation (MAD). Speranza [3] advanced a linear programming model, in which the risk is calculated by the semiabsolute deviation method. Gupta et al. [4] projected the hybrid approach for portfolio selection using a combination of multiple methodologies like investor’s behavioral survey, cluster analysis, analytical hierarchy process, and fuzzy mathematical programming. Ganasekaran and Ramaswami [5] proffered a portfolio optimization model applying the neurofuzzy framework. Gupta et al. [6] obtained ethical stock performance using the AHP technique and portfolio selection done by the FMCDM technique. Mehlawat [7] presented a detailed computation procedure of the AHP and applied FMCDM technique. Sanokolaei [8] proposed the fuzzy method for portfolio optimization based on the mean absolute deviation risk function. Sadati and Doniavi [9] advocated their portfolio selection model based on the possibility model with the fuzzy random variable parameter and applied the harmony search algorithm. Konak and Bagei [10] applied fuzzy linear programming for portfolio optimization. Wang et al. [11] introduced a new risk index variable called equilibrium risk value (ERV) of the random fuzzy expected value (EV) and the EV-ERV model was used for portfolio selection.A literature survey revealed several drawbacks in the K-means algorithm used for clustering and improper scaling because it involves identification of the number of clusters. In AHP, the stocks are ranked based the criteria of return, risk, liquidity, dividend, alpha, beta and stock prices, etc.This study presents a hybrid approach for portfolio selection with multiple methodology. First, the X-means algorithm needs to be performed for cluster analysis, which is an extended version of the K-means clustering. The drawbacks have been improved in X-means. In X-means, the number of clusters does not need to be specified. Then by applying the AHP, the stocks for all three clusters must be ranked. In this paper, some new features for stock selection have been included, such as relative strength index (RSI), coefficient of variation (CV), earnings yield (EY), and price to earnings growth ratio (PEG ratio), which have not been used earlier in the AHP. Optimization is done using fuzzy multiobjective linear programming with seven objective functions viz., return, risk, relative strength index (RSI), coefficient of variation (CV), earnings yield (EY), price to earnings growth ratio (PEG ratio), and AHP weighted score. The daily closing price, number of shares, turnover rate, earning per share, price to earnings ratio, price to earnings growth ratio, and market cap for all the 15 stocks selected are taken from the BSE, Bombay Stock Exchange, Mumbai, India (https://www.bseindia.com), from February ’15 to January ’16.This paper is organized in four sections as follows: Section2 includes an account of the research methodology, the FMOLP algorithm, and its working process with reference to each of the seven objectives, viz., return, risk, relative strength index, coefficient of variation, earning yield, price to earnings growth ratio, and AHP weighted score. Section 3 presents the numerical illustrations, while Section 4 contains the concluding remarks.


## 2. Methodology

To solve the multiobjective linear programming problem, the following step-by-step strategy is used.


### 2.1. Investor Behavior Pattern

Investor behavior plays an important role in the selection of stocks as each individual stock-holder will have a specific decision-making style. Three main categories of investors can be identified, viz., money makers, liquidity lovers, and risk averse investors, according to their investment topology [12]. The survey done above is based only on the characteristics of return, risk, and liquidity. Return, risk, and liquidity are the basic factors used in stock selection; however, some more important features, as listed below, need to be considered prior to selecting the stocks:(i)
J. Welles Wilder introduced the relative strength index in 1978. This evaluates the current and historical performance of a stock based on today’s closing prices. RSI normally falls within the 30-70 range.(ii)
Coefficient of variation enables the evaluation of the value of instability relative to the return rate.(iii)
Earning yield is the percentage of each amount invested in the stock which the company has received.(iv)
A comparative calculation or relation between the stock price, EPS, and the growth of the companies is defined by the price to earnings growth ratio.(v)
Market cap is used to classify the company size, which is of greater importance than the stock price.


### 2.2. Cluster

For every investor, the approaches employed in stock selection are different. Generally, however, the investors predominantly observe all the three aspects of return, risk, and liquidity. Therefore, based on these three points, stocks can be better categorized under three groups, with qualities like high return, minimum risk, and liquid stocks. Cluster analysis is a technique used to divide data into groups by which similar objects are placed within the same cluster which is different from the other cluster objects. To formulate the clusters, the X-means [13] clustering algorithm is used. It is an extended version of the K-means which attempts to automatically determine the number of clusters. It starts with just one centroid and then iteratively increases the centroid, as required. If a cluster is divided into two subclusters, then the data distribution is done using the Bayesian Information Criteria (BIC) which is a statistical model.The proposed research includes investor topology, clustering, the AHP, and optimization technique for portfolio selection. Different investors employ different approaches for investing in the stock market. Based on the preferences, the investors are divided into three different clusters:(a)
Investors who are willing to take only higher returns(b)
Investors who are not interested in taking more risks, even if the returns are less(c)
Investors who are neither in favor of greater risk nor favor low returns and who only desire secure investment (liquidity lovers)Therefore, based on these three points, stocks are divided into three groups, with qualities like high return, minimum risk, and liquid stocks.


### 2.3. AHP

AHP technique developed by Thomas L. Saaty [14] is a multicriteria decision-making (MCDM) tool. It has a particular application in group decision-making. Hierarchy structure design, weight analysis, and consistency proof are the three main steps of AHP for ranking the object. Figure 1 shows the 4-level hierarchy structure of AHP. Firstly, form a pair wise comparison matrix for each criterion with respect to its parent criteria.Figure 1
Hierarchy structure.Each entry of the judgmental matrix A is formed by the following rules:(1)aij>0aij=1ajiaij=1,wheni=j,foralli,jTo compare two things, we have a well-defined 1-9 scale which is given by Satty.For the matrix A of order “n” the normalized Eigenvector is called Priority vector.(2)Aw=λmaxw, where “w” is known as weight of the objects. λmax is the highest Eigen value of the matrix A. The consistency index (CI) for each nth order matrix is calculated as(3)CI=λmax-nn-1The consistency ratio (CR) is calculated as(4)CR=CIRIwhere RI the random index is determined by on the order of the matrix.The matrix is consistent if CR ≤ 0.10. However, if CR> 0.10, inconsistencies exist and pairwise comparisons need revision.


### 2.4. Portfolio Selection Model

The fuzzy multiobjective linear programming (FMOLP) [15] technique is commonly used for optimization. The MOLP can be changed to a single objective utilizing the membership functions.


#### 2.4.1. Portfolio Selection Problem

The multiobjective portfolio selection problem with seven objective functions such as return, risk, relative strength index, coefficient of variation, earning yield, price to earnings growth ratio, and AHP weight and some notations are introduced as follows:ri: return of the ith stock,xi: the proportion of the total fund invested in the ith stock,bi: the binary variable indicating whether the ith stock is contained in the portfolio or not, i.e., bi={1,ifithstockcontainedinportfolio, 0,ifnotcontaininginportfolio}ki: risk of the ith stock,Ri: relative strength index of the ith stock,Ci: coefficient of variation of the ith stock,wAHPi: the AHP weight of the ith stock,Ei: earning yield of the ith stock,Pi: p/e growth ratio of the ith stock,ui: the maximum fraction of the ith stock,li: the minimum fraction of the ith stock,n: total number of stocks in each cluster (15),a: number of stock in a selected portfolio (5).


#### 2.4.2. Parameters Used

(i) Return. The return of the portfolio is written as(5)f1x=∑i=1nrixiWhere ri=1/12∑i=112rit.(ii) Risk. The semiabsolute deviation of return of the portfolio below the expected return over the past period t, t=1,2,…T, can be written as(6)ktx=min0,∑i=1nrit-rixi=∑i=1nrit-rixi+∑i=1nri-ritxi2Consequently, the expected semiabsolute deviation of return of the portfolio x=(x1,x2,x3,…xn) below the expected return becomes(7)f2x=kx=1T∑t=1Tktx=∑t=1T∑i=1nrit-rixi+∑i=1nri-ritxi2T,where k(x) represents portfolio risk.Above risk function converted into linear function as optimization technique is for linear problem(8)f2x=kx=1T∑t=1Tktx=∑t=1T∑i=1nrit-rixi+∑i=1nri-ritxi2f2v=kv=1T∑t=1Tvt,wherevt+∑t=1Trit-rixi≥0.(iii) Relative Strength Index (RSI). The RSI of the portfolio is written as(9)f3x=∑i=1nRixi,where Ri=100-100/1+avgi and avgi=avggain/avgloss.(iv) Coefficient of Variation (CV). The CV of the portfolio is written as(10)f4x=∑i=1nCixi,where ci=SDi/returni of the ith stock.(v) Earning Yield (EY). The EY of the portfolio is written as(11)f5x=∑i=1nEixi,where Ei=1/p/eratio of the ith stock.(vi) Price to Earnings Growth Ratio (PEG Ratio). The PEG ratio of the portfolio is written as(12)f6x=∑i=1nPixi,where Pi=p/eratio/growthratio of the ithstock.(vii) AHP Weight. The AHP weight of the portfolio is written as(13)f7x=∑i=1nwAHPixiwhere wAHPi is weight of ith stock.


#### 2.4.3. Constraints

Investment economical restriction on the stocks:(i) Sum of proportion of stocks should be 1(14)∑i=1nxi=1(ii) Number of stocks held in a portfolio is(15)∑i=1nbi=a(iii) The maximum percentage of the investment which can be invested in a stock:(16)xi≤uibi,i=1,2,…,n,(iv) The minimum percentage of the investment which can be invested in a stock is(17)xi≥libi,i=1,2,…,n,The upper and lower bounds have been taken to avoid too many large investments and in the same manner too many small investments.


#### 2.4.4. The Decision Problem

(18) m a x  f 1 x = ∑ i = 1 n r i x i (19) m i n  f 2 v = ∑ i = 1 n k i x i (20) m a x  f 3 x = ∑ i = 1 n R i x i (21) m i n  f 4 x = ∑ i = 1 n C i x i (22) m a x  f 5 x = ∑ i = 1 n E i x i (23) m a x  f 6 x = ∑ i = 1 n P i x i (24) m a x  f 7 x = ∑ i = 1 n w A H P i x i (25) s u b j e c t   t o  v t + ∑ t = 1 T r i t - r i x i ≥ 0 , (26) ∑ i = 1 n x i = 1 , (27) ∑ i = 1 n b i = a , (28) x i ≤ u i b i ,  i = 1,2 , … , n , (29) x i ≥ l i b i ,  i = 1,2 , … , n , (30) x i ≥ 0 ,  i = 1,2 , … , n , (31) b i ϵ 0,1 ,  i = 1,2 , … , n .Assuming that, after solving (18) with the constraints ((25)–(31)), the solution is X1, then the other objective functions are also similarly calculated at X1.When this process is repeated for (19) through to (24) you will get seven solutions with respect to each objective.Next, identify the best upper bound (ub) and worst lower bound (lb) for all the objectives.The membership function forf(x)1, f(x)3, f(x)5, f(x)6, and f(x)7 is defined by(32)μfkx=1,iffkx≥ubfkx-lbub-lb,lb≤fkx≤ub0,iffkx≤lb,k=1,3,5,6,7(33)μfhx=0,iffhx≥ubub-fhxub-lb,lb≤fhx≤ub1,iffhx≤lb,h=2,4where μf(x) is the satisfaction degree of the objective function for a given solution X.Convert the multiobjective problem into a single objective using “weighted adaptive approach” based on AHP-criteria weight in respect of each objective.Model I (34) m a x  0.1961 a 1 + 0.1569 a 2 + 0.0642 a 3 + 0.0642 a 4 + 0.1176 a 5 + 0.1176 a 6 + 0.0712 a 7 S u b j e c t   t o  f k x - u p p e r   b o u n d - l o w e r   b o u n d a k ≥ l o w e r   b o n d ,  k = 1,3 , 5,6 , 7 f h x + u p p e r   b o u n d - l o w e r   b o u n d a h ≤ u p p e r   b o n d ,  h = 2,4 0 ≤ a j ≤ 1 ,  j = 1,2 , … , 7 ∑ i = 1 n x i = 1 ∑ i = 1 n b i = a x i - u i b i ≥ 0 ,  i = 1,2 , … n , x i - l i b i ≤ 0 ,  i = 1,2 , … n , x i ≥ 0 ,  i = 1,2 , … n .The solution obtained on solving Model I is the first iteration. The old lower bound will be replaced by the first iteration, only when improvement is required. This process must be repeated until the investors are satisfied with the solution.


## 2.1. Investor Behavior Pattern

Investor behavior plays an important role in the selection of stocks as each individual stock-holder will have a specific decision-making style. Three main categories of investors can be identified, viz., money makers, liquidity lovers, and risk averse investors, according to their investment topology [12]. The survey done above is based only on the characteristics of return, risk, and liquidity. Return, risk, and liquidity are the basic factors used in stock selection; however, some more important features, as listed below, need to be considered prior to selecting the stocks:(i)
J. Welles Wilder introduced the relative strength index in 1978. This evaluates the current and historical performance of a stock based on today’s closing prices. RSI normally falls within the 30-70 range.(ii)
Coefficient of variation enables the evaluation of the value of instability relative to the return rate.(iii)
Earning yield is the percentage of each amount invested in the stock which the company has received.(iv)
A comparative calculation or relation between the stock price, EPS, and the growth of the companies is defined by the price to earnings growth ratio.(v)
Market cap is used to classify the company size, which is of greater importance than the stock price.


## 2.2. Cluster

For every investor, the approaches employed in stock selection are different. Generally, however, the investors predominantly observe all the three aspects of return, risk, and liquidity. Therefore, based on these three points, stocks can be better categorized under three groups, with qualities like high return, minimum risk, and liquid stocks. Cluster analysis is a technique used to divide data into groups by which similar objects are placed within the same cluster which is different from the other cluster objects. To formulate the clusters, the X-means [13] clustering algorithm is used. It is an extended version of the K-means which attempts to automatically determine the number of clusters. It starts with just one centroid and then iteratively increases the centroid, as required. If a cluster is divided into two subclusters, then the data distribution is done using the Bayesian Information Criteria (BIC) which is a statistical model.The proposed research includes investor topology, clustering, the AHP, and optimization technique for portfolio selection. Different investors employ different approaches for investing in the stock market. Based on the preferences, the investors are divided into three different clusters:(a)
Investors who are willing to take only higher returns(b)
Investors who are not interested in taking more risks, even if the returns are less(c)
Investors who are neither in favor of greater risk nor favor low returns and who only desire secure investment (liquidity lovers)Therefore, based on these three points, stocks are divided into three groups, with qualities like high return, minimum risk, and liquid stocks.


## 2.3. AHP

AHP technique developed by Thomas L. Saaty [14] is a multicriteria decision-making (MCDM) tool. It has a particular application in group decision-making. Hierarchy structure design, weight analysis, and consistency proof are the three main steps of AHP for ranking the object. Figure 1 shows the 4-level hierarchy structure of AHP. Firstly, form a pair wise comparison matrix for each criterion with respect to its parent criteria.Figure 1
Hierarchy structure.Each entry of the judgmental matrix A is formed by the following rules:(1)aij>0aij=1ajiaij=1,wheni=j,foralli,jTo compare two things, we have a well-defined 1-9 scale which is given by Satty.For the matrix A of order “n” the normalized Eigenvector is called Priority vector.(2)Aw=λmaxw, where “w” is known as weight of the objects. λmax is the highest Eigen value of the matrix A. The consistency index (CI) for each nth order matrix is calculated as(3)CI=λmax-nn-1The consistency ratio (CR) is calculated as(4)CR=CIRIwhere RI the random index is determined by on the order of the matrix.The matrix is consistent if CR ≤ 0.10. However, if CR> 0.10, inconsistencies exist and pairwise comparisons need revision.


## 2.4. Portfolio Selection Model

The fuzzy multiobjective linear programming (FMOLP) [15] technique is commonly used for optimization. The MOLP can be changed to a single objective utilizing the membership functions.


### 2.4.1. Portfolio Selection Problem

The multiobjective portfolio selection problem with seven objective functions such as return, risk, relative strength index, coefficient of variation, earning yield, price to earnings growth ratio, and AHP weight and some notations are introduced as follows:ri: return of the ith stock,xi: the proportion of the total fund invested in the ith stock,bi: the binary variable indicating whether the ith stock is contained in the portfolio or not, i.e., bi={1,ifithstockcontainedinportfolio, 0,ifnotcontaininginportfolio}ki: risk of the ith stock,Ri: relative strength index of the ith stock,Ci: coefficient of variation of the ith stock,wAHPi: the AHP weight of the ith stock,Ei: earning yield of the ith stock,Pi: p/e growth ratio of the ith stock,ui: the maximum fraction of the ith stock,li: the minimum fraction of the ith stock,n: total number of stocks in each cluster (15),a: number of stock in a selected portfolio (5).


### 2.4.2. Parameters Used

(i) Return. The return of the portfolio is written as(5)f1x=∑i=1nrixiWhere ri=1/12∑i=112rit.(ii) Risk. The semiabsolute deviation of return of the portfolio below the expected return over the past period t, t=1,2,…T, can be written as(6)ktx=min0,∑i=1nrit-rixi=∑i=1nrit-rixi+∑i=1nri-ritxi2Consequently, the expected semiabsolute deviation of return of the portfolio x=(x1,x2,x3,…xn) below the expected return becomes(7)f2x=kx=1T∑t=1Tktx=∑t=1T∑i=1nrit-rixi+∑i=1nri-ritxi2T,where k(x) represents portfolio risk.Above risk function converted into linear function as optimization technique is for linear problem(8)f2x=kx=1T∑t=1Tktx=∑t=1T∑i=1nrit-rixi+∑i=1nri-ritxi2f2v=kv=1T∑t=1Tvt,wherevt+∑t=1Trit-rixi≥0.(iii) Relative Strength Index (RSI). The RSI of the portfolio is written as(9)f3x=∑i=1nRixi,where Ri=100-100/1+avgi and avgi=avggain/avgloss.(iv) Coefficient of Variation (CV). The CV of the portfolio is written as(10)f4x=∑i=1nCixi,where ci=SDi/returni of the ith stock.(v) Earning Yield (EY). The EY of the portfolio is written as(11)f5x=∑i=1nEixi,where Ei=1/p/eratio of the ith stock.(vi) Price to Earnings Growth Ratio (PEG Ratio). The PEG ratio of the portfolio is written as(12)f6x=∑i=1nPixi,where Pi=p/eratio/growthratio of the ithstock.(vii) AHP Weight. The AHP weight of the portfolio is written as(13)f7x=∑i=1nwAHPixiwhere wAHPi is weight of ith stock.


### 2.4.3. Constraints

Investment economical restriction on the stocks:(i) Sum of proportion of stocks should be 1(14)∑i=1nxi=1(ii) Number of stocks held in a portfolio is(15)∑i=1nbi=a(iii) The maximum percentage of the investment which can be invested in a stock:(16)xi≤uibi,i=1,2,…,n,(iv) The minimum percentage of the investment which can be invested in a stock is(17)xi≥libi,i=1,2,…,n,The upper and lower bounds have been taken to avoid too many large investments and in the same manner too many small investments.


### 2.4.4. The Decision Problem

(18) m a x  f 1 x = ∑ i = 1 n r i x i (19) m i n  f 2 v = ∑ i = 1 n k i x i (20) m a x  f 3 x = ∑ i = 1 n R i x i (21) m i n  f 4 x = ∑ i = 1 n C i x i (22) m a x  f 5 x = ∑ i = 1 n E i x i (23) m a x  f 6 x = ∑ i = 1 n P i x i (24) m a x  f 7 x = ∑ i = 1 n w A H P i x i (25) s u b j e c t   t o  v t + ∑ t = 1 T r i t - r i x i ≥ 0 , (26) ∑ i = 1 n x i = 1 , (27) ∑ i = 1 n b i = a , (28) x i ≤ u i b i ,  i = 1,2 , … , n , (29) x i ≥ l i b i ,  i = 1,2 , … , n , (30) x i ≥ 0 ,  i = 1,2 , … , n , (31) b i ϵ 0,1 ,  i = 1,2 , … , n .Assuming that, after solving (18) with the constraints ((25)–(31)), the solution is X1, then the other objective functions are also similarly calculated at X1.When this process is repeated for (19) through to (24) you will get seven solutions with respect to each objective.Next, identify the best upper bound (ub) and worst lower bound (lb) for all the objectives.The membership function forf(x)1, f(x)3, f(x)5, f(x)6, and f(x)7 is defined by(32)μfkx=1,iffkx≥ubfkx-lbub-lb,lb≤fkx≤ub0,iffkx≤lb,k=1,3,5,6,7(33)μfhx=0,iffhx≥ubub-fhxub-lb,lb≤fhx≤ub1,iffhx≤lb,h=2,4where μf(x) is the satisfaction degree of the objective function for a given solution X.Convert the multiobjective problem into a single objective using “weighted adaptive approach” based on AHP-criteria weight in respect of each objective.Model I (34) m a x  0.1961 a 1 + 0.1569 a 2 + 0.0642 a 3 + 0.0642 a 4 + 0.1176 a 5 + 0.1176 a 6 + 0.0712 a 7 S u b j e c t   t o  f k x - u p p e r   b o u n d - l o w e r   b o u n d a k ≥ l o w e r   b o n d ,  k = 1,3 , 5,6 , 7 f h x + u p p e r   b o u n d - l o w e r   b o u n d a h ≤ u p p e r   b o n d ,  h = 2,4 0 ≤ a j ≤ 1 ,  j = 1,2 , … , 7 ∑ i = 1 n x i = 1 ∑ i = 1 n b i = a x i - u i b i ≥ 0 ,  i = 1,2 , … n , x i - l i b i ≤ 0 ,  i = 1,2 , … n , x i ≥ 0 ,  i = 1,2 , … n .The solution obtained on solving Model I is the first iteration. The old lower bound will be replaced by the first iteration, only when improvement is required. This process must be repeated until the investors are satisfied with the solution.


## 2.4.1. Portfolio Selection Problem

The multiobjective portfolio selection problem with seven objective functions such as return, risk, relative strength index, coefficient of variation, earning yield, price to earnings growth ratio, and AHP weight and some notations are introduced as follows:ri: return of the ith stock,xi: the proportion of the total fund invested in the ith stock,bi: the binary variable indicating whether the ith stock is contained in the portfolio or not, i.e., bi={1,ifithstockcontainedinportfolio, 0,ifnotcontaininginportfolio}ki: risk of the ith stock,Ri: relative strength index of the ith stock,Ci: coefficient of variation of the ith stock,wAHPi: the AHP weight of the ith stock,Ei: earning yield of the ith stock,Pi: p/e growth ratio of the ith stock,ui: the maximum fraction of the ith stock,li: the minimum fraction of the ith stock,n: total number of stocks in each cluster (15),a: number of stock in a selected portfolio (5).


## 2.4.2. Parameters Used

(i) Return. The return of the portfolio is written as(5)f1x=∑i=1nrixiWhere ri=1/12∑i=112rit.(ii) Risk. The semiabsolute deviation of return of the portfolio below the expected return over the past period t, t=1,2,…T, can be written as(6)ktx=min0,∑i=1nrit-rixi=∑i=1nrit-rixi+∑i=1nri-ritxi2Consequently, the expected semiabsolute deviation of return of the portfolio x=(x1,x2,x3,…xn) below the expected return becomes(7)f2x=kx=1T∑t=1Tktx=∑t=1T∑i=1nrit-rixi+∑i=1nri-ritxi2T,where k(x) represents portfolio risk.Above risk function converted into linear function as optimization technique is for linear problem(8)f2x=kx=1T∑t=1Tktx=∑t=1T∑i=1nrit-rixi+∑i=1nri-ritxi2f2v=kv=1T∑t=1Tvt,wherevt+∑t=1Trit-rixi≥0.(iii) Relative Strength Index (RSI). The RSI of the portfolio is written as(9)f3x=∑i=1nRixi,where Ri=100-100/1+avgi and avgi=avggain/avgloss.(iv) Coefficient of Variation (CV). The CV of the portfolio is written as(10)f4x=∑i=1nCixi,where ci=SDi/returni of the ith stock.(v) Earning Yield (EY). The EY of the portfolio is written as(11)f5x=∑i=1nEixi,where Ei=1/p/eratio of the ith stock.(vi) Price to Earnings Growth Ratio (PEG Ratio). The PEG ratio of the portfolio is written as(12)f6x=∑i=1nPixi,where Pi=p/eratio/growthratio of the ithstock.(vii) AHP Weight. The AHP weight of the portfolio is written as(13)f7x=∑i=1nwAHPixiwhere wAHPi is weight of ith stock.


## 2.4.3. Constraints

Investment economical restriction on the stocks:(i) Sum of proportion of stocks should be 1(14)∑i=1nxi=1(ii) Number of stocks held in a portfolio is(15)∑i=1nbi=a(iii) The maximum percentage of the investment which can be invested in a stock:(16)xi≤uibi,i=1,2,…,n,(iv) The minimum percentage of the investment which can be invested in a stock is(17)xi≥libi,i=1,2,…,n,The upper and lower bounds have been taken to avoid too many large investments and in the same manner too many small investments.


## 2.4.4. The Decision Problem

(18) m a x  f 1 x = ∑ i = 1 n r i x i (19) m i n  f 2 v = ∑ i = 1 n k i x i (20) m a x  f 3 x = ∑ i = 1 n R i x i (21) m i n  f 4 x = ∑ i = 1 n C i x i (22) m a x  f 5 x = ∑ i = 1 n E i x i (23) m a x  f 6 x = ∑ i = 1 n P i x i (24) m a x  f 7 x = ∑ i = 1 n w A H P i x i (25) s u b j e c t   t o  v t + ∑ t = 1 T r i t - r i x i ≥ 0 , (26) ∑ i = 1 n x i = 1 , (27) ∑ i = 1 n b i = a , (28) x i ≤ u i b i ,  i = 1,2 , … , n , (29) x i ≥ l i b i ,  i = 1,2 , … , n , (30) x i ≥ 0 ,  i = 1,2 , … , n , (31) b i ϵ 0,1 ,  i = 1,2 , … , n .Assuming that, after solving (18) with the constraints ((25)–(31)), the solution is X1, then the other objective functions are also similarly calculated at X1.When this process is repeated for (19) through to (24) you will get seven solutions with respect to each objective.Next, identify the best upper bound (ub) and worst lower bound (lb) for all the objectives.The membership function forf(x)1, f(x)3, f(x)5, f(x)6, and f(x)7 is defined by(32)μfkx=1,iffkx≥ubfkx-lbub-lb,lb≤fkx≤ub0,iffkx≤lb,k=1,3,5,6,7(33)μfhx=0,iffhx≥ubub-fhxub-lb,lb≤fhx≤ub1,iffhx≤lb,h=2,4where μf(x) is the satisfaction degree of the objective function for a given solution X.Convert the multiobjective problem into a single objective using “weighted adaptive approach” based on AHP-criteria weight in respect of each objective.Model I (34) m a x  0.1961 a 1 + 0.1569 a 2 + 0.0642 a 3 + 0.0642 a 4 + 0.1176 a 5 + 0.1176 a 6 + 0.0712 a 7 S u b j e c t   t o  f k x - u p p e r   b o u n d - l o w e r   b o u n d a k ≥ l o w e r   b o n d ,  k = 1,3 , 5,6 , 7 f h x + u p p e r   b o u n d - l o w e r   b o u n d a h ≤ u p p e r   b o n d ,  h = 2,4 0 ≤ a j ≤ 1 ,  j = 1,2 , … , 7 ∑ i = 1 n x i = 1 ∑ i = 1 n b i = a x i - u i b i ≥ 0 ,  i = 1,2 , … n , x i - l i b i ≤ 0 ,  i = 1,2 , … n , x i ≥ 0 ,  i = 1,2 , … n .The solution obtained on solving Model I is the first iteration. The old lower bound will be replaced by the first iteration, only when improvement is required. This process must be repeated until the investors are satisfied with the solution.


## 3. Numerical Illustration

The results of an experimental study built on a data set of 147 assets registered in the BSE, Mumbai, India (from February-’15 to January-’16), are as follows.


### 3.1. Cluster Analysis

For performing cluster analysis, X-means tool of the Rapid Miner version 5.2 software is used. And the initial distribution of first centroid is performed by K-means clustering. The result of the X-means algorithm is shown in Table1.Table 1
Cluster result.
Parameters Cluster 1 (46 stocks) Cluster 2 (78 stocks) Cluster 3 (23stocks) average return 0.0441 0.0154 0.0731 average risk 0.0547 0.0344 0.0744 turnover rate 0.0010 0.0005 0.0010 Category Liquid less risky high returnAs per the topology of investors discussed in Section2.1,(i)
Cluster 1 includes liquid stocks as the liquidity is highest when compared with the other clusters, and risk is medium. This cluster is suitable for those investors who are interested in liquid stocks and medium risk.(ii)
Cluster 2 includes high return stocks, as the average value of return is higher in comparison to the other clusters. This cluster is meant for those investors who are focused only on maximum returns.(iii)
Cluster 3 contains less risky stocks, as the average risk value is low when compared with the other clusters. This cluster is good for those investors who are risk averse.Symbolic representations of stocks from each cluster are shown in Table2.Table 2
Stocks for each cluster.
Symbol Cluster 1 Cluster 2 Cluster 3 S1 Whbrady Blue Star Kinetic Eng. S2 Nelco Ltd. Great Estate Tokyo Plast S3 Nocil Ltd Swaraj Engine Force Motor S4 Ceat Limited Bajfinance Kg Denim S5 Nucleus S/w Exports Ltd. Finolex Ind. Zenith Fiber S6 Sauras.Cem. Bharat Pet. Jenson Nicolson S7 Fedder.Llyod Lakshmi Mill NIIT Ltd. S8 Dcw Ltd. Jsw steel Tata Elxsi S9 Eveready Ind. India Ltd. Pel Century Ext S10 Himachal Fertilizer Swan Eng Jasch Indust S11 Timex Group Pfizer Ltd. Medi-caps S12 Camph.& All Sri Adhikari Brothers Tel. Net. Ltd. Pas.Acrylon S13 Andhra Petro Kajaria Cer. Modi Rubber S14 Sha Eng Pla Asian Paints Mafatlal Ind S15 Majestic Aut Lic Housing Finance Panyam Cement


### 3.2. Numerical Calculation of AHP Weights

In this segment under the criteria and subcriteria in AHP, stocks are ranked according to the investor preference. The weights are given in Table3.Table 3
Weight of criteria and subcriteria.
Criteria weight sub-criteria Weight Basic factor 0.3529 Risk 0.1569 Return 0.1961 Growth factor 0.2353 PEG Ratio 0.1176 Earning Yield 0.1176 Variation factor 0.2353 Relative Strength Index 0.0642 Coefficient of Variation 0.0642 Liquidity 0.1070 Market cap 0.1765Tables4–6 represent the input data for all three clusters.Table 4
Input data for Cluster 1.
Stocks Return Risk RSI CV EY PEG AHP weight S1 0.0628 0.0402 56.897 1.5707 5.61 3.24 0.1126 S2 0.0296 0.0638 50.689 5.2433 5.29 0.51 0.0466 S3 0.0369 0.0471 52.540 3.0668 6.88 0.75 0.0592 S4 0.0314 0.0544 50.626 4.8066 4.757 1.72 0.1101 S5 0.0326 0.0534 51.306 3.7950 6.04 2.5 0.0553 S6 0.0512 0.0620 51.430 3.8625 5.9 -0.97 0.0501 S7 0.0323 0.0661 50.823 4.9407 13.41 -1.23 0.0534 S8 0.0417 0.0366 52.184 2.2204 2.73 0 0.0486 S9 0.0371 0.0534 54.159 3.4756 3.86 1.3 0.0655 S10 0.0301 0.0522 50.700 3.8591 5.05 1.68 0.0713 S11 0.0673 0.0483 55.167 1.9584 0.94 39.64 0.1287 S12 0.0653 0.0486 53.923 1.7890 6.86 0.51 0.0532 S13 0.0308 0.0592 49.333 4.6172 6.6 3.12 0.0475 S14 0.0628 0.0402 56.897 1.5707 3.99 1.43 0.0552 S15 0.0556 0.0669 49.732 3.1074 5.04 0 0.0428Table 5
Input data for Cluster 2.
Stocks Return Risk RSI CV EY PEG AHP weight S1 0.0110 0.0151 51.256 3.701 2.8 1.53 0.0443 S2 0.0008 0.0161 50.669 55.590 10.91 -4.49 0.0617 S3 0.0111 0.0172 51.076 3.810 4.99 4.94 0.0609 S4 0.0342 0.0174 44.017 1.320 5.27 1.3 0.0880 S5 0.0048 0.0188 49.703 10.388 5.76 0.86 0.0416 S6 0.0192 0.0190 52.799 2.506 7.91 0.88 0.0904 S7 0.0017 0.0197 49.607 31.856 5.33 7.06 0.0712 S8 0.0058 0.0203 50.001 9.464 8.16 1.49 0.0739 S9 0.0102 0.0213 51.640 5.128 5.87 0.51 0.0489 S10 0.0186 0.0215 52.378 2.956 0.47 -20.5 0.0739 S11 0.0128 0.0218 50.822 4.196 6.37 -3.28 0.0447 S12 0.0349 0.0220 56.296 1.718 5.08 0 0.0702 S13 0.0207 0.0229 53.777 2.986 3.47 2.36 0.0590 S14 0.0073 0.0232 51.026 7.824 2.69 4.74 0.1038 S15 0.0052 0.0235 50.436 12.596 9.08 1.03 0.0674Table 6
Input data for Cluster 3.
Stocks Return Risk RSI CV EY PEG AHP weight S1 0.0924 0.0668 53.803 1.796 -5.03 0.00 0.0379 S2 0.0915 0.0721 53.639 1.891 6.19 0.84 0.0717 S3 0.0907 0.0873 55.482 2.482 4.68 -0.99 0.1235 S4 0.0892 0.0650 53.335 1.778 14.13 -6.02 0.0592 S5 0.0883 0.0478 59.405 1.269 15.15 -13.41 0.0569 S6 0.0860 0.0669 51.985 2.230 -8.41 0.00 0.0385 S7 0.0807 0.0797 54.253 2.446 1.27 0.00 0.0578 S8 0.0805 0.0647 56.497 1.887 5.34 0.56 0.1658 S9 0.0779 0.1038 49.735 3.147 12.92 -2.13 0.0446 S10 0.1027 0.0758 54.414 1.989 10.03 1.70 0.1056 S11 0.0740 0.0602 52.276 2.414 3.79 -3.25 0.0417 S12 0.0734 0.0689 50.528 4.414 17.19 0.41 0.0703 S13 0.0704 0.0790 52.559 2.946 1.16 -1.34 0.0328 S14 0.0701 0.0537 55.462 1.839 -1.57 0.00 0.0397 S15 0.0698 0.0794 52.630 3.182 15.67 -217.43 0.0540


### 3.3. FMOLP Calculation

Upper and lower bound for each cluster are given by Table7.Table 7
Upper bound and lower bound.
cluster 1 cluster 2 cluster 3 Objective Ub Lb Ub Lb Ub Lb Return 0.0661 0.0311 0.0333 0.0014 0.0980 0.0725 Risk 0.0643 0.0385 0.0222 0.0156 0.0743 0.0511 RSI 56.7298 50.7244 55.0748 50.2265 58.0193 51.4598 CV 5.0742 1.6272 43.6201 1.7508 3.8445 1.5350 EY 10.4818 2.9446 9.9886 4.4508 16.4061 4.2379 PEG ratio 23.4067 -0.2626 5.9784 -2.1260 1.2829 -82.2285 AHP weight 0.1195 0.0507 0.0971 0.0522 0.1442 0.0526The iterations for each cluster are given by Tables8–10.Table 8
Iterations for Cluster 1.
Cluster 1 f1 f2 f3 f4 f5 f6 f7 iteration 1 0.0661 0.0643 56.7298 5.0742 10.4818 23.4067 0.1195 iteration 2 0.0311 0.0385 50.7244 1.6272 2.9446 -0.2626 0.0507 iteration 3 0.0647 0.0455 55.7323 1.8668 3.1830 23.2575 0.1176 iteration 4 0.0643 0.0451 55.2060 1.7007 6.1264 2.4610 0.0774 iteration 5 0.0643 0.0451 55.2060 1.7007 6.1264 2.4610 0.0774Table 9
Iterations for Cluster 2.
Cluster 2 f1 f2 f3 f4 f5 f6 f7 iteration 1 0.0333 0.0222 55.0748 43.6201 9.9886 5.9784 0.0971 iteration 2 0.0014 0.0156 50.2265 1.7508 4.4508 -2.1260 0.0522 iteration 3 0.0332 0.0202 51.3810 2.2922 5.1847 0.7219 0.0772 iteration 4 0.0329 0.0202 51.3191 2.4010 5.1672 0.7754 0.0782 iteration 5 0.0329 0.0202 51.3191 2.4010 5.1672 0.7754 0.0782Table 10
Iterations for Cluster 3.
Cluster 3 f1 f2 f3 f4 f5 f6 f7 iteration 1 0.0980 0.0743 58.0193 3.8445 16.4061 1.2829 0.1442 iteration 2 0.0725 0.0511 51.4598 1.5350 4.2379 -82.2285 0.0526 iteration 3 0.0888 0.0562 56.8504 1.5181 14.1940 -9.6902 0.0628 iteration 4 0.0849 0.0737 55.8689 2.1708 5.4825 0.0042 0.1442 iteration 5 0.0901 0.0734 54.3120 2.0392 10.3194 -3.9619 0.0847


## 3.1. Cluster Analysis

For performing cluster analysis, X-means tool of the Rapid Miner version 5.2 software is used. And the initial distribution of first centroid is performed by K-means clustering. The result of the X-means algorithm is shown in Table1.Table 1
Cluster result.
Parameters Cluster 1 (46 stocks) Cluster 2 (78 stocks) Cluster 3 (23stocks) average return 0.0441 0.0154 0.0731 average risk 0.0547 0.0344 0.0744 turnover rate 0.0010 0.0005 0.0010 Category Liquid less risky high returnAs per the topology of investors discussed in Section2.1,(i)
Cluster 1 includes liquid stocks as the liquidity is highest when compared with the other clusters, and risk is medium. This cluster is suitable for those investors who are interested in liquid stocks and medium risk.(ii)
Cluster 2 includes high return stocks, as the average value of return is higher in comparison to the other clusters. This cluster is meant for those investors who are focused only on maximum returns.(iii)
Cluster 3 contains less risky stocks, as the average risk value is low when compared with the other clusters. This cluster is good for those investors who are risk averse.Symbolic representations of stocks from each cluster are shown in Table2.Table 2
Stocks for each cluster.
Symbol Cluster 1 Cluster 2 Cluster 3 S1 Whbrady Blue Star Kinetic Eng. S2 Nelco Ltd. Great Estate Tokyo Plast S3 Nocil Ltd Swaraj Engine Force Motor S4 Ceat Limited Bajfinance Kg Denim S5 Nucleus S/w Exports Ltd. Finolex Ind. Zenith Fiber S6 Sauras.Cem. Bharat Pet. Jenson Nicolson S7 Fedder.Llyod Lakshmi Mill NIIT Ltd. S8 Dcw Ltd. Jsw steel Tata Elxsi S9 Eveready Ind. India Ltd. Pel Century Ext S10 Himachal Fertilizer Swan Eng Jasch Indust S11 Timex Group Pfizer Ltd. Medi-caps S12 Camph.& All Sri Adhikari Brothers Tel. Net. Ltd. Pas.Acrylon S13 Andhra Petro Kajaria Cer. Modi Rubber S14 Sha Eng Pla Asian Paints Mafatlal Ind S15 Majestic Aut Lic Housing Finance Panyam Cement


## 3.2. Numerical Calculation of AHP Weights

In this segment under the criteria and subcriteria in AHP, stocks are ranked according to the investor preference. The weights are given in Table3.Table 3
Weight of criteria and subcriteria.
Criteria weight sub-criteria Weight Basic factor 0.3529 Risk 0.1569 Return 0.1961 Growth factor 0.2353 PEG Ratio 0.1176 Earning Yield 0.1176 Variation factor 0.2353 Relative Strength Index 0.0642 Coefficient of Variation 0.0642 Liquidity 0.1070 Market cap 0.1765Tables4–6 represent the input data for all three clusters.Table 4
Input data for Cluster 1.
Stocks Return Risk RSI CV EY PEG AHP weight S1 0.0628 0.0402 56.897 1.5707 5.61 3.24 0.1126 S2 0.0296 0.0638 50.689 5.2433 5.29 0.51 0.0466 S3 0.0369 0.0471 52.540 3.0668 6.88 0.75 0.0592 S4 0.0314 0.0544 50.626 4.8066 4.757 1.72 0.1101 S5 0.0326 0.0534 51.306 3.7950 6.04 2.5 0.0553 S6 0.0512 0.0620 51.430 3.8625 5.9 -0.97 0.0501 S7 0.0323 0.0661 50.823 4.9407 13.41 -1.23 0.0534 S8 0.0417 0.0366 52.184 2.2204 2.73 0 0.0486 S9 0.0371 0.0534 54.159 3.4756 3.86 1.3 0.0655 S10 0.0301 0.0522 50.700 3.8591 5.05 1.68 0.0713 S11 0.0673 0.0483 55.167 1.9584 0.94 39.64 0.1287 S12 0.0653 0.0486 53.923 1.7890 6.86 0.51 0.0532 S13 0.0308 0.0592 49.333 4.6172 6.6 3.12 0.0475 S14 0.0628 0.0402 56.897 1.5707 3.99 1.43 0.0552 S15 0.0556 0.0669 49.732 3.1074 5.04 0 0.0428Table 5
Input data for Cluster 2.
Stocks Return Risk RSI CV EY PEG AHP weight S1 0.0110 0.0151 51.256 3.701 2.8 1.53 0.0443 S2 0.0008 0.0161 50.669 55.590 10.91 -4.49 0.0617 S3 0.0111 0.0172 51.076 3.810 4.99 4.94 0.0609 S4 0.0342 0.0174 44.017 1.320 5.27 1.3 0.0880 S5 0.0048 0.0188 49.703 10.388 5.76 0.86 0.0416 S6 0.0192 0.0190 52.799 2.506 7.91 0.88 0.0904 S7 0.0017 0.0197 49.607 31.856 5.33 7.06 0.0712 S8 0.0058 0.0203 50.001 9.464 8.16 1.49 0.0739 S9 0.0102 0.0213 51.640 5.128 5.87 0.51 0.0489 S10 0.0186 0.0215 52.378 2.956 0.47 -20.5 0.0739 S11 0.0128 0.0218 50.822 4.196 6.37 -3.28 0.0447 S12 0.0349 0.0220 56.296 1.718 5.08 0 0.0702 S13 0.0207 0.0229 53.777 2.986 3.47 2.36 0.0590 S14 0.0073 0.0232 51.026 7.824 2.69 4.74 0.1038 S15 0.0052 0.0235 50.436 12.596 9.08 1.03 0.0674Table 6
Input data for Cluster 3.
Stocks Return Risk RSI CV EY PEG AHP weight S1 0.0924 0.0668 53.803 1.796 -5.03 0.00 0.0379 S2 0.0915 0.0721 53.639 1.891 6.19 0.84 0.0717 S3 0.0907 0.0873 55.482 2.482 4.68 -0.99 0.1235 S4 0.0892 0.0650 53.335 1.778 14.13 -6.02 0.0592 S5 0.0883 0.0478 59.405 1.269 15.15 -13.41 0.0569 S6 0.0860 0.0669 51.985 2.230 -8.41 0.00 0.0385 S7 0.0807 0.0797 54.253 2.446 1.27 0.00 0.0578 S8 0.0805 0.0647 56.497 1.887 5.34 0.56 0.1658 S9 0.0779 0.1038 49.735 3.147 12.92 -2.13 0.0446 S10 0.1027 0.0758 54.414 1.989 10.03 1.70 0.1056 S11 0.0740 0.0602 52.276 2.414 3.79 -3.25 0.0417 S12 0.0734 0.0689 50.528 4.414 17.19 0.41 0.0703 S13 0.0704 0.0790 52.559 2.946 1.16 -1.34 0.0328 S14 0.0701 0.0537 55.462 1.839 -1.57 0.00 0.0397 S15 0.0698 0.0794 52.630 3.182 15.67 -217.43 0.0540


## 3.3. FMOLP Calculation

Upper and lower bound for each cluster are given by Table7.Table 7
Upper bound and lower bound.
cluster 1 cluster 2 cluster 3 Objective Ub Lb Ub Lb Ub Lb Return 0.0661 0.0311 0.0333 0.0014 0.0980 0.0725 Risk 0.0643 0.0385 0.0222 0.0156 0.0743 0.0511 RSI 56.7298 50.7244 55.0748 50.2265 58.0193 51.4598 CV 5.0742 1.6272 43.6201 1.7508 3.8445 1.5350 EY 10.4818 2.9446 9.9886 4.4508 16.4061 4.2379 PEG ratio 23.4067 -0.2626 5.9784 -2.1260 1.2829 -82.2285 AHP weight 0.1195 0.0507 0.0971 0.0522 0.1442 0.0526The iterations for each cluster are given by Tables8–10.Table 8
Iterations for Cluster 1.
Cluster 1 f1 f2 f3 f4 f5 f6 f7 iteration 1 0.0661 0.0643 56.7298 5.0742 10.4818 23.4067 0.1195 iteration 2 0.0311 0.0385 50.7244 1.6272 2.9446 -0.2626 0.0507 iteration 3 0.0647 0.0455 55.7323 1.8668 3.1830 23.2575 0.1176 iteration 4 0.0643 0.0451 55.2060 1.7007 6.1264 2.4610 0.0774 iteration 5 0.0643 0.0451 55.2060 1.7007 6.1264 2.4610 0.0774Table 9
Iterations for Cluster 2.
Cluster 2 f1 f2 f3 f4 f5 f6 f7 iteration 1 0.0333 0.0222 55.0748 43.6201 9.9886 5.9784 0.0971 iteration 2 0.0014 0.0156 50.2265 1.7508 4.4508 -2.1260 0.0522 iteration 3 0.0332 0.0202 51.3810 2.2922 5.1847 0.7219 0.0772 iteration 4 0.0329 0.0202 51.3191 2.4010 5.1672 0.7754 0.0782 iteration 5 0.0329 0.0202 51.3191 2.4010 5.1672 0.7754 0.0782Table 10
Iterations for Cluster 3.
Cluster 3 f1 f2 f3 f4 f5 f6 f7 iteration 1 0.0980 0.0743 58.0193 3.8445 16.4061 1.2829 0.1442 iteration 2 0.0725 0.0511 51.4598 1.5350 4.2379 -82.2285 0.0526 iteration 3 0.0888 0.0562 56.8504 1.5181 14.1940 -9.6902 0.0628 iteration 4 0.0849 0.0737 55.8689 2.1708 5.4825 0.0042 0.1442 iteration 5 0.0901 0.0734 54.3120 2.0392 10.3194 -3.9619 0.0847


## 4. Assets Allocation

The numerical results for each cluster are shown in Table11.Table 11
Results for each cluster.
Stock cluster 1 cluster 2 cluster 3 S1 0.0225 0 0 S2 0 0 0.3435 S3 0 0 0.0560 S4 0 0.0225 0.0225 S5 0 0 0.0225 S6 0 0.5555 0 S7 0 0.0225 0 S8 0 0 0 S9 0 0 0 S10 0 0 0.5555 S11 0.5555 0 0 S12 0.3770 0.3770 0 S13 0 0 0 S14 0.0225 0.0225 0 S15 0.0225 0 0Thus, from the results it is clear that Cluster 1 contains the high liquidity stocks, Custer 2 includes the low risk stocks, and Cluster 3 has the high return stocks, although the main objective of minimization of risk and maximization of return is to be preserved.


### 4.1. Comparison

Risk/return ratio (CV) is very helpful to choosing the stocks. Investors are risk averse, as they want to consider stocks with a low risk and a high degree return.Proposed approach gives better results as compared to the approach presented in Gupta et al. [4] as the CV (risk/return) is 0.81, 0.70, and 0.61 for Cluster 1, Cluster 2, and Cluster 3, respectively, in this model while Dr. Gupta’s [4] methodology results are 0.77, 1.12, and 0.65 for the same clusters. Based on the above results, the investor would like to invest with lower CV, since the lower value of risk/return ratio indicate a better risk-return trade-off.


## 4.1. Comparison

Risk/return ratio (CV) is very helpful to choosing the stocks. Investors are risk averse, as they want to consider stocks with a low risk and a high degree return.Proposed approach gives better results as compared to the approach presented in Gupta et al. [4] as the CV (risk/return) is 0.81, 0.70, and 0.61 for Cluster 1, Cluster 2, and Cluster 3, respectively, in this model while Dr. Gupta’s [4] methodology results are 0.77, 1.12, and 0.65 for the same clusters. Based on the above results, the investor would like to invest with lower CV, since the lower value of risk/return ratio indicate a better risk-return trade-off.


## 5. Conclusion

This paper presented a hybrid approach that was adopted while investigating the problem of portfolio selection. The hybrid approach involved important components such as Behavior Survey, Cluster Analysis, AHP, and FMOLP. Cluster analysis is done using the X-means algorithm, which gives a better fit to the data in the clusters as the number of clusters was decided by itself. In this paper, a few new and important criteria like RSI, CV, EY, and the PEG ratio have been considered, which are very helpful for beginners and a good start for stock selection. The FMOLP transforms a multiobjective problem to a single objective one using the “weighted adaptive approach” in which the weights are calculated by the AHP or chosen by the investors. The FMOLP model permits choices in solution. The main advantage of the model proposed is—if the investor is not satisfied with the portfolio he/she can change the weights of objective functions or recalculate the AHP model—based on the preferences of the decision-maker and thus achieves improved results. This approach gives better results as risk/return ratio is lower which indicates better risk-return trade-off.



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