Fuzzy MADM Approach of Stock Ranking and Portfolio Selection in Tehran Stock Exchange

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1 Huma Resource Maagemet Research 2016, 6(3): DOI: /j.hrmr Fuzzy MADM Approach of Stock Rakig ad Portfolio Selectio i Tehra Stock Exchage Ebrahim Abbasi 1,*, Sajad Pishghadam 2, Sama Ghasemi 2 1 Professor Assistat of Faculty of Social ad Ecoomic Sciece, Alzahra Uiversity, Deh vaak, Tehra, Ira 2 Ph.D studet of Fiacial Maagemet, Islamic Azad Uiversity, Electroic Brach of Tehra, Ira Abstract Proper portfolio selectio is oe of the most importat subject i fiacial literature that follows maximized expected retur ad miimized the risk aim of ivestmet. Fuzzy multi-attribute decisio makig (FMADM) method is used to selectig ad rakig the stocks of portfolio with fuzzy aalytic hierarchy process (FAHP) ad fuzzy simple additive weightig (FSAW) as well. Fially, stocks i portfolio raked based o FAHP ad FSAW methods, ad showed that the computed rak of selected stocks i portfolio with fuzzy aalytic hierarchy process (FAHP) method i compariso with computed rak of selected stocks i portfolio with fuzzy simple additive weightig method did t have differetial result ad the ivestors ca select the criteria for portfolio selectio whether FAHP or FSAW. Keywords Portfolio Theory, Multi-Attribute Decisio Makig, Fuzzy Theory, Fuzzy Aalytic Hierarchy Process (FAHP), Fuzzy Simple Additive Weightig (FSAW) 1. Itroductio Oe of the importat features of idustrialized ad developig coutries is the existece of active ad dyamic moey ad ivestmet markets. I other words, if the savigs of idividuals with correct mechaism are directed toward the productio sector, i additio of efficiecy that brigs for the capital owers, it ca be useful as a importat fudig, to lauch ecoomic projects i the commuity ad if they eter ito uhealthy ecoomic treds, it have udesirable effects for society. Accordig to expert opiio, oe of the reasos for the uderdevelopmet of the developig coutries is low levels of fixed ivestig i these coutries. The major problem of third world coutries is the lack of appropriate structure for ivestmets of idividuals ad orgaizatios. O the other had the active participatio of ivestors i the capital market is such that the essece of the existece of Stock Exchage depeds o idividuals ivestig. Selectig a suitable portfolio is always cosidered to be oe of the most importat issues i fiacial literature associated with the aim to maximize future returs ad to miimize ivestmet risk. I this regard, differet techiques ad approaches are used, each havig advatages ad disadvatages. I additio, due to ivestmet market dyamism, i relatio to the models portfolio selectio * Correspodig author: abbasiebrahim2000@yahoo.com (Ebrahim Abbasi) Published olie at Copyright 2016 Scietific & Academic Publishig. All Rights Reserved processes ad ew eeds are always idetified. Curret methods of selectig the optimal portfolio do ot have ecessary performace. Therefore, to solve these problems, iovative approaches are of great cocer. Fuzzy Multi Attribute decisio makig method (FMADM( used i this study to rak ad to choose stocks portfolio, is oe of the iovative ways that ca assess the issue of selectio portfolio ad its performace by cosiderig the Aalytic Hierarchy Process (AHP) ad Simple Additive Weightig (SAW). I this study, we ited to aswer the questio of how fuzzy theory ca be cosistet with the ucertaity caused by the fiacial markets ad the behavior of ivestmet decisio ad how Fuzzy Multi Attribute decisio makig method ca act i lie with the stock ratigs as well as lead to selectig the stock portfolio ad determiig the portfolio based o its performace. 2. Literature Review The word "portfolio", i simple words, is referred to a combiatio of assets which is formed by a ivestor to Ivest ad to gai greater efficiecies ad to reduce risk. (Noorbakhsh, 2010) Moder stock portfolio theory is a holistic approach to the stock market. This theory, ulike the other methods (techical ad fudametal), pays attetio to the stock selectio or market basket. I other words, the macro perspective is agaist the microecoomic perspective. As well as i creatig a stock portfolio, relatioship betwee risk ad retur stock with each other as a whole is importat.

2 56 Ebrahim Abbasi et al.: Fuzzy MADM Approach of Stock Rakig ad Portfolio Selectio i Tehra Stock Exchage This view is based o mathematical ad statistical calculatios ad usig the moder portfolio theory ad optimizatio models, we ca costruct portfolios with the lowest risk relative to expected retur or the highest returs relative to expected risk (Jabbari, 2012). Moder portfolio theory (MPT), is a ivestmet theory that tries to maximize portfolio expected retur for acceptig a amout of portfolio risk, to equate or to miimize the risk to a level of expected efficiecy that this matter is fulfilled through careful choose of differet proportios obtaied by diverse assets. There are may models i the optimal portfolio selectio. The first study i formig of a optimal portfolio is semi-variace model that is preseted by Harry Markowitz i This model is based o the ormal distributio of expected stock returs. Sice with the escalatig crisis i the corporate operatig eviromet, ucertaity is also icreased, therefore the use of statistical criteria does ot give acceptable, reasoable ad adequate results. (Ustu S. K., 2010) Markowitz s view is the relatioship betwee risk ad retur. Markowitz, for the first time, proposed the optimizig of portfolio decisio makig accordig to the mea ad variace criteria. He defied variace portfolio as the sum of the weighted variace ad covariace of stock i the portfolio ad showed that the stock portfolio diversificatio ca reduce risk. He defied property of efficiet stock portfolio as follows: the property of havig miimum variace for a give yield or havig maximum retur for a give risk. He defied locus of efficiet portfolios (the differet risks) as efficiet frotier ad used a mathematical secod grade programmig for obtaiig the miimum variace for a give yield. (Abiri, 2011) Rakig (i idustry or the overall market) is oe of the strategies to achieve the cocept of coversio of raw iformatio ito relevat iformatio for decisio-makig. Rakigs ca be doe based o various parameters which are icluded as fudametal elemets of aalysis. (Ustu S. K., 2010) The aim of researchers is to use more accurate Word "portfolios egieerig" istead of the term "portfolio optimizatio". The term portfolio egieerig was first itroduced i the semial work of Jacobs ad Levy (1995), i which they proposed that equity maagers use a uified approach whe structurig their portfolios, focusig o the widest possible stock uiverse, ot o preselected groups or particular subsets of equity securities. As a effort to implemet the use of this word ad dispel doubts rooted i uiversities ad educatioal eviromet or fiacial idustry, we express the followig defiitio of egieerig portfolio: «Portfolio egieerig is a cross-discipliary field that relies o the techiques ad methods of mathematical optimizatio (sigle or multi-objective), portfolio theory, ad computer sciece to structure high-yield, well-diversified ivestmet portfolios.»(p. Xidoas, 2012) Some mathematical optimizatio methods ad their applicatio i egieerig portfolio, based o fuzzy theory ad its methods are metioed followig: Multi-criteria decisio makig The multi-criteria decisio-makig i recet decades has bee of iterest to researchers. Istead of usig the optimality criterio, the multi-criteria is used. I most cases, decisios are desired ad satisfactory for the decisio maker if decisio is evaluated accordig to multi-criteria. The criteria may be quatitative or qualitative. Multi-criteria decisio-makig models (MCDM) which are oe of the first obvious aspects of the decisio makig are classified ito two major categories of multi-objective decisio models (MODM) ad Multiple Attribute Decisio Makig model (MADM). (P. Xidoas, Multicriteria Portfolio Maagemet, 2012). Multi-objective decisio models I these models to optimize several targets simultaeously, are cosidered. The criteria for each goal may be differet with scale for other purposes. The mai differece betwee multi-objective decisio models ad multi-criteria decisio-makig models is that the former is defied i the decisio makig cotiuous space ad the secod is defied i discrete space. (P.Xidoas, Multicriteria Portfolio Maagemet, 2012) Multi-attribute decisio makig models Multiple Attribute Decisio Makig, meas decidig o the criteria which are usually i coflict. I these models, selectig a optio from amog the available optios is cosidered. I a geeral defiitio, Multi Attribute decisio makig is referred to certai decisios (the preferred type) such as assessmets, priorities, or choosig from the available optios (which sometimes should be doe betwee several coflictig attributes). (T. Hester, 2013) Fuzzy theory Fuzzy logic is a strategy by which complex systems that their modelig usig classical mathematics ad modelig methods is impossible or very difficult ca be modeled easily ad with greater flexibility. Fuzzy set theory is a extesio of traditioal set theory "which solves may of the problems associated with ucertai ad ambiguous data. This will cause that it iclude very complex problems with law low. This theory provides a strog mathematical framework ad studies pheomea vague coceptual that ca be precise. Fuzzy set theory is a valuable tool for stregtheig comprehesiveess ad reasoableess of the decisio-makig process. (Patrick T. Hester, 2013) Fuzzy Multiple Attribute Decisio Makig The cocept of combiig theory ad fuzzy Multi Attribute decisio-makig are expressed as Fuzzy Multiple Attribute Decisio Makig (FMADM). (Kiris, 2010) Fuzzy hierarchical aalysis process (FAHP) Aalytic Hierarchy Process or AHP is oe of the most

3 Huma Resource Maagemet Research 2016, 6(3): popular multi-criteria decisio-makig techiques that ca be used, whe the decisio maker is faced with several competitor optios ad decisio-makig criteria. Approach AHP (Aalytical Hierarchy Process fuzzy) is used for determiig the bechmark i terms of subjective ad idividual judgmet of each decisio-maker. (Ustu S. K., 2010) Fuzzy SAW (SAW) Simple Additive Weightig (SAW) is a valued fuctio formed by additioal simple cocessios that expresses the target achievemet uder ay stadard ad is multiplied by special weight. This method ca compesate criteria. It also gives isights for decisio-makig for the selectio of suitable alteratives ahead. (Lazim Abdullah, 2014) I this sectio we focus o providig the curret research activities i the field of portfolio maagemet ad multi-criteria decisio through the poderig the source bibliography ad literature of covergig ad similar studies. Ivestig approach i the framework of portfolio selectio, i the light of the ideas of Markowitz ad Sharpe crossed the evolutioary process ad applicatio of mathematical programmig has icreased the accuracy of ivestig decisio i the portfolio selectio. Various models for the leadig the ivestig withi the framework of portfolio selectio usig mathematical programmig is provided. Because of the developmet of fuzzy mathematics i may differet scieces ad its icreasig importace i fiacial matters, differet models are preseted to solve the fiacial problems by usig the fuzzy mathematics sciece. I this study, coducted research i the fiacial field usig portfolio theory is first discussed, ad the research i the field of portfolio selectio usig the portfolio theory ad the fuzzy set theory is expressed: Hamid Shahristai et al i their research bega to study o the geeralized theory of Markowitz portfolio optimizatio through the presetatio of their proposed model of optimal stock selectio. Fidigs of this study show that i spite of usig the CAPM theory i fiacial circles by ivestors, it was observed that this model caot be used for retail ivestors because portfolio selected by them are a subset of the market portfolio. Mohammadi ad Molaei i a research etitled the use of multi-criteria decisio gray i evaluatig the performace of compaies, usig their etropy method, raked the ivestmet ad Mother compaies listed o Tehra Stock Exchage based o fiacial ratios ad criteria. They have used the cocept of gray theory to overcome the ureliable coditios due to the lack of iformatio. Zohouri ad Fazli (2009), usig the approach of decisio-makig ad the fuzzy multi-criteria optimizatio, have preseted a ew approach for stock screeig. Two geeral criteria are used i this aalysis: The firms health criteria ad the compaies market success criteria. Fially, the basic criterio for selectig the suitable compaies to ivest is proximity of the two criteria of fiacial health ad success i the corporate market. Yahyazadehfar et al (2011) could select ad form portfolio usig the historical data ad statistical techiques of fuzzy set theory i the ew model portfolio selectio of mea-variace λ to estimate future returs. I this model, they attempted to Portfolio usig the fiacial expert judgmet ad the optimism - pessimism metality of ivestors with respect to the expected retur ad with the assumptio that stock returs are fuzzy radom variable. Heibati Farshad et al (2011) i research for optimal stock portfolio usig Aalytic Hierarchy Process (AHP), gray relatioal aalysis (GRA), ad goal programmig (GP), i ew ways, could select the shares of compaies based o ivestmet criteria based o expert opiios, the prioritize them usig gray relatioal aalysis. They the selected the optimal portfolio accordig to the priorities obtaied by goal programmig. Kav et al i a research, etitled usig the gray relatioship aalysis for solvig multi-criteria decisio, described this model. I this study, two case studies are solved usig gray relatioal aalysis ad the resultig solutio is compared with the solutios that is obtaied by solvig the problems usig the data coverig aalysis method, usig the TOPSIS method ad usig the weighted simple sum method (SAW). Rakig the optios by gray relatioal aalysis is closer to the results of TOPSIS ad simple weighted average. Seffak Kirish et al (2010) preseted the fuzzy multi-criteria decisio-makig approach to portfolio selectio uder ucertaity. They bega to rak the stock portfolio of compaies listed o the Istabul Stock Exchage by a accurate defiitio of the circumstaces ad patters of liguistic (sematic). I their research, they also preseted a flexible decisio-makig method for the itegratig of ivestor s prefereces. The hypothesis Preseted the fuzzy multi-criteria decisio-makig approach to portfolio selectio uder ucertaity. They bega to rak the stock portfolio of compaies listed o the Istabul Stock Exchage by a accurate defiitio of the circumstaces ad patters of liguistic (sematic). I their research, they also preseted a flexible decisio-makig method for the itegratig of ivestor s prefereces. 3. Methodology I this research, the attitude FMADM (Fuzzy Multi Attribute Decisio Makig) is preseted to rak the stock ad to select the portfolio to establish a performace evaluatio method for ivestors. Geeral steps of proposed method are as followig: A) Idetifyig idicators for the stock performace to determie criteria for the costructig a framework for performace evaluatio. B) The use of fuzzy AHP to fid the weight criteria usig idividual ad subjective guesses.

4 58 Ebrahim Abbasi et al.: Fuzzy MADM Approach of Stock Rakig ad Portfolio Selectio i Tehra Stock Exchage C) Applicatio of SAW fuzzy (FSAW) to the stock performace rakig. D) Create a portfolio usig the stock performace. (Ustu S. K., 2010) As this study states rakig the stock ad portfolio selectio based o multi-criteria decisio, so accordig to the defiitio of multi-criteria decisio makig, idetifyig ad desigatig them as studied variable criteria (criteria) is carried out. Oe of the mai criteria that ca be take ito accout as a criterio i hypothesis of study, is the fiacial structure criteria, which ca be defied as expected returs, the ratio of price to earigs per share, profit (loss) et, the market price (value office) etc. give i the hierarchical structure evaluatio i fuzzy portfolio selectio problem. That is, whe the ivestmet strategy for portfolio selectio is evaluated accordig to the metioed aspects, they ca be proposed as a fuzzy multi-attribute decisio problem. After that the decisio-makig criteria problem is determied, weightig the variables should be doe based o AHP fuzzy ad aaccordig to the metal guesses ad the liguistic variables. Afterwards, based o the weighted simple fuzzy sum ad accordig to the three steps of evaluatio (evaluatig the variables), fuzzy combied decisio ad rakig the fuzzy umber by without scalig the data, the performace of the stock is raked. Fially, the stock portfolio is expressed by stock performace ad its rakig. (Ustu S. K., 2010) Software MATLAB will be used for performig the data aalysis for the implemetatio of fuzzy multi-criteria decisio-makig methods, to rak ad select the stock portfolio. The populatio i this study is related to all public compaies listed o the Tehra Stock Exchage which were active i the cosidered period. The sample based o the limitatios of the statistical populatio is determied as follows: 1. All active compaies i leadig idustries that are accepted by the ed of 2009 at Tehra Stock Exchage, ad by the ed of 2012 still exist i the markets list of security exchage. 2. They fiscal year eded March each year. 3. Durig this period, ot havig more tha three moth trasactioal iterruptio. 4. Compaies that their atures are ot ivestig compaies, because we do ot ited to form ew portfolio from the stock portfolio. 5. Durig this period the three-year fiacial iformatio is available ad their iformatio ca be accessed. O this basis, ad cosiderig the limitatios metioed, amog all compaies i the Tehra Stock Exchage, 62 compaies were selected from the commuity. Do ay research to aswer the research questios ad hypothesis testig ivolves detectig, idetifyig ad defiig each of the variables is accurate. Variables based o hierarchical structure charts as well Multiple Attribute Decisio Makig Based o Fuzzy approach to the problem of fuzzy portfolio selectio are expressed as follows: The FAHP method Hierarchical fuzzy cocepts used i the aalysis are briefly described below: Fuzzy umber (a qualitative criteria to quatitative criteria) The cocepts used i fuzzy hierarchical aalysis is briefly described below: Fuzzy umber (covertig the qualitative criteria to quatitative oe): Fuzzy umbers are a fuzzy subsets of real umbers that idicates the extet of the cofidece iterval. This umber is proposed i the form of a triagular fuzzy umber (TFN), is used for weightig ad value of success of predictio. Accordig to the defiitio Larhuve ad Pdrych i 1983, a triagular fuzzy umber (TFN) must have the followig basic characteristics: (Kiris, 2010) Target Stock valuatio Dimesios :The purpose of ivestor : Fiacial Structure : Sustaiable Developmet Structure Criteria : egative deviatio : profit / et loss : R&D : positive deviatio : Equity : Quality Maagemet System : Retur : Market price / book : corporate image : Costs value Figure 1. The hierarchical structure evaluatio

5 Huma Resource Maagemet Research 2016, 6(3): A fuzzy umber oe o the R, if a member of the μ A (x): R 0,1, a triagular fuzzy umber (TFN) ad is equal to: X L M L, L x M μ A (x) = U X U M, M x U 0, otherwise Where L ad U respectively as upper ad lower boudaries are fuzzy umber A, ad M is defied as the omial value. Triagular fuzzy umber (TFN) ca be expressed as A = (L, M, U) be show ad operatioal rules two triagular fuzzy umber A 1 = (L 1, M 1, U 1 ) ad A 2 = (L 2, M 2, U 2 ) is show as follows: A) Collectig a fuzzy umber: A 1 + A 2 = L 1, M 1, U 1 + L 2, M 2, U 2 = (L 1 + L 2, M 1 + M 2, U 1 + U 2 ) B) Subtract a fuzzy umber: A 1 A 2 = L 1, M 1, U 1 L 2, M 2, U 2 = (L 1 L 2, M 1 M 2, U 1 U 2 ) C) Multiplied by a fuzzy umber: A 1 A 2 = L 1, M 1, U 1 L 2, M 2, U 2 = (L 1 L 2, M 1 M 2, U 1 U 2 ) D) For U i > 0 ad M i > 0 ad L i > 0 ito a fuzzy umber will be as follows: A 1 A 2 = L 1, M 1, U 1 L 2, M 2, U 2 = ( L 1 L 2, M 1 M 2, U 1 U 2 ) E) For U i > 0 ad M i > 0 ad L i > 0 reverse a fuzzy umber will be as follows: A 1 1 = (L 1, M 1, U 1 ) 1 = ( 1 1 1,, ) L 1 M 1 U 1 Liguistic variables (mea) I this research, computatioal techiques are based o fuzzy umbers defied by Karama ad his colleagues i 2006, accordig to Table 1: Reverse triagular fuzzy umbers Table 1. Liguistic variables Triagular fuzzy umbers Prefereces Num (1,1,1) (1,1,1) Equally importat 1 (2 1 3/2) (0.5,1,1.5) Almost idetical 2 (1 3/2 2/1) (1,1.5,2) Slightly more importat (3/2 2/1 5/2) (1.5,2,2.5) More importat 4 (2/1 5/2 3/1) (2,2.5,3) Much more importat (5/2 3/1 7/2) (2.5,3,3.5) Quite importat Liguistic variables have first bee used for evaluatig of sematic ratios that by ivestors to compare biary pairs ad the importace of stadards i AHP was give. As well as the performace of variables for each criterio are used, as a method to calculate by the liguistic coditios as "very good", "good", "relatively good", "weak" ad "very weak". Procedure ad methods to determie ad evaluate weights of criteria By FAHP ca be classified accordig to the followig steps: First stage: Formatio of Comparative matrix pairs (pairs) amog all elemets (criteria) i the size of hierarchical system ad to idetify ad assig coditios ad patters of liguistic (sematic) to compare pairs of double stadards by usig this questio which of the two elemets (criteria) are more importat. Secod stage: The use of geometric mea techique i the expressio of geometric mea fuzzy ad fuzzy weight of each criterio based o the model of Buckley is accordig to the followig relatioship: r i = (a i1 a i2 a i ) 1 w i = r i (r1 + + r) 1 Where a i is i comparative value of criterio i relative to the criterio. Therefore, r i is fuzzy geometric mea comparative value criterio i compared to other criteria ad where w i fuzzy weight of i-th criterio, ca be deoted by a triagular umber fuzzy as w i = (Lw i, Mw i, Uw i ). Here Lw i, Mw i ad Uw i are respectively low, average ad up value of i-th criterio. (Ustu S. K., 2010) SAW phase (Fuzzy SAW): Additioal simple weightig (SAW) is the valued fuctio based o additioal cocessios simple form that expresses the target achievemet uder ay criterio ad is multiplied by special weight. This method is able to compesate criteria. It also provided isight ito the decisio-maker for creatig the selectio of available suitable alteratives. Its calculatio is simple ad ca be doe without the aid of complicated computer programs. FSAW ca be expressed as follows: Measure the variables: By usig the assessmet of liguistic variables (sematic) to show the performace criteria by such phrases as "very good", "good", "relatively good", "weak" ad "very weak", ivestors will be asked to judge their metal trasfer ad each variable liguistic (sematic) ca be show by a TFN (triagular fuzzy umbers) with the scale betwee the Cosider E ij k for deotig the value of fuzzy performace of ivestor k which states variable i uder criterio j, ad all the evaluatio criteria will be show as follows: E ij k = LE ij k, ME ij k, UE ij k

6 60 Ebrahim Abbasi et al.: Fuzzy MADM Approach of Stock Rakig ad Portfolio Selectio i Tehra Stock Exchage This study uses the cocept of mea value, for itegratig the value of judgmet fuzzy ivestor m that is as follow: E ij = 1 m E ij 1 + E ij E ij m The value of edpoit LE ij, ME ij ad UE ij of the average fuzzy umber ( E ij ) ca be solved by method of Buckley as follow: m LE ij = ( LE k ij ) /m ; ME ij = ( UE ij = ( k=1 k k=1 UE ij k Fuzzy sythetic decisio )/m m k=1 ME ij k )/m ; Accordig to the weight of each of the above criteria, the weight w j is derived from FAHP that directio (vector) w = weight criteria ca be deoted w = (w 1,, w j,, w ) t, while fuzzy performace matrix E i each of the variables ca be obtaied by each fuzzy performace value of variable uder criterio ad is equal to E = (E ij ). (Ustu S. K., 2010) Approximatio of fuzzy umber R i of Fuzzy Sythetic Decisio variables ca be deoted R i = (LR i, MR i, UR i ). Cosiderig the fact that LR i, MR i ad UR i are respectively the value of low, medium ad high combied performace of variable i which is as follow: LR i = UR i = j=1 j=1 LE ij UE ij Lw j ; MR j = ME ij Mw j ; Uw j Rakig the fuzzy umbers I stock ratig, oe ca try to form the portfolio accordig to the BNP values calculated for each variable. (Ustu S. K., 2010) I this study, procedure of defuzzificatio is defied i the form of determiig the value of the best o-fuzzy performace (BNP), which is simple ad practical method ad the does ot eed to cosider the prefereces of ivestors. The best o-fuzzy performace (BNP) of fuzzy umber R i ca be obtaied by the followig equatio: j=1 BNP i = UR i LR i + MR i LR i /3 + LR i, i Problem of stability ad determiig the fuzzy biary compariso matrix i FAHP is aother issue that eeds to be addressed. Stability ad determiatio of compariso matrix i AHP is evaluated by the ratio stability. But the results of fuzzy sythetic decisio obtaied by fuzzy umbers are due to the liguistic judgmets. Hece, it is ecessary that o-fuzzy rakig method or i other words defuzzificatio or makig o- fuzzy techique is used. Defuzzificatio is a techique for covertig fuzzy umbers to the real oes. There are several methods for this purpose. I this study, we have used the best o-fuzzy performace method. Implemetatio of fuzzy AHP method Step Oe: outliig the model Because i this study, the AHP method is used for rakig ad selectio of the portfolio, so the first step is to create a hierarchical structure. At this poit, usig the iformatio obtaied from the previous steps, the hierarchy was established. The, to outlie hierarchy, depedecy relatioships betwee criteria was evaluated by a team of experts accordig to correspodig idices ad the compaies icluded i the portfolio, through omial group method. The research variables by hierarchical structure diagram as well Multi Attribute Decisio Makig Based o fuzzy approach to solve the problem fuzzy portfolio selectio are expressible accordig figure 1. Step Two: Pairwise comparisos ad calculatig the idexes Compatibility: After collectig questioaires paired comparisos, Compatibility idices were calculated for all matrices, to esure paired comparisos umbers. CI = max 1 ; CR = CI RI where i: CI: Compatibility Idex CR: Rate Adjustmet RI: Radom Idex : umber of factors compared Icosistecy rate of matrices less tha 0.1 is certified ad reliable Accordig to the carried out calculatios ad based o collected the data, fially, the rate of icompatibility matrices was that is less tha 0.1. Step Three: formig the judgmet matrix To calculate the weighted geometric mea of commets, it is operated accordig to followig relatio: a ij G = K=1 a ijk β K 1 β K = K=1 a ijk β K = 1,, m, K = 1,,, i, j Step Four: Calculate the weight of evaluatio criteria I the followig steps, the calculatios of weight of evaluatio criteria accordig to the fuzzy aalytic hierarchy process are preseted: Calculatig S i for each pair-wise compariso matrix row: After calculatig the weighted geometric mea compariso matrix, S i, which is a triagular fuzzy umber for each of the rows of the matrices, is calculated from the followig relatio: S i = m j =1 j M gi m i=1 j =1 j M gi I this relatio i represets the i-th row ad j represets 1

7 Huma Resource Maagemet Research 2016, 6(3): j the j-th colum. I this relatio M gi are triagular fuzzy umbers are pair-wise compariso matrix. Calculatio of magitude of S i relative to each other: I geeral, if S 1 = l 1, m 1, u 1 ad S 2 = l 2, m 2, u 2 are two triagular fuzzy umbers, magitude S 2 relative to the S 1 is defied as follows: = V S 2 S 1 = hgt S 1 S 2 = μ S2 d 1, if m 2 m 1 0, if l 1 u 2 l 1 u 2 m 2 u 2 m 1 l, 1 otherwise Calculatig the magitude degree of a covex fuzzy umber: Calculate the magitude of a covex fuzzy umber S that is greater tha K covex fuzzy umber S i that i = 1,2,3,, k is obtaied as follows: mi V S S i = V S S 1, S 2,, S k = V S S 1 ad S S 2 ad ad S S k i = 1,2,3,, k Calculatig the weight or importace of criteria: Followig relatio is used to calculate the weight of the criteria: d A i = mi V S i S k k = 1, 2,, k i So ot ormalized weight vector will be as follows: W = d A 1, d A 2,, d A T ; A i i = 1,2,, Calculatio of the fial weight i the form of ormalized criteria: By help of the above relatio, o-ormal relatioship results obtaied from the previous step is ormalized, obtaied ormalized results is called W. W = d A 1, d A 2,, d A T Table 2. Calculate the relative weights of the criteria Criteria Ivestmet objectives Fiacial structure Sustaiable developmet structure The relative weight Step Five: calculatio of the fial weight of criteria I this step, depedece matrix was formed usig the matrices of fial weight of weighted average of pairwise compariso of the total weight iterdepedece criteria. The fial weight of each criterio was obtaied accordig to followig relatio from the depedecy matrix multiplicatio multiplied by the relative weightig matrix criteria, obtaied from a weighted average pairwise compariso matrices criteria. I the followig the way of calculatig the fial weight of criteria is give: N j = B. j N j : fial weight of the j-th criterio B: depedecy matrix criteria j : the relative weight of the j-th criterio Table 3. Calculate the fial weight of the criteria Criteria Ivestmet objectives Fiacial structure Sustaiable developmet structure = The fial weight 0,3758 0,3501 0,2741 Step Six: calculatig the relative weight of idices The first idex icludes 4 sub-criteria ad secod ad third criteria, each iclude three sub-criteria. To determie the relative weights of the sub-criteria, we act similar the steps 1 to 5 which was metioed i step 4. The relative weight of ivestor s purposes: I this step usig paired comparisos of the quadruple criterio idices of ivestors purposes relative to each other, the relative weight of each was obtaied accordig to the correspodig judgmet matrix. Table 4. Compute the relative weights ivestmet goals Idices Negative deviatio (D 11 ) Positive deviatio (D 12 ) Retur (D 13 ) Costs (D 14 ) The relative weight The relative weight of the fiacial structure: With paired comparisos of triple idices of criterio of fiacial structure relative to each other, the relative weight of each was obtaied accordig to the correspodig judgmet matrix. Table 5. Compute the relative weights i the fiacial structure Idices Profit (D21) Equity (D22) Market price / book value (D23) The relative weight The relative weight of the sustaiable developmet idices: Triple idices of sustaiable developmet criterio were compared relative to each other ad relative weight of each was obtaied accordig to the correspodig judgmet matrix. Table 6. Compute the relative weights i the Idex of Sustaiable Developmet Idices R & d (d31) Quality maagemet system (d32) Corporate image (d33) Relative weight

8 62 Ebrahim Abbasi et al.: Fuzzy MADM Approach of Stock Rakig ad Portfolio Selectio i Tehra Stock Exchage Step Seve: The fial weight calculatio idices I this step, the fial weight of each of the idices has bee obtaied accordig to followig relatio by multiplyig its relative weight by the fial weight of its leaders. Accordig to the above defiitio, the fial weight of each of the idices is calculated i the followig table: W i = w i N ij W i : fial weight of the i-th idex w i : relative weights of the i-th idex, N ij : fial weight of the j-th criterio, leaders of the i-th idex Criteria Table 7. Calculate the fial weight i the idices Weight Idices D 11 Relative weight Fial weight D D D 13 D 14 D D D 22 D 23 D D D 32 D Step Eight: calculatio of relative weight of portfolio compaies After calculatig the fial weight of each of the idices, the relative weight of each of the compaies i the portfolio was calculated accordig to the paired comparisos. I this poit, every compay existig i the portfolio (optios) relative to the idividuals sub-criteria are compared with each other. Step Nie: Calculate the weight of portfolio compaies Sice the fial weight of each of the portfolio compaies is obtaied by a combiatio of their ratigs with respect to the idices, the weight of each of the portfolio compaies is obtaied by the followig equatio. R k = i W i r ik, k R k : fial weight of the k-th portfolios compay W i : fial weight of the i-th subcriteria r ik : the relative weight of the k-th portfolio compay with respect to the i-th sub-criteria. Step Te: Prioritizatio of portfolio compaies Accordig to the fial weights obtaied from existig compaies i the portfolio i the previous step ad accordig to the criteria ad existig coditios, Rakig the selected compaies carried out. Implemetatio of SAW (SAW fuzzy) Step Oe: weightig idices As previously metioed, first step i the SAW method is toweigh idices. Similar to the fuzzy AHP method for weightig the idices, we utilized paired comparisos. Step Two: The use of fuzzy approach to form decisio-makig matrix After determiig the amout of weights, criteria, i the ext step it is ecessary that a value is assiged to each of the ivestors (portfolio compaies) accordig to the above-metioed idices. As previously metioed, sice the decisio-maker provides values of the decisio matrix i the form of subjective ad qualitative based o idices, because of the beig verbal ature of the variables. Fuzzy approach ca be used to iitialize them. fuzzy umbers used i this research that are itroduced to assess ivestor j agaist the idex i, is deoted i the form of a ij = (LE ij, ME ij, UE ij ). I this approach, Valuatio spectrum for each idex is 5-poit Likert spectrum ad for each optio (very low, low, medium, high ad very high) has bee defied to be a triagular membership fuctio ragig from zero to 100. I this way, for the optio quite low, fuzzy umber (60, 30, 0) is defied, for the bottom optio, fuzzy umber (70, 40, 10) is defied, for medium optio, fuzzy umber (80, 50, 20) is defied, for the top optio, fuzzy umber (90, 60, 30) ) is defied ad for high optio, fuzzy umber (100, 70, 40) is defied. Step Three: defuzzificatio decisio matrix with the ceter area method After determiig the fuzzy values of ivestors, for the coveiece of calculatio it is ecessary that all fuzzy umbers are coverted to certai umbers or so-called defuzzy. For defuzzificatio of the fuzzy values, the ceter area method is used. Followig relatio shows formula of ceter area for the defuzzificatio: CA ij = UE ij LE ij + ME ij LE ij + LE 3 ij, i, j where CA ij idicates the defuzzificated value of a fuzzy umber. Step Four: evaluatig ad prioritizig short-term goals usig SAW techique SAW techique is oe of the oldest methods applyig i the approach MADM. I this method, by havig a vector W (weight idex) ad the value o-scaled of ay optio with respect to each Idex ( ij ), the most suitable optio for (A ) ca be calculated accordig to followig relatio. A = A i max j =1 ij w j I other words, i this way, a better optio is that the sum of the weighted values Scale ( j =1 ij w j ) is greater tha the other optios. This method requires similar scales or o-scaled measuremets that ca be compared with each other optio. Fidigs of research usig the methods of AHP ad SAW

9 Huma Resource Maagemet Research 2016, 6(3): I each study, proceedig the fidigs ad its results are cosidered to be importat part. I fact, the detailed aalysis ad correct coclusios of the collected data, because they are ivestigated as the basis for future plaig i society, are of much more importat. The obtaied results should be documeted ad be explaiable to the scietific commuity to esure its accuracy, ad ca be used for rectifyig the defects ad shortcomigs ad typically movig to the commercializatio of research. The results obtaied by the applicatio of AHP I this study, we were to apply other criteria i additio to the traditioal criteria for stock selectio ad portfolio risk ad retur i the rakig. These criteria were idetified by the hierarchical structure diagram ad by the goals of ivestors ad by the relatioships betwee criteria ad sub criteria, after collectig the opiio of experts, through completig the questioaires desiged ad paired comparisos based o liguistic variables. The, to esure the umbers pairwise comparisos, compatible stadards were calculated for all matrices. Icosistecy rate obtaied for each matrix should ot be greater tha 0.1, otherwise the matrix is icompatible. Calculatio of the rate of icompatibility Showed that this rate is 0.081, which is less tha 0.1, ad is ot a sig of icompatibility of matrices. I the ext step, the relative weight of criteria was obtaied ad showed that the relative weight of stadard ivestor purposes ( D 1 ) weight ad criteria of fiacial structure (D 2 ) with a weight of As well as the criteria of sustaiable developmet with the relative weight of , are respectively i the secod ad the third priority ad reflects the fact that idividuals are payig more attetio to the purpose criteria of ivestig. I determiig the fial weightig criteria, it was foud that the fial weight of ivestor s purposes criteria of weigh , has the most weight ad fiacial structure criteria ad the structure of sustaiable developmet of weights ad , respectively, are ext i rak. Oe of the mai reasos for the rakigs i determiig the fial weight is ivestors desire to get more average returs from the ivestig doe i stocks of selected compaies which the sub-criteria (average retur) is related to criteria of purpose of the ivestor s ivestig. Like the relative weight of the criteria, the relative weight of their sub criteria was also tested i a hierarchical diagram. Calculatios show that relative weight of the average returs, amog the sub criteria related to the ivestig purpose criteria i a hierarchical diagram, it has more weight ad its weight is equal to Similarly, the stadard deviatio of the egative (less risk) of the value is i secod priority ad sub positive deviatio (risk) ad the cost respectively with values of ad , are i the ext raks. This rakig idicates that from the viewpoit of experts, amog the sub-criteria related to the mai criterio for the purpose of ivestig, ivestors pay more attetio to average returs sub-criteria. Calculatios show that i determiig the relative weight criteria, the fiacial structure criterio (D 2 ), the relative weight of the profit criterio, by assigig weight , has the highest weight ad is i first rak. This rakig idicates that from the view of experts, amog the sub-criteria related to the mai criterio of the compay's fiacial structure, ivestors pay more attetio to sub-criteria profit. Calculatios show that i determiig the relative weights of the sub-criteria of sustaiable developmet criterio (D 3 ), the relative weight of the stadard compay image by assigig relative weights of is i the first rak ad the criteria for R & D ad quality maagemet system respectively with assigig the weight of ad are respectively i secod ad third. This rakig idicates that from the view of experts, amog the sub-criteria related tomai criterio of sustaiable developmet of a compay structure, ivestors pay more attetio to compay image sub-criteria. I determiig the fial weight of all the sub-criteria ad accordig to the fial weight criteria, It was foud that sub-criteria et profit by assigig a weight of is i first rak with the highest weight Ad sub-criteria of average efficiecy, equity ad corporate image respectively by assigig weights to , ad are i secod up fourth ad the rest of the criteria, likewise were ext i rak. The results obtaied by the applicatio of SAW I implemetatio of FSAW method due to the eormous calculatio i the FAHP method, this method has bee used as a alterative ad compariso method we acted i followig way: The idex weight ifluecig the ivestor selectio was doe by pairwise compariso method. I the ext step, the decisio matrix is formed ad assiged by the fuzzy umbers. After the makig matrix usig BNP which is oe of the ceter area method i fuzzy theory ad method FSAW, fuzzy umbers became defiitive. 4. Coclusios I this study, we proceed to review the issues related to the selectio ad stock ratigs based approach FMADM i the Tehra Stock Exchage. Topics such as ew portfolio theory, ad fuzzy multi-criteria decisio-makig methods i its implemetatio, fuzzy hierarchical aalysis process (FAHP) ad how to implemet it i determiig ad rakig the stocks i the portfolio, fuzzy SAW method (FSAW) ad how to implemet it i the selectio ad rakig of portfolios i a effort comparable with fuzzy hierarchical aalysis process (FAHP) was expressed. Although the aalytic hierarchy process (AHP) was developed by hour i 1980, a very useful tool for decisio aalysis i relatio to the issue of multi-criteria decisio takes ito accout, but i this study, fuzzy Aalytic Hierarchy Process ( FAHP) based o the method of Buckley

10 64 Ebrahim Abbasi et al.: Fuzzy MADM Approach of Stock Rakig ad Portfolio Selectio i Tehra Stock Exchage i years 1985, ad also the method of Karama ad their colleagues i the year 2006 for the aalysis of the fuzzy hierarchical stated ad type of expaded AHP model is hour model, has bee used i order that i some cases, the creditors would be allowed to make use of ratio fuzzy istead of ratio accurate. The preset research activity is i the field of coectivity portfolio maagemet (PM) ad Multiple Attribute Decisio Makig which has bee developed with valuable suggestios to improve the efficiecy of people ivolved i the ivestmet activities either directly or idirectly. Fuzzy multi attribute Method (MADM), used i the research to rak ad to select stock portfolio, is oe of the iovative methods that ca evaluate the stock portfolio selectio problem ad its performace by cosiderig the techique of aalytic hierarchy process (AHP) ad also Simple Additive Weightig (SAW). Accordig to the obtaied results, it was determied that the results of these two are almost similar to each other ad there is o discerible differece i the rakig of ivestig by the two methods. Prioritizig the existig firms i the stock portfolio was also carried out by two ways of fuzzy SAW ad fuzzy AHP. Therefore Accordig to the results, ivestors are able for selectig ad rakig the stock portfolio, use separately each of the fuzzy multi-criteria decisio-makig methods approach, cosisted of two methods fuzzy hierarchical aalysis process (FAHP) ad SAW phase (FSAW). Researchers suggestio to ivestors is to use both techiques simultaeously so that they ca obtai better results ad comparable by usig the two methods of FAHP ad FSAW i decisio-makig. REFERENCES [1] Abbasi Ebrahim, Moghadasi motahare. (2010). Selectio ad portfolio optimizatio usig geetic algorithms based o differet defiitios of risk. Joural of Idustrial Maagemet, Islamic Azad Uiversity. Fifth year. No. 11. pp [2] Chig-Lai Hwag, Sho-Je Che. (1992). Fuzzy Multiple Attribute Decisio Makig. Berli: Spriger. Book Volume 375. [3] Erique Ballestero, Mila Bravo, Blaca Pérez-Gladish, Mar Areas-Parra, David Plà-Satamaria. (2012). Socially Resposible Ivestmet: A multicriteria approach to portfolio selectio combiig ethical ad fiacial objectives. Europea Joural of Operatioal Research, Volume 216, Issue 2, 16 Jauary 2012, Pages [4] Hasa Selim, Mualla Goca Yuusoglu. (2013). A fuzzy rule based expert system for stock evaluatio ad portfolio costructio : A applicatio to Istabul Stock Exchage. Expert System with Applicatio, Volume 40, Issue 3, 15 February 2013, Pages [5] Heibati Farshad, Rahama Roodposhti Fereidoo, M.A Afsharkazemy, ad A.H Abiry, (2011). Portfolio selectio model evaluatio usig Aalytical Hierarchy Process (AHP), gray relatioal aalysis (GRA) ad goal programmig (GP). Joural of Fiacial Egieerig ad Maagemet portfolios. Volume 2. (6). Pages [6] Lazim Abdullah, C.W. Rabiatul Adawiyah. (2014). Simple Additive Weightig Methods of Multicriteria Decisio Makig ad Applicatio: A Decade Review. Iteratioal Joural of Iformayio Processig ad Maagemet (IJIPM), Volume 5, Number 1, February [7] Nese Yalci Secme, Ali Bayrakdaroğlu, Cegiz Kahrama. (2009). Fuzzy performace evaluatio i Turkish Bakig Sector usig Aalytic Hierarchy Process ad TOPSIS. Expert System with Applicatio, Volume 36, Issue 9, November 2009, Pages [8] Ozde Ustu, Safak Kiris. (2010). FUZZY MCDM APPROACH OF STOCKS EVALUATION AND PORTFOLIO SELECTION. 24th Mii EURO Coferece Cotiuous Optimizatio ad Iformatio-Based Techologies i the Fiacial Sector (pp ). Izmir - Turkey: Izmir Uiversity of Ecoomics, Turkey [9] Paos Xidoas, Mavrotas, G., Kritas, T., Psarras, J., Zopouidis, C. (2012). Multicriteria Portfolio Maagemet. Spriger Optimizatio ad its Applicatio, Spriger Sciece + Busiess Media New York 2012, Pages [10] Patrick T. Hester, Mark Velasquez. (2013). A Aalysis of Multi-Criteria Decisio Makig Methods. Iteratioal Joural of Operatio Research, Volume 10, No. 2, Pages [11] Safar Fazli ad Taghizadeh Rasoul. (2010). Fuzzy rakig method for the optimum portfolio selectio i Tehra Stock Exchage. Joural of Idustrial Maagemet Studies, Volume 8 Issue 19, Pages [12] Yahyazadehfar Mahmoud, Safaei Qadikolaei Abdul Hamid ad Khakpoor Mehdi, (2011). Compare the models at radom ad fuzzy radom stock portfolio based o the expected retur o the Tehra Stock Exchage. Shiraz Uiversity of accoutig developmets. Volume 3. (1). Pages