Design and implementation of fuzzy expert system for Tehran Stock Exchange portfolio recommendation

Design and implementation of fuzzy expert system for Tehran Stock Exchange portfolio recommendationMehdi Fasanghari,Gholam Ali Montazer *Faculty of Engineering,IT Department,Tarbiat Modares University,P.O.Box 14115-179,Tehran,Irana r t i c l e i n f o Keywords:Fuzzy expert systemMultiple Criteria Decision Making (MCDM)Portfolio recommendation Tehran Stock Exchange (TSE)Fuzzy Delphi Methoda b s t r a c tThe key issue for decision making in stock trading is selection of the right stock at the right time.In order to select the superior stocks (alternatives)for investment,a finite number of alternatives have to be ranked considering several and sometimes conflicting criteria that often are vague and have uncertainty conditions.Therefore,we are faced with a special Multiple Criteria Decision Making (MCDM)problem.The purpose of this paper is to develop a fuzzy expert system for selecting superior stocks in order to encounter the uncertainty of stock portfolio recommendation and model the recommendation rules which experts at Tehran Stock Exchange (TSE)use for portfolio recommendation.The results of imple-menting the designed fuzzy expert system at TSE were affirmative.Ó2010Elsevier Ltd.All rights reserved.1.IntroductionFinance portfolio recommendation is a collection of investment suggested to an institution or private individual.Selection of the assets and their proportion is a key problem in portfolio recom-mendation.It is difficult to decide which securities should be se-lected because of the existence of uncertainty on their returns;hence,it is necessary to balance between the maximizing expected return and minimizing the risk of the selected portfolio for better recommendation (Bermúdez,Segura,&Vercher,2007).To smooth portfolio recommendation,computerized and intelligent decision support has been used since past decades (Edward &Magee,2001).There are many analytical approaches for decision making in stock exchange,which are categorized in two groups of technical analysis and fundamental analysis (Edward &Magee,2001).Tech-nical analysts believe that the stock profitability prediction is pos-sible through studying stock prices in the past (Edward &Magee,2001).There are technical indicators for studying price trends of each stock such as moving averages,relative strength index (RSI),and moving average convergence divergence (MACD)(Abdollah-zadeh,2002).Fundamental analysts analyze audit reports,income statement,quarterly balance sheets,dividend records,sales re-cords,management capabilities,and competitive situation of the company and then calculate intrinsic value of each stock based on prediction of cash flow for next few years.If the market price of a stock is lower than its intrinsic value,its price is expected to rise and they decide to buy it (Alexander,Sharpe,&Bailey,1993).Stock portfolio recommendation is a complex multiple attribute bination of scientific methodology and personal experience of the field of the stock portfolio recommendation is the vital point to be succeeded in this field.Thus,all advanced tools will be utilized in order to ensure the integration of the know-how of professionals in the field of portfolio management.A fundamen-tal portfolio theory using model of mean–variance was developed by Markowitz (1952);anticipated return related to the undertaken risk is the main idea for portfolio recommendation.Then Sharpe (1964)developed a capital asset pricing model developing the mean–variance model and the multiple attribute utility theory,un-der performance conditions towards risk.Ross (1976)also devel-oped arbitrage pricing theory,where the stock return common component is expressed through a number of effect factors.Siskos and Despotis (1989)use multiple objectives interactive linear programming method to achieve a portfolio composition.Then Siskos,Spyridakos,and Yannacopoulos (1993)ranked the stocks with Minora method of preference analysis.Many methods are applied based on one of technical or fundamental approaches or the combination of them.For example,Sycara,Decker,Pannu,Williamson,and Zeng (1996)applied intelligent agents’technology for retrieving stock market data from distributed Internet sources.Hurson and Zopounidis (1997)examined two types of problems:stock evaluation and portfolio composition;Spronk and Hallerbach (1997)proposed a system based on multiple criteria analysis for supporting individual financial decision making in portfolio management for any type of investor.Zopounidis,Despotis,and Kamaratou (1998)proposed a portfolio selection in Athens Stock Exchange (ASE),for the two-year period of 1989–1990.A proactive system for expressing and implementing high-level stock tradingstrategies was developed by Garc´ıa,Gollapally,Tarau,and Simari 0957-4174/$-see front matter Ó2010Elsevier Ltd.All rights reserved.doi:10.1016/j.eswa.2010.02.114*Corresponding author.Tel.:+982182883990.E-mail addresses:fasanghari@ (M.Fasanghari),montazer@modares.ac.ir (G.A.Montazer).Expert Systems with Applications 37(2010)6138–6147Contents lists available at ScienceDirectExpert Systems with Applicationsjournal homepage:/locate/eswain2000which was suitable for implementing a deliberative multi-agent system in portfolio recommendation with technical and fun-damental approaches.Zopounidis and Doumpos(2000)presented a multiple criteria expert system which deals with the problem of portfolio selection and composition.Kwas´nicka and Ciosmak (2001)also applied a fuzzy expert system for technical analysis and artificial neural network for fundamental analysis in order to analyze the stock market and calculate the attractiveness of the companies and identifying buy signals.Luo,Liu,and Davis(2002) designed a decision support system for buy or sell decisions based on fundamental analysis principles and technical indicators.The most important elements of multiple criteria expert sys-tems,an important category of expert systems,were developed by Matsatsinis,Samaras,Siskos,and Zopounidis(2002)for stock evaluation and portfolio management.Therefore,Matsatsinis et al.(2002)can consider many important criteria that derive from fundamental and technical analysis,as well as the investor’s profile that represents his goals,preferences and policies besides the two basic criteria of return and risk in portfolio management with an MCDM approach.In addition,Samaras and Matsatsinis(2003) use a multiple criteria expert system for which uses MCDM meth-ods in order to rank the ASE stocks based on fundamental analysis, technical analysis and stock exchange analysis.Furthermore,Pohe-kar and Ramachandran(2004)used a genetic programming for dis-covering appropriate technical trading rules in stock exchange. Besides,Vranes,Stanojevic,Stevanovic,and Lucin(1996)suggested an investment program combiningfive different techniques:heu-ristics,expert system,fuzzy logic,investor risk model,and PROM-ETHEEII(a MCDM method)as an intelligent multiple criteria expert systems,in which multiple criteria methodologies and at least one artificial intelligent technology are combined.The Fineva system(Zopounidis&Doumpos,1999)and Intelligent Investor Samaras and Matsatsinis(2003)in the ASE are the other example of intelligent multiple criteria expert systems.In the stock portfolio recommendation problem,there are afi-nite number of stocks existing in stock exchange as alternatives which have to be ranked considering many different and conflict-ing criteria.Accordingly,this problem is considered as a MCDM problem.Fuzzy logic is initially introduced to deal with limitation in stock value reorganization and stock portfolio recommendation in TSE to encounter the uncertainty of criteria of decision making for stock portfolio recommendation.In this case,fuzzy decision making have been used in fuzzy expert system to recommend stock portfolio in TSE.The outline of the paper is as follows:Next section,fuzzy expert system for portfolio recommendation has been described.After-wards,the architecture of fuzzy expert system has been introduced and design of the proposed system for portfolio recommendation at TSE has been presented;then the designed system for stock portfolio recommendation at TSE is explained.Subsequently,the designed system has been evaluated and the obtained results have been described.Finally,conclusion has been showed the benefits of proposed system.2.Fuzzy expert systemProblem solving mechanism is only a small part of intelligent computer system(Turban&Aronson,1998).Thus,the necessity of expert system,a computing system capable of representing and reasoning about some knowledge-rich domain with a view to solving problems and giving advice(Jackson,1990),was increased.Usage of expert systems or knowledge-based system has exten-sively increased during last decade.The main difference of these systems with other software is that they process knowledge in-stead of data or information(Darlington,2000).Expert systems provide powerful means for solving different problems which are impossible to solve by conventional methods.Expert system is also one of the successful branches of artificial intelligence from com-mercial point of view(Metaxiotis,Psarras,&Askounis,2002).From another point of view,fuzzy set theory provides a frame-work for handling the uncertainties.Zadeh(1965)initiated the fuz-zy set theory.Bellman and Zadeh(1970)presented some applications of fuzzy theories to the various decision making pro-cesses in a fuzzy environment.In non-fuzzy set every object is either a member of the set or it is not a member of the set,but in fuzzy sets,every object is to some extent member of a set and to some extent it is member of another set.Thus,unlike the crisp sets membership is a continuous concept in fuzzy sets.Fuzzy is used in cases which there are linguistic variables.Fuzzy theory is widely applicable in information gathering,modeling,analysis,optimiza-tion,control,decision making and supervision(Bellman,1970).Fuzzy expert decision support system is such an expert system that uses fuzzy logic instead of Boolean logic.It can be seen as spe-cial rule-based systems that uses fuzzy logic in its knowledge-base and derives conclusions from user inputs and fuzzy inference pro-cess(Kandel,1992)while fuzzy rules and the membership func-tions make up the knowledge-base of the system.In other words a‘‘fuzzy if–then”rule is an‘‘if–then”rule which some of the terms are given with continuous functions(Wang,1994).It is obvious that it lacks a system that can combine the advan-tages of the aforementioned systems.Thus,it becomes necessary to develop a decision support system which can support the proce-dure of stock portfolio recommendation considering the investors parameters,stock market situation and stored knowledge of the last recommendations.3.Architecture of fuzzy expert systemA fuzzy expert system is comprised of four components:fuzzifi-cation unit,knowledge base,decision making logic,and defuzzifi-cation unit which should be embedded in the architecture detail for fuzzy expert system construction.The big picture of system architecture is composed of three main blocks as shown in Fig.1.3.1.Fuzzy inference engineA program which analyzes the rules and knowledge aggregated in the database andfinds the logical result.There are different selection for the fuzzy inference engine depending on the aggrega-tion,implication and operators used for s-norm and t-norms (Wang,1994).er interfaceUsers of this system are organizational decision makers that enter the real number of all linguistic variables via user interface. Also,user interface shows the result scoring of all stocks;therefore, as providing this aim in the designed system,MATLAB user inter-face is used.3.3.Fuzzy rule baseExperts’experience is used to build up the fuzzy rules. These rules are conditional statements and in general can be repre-sented asIF x is X and y is Y and...THEN o is O;where x and y are linguistic input variables.X and Y are possible lin-guistic values for x and y;respectively.They are modeled as fuzzyM.Fasanghari,G.A.Montazer/Expert Systems with Applications37(2010)6138–61476139sets based on reference sets containing x and y:Similarly the output or decision variable,o is a linguistic variable with a possible value,O modeled as a fuzzy set.The clause x is X and y is Y can be inter-preted as fuzzy propositions delivering partial set membership or partial truth.Consequently the partial truth of the rule premise can be evaluated,modifying the fuzzy set parameters of the output fuzzy sets(Matthews,2003).4.Design the fuzzy expert system for portfolio recommendationThe proposed fuzzy expert system aims at evaluating stocks of TSE so that make the portfolio and recommend it to the target cus-tomers at TSE according to their preferences and stocks pay off.For stock portfolio recommendation,the proposed system ranks the stocks by starting with the best one towards the worst.Ranking criteria are fundamental analysis ratios and qualitative criteria from the TSE.The undertaken risk is also incorporated in the rank-ing process.Thus,the portfolio recommendation is adapted to the investor’s preferences.The eligible stocks to be included in the portfolio are picked out from the top of every ranking list.The basic goals of developing the system are:The support of the decision-investor to the procedures of ill-structured decision making problems,such as the evaluation-ranking of stocks/companies.High level of interaction between the decision maker and the system.The completed formulation of the preferences of the decision maker and the absolute specialization of his preference profile. The incorporation of a huge volume of‘‘active information”in order for the system to be always updated and to support valu-able decisions.Architecture that gives the possibility of incorporation of any environment of work and development.The fuzzy expert system is intended to provide support for investment decisions regarding TSE stocks.However,it is fully parameterized and can be used in other stock exchanges,too,pro-vided it is equipped with the respective databases.The intention of this section is to describe the development methodology of fuzzy expert system which can propose effective-ness and efficient recommendation in TSE.The overview of the framework is shown in Fig.2.There are seven fundamental steps in the development of a fuzzy system that consist of the fuzzy inference process.The detailed description of these steps is as follows:4.1.Identification of critical factorsIn Iran’s stock exchange,there are several important factors can be used by experts to evaluate stocks and make a portfolio of stocks.Hence,to recognize the most important factors from tacit knowledge of experts,a questionnaire was distributed among ex-perts,investment companies,and brokerage companies doing stock transactions at TSE.This questionnaire had been distributed among42selected experts with the presented academic level in Table5,in which35of these experts completed the questionnaire. Moreover,the Fig.3illustrates the combination of the experts who were participated in the survey.Factors which were more important for stock portfolio recom-mendation at TSE have been chosen by experts:1–market ofFig.2.Fuzzy inference process for stock portfolio recommendation in TSE. 6140M.Fasanghari,G.A.Montazer/Expert Systems with Applications37(2010)6138–6147stocks,2–sale’s rules,3–Earned Per Share (EPS),4–projects,5–stockholders,6–legal audit report,and 7–float share.It was assumed that decision makers could assign ratings to dif-ferent stocks under different selection criteria using common lin-guistic terms,for example,‘‘low”,‘‘medium”and ‘‘high”.Each linguistic term is defined by a membership function which helps us to take crisp input values and map them into degrees of mem-bership in fuzzy data space.The most commonly used membership function has three types:bell-shaped,triangle-shaped and trape-zoid-shaped (Ngai,2003).Due to our experience in the present study,we assume the input and output fuzzy numbers are trape-zoidal forms and these forms approximate human thought pro-cesses.Trapezoidal membership functions,hence,have been used to define the fuzzy sets for the linguistic values describing factors.The most important factors which are critical for decision mak-ing for portfolio recommendation in TSE were obtained by litera-ture review:‘‘The future projects of the company (projects)”,‘‘The management team reputation”,‘‘Market of stocks,Earned Per Share (EPS)”,‘‘Stockholders,Effectiveness in general situation of the stock market”,‘‘The history of company transactions”,‘‘The sale’s rules”,‘‘The logical balance sheet”,‘‘The legal audit report”,‘‘The internal and external political situation”,‘‘The world position and comparative advantage”,‘‘The direct of the market (total in-dex)”,‘‘The size of the float stock of the company (Float Share)”,‘‘The supplement cost of the company”,‘‘Economic Order of Quan-tity (EOQ)”,and ‘‘The trend of stockholders due”.In the first round of Fuzzy Delphi Method the 17factors,the most important factors for decision making in TSE,were scored.The mean value of obtained scores for each proposed factors,after defuzzification,has been illustrated in Tables 1and 2.For the second round of Delphi approach,the final results of the first one were sent for the 35experts who contributed in the first round.All of them answered the questionnaire.The results of this step have been shown in Table 3,after deffuzification of the aggre-gated fuzzy numbers for each of the 17factors.Finally,in the third round,the results of the second one were sent to the 35experts who answered in the last round.In this round d ¼0:139<0:2(see Appendix B ).Thus,this step was the fi-nal step and the Delphi rounds were cut.The results of the thirdstep have been shown in Table 4while the number of critical fac-tors of stock portfolio recommendation in TSE have been decreased to 7:‘‘The future projects of the company (projects)”,‘‘Stockhold-ers”,‘‘Earned Per Share (EPS)”,‘‘Market of stocks”,‘‘The sale’s rules”,‘‘The size of the float stock of the company (Float Share)”,and ‘‘The legal audit report”.It is obvious from Table 4that the most effective factor for decision making in stock portfolio recom-mendation in TSE is ‘‘Projects”by mean score of 4.8and variance of 0.82.The lowest variance also show that most of the experts be-lieved that the factor of ‘‘Project”is an effective critical factor for portfolio recommendation in TSE.4.2.Fuzzy rules constructionFuzzy system makes decisions and generates output values based on knowledge provided by the designer in the form of IF_TH-EN rules.The rule base specifies qualitatively how the output parameter of the stock ranking is determined for various instances of the input factors:market of stocks,sale’s rules,(EarnedPerbination of experts who filled out questionnaires.Table 1The academic level of the experts who complete the questionnaire.Academic level Frequency Percentage of frequency (%)BSc 2764MSc 1126PHD 410Total42100Table 2The results (mean and variance)of the first round of Fuzzy Delphi.FactorMean Variance The future projects of the company (projects)91 1.99The management team reputation 89 2.33Market of stocks88 4.55Earned Per Share (EPS)85 3.21Stockholders76 4.44Effectiveness in general situation of the stock market 74 3.65The history of company transactions 72 5.87The sale’s rules65 6.7The logical balance sheet 657.54The legal audit report589.65The internal and external political situation 57 6.98The world position and comparative advantage 53 6.36The direct of the market (total index)44 6.94The size of the float stock of the company (Float Share)38 5.45The supplement cost of the company 36 4.25Economic Order of Quantity (EOQ)34 3.66The trend of stockholders due223.76Table 3The results (mean and variance)of the second round of Fuzzy Delphi.FactorMean Variance The future projects of the company (projects) 4.800.74Stockholders4.550.79Earned Per Share (EPS) 4.120.96Market of stocks 3.840.89The sale’s rules3.120.95The size of the float stock of the company (Float Share) 2.86 1.01The legal audit report2.80 1.23The internal and external political situation2.760.89Effectiveness in general situation of the stock market 2.620.94The trend of stockholders due 2.60.87The logical balance sheet2.560.96Table 4The results (mean and variance)of the third round of Fuzzy Delphi.FactorMean Variance The future projects of the company (projects) 4.80.82Stockholders4.610.89Earned Per Share (EPS) 3.960.93Market of stocks 3.430.90The sale’s rules3.27 1.06The size of the float stock of the company (Float Share) 2.91 1.12The legal audit report2.810.96M.Fasanghari,G.A.Montazer /Expert Systems with Applications 37(2010)6138–61476141Share)EPS,projects,stockholders,legal audit report,and float share.These degrees were formulated in term of their linguistic variables:Low,Medium,and High.For instance,‘‘If market of stocks is medium,sale’s rules is high,EPS is medium,projects is low,stockholders is low,legal audit report is medium,and float share is low,the stock ranking in portfolio is low”.To develop a knowledge-based system,it is very difficult to eli-cit and integrate knowledge from multiple experts (Hwang,Chen,Hwang,&Chu,2006),especially the application domains in which various time scales of elements need to be taken into account.To deal with this problem,Fuzzy Delphi Method is proposed in this section,which takes time scales into consideration whileelicitingFig.4.Fuzzy rule base view of the fuzzy expert system for stock portfoliorecommendation.Fig.5.The sensitive analysis of fuzzy expert system for stock portfolio recommendation.6142M.Fasanghari,G.A.Montazer /Expert Systems with Applications 37(2010)6138–6147expertise from multiple experts and model the vague expert’s opinion (Appendix B ).There were potentially 37=932rules be-cause each rule is consisted of seven inputs,and input parameters are evaluated with three linguistic variables:Low,Medium,and High.In the first round it was asked of 80experts of TSE to determine the importance of all of the 932rules with a trapezoidal numbers (Appendix A ).It is not necessary to increase the number of rules in rule base of fuzzy expert system.The rules with high degree of importance should be selected for the speed and accuracy of thefuzzy expert system is not increased with the number of rules necessarily.As the first step in Fuzzy Delphi Method,the first round of Fuzzy Delphi Method has been run;furthermore,the average of the re-sults has been computed according to step 2of Fuzzy Delphi Meth-od.Then in step 3,the output of the last step has been returned to the experts.The new scores of the 932rules have been collected.Likewise,the Fuzzy Delphi Method has been run for three rounds owing to the fact that,in the third round,all of the 932rules approximately have d i 60:2.Only were the 119rules selected since the experts believed that the rules with importance degree higher than 7.5,the threshold point of importance degree for rule selection,should be considered in rule base of fuzzy expert system for stock portfolio recommendation.The rule base of designed fuz-zy expert system for stock portfolio recommendation in TSE is illustrated in Fig.4.As shown in Fig.5,the sensitive analysis can be done in the rule base to extract the relationship among the effective parameters in stock portfolio recommendation at TSE.As an instance,in Fig.5,if the score of the Projects,ad effective parameters in portfolio recommendation at TSE,be less than 5,the score of Stockholders parameter has no sensitive effect on the created value of the recommended portfolio.4.3.FuzzificationFuzzification simply refers to the process of taking a crisp input value and transforming it into the degree required by the terms.If the form of uncertainty happens to arise because of imprecision,ambiguity,or vagueness,the variable is probably fuzzy and can be represented by a membership function.If the inputs generally originate from a piece of hardware,or drive from sensor measure-ment,then these crisp numerical inputs could be fuzzified in order for them to be used in a fuzzy inference system (Ross,2005).SinceTable 5Membership functions for linguistic values of stock portfolio recommendation factors.Linguistic variables Linguistic values Fuzzy trapezoidal number Market of stockLow(0,0,0.5,7.25)Medium (5.25,6.25,6.5,7.75)High (6.25,9.75,10,10)Sale’s rulesLow(0,0,2,4.5)Medium (3,6.25,7.25,8)High (7.25,9.25,10,10)EPSLow(0,0,0.25,6.25)Medium (3.75,5.75,6.5,8.5)High (5.25,9,10,10)ProjectsLow(0,0,7,7.75)Medium (5.5,6,6.25,8)High (5,9.25,10,10)StockholdersLow(0,0,1.75,7.25)Medium (4.75,5.25,6,7.75)High (6,8.75,10,10)Legal audit reportsLow(0,0,1.25,5.5)Medium (3.25,5.75,6.25,8.25)High (7.25,8.5,8.9,9.1)Float sharesLow(0,0,1.5,5.75)Medium (5,5.75,6,7.25)High(7,8.25,10,10)Fig.6.Market of stock input variable in fuzzy expert system for stock portfolio recommendation.M.Fasanghari,G.A.Montazer /Expert Systems with Applications 37(2010)6138–61476143inputs data are manually,singleton fuzzification method has been used to benefit of its simplicity and speed of calculations in fuzzy expert system.Defined linguistic variables of factors have been illustrated in Table5,in which fuzzification inputs are in trapezoidal fuzzy num-bers and their range is between0and10.For instance,Market of stock input variable in fuzzy expert system is illustrated in Fig.6.4.4.Fuzzy inference engineFuzzy inference is guided by the fuzzy rules.The standard max–min inference algorithm was used in the fuzzy inference process, as it is a commonly used fuzzy inference strategy(Ngai,2003). The max–min composition applied to two association a(h,l)and b(l,b)defined over Cartesian products HÂL and LÂW yields a new association c defined over the product space HÂW.The new association,which is also a fuzzy set,is expressed by:c¼[HÂW_l½l aðh;lÞ^l bðl;wÞ =ðh;wÞ;h2H;l2L;w2W;ð1Þwhere symbols^and_denote min and max operators,respec-tively.Note that singleton in Eq.(1)indicates the membership func-tion of the new association produced by max–min composition.lcðh;wÞ¼_l½l aðh;lÞ^l bðl;wÞ ;h2H;l2L;w2W:ð2ÞIn the max–min composition,the min operator is used for the AND conjunction(set intersection)and the max operator is used for the OR disjunction(set union)in order to evaluate the grade of mem-bership of the antecedent clause in each rule.Mamdani inference is used as Eq.(3):lB¼MaxMi¼1½Sup minðl AðxÞ;l Aðx1Þx&U;...;l Aðx nÞ;l BðyÞÞ :ð3Þ4.5.DefuzzificationThe process of computing a single number which represents the best outcome of the fuzzy set evaluation is called defuzzification (Ngai,2003).There are several methods that can be used for defuzzification.These include the methods of maximum or the average heights methods,and others.These methods tend to jump erratically on widely non-contiguous and non-monotonic input values(Diego,1999).We chose the centroid method,also referred to as the center-of-gravity(COG)method,as it is frequently used and appears to provide a consistent and well-balanced approach (Klir&Folger,1998).For each output using this defuzzification method illustrated in Eq.(4),the resultant fuzzy sets are merged into afinal aggregate shape and the centroid of the aggregate shape computed.x0¼R lðxÞx dxR lðxÞdx:ð4Þparison of the overall rating for all stocksDue to the risk of each customer at TSE,a portfolio of stock ex-change has been constructed.Parameters of this assignment have been obtained through the questionnaires of critical factors deter-mination.As illustrated in Table6,customers having minimum of risk and prefer to have more investment on bonds(60%)in theirTable6Customer preferences for recommended portfolio based on their risks.The lowest risk Stock The highest risk15%High risk5%Medium risk5%Low risk2%The lowest risk1%Bond60%Cash10%Low risk Stock The highest risk26%High risk6%Medium risk5%Low risk4%The lowest risk2%Bond47%Cash10%Medium risk Stock The highest risk18%High risk2%Medium risk8%Low risk4%The lowest risk18%Bond40%Cash10%High risk Stock The highest risk2%High risk4%Medium risk12%Low risk15%The lowest risk42%Bond20%Cash5%The highest risk Stock The highest risk2%High risk2%Medium risk8%Low risk16%The lowest risk52%Bond15%Cash5%Table7The mean and variance of the stock ranking having been defuzzified.Factor Parameters Selected stocks at TSEBama IKC Sepahan A.Petrolium Jabber Qadir Azarab Pars qand Payam Metal mineMarket of stock Mean9536615238 Variance0.440.760.66 1.020.230.780.7810.850.32 Sale’s rules Mean7864543326 Variance0.450.840.55 1.030.36 1.30.960.270.750.54 EPS Mean9645711247 Variance0.350.690.69 1.530.54 1.240.680.980.480.36 Projects Mean8746632256 Variance0.62 1.01 1.210.660.88 1.08 1.020.650.680.25 Stockholders Mean6817514458 Variance0.530.990.880.890.66 1.25 1.350.750.680.42 Legal audit reports Mean7867784316 Variance0.50.570.21 1.030.640.950.580.760.840.37 Float shares Mean4756467258 Variance0.31 1.10.680.970.65 1.010.610.710.640.586144M.Fasanghari,G.A.Montazer/Expert Systems with Applications37(2010)6138–6147。