Decision Science Letters

Transcription

1 Decisio Sciece Letters 3 (214) Cotets lists available at GrowigSciece Decisio Sciece Letters homepage: Possibility theory for multiobective fuzzy radom portfolio optimizatio Mir Ehsa Hesam Sadati a, Ali Doiavi a ad Abbas Samadi b a Departmet of Idustrial Egieerig, Urmia Uiversity, Urmia, Ira b Departmet of Idustrial Egieerig, Payame Noor Uiversity of Tabriz, Tabriz, Ira C H R O N I C L E A B S T R A C T Article history: Received October 15, 213 Received i revised format March Accepted March 24, 214 Available olie April Keywords: Multi-obective portfolio optimizatio model Possibility ad Necessity-based model Fuzzy radom variables The problem of portfolio optimizatio is a stadard problem i fiacial world ad it has received tremedous attetios. Portfolio optimizatio plays essetial role i determiig portfolio strategies for ivestors. Portfolio optimizatio is itrisically a discrete optimizatio problem whose decisio criteria are i coflict ad the proposed study of this paper cosiders a portfolio optimizatio problem ivolvig fuzzy radom variables. To solve the proposed model, we first preset the possibility ad ecessity-based model to reformulate the fuzzy radom portfolio selectio model ito liear programmig models ad usig the resulted liear programs, a multi-obective problem is costructed. To solve the multi-obective problem we propose some methods to cosider decisio makers optimistic ad pessimistic views. A umerical eample illustrates the whole idea o multiobective fuzzy radom portfolio optimizatio by possibility ad ecessity-based model. 214 Growig Sciece Ltd. All rights reserved. 1. Itroductio Portfolio selectio discusses the problem of how to choose appropriate combiatios of assets amog available oes so that the ivestmet ca brig the maimum ivestmet retur. I other words, portfolio selectio helps maagers cocetrate betwee product ad customer to ivest resources i the best effective way. The ivestmet retur ad risk always appear, simultaeously, i other words, portfolio optimizatio cosists of the portfolio selectio problem to determie the optimum approach of ivestmet o a particular amout of moey i a give set of securities or assets (Markowitz, 1952). Markowitz (1952) proposed mea-variace models for the portfolio selectio problem ad he formulated them mathematically i differet ways such as miimizig variace for a give epected value, or maimizig epected value for a give variace. Sice the, the mea variace models have bee well developed i both theory ad algorithm (Crama & Schys 23; Xia et al. 2). The mea-variace models for the portfolio selectio problem has bee cetral to research activities of Correspodig author. Tel.: addresses: e.hesam136@gmail.com (M. E. Hesam Sadati) 214 Growig Sciece Ltd. All rights reserved. doi: /.dsl

2 36 this area ad it has served as a basis for the developmet of moder fiacial theory for may years (Hao & Liu 28). I the term of developmet of moder fiacial theory, two eamples are collected here: Yu & Lee (211) cosidered trasactio cost, short sellig ad higher momets to achieve bigger fleibility i portfolio selectio. They cosidered five portfolio rebalacig models, by cosiderig trasactio cost ad by cosiderig criteria such as icludig risk, retur, short sellig, skewess, ad kurtosis to determie the importat desig criteria for a portfolio model. Woodside- Oriakhi et al. (213) cosidered the problem of rebalacig a eistig fiacial portfolio, where trasactio costs have to be paid if there is chage i amout held of ay asset ad they modeled the problem as a mied iteger quadratic with a eplicit o the amout that ca be paid i trasactio cost. Markowitz (1959) defied a semi-variace for as symmetric radom returs sice researchers poited out that the asymmetric returs is ot a appropriate method for measurig the risk. May researchers studied the properties ad computatio problem about semi-variace (Grootveld & Hallerbach, 1999; Markowitz, 1993) ad developed the mea semivariace models (Chow & Deig, 1994). Koo ad Yamazaki (1991) itroduced a advaced model i which a mea-absolute deviatio (MAD) model ad absolute deviatio were utilized as a measure of risk. Sice Markowitz (1952), portfolio theory has bee greatly improved. The researches maily focused o two aspects. O oe had, portfolio selectio was ivestigated uder stochastic eviromet, (see e.g. Harlow & Rao, Plat (29) proposed stochastic model for portfolio specific mortality eperiece, where the proposed stochastic process was applied to two isurace portfolios, ad the impact o the Value at Risk for logevity risk was quatified. O the other had, the portfolio problem was hadled i fuzzy eviromet (e.g. Parra & Terol, 21). Durig the last few years, fuzzy portfolio selectio have emerged ad received a great deal of researchers (Sadadi et al., 211; Gharakhai & Sadadi, 213; Dastkha et al., 213). Sadadi et al. (211) proposed a method i a form of fuzzy liear programmig, which was capable of determiig the amout of ivestmet i differet time cycles whe determiig of ivestmet i differet plaig areas was primary cocer ad they cosidered retur ad borrowig/leadig rate as fuzzy triagular umbers istead of crisp represetatios. Gharakhai ad Sadadi (213) eamied advaced optimizatio approach for portfolio problem itroduced by Black ad Litterma (1991) which proposed a ew approach to estimate asset retur to cosider the shortcomigs of Markowitz stadard Mea-Variace optimizatio ad they represeted ivestor s view about future asset retur by usig fuzzy umbers. I additio, Dastkha et al. (213) dealt with applicatio of three differet operators of fuzzy mathematical programmig i a meaabsolute deviatio portfolio selectio problem with real features of miimum trasactio lots, fied ad proportioal trasactio cost, cardiality costrait ad bouds o holdig costrait. Traditioally, returs of idividual securities were cosidered as radom variables uder the assumptios that ivestors have eough historical data about security returs ad the situatio of asset markets i future ca be correctly reflected by asset data i the past. However, these assumptios will be violated whe ew securities are listed i the market, or the real asset market is chaged. To deal with this problem, researchers have made use of fuzzy set theory (Zadeh, 1965). I real world, the possibility distributio fuctios of security returs may be partially kow. The, fuzzy returs with radom iformatio appear. I fact, for a ivestor, the fuzziess ad radomess of security returs are ofte mied up with each other. Thus, the ivestor will be faced with fuzzy returs with radom parameters. I such situatios, we may employ fuzzy radom theory (Liu, 29) to deal with this twofold ucertaity of fuzziess ad radomess. Radom ad fuzzy optimizatio models provide useful methods for ivestors to hadle ucertaity. Accordig to Sadadi et al. (212), portfolio problem becomes more complicated whe the retur of all risky assets are subect to ucertaity. They proposed a ew portfolio modelig approach with ucertai data ad it was also aalyzed usig differet robust optimizatio techiques ad Zhag et al. (213) cosidered a multi-period portfolio selectio problem imposed by retur demad ad risk cotrol i a fuzzy ivestmet eviromet, i which the returs of assets are characterized by fuzzy umbers. Liu

3 M. E. Hesam Sadati et al. / Decisio Sciece Letters 3 (214) 37 et al. (212) dealt with multi-period portfolio selectio problems i fuzzy eviromet by cosiderig some or all criteria, icludig retur, trasactio cost, risk ad skewess of portfolio. I reality, however, sometimes ivestors have to deal with the ucertaity of both radomess ad fuzziess simultaeously. For eample, security returs are usually regarded to be ormally distributed radom variables, but the epected values may be fuzzy. Hao ad Liu (29) developed two ovel types of mea-variace models for portfolio selectio problems, i which the security returs are assumed to be characterized by fuzzy radom variables with kow possibility ad probability distributios. I their proposed models, epected retur of a portfolio was as the ivestmet retur ad the variace of the epected retur of a portfolio was as the ivestmet risk. I fact, from a practical viewpoit the fuzziess ad radomess of security returs are ofte mied up with each other, so fuzzy radom variable ca be a ew useful approach to solve this kid of problem (Li & Xu, 29). A fuzzy radom variable was first itroduced by Kwakeraak (1978), ad its mathematical basis was costructed by Puri ad Ralescu (1986). A overview of the developmets of fuzzy radom variables was foud i the recet article of Gil et al. (26). Shapiro (29) implies that Radomess models the stochastic variability of all possible outcomes of a situatio, ad fuzziess relates to the usharp boudaries of the parameters of the model. I this paper, the assets retur i portfolio selectio problem are fuzzy radom variables ad we use the cocept of possibility ad ecessity-based model to develop a solutio method for the fuzzy radom portfolio optimizatio problem ad reformulate the portfolio optimizatio problem to the liear programmig (Sadati & Nematia, 213). We costruct multiobective programmig model by usig these two types of liear programmig to fid the obective fuctio value. The rest of the paper is orgaized as follow: Sectio 2 icludes basic cocept o fuzzy ad fuzzy radom theory. I Sectio 3, the problem formulatio is preseted. I sectio 4, a umerical eample is solved to show how our ew method works. Fially, coclusio ad future work will preset i sectio Basic cocepts A Fuzzy radom variable was first itroduced by Kwakeraak (1978), ad its mathematical basis was costructed by Puri ad Ralescu (1986). A overview of the developmets of fuzzy radom variables was foud i the recet article of Gil et al. (26). I geeral, fuzzy radom variables ca be defied i a dimesioal Euclidia space R. We preset the defiitio of a fuzzy radom variable i a sigle dimesioal Euclidia space R. Defiitio 1 (Sakawa 1993) Let (Ω, A, P) be a probability space, where Ω is a sample space, A is a σ-field ad P is a probability measure. Let F N be the set of all fuzzy umbers ad B a Borel σ-field of R. The a map Z : F is called a fuzzy radom variable if it holds that, R Z A B,,1 (1) where Z Z, Z R (2) Z is a α-level set of the fuzzy umber Z for. Ituitively, fuzzy radom variables are cosidered to be radom variables whose realized values are ot real values but fuzzy umbers or fuzzy sets.

4 38 Defiitio 2 LR fuzzy umber Z is defied by followig membership fuctio: Z L if Z Z 1 1 (3) 1 Z 1 1 R if Z Z Z X if Z Z 1 where Z, Z shows the peak of fuzzy umber Z ad, represet the left ad right spread respectively; LR with L() R() 1 ad L(1) R(1) are strictly decreasig, cotiuous fuctios.,,1,1. A possible represetatio of a LR fuzzy umber is Z Z, Z 1,, 3. Problem formulatio Ivestors are ratioal ad behave i a maer of maimizig their utility with a give level of icome or moey. I the followig problem, called Fuzzy Radom Optimal Portfolio selectio problem, the retur rate of assets are fuzzy radom variables: Problem 1 (4) 1 ma Z R M 1 (5) R R (6) 1 U ; 1, 2,...,. (7) subect to, The parameters ad variables are defie as follow, for =1, 2,, : 1 R R, R,, LR 1 R R R R, R R tr, R tr represet fuzzy radom variables whose observed value for each is fuzzy umber,,,. LR is a radom vector i which t is a radom variable with cumulative distributio fuctio T. : the umber of assets for possible ivestmet M : available total fud R R : the rate of retur of assets ( per period) : the retur i dollars : decisio variables which represet the dollar amout of fud ivested i asset U : the upper boud of ivestmet i asset. LR

5 3.1 Possibility-based model M. E. Hesam Sadati et al. / Decisio Sciece Letters 3 (214) 39 By Zadeh's etesio priciple for obective fuctio i problem 1, its membership fuctio is give as follows for each : Z t L if t Z 1 t 1 if Z t Z Z (8) 1 t Z R if otherwise where Z Z, Z 1,,, Z R, ad 1 1 Z R. The degree of f uder the possibility distributio Z t Z ( ) f supmi y, y y y. possibility Z y, y 1 2 LR 1 Z( ) 1 f is give as follows: The possibility degree of fuzzy costrait R R uder the possibility distributios is 1 defied as follows: (9) R R sup mi, y 1 R y y y y, y R (1) We maimize the degree of possibility Z f ad the degree of possibility R R, our portfolio selectio model i Problem 1 comes by the followig model: 1 Problem 2 ma f (11) subect to Pr Z f, (12) M, (13) 1 R R (14) 1 Pr, ; U 1, 2,...,. (15) where λ is a predetermied probability level ad η is a predetermied possibility level. A feasible solutio of portfolio selectio problem is called a possibility solutio. I order to trasform the above model to a liear programmig model, we eed to reformulate Eq. (12) ad Eq. (14) i problem 2. Cosider the followig theorem: Theorem 1: (Katagiri et al. 28) For ay decisio variable, it holds that:

6 31 1) Pr Z f R T R R f 1 1 2) Pr R R R T 1 R R R T 1 R L where T, L ad R are pseudo iverse fuctios defied as: T if t Tt, L sup t Lt ad R sup t Rt. Now the optimal solutio of Problem 2 is equal to the followig liear fractioal programmig problem: Problem 3 ma R T R R (16) M (17) 1 subect to, R 1 T R 2 R R T R 2 L 1 1, (18) 1 1 (19) U ; 1, 2,...,. This ca be solved by oe of the LP solver (such as LINGO) ad obtaied a optimal solutio for portfolio selectio problem. 3.2 Necessity-based model The possibility-based model may be improper sice the obtai solutio will be too optimistic, so ecessity-based model ca be suitable for pessimistic decisio maker who wishes to avoid risk. The degree of ecessity N Z f for fuzzy costrait Z f uder the possibility distributio Z t is defied as follows: ( ) if ma 1,1 Z ( ) 1 f (2) N Z f y y y y y, y 1 2 The ecessity degree of fuzzy costrait R R uder the possibility distributios is 1 defied as follows: if ma 1,1 1 R R 1 y, y 1 (21) N R R y y y y 1 2 We maimize the degree of ecessity NZ f ad the degree of ecessity R R, 1 therefore our portfolio selectio model i problem 1 come by followig model:

7 Problem 4 M. E. Hesam Sadati et al. / Decisio Sciece Letters 3 (214) 311 ma f (22) subect to Pr N Z f, (23) M, (24) 1 N R R (25) 1 Pr, ; U 1, 2,...,. (26) where λ is a predetermied probability level ad η is a predetermied possibility level. A feasible solutio of portfolio selectio problem is called a ecessity solutio. I order to trasform the above model to a liear programmig model, we eed to reformulate Eq. (23) ad Eq. (25). Cosider the followig theorem: Theorem 2: (Katagiri et al. 28) Let be a positive decisio vector the: NZ f ) Pr R T R L f 1 1 2) Pr N R R R T 1 R L 1 R T 1 R L 1 where T, L ad R are pseudo iverse fuctios defied as: T if t Tt, L sup t Lt ad R sup t Rt. Now the optimal solutio of Problem 4 is equal to the followig liear parametric programmig problem: Problem 5 ma R T R L (27) 1 1 M (28) 1 subect to, R T 1 R 2 L 1 R T 1R 2 L 1 (29) 1 1 (3) U ; 1, 2,...,. This ca be solved by oe of the LP solver (such as LINGO) to determie the optimal solutio of portfolio selectio problem. Sadati ad Nematia (213) itroduced fuzzy radom portfolio problem as a two-level liear programmig to calculate the upper boud ad lower boud of the obective fuctio value separately based o the decisio maker opiio ad accordig to the possibility ad ecessity-based model we reformulated the fuzzy radom portfolio optimizatio to the liear programmig. I this paper, our purpose is to fid the optimum solutio of portfolio optimizatio

8 312 whe we wat to use possibility ad ecessity (optimistic ad pessimistic approach) i oe problem, so we have to reformulate our problem as a multiobective problem ad put both obective fuctios of problem 3 ad 5 i oe problem. Therefore, our multiobective portfolio optimizatio formulates as: Problem (31) 1 1 ma R T 1 R R 2 (32) 1 1 ma R T 1 R L 1 M (33) 1 subect to, R 1 T R 2 R R T R 2 L 1 1, (34) 1 1 R T 1 R 2 L 1 R T 1R 2 L 1 (35) 1 1 U ; 1, 2,...,. (36) To solve this multiobective problem we use the cocepts of optimistic ad pessimistic. As we kow that possibility based model may be improper sice the obtai solutio will be too optimistic, so ecessity based model ca be suitable for optimistic decisio maker who wishes to avoid risk. To fid the optimum solutio for this multiobective portfolio optimizatio we preset two ways: first we fid the optimum solutio ust for sigle obective of possibility-based model with all costraits (possibility ad ecessity-based model costraits) after we fid the solutio, we use this optimum solutio as ecessity-based model s costrait. It meas the obective fuctio value for sigle obective possibility-based model becomes as a costrait for ecessity-based model i the et step ad fially we fid the optimum solutio for sigle obective of ecessity based model. This way completely is illustrated by followig model: First, we will fid the optimum solutio for sigle obective of possibility-base model without computig the obective fuctio of ecessity-based model: Problem (37) 1 1 ma R T 1 R R M (38) 1 subect to, R 1 T R 2 R R T R 2 L 1 1, (39) 1 1 R T 1 R 2 L 1 R T 1R 2 L 1 (4) 1 1 U ; 1, 2,...,. (41)

9 M. E. Hesam Sadati et al. / Decisio Sciece Letters 3 (214) 313 After solvig this problem by Ligo we fid the obective fuctio value(ofv) of problem 7 the with this result we determie the ecessity-based model with etra costrait stated i Eq. (37) resulted accordig to problem 7,Therefore, we have: Problem 8 2 (42) 1 1 ma R T 1 R L (43) 1 1 subect to R T 1 R R OFV(problem 7) M, (44) 1 R 1 T R 2 R R T R 2 L 1 1, (45) 1 1 R T 1 R 2 L 1 R T 1R 2 L 1 (46) 1 1 U ; 1, 2,...,. (47) By solvig this problem we fid the optimum solutio of multiobective portfolio optimizatio. The secod way to fid the optimum solutio for this multiobective portfolio optimizatio is same as first way but at first, we fid the sigle obective of ecessity-based model ad after the fidig of optimum solutio of ecessity-based model we use this result as a costrait for possibility based model ad fially same as above we will fid the optimum solutio for possibility-based model. Therefore, we have: Problem 9 2 (48) 1 1 ma R T 1 R L 1 M (49) 1 subect to, R 1 T R 2 R R T R 2 L 1 1, (5) 1 1 R T 1 R 2 L 1 R T 1R 2 L 1 (51) 1 1 U ; 1, 2,...,. (52) Similar to problem 7, we fid the obective fuctio value(ofv) the with this result we determie the possibility-based model with etra costrait (48) which has bee resulted accordig problem 1,Therefore, we have:

10 314 Problem (53) 1 1 ma R T 1 R R 2 (54) 1 1 subect to R T 1 R L 1 OFV(problem 9) M, (55) 1 R 1 T R 2 R R T R 2 L 1 1, (56) 1 1 R T 1 R 2 L 1 R T 1R 2 L 1 (57) 1 1 U ; 1, 2,...,. (58) By solvig this problem, we obtai the secod type of optimum solutio for our portfolio selectio. I et sectio with a umerical eample, we will illustrate whole aspects of our approach. 4. Numerical eample I this sectio, a eample is give to illustrate the proposed possibility ad ecessity-based model for portfolio optimizatio selectio. We believe that a ivestmet pla eeds to cosider ot oly the historical data, but also ew iformatio. Therefore, we decided to use the secod type of data, which have bee received after startig the first decisio. Let us cosider 5 securities whose returs are fuzzy radom variables ad their values are give i Table 1. t is a ormal radom variable whose mea ad variace 1. The upper boud of ivestmet amout i each stock is set to o more tha 6 uits of the total available fud. Give a total allocatio budget of 2 uits ad aual retur which is fuzzy radom variable is show as R M r where r 1.3 t,1.3 t,.3,.3. Now we wat to kow what is the optimal solutio for our portfolio selectio problem for the differet levels of probability ad possibility {.1,.4,.7,.9}. We apply the possibility ad ecessity-based model based o theorems 1ad 2 to obtaied fuzzy radom portfolio selectio problem. First of all we calculate possibility ad ecessity-based model separately, ad the calculate the multiobective portfolio optimizatios accordig to the two ways have bee preseted. All the results are collected i tables 2 to 5. Table 1 Parameters of the eample R R R

11 M. E. Hesam Sadati et al. / Decisio Sciece Letters 3 (214) 315 Table 2 Numerical results (possibility-based model), OFV a Table 3 Numerical results (ecessity-based model), OFV a a Obective fuctio value. Table 4 Numerical results (multiobective portfolio optimizatio accordig to first way), OFV b P OFV b N b Obective fuctio value of Possibility ad Necessity-based model Table 5 Numerical results (multiobective portfolio optimizatio accordig to secod way), OFV b P OFV b N

12 316 Clearly, the greater the, value, the greater the level of possibility ad the lower the obective fuctio value is. The umerical results of Table 2 ad Table 3 have bee calculated separately but results of Table 4 ad Table 5 are based o the multiobective problem, which show how possibility ad ecessity approach ifluece each other ad these results are helpful for decisio makers who are i ambiguity to choose which approach to fid their solutio i portfolio selectio. I other words, whe they use oly possibility-based model for their portfolio selectio, after receivig their optimum solutio, they foud their solutio too optimistic ad the try to use pessimistic approach of ecessity-based model ad it keeps them i ambiguity to choose which approach. Therefore, for these kids of decisio makers, the best approach is to use multiobective portfolio selectio, which has bee idicated i problem 6. The comparisos betwee sigle obective ad multiobective are depicted i Fig 1 ad Fig 2. Possibility_based model Necessity-based model Possibility-based model Necessity-based model OFV of Portfolio retur OFV of Portfolio retur Predetermied Levels Predetermied Levels Fig. 1. Compariso of Possibility ad Necessity-based model i sigle obective Fig. 2. Compariso of Possibility ad Necessity-based model i multiobective 5. Coclusio Portfolio optimizatio has bee oe of the most importat fields of research i ecoomic ad fiace. Sice the prospective returs of assets used for portfolio optimizatio problem are forecasted values, cosiderable ucertaity is ivolved. I this paper, Markowitz s mea-variace idea was eteded to portfolio selectio by possibility ad ecessity-based model. This paper proposed a solutio method for portfolio selectio model whose parameters were fuzzy radom variables. The idea was based o possibility ad ecessity-based model. We first preseted the possibility ad ecessity-based model to reformulate the fuzzy radom portfolio selectio model to liear programmig, the with these two kid of liear programs we costructed the multiobective problem. The optimum solutio of multiobective problem based o the illustrated eample was helpful for decisio makers who are i ambiguity to choose which approach to fid their solutio i portfolio selectio (optimistic or pessimistic approach). For future research, we will apply the other methods for fuzzy radom portfolio selectio model ad improve our portfolio selectio problem. Refereces Areas Parra, M., Bilbao Terol, A., & Rodrıguez Urıa, M. V. (21). A fuzzy goal programmig approach to portfolio selectio. Europea Joural of Operatioal Research, 133(2), Black, F., & Litterma, R. B. (1991). Asset allocatio: combiig ivestor views with market equilibrium. The Joural of Fied Icome, 1(2), Chow, K. V., & Deig, K. C. (1994). O variace ad lower partial momet betas the equivalece of systematic risk measures. Joural of Busiess Fiace & Accoutig, 21(2), Crama, Y., & Schys, M. (23). Simulated aealig for comple portfolio selectio problems. Europea Joural of operatioal research, 15(3),

13 M. E. Hesam Sadati et al. / Decisio Sciece Letters 3 (214) 317 Dastkha, H., Golmakai, H. R., & Ghareh, N. S. (213). How to obtai a series of satisfyig portfolios: a fuzzy portfolio maagemet approach.iteratioal Joural of Idustrial ad Systems Egieerig, 14(3), Gharakhai, M., & Sadadi, S. (213). A fuzzy compromise programmig approach for the Black- Litterma portfolio selectio model. Decisio Sciece Letters, 2(1), Gil, M. Á., López-Díaz, M., & Ralescu, D. A. (26). Overview o the developmet of fuzzy radom variables. Fuzzy sets ad systems, 157(19), Grootveld, H., & Hallerbach, W. (1999). Variace vs dowside risk: Is there really that much differece?. Europea Joural of operatioal research, 114(2), Hao, F. F., & Liu, Y. K. (28). Portfolio Selectio Problem i Fuzzy Radom Decisio Systems. I Iovative Computig Iformatio ad Cotrol, 28. ICICIC'8. 3rd Iteratioal Coferece o (pp ). IEEE. Hao, F. F., & Liu, Y. K. (29). Mea-variace models for portfolio selectio with fuzzy radom returs. Joural of Applied Mathematics ad Computig,3(1-2), Harlow, W. V., & Rao, R. K. (1989). Asset pricig i a geeralized mea-lower partial momet framework: Theory ad evidece. Joural of Fiacial ad Quatitative Aalysis, 24(3), Katagiri, H., Sakawa, M., Kato, K., & Nishizaki, I. (28). Iteractive multiobective fuzzy radom liear programmig: Maimizatio of possibility ad probability. Europea Joural of Operatioal Research, 188(2), Koo, H., & Yamazaki, H. (1991). Mea-absolute deviatio portfolio optimizatio model ad its applicatios to Tokyo stock market. Maagemet sciece, 37(5), Kwakeraak Kwakeraak, H. (1978). Fuzzy radom variables I. Defiitios ad theorems. Iformatio Scieces, 15(1), Li, J., & Xu, J. (29). A ovel portfolio selectio model i a hybrid ucertai eviromet. Omega, 37(2), Liu, B. (29). Theory ad practice of ucertai programmig (Vol. 239). Spriger. Liu, Y. J., Zhag, W. G., & Xu, W. J. (212). Fuzzy multi-period portfolio selectio optimizatio models usig multiple criteria. Automatica, 48(12), Markowitz, H. (1952). Portfolio selectio. The oural of fiace, 7(1), Markowitz, H. (1959). Portfolio selectio: efficiet diversificatio of ivestmets (No. 16). Yale uiversity press. Markowitz, H., Todd, P., Xu, G., & Yamae, Y. (1993). Computatio of mea-semivariace efficiet sets by the critical lie algorithm. Aals of Operatios Research, 45(1), Plat, R. (29). Stochastic portfolio specific mortality ad the quatificatio of mortality basis risk. Isurace: Mathematics ad Ecoomics, 45(1), Puri, M. L., & Ralescu, D. A. (1986). Fuzzy radom variables. Joural of mathematical aalysis ad applicatios, 114(2), Sadati, M. E. H., & Nematia, J. (213). Two-level liear programmig for fuzzy radom portfolio optimizatio through possibility ad ecessity-based model. Sadadi, S. J., Seyedhosseii, S. M., & Hassalou, K. (211). Fuzzy multi period portfolio selectio with differet rates for borrowig ad ledig. Applied Soft Computig, 11(4), Sadadi, S. J., Gharakhai, M., & Safari, E. (212). Robust optimizatio framework for cardiality costraied portfolio problem. Applied Soft Computig,12(1), Shapiro, A. F. (29). Fuzzy radom variables. Isurace: Mathematics ad Ecoomics, 44(2), Sakawa, M. (1993). Fuzzy sets ad iteractive multiobective optimizatio. New York: Pleum. Woodside-Oriakhi, M., Lucas, C., & Beasley, J. E. (213). Portfolio rebalacig with a ivestmet horizo ad trasactio costs. Omega, 41(2), Xia, Y., Liu, B., Wag, S., & Lai, K. K. (2). A model for portfolio selectio with order of epected returs. Computers & Operatios Research, 27(5), Yu, J. R., & Lee, W. Y. (211). Portfolio rebalacig model usig multiple criteria. Europea Joural of Operatioal Research, 29(2),

14 318 Zadeh, L. A. (1965). Fuzzy sets. Iformatio ad cotrol, 8(3), Zhag, W. G., Liu, Y. J., & Xu, W. J. (213). A ew fuzzy programmig approach for multi-period portfolio optimizatio with retur demad ad risk cotrol. Fuzzy Sets ad Systems.