THE WEIGHTED POSSIBILISTIC MEAN VARIANCE AND COVARIANCE OF FUZZY NUMBERS

Transcription

1 Quattatve Methos Iques THE WEIGHTED POSSIBILISTIC MEAN VARIANCE AND COVARIANCE OF FUZZY NUMBERS Supa SUDRADJAT PhD Depatmet of Mathematcs Faculty of Mathematcs a Natual Sceces Paaaa Uvesty Jl Raya Baug-Sumeag Jatago 6 Baug Ioesa E-mal: aat@yahoocom Astact: I the cotet of Makotz potfolo selecto polem Makotz959 ths pape evelops a eghte posslstc mea-vaace a covaace potfolo selecto moel I the pocess of selectg a potfolo ths ok s cocee th the choce of the potfolo Makotz95 Key os: Potfolo selecto; VaR; Posslstc mea value; Posslstc theoy; Weghte posslstc mea; Fuzzy theoy Itoucto Ths atcle s ogaze the follog ay: Secto popose a oveve of popotoal tasacto costs moel; Secto e cose the posslstc theoy popose y Zaeh [5] a e peset the ate of etu o secuty gve y a tapezoal fuzzy ume; Secto e state the eghte posslstc mea vaace a covaace of fuzzy umes Thus ae etee some ecetly esults ths fel [85] The popotoal tasacto costs moel Tasacto cost s oe of the ma souces of coce to maages see [ 6] Assume the ate of tasacto cost of secuty a allocato of k assets s c thus the tasacto cost of secuty a allocato of assets s c The tasacto cost of potfolo s c k Coseg the popotoal tasacto cost a the shotfall poalty costats e pupose the follog mea VaR potfolo selecto moel th tasacto costs []: Ma k k R E v c E v c Suect to P{ v < VaR } k 9

2 Quattatve Methos Iques M M Tagula a tapezoal fuzzy umes We cose the posslstc theoy popose y Zaeh [5] Let a a e to fuzzy umes th memeshp fuctos μ a a μ espectvely The posslty opeato Pos s efe [] Let the ate of etu o secuty gve y a tapezoal fuzzy ume hee < < The the memeshp fucto of the fuzzy ume ca e eote y: μ othese We meto that tapezoal fuzzy ume s tagula fuzzy ume f Let us cose to tapezoal fuzzy umes a If the e have Pos sup m{ μ μ y} y { } { μ μ } m { } m hch mples that Pos If a the the supemum s acheve at pot of tesecto δ of the to memeshp fucto μ a μ A smple computato shos that Pos δ a δ δ If > the fo ay < y at least oe of the equaltes μ μ y hol Pos No e summaze the aove esults as Thus e have Pos δ 5

3 Quattatve Methos Iques Especally he s the csp ume the e have hee Pos δ We o tu ou atteto the follog lemma δ Lemma [] Assume that tapezoal fuzzy ume cofece level α th α α The level set of a fuzzy ume The fo ay gve Pos f a oly f α α s a csp suset of R a eote y ] [ { μ R} the accog to Calsso et al [] e have ] [ { μ R} [ ] Gve ] [ a a ] s [ the csp posslstc mea value of E a a hee E eotes fuzzy mea opeato s a tapezoal fuzzy ume the We ca see that f E 5 6 The eghte posslstc mea vaace a covaace of fuzzy umes The classcal mea-vaace potfolo selecto polem uses the vaace as the measue fo sk hch puts the same eght o the o se a upse of the etu I ths secto e stuy the eghte posslstc mea-vaace a covaace potfolo selecto moel Defto [6] Let -eghte posslstc vaace of s F e a fuzzy ume th ] [ ] [] Va hee eghtg fucto s o-eceasg a satsfes [ The 5

4 Quattatve Methos Iques 5 The staa evato of s efe y Va σ Let fuzzy ume a e a eghtg fucto e efe the eghte posslstc vaace of y Va a the eghte covaace of a s efe as Cov If ] [ Va a Cov Let a e fuzzy umes of tapezoal fom Let hee e a eghtg fucto the the poe-eghte vaace a covaace a ae compute y Va 6 α α 5 Cov 6 6 Theoem [] The mea-vaace effcet potfolo moel s q R c E ma 7 suect to Pos < q 8

5 Quattatve Methos Iques M 9 M I the et theoem e ete [] Theoem to the case eghte posslty mea- vaace appoach th a specal eghte Theoem Let the eghte posslty mea vaace covaace of the tapezoal ume hee < < a ato s a tapezoal ume fo VaR level q Fo > q the the posslstc mea vaace covaace potfolo selecto moel s q ma R suect to 6 c q M M Poof : Fom the equato 5 e have 5

6 Quattatve Methos Iques 5 E 6 q Fom Lemma e have that Pos < q s equvalet th q Futhemoe fom - gve y Theoem e get the follog fom : ma q R c 6 5 suect to q 6 7 M M 8 Ths completes the poof Polem 5-8 s a staa mult-oectve lea pogammg polem Also e ca ota a optmal soluto y usg some algothms of mult-oectve pogammg [ ]

7 Quattatve Methos Iques Fo e see that lm E 8 6 Refeeces Aott R D a Wage WH The measuemet a cotol of tag costs Facal Aalysts Joual pp 7-8 Bellma R a Zaeh L A Decso makg a fuzzy evomet Maagemet Scece 7 97 pp -6 Caalleo R Cea E Muoz M M Rey L a Stacu Masa I M Effcet soluto cocepts a the elatos stochastc mult-oectve pogammg Joual of Optmzato Theoy a Applcatos pp 5-7 Calsso C a Fulle R O posslstc mea value a vaace of fuzzy umes Fuzzy sets a systems pp Duos D a Pae H Posslty theoy Pleum Pess Ne Yok Fulle R a Malee P O eghte posslstc mea value a vaace of fuzzy umes Tuku Cete fo Compute Scece 7 Guohua C et al A posslstc mea VaR moel fo potfolo selecto AMO-Avace Moelg a Optmzato Vol 8 No 6 8 Iuguch M a Ramk J Posslstc lea pogammg: a ef eve of fuzzy mathematcal pogammg a a compaso th stochastc pogammg potfolo selecto polem Fuzzy Sets a Systems pp -8 9 Makotz H M Potfolo Selecto Joual of Face 7 95 pp 77-9 Makotz H M Potfolo Selecto Joh Wley & Sos Ne Yok 959 Pea V Some optmalty cotos fo multoectve pogammg polems th set fuctos Rev Roum Math Pues Appl 99 Suaat S O posslstc appoach fo a potfolo selecto Mathematcal Repots accepte Feuae 7 Suaat S a Pea V O potfolo optmzato usg fuzzy ecsos 6 th Iteatoal Cogess o Iustal a Apple Mathematcs Zuch Stzela 6- July 7 Suaat S Popescu C a Ghca M A potfolo selecto polem th a posslstc appoach ND Euopea Cofeece o opeatoal eseach Pague July Zaeh LA Fuzzy sets as a ass fo a theoy of posslty Fuzzy sets a systems 978 pp -8 6 Zhou X Y a L D Cotouos tme mea- vaace potfolo selecto A stochastc LQ fameok Apple Mathematcs a Optmzato pp 9- Supa Suaat as o Taskmalaya Ioesa o May Afte completg secoay school at SMA St Maa Baug 977 he cotue hs stuy at the Depatemet of Mathematcs Uvestas Paaaa Ioesa hch he gauate Mach 98 I 986 he as accepte th the Depatemet of Mathematcs Uvestas Paaaa Ioesa as a uo lectue I July 986 he et o th hs stuy at Depatmet of Iustal Egeeg a Maagemet Isttute of Techology of Baug ITB Ioesa th the pupose of otag a maste egee Iustal Egeeg a Maagemet hch he succeee Octoe 989 I Vce Dea fo Amstato a Face Faculty of Mathematcas a Natual Sceces Paaaa Uvesty Baug Ioesa I Septeme he got the oppotuty to pusue PhD egee at the Uvesty of Buchaest He state PhD Septeme at the Opeatos Reseach a Statstcs Faculty of Mathematcs a Ifomatcs Uvesty of Buchaest ue supevso of Pof Vasle Pea hch he succeee Jue 7 The suect of he eseach s potfolo optmzato moels a fuzzy stochastc moels facal moelg hch le to thess 55

8 Quattatve Methos Iques Cofcatos of efeeces: [] Aott R D a Wage WH The measuemet a cotol of tag costs Facal Aalysts Joual pp 7-8 [] Bellma R a Zaeh L A Decso makg a fuzzy evomet Maagemet Scece 7 97 pp -6 [] Caalleo R Cea E Muoz M M Rey L a Stacu Masa I M Effcet soluto cocepts a the elatos stochastc mult-oectve pogammg Joual of Optmzato Theoy a Applcatos pp 5-7 [] Calsso C a Fulle R O posslstc mea value a vaace of fuzzy umes Fuzzy sets a systems pp 5-6 [5] Duos D a Pae H Posslty theoy Pleum Pess Ne Yok 998 [6] Fulle R a Malee P O eghte posslstc mea value a vaace of fuzzy umes Tuku Cete fo Compute Scece [7] Guohua C et al A posslstc mea VaR moel fo potfolo selecto AMO-Avace Moelg a Optmzato Vol 8 No 6 [8] Iuguch M a Ramk J Posslstc lea pogammg: a ef eve of fuzzy mathematcal pogammg a a compaso th stochastc pogammg potfolo selecto polem Fuzzy Sets a Systems pp -8 [9] Makotz H M Potfolo Selecto Joual of Face 7 95 pp 77-9 [] Makotz H M Potfolo Selecto Joh Wley & Sos Ne Yok 959 [] Pea V Some optmalty cotos fo multoectve pogammg polems th set fuctos Rev Roum Math Pues Appl 99 [] Suaat S O posslstc appoach fo a potfolo selecto Mathematcal Repots accepte Feuae 7 [] Suaat S a Pea V O potfolo optmzato usg fuzzy ecsos 6 th Iteatoal Cogess o Iustal a Apple Mathematcs Zuch Stzela 6- July 7 [] Suaat S Popescu C a Ghca M A potfolo selecto polem th a posslstc appoach ND Euopea Cofeece o opeatoal eseach Pague July 8-7 [5] Zaeh LA Fuzzy sets as a ass fo a theoy of posslty Fuzzy sets a systems 978 pp -8 [6] Zhou X Y a L D Cotouos tme mea- vaace potfolo selecto A stochastc LQ fameok Apple Mathematcs a Optmzato pp 9-56