Proposing a Mixed Model Based on Stochastic Data Envelopment Analysis and Principal Component Analysis to Predict Efficiency

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1 J. Bac. pp. Sc. Re TextRad Pcat ISSN Ja f Bac ad pped Scetfc Reeach Ppg a Mxed Mde Baed Stchatc Data epet a ad Pcpa Cpet a t Pedct ffcec Yagh Mehd Bah Depatet f Idta geeg Teha Paae N Uet Teha Ia. Depatet f Idta geeg Shahed Uet Teha Ia. BSTRCT Data epet a a techqe aed ea pgag ethd t ctct face paaetc da e the data. Th da ed t eaate ppta effcec. I th pape a xed de wa pped aed Stchatc Data epet a SD ad Pcpa Cpet a f pedctg the a t' effcece a gazat. It wa ted t aate the t ptat htcg f D g: eg ae t etate the effcec the ea dtt f weght f pt ad tpt f the de ad aet effcet ache. The fwg de ceed the eted pe caed eteg the tchatc effect ad cdeg the effect f fzz weght the pt ad tpt the e f SD techqe wth fzz weght. wa ed t detee effcec ea f t wth a k. Fa de t each a ette detadg f the pped de t wa apped t pedct effcece f e Iaa Bak ache. The hgh ceat etwee ea ad pedcted effcece wa taed whch epeeted the adt f the pped SD/ de. KYWORDS: Stchatc Data epet a Pcpa Cpet a ffcec Dec akg Ut Fzz the.. INTRODUCTION aatg ad cpag the peface f a t f a gazat a ptat pat f the epte f gazat aageet. Oe f the t ptat t f eate peface cpag thee t a qattate pece ad pwef appach caed Data epet a D. Th techqe cdeed t peface eaat t a aageet; t e pece ecgzed t de ct. Th ethd a ha e a htcg the t ptat f whch ae pt f pedctg effcec at deteg acceptae k ee f the aage the dect f acheg the pedcted effcece each t ad ea weght dtt t the pt ad tpt. F pepag the pt f pedctg effcec ad the ee f t depedet k e ca eeft f a atheatca de whch aed Stchatc Data epet a SD eteg tchatc effect eeta fact ke ecc cdt the pt ad tpt f the t de ct. f g the pe f ack f ea dtt f weght t pt ad tpt f the D de we ed fzz weght accdg t expet gget f the weght f the de pt ad tpt. O the the had th pape a xed de wa pped aed Pcpa Cpet a ad Stchatc Data epet a SD de t pedct effcec ea f t wth a k ee. Stchatc ctat pgag a e ptat ad ef ethd tchatc pgag. Chae ad Cpe 989 eteed the chace ctaed pgag eeach peat teate f the ft te []. The ag wth Rd 98 dced data eepet aa f cacatg effcec. Segpta et a. 98 pped the tchatc D de. I the wd thee tde ced the de f data eepet aa wth chace ctat pgag CCP ad ed the taed tchatc de f etatg effcec ad cdeg the eag e f pt aae. The tde cdcted the weght f pt ad tpt D ae ted ad the t ptat f the ae the atce f Da ad Thaa 988 Chae et a. 989 R ad Ga 99 ad Jahahah et a. 99. Lad et a. 99 pped a de kw a LLT. I th de the cdeed th ctat f the eepet f f CCR de a the tchatc aae. fte ppg LLT de Cpe et a. 99 pped a ew de appg Sa Satfact de. Th ew de a cat f the ccept f atfact dec akg wth CCD de data eepet aa wth tchatc ctat. Jack 00 etated the effcec fee aket g data eepet aa. Cpe et a. 00 pceeded t *Cepdg th: Ue Ghaeak Depatet f Cca Pchg Paa-e-N Uet f Kh. a: gaeak@ga.c 0

2 Yagh ad Bah.0 the aa f techca effcec g tchatc ctat pgag appach. Saat et a. 00 peeted a ethd f tag a c et f fzz pt ad tpt weght. The ft ggeted the de f detetc data ad the deeped t f fzz data. Hag et a. 00 pped the ced de f SD ad chace ctat pgag. Cpe et a. 00 peeted the at pped de the SD gd. I th pape the pped the tpt-aed BCC ad de ad ped t appg the tated ccept BCC de ad the hpthe f ad pt ad tpt ad a dtt f the. de et a. 00 epaced paaete wth pt/tpt eted g pcpa cpet aazg. Bce et a 008 ed D ad techqe f peface eaat teet ak dt g a appach a t Cca' appach. Shaga et a 00 pped a de at a cad f data eepet ad Pcpa cpet aa t t akg. Ka et a. 0 pped a tw-tage appach f tegatg depedet cpet aa IC ad data eepet aa D t ece dcat etwee effcet ad effcet dec-akg t the D. The et f th pape gazed a fw: a ef decpt ge at Stchatc D de ad fzz et the ect. Fat pped SD/ de deced deta ect. pactca exape f appcat deed f th epca td dceted ect. Dc ad fte wk ae azed the at ect ect.. Stchatc D ad Fzz et the:. Stchatc D tdct Data epet a D ae that thee ae DMU Dec Makg Ut DMU whe whe et deted =. The peface f each DMU chaactezed t pdct pce f pt X f = t ed tpt Y f =. It a aed that a DMU hae pt ad tpt ect ad a the cpet f thee ect ae pte. D de: SD de: t : k x k x k t : k x k k P [ ] x Whee the ae tw de ae deged t eae the peface D effcec f the pecfc k-th DMU a k. The ad U epeet weght tpe eated t the -th pt ad the -th tpt epecte. I q. P tad f a pat ad the pecpt "^" peet that ŷ a tchatc aae. It ptat t et that th td teeted fte pag whee the qatt f pt ca e cted a dec aae wht eg ae t ct tpt ecae thee qatte deped p extea fact ch a a ecc cdt. Hece the pt ae cdeed a detetc aae ad the tpt ae cdeed a tchatc aae. T dece the aatca tcte f SD de t cpaed wth a tadta D de fte efeed t a "D at f''. Matheatca the tw de hae the fwg fat: Mde fated de the cdt that each DMU eaated a at f t tta weghted tpt t tta weghted pt. The ga D de detee the at f a DMU t e e tha eqa t t. Ceqet t eg t a effcec age etwee 0 ad 00%. Meawhe q. fate the at t e e tha eqa t a peced ae the age etwee 0 ad 00% that epeet a expected effcec ee f the -th DMU. Cpe et a. 00 cde the expected effcec ce a a "apat ee'' that a eqeted a tde atht ad/ a dgeta tat. Sce et t e t q. the detetc de ca e cdeed a a peca cae f the SD de. The the tad f the pat that tpt/pt at ece e tha wth a chce f weght tpe. Th cdeed a a k cte epeetg tt f a aage. O the the had - hw the pat f attag the eqeet. Lke the k cte a a deced ae that 0

3 J. Bac. pp. Sc. Re eaed the age etwee 0 ad. Whe =0 q. t ceta eqed that the t-pt/pt at ece e tha eqa t. Cee = t the eqeet de a eect f weght tpe. The ecte f q. fated kwhe that f q. expeed k whee the "'' tad f a expected ae f the f weghted.. Fzz et the The f fzz et qte a t a atttde whe facg cetate t expe accate wd ch a appxate e ea etc. a we a f ctec wth ecte dget f dffeet pepe de t a tepetat f a ect. adeh 9 tdced fzz et the f the ft te expeg t the e f dec-akg. I fzz et eehp degee f a eeet etwee 0 ad whe cac et thee ae tw tate: a eeet wth the degee de the et t t wth degee 0. I de t eaate the ad atte cde the dc th pape whch MC-BC et cafcat caed t g et aage ecte dget ad tdcg fzz ccept f ptzg the ctea. T achee thee ed fzz et fzz e ad gtc aae hd ft e tdced Che 00. Deft.. fzz et ~ a ee f dce X defed a eehp fct ~ X whch acate x X a ea e the tea [0]. expe ŷ k ~ X eehp degee f x ~. Deft.. The -ct f fzz et ~ a cp et ~ x ~ x Spp ~ x ~ x 0. ~ a f ad f Sppx X ~ x.. The ppt ~ the cp et Deft.. fzz et ~ f ee et X cex f ad f ~ x ~ x ~ x X [0] whee dete the peat. Deft.. ~ a fzz e f ad f ~ a ad cex fzz et f X. Deft.. taga fzz e ~ defed wth pecewe ea eehp fct ~ x a fw: ~ x a a x a a a a x x a x a a a 0 thewe d a a tpet a a a dcated whee a a the we ad ppe d epecte ad a the t ke ae f ~ Shaad 00.. Fat Pped SD/ de. Refat Stchatc D de I th td the ctat ad ecte f q. ae efated CCP pped Cpe 00. Reeach Cpe et a. 00 hw hw t cpate the CCP techqe t the D at f. I the SD de f thee pape th pt ad tpt ae tchatc aae. Hece fat peeted th td ca e cdeed a a peca cae f the SD. The ctat f q. cdg the tchatc pce ca e ewtte a fw: P x 0

4 Yagh ad Bah.0 Ẑ... q. eqaet t: whee the expected ae f ŷ ad : c c c c c... Hee dcate the aace-caace atx f the -th DMU whch the "'' tad f a aace ad the "c'' efe t a caace peat. T efate q. CCP th td tdce the fwg ew aae : whch fw the tadad a dtt wth ze ea ad t aace. Sttt f q. q. pdce: Sce fw the tadad a dtt the aat f q. 8 exected a fw: Hee tad f a cate dtt fct f the a dtt ad dcate t ee fct. The SD de taed epacg q. q. 9 ad t etg fat ece: x P x P 8 x x.. : x k t k K 0 0

5 J. Bac. pp. Sc. Re ŷ 0 N 0 K K K Th eeach ae that a tchatc aae ŷ f each tpt expeed : Whee a expected ae f ad t' tadad deat. Sect f th pape dece hw t detee the aeage ad the tadad deat f h/he pedct f a dec ake. Cpe et a. 00 pped the apt ag wth a pactca atae. It a aed that a ge ad aae fw a a dtt. Ude ch a apt ece:... Icpat f q. t q. 0 pde: Next pag attet t we efate the ecte f q. a fw: It aed that the ad aae fw a a dtt N0 q.. Ude ch a apt = ad ecae f =0 ceqet the SD de ca e wtte the fwg de:. Ipt/tpt weght deteat wth g fzz appach g ;... D D D D e a gp f dec ake expeg ecpca dget atxe k R K... ; cepdg t pa we cpa f a et f cte... whee K K R a pte qaed atx whch adate f k : x k t x k k k : k k k U : x k t x } { 0

6 Yagh ad Bah.0 The dget epeet the eate ptace t the dec ake Dk f cpaed t. The cpa atx ge the k-th dec ake deted a fw: R K k k k Wth aze the cpa atx ad g w geetc ea ethd the weght f cte ae taed Saat et a. 00. I e td f deteat ptace f weght D de ed gp aatca heach pce GHP techqe. I de t g th techqe wth cdeg expet gget ft the pt ad tpt weght ta wth g qetae f ee cpag ad t dtt ag dda f expet de t kw the ecte pefeece f expet at the at f the weght f each tpt ad fa the pefeed weght f each tpt wa taed g eq. a fwg. The wth cdeg th apt that ad ax f weght eqa ze ad de 0 w f weght epecte we ca hw w. the weght de a fzz e If fzz weght pt de the effcec at w ta wth cdeg expet gget f each t. Bt the t f de a e feae pttg fzz weght t we hd defe aace eg f taed weght. It ette ft we defe the ad aace eg f weght t peet f feae t that th eg ca e chage wth paaete -ct. I the wd whatee -ct ce t the expet' gget eg apped e accate ad whatee -ct ce t 0 the taed effcec w e accate. F exape the aace eg f -th pt wth paaete -ct a fw: q. ca e ewtte a fw: w w w w w w w Wth a pce the aace eg f -th tpt U a fw: w w w 8 The the ea f g fzz weght effcec eaat ae ceag de fext ad deceag pat f de' feat. Nw wth addg ae fzz ctat t eq. fzz SD de defe a fw: t : x k k x { }.. w. w w. w w w

7 J. Bac. pp. Sc. Re F g the ae de ft we cde f the t feae at eat f e t we deceae at f ad e t aga. The adatage f th wk ceag age f paaete' weght eect t the faw deceag accac f expet' gget La Yg et a 99.. tat f Otpt T detee ad f ŷ th td tze thee dffeet kd f tpt etate. dec ake wh ed fte pag aked t fecat the fwg thee etate each tpt f the -th DMU: the t ke etate ML the pttc etate OP ad the petc etate P. The ML the t eatc etate f ŷ. F a tattca ewpt t cdeed a the de the hghet pt f the pat dtt f each tpt. The OP aed t e the ke t pe tpt qatt f eethg ge we. It ca e ee a a etate f the ppe d f the pat dtt. The P teded t e the ke t pe tpt qatt f eethg ge wg. It a etate f the we d f the pat dtt. g that the data fw the eta pat dtt th td cet the thee etate t the expected ae ad aace f each t- pt. The expected ae f t dtt appxate: The aace ece: OP ML P / 0 Whee ML a de ad OP + P/ hw a dage etwee OP ad P the expected ae ca e ee a a weghted athetc ea f the de ad the dage. The de ha tw-thd f the ete weght. It ptat t te that the ae tpe f etat wde ed PRT/CPM Pga aat ad Reew Techqe/Ctca Path Methd. PRT/CPM a aageet cece techqe f pag actt te ad chedg whe th td e the techqe t etate the expected ae ad aace f each tpt. Ug the pped appach fte cetat egadg each tpt whch a fctate de t a ecc fact ca e cpated t D fat Sazehpaa 00.. SD/ de: Sppe we hae depedet hgee dec akg t that the ppe eaatg ad akg t aed taed effcec =... ad a k ee f the wth q. 9 =... p. S the effcec atx ca e epeeted a fw: OP P / D p I de t take the effcec data eea tep ae caed t a fwg: Step : Cacate the ape ceat atx R p p Whee /. ad ape caace. S 0

8 Yagh ad Bah.0 p Step : Cpte egeae... p ad cepdg azed egeect.... p p Step : Cpte ad Seect the pcpa cpet Hee t eakae that a tw dffeet pcpa cpet ae ceated wth each the whch hw that thee fat pept etwee the tw. The aace f a tw pcpa cpet. The eate ptace f pcpa cpet ca e expeed the ppt / p. Geea the pcpe t che the ft M pcpa cpet a e t atf cate ctt tt f dpe M e.g P Shaga & Jh 00 pted t that t wa ge apppate t dect ate DMU a f the tpt pt at dd t eet a Gaa dtt a eqed. Hwee de D tcte t de t eed a dtt apt f fact at pt tpt. Fthee a the che pcpa cpet ca e teated a deae tpt D de f the e pefe each DMU the gge the te f th e. Cepdg a d e ca e ee a a ta pt f a DMU. Hwee tpt f ga D de eed t e tct pte whe the eeet f the che pcpa cpet ca e egate. S a ea te ceag data tafat ade t the egate et f t ad eg egate the eqat: Whee Q a c chce t ee that a tafed ae ae pte. a et a the eeet f the che pcpa cpet ae eqa ceaed the ae ceet. Pat 00 ped that the pt-eted BCC de wa tpt taat aat. Kx Le & Pat 00 fthe ped that a pt-eted CCR de wth a ge ctat pt d ccded wth the pt-eted BCC de. T eaate peata effcec f DMU0 a pfed pt-eted CCR de pped a fw: t : M P P P P 0 whee P the weght attached t the tpt... D... p p W 0 M P M... M M p Q =... ad the weght ctat the fact that the th pcpa cpet cae the tta dpe e tha the + th e de. P epeet. Pactca exape Cde a ak whch ha 0 ache ad the ak pe gg t pedct the effcec f t ach ad a the k that ee ach aage hd accept f eachg the pedcted effcec f the epect f the acated dget f the fte faca ea. It hd e ad that th te the ache pt pee expee ffca expee ad the ct f the pace etg ppe the ache pace ae eg eaed ae pgaed at the ed f each ea f the fte faca ea the faewk f dgetg te. Bt at the ache tpt the at f gated facte ad the fw f te-ak ece pedctg cdcted aed PRT/CPM techqe g the tpt ae f each ach the at faca ea. The ache P 08

9 J. Bac. pp. Sc. Re pt ad the etated tpt ae defed tae the pe a the f f the pttc etate OP the t ke etate ML ad petc etate P. The pe gg t d the efg eae f pg the effcec f h de ct et etatg the effcec ad t depedet k f each ach ad f the fte faca ea. The expected effcec f the pe cdeed f a the dec akg t. Tae. The dgeted pt ad tpt etate Bach cde Offca expee 9 0 Ipt pea expee Ret ct gated facte OP ML P Otpt te-ak ece OP ML P Cptata et Ft we etate the expected ae ad eated tadad deat f tpt f each ach f ak wth PRT/CPM techqe g eq. 0 &. Ret ae taed a hw tae. Tae. Mea ad tadad deat f tpt etate Bak ach Sce the pped de aed the a dtt apt f tpt a pat pt ed wth Stat Gaphc p. ftwae f tpt f ache that et ae hw Fg Fge. Na pt f tpt ad I th exape data et f ak' tpt e pte ad a e egate f pt heet. Ceqet pt ad tpt ae wee azed. de t kw the ecte pefeece f ak 09

10 Yagh ad Bah.0 expet at the weght f each tpt the qetae wee dtted ag f dda f depte f the ak peg ad fa the pefeed weght f each tpt wa taed g eq. a fwg: Tae. Otpt weght Otpt The ded ctat f each tpt g eq. 8 a fwg: weght gated facte 0. te-ak ece Ctat f ŷ : Ctat f ŷ The pped SD/ de ect e aae W P ad. The ga f the ptzat f th de etate effcec ea W f each ach f ak. Sce Wk epeet pedcted effcec ea f k-th ach f ak. Ft we eed t ta a effcec ee wth a k ee g pped de eq. 9 wth cdeg ae ctat. We et the paaete = th de. The et f 0 ta pped de f each ach f ak peeted Tae. Thee et hae taed accdg t dffeet k ee f ache. Tae. t f effcec wth a k ee pped de Bak' ache Bak ach Tae ge the et a the pce a ggeted Sect. ccdg t the pcpe M egeae the the ae tted f eg ae tha hae ee eected whe cate ctt tt f dpe 9%. Tae. geae ad pcpa cpet f k ee Rk ee geae

11 J. Bac. pp. Sc. Re fte the tafat the egate pcpa cpet g eq. the ew tafed tpt f the ft pcpa cpet hae ee che a ee the Tae. Bak ach The et f pedcted effcece ea wee taed wth pped SD/ de g eq. whch tae epeeted t. Bach ffcec W I de t ef the de peface the ea effcece wee taed wth D de ad ea tpt eq. f a f ache fh the pedcted faca ped ad et f thee wee cpaed wth et f pedcted effcece wee taed wth pped SD/ de that tae 8 epeeted t. Bach Tae. The effcec ea aed p the SD/ de Tae. The effcec ea aed p the SD/ de Tae 8. Cpa etwee ea effcece wth D Mde ad pedcted effcece wth pped SD/ de gated facte 8 te-ak ece Rea effcec Pedcted effcec Ceat ate etwee ea effcece ad pedcted effcece = 0.99 The ceat ate etwee ea effcece ad pedcted effcece cacated wth Stat Gaphc p. ftwae. The hgh ceat ate 0.99 ha taed epeet the adt f pped SD/ de. 0

12 Yagh ad Bah.0. DISCUSSION ND CONCLUSION Th td pped a xed de aed Stchatc Data epet a SD ad Pcpa Cpet a f pedctg effcece f a t a gazat. It cpated fte fat t' tpt t aatca faewk. Th de wa ted t ce the t ptat htcg f Data epet a D eteg the tchatc effect f tpt aae ad fzz weght f pt/tpt weght de t peet the ea dtt f weght wth g f Stchatc D SD techqe. Pcpa Cpet a wa ed t detee effcec ea f t wth a k. I de t each a ette detadg f the pped SD/ de t wa apped t pedct effcece f a e f Iaa Bak ache. I de t ef the pped de peface the ea effcece wee taed wth ea tpt ad D de f a f ache fh the pedcted faca ped. Th et wee cpaed wth the et f pedcted effcece wee taed wth pped de. The hgh ceat 0.99 etwee ea ad pedcted effcece wa taed whch epeeted the adt f the pped SD/ de. I th pped de a dtt wa apped f tpt tchatc aae. It a taght fwad atte t cdct a tattca tet the faewk f SD/ aa ad the a dtt. It eceded t exae whethe the dtt ca e ed f fte aa ad app the f fg the ew de. RFRNCS. de N. & Ga B. 00. Icdg pcpa cpet weght t pe dcat data eepet aa. Ja f the Opeat Reeach Scet f Japa pp... Bce H C. W D.D Oe akg peface eaat g data eepet aa ad pcpa cpet aa. Ja Cpte & Opeat Reeach pp. -.. Cha Cpe Rd K 98. Data epet a Refeece ad D e ftwae Kwe cadec Pe.. Chae. W. W. Cpe Q. L. We ad. M. Hag 989 Ce-Rat data epet a ad Mt-Oecte Pgag 0 pp Che C. T. 00. xte f the TOPSIS f gp dec-akg de fzz eet. Fzz Set ad Ste 9.. Cpe WW. Hag.M. L SX. 99. Satfg D de de chace ctat. a f peat eeach PP Cpe WW. Deg H. Hag. ad L SX. 00. Chace ctaed pgag appach t techca effcece. Ja f the Opeata Reeach Scet PP Cpe WW. Hag M Lea. L SX. Oee OB. 00. Chace ctaed pgag fat f tchatc chaactezat f effcec ad dace D. Ja f pdctt aa 9 PP D R. G. ad. Thaa Redcg Weght Fext Data epet a. Ja f Opeata Reeach Scet 9 PP Hag M. L SX 00. Chace Ctaed Pgag ad Stchatc D Mde. Pceedg f the Dec Scece Ittte PP -9.. Jahahah G. M. ezaee S. Saat ad S. Mehaa. 99. The Re f Bd Mtpe D; wth a pca Std. Ja f Scece Iac zad Uet 9 PP. -.. Ka L.J. L C.J. Ch C. 0. ffcec eaeet g depedet cpet aa ad data 0

13 J. Bac. pp. Sc. Re eepet aa. Ja f the Opeat Reeach 0 pp. 0.. Kx Le C.. & Pat J. T. 00. Rada D de wtht pt wtht tpt. pea Ja f Opeata Reeach 8.. La Yg-J Hwag chg-a. 99. Fzz atheatca pgag ethd ad ppcat. Spge-eag.. Lad K. Le C..K. The S 99. Chace ctaed data eepet aa. Maagea ad deca ecc. PP. -.. Pat J. T. 00. Taat aace data eepet aa: geeazat. a f Opeat Reeach Peachada I. M. 00. te D. pcpa cpet aa: peet t Je h appach. pea Ja f Opeata Reeach R Y. ad B. Ga 99. teate Methd f Teatg Fact Weght D. Oega PP Saat S.. Meaa ad G. R. Jahahah 00. Pcede f Fdg a C et f Weght Fzz D. Fzz Set ad Ste Sazehpaa M. 00 Pect ct. Teha Khaa cat.. Segpta JK. 98. tchatc pgag. Iteata a f te cece PP Shaga R. Jh C.00. t a cad f data eepet ad pcpa cpet aa. Oega Shaad H. 00. The f fzz et dta egeeg ad aageet. Teha Gtaeh O Paeh. adeh L.. 9. Fzz et. If ad Ct h J. 00. Data eepet aa. pcpa cpet aa: tate td f ecc peface f Chee cte. pea Ja f Opeata Reeach 0. 0