Selective Convexity in Extended GDEA Model

Transcription

1 Appled Mathematcal Scences, Vl. 5, 20, n. 78, Selectve nvet n Etended GDEA Mdel Sevan Shaee a and Fahad Hssenadeh Ltf b a. Depatment f Mathematcs, ehan Nth Banch, Islamc Aad Unvest, ehan, Ian b. Depatment f Mathematcs, Scence and Reseach Banch, Islamc Aad Unvest, ehan, Ian Abstact In ths pape, an etensn f the genealed data envelpment analss (GDEA mdel has been ntduced. In ths genealatn dffeent cases f pductn pssblt set (PPS f the etended genealed data envelpment analss (EGDEA mdel ae nvestgated. he ppsed EGDEA mdel s used t evaluate decsn s makng unts (DMUs ethe wth full cnvet assumptn mpsed n nputs and utputs (R and B mdels wthut cnsdeng cnvet assumptn f each nput and utput (FDH mdel. We have ppsed a mdel n the EGDEA, whch can teat each nput and utput ndvduall wth espect t the cnvet assumptn. Kewds: Genealed Data Envelpment Analss; Selectve nvet. Intductn GDEA mdel was suggested b Yun et al. [6] s a methd, whch can teat the basc DEA [,2] mdels, specfcall, the R mdel, the B mdel and the FDH mdel n a unfed wa. hee ae theetcal ppetes n elatnshps amng the GDEA mdel and thse DEA mdels. he GDEA mdel makes t pssble t calculate the effcenc f DMUs ncpatng vaus pefeence stuctues f decsn makes. In de t fnd a bette appmatn f evaluatn f DMUs, the GDEA mdel has been develped t etended GDEA (EGDEA mdel. hs develpment has been pefmed b the ntepetatn f dual f the GDEA mdel. Als, ths mpvement leads us t defne the defntn f the PPS f the EGDEA mdel. he defntn f the PPS f the EGDEA espndng auth. Shaee@ah.cm (Sevan Shaee P.O.B: 455/775 and 455/4933, Pst de:

2 3862 S. Shaee and F. Hssenadeh Ltf mdel defned n ths pape has specal stuctue: unde sme specal cndtns ethe the full cnvet assumptn f the PPS nn-cnvet assumptn can be btaned b ts defntn. But ths defntn causes sme pblems t the analst. Snce, he she has t accept ethe the full cnvet assumptn f the PPS f the EGDEA (R and B mdels negate the full cnvet assumptn at all (FDH mdel. In ethe case sme shtcmngs and msleadng estmates ae undestandable. In de t vecme the dlemma, a new defntn f the PPS f the EGDEA has been pesented that allws each nput and utput can be udged ndvduall wth espect t the cnvet assumptn. We have used the cncept f selectve cnvet n DEA [5] n de t ppse the abve mentned PPS. hs fleblt f the defntn n the ppsed PPS f the EGDEA mdel slves the shtcmngs f the pevus defntn. he abve assumptn f selectve cnvet leads t the develpment f a ange f new EGDEA mdel, f whch the FDH and B mdels ae tw specal cases. Als the eact defntn f PPS f the EGDEA mdel has been pesented n de t mpse the selectve cnvet assumptn n t. In the defntn f the PPS f the EGDEA mdel dffeent cases f PPS has been cnsdeed f whch the PPS f the R, the B and the FDH mdels ae specal cases f the mentned defntn. he pape s gven n 7 Sectns. In Sectn 2, we have nvestgated basc GDEA and etended GDEA mdels. he mtvatn f etended mdels has been descbed n sectn 3. he dffeent ppetes f PPS have been cnsdeed n Sectn 4. he defntn f new ange f etended GDEA mdels that allw selectve cnvet n the fmulatn has been ntduced n Sectn 5. A numecal eample s gven n Sectn 6. Sectn 7 nvestgates a mdel t fnd the full effcent DMUs n the ppsed mdel. 2. Basc GDEA and etended GDEA mdels In the fllwng dscussn, we assume that thee est n DMUs t be evaluated. Each DMU cnsumes vang amunts f m dffeent nputs t pduce p dffeent utputs. Specfcall, DMU cnsumes amunts = f nputs (=,,m and pduces amunts = f utputs (k=,,p. f these cnstants, whch geneall take the ( fm f bseved data, we assume > 0 f each (=,,m and > 0 f each k=,,p. Futhe, we assume that thee ae n duplcated n bseved data. he p n utput mat f the n DMUs s dented b Y, and the m n nput mat f the n DMUs dented b X. =,..., and =,..., ae amunts f nputs and ( m ( ( p utputs f DMU, whch s evaluated. In addtn, ε s a small pstve numbe (nn- Achmedean and =(,, s a unt vect. he fmulatn f GDEA mdel s n the fllwng manne Yun [6]:

3 Selectve cnvet n etended GDEA mdel 3863 Mame Δ S. t. Δ,μ k,ν ~ Δ d p p = = μ m v m = = μ + + α( p μ ( v = = =, = 0, + μ, ν ε, =,..., p, =,..., m m ν ( +, =,..., n, ( Nw we can fnd the dual f the mdel ( as fllws: Mnme w, κ, λ, s S. t. ω ε { α( Z λ 0, s 0. n whch an utput-nput vect DMUs espectvel, dented b Y =, =,..., n and Z = X λ =, s Z + D } λ w + s + κ = 0, (2 f a DMU, =,,n, and utput-nput mat Z f all In addtn, we dente a ( p + m n mat Z b Z = (,,..., and w = ( ω,..., ω and α s a gven pstve numbe. A ( p + m n mat D = ( d,..., d n s a mat ( Z Z eplaced b 0, ecept f the mamal cmpnent (f thee est plual mamal cmpnents, nl ne s chsen fm amng them n each clumn. he mdel (2 fstl tes t pect DMU unde the dectn =(,, whch s a ( p + m unt vect. If the pectn s n the weakl effcent fnte the mdel (2 tes t pect the DMU 0 n the stngl effcent fnte b usng the cncept f nn- Achmedean wth slack vaables vect s whch s a ( p + m -vect. In de t use me genealed mdel we etend the GDEA mdel n whch the ppsed mdel can pect a DMU unde evaluatn alng an feasble dectn such as δ δ = 0, δ 0. heefe we ppse the etended GDEA mdel n the fllwng δ manne:

4 3864 S. Shaee and F. Hssenadeh Ltf Mame S. t. Δ,μ,ν k Δ ~ Δ d p m μ δ + = = p μ = = + α( m p μ ( v δ v = = =, = 0, + μ, ν ε, =,..., p, =,..., m m ν ( +, =,..., n, (3 and the dual f mdel (3: Mnme w, κ, λ, s S. t. ω ε { α( Z s λ =, δ Z + D } λ ω + s δ + κ = 0, λ 0, s 0. In mdel (4 we fstl mve t each t effcent fnte alng the dectn δ δ = 0, δ 0 and secndl, t t vansh slack vaables n de t fnd stngl δ effcent unts. We defne the pductn pssblt set (PPS f etended GDEA technlg n the fllwng manne: (4 PPS EGDEA { ˆ ˆ = ( ˆ, ˆ ( α( Z Z + D λ + κˆ 0, λ =, 0} (5 = λ heem. We have the fllwng stuctues: a If α s suffcentl small pstve numbe and κ s fed at e then (5 wll be the PPS f FDH technlg. b If α s suffcentl lage pstve numbe and κ s fed at e then (5 wll be the PPS f B technlg. c If α s suffcentl lage pstve numbe then (5 wll be the PPS f R technlg. Pf: We knw that the PPS f R, B and FDH mdels ae espectvel as fllws:

5 Selectve cnvet n etended GDEA mdel 3865 PPS PPS PPS R B FDH = {ˆ = ( ˆ, ˆ = {ˆ = ( ˆ, ˆ = {ˆ = ( ˆ, ˆ Zλ ˆ, λ 0}, Zλ ˆ, λ =, λ 0}, Zλ ˆ, λ =, λ {0,}, =,.., n} a If α s suffcentl small pstve numbe and κ s fed at e then fm (5 we wll have D ˆ λ 0. Fm the defntn f D ˆ the fllwng nequaltes wll be satsfed wth λ, = 0 : = λ k ˆ and ˆ, =,..., n. heefe, we wll have Zλ ˆ f λ = that s the eact defntn f PPS f FDH technlg. and λ {0, } b If α s suffcentl lage pstve numbe and κ s fed at e then fm (5 we wll have α ( Zˆ Z λ 0 but then ( Zˆ Z λ 0 and ths mples that Zλ Zˆ λ and als because λ =, we wll have Zλ ˆ and ths s the eact defntn f PPS f B technlg. c If α s suffcentl lage pstve numbe then fm (5, It s bvus that n n ˆ α α ( Z Z λ + κˆ 0 and t s elevant that X Xˆ α λ and λ Y Yˆ but = α + κ = α + κ n α then f we set λ = λ, =,..., n, t wll be btaned that λ X Xˆ and α + κ n = λ Y Yˆ whch ae the eact defntn f PPS f R technlg. = 3. Mtvatn f mdfcatn In GDEA mdel, the cncept f adal effcenc s cmpehensble b ntng the stuctue f GDEA mdel. It s undestandable the DMU unde evaluatn appaches effcent fnte alng the dectn (,, that s a (m+p-vect. Appachng effcent fnte nl alng the abve mentned dectn causes sme shtcmngs and msleadng estmates, because smetmes the DMU t be evaluated has t appach effcent fnte alng a cetan and specfc dectn. F eample, f mantanng sme specal cndtns the DMU unde evaluatn s nt able t ncease decease sme specal utputs nputs espectvel. heefe we need t make sme changes n the fmulatn f GDEA mdel t vecme shtcmngs and weaknesses. In ppsed EGDEA mdel the DMU t be evaluated appaches effcent fnte alng an abta dectn. hs fleblt f fundatn f EGDEA mdel nt nl has me genealed

6 3866 S. Shaee and F. Hssenadeh Ltf stuctue but als pvdes bette appmatn. B the wa EGDEA mdel slves the weaknesses and dsadvantages f GDEA mdel. he stuctue f PPS f EGDEA mdel s based n paametc dmnatn stuctue. he PPS f dffeent technlges such as B and FDH technlg can be acheved b changng paamete α fm suffcentl lage pstve numbe t suffcentl small pstve numbe espectvel. On ne hand sme ambguus pnts ae appaent n the chce f paamete, f eample, f paameteα s nethe suffcentl lage pstve numbe n suffcentl small pstve numbe, what s the pductn technlg? Is t FDH technlg B technlg a mtue f FDH and B technlg? On the the hand almst all the tmes, ethe the full cnvet assumptn f B technlg elaatn f the cnvet assumptn f FDH technlg mght be pblematc. F eample, cnsde a stuatn that a specal nput f DMUs s the numbe f emplee. It s bvus that the abve mentned nput ften s nt a pat f cnvet assumptn because the cmpste unt f DMUs ma be nfeasble. Als lts f dffeent eamples ma be addessed n dffeent felds [5] wth pevus stuctue. heefe the use f selectve cnvet t vecme the dffcultes caused b EGDEA mdel s unavdable. Based n the abve dscussn, we need t mdf the defntn f the PPS f EGDEA technlg n a wa that allws each nput and utput t be evaluated ndvduall wth espect t cnvet assumptn that cause the cncept f selectve cnvet. he pblem f selectve cnvet s gven n emaned sectns. 4. Ppetes f the PPS f EGDEA mdel In de t pesent a sutable defntn t the PPS f the EGDEA mdel that can teat wth each nput and utput ndvduall wth espect t the cnvet assumptn, we need t defne sme ppetes f the PPS f EGDEA mdel. Als we need t pesent a ppet that can develp the teatment f PPS unde the cnsdeatn f abta cnvet assumptn f each nput and utput. Wth cnsdeng last assumptn, t s pssble t defne a PPS based n the mentned ppetes that can behave wth each nput and utput ndvduall wth espect t ts cmplance wth the cnvet assumptn. hs assumptn s called Selectve cnvet and has been defned b Pdnvsk [5]. We use the cncept f selectve cnvet n de t mpve the defntn f the PPS f EGDEA mdel that can cnsde the cnvet assumptn t abta nput and utput. Wth the defntn f the PPS f EGDEA based n the stated ppetes, a new mdel can be btaned that B and FDH mdels ae specal cases f ppsed mdel. We pstulate the fllwng ppetes f P (the pductn pssblt set: (A Nn-empt assumptn. DMU = (, P, =,..., n. (A2 Pssblt assumptn. = (, P, 0 and mples = (, P.

7 Selectve cnvet n etended GDEA mdel 3867 Ou thd ppet takes n t accunt the stuatns whee nl sme f the nputs and utputs ae suppsed t satsf the cnvet pncple, whle the thes ae nt. state ths ppet fmall, cnsde the fllwng pattn f the nput and utput sets I and O: N N I = I I, O = O O N N whee the subsets I and I, as well as O and O, ae mutuall dsnt. he fllwng ppet states that a cnve cmbnatn f an tw feasble DMUs = (, and = (, s feasble pvded the nputs and utputs f these tw N N DMUs fm the sets I and O ae dentcal. (A3 Selectve cnvet assumptn. = (, P and = (, P. N N Assume that = f all I and = f all O. hen, f an λ [0,], the unt λ + ( λ P. he cncept f the ppet (A3, s llustated b Pdnvsk [5]. Me genealed nfmatn can be fund n Pdnvsk [5]. We llustate the selectve cnvet assumptn n the fllwng eample. Eample. In de t llustate the cncept f the assumptn f selectve cnvet, cnsde the DMUs A and B n able 2. able 2. he data set f Eample 2 Output Input Input 2 Input 3 DMU A DMU B We have llustated n Fgue, the PPS f mdfed EGDEA technlg f tw DMUs n the thee dmensnal space, wth egad t the selectve cnvet assumptn. he unn f tw tanspted unbunded cnes epesents the FDH technlg btaned b the tw unts. akng n t accunt the selectve cnvet assumptn wth egad t set I = {,3 }, the unbunded psm BDqt s added t the FDH technlg acheved b the tw DMUs. In patcula, because f cnsdeatn (A3, cnve cmbnatn f feasble unts ae feasble pvded the nput 2 s dentcal. F me llustatn f nput 2 s equal t 4, the applcatn f the assumptn f selectve cnvet leads t the tangle BD beng egaded as feasble.

8 3868 S. Shaee and F. Hssenadeh Ltf Input 3 Input 2 Input Fg..he pductn pssblt set n Eample. As we can see n the assumptn f selectve cnvet (A3, t s an cnceptual assumptn and we can nt use t n pactcal pblems and we need t make sme mdfcatn n (A3. ths end, we utle the genealatn f selectve cnvet assumptn as eplaned n Pdnvsk [5]. We use the genealatn f selectve cnvet assumptn n the fllwng manne: ( A 3 Genealed selectve cnvet asumptn. nsde an K DMUs k = ( k, k P, k =,..., K. F an vect the unt as fllws: K λk, λ k = k = ma{ k k =,..., K}, f I f I N (6 K λ R + such that K k = λ = defne K λk k, f O λ = k = (7 N ma{ k k =,..., K}, f O λ λ λ hen = (, P. It s clea that the assumptn ( A3 mples (A3. In the pesence f pssblt assumptn (A2, the cnvese s als tue. Me nfmatn can be fund n Pdnvsk [5]. heefe the set f assumptns (A, (A2 and (A3 s equvalent t the set (A, (A2 and ( A 3. k

9 Selectve cnvet n etended GDEA mdel 3869 We have used the cncepts f selectve cnvet, n de t ppse a slutn and fnd a sutable defntn t the PPS f etended GDEA technlg. In the fllwng sectn the defntn f the PPS f etended GDEA technlg has been pesented. 4. he PPS f EGDEA wth egad t selectve cnvet assumptn Based n the ppetes eplaned n the pevus sectn, we epesent a mdfed defntn f the PPS f EGDEA that nt nl satsfes the nn-empt assumptn and pssblt assumptn but als the selectve cnvet assumptn. Wthut lse f genealt we can assume that the set I cntans the fst,..., m N nputs whle the set I cntans the m +,..., m nputs and als the set O cntans the N fst,..., whle the set O s the set +,...,. Wth these ntatn an vect f nputs can be epesented n the fm N = [, ] and als N = [, ] whee and ae the fst m and cmpnents f nputs and utput vects espectvel. he same ntatn etends t the, Z and Z matces whee s the fst cmpnents f utputs and fst m cmpnents f nputs, Z s the fst ws f Z and ws + untl + m and Z smla t Z althugh wth these ntatn we wll have N, Z N N and Z the emaned pats f the mentned vect and matces espectvel. hee fe, the fllwng heem s epesented based n the pncples (A, (A2 and (A3. heem 2. he mnmal PPS f etended GDEA technlg P, whch satsfes the ams (A-(A3, s the set f all pas ˆ = ( ˆ, ˆ such that ˆ 0 and ˆ 0 and thee n ests a vect λ R f whch the fllwng elatns ae tue (because we use the FDH and B technlg κ s fed at e: ˆ ( α ( Z Z + Dˆ λ 0 (8 D N ˆ λ 0 (9 λ = and λ 0 (0 Pf. nsde = (, that s an bseved DMU. It s bvus that ( α( Z Z + D λ 0, D N λ 0 f λ =, λ k = 0 and theefe (A s satsfed. see (A2, cnsde ˆ = ( ˆ, ˆ P, ˆ 0and ˆ. Because, ˆ = ( ˆ, ˆ P, thee est a vect λˆ that satsfes the cndtns (8-(0, theefe ˆ ( α ( Z Z + Dˆ λ 0, D N ˆ λ 0. B the defntn f mat D and u assumptn t s tval that ˆ ˆ ( λ ( ˆ Z Z Z Z λ and ˆ λ ˆ N D ˆ D λ and ˆ N D λ ˆ ˆ D λ, theefe = (, P and the pncple (A2 s satsfed.

10 3870 S. Shaee and F. Hssenadeh Ltf In de t see that P satsfes (A3, cnsde an tw unts = (, P and N N = (, P such that = f I and = f O. hen these tw unts satsf (8-(0 wth sme vects λ and λ espectvel. F an β [0,] β β β cnsde = (, = β (, + ( β (, and defne λ β = βλ + ( β λ. hen β β β β the cndtns (8-(0 ae bvusl satsfed b the unt = (, and λ = λ. heefe (A3 s satsfed. pve that P s the mnmal set, assume that anthe set P als satsfes (A-(A3. We need t shw that, f (, = P, then = (, P. nsde epesentatn (8-(0 f the unt = (, P. F the vect λ fm ths λ λ λ epesentatn, defne = (, accdng t (6 and (7, whee K shuld be taken equal t n, the numbe f bseved DMUs. Snce P satsfes ( A 3 the unt λ λ λ = (, P. Snce ths unt dmnates = (, n the Paet sense, accdng t (A2, = (, P. heem 2 shws that the PPS defned b the elatns (8-(0 s a subset f an the set P whch satsfes the pncples (A, (A2 and (A3. heefe the PPS P s the mnmal set based n the abve thee pncples. 5. he etended GDEA mdel based n selectve cnvet assumptn Accdng t heem 2, mdel (4 wthut usng slack vaables and κ fed at e, we ppse the EGDEA mdel based n selectve cnvet assumptn n the fllwng manne: Mnme ω w,λ S. t. δ = δ { α( Z D N λ =, λ 0 Z δ λ ω δ + D N N N N whch ( δ δ δ 0, δ } λ ω 0, δ ( s the decmpstn f vect δ t the acceptance f cnve and nn-cnve paametes. heefe, we use mdel ( n de t slve sme pblems that allw cnve and nn-cnve data.

11 Selectve cnvet n etended GDEA mdel Numecal eample llustate the utcmes f the ppsed mdel, we used the mdfed mdel ( f the data set n able 2. he esults f u mdel ae pesented amng the clumns f able 3. able 3 shws the effcenc f each f DMUs btaned b slvng mdel (4 wthut 0 slack vaables,κ fed at e, 3 δ = and ne tme f α = 0 suffcentl lage that causes the B effcenc and the the tme f α = that causes the FDH effcenc. Als dffeent cases slved b mdel ( wth ncpatng selectve cnvet pncple based n dffeent chce f the sets I and O. We knw that n mdel (4 wthut slack vaables and (, f ω = 0 then the DMU unde evaluatn s effcent (but nt necessal full effcent the wse s nt effcent. We can see that the me mdel ( eaches t the B technlg amng the clumns f able 3, the less the DMUs ae effcent and t s bvus b the defntn f PPS f B and FDH technlg. able 2. he data set f numecal eample Input Input 2 Output Output 2 Output 3 DMU DMU DMU DMU DMU able 3. he effcenc f DMUs n dffeent mdels DMU FDH Mdel (, Mdel (, Mdel (, B α = 3 α = 0 3 α = 0 3 α = 0 3 α = 0 I = { }, O = {} I = {}, I = {}, O = {,2} O = {,2,3}

12 3872 S. Shaee and F. Hssenadeh Ltf 6. Fndng full effcent DMUs In ths sectn we use a new mdel n de t vef whethe the taget DMU = (, f DMU s effcent n Paet sense. F eample n mdel (4 the taget f DMU n the case when α s suffcentl lage s n the fllwng manne: α + κ = ωδ (2 α α We ntduce the unt ( + s, + s wth espect t taget DMU = (, btaned fm (2, whee s and s ae nn-negatve vects f dmensn p and m, espectvel. Mdel (3 has been suggested t fnd the full effcent (effcent n Paet sense DMUs. he sum f vects s and s s mamed n mdel (3, whle the cnstants ensue that the esultng unt ( + s, + s s feasble n technlg P, as defned n heem 2. Mdel (3 can be slved wth espect t the vaable vects N λ, s and s. Ntatns s, s, s and s ae defned as pevus sectns. Mame s S. t. { α( Z D N λ + s λ =, λ 0, s Z N 0. + D 0 } λ + κ N + s = 0, (3 N s s n whch = N s and s =. s N s Suppse that λ and s s an ptmal slutn t mdel (3. hen t s eas t see that the DMU ( + s, + s s effcent n Paet sense and can be cnsdeed as the full effcent taget f DMU. In patcula, f the ptmal slutn n mdel ( s equal t e and s = 0 n the ptmal slutn f mdel (3, then DMU s tself effcent n Paet sense. 7. nclusn In ths pape, an etensn f GDEA mdel based n paametc dmnatn stuctue has been suggested. In addtn, we defned the PPS f EGDEA technlg. We used the

13 Selectve cnvet n etended GDEA mdel 3873 cncept f selectve cnvet assumptn n DEA t ppse a ange f new EGDEA mdel that allws each nput and utput can be udged ndvduall wth espect t cnvet assumptn. F fndng the abve mdel, we ntduced a new stuctue f PPS f EGDEA mdel unde cnsdeatn f selectve cnvet assumptn. Als, the ppsed methd has been utled n a numecal eample and we encunteed wth the epected esults. Futheme, we nvestgated the pblem f fndng full effcent DMUs wth a mdel btaned b the secnd ptmatn stage n DEA. Refeences [] Allen, R., Athanasspuls, A., Dsn, R.G., hanassuls, E., Weghts estctns and value udgements n data envelpment analss: Evlutn, develpment and futue dectns. Annals f Opeatns Reseach 997; 73, [2] Banke, R.D., hanes, A., pe, W.W., sme mdels f estmatng techncal and scale neffcences n data envelpment analss. Management Scence 984; 30, [3] Pdnvsk, V.V., Pductn tade-ffs and weght estctns n data envelpment analss. Junal f peatnal Reseach Scet 2004; 55, [4] Pdnvsk, V.V., he eplct le f weght bunds n mdels f data envelpment analss. Junal f Opeatnal Reseach Scet2005; 56, [5] Pdnvsk, V.V., Selectve cnvet n DEA mdels. Eupean Junal f Opeatnal Reseach 2005; [6] Yun, Y.B., Nakaama, H., ann,., A genealed mdel f data envelpment analss. Eupean Junal f Opeatnal Reseach 2004; Receved: August, 20