A Weight Restriction Approach for Evaluating Two-Stage Decision Making Units

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1 A Weght Rettn Appoah fo Evaluatng Two-Stage en Makng Unt A. Paan, A.R Hajhoen Abtat In th pape, the elatve effen of two-tage en akng unt (MU) etate b ofng the pout oel popoe b Kao an Hwang (2008) an the atve oel b Chen et al. (2009). Atle that have o fa aee the evaluatn of two-tage unt have been anl unable to alulate elatve effen. When no alulatn of elatve effen avalable, t not poble to fo effen fonte, etene benhak unt, etate etun to ale an o on. Bae on the natue of two-tage oel, we popoe to one the a ata envelopent anal oel wth the auane egn tpe II. In th etn, the elatve effen of two-tage MU etate. The valt of the etho alo pove. An eaple peente to eplan the etho an aw a opan between th etho an othe avalable etho of two-tage unt. Kewo ata envelopent anal; auane egn tpe ; elatve effen; two-tage MU. I. INTROUCTION ATA envelopent anal (EA), whh wa popoe b Chane et al. [], a atheatal pogang tehnque ue to eaue the elatve effen of a goup of en akng unt (MU). It abe a weght to eve nput an output. In the EA etho, the effen of MU epee a the at of weghte u of output to the weghte u of nput. Weght ae onee a en vaable n EA oel. MU o not alwa follow the ple nput-output patten. Mult-tage an ult-oponent te ae oon eaple of en akng unt wth oe ople tutue. In oe en akng unt, ung a ee of nput lea to poutn of output that ae not onee the fnal output of the te. Thee output, whh ae known a the le ata, fo the nput of the eon tage an el fnal output. Suh en akng unt ae alle two-tage en akng unt. The oveall tutue of twotage unt hown n Fg.. Th wok wa uppote b the eeah fel of Zahean Banh, Ila Aa Unvet, Zahean, Ian. A. Paan wth the epatent of Matheat, Zahean Banh, Ila Aa Unvet, Zahean, Ian (oeponng autho to pove phone: ; fa: ; e-al: paan_au@ahoo.o, a.paan@ auah.a.). A. R. Hajhoen wth the epatent of Matheat, Zahean Banh, Ila Aa Unvet, Zahean, Ian (e-al: hajhoen54@ahoo.o). Fg. Two-tage en akng unt Two-tage en akng unt ae wel ue fo eal wol pupoe. Fo ntane, the ae eploe n bank, oal nuane te, et. [2]. Nueou tue have been ntate ne 999 to tu two-tage en akng unt. Kao an Hwang [3] ntoue a oel fo alulatn the ft an eon tage effene a well a oveall te effen ung the ultple fo of CCR oel. The efne oveall te effen a a pout of the ft an eon tage effene. The ntoue a oel, known a the pout oel, fo alulatn the effen of two-tage unt. Chen et al. [4] onee oveall te effen to be a onve obnatn of the ft an eon tage effene an popoe a oel known a the atve oel. In atn to the afoeentne etho, whh wee bae on the ultple fo of EA oel, eeahe alo have foue on the aeent of pefoane of two-tage unt bae on the envelopent fo of EA. Chen an Zhu [5] ntoue an envelopent fo fo the aeent of pefoane of two-tage unt wth vaable etun to ale auptn. Chen et al. [6] alo pove equvalene of the oel evelope b Chen an Zhu [5] n ontant etun to ale tuatn an ual oel b Kao an Hwang [3]. Chen et al. [7] epote that the etng envelopent oel ae not apable of etenng effent fonte an fonte pojetn (benhak unt) an th a ba poble n evaluatng two-tage unt. Fo oe nfoatn on two-tage EA oel ee the atle b Cook et al. [8]. One of the ue wth ata envelopent anal the tu of the effet of auane egn on EA oel. In EA, auane egn ae ve nto two goup: ) auane egn tpe I, whh onl poe ontant on nput o output; 2) auane egn tpe II, whh poe ontant on both nput an output. Applatn of the auane egn tpe II on EA oel often lea to poblt of alulatn of elatve effen [9]. The ot potant poble n uh ae ae lak of foatn of an effent fonte an the nablt to poue a fonte pojetn. Reentl, Khall et al. [0] have ntoue a ISSN:

2 etho fo alulatn of elatve effen unt wth lnea auane egn tpe II. The oel a non-lnea pogang poble. The ultple oel evelope fo alulatn the effen of two-tage MU nlue two tpe of ontant, whh have the natue of the auane egn tpe II. Hene, t poble fo thee oel to fal to alulate elatve effen. A a eult, the afoeentne poble eege. Soe eaple ae pove n the eon etn of th pape to laf the pont. B oneng twotage ultple oel a EA oel wth auane egn tpe II, thee oel (poutve an atve oel) ae then ofe uh that the beoe apable of alulatng elatve effen. Hene, the peent pape nlue the followng etn. The eon etn ue two-tage EA oel an the avantage. The th etn popoe a etho fo alulatn of elatve effen of two-tage en akng unt. An eaple alo pove fo a bette unetanng of the etho. The fnal etn of the pape peent a onlun. II. TWO-STAGE EA Cone a two-tage poe an aue that thee ae n en akng unt. Eve ue ne a t nput to poue ne a output fo the ft tage. Net, the output ae ue a the nput fo the eon tage an poue the output of th tage, whh ae hown b. Th poe epte n Fg.. A. Poutve two-tage EA Kao an Hwang [3] evelope a oel fo alulatn the effen n two-tage EA. The etho able to alulate oveall effen a well a ft an eon tage effene b an objetve funtn a: u w * w v u v o Th oel known a pout oel an epee a follow: a u v.t u w,, j j j,n,, (-) (-2) Reult of the pout oel () apple to eaue the effen of 5 two-tage unt peente n Table I ae hown n the eon olun of Table II. Aongl, none of the MU whee hown to be effent. That to a, the pout oel () not apable fo alulatng the elatve effen of two-tage en akng unt. Table I Fve Two-tage MU B. Atve two-tage EA Chen et al. [4] tue the oveall effen of two-tage en akng unt a the onve obnatn of the ft an eon tage effene. Hene, the oveall effen of epee a follow: 2 o (2) w w v +w u w whee, w an w ae weght atfng the elatnhp 2. Chen et al. [4] evelope the followng oel bae on Relatn (2). Th oel known a the atve oel an egne to ae the pefoane of two-tage en akng unt. a w w o v +w 2u o w o,.t u w, j,,n, j j w j v j, j,,n, (3) The efne w an w2 a follow n oe to olve the above oel [4]: w v v + w w w v + w 2 Hene, oel (3) tanfoe nto the followng oel a: a w o o v + w o, j (4-) j.t u w, j,,n, (4-2) () (4) ISSN:

3 A een n the fouth olun of Table II, none of the MU n Table I ae hown to be effent b olvng oel (4). That to a, the atve oel (4) not apable of alulatng the elatve effen bae on the ata peente n Table I. ( ) u w, j,,n, j j Table II Meaung oveall effen of MU n Table I Theefoe, thee a ba poble n two-tage oel whh the ae not able to alulate elatve effen, whle the ft a of EA opang MU togethe. III. RELATIVE EFFICIENCY A entne, ontant (-) an (-2) n oel () an ontant (4-) an (4-2) n oel (4) have the natue of the auane egn tpe II. Peene of auane egn tpe II n thee oel the an aue of the nablt to alulate elatve effen. Hene, thee oel ae onee a EA oel wth auane egn tpe II an ofatn wee pefoe to alulate the elatve effen of twotage unt. (6) The above oel ewtten bae on the vaable tanfoatn ntoue b Chane an Coope [2] a follow: k v,u,w, a u,.t v, j j u v 0, j,,n, w v 0, j,,n, u w 0, j,,n, j j o A. Relatve effen n pout two-tage EA B applng the etho evelope b Thopon an Thall [] to oel (), the followng fatnal oel obtane fo alulatn of elatve effen of two-tage en akng unt: u o v f ( v,u,w ) a, a u j v j j,..., n j (5-) j.t u w, j,,n, (5-2) We efne a u v j j,..., n (5). Aong to ontant (5-) an (5-2),. The value of all nput an output fato a well a le ata ae lage than eo an thu 0. Hene, bae on Khall et al. [0], the followng oel evelope a: g v,u,w, a u o v,. t u j v j, j,,n, (7) efntn : an effent two-tage en akng unt f the optal value of oel (7) an all vaable have potve value, fo at leat one optal olutn. Theoe 2: Moel (5) an (6) ae equvalent a the have equal optal objetve value. v,u,w, an optal olutn to oel Poof: Aue (6). Theefoe, ( v,u,w ) a feable olutn to oel (5). Moeove, ne a u j,.., n v j j then: a u v j j j,..., n an 0 { } u o v u o v u j v j a,..., n g ( v,u,w, ) f ( v,u,w ) Hene, the optal value of oel (6) le than o equal to the optal value of oel (5). On the othe han, f an optal olutn to oel (5) an a u j,.., n v j j a feable olutn to oel (6). In atn,, then ISSN:

4 a j,..., n u j v j ( u v ) a{ u v } ( u v ) g ( v,u,w, ) f ( v,u,w ),..., n j j Theefoe, f ( v,u,w, ) an optal olutn to oel (6), then an. Hene, the optal value of oel (5) le than o equal to the optal value of oel (6). It onlue that the optal value of (5) an (6) ae equal. Aongl, the effen value obtane b oel (5) an (6) ae equal. Theefoe, ntea of olvng oel (5), whh gve the elatve effen of two-tage en akng unt, t poble to ue oel (6) to alulate the elatve effen of two-tage unt. Theoe 3: Moel (6) equvalent to oel (7). Poof: Aue ( v,u,w, ) an optal olutn to oel (6). Ift v, then ( V,U,W, ) ( tv,tu,tw, ) a feable olutn to oel (7). On the othe han, k ( V,U,W, ) U t u o o u o v g ( v,u,w, ) Hene, the optal value of oel (6) le than o equal to the optal value of oel (7). If the optal olutn to oel (7), t a feable olutn to oel (6) too. Moeove, ( ) U k V,U,W,C o g V,U,W,C U C V Theefoe, the optal value of oel (7) le than o equal to the optal value of oel (6). In th ae, the optal value of oel (6) an (7) ae equal an thee oel ae equvalent aongl. Reult 4: Moel (5) an (7) ae equvalent n te of the optal value. Reult 5: Moel (6) alulate the elatve effen of two-tage unt. Reult 6: Moel (7) alulate the elatve effen of two-tage unt. Bae on the above entne theoe an eult, we an efne the onept of efeene pont (MU) fo non-effent unt. efntn 7: Cone ( v,u,w, ) a an optal olutn fo evaluatng b oel (0). a efeene fo, f u v 0 p p B. Relatve effen n atve two-tage EA In oe to alulate elatve effen b the atve twotage EA oel the above poe teate an oel (4) tanfoe nto the followng fatnal oel a: w o o w j j F ( v,u,w ) a a j,..., n w o + v w j + v j.t u j w j, j,,n, If a w j j w j + v j, the j,..., n followng fatnal oel eve fo oel (8): G( v,u,w, ) a w o o w o + v, t. w + u w + v, j j j,,n, u w, j,,n, j j The above oel an be ewtten a follow aong to the etho ntoue b Chane an Coope [2]: K v,u,w, a w + u, o t. w o + v, w + u ( w + v ) 0, j j j,...,n, w v 0, j,,n, u w 0, j,,n, j j (8) (9) ISSN:

5 v 0,,,, u 0,,,, w 0,,,, (0) efntn 8: MU an effent two-tage unt f the o optal value of oel (0) equal to an all vaable have potve value, fo at leat one optal olutn. Theoe 9: Moel (8) an (9) ae equvalent. v,u,w, the optal olutn to oel (9), Poof: If then ( v,u,w ) a feable olutn to oel (8). In atn, f ( v,u,w ) an optal olutn to oel (8), then F ( v,u,w) F ( v,u,w). In oel (9) a w j j w j + v j an j,..., n 0, hene: a w j j w j + v j j,..., n w o o w o + v a w j j w j + v j j,..., n w o o w o + v Theefoe, G ( v,u,w,) F ( v,u,w) F ( v,u,w) an the optal value of oel (9) le than o equal to the optal v,u,w the optal value of oel (8). On the onta, f value of oel (8), then ( v,u,w, ) a feable olutn to oel (9) wth a w j j w j + v j. j,..., n Moeove, f ( v,u,w, ) the optal olutn to oel (9), then G ( v,u,w,) G ( v,u,w, ). Sne a w j j w j + v j j,..., n w o o w o + v a w j j w j + v j,..., n (( w + u o ) ( w + v )), the followng elatn tue:. Hene, the optal value of oel (8) le than o equal to the optal value of oel (9) an thu the optal value of oel (8) an (9) ae equvalent. The effen value obtane fo oel (8) an (9) ae equal. Theefoe, t poble to ue oel (9) to alulate the elatve effen of two-tage unt ntea of oel (8). Theoe 0: Moel (9) an (0) ae equvalent. Poof: Aue the optal olutn to oel (9). If t w + v, then a feable olutn to oel (0). Moeove, f the optal olutn to oel (0), then K ( V,U,W, ) W + U. On the othe han, w + u o t w + u o w + v G ( v,u,w, ) Hene, an the optal value of oel (9) le than o equal to the optal value of oel (0). On the onta, f the optal olutn to oel (0), then t onee a a feable olutn to oel (9). In atn, f an optal olutn to oel (9), then. On the othe han, G ( V,U,W,C ) W + U o. C W + V W + U K V,U,W,C Hene, o o an the optal value of oel (0) le than o equal to that of oel (9). It theefoe onlue that the optal value of oel (9) an (0) ae equal an thee oel ae equvalent aongl. Reult : Moel (8) an (0) ae equvalent fo havng equal optal value. Reult 2: Moel (9) alulate the elatve effen of two-tage unt. Reult 3: Moel (0) alulate the elatve effen of two-tage unt. efntn 4: Cone ( v,u,w, ) a an optal olutn fo evaluatng b oel (0). a efeene fo, f w + u ( w + v )0 p p p p ISSN:

6 IV. EXAMPLE Hee, we evaluate the pefoane of 5 two-tage unt n Table I b ung oel (7) an (0). Aong to oel (7), 5 two-tage en akng unt n Table I ae aee whh ae hown n the th olun of Table II. B th etn to eaue pefoane, unt 2 an 5 ae evaluate effent. Alo, a een n the lat olun of Table II, the elatve effen of two-tage unt epee bae on oel (0). Aong to the eult, unt, 2, 3 an 5 ae effent. A opan between la oel of two-tage EA an new popoe oel ae pove n Fg. 2 an 3. Table V epot efeene two-tage MU fo non-effent two-tage unt. Fo eaple, unt 5, 2 an 22 ae efeene of unt 8, bae on oel (7). Table III ata of 24 Tawanee opane Table IV Reult of ffeent etho Fg. 2 Copan between abolute effen of oel () an elatve effen of oel (7). Table V Refeene pont of neffent two-tage unt Fg. 3 Copan between abolute effen of oel (4) an elatve effen of oel (0). ata on 24 Tawanee opane, whh wa pepae b Kao an Hwang [3], ae hown n Table III. Inuane ot an eeah ot fo the nput of the ft tage, whle et peu an elf-nuane peu ae the le ata. In atn, otent poft an nvetent noe ae onee a the fnal output of the te. Table IV how the effen oe of unt ae obtane ung oel (7) an (0) a well a the oel popoe b Kao an Hwang [3] an Chen et al. [4]. Aong to the oel popoe n th eeah, oe unt ae effent but aong to pevu oel, none of the unt ae effent. B olvng oel (7), unt 2, 5, 2 an 22 ae effent an b oel (0), unt, 2, 5, 9, 2, 5, 9, 20, 22 an 24 ae effent. ISSN:

7 V. CONCLUSION In th eeah, the oel evelope b Kao an Hwang [3] an Chen et al. [4] wee ofe to be able to alulate the elatve effen of two-tage en akng unt. Soe theoe wee ue to nate that the popoe oel ae alwa apable fo alulatng the elatve effen n two-tage EA. B etenng the ual oel of ultple fo etene n th pape, envelopent oel an be obtane fo poutn fonte pojetn of two-oponent neffent unt. Moeove, t poble to tet ankng, poutvt, benhakng bae on the popoe oel. Sla etn an be one to eaue the elatve effen n two-tage EA wth vaable etun to ale auptn. REFERENCES [] A. Chane, W. W. Coope an E. Rhoe, Meaung the effen of en akng unt, Euopean Jounal of Opeatnal Reeah, Vol. 2, pp , 978. [2] Y. L, Y. Chen, L. Lang an J. Xe, EA oel fo etene twotage netwok tutue, Oega, Vol. 40, pp. 6-68, 202. [3] C. Kao an S. N. Hwang, Effen eopotn n two-tage ata envelopent anal: An applatn to non-lfe nuane opane n Tawan, Euopean Jounal of Opeatnal Reeah, Vol. 85, pp , [4] Y. Chen, W.. Cook, N. L an J. Zhu, Atve effen eopotn n two-tage EA, Euopean Jounal of Opeatnal Reeah, Vol. 96, pp , [5] Y. Chen, an J. Zhu, Meaung nfoatn tehnolog net pat on f pefoane, Infoatn Tehnolog an Manageent Jounal, Vol. 5, pp. 9 22, [6] Y. Chen, L. Lang an J. Zhu, Equvalene n two-tage EA appoahe, Euopean Jounal of Opeatnal Reeah, Vol. 93, pp , [7] Y. Chen, W.. Cook an J. Zhu, evng the EA fonte fo twotage poee, Euopean Jounal of Opeatnal Reeah, Vol. 202, pp , 200. [8] W.. Cook, L. Lang an J. Zhu, Meaung pefoane of two-tage netwok tutue b EA: A evew an futue pepetve, Oega, Vol. 38, pp , 200. [9] V. V. Ponovk, Sutablt an eunan of non-hoogeneou weght ettn fo eaung the elatve effen n EA, Euopean Jounal of Opeatnal Reeah, Vol.54, pp , [0] M. Khall, A. S. Caanho, M. C. A. S. Potela, an M. R. Aleaee, The eaueent of elatve effen ung ata envelopent anal wth auane egn that lnk nput an output, Euopean Jounal of Opeatnal Reeah, Vol. 203, pp , 200. [] R. G. Thopon an R. M. Thall, Polheal auane egn wth lnke ontant, New etn n Coputatnal Eono, Vol. 4, pp. 2-33, 994. [2] A. Chane an W. W. Coope, Pogang wth lnea fatnal funtnal, Naval Reeah Logt, Vol. 9, pp. 8-86, 962. ISSN: