CEEP-BIT WORKING PAPER SERIES. Efficiency evaluation of multistage supply chain with data envelopment analysis models

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1 CEEP-BIT WORKING PPER SERIES Effcency evaluaton of multstage supply chan wth data envelopment analyss models Ke Wang Wokng Pape 48 Cente fo Enegy and Envonmental Polcy Reseach Bejng Insttute of Technology No.5 Zhongguancun South Steet Hadan Dstct Bejng 181 July 213 Ths pape can be cted as: Wang K. Effcency evaluaton of multstage supply chan wth data envelopment analyss models. CEEP-BIT Wokng Pape. The vews expessed heen ae those of the authos and do not necessaly eflect the vews of the Cente fo Enegy and Envonmental Polcy Reseach. 213 by Ke Wang. ll ghts eseved.

2 The Cente fo Enegy and Envonmental Polcy Reseach Bejng Insttute of Technology (CEEP-BIT) was establshed n 29. CEEP-BIT conducts eseaches on enegy economcs clmate polcy and envonmental management to povde scentfc bass fo publc and pvate decsons n stategy plannng and management. CEEP-BIT seves as the platfom fo the ntenatonal exchange n the aea of enegy and envonmental polcy. Cuently CEEP-BIT Ranks 121 top1% nsttutons n the feld of Enegy Economcs at IDES ( and Ranks 157 top1% nsttutons n the feld of Envonmental Economcs at IDES ( top/top.env.html). Y-Mng We Decto of Cente fo Enegy and Envonmental Polcy Reseach Bejng Insttute of Technology Fo moe nfomaton please contact the offce: ddess: Decto of Cente fo Enegy and Envonmental Polcy Reseach Bejng Insttute of Technology No.5 Zhongguancun South Steet Hadan Dstct Bejng 181 P.R. Chna ccess: Tel: Fax: Emal: ceepe@vp.163.com Webste: 2

3 Effcency evaluaton of multstage supply chan wth data envelopment analyss models Ke Wang a Cente fo Enegy and Envonmental Polcy Reseach BIT 5 S. Zhongguancun Steet Bejng 181 Chna b School of Management & Economcs Bejng Insttute of Technology 5 S. Zhongguancun Steet. Bejng 181 Chna bstact: In ode to evaluate the multstage supply chan effcency an appopate pefomance evaluaton system s mpotantly equed. In pactce a epesentatve multstage supply chan has thee membes composng a supple-manufactue-etale stuctue and has ntemedate measues connectng these thee supply chan membes. Exstng Data Envelopment nalyss (DE) models have dffcultes n measung these knds of supply chan effcences dectly. In ths pape we develop seveal DE based models fo chaactezng and measung these multstage supply chan effcences wth the consdeaton of the ntemedate measues. We llustate the models n a thee-stage supply chan context whch can epesent dffeent elatonshps between the supple manufactue and etale when they ae teated as n dffeent supply chan stuctues: ) the non-coopeatve ) the patal-coopeatve and ) the coopeatve supply chan stuctue. Moeove the geneal DE famewoks fo multstage supply chan models ae poposed and these models ae demonstated wth an llustatve example. Keywods: Data envelopment analyss; Effcency evaluaton; Multstage supply chan 1. Intoducton Supply chan pefomance evaluaton s a basc wok fo an oganzaton to pomote ts supply chan effcency. In ode to evaluate the multstage supply chan effcency an appopate pefomance evaluaton system s mpotantly equed. One dffculty pesented n evaluatng the pefomance of a multstage supply chan and ts membes s the exstence of multple measues that chaacteze the pefomance of chan membes. Besdes the othe dffculty s the exstence of conflcts between the supply chan membes wth egad to specfc measues. Data envelopment analyss (DE) fst poposed by Chanes Coope and Rhodes (1978) s a lnea pogammng based methodology fo evaluatng the elatve effcency of each membe of a set of oganzatonal unts whch ae called decson makng unts (DMUs). The DMUs consume vaous specfed nputs to poduce vaous specfed outputs. DE evaluates the effcency of each DMU elatve to an empcal poducton possblty fonte whch s detemned by all DMUs unde appopate assumptons wth espect to the etuns to scale and the oentaton (Banke Chanes and Coope 1984). The tadtonal standad DE model does not consde the ntenal stuctue of the DMUs.e. t teats each DMU as a black box by consdeng only the nput consumed and the output poduced by each DMU. Ths DE appoach has dffcultes n measung supply chan effcences whch nclude the supply chan s effcency and t membes effcences because t nethe povdes nsght of the ntenal opeatons of each DMU no makes locatons of effcency o neffcency to the multple sub-stages n each DMU(Rene and Hofmann 26; Xu L and Wu 29). Thus the standad DE appoach should be mpoved to emphasze the stages of the supply chan poducton pocess.e. the complcated sees supply chan poducton pocess should be dvded nto sub pocess n whch the ntemedate poducts ae consdeed. Especally some of the ntemedate poducts ae outputs fom a sub-pocess on the one hand and ae nputs to anothe sub-pocess on the othe hand. The eseaches of Sefod and Zhu (1999) Zhu (2) Fae and Gosskopf (2) and Sexton and Lews (23) ae some examples of ths appoach. In the papes the complcated poducton pocess s composed of two sub pocesses connected n sees. Sefod and Zhu (1999) consde a poducton pocess of commecal bank as two stages of poftablty and maketablty. The nputs of the fst stage denote the nputs of the bank poducton pocess and the outputs of the second stage 3

4 denote the outputs of the bank poducton pocess. In addton thee ae ntemedate poducts whch ae the outputs of the fst stage as well as the nputs of the second stage. The effcences of the fst stage second stage and the bank whole poducton pocess ae calculated though thee ndependent DE models. Wth the smla dea Zhu (2) analyzes the effcences of the 5 fotune companes. Fae and Gosskopf (2) develop a netwok DE appoach to model geneal multple stage pocesses wth ntemedate nputs and outputs. The netwok model has a geneal stuctue whch allows to be appled to a vaety of stuatons nclude the ntemedate poducts the allocaton of fxed factos and cetan dynamc systems. The dynamc netwok model s geneally conssted of two dstnct sub technologes P(t) and P(t1) one fo each evaluaton peod. The outputs fom P(t) ae dvded nto two pats: nomal outputs whch ae the conssts of the fnal outputs of the whole system and ntemedate outputs whch ae used as nputs at P(t1). The nputs to P(t1) ae also dvded nto two pats: ntemedate nputs whch ae same to the ntemedate outputs fom P(t) and nomal nputs whch dectly come fom the outsde of the system. Sexton and Lews (23) show n the pape how to use DE appoach to model two stage DMUs and apply the model to majo league baseball. They demonstate the model s advantages ove standad one stage DE as: ) the model can detect the neffcency that may be undetected by the one stage DE model; ) the model can povde geate manageal nsght nto the locatons of neffcency wthn the whole system. The two stage sub poducton pocess concept has also been appled to evaluate the pefomance of nfomaton technology (Chen and Zhu 24; Chen Lang Yang and Zhu 26a) nsuance company (Kao and Hwang 28; Chen Cook L and Zhu 29a; Chen Lang and Zhu 29b) and supply chan (Lang Yang Cook and Zhu 26; Chen Lang and Yang 26b). Chen and Zhu (24) poposed a DE famewok whch consdes a two stage pocess as effcent when each stage s effcent. In the two stage DE model of Kao and Hwang (28) the effcency of the oveall pocess s the poduct of the effcences of the two stages and the multples on the ntemedate measues ae the same fo the two stages. The assumpton of same multples also lnks the two stages because f one assumes that the multples on the ntemedate measues ae not the same then the two stage DE model s equvalent to teatng each stage ndependently of the othe stage by usng the standad DE model. Chen Cook L and Zhu (29a) pont out the lmtaton of the Kao and Hwang s model and develop an addtve two stage DE appoach n whch the oveall effcency s expessed as a weghted sum of the effcences of the ndvdual stages. Chen et al. (29b) also pove the equvalence between the Kao and Hwang (28) model and the Chen and Zhu (24) model. Lang Yang Cook and Zhu (26) develop seveal DE based appoaches fo chaactezng and evaluatng supply chan and ts membe effcences when the ntemedate measues ae consdeed n the pefomance evaluaton. The two stages of the supply chan n the models ae llustated as a selle-buye stuctue and the elatonshp between them s teated as a leade-followe and a coopeatve elatonshp. Wth the same model stuctue Chen Lang and Yang (26b) popose a DE game model appoach fo supply chan effcency evaluaton. Othe applcaton of the two stage DE concept can be found n the eseach of pefomance evaluaton of economcs of OECD countes (Peto and Zofo 27) and manufactung ndustal (Lu and Wang 29). In ou cuent eseach we consde a epesentatve multstage supply chan whch has thee membes composng a supple-manufactue-etale stuctue and has ntemedate measues connectng these supply chan membes. Expendng the dea fo two stage DMU effcency measung n Lang Yang Cook and Zhu (26) and consdeng not only ntemedate nputs and outputs but nomal nputs and outputs n Fae and Gosskopf (2) we develop seveal DE based models fo chaactezng and measung these multstage supply chan effcences. nd we llustate the models n a thee stage supply chan context whch can epesent dffeent elatonshps between the supple manufactue and etale when they ae teated as dffeent chan stuctues: ) the non-coopeatve supply chan stuctue; ) the patal-coopeatve supply chan stuctue; and ) 4

5 the coopeatve supply chan stuctue. In the non-coopeatve stuctue the supple s fst evaluated the manufactue s second evaluated usng the nfomaton elated to supple s effcency and the etale s thd evaluated usng both of the supple s and manufactue s nfomaton about effcences. We name ths stuctue backwad non-coopeatve stuctue. The evaluaton ode also can be evesed and we call t fowad non-coopeatve stuctue. In the patal-coopeatve stuctue an allance s constucted between supple and manufactue o manufactue and etale. The allance s fst evaluated and then wth the allance s nfomaton about effcency whch s defned as the athmetc mean of the allance membes effcences the supple o etale s evaluated. nd n the coopeatve stuctue all the supply chan membes ae evaluated smultaneously usng a jont effcency whch s defned as the athmetc mean of the supple s manufactue s and etale s effcences. It s mpotant to emphasze that the pmay wok of ou study s to povde an analytcal famewok wthn whch to measue the effcency of multstage supply chan. Whle t s the case that at pesent n many supply chans data may not be complete enough to pemt one to conduct such models the models do seve seveal mpotant puposes. Fstly these models povde a methodology fo pefomng what f analyses on a geneal multstage supply chan and dffeent supply chan stuctues ae assumed to smulate the opeatons. Secondly n those specfc supply chans whee elevant evaluaton data ae complete these models wll play a useful ole. Futhemoe these models can povde mpotant nsghts nto supply chan opeatons snce the manages may have stong motvatons to dentfy the supply chan pefomance and wok towad a moe compettve supply chan stuctue fo managng the supply netwoks. The stuctue of the pape s oganzed as follows. In secton 2 3 and 4 we develop models fo the non-coopeatve patal-coopeatve and coopeatve supply chan stuctues espectvely. In secton 5 the geneal famewoks fo multstage supply chan DE models ae pesented. These models ae then demonstated wth an llustatve example n Secton 6. Secton 7 concludes the pape. 2. The non-coopeatve multstage supply chan models The multstage supply chan s descbed n Fgue 1 n whch the supple manufactue and etale ae denoted by B and C espectvely. The whole supply chan s consdeed as a DMU whee X XB and XC ae the nomal nput vectos (e.g. labo opeaton cost fxed cost shppng cost mateal etc.) of B and C espectvely and Y2 YB2 and YC ae the nomal output vectos (e.g. numbe of fnal poducts shpped sales poft etc.) of B and C espectvely. Y1 s the output vecto of and also the nput vecto of B then YB1 s the output vecto of B and also the nput vecto of C. Theefoe Y1 and YB1 (e.g. numbe of vaous knds of ntemedate poducts shpped) ae consdeed as the ntemedate measues whch lnk the supply chan membes. X B X C X DMU Y 1 Y B1 B C Y C Y 2 Y B2 Fgue 1. Thee stage supply chan stuctue Suppose thee ae n homogenous supply chans (DMUs) as above. The CCR DE model (Chanes Coope and Rhodes 1978) fo the supply chan oveall effcency measue s the followng model (1). 5

6 u y 2 2 B2 B2 j j j B B v v x j v u y st B2 B2 j j j B B v v v u v u v u v j n. 2 B2 B Model (1) only consdes the nputs and outputs of the whole supply chan but msses the ntemedate measues whch lnk the supply chan membes. Model (1) also can not ndcate the ndvdual effcency of each supply chan membe. Theefoe t s consdeed as a black box model. In ode to evaluate the supply chan pefomance as well as ts membes pefomance and consde the elatonshp between the supple manufactue and etale. We fst popose the non-coopeatve models n whch the supple s fst evaluated the manufactue s second evaluated usng the nfomaton elated to supple s effcency and the etale s thd evaluated usng both of the supple s and manufactue s nfomaton of effcences. Fstly the effcency of the supple s measued n model (2) whch can be easly tanslated nto the standad DE multple model. j j v st.. 1 j j v u u v j n u u The optmal soluton of model (2) ae and and the optmal value supple s effcency. Then the effcency of the manufactue s measued n model (3). u yj u yj B B 1 1 B v x u y j j j j B B 1 1 v x j u yj j j v x j j j v v st (1) (2) s the 1 2 B1 B u u v u 2 B u v j n. In model (3) the manufactue s effcency s measued unde the condton that the supple s effcency emans at 1 2 B1 B2. The optmal soluton of model (3) ae u u v u u and v and the optmal value B s the manufactue s effcency when the supple has fst acheved ts best pefomance. Model (3) s equvalent to the followng standad DE multple Model (4) accodng to Chanes and Coope tansfomaton (Chanes and Coope 1962): (3) B 6

7 yj yj B B B yj yj yj yj yj 1 s t y y x y B B j j j j B B B j n. t last the effcency of the etale s measued n model (5). u y j B1 B1 BC v x u y j j u y st.. 1 v x u y j B1 B1 j j j j B B 1 1 v u yj j j B B 1 1 v x j u yj j j v j j v B B1 B2 B u u v u u v u v j n. Unde the condton that the supple has fst acheved ts best pefomance and the manufactue s effcency emans at model (5) gves the etale s effcency BC as the 1 2 B1 B2 B C optmal value and the optmal soluton as u u v u u v u C and v. Model (5) s equvalent to the followng standad DE multple Model (6) accodng to Chanes and Coope tansfomaton (Chanes and Coope 1962): B 1 (4) (5) 7

8 yj BC B1 B1 1 yj yj yj B s t y x y B1 B1.. j j j y y x y B B 1 1 j j j j B B yj y y x j j j 1 y y x j j j B1 B1 B2 B j n. Then the effcency of the whole supply chan can be defned as the athmetc mean of the supply chan membes effcences n (7). EBC w1 w2 B w3 BC and w1 w2 w3 1. (7) whee w1 w2 and w3 ae decson make specfed weghts of supple manufactue and etale espectvely whch can eflect the mpotance n the supply chan o the powe to nfluence the supply chan. We name the model stuctue above the backwad non-coopeatve stuctue because the evaluaton ode s the same wth the goods flow sequence n a supply chan. The evaluaton ode also can be evesed and we call t the fowad non-coopeatve stuctue. In ths stuctue the etale s fst evaluated the manufactue s second evaluated usng the nfomaton elated to etale s effcency and the supple s thd evaluated usng both of the etale s and manufactue s nfomaton about effcences. The models of fowad non-coopeatve stuctue ae smla to the models of backwad non-coopeatve stuctue and ae omtted hee. 3. The patal-coopeatve multstage supply chan models In the backwad non-coopeatve stuctue the fst evaluatng membe has contol ove the second and the thd evaluatng ones the second evaluatng membe has patal contol ove only the thd one and the thd evaluatng membe has no contol ove any othe ones. In ths secton we popose the patal-coopeatve models n whch an allance s constucted between supple and manufactue o manufactue and etale. The allance s fst evaluated and then wth the allance s nfomaton about effcency whch s defned as the athmetc mean of the allance membes effcences the supple o etale s evaluated. Suppose supple and manufactue ae alled and seek to smultaneously mze both of the effcences fst. The jont effcency of the allance s measued n model (8). w j j j j w B B B 1 1 ( B) v x j v u yj st.. 1 u j j v j j B B 1 1 v x j u yj B1 B2 B u v u u v j n (6) (8) 8

9 whee w wb 1 and w wb ae decson make specfed weghts of supple and manufactue espectvely. The weghts ndcate the mpotance o the powe of nfluence of each allance membe n the supply chan whch ae not the optmzaton vaables of the pogam. Chen Cook L and Zhu (29a) pont out that model (8) cannot be tuned nto a lnea pogam usng the usual Chanes and Coope tansfomaton (Chanes and Coope 1962). To solve the poblem they ague that the weghts should eflect the sze of each DMU sub stage and one easonable pesentaton of the sze s the poton of the total esouces devoted to each stage. Based upon the weght choce appoach to convet model (8) nto a lnea pogam w and wb can be defned n (9): B B 1 1 v v u yj w w B B 1 1 B. (9) B B 1 1 v x v x u y v x v x u y ( B) j j j j j j Then the objectve functon of model (8) can be tansfomed nto (1). B B B B The optmal soluton of model (8) ae s the allance s effcency j j j j B B 1 1 v v u yj w and. (1) 1 2 B1 B2 u u v u u B and v and the optmal value epesent the optmal weghts obtaned fom model (8) by though (9). The altenatve standad DE multple model of (8) s followng: y y y y w B B B yj j j j j ( B) s t y y x.. j j j 1 y y x y 1 B B 1 1 j j j j B B B j n. Then the effcency of the etale s measued n model (12). u y j B1 B1 ( B) C v x u y j j j B1 B1 j j u y st.. 1 v x u y j j j j B B 1 1 v v u yj j j v x j j j B B 1 1 v x j u yj 1 1 ( B) 1 2 B1 B2 B u u v u u v u v j n. In model (12) the etale s effcency s measued unde the condton that the allance s effcency emans at 1 2 B1 B2 B C ( B). The optmal soluton of model (9) ae u u v u u v u and C* v and the optmal value ( B) C (11) (12) s the etale s effcency when the allance has fst acheved ts best pefomance. The altenatve standad DE multple model of (12) s followng: 9

10 yj ( B) C B1 B1 1 yj j j j B B yj s t y x y B1 B1.. j j j y y y y j j j j ( B) y y x y y x y B1 B1 B2 B2 B B 1 1 j j j j 1 2 B1 B1 B2 B j n. Each allance membe s effcency also can be measued ndvdually by usng the optmal soluton of model (8) as n (14) u y j B1 B1 B2 B2 u y j u yj u y j /( B) B /( B). (14) B B 1 1 v x v x u y j j j Consdeng the defnton n secton 3 the effcency of the whole supply chan can be defned as the athmetc mean of the allance s and etale s effcences as follow: E w w and w1 w2 1. (15) ( B) C 1 ( B) 2 ( B) C whee w1 and w2 ae decson make specfed weghts. The allance also can be bult up between the manufactue and etale and the effcency of the allance s fst measued. Then the effcency of the supple s measued wth the allance s nfomaton of effcency. The models unde ths supply chan stuctue ae smla to model (8) and (12) and ae omtted hee. 4. The coopeatve multstage supply chan models In ths secton we wll consde the coopeatve stuctue of supply chan n whch all of the membes ae evaluated smultaneously. The coopeatve model seeks to mze the jont effcency defned as the athmetc mean of supple s manufactue s and etale s effcences n model (12). j j v u yj u yj 1 B B 1 1 v u yj u yj B1 B1 v u yj 1 u y u y u y w w w j j j j j B 1 1 C B B B1 B1 ( BC ) v v u yj v u yj st.. 1 w w w 1 B C u u v u u v u v j n. 1 2 B1 B2 B whee w wb and wc ae also decson make specfed weghts fo supple manufactue and etale espectvely and eflect the mpotance o the powe of nfluence of each membe n the supply chan. Each weght also can be defned accodng to the poton of the total esouces devoted to each membe as n (17). (13) (16) 1

11 w w w v x j B B 1 1 B1 B1 v v v u yj u yj v x u y B B 1 1 j j B B B 1 1 B1 B1 v v v u yj u yj v x u y B1 B1 j j C B v x j v x. B 1 1 B1 B1 j v u yj u yj The optmal soluton of model (12) ae (17) 1 2 B1 B2 B C u u v u u v u C* and v and the s the effcency of the whole supply chan. The altenatve standad DE optmal value multple model of (16) s followng: u y BC B B 1 1 u yj u yj v u yj j j j j j ( BC ) s t u y u y v x.. j j j B1 B1 u yj v u yj v x v x u y v x u y 1 B B 1 1 B1 B1 j j j j j 1 2 B1 B2 B u u v u u v u v j n. By usng the optmal soluton of model (16) each supply chan membe s effcency could be measued ndvdually n (19) u yj u y j /( BC ) v x B1 B1 B2 B2 j j B /( BC ) B B 1 1 v u yj C C j C /( BC ) C C B1 B1 v u yj u y u y 5. The geneal famewoks fo multstage supply chan models Now we consde the extenson to the geneal famewoks fo multstage supply chan wth P stages unde the non-coopeatve and the coopeatve stuctues. Consde a P stage supply chan llustated n Fgue 2. The nput of stage 1 s denoted by M. Thee ae two foms of outputs fom stage p (p1 P) whch s M p(1) and M p(2). The fome epesents the output that leaves ths stage and wll not become the nput to the p1 stage and the late epesents the output that becomes the nput to the next stage. Futhemoe thee ae two foms of nputs to stage p. M p-1(2) epesent the ntemedate measue fom stage p-1 and M p-1(3) epesent a new nput ente the supply chan at stage p. u y j. (18) (19) 11

12 p 1(3) M P 1(3) M M stage 1 stage p stage P M 1(2) p 1(2) M M P 1(2) M p(2) M P(1) 1(1) M p(1) M Fgue 2. Multstage supply chan stuctue When p2 P p(1) m j (1 R) s the th component of output fom stage p of DMUj that leaves the supply chan at ths stage and wll not become the nput to stage p1; p(2) m kj the kth component of output fom stage p of DMUj that becomes the nput to stage p1; and (k1 K) s p 1(3) m j (1 I) s the th component of nput to stage p of DMUj that entes the supply chan at ths stage. t the fst stage 1 all nputs ae denoted as and at the last stage P all outputs ae seen as. p u w p k and p 1 v m j ae the weghts assocated wth outputs m m and p(1) p(2) j kj p 1(3) m j espectvely. P(1) m j Theefoe when p2 P the geneal model fo the effcency measue of stage p n a multstage supply chan wth P stages unde the non-coopeatve stuctue could be expessed as n model (2): p p(1) p p(2) u mj wk m k kj p 1 p 1(2) p 1 p 1(3) p w m v m whee k k k k kj j u m w m st.. 1 w m v m k p p(1) p p(2) j k k kj p 1 p 1(2) p 1 p 1(3) k kj j u m u m q q(1) q q(2) j k k j q 1 q 1(2) q 1 q 1(3) k kj j w m v m q q(1) q q(2) j k k j 1 q 1 q 1(2) q 1 q 1(3) k kj j p p p u w v 1... R k 1... K 1... I k u m u m w m v m j 1... n p 1... P q 1... p 1. p s the effcency of stage p whch s measued unde the condton that the effcences of stage 1 to stage q(p-1) eman at q. Smla to Cook Zhu B and Yang (21) the geneal model fo the oveall effcency measue of multstage supply chan wth P stages unde the coopeatve stuctue could be expessed as n model (21): q (2) 12

13 p p(1) p p(2) p ( u mj ) wk m k kj p 1 p 1(2) p 1 p 1(3) v mj ( ) p w k k mkj v mj 1 1(1) 1 1(2) u mj w k kmkj vm j p p(1) p p(2) u mj w k k mkj 1 p 1 p 1(2) p 1 p 1(3) wk mkj v yj st.. 1 k p p p u wk v 1... R k 1... K 1... I p 1... P j 1... n. ae the nput and assocated weght fo stage 1 and s the oveall whee m and j effcency of the multstage supply chan. v oveall oveall (21) 6. n llustatve example We now apply the poposed multstage supply chan DE models to the numecal example used n Tone and Tsutsu (29). Table 1 povdes the data fo the effcency measuement of ten vetcally ntegated electc powe companes consst of thee dvsons of powe geneaton tansmsson and dstbuton. The nputs outputs and ntemedate measues ae as follows. The nput of the powe geneaton stage s x (labo) and the output of ths stage s y (electc powe geneated) whch s also the ntemedate nput of the powe tansmsson stage. The tansmsson stage have anothe nput x B (labo) one output y B2 (electc powe sold to lage customes) and one ntemedate output y B1 (electc powe sent) whch s also the ntemedate nput of the powe dstbuton stage. nothe nput of the powe dstbuton stage s x c (labo) and the output of ths stage s y C (electc powe sold to small customes). Table 2 epots the effcency scoes obtaned fom the black box model backwad non-coopeatve model patal-coopeatve model and coopeatve model. We fst solve the black box model (1) usng nput x x B and x c and output y B2 and y C whee the ntemedate nput and output ae neglected. The black box oveall effcency of the supply chan s epoted n Table 2 and half of the DMUs ae measued as oveall effcent. We then solve non-coopeatve model (2) (4) and (6) patal-coopeatve model (11) and (13) and coopeatve model (18). E-BC E-(B)C and E-(BC) ae the oveall effcency scoes of the supply chan unde these thee dffeent models espectvely. We pont out that n the backwad non-coopeatve model when calculatng the oveall effcency the weghts assgned to the powe geneaton tansmsson and dstbuton stages ae.4.2 and.4 espectvely and these weghts selecton ae just fo llustatve pupose (same wth Tone and Tsutsu s (29) example). Futhemoe n the patal-coopeatve model the weghts assgned to the allance of powe geneaton and tansmsson stages and the powe dstbuton stage ae.6 and.4 espectvely. No supply chan s measued effcent unde the non-coopeatve and patal-coopeatve model and only one supply chan (DMU 3) acheves effcent unde coopeaton model. Table 2 also epots that unde the backwad non-coopeatve stuctue no DMU s ated as oveall effcent. Howeve the powe geneaton and tansmsson stages fo DMU 3 the powe dstbuton stage fo DMU 4 and the powe tansmsson stage fo DMU 9 ae measued as effcent. Smla evaluaton esults also can be found unde the patal-coopeatve and coopeatve models. The effcency scoes ae compaed n Fgue 3. It can be seen that the scoes of the black box model tend to be hghe than the scoes of coopeaton model and the scoes of non-coopeatve model ae the lowest. Ths fgue clealy ndcates that n ths example the multstage DE models have hghe dscmnate powe than that of the black box DE model. Fgue 3 also shows that the 13

14 tends of the effcency scoes of thee multstage DE models ae oughly smla and on aveage the effcency unde the coopeatve model appeas the hghest and the effcency unde the non-coopeatve model lowest. Table 1. Data of nputs and outputs fo the multstage supply chan DMU x y 1 x B y B2 y B x C y C Table 2. Data of effcences fo the multstage supply chan and ts membes DMU Black box oveall effcency.6999 (8).587 (9) 1. (3) 1. (3).5748 (1).7621 (7) 1. (3) 1. (3) 1. (3).953 (6) E-BC Backwad (7) (1) (1) (4) (9) (6) (2) (5) (3) (8) noncoopeatve stuctue Patalcoopeatve stuctue Coopeatve stuctue B BC E-(B)C ( B) /( B).4839 (8).3619 (1).826 (2).8222 (3).3944 (9).5333 (7).868 (4).584 (6).9938 (1).6197 (5) B /( B) ( B) C E-(BC) ( BC ) /( BC ).5793 (7).3863 (1) 1. (1).9965 (2).4782 (9).5977 (6).7617 (4).746 (5).9934 (3).518 (8) B /( BC ) C /( BC ) Note: paentheses unde the each effcency scoe denote the ank of each DMU 14

15 Fgue 3. Effcency of supply chan unde dffeent models 7. Concluson In ths pape we developed seveal multstage supply chan DE models fo measung the effcences of the whole supply chan and t membes unde dffeent supply stuctues. We consde a epesentatve multstage supply chan whch has thee membes: supple manufactue and etale. The non-coopeatve stuctue model s fst poposed to evaluate the effcency of each supply chan membe accodng to a specfed sequence (backwad and fowad pocess). The patal-coopeatve stuctue model s second poposed to evaluate the effcency of the allance whch s constucted between two of the supply chan membes and the othe membe whch s not n the allance. The coopeatve stuctue model s thd poposed to evaluate the effcences of all of the supply chan membes smultaneously. The ntemedate measues between each supply chan membe ae consdeed whch lnk the membes and the whole supply chan. Two dffeent knds of nputs/outputs (nomal and ntemedate) ae also consdeed n the models n ode to make the multstage supply chan models becomng moe geneal. lthough some of these models ae nonlnea pogammng poblems they can be solved as lnea pogammng poblems by choosng decson make specfed weghts. Moeove we extend these models to the geneal famewoks fo multstage supply chan wth P stages unde the non-coopeatve and the coopeatve stuctues. Because tadtonal DE models cannot be dectly appled to evaluatng multstage supply chan the models poposed n ths pape could be seen as effcent tools fo the pefomance evaluaton of ths knd of supply chan. lthough n ths study we use the coopeatve and non-coopeatve concepts fo modelng the multstage supply chna we ae not tyng to examne whethe the supple manufactue and etale of a specfc supply chan ae behavng n a coopeatve non-coopeatve o patal-coopeatve manne. lso we have to pont out that the cuent study s not an empcal eseach and we only popose an analytcal famewok fo measung the effcency of multstage supply chan. Theefoe an example of effcency measuement of vetcally ntegated electc powe companes s used to demonstate ou wok of cuent pape. The example llustates that the multstage models could povde moe detaled nfomaton about the effcency o

16 neffcency decomposton to each supply membe. Theefoe these models may become useful tools fo the manages n montong and plannng the supply chan opeatons so as to sgnfcantly ad them makng multstage supply chans moe effcent. Whle the majo wok of the cuent study les n the methodologcal developments t has potental fo applcaton n vaous supply chan opeatons when enough elated evaluaton data do exst o can be acqued. Fnally we pont out that the models poposed n ths pape ae based on constant etuns to scale (CRS) DE. The vaable etuns to scale (VRS) DE fo multstage supply chan pefomance evaluaton s a subject fo futue eseach. Futhemoe the pefomance evaluaton of specfc multstage supply chans whch have smla stuctues poposed n ou models s anothe subject fo futhe eseach. Refeences Banke R.D. Chanes. Coope W.W. (1984). Some models fo the estmaton of techncal and scale neffcences n data envelopment analyss. Management Scence 3(9) Cook W.D. Zhu J. B G. Yang F. (21). Netwok DE: ddtve effcency decomposton. Euopean Jounal of Opeatonal Reseach 27(2) Chanes. Coope W.W. (1962). Pogammng wth lnea factonal functonal. Naval Reseach Logstcs Quately 9(3-4) Chanes. Coope W.W. Rhodes E. (1978). Measung the effcency of decson makng unts. Euopean Jounal of Opeatonal Reseach 2(6) Chen Y. Cook W.D. L N. Zhu J. (29a). ddtve effcency decomposton n two-stage DE. Euopean Jounal of Opeatonal Reseach 196(3) Chen Y. Lang L. Yang F. Zhu J. (26a). Evaluaton of nfomaton technology nvestment: data envelopment analyss appoach. Computes and Opeatons Reseach 33(5) Chen Y. Lang L. Yang F. (26b). DE game model appoach to supply chan effcency. nnals of Opeatons Reseach 145(1) Chen Y. Lang L. Zhu J. (29b). Equvalence n two-stage DE appoaches. Euopean Jounal of Opeatonal Reseach 193(2) Chen Y. Zhu J. (24). Measung nfomaton technology s ndect mpact on fm pefomance. Infomaton Technology and Management Jounal 5(1-2) Fae R. Gosskopf S. (2). Netwok DE. Soco-Economc Plannng Scences 34(1) Kao C. Hwang S. N. (28). Effcency decomposton n two-stage data envelopment analyss: an applcaton to non-lfe nsuance companes n Tawan. Euopean Jounal of Opeatonal Reseach 185(1) Lang L. Cook W.D. Zhu J. (28). DE Models fo Two-Stage Pocesses: Game ppoach and Effcency Decomposton. Naval Reseach Logstcs 55(7) Lang L. Yang F. Cook W.D. Zhu J. (26). DE models fo supply chan effcency evaluaton. nnals of Opeatons Reseach 145(1) Lu S.T. Wang R.T. (29). Effcency measues of PCB manufactung fms usng elatonal two-stage data envelopment analyss. Expet Systems wth pplcatons 36(3)

17 Peto.M. Zofo J.L. (27). Netwok DE effcency n nput output models: wth an applcaton to OECD countes. Euopean Jounal of Opeatonal Reseach 178(1) Rene G. and Hofmann P. (26). Effcency analyss of supply chan pocesses. Intenatonal Jounal of Poducton Reseach 44(23) Sefod L.M. Zhu J. (1999). Poftablty and maketablty of the top 55 US commecal banks. Management Scence 45(9) Sexton T. R. Lews H. F. (23). Two-stage DE: n applcaton to majo league baseball. Jounal of Poductvty nalyss 19(2-3) Tone K. Tsutsu M. (28). Netwok DE: slacks-based measue appoach. Euopean Jounal of Opeatonal Reseach 197(1) Xu J. L B. Wu D. (29). Rough data envelopment analyss and ts applcaton to supply chan pefomance evaluaton. Intenatonal Jounal of Poducton Economcs 122(2) Zhu J. (2). Mult-facto pefomance measue model wth an applcaton to Fotune 5 companes. Euopean Jounal of Opeatonal Reseach 123(1)