The Efficiency Measurement of Parallel Production Systems: A Non-radial Data Envelopment Analysis Model

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1 Jural f muter Sciece 7 (5): , 20 ISSN Sciece Publicatis The Efficiecy Measuremet f Parallel Prducti Systems: A N-radial Data Evelmet Aalysis Mdel,2 Ali Ashrafi,,3 Azmi Bi Jaafar,,4 Mhd Rizam Abu Baar ad,4 Lai S Lee Istitute fr Mathematical Research, Uiversity Putra Malaysia, UPM SERDANG, Selagr, Malaysia 2 Deartmet f Mathematics, Faculty f Mathematics, Statistics ad muter Sciece, Uiversity f Sema, Sema, Ira 3 Deartmet f Ifrmati Systems, Faculty f muter Sciece ad Ifrmati Techlgy, 4 Deartmet f Mathematics, Faculty f Sciece, Uiversity Putra Malaysia, UPM SERDANG, Selagr, Malaysia Abstract: Prblem statemet: Data Evelmet Aalysis (DEA) is a -arametric techique fr measurig the relative efficiecy f a set f rducti systems r Decisi Maig Uits (DMU) that have multile iuts ad ututs. Smetimes, DMUs have a arallel structure, i which systems cmsed f arallel uits wr idividually; the sum f their w iuts/ututs is the iut/utut f the system. Fr this tye f system, cvetial DEA mdels treat each DMU as a blac bx ad igre the erfrmace f its uits. Arach: This study itrduces a DEA mdel i Slacs-Based Measure (SBM) frmulati which csiders the arallel relatishi f the uits withi the system i measurig the efficiecy f the system. Uder this framewr, the verall efficiecy f the system is exressed as a weighted sum f the efficiecies f its uits. Results: As a SBM mdel, the rsed mdel is -radial ad is suitable fr measurig the efficiecy whe iuts ad ututs may chage -rrtially. A theretical result shws that if ay uit f a arallel system is iefficiet the the system is iefficiet. clusi: This study itrduces a -radial DEA mdel, taes it accut the erati f idividual cmets withi the arallel rducti system, t measure the verall efficiecy as well as the efficiecies f sub-rcesses. This hels the decisi maers recgize iefficiet uits ad mae later imrvemets. Key wrds: Data evelmet aalysis, decisi maig uit, rducti ssibility set, arallel rducti system, slacs-based measure INTRODUTION Data Evelmet Aalysis (DEA) is a liear rgrammig methdlgy i Oeratis Research ad Ecmics that is extesively alied by varius research cmmuities (Sh ad M, 2004; Sel et al., 2007; Rayei ad Salghi, 200 Zreia ad Ela, 20). The dmai f iquiry f the DEA is a set f rducti systems r decisi maig uits (DMU), which use multile iuts t rduce multile ututs. The aim f the DEA is t measure the relative efficiecy f each DMU withi a data set. The results secify hw efficiet each DMU has erfrmed as cmared t ther DMUs i cvertig iuts t ututs. A issue which is f greater ccer t the iefficiet DMUs is what factrs cause the iefficiecy. Several studies have assiged t breaig dw the verall efficiecy it cmets s that the surces f iefficiecy ca be idetified. Oe tye f decmsiti emhasizes the sub-rcesses f the rducti rcess. I recet years, a great umber f DEA studies have fcused tw-stage rducti systems, where all ututs frm the first stage are itermediate rducts that mae u the iuts t the secd stage. Fr examle see (Sext ad Lewis, 2003; he ad Zhu, 2004; Liag et al., 2006; Ka ad Hwag, 2008; he et al., 2009) amg thers. Recetly, Te ad Tsutsui (2009) ad et al., (200) have rsed DEA mdels fr measurig the efficiecy f etwr systems cected i series with liig activities. rresdig Authr: Ali Ashrafi, Istitute fr Mathematical Research, Uiversity Putra Malaysia, UPM SERDANG, Selagr, Malaysia Tel: Fax:

2 I sme situatis, DMUs have a arallel structure that is cmsed f a set f uits that wr idividually; the sum f their w iuts/ututs is the iut/utut f the DMUs. A tyical examle f these rducti systems is a uiversity with faculties. The verall efficiecy f the uiversity ca be calculated by the ttal iuts used ad ttal ututs rduced by all faculties. Each secific faculty ca have a efficiecy measured by cmarig it with the equivalet faculties f ther uiversities. The study f Färe ad Primt (984), which discusses the efficiecy f firms with multile lats, is rbably the first study f such DMUs. Ka (998) alied the Färe ad Primt s methdlgy fr measurig the efficiecy f frest districts with multile wrig circles i Taiwa. astelli et al. (2004) discussed a hierarchical structure i which if there is ly e layer, it becmes a arallel system. The wrs f Färe et al. (997), Tsai ad Mlier (2002) ad Yu (2008) exted f the ideedet arallel system where certai resurces are shared by sme uits. vetial DEA mdels view this tye f rducti system as a blac bx ad igre the eratis f its uits. Recetly, Ka (2009) has mdified a stadard DEA mdel ad itrduced a radial DEA mdel that evaluates the verall efficiecy f the system as well as the efficiecies f its uits. His methd decmses the iefficiecy slac f a DMU it the iefficiecy slacs f its sub-dmus. As a alicati f Ka s arallel mdel, Rayei ad Salghi (200) examie the erfrmace f the uiversities i Ira via a arallel rducti rcess. This study resets a alterative methd fr estimatig the efficiecy f a arallel rducti system ad the efficiecy f its uits. Sice DEA mdels imlicitly use Prducti Pssibility Set (PPS) t evaluate the efficiecy f DMUs, we first defie the PPS f the arallel rducti systems. The, based this PPS, we itrduce a -radial DEA mdel i Slacs-Based Measure (SBM) frmulati fr aggregatig the uits i a arallel rducti system. Uder this framewr, the verall efficiecy f the system is exressed as a weighted sum f the efficiecies f its uits. With decmsiti f the verall efficiecy, the uits which cause the iefficiet erati f the system ca be idetified fr future imrvemets A examle frm the frest rducti idustry i Taiwa is alied t cmare the ew arach with Ka s arallel mdel. MATERIALS AND METHODS that all iuts ad ututs are sitive. We dete the DMU by (x, y ), where x = (x, x 2,, x m ) T ad y = (y, y 2,, y m ) T are iut ad utut vectrs, resectively. The Prducti Pssibility Set (PPS) T is defied as a set f all iuts ad ututs f a rducti techlgy i which ututs ca be rduced frm iuts. Uder the assumti f stat Returs t Scale (RS) the PPS ca be rereseted as fllws: T = (x,y) x λx, y λy, λ 0, =,..., = = where, λ = (λ,...,λ) R is the itesity vectr. The PPS uder Variable Returs t Scale (VRS) assumti ca be defied by addig the cvexity cstrait λ = = it T c. Defiiti : (Dmiace). We say that DMU (x, y ) dmiates DMU q (x q, y q ) if ad ly if x x q ad y y q with strict iequality hldig fr at least e cmet i the iut r the utut vectr. Thus, a DMU f the PPS is t dmiated if ad ly if there is ther DMU (rigial r virtual) i the PPS which satisfies Defiiti. Defiiti 2: (Efficiecy). DMU =(x,y ) is efficiet if ad ly if there is (x,y) f PPS such that (x,y) dmiates (x,y ). Radial ad radial DEA mdel: DEA rvides fr tw tyes f measure: radial ad -radial. Radial mdels assume rrtial chage f iuts r ututs ad usually disregard the existece f slacs i measurig efficiecy scres. Radial measures are rereseted by R (hares et al., 978) ad B (Baer et al., 984) N-radial mdels, the ther had, regard the slacs f each iut r utut ad the variatis f iuts ad ututs as t rrtial; i ther wrds i -radial mdels the iuts/ututs are allwed t decrease/icrease at differet rates. N-radial mdels iclude Russell measure (Färe ad Lvell, 978) ad Slacs-Based Measure (SBM) (Te, 200). Fr evaluatig the efficiecy f DMU ( {,,}) the iut-rieted R mdel, whse urse is t miimize iut custm while eeig the Prducti ssibility set: Suse we have DMUs, where each DMU (=,,) uses m iuts x i (i =,, m) t rduce s ututs y r (r =,, s). It is assumed level f curret ututs, is set as fllws: 750

3 θ * = mi θ ε( s s.t. θx y λ,s m i i= = λ x = λ =,s = s y s r r= s s, ), 0, θ free, () where, ε is -Archimedea small value ad the timal sluti f θ*is efficiecy scre. Als egative vectrs s = (s m,...,s m) R ad s s = (s,...,s s) R idicate iut excess ad utut shrtfall slacs, resectively. Liewise the ututrieted R mdel ca be defied. Suse a timal sluti fr mdel () t be * * * * ( θ, λ,s,s ). Defiiti 3: (R-efficiecy). DMU is Refficiet if ad ly if θ* = ad s * = s *= 0. The mdel reseted i () is called the R evelmet mdel. The dual f mdel () (withut ε, i.e., ε = 0), r the R multilier mdel, is give by: θ * = max s.t. uy vx =, uy - vx 0, =,...,, u, v 0, m (2) where, v = (v,..., v m ) R ad u = (u,...,us ) R are dual variable vectrs crresdig t the cstraits f mdel (). Fig. : The arallel rducti system s The SBM mdel, as a -rieted ad -radial DEA mdel, fr evaluatig the efficiecy f DMU is defied as fllws: ρ m si * m i= x i = mi ρ = s sr s r= y r = = y = λ y s, = s.t. x λ x s, λ, s, s 0 system DMU, resectively. I ther wrds, we have: 75 (3) The timal sluti f ρ* is the SBM efficiecy scre. It ca be bviusly idetified that 0<ρ* ad surts the rerties f uit ivariace ad mte. Suse a timal sluti fr mdel (3) t be * * * * ( ρ, λ,s,s ). Defiiti 4: (SBM-efficiecy). DMU is SBMefficiet if ad ly if ρ* =. This defiiti is equivalet t s * * = s = 0. It meas that there are iut excesses ad utut shrtfalls i ay timal sluti. The iut (utut)-rieted SBM mdel ca be defied by igrig the demiatr (umeratr) f the bective fucti. Here we emhasize that, as demstrated by Te (200), the efficiecy scre measured by the SBM mdel is t greater tha the efficiecy scre measured by the R mdel. Mrever, a DMU is SBM-efficiet if ad ly if it is R-efficiet. Parallel rducti system: sider a arallel rducti rcess as shw i Fig.. Suse we have DMUs, f which each DMU (=,,) is cmsed f t uits (sub-dmus) cected i arallel. Each sub-dmu uses the same iuts t rduce the same ututs, idividually. Sub-DMU (=,,t ) has m iuts x i (i=,,m) ad s ututs r t x = i ad the sum f all r y (r=,...,s). The sum f all x i ver, amely, t y = r y ver, amely,, are the ith iut ad the rth utut f the

4 t = t y = x = x, = y (4) Ka s arallel mdel: Ka (2009) has develed a DEA mdel based the iut-rieted R multilier mdel such that by miimizig the iefficiecy slac istead f maximizig the efficiecy, we are able t decmse the iefficiecy slac f a DMU it the iefficiecy slacs f its sub-dmus. Ka s mdel fr measurig the iefficiecy f DMU is give by: mi t s = s.t. vx =, uy vx s = 0, uy vx s = 0, =,...,t, uy vx 0, =,...,,, =,...,t, uy -vx 0, =,...,,, u,v 0 (5) Suse, DMU ( {,.,}) t be the DMU uder evaluati. I a effrt t measure the verall efficiecy f DMU, first by usig the iut-rieted SBM mdel, we calculate the efficiecy scre f each sub-dmu (=,,t ) as fllws: s E = mi - m i m i= xi t = = = t λy s = y, = = s.t. λ x s x, λ, s, s 0, =,...,t, =,..., (7) After evaluatig the efficiecy scres f all Sub- DMUs, we defie the verall efficiecy scre f DMU as a weighted cmbiati f the efficiecy scres f its sub-dmus. This is shw i the fllwig equati: t 2 2 t t = E = w E w E... w E = w E (8) where, s ad s ( =,...,t ) are the iefficiecy slacs f DMU ad its sub-dmus, resectively. O timality, the efficiecy scres fr DMU ad sub-dmu ( =,,t ) ca be calculated as fllws: where, the weight w f each sub-dmu is m xi (9) m i= xi w =, =,...,t t * * = = = E s s s E =, =,...,t, vx * * (6) Hece w is the arithmetic mea f the rti f resurces devted t each sub-dmu by DMU. Frm (4), it ca be verified that w =. t = where, (*) shws the timal value frm mdel (5). The arallel SBM mdel: The PPS T arallel uder the RS assumti fr t sub-dmus f arallel rducti systems, is defied by: Defiiti 5: DMU is said t be efficiet i sub DMU, if E =. Defiiti 6: DMU is said t be efficiet if its verall efficiecy scre is equal t e, i.e., E =. Decmsig the verall efficiecy f a system it the weighted cmbiati f its uit s efficiecies t t arallel T = (x,y) x λx, y λy, λ 0 hels us t idetify the uits that cause iefficiecy. By = = = = usig the mdel (7), we are able t recgize the iefficiet sub-dmus ad mae later imrvemets. Als, usig Eq. 8 we ca evaluate the verall efficiecy Nte that if all λ, =,...,t assciated with the f the DMU i a way that taes it accut the arallel eratis f all its sub-dmus. sub-dmus withi the DMU are the same, the T Nte that, similarly, we ca itrduce a ututrieted SBM mdel fr arallel rducti systems cverts t the cvetial PP, amely T. 752

5 such that the weights assciated t su-dmu f DMU ca be defied as: * * x = x s,y = y s (5) s yr (0) s r= yr w =, =,...,t RESULTS * * Sice (s,s ) (0,0), (x, y ) is dmiated by (x,y). Hece, accrdig t Defiiti 2, DMU is RSiefficiet. As a ctrasiti t Therem, we have. The fllwig therem exlais the relatishi betwee the efficiecy f a arallel rducti system ad its rducti uits. Therem : If ay f sub-dmu = (x, y ) ( =,...,t ) f DMU is RS-iefficiet, the DMU = (x, y ) is RS-iefficiet. Prf: Suse ay f sub-dmu = (x, y ) (=,,t ) t be RS-iefficiet. We will shw that there is arallel (x,y) T such that (x, y ) is dmiated by ( x, y). Withut lss f geerality, we assume that sub-dmu = (x, y ) is RS-iefficiet. The, the fllwig system has a sluti * * * * * {λ, =...,t, ;s ;s } with (s,s ) > (0,0) : t * * = = = t * * = = = x λ x s, y λ y s We set: t t * * * * = = λ = = λ = = = = x x s x,y y s y () (2) Frm (4) ad (2) we have t t * * = = = 2 = 2 (3) x x x s,y y y s Nw we defie: t t = 2 = 2. (4) x = x x,y = y y Sice we have: arallel (x,y ) P, thus (x,y) T arallel rllary : If DMU (x, y ) is RS-efficiet, the each sub-dmu = (x, y ) (=,,t ) f DMU is RS-efficiet. Emirical examle: Nw, we aly the rsed mdel t the atial frests f Taiwa as studied by Ka (2009). I Taiwa, the frest lads are divided it eight regis, each f which is divided it fur r five sub-regis called wrig circles (Ws). These Ws are the basic cmet i the maagemet f the frest. The frest rducti rcess is a characteristic arallel rducti rcess, i that each regi has several subrdiated Ws eratig idividually. There are fur iuts: Lad (x ): area i thusad hectares Labr (x 2 ): umber f emlyees i erss Exeditures (x 3 ): mey set each year i te thusad ew Taiwa dllars Iitial stcs (x 4 ): vlume f frest stc befre the erid f evaluati i 0000 m 3 The ututs are Timber rducti (y ): timber rduced each year i cubic meters Sil cservati (y 2 ): vlume f frest stc i 0000 m 3, as higher stc level leads t less sil ersi; ad Recreati (y 3 ): visitrs serviced by frests every year i thusads f visits The data are shw i Table. Fr each iut (utut) the quatity f a regi is the sum f its sub-regis. The results f this measuremet f efficiecy are rerted i Table 2, where the secd clum shws the weights calculated frm (9) fr each sub-dmu, the third clum is the efficiecy scre calculated usig mdel (7) ad Eq. 8. The efficiecy scres f the eight DMUs calculated by the iut-rieted f mdel (3) withut taig it accut the eratis f sub-dmus are. Hece shw i the last clum uder the headig cvetial SBM mdel. 753

6 Table : Taiwa frest data Iuts Oututs Wrig circles Lad Labr Exeditures Iitial stcs Timber Sil cs. Recreati Ltug Regi Taiei Tai-ig-sha ha-chi Na-au H-ig Hsichu Regi Guay-sha Ta-chi hu-tug Ta-hu Tugshi Regi Sha-chi A-ma-sha Li-yag Li-sha Natu Regi Tai-chug Ta-ta Pu-li Shui-li hu-sha hiayi Regi A-li-sha Fa-chi-hu Ta-u Tai-a Pigtug Regi hih-sha ha-chu Liu-guay Heg-chu Taitug Regi Kua-sha hi-be Ta-wu ha- g Hualie Regi Shi-cha Na-hua Wa-yg Yu-li DISUSSION Frm Table 2, it ca be see that six DMUs are efficiet uder the cvetial SBM mdel while As ited ut by may authrs icludig Ka ad accrdig t Therem, uder the arallel SBM mdel, Hwag (2008), Ka (2009), he et al. (2009), Te sice e f DMUs erfrms efficietly i all its w ad Tsutsui (2009) ad et al. (200), the sub-dmus, e f them erfrms efficietly as a whle. cvetial DEA mdels aly a sigle rcess t Thus, by usig the results f this efficiecy measuremet evaluate the trasfrmig efficiecy f multile iuts we are able t idetify the iefficiet sub-dmus ad mae ad ututs such that they fail t measure the effrts f future imrvemet. The raigs f the verall differet rcesses ad sub-rcesses withi the efficiecy scres f the eight regis taig ur rducti systems. Thus, we cat evaluate the arach ad taig Ka s arach are shw i imact f sub-rcess iefficiecies the verall Table 3. maris f the tw sets f scres shws efficiecy f the system as a whle. I these cases, it is the t have almst idetical raig. The Searma ssible that the cvetial DEA mdels evaluate a Ra rrelati cefficiet fr the raigs i system as efficiet eve if e f its cmet Table 3 is 0.976, shwig that the crrelati rcesses is efficiet betwee ur results ad Ka s results is very high. 754

7 Table 2: Efficiecy scres Wrig Weight Parallel vetial circles (w ) SBM mdel SBM mdel Ltug Regi Taiei Tai-ig-sha ha-chi Na-au H-ig Hsichu Regi Guay-sha Ta-chi hu-tug Ta-hu Tugshi Regi Sha-chi A-ma-sha Li-yag Li-sha Natu Regi Tai-chug Ta-ta Pu-li Shui-li hu-sha hiayi Regi A-li-sha Fa-chi-hu Ta-u Tai-a Pigtug Regi hih-sha ha-chu Liu-guay Heg-chu Taitug Regi Kua-sha hi-be Ta-wu ha- g Hualie Regi Shi-cha Na-hua Wa-yg Yu-li ONLUSION I a earlier study, a radial DEA mdel was itrduced by Ka (2009) fr measurig the efficiecy f a system cmsed f arallel uits eratig ideedetly ad where the sum f iuts/ututs fr all uits is equal t the iut/utut f the system. I this study, we have itrduced a -radial mdel based a Slacs-Based Measure (SBM) framewr that evaluates the verall efficiecy f the system by csiderig the eratis f its uits. Uder this framewr, the verall efficiecy f the system is exressed as a weighted sum f the efficiecies f its uits. With decmsiti f the verall efficiecy, the uits which cause the iefficiet erati f the system ca be idetified fr future imrvemets The rsed mdel is based the assumti f stat Returs t Scale (RS). By addig the cvexity cstrait it the PPS which is built by t sub-dmus, the discussi ca be exaded t use the Variable Returs t Scale (VRS) assumti. It is tewrthy that real systems are geerally mre cmlex tha the arallel system discussed i this study. Te ad Tsutsui (2009) develed a etwr DEA mdel based a weighted SBM arach that ca be alied i series systems. Sice the series ad arallel structure are tw basic structures f a etwr system, we ca trasfrm a etwr system it a cmbiati f series ad arallel structures t evaluate the verall efficiecy ad the efficiecies f sub-rcesses. AKNOWLEDGMENT The researchers tha Prfessr Azizllah Memariai fr his cmmets ad suggestis. This wr is surted by Graduate Research Assistace f Uiversity f Putra Malaysia (Grad ). Table 3: Raig f efficiecy scres Ka s results Regis Our raig Raig Overall efficiecy Ltug Regi Hsichu Regi Tugshi Regi Natu Regi hiayi Regi Pigtug Regi Taitug Regi Hualie Regi Thus the ew arach is suitable fr measurig the verall efficiecy f the whle system with the added beefit f allwig iefficiet sub-dmus t be idetified ad tetially rectified. 755 REFERENES Baer, R.D., A. hares ad W.W. er, 984. Sme methds fr estimatig techical ad scale efficiecies i DEA. Maage. Sci., 30: DOI: 0.287/msc astelli, L., R. Peseti ad W. Uvich, DEAlie mdels fr the efficiecy evaluati f hierarchically structured uits. Eur. J. Oerat. Res., 54: DOI: 0.06/S (03) hares, A., W.W. er ad E. Rhdes, 978. Measurig the efficiecy f decisi maig uits. Eur. J. Oerat. Res., 2: htt:// t/measurig%20the%20efficiecy%20f%20desc isi%20maig%20uits.df.

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