Tuning SFA Results for Use in DEA. Kaoru Tone a,*, Miki Tsutsui a,b

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1 Tunng SF Reult for Ue n DE Kaoru Tone a,*, Mk Tutu a,b a Natonal Graduate Inttute for Polc Stude Roppong, Mnato-ku, Toko , Japan b Central Reearch Inttute of Electrc Power Indutr Iwato Kta, Komae-h, Toko , Japan Th reearch upported b Grant-n-d for Scentfc Reearch (c) Japan Socet for the Promoton of Scence. * Correpondng author. E-mal addree: tone@grp.ac.jp (K. Tone), mk@crep.denken.or.jp (M. Tutu). 1

2 Tunng SF Reult for Ue n DE btract fter pontng out hortcomng of the tradtonal adjutment cheme for combnng SF reult for ue n DE n the three tage approach, we propoe a new cheme. We demontrate the effect of th adjutment formula ung an electrc utlt data et. Keword: DE, SF, data adjutment, mult-tage approach 2

3 Tunng SF Reult for Ue n DE 1. Introducton Data envelopment anal (DE) ha been wdel utlzed for evaluatng relatve effcenc of organzaton wth multple nput reource and output product. DE emplo mathematcal programmng technque and manl deal wth data et that are uppoed determntc. Snce the objectve organzaton, called Decon Makng Unt (DMU), ma belong to everal dfferent operatonal envronment and ther data ma ubject to tattcal noe, t trongl demanded that the true manageral effcenc hould be dentfed after accountng (deletng) the operatng envronment effect and tattcal noe on the data. For th purpoe, Fred et al. (2002) propoed a three-tage procedure that combne DE and tochatc fronter anal (SF) a follow. t the frt tage, the emplo DE for fndng lack of each DMU that conttute the element of neffcenc. t the econd tage, the appl SF to eplan thee lack n term of the operatng envronment, tattcal noe and manageral effcenc. Then, the adjut the frt-tage data et b purgng the nfluence of the operatng envronment and tattcal noe. Latl, the appl DE to the adjuted data et at the thrd tage. vkran and Rowland (2006) further developed Fred et al. (2002) wthn the non-radal DE model,.e., the lack-baed meaure (SBM) ntroduced b Tone (2001). Th paper focue on ther data adjutment cheme. Frtl, we pont out rratonalt of ther adjutment formulae n that ther adjutment cont of potve tranlaton of the regreed term o that the adjuted data hould be non-negatve, nce mot DE model requre non-negatve data et. However, th operaton caue erou ba n the thrd tage DE core. We wll demontrate th fact ung 3

4 eample. Then we propoe a new procedure for tunng SF reult for ue n the thrd tage DE. Th paper unfold a follow. In Secton 2, we brefl urve the mult-tage ue of DE and SF. Reader are recommended to refer to Fred et al. (2002) and vkran and Rowland (2006) for detaled dcuon on the motvaton of the mult-tage approach. In Secton 3, we wll demontrate the rratonalt of ther adjutment cheme that combne the SF reult wth the orgnal data et. Then, we propoe a new tunng cheme for adjutng the SF reult for ue n the thrd tage DE n Secton 4. Comparon of our propoed cheme wth the prevou one preented n Secton 5. Some concludng remark follow n Secton Mult-tage Ue of DE and SF 2.1 Mult-tage approach We deal wth n DMU wth the nput matr X R and output matr m n Y R n, where m and are number of nput and output, repectvel. For the target DMU, ), where ( o o m o R and o R are nput and output of the DMU, we epre them n term of X, Y, the ntent vector n λ R, the nput lack m R and the output lack R a follow: o o = Xλ = Yλ (1) Both Fred et al. (2002) and vkran and Rowland (2006) evaluate the nput lack m R and output lack, whch repreent neffcenc of DMU, ), R ( o o b mean of DE model. Dfference et n the DE model utlzed a follow. Fred et al (2002) emplo the nput-orented BCC model (Banker et al. (1984)): 4

5 mn θ ubject to θ eλ = 1 λ 0, = Xλ = Yλ o o 0, 0, (2) where n e R denoted a row vector n whch all element are equal to 1. vkran and Rowland (2006) utlze the non-radal lack-baed model (SBM) ntroduced b Tone (2001): 1 1 m mn ρ = 1 1 ubject to o o eλ = 1 = 1 m r= 1 = Xλ = Yλ λ 0, o r ro 0, 0. (3) Refer to vkran and Rowland (2006) for comparon of thee two approache. We wll not go nto the detal but jut denote the optmal lack obtaned b and. Both paper regard thee lack a the ource of neffcence. However, actual performance are lkel to be attrbutable to ome combnaton of manageral neffcence, envronmental effect and tattcal noe. Thu, the tred to olate thee three effect ung tochatc fronter anal (SF) n the econd tage. The general functon of the SF regreon repreented n Eq. (4) below for the cae of nput lack. j = f ( z ; β ) v u, = 1, K, m; j = 1, K, n, (4) j j j where j the tage 1 lack n the th nput for the jth unt, z j the envronmental 5

6 varable, β the parameter vector for the feable lack fronter and v u the j j 2 compounded error tructure where v N(0, σ ) repreent tattcal noe and u 0 repreent manageral neffcenc. j j 2.2 djutment of Orgnal Data b SF Reult: Prevou Stude Fred et al. (2002) and vkran-rowland (2006) propoed the followng adjutment cheme. (a) Fred et al. (2002) adjut the nput data b deletng gnfcant envronmental effect and tattcal noe a follow: v Input adjutment j ) ) ) ) [ ma { z } z β ] [ ma { v } v ] = j k j β (5) k j (b) vkran and Rowland (2006) adjut the output data a follow: Output adjutment ) ) r r ) ) [ z mn { z β }] [ v { v }] = β mn (6) j k rk The role of ma and mn n the above formula to enure the adjuted data { } and { } j to be potve, nce mot DE model demand the data et to be potve. Th operaton a tranlaton of the SF reult. ctuall, n the nput adjutment cae, let u defne zˆ ma { z βˆ } ẑ and wrtten a and vˆ ma { vˆ } k k. Then vˆ are fed (contant) for all DMU wthn the nput tem. Thu, (5) can be j = z β j ˆ j vˆ j zˆ vˆ th formula ndcate, the SF reult are tranlated b zˆ vˆ for each. In the net ecton, we pont out the trouble that th tranlaton nduce. 6

7 3. Shortcomng of Prevou djutment We wll demontrate rratonalt of the above adjutment cheme ung two eample a follow. 3.1 Two DMU wth ngle nput and ngle output cae The adjutment formulae (5) and (6) are ntroduced o that the adjuted value are aured to be non-negatve or potve. Th mean a potve tranlaton of the adjuted data. Now, we nvetgate how a potve tranlaton effect DE effcenc core ung a mple eample. Th eample deal onl wth tranlaton ue but not wth envronmental and noe ue. Table 1 ehbt two DMU and B wth a ngle nput and a ngle output. We tranlate the nput b k. Thu, nput 1k whle B 2k. Fgure 1 depct thee hft from to and from B to B. We tranlate onl nput value but keep the output value unchanged. Table 1. mple eample Input Output Tranlated Output Input k Y 1 2 1k 2 B 2 1 2k 1 7

8 2 1 B B 1 2 1k 2k Fgure 1. Input Tranlaton In both cae,.e., the orgnal and the tranlated cae, and are effcent and B and B are neffcent compared wth and, repectvel. The radal and nput-orented DE effcenc core of B are calculated n term of k a follow: Under the contant return-to-cale aumpton (CRS) (CCR-I) 1 k θ C ( k) =. (7) 2(2 k) Under the varable return-to-cale aumpton (VRS) (BCC-I) 1 k θ V ( k) =. (8) 2 k We notce that under th ngle nput and ngle output cae the nput-orented SBM model gve the ame effcenc value wth the radal model. The are monotone ncreang n k and hence the dfference n effcenc between and B monotone decreang n k. ctuall, the BCC-I core of B tend to unt (that of ) a k tend to nfnt. Th mple eample demontrate that the nput 8

9 tranlaton factor k effect the effcenc core gnfcantl and ndcate that the adjutment formulae (5) and (6) uffer from the ma and mn value ncluded that are tranlaton term n the repectve formula. The net eample wll evdence th fact. 3.2 mult-tage eample We demontrate rratonalt of the adjutment formula (5) ung an actual data et Data and tattc We emploed the data from U.S. and Japan electrc utlte (48 U.S. and 8 Japan) durng the ear We count a utlt at a certan ear a an ndependent DMU and, after deleng outler, we obtaned 351 utlte a our DMU. We emploed three nput and one output a follow: Input Input 1: The total nameplate capact of electrc power plant meaured n Mega Watt (MW) Input 2: The conumed fuel converted to Brth Thermal Unt (BTU) Input 3: The number of emploee Output Output 1: The generated electrc power meaured n Mega Watt hour (MWh) Stattc on the data are dplaed n Table 2. 9

10 Table 2 Stattc of the Data Input 1: Name Input 2: Input 3: Output 1: Plate Capact Fuel (BTU) Emploee Generaton (MW) (1/10,000) (MWh) verage Mn Ma S. D DE model We emploed the nput-orented SBM under the varable return-to-cale (VRS) aumpton Frt tage DE The reult of the 1 t tage nput-orented SBM are ummarzed n Table 3. Table 3: 1 t Stage SBM Reult verage Mn Ma S.D. SBM core Second tage SF We appled SF for the optmal nput lack obtaned n the 1 t tage SBM. We emploed everal envronmental factor contng of non-dcretonar, dcretonar and dumm varable whch are out of control of DMU. We utlzed LIMDEP 8.0 for th purpoe. 10

11 3.2.5 djutment We adjuted the lack and hence the nput ung the SF reult b mean of the formula (5). In th formula, the term zˆ ma { z βˆ } and k vˆ { vˆ } ma are fed (contant) for all DMU wthn the nput. k Hence, the adjutment formula (5) become to a tranlaton a we denoted n the precedng ecton. We record thee ma term for each nput tem n Table 4. Table 4: The Ma Value zˆ vˆ { z βˆ } k Input 1 Input 2 Input 3 ma { vˆ } ma k Stattc of the adjuted data are ummarzed n Table 5. Table 5: Stattc of the djuted Data Input 1: Input 2: Input 3: Output 1: Name Plate Fuel (BTU) Emploee Generaton Capact (1/10,000) (MWh) (MW) verage Mn Ma S. D

12 3.2.6 Thrd tage DE We appled the nput-orented SBM under varable return-to-cale aumpton to the adjuted data et. Stattc of the effcenc core are recorded n Table 6. Table 6: 3 rd Stage SBM Reult verage Mn Ma S.D. SBM core Comparon of Table 3 and Table 6 demontrate a bg change n the average core: from to Fgure 2 compare the dtrbuton of the effcenc core at the 1 t and 3 rd tage SBM. Th level up mght be caued b the adjutment formula (5) ung the ma value for preventng negatve nput value. The reult of the 3 rd tage SBM almot lot the dcrmnatng power n effcenc evaluaton and are unacceptable. lthough we decrbed our eperence wth the VRS model, we have eperenced mlar odd reult under the contant return-to-cale (CRS) aumpton Effcenc Stage 1 Stage 3 (dj Ma) DMU 12

13 Fgure 2: Comparon of Stage 1 and Stage 3 Effcenc Score 4. New Tunng of SF Reult In th ecton, we propoe a new adjutment cheme. 4.1 Re-adjutment Frt, we emplo the SF formula for adjutment wth no recoure to ma or mn a follow. Input adjutment j ) ) = j z j β vj (9) Output adjutment ) r j ) = z β v (10) Then we re-adjut them nto or ung the followng formula. j Re-adjutment Input j ma mn = ( j mn ) mn ( = 1, K, m : j = 1, Kn) (11) ma mn where ma = ma { }, = mn { }, = ma { }, mn { }. k N k mn k ma k mn = k Output r ma r mn = ( r mn ) r mn ( r = 1, K, : j = 1, Kn) (12) r ma r mn where r ma = ma { }, = mn { }, = ma { }, mn { }. k N rk r mn rk r ma rk r mn = rk 13

14 4.2 Ratonale The propoed re-adjutment cheme ha the followng properte: (1) j ncreae n j. Thu, the re-adjuted data have the ame rankng wth the adjuted data. ctuall a lnear tranformaton of wth a potve j j coeffcent. The coeffcent and the contant term of th lnear tranformaton are contant wthn the repectve nput tem. (2) t ma, ma attan the mamum value =. ma ma (3) t mn, mn attan the mnmum value Hence, the re-adjuted data et { } j =. mn mn reman n the range [ ]( ) mamum and mnmum value are the ame between { } and { } j mn, ma, and the For the output de, we have the ame propert: the re-adjuted data et { } reman n the range [ ]( r) the ame between { } and { }. r mn, r ma, and the mamum and mnmum value are. Thee properte are appealng n that the elmnate ambgut regardng the range of adjuted nput and output value that effect the DE core gnfcantl a we have hown n the prevou eample. Furthermore, when we tart the frt tage DE, we uuall confrm that the range of nput and output value are approprate for the choen DE model. (We delete outler before gong nto the frt tage.) Therefore, t not odd to keep the range tatu quo and re-evaluate the DE effcenc core at the thrd tage ung the re-adjuted data et. 5. Numercal Comparon We re-adjut the US electrc utlt data et and compare the reult. j 14

15 Ung the formula (9) (but not ung the ma n (5)), we adjuted the nput data, and then re-adjuted the data b the formula (11). Table 7 dpla the tattc of the re-adjuted data. epected, the mn and ma value are the ame wth the orgnal data n Table 2. Table 7 Stattc of the Re-adjuted Data Input 1: Input 2: Input 3: Output 1: Name Plate Fuel (BTU) Emploee Generaton Capact (1/10,000) (MWh) (MW) verage Mn Ma S. D The 3 rd tage SBM wa appled to th data et and the reult are ummarzed n Table 8. Table 8: Reult of 3 rd Stage SBM ung the Re-adjuted Data verage Mn Ma S.D. SBM core Fgure 3 compare the effcenc core of the 1 t and the new 3 rd tage SBM. The upgrade of the average core from (1 t tage) to (New 3 rd tage) reflect the effect of envronmental factor and tattcal noe dentfed n the 2 nd tage SF. Compared wth the Fgure 2 whch reulted from the adjutment ung 15

16 ma, the new 3 rd tage reult are more acceptable for effcenc evaluaton Effcenc Stage 1 New Stage DMU Fgure 3: Comparon of Stage 1 and New Stage 3 Score 6. Concludng Remark In the DE tude, man author have tred to dentf the true manageral effcenc after accountng for the operatonal envronment effect and tattcal noe on the data. The three tage approach propoed b Fred et al. (2002) a remarkable advance on th lne. The combned DE wth SF n the manner that the lack obtaned n the 1 t tage DE wa regreed b mean of the envronmental effect, tattcal noe and manageral effcenc n the data. Then the adjut the orgnal nput data ung the regreon reult. In th paper, we have ponted out hortcomng n ther data adjutment and propoed a new adjutment cheme of SF reult for ue n DE. Th cheme wa appled to U.S. and Japan electrc utlte and proved t uperort over the tradtonal one. Combnng non-parametrc DE wth parametrc SF ma aroue everal fundamental problem. The data adjutment problem an mportant ue 16

17 among them. We hope our method erve a a teppng tone to the fnal reoluton. Reference Fred HO, Lovell CK, Schmdt SS, Yaawarng S. ccountng for envronmental effect and tattcal noe n data envelopment anal. Journal of Productvt nal 2002; 17: vkran NK, Rowland T. How to better dentf the true manageral performance: State of the art ung DE, OMEG 2006 (forthcomng). Tone K. lack-baed meaure of effcenc n data envelopment anal. European Journal of Operatonal Reearch 2001; 130: Banker RD, Charne, Cooper WW, Some model for etmatng techncal and cale neffcence n data envelopment anal. Management Scence 1984, 30: