A Generalized Cost Malmquist Index to the Productivities of Units with Negative Data in DEA

Transcription

1 Proceedigs of he 202 Ieraioal Coferece o Idusrial Egieerig ad Operaios Maageme Isabul, urey, July 3 6, 202 A eeralized Cos Malmquis Ide o he Produciviies of Uis wih Negaive Daa i DEA Shabam Razavya Deparme of Mahemaics Islamic Azad Uiversiy Souh ehra Brach, ehra-ira hasem ohidi Deparme of Mahemaics Islamic Azad Uiversiy, Ceral ehra Brach, ehra-ira Absrac I some daa evelopme aalysis (DEA) applicaios, some ipus of DMUs have egaive values wih posiive cos. his paper geeralizes he global cos Malmquis produciviy ide o compare he produciviy of differe DMUs wih egaive ipus i ay wo periods of imes uder variable reurs o scale (VRS) echology, ad he he geeralized ide is decomposed o several compoes. he obaied compoes are compued usig oparameric liear programmig models, ow as DEA. o illusrae he geeralized ide ad is compoes, a umerical eample a hree successive periods of ime is give. Keywords Malmquis ide; Circulariy; Daa evelopme aalysis (DEA); Cos efficiecy; Reurs o scale (VRS); Negaive daa.. Iroducio Daa Evelopme Aalysis (DEA) is a liear programmig for evaluaig he relaive efficiecy of each member of a se of orgaizaioal uis (Chares e al. (978)). he uis, called Decisio Maig Uis (DMUs), cosume various levels of each specified ipus ad produce various levels of each specified oupu. DEA evaluaes he efficiecy of a DMU relaive o a empirical producio possibiliy froier deermied by all DMUs uder assumpios regardig reurs o scale ad orieaio. I rece years he measureme ad aalysis of produciviy chage has eoyed a grea deal of ieres amog researchers sudyig firm performace ad behavior. Research effor has focused o he ivesigaio of he causes of produciviy chage ad o is decomposiio. I his framewor, he Malmquis ide was firs iroduced i produciviy lieraure by Caves e al. (982). Färe e al. (994) decomposed produciviy chage io a par aribuable o chage of echical efficiecy ad echical chage ad used o-parameric mahemaical programmig models for is compuaio. he cos Malmquis ide proposed by Maiadais ad haassoulis (2004) was eeded by ohidi e al. (200) io he Profi Malmquis ide (PM) ha ca be used whe ipu ad oupu prices are available ad producers wa o maimize oal profi of DMUs. o compare he cos efficiecy of DMUs i he differe periods of ime, ohidi e al. (20) used he cove combiaio (weighed average) of ipus coss i differe periods of ime ad obaied a commo cos for ipus. hey used commo cos ad obaied a global cos efficie froier, as sigle base, ad proposed a global cos Malmquis ide. his ide is circular ad i gives a sigle measure of produciviy chage, ad is models are always feasible. 407

2 radiioal DEA model ha is applied for compuig Malmquis idees cao deal wih egaive daa. he approach amed rage direcioal model (RDM) was developed by Porela e al. (2004) o measure efficiecy uder egaive daa. hey iroduced a model based o a ideal poi ad a direcioal vecor ha will be described laer. Oher approaches are also iroduced o measure efficiecy of a DMU i Sharp e al. (2007) ad Emrouzead e al. (200) for he case where daa are egaive. For eample Sharp e al. (2007) preseed a modified slacs-based measure (MSBM) model ha ca deal wih boh egaive ipus ad egaive oupus. he direcioal vecor is amed he SP rage i he MSBM model ad he efficiecy score of a DMU ha is calculaed by his approach cao be greaer ha he RDM efficiecy of Porela e al. (2004). Porela ad haassoulis (200) calculaed he mea-malmquis ide usig he RDM model. his paper preses a global cos Malmquis ide for comparig he produciviy of a DMU i wo ime periods whe some ipus have egaive values. Decomposiio of he proposed ide ad heir ierpreaios is preseed. For compuig he ide ad is compoes he RDM is used. his paper is orgaized as follows. Secio 2 preses a bacgroud. I Secio 3 he global cos Malmquis ide ad is compoes are preseed uder egaive daa. Secio 4 provides a umerical eample. Secio 5 cocludes. 2. Bacgroud he RDM iroduced by Porela e al. (2004) ca be used for comparig DMUs whe some ipus ad/or oupus are egaive. Assume ha i ime period, =,,, h ui, =,,, uses a ipu vecor = (, 2,, m ) o produce a oupu vecor y = ( y, y 2,, y s ). Cosider a poi wih maimum oupus ad miimum ipus observed i period as a ideal poi of ime period (for each ipu i, i =,, m, ad for each oupu r, r =,, s, r ma y r ) ad he direcioal vecor as ( g, g ) = ( g,, g, g,, g ). I he RDM model his direcioal vecor for DMU i a give ime period IP i is as mi { i } y m s yr yr r r IP is as { } is as, g = R = ma { y } y ; r =,, s ad g R i mi { i } i = = ; i =,, m, ha reflecs rages of possible improveme for his DMU. he RDM model for DMU i ime period is as follows (Porela e al. (2004)): where ( ) {, ca produce } i { β β β } DR (, y, R, R ) = sup ( R, y + R ), y y = y y is he producio echology of period ad ehibis variable reurs o scale (VRS). his paper applies he ipu-orieed RDM model ha is a paricular ype of model (). I his case R is y se o he zero vecor. For DMU, i ime period, =,,, he ipu orieed RDM model is as follows (Porela e al. (2004)): () β = ma β s.. λ β R, i =,, m, = = = λ y y, r =,, s, λ =, λ 0, i i i r r =,,. (2) 408

3 he value of β i model (2) is a iefficiecy measure ad is equal o β = DR (, y, R,0). herefore he measure of ipu efficiecy of ui is RDM (, y, R, 0) = β. Also for each ipu i we have β = mi { } { } i i i mi i ad his ipu efficiecy measure reflecs he disace bewee observed ad arge ipu levels ( ). I fac he observed ad arge levels are compared by he ideal uobserved DMU. i 3. A Malmquis ide o compare he produciviies of uis wih egaive dad i DEA =, uses a vecor (, q l = ),,, q l m Assume DMU,,, P N P N + =, ha variables P idicae he posiive ipus ad variables N idicae he egaive ipus of DMU for producig a oupu s vecor y i ime period. Also assume w m, w = ( w, w ) is he ipu price vecor of period, where w P is he prices of posiive ad P N w N C ( y, w, w ) usig he followig model: P N is he prices of egaive ipus of DMUs. We compue he cos fucio + ( β ) ( β ) q q l CF CF (, P, N ) = mi ip i in i i i i = i = q+ C y w w w R w R s.. λ y y, r =,, s, = = = r r CF i i i λ βr, i =,, m, λ =, λ 0, =,, J. (3) Similarly, we have + ( β ) ( β ) q q l CF CF (, P, N ) = mi ip i in i i i i = i = q+ C y w w w R w R s.. λ y y, r =,, s, = = = = = = λ 0, r r CF i i λ βr, i =,, m, λ =, i (4) CF where R i mi { mi { i }}, i,, m i global echology. ha is, for each ipu { { i i }} = =. Here, he ideal poi is a global ideal poi defied over he IP = mi mi, i =,, m. 409

4 Now we ca defie he ide uder VRS echology i he presece of egaive daa o evaluae he produciviy of DMU as ( ) ( ) P N P P N N C ( y, w, w ) w w = C y w w w w (, P, N ) P P N N. (5) he erm w P P w N N i (5) is he observed cos of DMU i ime period. he value greaer ha for reflecs he produciviy regressed from period o Decomposiio of he proposed ide he proposed ide ca be decomposed as follows: C ( y, w P, w N ) ( w P P w N N ) = C ( y, w P, w N ) ( w P P w N N ) C ( y, w P, w N ) ( w P P w N N ) C ( y, w P, w N ) ( w P P w N N ) C ( y, w P, w N ) ( w P P w N N ) C ( y, w, ) ( P w N w P P w N N ). (6) he erm ouside he braces i (6) is he overall efficiecy chage compoe of ui from o + ( OEC ) ad he erm iside he braces provides cos froier shif bewee periods ad + uder VRS producio echologies of wo ime ad + ( CC ). he value greaer ha of compoe OEC reflecs he overall efficiecy has regressed ad he value uder for CC compoe idicaes here is a echical progress for DMU from period of ime o Numerical Eample I able we have si DMUs wih wo ipus ad oe oupu available. I period 2 DMU improves is ipu levels by 0% ad he oher DMUs improve heir ipu levels by 25%. able. Ipu-oupu daa of period DMU Ipu level Ipu 2 level Oupu level Ipu price Ipu 2 price DMU DMU DMU DMU DMU DMU he price of ipu 2 is reduced by 50% i period 2. I order o evaluae he produciviy chage of DMU bewee wo ime periods ad 2 by compuig he 2 ide ad is compoes we assume ha w = w + w. 2 2 Based o he daa i able ad he daa of period 2, he values of he are show i able ide ad is compoes for DMU

5 able 2. he proposed ide ad is compoes OEC CC Because he value of ide is less ha, he produciviy of DMU has improved from period o period 2. he value of OEC is.0755, i meas ha he overall efficiecy of DMU has regressed bewee wo ime ad 2. Similarly, he produciviy of he oher DMUs ca be aalyzed. 5. Coclusios ohidi e al.'s (20) cos Malmquis produciviy ide is a circular ide for measurig he produciviy chage of a DMU i he case ha he prices of ipus are ow ad he ipus are posiive. Bu, his ide cao be used whe egaive ipus occur. o solve his problem, his paper preseed he ide ad decomposed he proposed ide. he ide ad is compoes preseed here saisfy circulariy ad are calculaed by usig he RDM model iroduced by Porela e al. (2004) ad some modified DEA models. Refereces Caves, D.W., Chrisese, L.R., ad Diewer, W.E., he ecoomic heory of ide umbers ad he measureme of ipu, oupu ad produciviy, Ecoomerica 50, , 982. Chares, A., Cooper, W. W., ad Rhodes, E., Measurig he efficiecy of decisio maig uis, Europea Joural of Operaioal Research, 2, , 978. Emrouzead, A., Aouze, A.L., ad haassoulis, E., A semi-orieed radial measure for measurig he efficiecy of decisio maig uis wih egaive daa, usig DEA, Europea Joural of Operaioal Research, 200, , 200. Färe, R., rossopf, S., Norris, M., ad Zhag, Z., Produciviy growh, echical progress ad efficiecy chages i idusrialized couries, America Ecoomic Review, 84, 66-83, 994. Maiadais, N., ad haassoulis, E., A cos Malmquis produciviy ide, Europea Joural of Operaioal Research, 54, , Porela, M.C.A.S., haassoulis, E., ad Simpso,.P.M., Negaive daa i DEA: A direcioal disace approach applied o ba braches, Joural of he Operaioal Research Sociey, 55, -2, Porela, M.C.A.S., ad haassoulis, E., A circular Malmquis-ype ide for measurig produciviy, Aso Worig Paper RP08-02., Aso Uiversiy Birmigham B47E, UK, Porela, M.C.A.S., ad haassoulis, E., Malmquis-ype idices i he presece of egaive daa: A applicaio o ba braches, Joural of baig & Fiace, 34, , 200. Sharp, J.A., Meg, W., ad Liu, W., A modified slacs-based measure model for daa evelopme aalysis wih 'aural' egaive oupus ad ipus, Joural of he Operaioal Research Sociey, 58, , ohidi,., Razavya, S., ad ohidia, S., A global cos Malmquis produciviy ide usig daa evelopme aalysis, Joural of he Operaioal Research Sociey, doi: 0.057/ors.20.23, 20. ohidi,., Razavya, S., ad ohidia, S., A profi Malmquis produciviy ide, Joural of Idusrial Egieerig Ieraioal, 6(0), 23-30,