The lower limit of interval efficiency in Data Envelopment Analysis

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1 Jurnal f aa nelpmen nalysis and ecisin Science 05 N. (05) ailable nline a lume 05, Issue, ear 05 ricle I: dea-00095, 9 Pages di:0.5899/05/dea Research ricle aa nelpmen nalysis and ecisin Science The lwer limi f ineral efficiency in aa nelpmen nalysis ian Rahmani Parchiklaei eparmen f Mahemaics, Nur ranch Islamic zad niersiy, Nur, Iran. pyrigh 05 ian Rahmani Parchiklaei. This is an pen access aricle disribued under he reaie mmns ribuin License, which permis unresriced use, disribuin, and reprducin in any medium, prided he riginal wrk is prperly cied. bsrac In daa enelpmen analysis echnique, he relaie efficiency f he hmgenus decisin making unis is calculaed. These calculains are dne based n he classical mdel f linear prgramming such as R,,. ecause f maximizing he weighed sum f upus ha in inpus f ne uni under cerain cndiins, he bained efficiency in all f hese mdels is he upper limi f exac relaie efficiency. In her wrds, he efficiency is calculaedfrm he pimisic iewpin. T caculaed he lwer limi f efficiency, i.e. he efficiency bained frm a pessimisic iewpin fr cerain weighs, he exising mdels cann calculae he exac lwer limi and in sme cases, here exis sme mdels ha shw an incrrec lwer limi. Thrugh he mdel inrduced in he presen sudy, we can calculae he exac lwer limi f he ineral efficiency. The designed mdel can be bained by minimizing he rai f weighed sum f upus ha f inpus fr eery uni under cerin cndiins. The exac lwer limi can be calculaed in all saes hrugh ur adped mdel. Keywrds: aa enelpmen analysis, Ineral efficiency, Ineral inefficiency, lwer limi,upper limi. Inrducin The daa enelpmen analysis () was inrduced by harnes e al.( 978) fr he firs ime. I is an effecie l fr decisin making and managemen. is a nnparameric echnique fr measuring and ealuaing he relaie efficiency f decisin making unis(m) wih he cmmn inpus and upus. In mdels, efficiency is he weighed sum f upus ha f inpus. If he relaie efficiency f he M under ealuain equal ne, i is said be efficien ; herwise, i is said be inefficien. ecause eery decisin making uni can be ealuaed frm bh pimisic and pessimisic iewpins, i is beer ha he efficiencies be bainedas inerals and in his way, he bained efficiency wuld encmpass he pssible range f he efficiency alues fr each M. If he range f he ineral efficiency is large, i means ha he M can be gd frm he pimisic iewpin and can be bad frm he pessimisic ne. The ineral sudies he ealuain f he decisin making unis in mre deails. The cneninal mdels, R and examine he mdel frm pimisic iewpin. nher mdel called as Inered (I), examines each M frm pessimisic iewpin. The becie funcin f I is he rai f he weighed sum inpus ha f upus. yle e al. (995) sudied he efficiency measuremen bh rrespnding uhr. mail address: bianrah40@gmail.cm; Tel:

2 Jurnal f aa nelpmen nalysis and ecisin Science 05 N. (05) hp:// frm pimisic and pessimisic iewpins, and s nani e al. (00) bain he lwer limi f ineral efficiency, They inrduced a mdel which was differen frm he I mdel. Recenly, Wang e al. (007) hae presened he bunded inpu riened mdels. In hese mdels, he perfrmance f he decisin making unis is ealuaed in a range f he ineral and he maximum inpu and upu is used bain he bes and wrs relaie efficiency. The presen paper is shw he drawbacks f he nani e al. mdel (00) and inrduce a mdel which is free f hese drawbacks. The adenage f he presen auhrs, mdel er hire mdel is represened by an example. The res f he paper cers 3 secins. In he secin, he mdels, In he secin 3, he numerical examples will be represened,and he las secin is he cncluin par f he paper. mdels In, he maximum rai f upu inpu is assumed as he efficiency which is calculaed frm he pimisic iewpin fr each M. nsider n decisin making unis (M : J={,,n}) each f which M is using inpus x i, i=,,m, prduce upus y r, r=,,s. Le he inpu and upu ecrs fr M be = (x,,x m ) and = (y,,y s ), respeciely. Fr M i has been assumed ha 0, 0 and 0, 0. The efficiency f he M,(J) relaie he hers is bained hrugh he fllwing fracinal mdel: S..,,..., n (.) 0, 0. where and are he weigh ecrs fr he inpu ecrs and upu ecrs, respeciely. This fracinal mdel will be changed in he fllwing linear mdel call R by he harnes-per cnersin: S.., (.) 0, 0, 0. mn sn where R, R are he inpu and upu marix respeciely. When he pimal alue f he becie funcin equals ne, M is raed as efficien and herwise is raed as inefficien. nani e al.(00), he upper limi f ineral efficiency is frmulaed as fllws: S... 0, 0. wih he change f ariable, cnsrains are creaed. (.3) Inernainal Scienific Publicains and nsuling Serices

3 Jurnal f aa nelpmen nalysis and ecisin Science 05 N. (05) hp:// The cnsrains r 0 fr any, his guaranees ha he relaie efficiency is calculaed. Tha is, he hyperplane we bain be suppred n prducin pssibiliy se (PPS). Oherwise, he relain (.3) will n be relaie efficiency. Then, by he harnes-per cnersin, we hae: S.. (.4) 0, 0. They pred ha he pimal alue f (.) and (.4) are equialen. If he becie funcin f R mdel is minimizain, bain he efficiency frm he pessimisic iewpin, i happens ha =0, 0 and he alue f he becie funcin becames zer (0) fr all M s. This is why nani e al.(00) haed cnsidered he minimizain prblem f (.3) bain he lwer limi f ineral efficiency fr M : Min S.. 0, 0. In fac, by mdel (.5) we bain weighs hrugh which he relaie efficiency is calculaed and he wrs sae ccurs he uni under ealuain (M ). They bained he fllwing sluins prblem by using he harnes-per cnersin which can be represened as fllws: Min S.., (.6) 0, 0. The prblem (.6) can n be replaced wih he equialen linear prgramming prblem. y assuming ha fr each, (.6) can be diided in he fllwing n prblem : Min S.., 0, 0. The cndiin (,..., n) fr a gien cann be replaced wih { } (.5) (.7) Inernainal Scienific Publicains and nsuling Serices

4 Jurnal f aa nelpmen nalysis and ecisin Science 05 N. (05) hp:// becames by cnsidering her cnsrains, his can be a separable prblem. T clarify he pin, le, s assume ha: {a,a,,a n }=α I is bius ha hrugh his relain, we can bain n relain wih he fllwing disuncin: { a, a,..., an} a a... an (.8) Hweer, i is eiden ha if a a... a n is rue, hen we can n cnclude ha {a,a,,a n }=α. Fr example, i is bius ha if is a rue prpsiin, hen ne can n cnclude ha {5,6,3}=5. The relain (.8) is recursin relain if he prpsiin a a... an is rue, hen he prpsiin a, a,..., a n is rue. Thus, we can wrie : {,..., n } { i i,..., n... n { i i,..., n In rder he pimal alue f prblem (.7), hey belied ha (.7) can be reduced he fllwing n LP prblems: Min S.., 0, 0. (,..., n) 0, They cnsidered he pimal sluin f (.9) as he lwer limi f ineral efficiency. ecause when is, biusly. Thus, he lwer limi f ineral efficiency can be mahemaically wrien as fllws : 0 Min (.0) where a b Min a, b las, hey pred ha he lwer limi f ineral efficiency can be bained as fllws: p r Min pr, (.) p r The abe mdel can be criicized because f he fllwing drawbacks, which Wang e al. (007) hae referred in heir aricale.. I is bius ha he mdels (.5) and (.7) can n be equialen. In mdel (.5), he denminar f he becie funcin can be equal ne fr geing sme pimal sluins nly when he prblem is in maximizain ype.. This mdel can specify nly a single decisin making uni wih he smalles lwer limi, hus he inefficiency This mdel can specify nly a single decisin making uni wih he smalles lwer limi, hus he inefficiency brder will n be deermined. The mdel uilizes nly ne decisin making uni, a M which is a surce cllecin fr M. S he mdel has nly w cnsrains. Therefre, he (.9) Inernainal Scienific Publicains and nsuling Serices

5 Jurnal f aa nelpmen nalysis and ecisin Science 05 N. (05) hp:// weighs f nly ne inpu and upu is n equal zer and he weighs f he her inpus and upus are zer. sequenly, he daa f nly ne inpu and upu are used. 3. The mdel (.7) guaranees ha:, I means ha i des n guaranee ha (.) is he maximum f all he decisin making unis. Tha is, i des n guaranee ha he res f he decisin making unis are in ne side f he hyperplane which passes he designaed uni. In her wrds, i des n guaranee ha he defining hyperplane is n PPS. We represen his inuiily: Figure : unerexamplefffffffffff ssume ha he uni () is under ealuain, and he cnsrain f mdel (.7) is uni (). We see ha he hyperplane passing he pin () des n guaranee ha he her M s are in ne side f he menined hyperplane. The uni () des n apply he cnsrain 0. Wang e al.(007) emply a irual M which cnsumes he ms inpus nly prduce he leas upus and called i ani-ideal M (M). They fund he bes relaie efficiency and base n ha presened a mdel frm which upper and lwer bunds n efficiency ineral are bained. They emplyed minimizain f becie funcin bain he lwer limi f efficiency ineral. Since his irual M may n exis in realiy and in pracical uses, a mdel is presens which lacks he irual. T sle hese prblems, we add cnsrain (.) he se f cnsrains (.9). In his case, he fllwing linear mdel is bained : Inernainal Scienific Publicains and nsuling Serices

6 Jurnal f aa nelpmen nalysis and ecisin Science 05 N. (05) hp:// Min s., 0, 0, p p 0, 0. p (.3) T ealuae M, his mdel need sle n- in reurn fr all unis. Ne ha his mdel may n be feasible fr sme M int c. Lemma.. If H is a suppr hyperplane n T c in, hen T. Lemma.. If in Tc hen (.3). Prf. ssume ha in Tc and (.3) is feasible. In his case here is (, ) 0 s ha p p 0 is fr all p and 0 p. ssume ha H={ : 0 } In his case T c H and Tc H, hen H n T c in is suppr ccrding Lemma. Tc and his is incnsisen wih in Tc. k Prpsiin.. If here is an M k ha > 0, ( k), (, ) 0, >, hen minimum and maximum efficiency M k is equal. Prf. if here is an M k ha k k hen : 0 k k and : 0 k k ; herefre (,). Frm his relain we hae k, k k hen maximum and minimum efficiency M k is equal. T ge he M s efficiency, in addiin he surce uni f M, he afremenined mdel, uses her decisin making unis. S i guaranees ha he her decisin making unis are in ne side f he hyperplane passing frm M. In her wrds, define hyperplane n PPS will be guaraneed. n example is shwn cmpare mdels. 3 Numerical example Le us assume an example which has 0 M, wih ne inpu and w upus. The daa are shwn in Table. Inernainal Scienific Publicains and nsuling Serices

7 Jurnal f aa nelpmen nalysis and ecisin Science 05 N. (05) hp:// M F G H I J Inpu Oupu Table.alue f inpu and upus Oupu When we bain he lwer limi f ineral efficiency accrding frmula (.0), he infrmain wih fie decimal digis are represened in Table. M F G H I J Table. The lwer limi f ineral efficiency accrding nani frmula (.) Nw, we find he pimal sluin accrding mdel (.9). The daa are shwn in Table 3. M F G H I J Table 3: The pimal alue f and he pimal alues f and. Inernainal Scienific Publicains and nsuling Serices

8 Jurnal f aa nelpmen nalysis and ecisin Science 05 N. (05) hp:// y sudying able 3, we see ha he cndiin 0 des n apply eery. Fr M in pimal sluin fr he secnd index, we hae Fr M in pimal sluin fr he secnd index, we hae These cndiins exis fr Ms f I,H,G,F,; ha is, heir alues, in he secnd index are als psiie. Nw, n he basis f mdel (.3), we bain he pimal sluin and pimal alue. Table 4 demnsraes hese pieces f infrmain. M F G H I J O Table 4: The pimal alue O and pimal alues f ,, y cmparing he ables and 3, we see ha he pimal sluins bain hrugh frmula (.9) and (.) are differen frm each her, which is incnsisen wih he herem represened by nani e al. paper. y sudying Table 3 and accrding he pimal alue f weigh ecrs, we see ha hese alues f M, I, d n apply her cnsrains (secnd cnsrain), bu ur mdel lacks his defec. y sudying Table 4, i is shwn ha he pimal alues f weigh ecrs fr each M apply all f he her cnsrains. In addiin, accrding he pimal sluin represened in Table 4, we see ha he Ms f G,H,I d n hae he feasible sluin, where accrding Table 3, hese hae he pimal sluin. 4 nclusin The preius mdels calculae he efficiency frm an pimisic iewpin. In her wrds, if is he sluin f he R mdel and R is a real efficiency f a uni, hen i will always be R. Therefre, he scieniss` are lking fr an (a) ha a R exsis. S, finding he lwer limi f ineral efficiency is ne f he scieniss effrs. nani e al. hae represened a mdel which had sme bscuriies. Their mdel cann d his impran prblem prperly. We hae presened a mdel which lacks hese defecs. Knwing hw classify he efficien and inefficien unis helps us reme inefficien nes. T pre ha ur Mdel is crrec, an example has been presened and he resuls hae been cmpared wih each her. References []. harnes, W. W. per,. Rhdes, Measuring he efficincy f decisin making unis, urpean Jurnal f Operainal Research, (987) [] T. nani,. Maeda, H. Tanaka, ual mdels f ineral and is exensin ineral daa, urpean Jurnal f Operainal Research, 36 (00) hp://dx.di.rg/0.06/s0377-7(0) Inernainal Scienific Publicains and nsuling Serices

9 Jurnal f aa nelpmen nalysis and ecisin Science 05 N. (05) hp:// [3]. M. Wang, J.. ang, Measuring he perfrmances f decisin-making unis using ineral efficiencies, Jurnal f mpuainal pplied Mahemaics, 98 (007) hp://dx.di.rg/0.06/.cam [4]. M. Wang, ang R. Greabanks, Ineral efficiency assessmen using daa enelpmen analysis, Fuzzy ses and sysems, 53 (005) hp://dx.di.rg/0.06/.fss [5] N. S. Wang, R. H. i, W. Wang, aluaing he perfrmances f decisin-making unis based n ineral efficiencies, Jurnal f mpuainal pplied Mahemaics, 6 (008) hp://dx.di.rg/0.06/.cam Inernainal Scienific Publicains and nsuling Serices