Finding Strong Defining Hyperplanes of Production. Possibility Set with Interval Data

Transcription

1 Aled Matheatcal Scence, Vol 6, 22, no 4, Fndng Stong Defnng Helane of Poducton Poblt Set wth Inteval Data F Hoenzadeh otf a *, G R Jahanhahloo a, S Mehaban b and P Zaan a a Deatent of Matheatc, Scence and Reeach Banch Ilac Azad nvet, Tehan, Ian b Deatent of Matheatc, Tabat Moalle nvet, Tehan, Ian Abtact We call a a of eotve nut and outut vecto an actvt and ee t b the notaton (, ) Poducton Poblt Set (PPS) defned a the et of feable actvte Data enveloent anal (DEA) odel lctl ue PPS to evaluate elatve effcenc of Decon Makng nt (DM) Although DEA odel ued fo evaluate the coe effcenc of a DM, the can not eent effcent fonte of PPS In th atcle, we ooe a ethod fo fndng all Stong Defnng Helane of PPS (SDHP) wth nteval data The ae equaton that fo tong effcent uface Thee equaton ae ueful n Sentvt and Stablt Anal, the tatu of Retun to Scale of a DM, ncooatng efoance nfoaton n to the effcent fonte anal and o on Kewod: Data enveloent anal; Inteval Data; Poducton Poblt Set Intoducton Data enveloent anal (DEA) a ueful ethod to evaluate elatve effcenc of ultle-nut and ultle-outut unt baed on obeved data It dchotoe the decon akng unt (DM): effcent o neffcent Howeve, uncetant uch a a * Coeondng autho, Fahad Hoenzadeh otf, E-al: fahad@hoenzadeh Adde: Deatent of Matheatc, Scence and Reeach Banch, Ilac Azad nvet, Ahaf Ifahan, Heaak, Tehan, Ian

2 98 F Hoenzadeh otf et al eaueent eo hould be ncooated n obeved data The ognal DEA odel aue that the level of nut and outut ae known eactl Recentl, Cooe et al [ 6 ] addeed the oble of ece data n DEA n t geneal fo Iece data ean that oe data ae known that the tue nut and outut value le wthn bounded nteval whle othe data ae known onl to atf defnte odnal elaton Effcent uface ae ueful n analzng DEA effcent DM and ncooatng efeence nfoaton to fnd efeence DM n ultobectve ogang The cuent atcle oceed a follow: In ecton 2, we gve a concet of nteval data and oe bac defnton, odel, and concet ae eented that wll be ued n othe ecton In ecton 3, we t to fnd all Stong Defnng Helane of Poducton Poblt Set (SDHP) wth nteval data In ecton 4, a nuecal eale condeed and ecton 5 gve ou concluve eak 2 Backgound Suoe we have n DM, each ung nut to oduce outut We denote b the level of the -th nut ( =,, ) and the level of the -th outut ( =,, ) fo the -th unt, ( =,, n ) J { } The tue nut and outut data ae known to le wthn bounded nteval, e [, ] and [, ], wth the ue and lowe of nteval gven a contant and aued tctl otve We ue a odel whch defned a follow: a St z = u [, ] v u [ ], = () [, ] v [, ], =,, n u, v,,,,,, Obvoul, above odel not lnea Now, ung of the followng odel, we obtan an ue and lowe bound of the effcenc coe fo DM ( J ) : a Z = u St v = (2)

3 Fndng tong defnng helane 99 and a z u v u v, =,, n, u, v,,,,,, = u v = St u v (3) u v, =,, n, u, v,,,,,, Suoe that the effcenc z * and z * be attaned b evaluatng DM n odel (2) and * odel (3) eectvel Suoe that z be otal obectve value n odel (), then fo * * each [, ] and [, ], we wll have z [ z ] *, z Fo evaluatng the effcenc of DM ( {,, n}) the nut-oented odel (4) can be aled n θ n St λ θ (4) = n = λ λ Λ n Whee λ a eotve vecto n R The above odel called CCR f Λ = { λ λ } and BCC 2 f Λ = { λ λ, e λ = } Whee e= (,,) ( a vecto of one ) The PPS of CCR odel T c and the PPS of BCC odel T v The odel (4) called enveloent fo DM ( {,, n) } Paeto Effcent f and onl f θ * = n (4) and the otal value of followng lnea ogang equal to zeo Chane, Cooe, Rhod See[4] 2 Banke, Chane, Cooe See[]

4 2 F Hoenzadeh otf et al a es + es + St n λ + S = (5) = n = λ S + = + λ Λ, S, S Fo DM, we defne t efeence et, E, b E ={ λ * (4) } {,, n oe otal oluton of n} The DM n E ae aeto effcent and an eotve cobnaton of the aeto effcent a well A a eult, efeence of a DM ae effcent DM that thee a cobnaton of the that donate t The dual of (4), whch n the followng fo, called ultle fo a u t + u St v t =, t t u v + u, =,,n (6) u, v u fee Note that f u = n (6) the odel CCR, ele t BCC * * DM Stong Effcent f n (6) u t + u = 3 * * and ( u, v ) fo oe otal * * oluton If u t * * + u = and no ( u, v ) et, then DM called Weak Effcent Th ean that, weak effcenc occu when the otal obectve of (6) one and at leat one coonent of each otal oluton zeo Effcent Fonte the et of all ont (actual o vtual) wth effcenc coe equal to unt 3 Fndng tong defnng helane of PPS wth nteval data At ft glance, t ee that ung ultle fo, all defnng helane of PPS can be obtaned Howeve n ealt th not tue, nce the tuctue of enveloent fo oe tong degeneac then the ultle fo a oduce altenatve otal oluton In Fg notce DM ng odel (6), t een that thee ae altenatve otal oluton whch defne nfnte nube of helane ang though A, fo whch onl two helane (H and H 2 ) ae defnng helane 3 (*) ued fo otal oluton

5 Fndng tong defnng helane 2 H 2 H H A DM Fg Helane H not defnng H and H 2 ae defnng To eove th dawback, the followng ethod fo fndng Stong 4 Defnng Helane eented Suoe aong all DM, Q of the ae tong effcent; then all of th DM ae on oe tong uotng 5 helane Becaue of tong effcenc, thee et oe * * * t * t * ( u, v ) oluton fo (6); then the tong effcent DM le on u v + u = ; and th helane tong, accodng to t defnton ng the followng theoe t can be detened whch DM le on the ae uotng helane Theoe et (, ) and ( q, q ) be obeved DM that le on a tong uotng helane, then each conve cobnaton of the on the ae helane[9] Coolla If, ) and, ) ae two effcent DM (tong o weak) and le on ( ( q q the ae helane then fo (,) helane μ, μ(, ) ( μ)(, ) + effcent and on that Theoe 2 Conde (, ) and ( q, q ) ae two obeved DM le on dffeent helane (ecludng the nteecton, f t not et) Then eve ont (vtual DM) whch obtaned b tct conve cobnaton of the an nteo ont of PPS In othe wod th vtual DM adal neffcent [9] 4 t Helane { P =, P } coonent of P ae zeo 5 Fo defnton and oete ee [3] α tong f none of coonent of P ae zeo; and weak f oe q q

6 22 F Hoenzadeh otf et al Now, we note that the DM wth the leat nut and ot outut Suoe aong the, Q DM ae tong effcent Wthout loe of genealt we can aue that thee effcent DM ae DM,,DM Q Conde the et F = {,,Q}, we take a dtnct a DM and DM q, whee and q belong to F, and contuct a vtual DM a follow: DM k = DM + DM q 2 2 ng the DEA odel we can detene DM k effcent o neffcent In the ft cae, b Theoe, DM and DM q ae on the ae helane ; n the econd cae the ae not (b Theoe 2) Fo each ebe (=,,Q) of F a new et F wll be contucted F a ubet of F that t ebe ae colana Th ean thee et oe helane contan DM and oe DM n F It obvou that Defnton H a tong defnng helane of PPS f and onl f t uotng, at leat + tong effcent DM of PPS le on H and t gadent coonent coeondng wth outut vecto ae nonnegatve and coonent coeondng wth nut vecto ae nonotve To decbe ou defnton, we can ake followng cteon fo DM wth the leat nut and ot outut We chooe an abta + ebe of F uch that none of the belong to oe othe F we note that when we deal wth T c, one of thee + DM can be ogn, theefoe onl +- ebe of F ae needed We call th et D = {,, + } ng D, a helane can be contucted a follow: F =, (7) Whee,,, ae vaable, t (,, ; t =,, + ),, = and q t ( q =,, ; t =,, + ) ae the leat of the th nut and the ot of the qth outut fo the th unt, eectvel Suoe that the equaton of above entoned helane that ae though fo the DM wth the leat nut and ot outut n the fo of P t z + α = whee z = (,,,,, ), P the gadent of the helane and α a cala

7 Fndng tong defnng helane 23 t Theoe 3 Conde H = { z P z + α = } w w w w Suoe w =,,,,, ) defned a follow: w = a ( { : =,, n} =,,, { : =,, n} =,, w = n call w Negatve Ideal (NI) If t Pz + α =, D, t Pz + α, F D, t Pw + α, that P t z + α = contucted b (7) e all effcent DM and NI ae n then H uotng[9] Now we ae n the oton to ut all togethe the ngedent of the ethod H 6 4 Sua of fndng all Stong Defnng Helane ' algoth fo DM wth the leat nut and ot outut Ste Evaluate n DM wth the leat nut and ot outut b utable fo of odel (4) and (5) Put ndee of tong effcent DM n F et F = Q Ste 2 Fo F q and q, q F that q F Ste 3 Fo each (=,,Q) F = F F, evaluate DM k = DM + DM q If t effcent 2 2 Ste 4 Chooe an abta + ebe of F uch that none of the belong to oe othe F when ou deal wth T c, one of thee + DM can be ogn Call th et D = {,, + } Contuct a helane ung E q (7) Suoe that the equaton of helane n the fo of P t z + α = whee z =,,,,, ) and α a cala ( Ste 5 If P ha an coonent le than o equal to zeo go to 6, ele let w w w w w =,,,,, ) defned a follow: If w w ( = a = n { : =,, n}, =,,, { : =,, n}, =,, 6 t H { z P z α } = +

8 24 F Hoenzadeh otf et al t Pz + α =, D, t Pz + α, F D, t Pw + α, then P t z + α = Stong Defnng Helane that ae though fo the DM wth the leat nut and ot outut Ste 6 If anothe ubet of F wth + ebe can be found go to 4, ele to Fndng all Stong Defnng Helane ' algoth fo DM wth the ot nut and leat outut eactl la to the entoned algoth 4 Nuecal eale Conde the nteval data ettng of the followng Table contan 8 DM wth one nut and one outut, whee and how lowe and ue bound fo nut eectvel A la notaton ued fo outut Table : Data of nuecal eale Now, we obtan the equaton of tong defnng helane fo DM wth the leat nut and ot outut Fndng Stong Defnng Helane wth the ot nut and leat outut la CCR Model: We obtan the tong effcent DM b alng (4) and (5) Hee F ={} H contucted on D = {o, } whee o ogn = that eld to =

9 Fndng tong defnng helane 25 BCC odel: The tong effcent DM ae F ={2, 6, } Snce a tct conve cobnaton of ( 2 و 2 ) and ( 6 ) 6, effcent, o we have lal, we have, F F F F { } F 6 6 = 2,6, F = {}, = {6,2,}, F = φ, = {,6}, F = {2}, Stong helane ae F6, 6 F2 2 (b Theoe ) and H : D = {2,6} =, that eld to = H 2 : D 2 = {6, } =, that eld = One can eal vef that condton of Theoe 3 ae held and both H and H 2 ae defnng Hee w =(55, 54) 5 Concluon In th ae a ethod fo fndng all Stong Defnng Helane of PPS wth nteval data wa ooed Ft, we found the fo DM wth the leat nut and ot outut and then vce vea B ung thee helane, all ebe of efeence et of a DM can be found Moeove, the can be ued n analzng aeto oluton n ultobectve ogang; whch th a oble fo futhe eeach

10 26 F Hoenzadeh otf et al Refeence [] RD Banke, A Chane, WW Cooe, Soe odel fo etatng techncal and cale neffcence n Data Enveloent Anal, Manageent Scence, 3, (984), [2] R D Banke, W W Cooe, M Sefod, R M Thall, J Zhu, Retun to cale n dffeent DEA odel, Euoean Jounal of Oeatonal Reeach, 54, (24), [3] M S Bazaaa, H D Sheal, C M Shett, Nonlnea Pogang: Theo and Algoth, John Wle & Son, (979) [4] A Chane, W W Cooe, E Rhode, Meaung the effcenc of decon akng unt, Euoean Jounal of Oeatonal Reeach, 2, (978), [5] W W Cooe, M Sefod, K Tone, Data Enveloent Anal: A Coehenve Tet wth Model, Alcaton, Refeence and DEA-Solve Softwae, Kluwe Acadec Publhe, (999) [6] W W Cooe, K S Pak, G Yu, IDEA and AR-IDEA: odel fo dealng wth ece data n DEA, Manageent Scence, 45, (999), [7] F Hoenzadeh oft, G R Jahanhahloo, F Reza Balf, H Zhan Reza, Rankng of DM on Inteval Data b DEA, Intenatonal Matheatcal Fou, 2, (27), [8] G R Jahanhahloo, F Hoenzadeh otf, N Shoa, M Sane, G Tohd, Sentvt and tablt anal n DEA, Aled Matheatc and Coutaton, 69, (25), [9] G R Jahanhahloo, F Hoenzadeh oft, H Zhan Reza, F Reza Balf, Fndng tong defnng helane of Poducton Poblt Set, Euoean Jounal of Oeatonal Reeach, 77, (27), [] G R Jahanhahloo, F Hoenzadeh oft, M Zohehbandan, Fndng the ecewe lnea fonte oducton functon n Data Enveloent Anal, Aled Matheatc and Coutaton, 63, (25),

11 Fndng tong defnng helane 27 [2] G Yu, Q We, P Bockett, Zhou, Contucton of all DEA effcent uface of the Poducton Poblt Set unde the genealzed Data Enveloent Anal odel, Euoean Jounal of Oeatonal Reeach, 95, (996), 49-5 Receved: Augut, 2