Fuzzy free disposal hull models under possibility and credibility measures

Transcription

1 286 Int. J. Data Analyi Technique and Strategie, Vol. 6, No. 3, 204 Fuzzy free dipoal hull odel under poibility and credibility eaure Rahed Khanjani Shiraz School of Matheatic, Iran Univerity of Science and Technology, Tehran, Iran E-ail: Madjid Tavana* Buine Syte and Analytic Departent, La Salle Univerity, Philadelphia, PA 94, USA Fax: E-ail: *Correponding author Khalil Paryab School of Matheatic, Iran Univerity of Science and Technology, Tehran, Iran E-ail: Abtract: The free dipoal hull (FDH) odel are ued a an alternative to data envelopent analyi (DEA) odel for perforance eaureent and efficiency aeent. The conventional FDH odel are ued to evaluate the perforance of a et of fir or deciion-aking unit (DMU) uing deterinitic input and output data. However, the input and output data in the real-life perforance evaluation proble are often iprecie and abiguou. The ipreciene and abiguity aociated with the input and output data in FDH can be repreented with fuzzy variable. In thi paper, the concept of chance-contrained prograing i ued to develop FDH odel with variou return to cale auption, including variable return to cale (VRS), variable non-increaing return to cale (NIRS), variable non-decreaing return to cale (NDRS), and contant return to cale (CRS), for efficient DMU with fuzzy data. We propoe two fuzzy FDH odel with repect to poibility and expected value (credibility approach) contraint. Finally, a nuerical exaple i preented to deontrate the efficacy of the propoed procedure and algorith. Keyword: data envelopent analyi; DEA; free dipoal hull; FDH; credibility; poibility; fuzzy et. Reference to thi paper hould be ade a follow: Shiraz, R.K., Tavana, M. and Paryab, K. (204) Fuzzy free dipoal hull odel under poibility and credibility eaure, Int. J. Data Analyi Technique and Strategie, Vol. 6, No. 3, pp Copyright 204 Indercience Enterprie Ltd.

2 Fuzzy free dipoal hull odel under poibility and credibility eaure 287 Biographical note: Rahed Khanjani Shiraz i a doctoral tudent in Applied Matheatic and Operation Reearch at the Iran Univerity of Science and Technology in Tehran, Iran. He received hi BS in Pure Matheatic fro the Univerity of Mohaghegh Ardabili in Ardabil and hi MS in Applied Matheatic and Operation Reearch fro the Iran Univerity of Science and Technology. He ha publihed everal paper in international journal including Expert Syte with Application. Hi reearch interet are in tochatic prograing, fuzzy prograing, atheatical prograing and data envelopent analyi. Madjid Tavana i a Profeor of Buine Syte and Analytic and the Lindback Ditinguihed Chair of Inforation Syte and Deciion Science at La Salle Univerity, where he erved a Chairan of the Manageent Departent and Director of the Center for Technology and Manageent. He i a Ditinguihed Reearch Fellow at Kennedy Space Center, Johnon Space Center, Naval Reearch Laboratory at Stenni Space Center, and Air Force Reearch Laboratory. He wa recently honoured with the pretigiou Space Act Award by NASA. He hold a MBA, PMIS, and PhD in Manageent Inforation Syte and received hi Pot-Doctoral Diploa in Strategic Inforation Syte fro the Wharton School at the Univerity of Pennylvania. He i the Editor-in-Chief of Deciion Analytic, International Journal of Applied Deciion Science, International Journal of Manageent and Deciion Making, International Journal of Strategic Deciion Science, and International Journal of Enterprie Inforation Syte. He ha publihed everal book and over 00 reearch paper in acadeic journal uch a Inforation Science, Deciion Science, Inforation Syte, Interface, Annal of Operation Reearch, Advance in Space Reearch, Oega, Inforation and Manageent, Expert Syte with Application, European Journal of Operational Reearch, Journal of the Operational Reearch Society, Coputer and Operation Reearch, Energy Econoic, Applied Soft Coputing, and Energy Policy. Khalil Paryab i an Aitant Profeor of Matheatic at Iran Univerity of Science and Technology in Tehran, Iran. He received hi BSc and PhD in Pure Matheatic, fro Tabriz Univerity. He received hi econd PhD in Applied Matheatic fro Iran Univerity of Science and Technology. He ha publihed everal paper and book in pure and applied atheatic. Hi reearch interet are in the application of graph theory, nuerical analyi, and atheatical prograing. Introduction Data envelopent analyi (DEA), introduced by Charne et al. (978), i a powerful atheatical ethod that utilie linear prograing (LP) to deterine the relative efficiencie of a et of functionally iilar deciion-aking unit (DMU). A DMU i conidered efficient when no other DMU can produce ore output uing an equal or leer aount of input. The DEA generalie the uual efficiency eaureent fro a ingle-input ingle-output ratio to a ultiple-input ultiple-output ratio by uing a ratio of the weighted u of output to the weighted u of input. A core of one i aigned to the frontier (efficient) unit. The frontier unit in DEA are thoe with axiu output level for given input level or with iniu input level for given output level.

3 288 R.K. Shiraz et al. The free dipoal hull (FDH) odel, introduced by Deprin et al. (984), i deigned a an alternative to DEA, where only the trong (free) dipoability of input and output i aued. It i ued to etablih a bet practice group aongt a et of oberved unit and to identify the unit that are inefficient when copared to the bet practice group. FDH relaxe the convexity auption in the variable return to cale (VRS) DEA odel. The original FDH odel propoed by Deprin et al. (984) ha been further explored by Tulken (993), Tulken and Vanden Eeckaut (995a, 995b), and Kerten and Vanden Eeckaut (999). FDH odel are traditionally baed on a VRS auption and are coputed via enueration algorith (Tulken, 993). Kerten and Vanden Eeckaut (999) introduced return to cale in the FDH technology but they did not develop any linear prograe for coputing the technical inefficiency. Agrell and Tind (200) derived a linear prograe for the FDH odel but without return to cale auption and with a radial output ditance function. Finally, Leleu (2006) provided a LP fraework to copute the technical inefficiency for FDH technology under variou return to cale auption. The conventional DEA and FDH evaluation ethod are baed on well-defined, precie and deterinitic data for the production et. However, thi auption ay not be true in the real-world ituation ince the precie eaureent of data i not poible or expenive in any application. A few reearcher have propoed variou ethod for dealing with iprecie and abiguou data in FDH. Jahanhahloo et al. (2004) conidered the efficiency analyi in the FDH odel where the aount of input and output were located within the bounded interval. They then converted the non-linear odel into a LP odel. Farnooh et al. (20) tudied FDH odel with tochatic data. However, any real-life proble ue linguitic data uch a good, fair or poor that cannot be apped to interval data. Fuzzy et theory can be ued to deal with the iprecie input and output in fuzzy DEA and FDH proble. Fuzzy et theory wa introduced by Zadeh (965) a a ean of repreenting and anipulating data that wa not precie, but rather fuzzy. It wa pecifically deigned to atheatically repreent uncertainty and vaguene and to provide foralied tool for dealing with the ipreciion intrinic to any proble. The fuzzy input and output variation in DEA have been tudied by any reearcher uch a Sengupta (992a, 992b), Trianti and Girod (998), Guo and Tanaka (200), Lertworairikul et al. (2003), León et al. (2003), Kao and Liu (2000a, 2000b, 2003, 2005), Pei-Huang (2006), Liu (2008), Liu and Chuang (2009), Zhou et al. (202), Wang and Chin (20), Majid Zerafat Angiz et al. (202), and Pendharkar (202). Sengupta (992a, 992b) wa the firt to introduce a fuzzy atheatical prograing approach in which fuzzine wa incorporated into the DEA odel by defining tolerance level on both the objective function and contraint violation. Guo and Tanaka (200) preented a fuzzy CCR odel by converting fuzzy contraint uch a fuzzy equalitie and fuzzy inequalitie into crip contraint by predefining a poibility level and uing the coparion rule for fuzzy nuber Kao and Liu (2000a, 2000b, 2003, 2005) tranfored fuzzy input and fuzzy output into interval by uing -level et and built a faily of crip DEA odel for the interval. Uing an -cut ethod propoed by Sakawa (993), Lertworairikul et al. (2003) propoed the poibility and neceity ethod for olving a fuzzy DEA-CCR odel. They introduced a poibility approach in which the contraint were treated a fuzzy event and tranfored fuzzy DEA odel into poibility DEA odel by uing

4 Fuzzy free dipoal hull odel under poibility and credibility eaure 289 poibility eaure of the fuzzy event (fuzzy contraint). The poibility theory i baed on two dual fuzzy eaure poibility and neceity eaure (Duboi and Prade, 988; Klir, 999; Zadeh, 978). Liu (2008) and Liu and Chuang (2009) developed a fuzzy DEA/aurance region odel for the election of flexible anufacturing yte and the aeent of univerity librarie, repectively. Zhou et al. (202) propoed a generalied fuzzy DEA odel with aurance region, whoe lower and upper bound at given level could be obtained. Wang and Chin (20) propoed a fuzzy expected value approach for DEA in which fuzzy input and fuzzy output were firt weighted repectively, and their expected value then ued to eaure the optiitic and peiitic efficiencie of DMU in fuzzy environent. Majid Zerafat Angiz et al. (202) introduced an alternative LP odel that included oe uncertainty inforation fro the interval within the -cut approach and propoed the concept of local a-level to develop a ulti-objective LP to eaure the efficiency of DMU under uncertainty. Pei-Huang (2006) ued ultiple criteria ranking by fuzzy DEA. León et al. (2003) developed oe fuzzy verion of the claical DEA odel (in particular, the BCC odel) by uing ranking ethod baed on the coparion of -cut. Pendharkar (202) developed a fuzzy claification yte uing DEA and illutrated it application uing a iple graduate adiion deciion-aking proble. Trianti and Girod (998) ued the approach propoed by Carlon and Korhonen (986) to forulate the fuzzy BCC and FDH odel which were radial eaure of efficiency. Although reearch on FDH with fuzzy data ha been liited, there are a nuber of recent tudie dealing with fuzzy data in the FDH literature. To the bet of our knowledge, o far, there i very little reearch in fuzzy FDH. In thi paper, the concept of chance-contrained prograing i ued to develop FDH odel with variou return to cale auption, including VRS, variable non-increaing return to cale (NIRS), variable non-decreaing return to cale (NDRS), and contant return to cale (CRS), for efficient DMU with fuzzy data. We propoe two fuzzy DEA odel with repect to the poibility and expected value (credibility) approache. The reainder of the paper i organied a follow. In the next ection, we preent oe preliinarie and definition for fuzzy et. In Section 3, we preent the baic FDH odel. In Section 4, we preent the fuzzy FDH odel. In Section 5, we preent the reult of a nuerical exaple to deontrate the efficacy of the procedure and algorith. Section 6 preent our concluion and future reearch direction. 2 Background on fuzzy et theory In thi ection, oe baic concept on fuzzy et are given a follow: Definition (Duboi and Prade, 980): Let U be a univere et involving a claical et of object. A fuzzy et A of U i defined by a eberhip function μa ( x) [0,], where μ ( x), x U, indicate the degree of eberhip of A to U. A Definition 2 (Duboi and Prade, 980): The -cut of a fuzzy et A, A, i the crip et of eleent belonging to A at leat to the degree defined a A = { x U μ ( x) }. A

5 290 R.K. Shiraz et al. Definition 3 (Zierann, 996): A fuzzy ubet A of real nuber R i convex if and only if μa ( λx + ( λ) y) in( μa ( x), μa ( y)), x, y R, λ [0, ]. Alternatively, a fuzzy et i convex if all -cut are convex. Definition 4 (Duboi and Prade, 980): A fuzzy nuber of generalied left and right type i denoted by A = (,,, ) LR where and are the (non-negative) left and right pread, repectively, and and 2 are the ean value of A. The eberhip function of A can be expreed a x L, x, μa( x) =, x x R, x. 2 where L and R are the left and right function, repectively. In particular, uppoe that: x, 0 x Lx = Rx = 0, otherwie A = (,,, ) LR = (,,, ) = (,,, + ) i then called a trapezoidal fuzzy nuber. Alo, if = 2 =, A = (,, ) = (,, ) = (,, + ) i called a triangular fuzzy nuber. Definition 5 (fuzzy arithetic) (Duboi and Prade, 980): Let A = (,,, ) LR and B = (,,, ) LR be two poitive fuzzy nuber. Then, the fuzzy arithetic of A and B can be defined a follow: Addition: (,,, ) + (,,, ) = ( +, +, +, + ) Subtraction: LR LR LR (,,, ) (,,, ) = ( +,,, + ) LR LR LR Definition 6 (Zadeh, 978; Zierann, 996): Let (Θ, P(Θ), Po) be a poibility pace where Θ i a non-epty et involving all poible potentially event, where P(Θ) i the power et of Θ. For each A P(Θ), there i a non-negative nuber Po(A), the o-called poibility eaure, with the following propertie: Po{ } =, Po{Θ} = 2 A B iplie Po(A) Po(B) for any A, B P(Θ) 3 Po{ A } = Sup Po{ A }. k k k k LR ()

6 Fuzzy free dipoal hull odel under poibility and credibility eaure 29 Definition 7(Zierann, 996; Duboi and Prade, 978, 988): The neceity eaure of A, denoted by Nec(A), i defined on (Θ, P(Θ), Po) a Nec{A} = Po{A c } where A c i the copleent et of A. For any et A and B, the propertie of the neceity eaure are preented a follow: Nec{ } = 0, Nec{Θ} = 2 Po(A) Nec(A) 3 A B iplie Nec(A) Nec(B) 4 Po(A) < Nec(A) = 0 5 Nec(A) > 0 Po(A) =. Definition 8 (Liu and Liu, 2002): Let ξ be a fuzzy variable on the poibility pace (U, P(U), Po). The poibility, neceity and credibility of a fuzzy event {ξ r} are repreented by: Po{ ξ r} = Sup μ ( t), t r Ne{ ξ r} = Sup μ ( t), ξ t< r ξ Cr( ξ r) = [ Po{ ξ r} + Nec{ ξ r} ] 2 where μ ξ : R [0, ] i the eberhip function of ξ and r i a real nuber. The credibility eaure i fored on the bai of the poibility and neceity eaure and the iplet cae i taken a their average. The credibility eaure i the elf-dual et function Cr(ξ r) = Cr(ξ < r) (ee, Liu and Liu, 2002). Definition 9 (extenion principle) (Zierann, 996): Aue that X i a Carteian product of univere X = X X r and A,, A r are r fuzzy et in X,,X r, repectively. The function f i a apping fro X to a univere Y, y = f(x,,x r ). The extenion principle enable u to introduce a fuzzy et B in Y a follow: where {(, (,, r),(,, r) } B = y μ y y = f x x x x X μ ( y) B B { up in μ x,, μ xr }, f ( y) θ A A r ( x ),, xr f = ( y) 0, ow.. where f i the invere of f. Duboi and Prade (980) odified the extenion principle by the algebraic u and product intead of up and in, repectively. 3 FDH odel The conventional FDH technology i defined under the VRS auption, that i, VRS-FDH technology i repreented by Tulken (993) a follow:

7 292 R.K. Shiraz et al. T n ( x, y): λjx xi j =,, n, j= = λjy yr, λj =, λj {0,} j= j= VRS FDH n n Tulken (993) obtained the input orientation of FDH analyi for DMUp by olving the following linear integer prograing for variou return to cale: in θ t.. n j= n j= n j= x λ x θ, i =,,, j ip y λ y, r =,,, j rp j =, λ {0,}, j =,, n. j λ Agrell and Tind (200) howed that there exit an equivalent LP proble to the traditional MILP proble to copute the FDH efficiency eaure under VRS. The following linear prograe i derived fro Agrell and Tind (200). (2) (3) in j= n j= ubject to: j x λ x θ, i =, ; j =,, n, j ip j y λ λ y, r =,, ; j =,, n, j j rp J j =, λ 0, j =,, n. j λ θ (4) Leleu (2006) introduced NIRS, NDRS and CRS pecification by reference to VRS technology a: { VRS FDH } Γ T = ( x, y ): ( x, y ) = ( + δ)( x, y), ( x, y) T, δ Γ (5) FDH where ( + δ) i a caling paraeter that i utilied to introduce variou return to cale auption. In (5), Г {VRS, NIRS, NDRS, CRS} with VRS = {δ: δ = 0}, NIRS = {δ: δ 0}, NDRS = {δ: δ 0} and CRS = {δ: δ }. The integration of return to cale auption into FDH perit the developent of four technologie. Leleu (2006) extended the etiation of the FDH uing the following LP odel with NIRS, NDRS and CRS technologie.

8 Fuzzy free dipoal hull odel under poibility and credibility eaure 293 where in j= n j= ubject to: j x λ + ω x θ i =,, ; j =,, n, j j ip j y λ + ω λ y r =,, ; j =,, n, j j j rp J j =, λ 0, j =,, n. j λ θ ω Γ, Γ { VRS, NIRS, NDRS, CRS} j j j { j : j 0 }, { j : j 0} = { j : j 0 }, = { j : j } VRS = ω ω = NIRS = ω ω NDRS ω ω CRS ω ω uncontrained (6) (7) In (7), the variable ω j i the caling factor for each DMU. Thi for i appealing becaue it ha the dual forulation and the hadow profit interpretation, a noted by Leleu (2006). The dual forulation of the FDH odel with variou return to cale auption a preented by Leleu (2006) enhance the econoic interpretation of the FDH technology in ter of hadow price. The dual of the above odel a preented by Leleu (2006) i a follow: ax π t.. and vx =, j=,, n, ip rp u y y v x + π 0, j =,, n, u, v 0, r =,, ; i =,, ; j =,, n. (8) u y v x 0, j =,, n; under NIRS, u y v x 0, j =,, n; under NDRS, u y v x = 0, j =,, n; under CRS, u y v x, j =,, n; uncontrained under VRS. (9)

9 294 R.K. Shiraz et al. where u (r =,,; j =,,n) and v (i =,,; j =,,n) are the weight aigned to the r th output and the i th input fro the j th DMU, repectively. Notice that the objective function of odel (4), (6) and (8) repreent the bet relative efficiency and the DMU * with θ p = (π * = ), are called the technically input efficient, and thoe unit with * θp (π * ) are called technically input (output)-inefficient. 4 Fuzzy FDH odel In thi ection, we develop an iprecie FDH-baed forulation for dealing with the fuzzy paraeter on a poibility pace (Θ, P(Θ), Po) through efficiency eaureent. Let u conider n DMU, indexed by j =,,n, where each of the DMU conue different fuzzy input, indexed by x (i =,,), to ecure different fuzzy output indexed by y (r =,,). The following fuzzy FDH odel reult fro conideration of the fuzzy input and output for DMU p : ax π t.. vx =, j=,, n, ip rp u y y v x + π 0, j =,, n, u, v 0, r =,, ; i =,, ; j =,, n. And the following hold for variou return to cale auption: u y v x 0, j =,, n; under NIRS, u y v x 0, j =,, n; under NDRS, u y v x = 0, j =,, n; under CRS. We contruct the following generic FDH odel, called the poibility contrained prograing odel a follow: ax π t.. Po v xip = δ, j =,, n, () Po u ( y yrp ) v x + π 0 δ, j =,, n, u, v 0, r =,, ; i =,, ; j =,, n. (0)

10 Fuzzy free dipoal hull odel under poibility and credibility eaure 295 And for variou return to cale auption we have the following: Po uy vx 0 δ, j =,, n; under NIRS, Po uy vx 0 δ, j =,, n; under NDRS, Po uy vx = 0 δ, j =,, n; under CRS. where δ i a pre-pecified acceptable level of poibility, and varie between [0, ]. Thee paraeter, aued known a priori, are alo called the threhold (or apiration) level. Let u aue that the input and output, x = ( x, x, x, x ), and (,,, ) LR y = y y y y are characteried by the left and right trapezoidal fuzzy nuber. Notice that in odel (0) the extenion principle (ee Definition 9) enable u to generalie the eberhip function of u ( y yrp ) and vx a follow: LR and μ μ v x u y vx t L t v x vx () t = t v x vx R t v x,. u ( y yrp ) t L t u y y ( + rp ) () t = t u ( y yrp ) u y y 2 ( rp ) R t u y y u ( y + yrp ) 2 ( rp )., (2) (3) Therefore, vx i and rp u ( y y ) can be denoted a:

11 296 R.K. Shiraz et al. and vx i, vx i, vx i, vx i LR u ( y + yrp ), u ( y yrp ), u ( y yrp ), u ( y + yrp ), LR repectively. In order to olve the poibility contrained prograing odel (), we convert the contraint in thi odel into their repective crip equivalent. We ue Theore to olve PCCP odel (). 4. The fuzzy FDH odel with poibility eaure Theore (Sakawa, 993): Let λ and λ 2 be two independent fuzzy nuber with continuou eberhip function. For a given confidence level [0, ], { } R L Po λ λ if and only if λ λ, 2, 2, L R where λ,, λ, and L R λ2,, λ2, are the left and the right ide extree point of the -level et [, L R λ, λ, ] and [ L R λ2,, λ2, ] of λ and λ 2, repectively, and Po{ λ λ 2 } ean that the degree of poibility λ i greater than or equal to λ 2. Conider the lat et of contraint in odel () for the deterinitic equivalent. Uing the firt part of Theore, thee contraint can be written a follow: Po u y y v x + π 0 δ ( rp ) L R ( rp ) δ δ u y yrp R δ y yrp u y y v x + π 0 ( ( )) + v x + R ( δ) x + π 0. Siilarly, Theore can be applied to the reaining contraint and ultiately odel () i tranfored into odel (4). ax π t.. v x + R ( δ) x, j =,, n, ip ip v x L ( δ) x, j =,, n, (4) ip ip

12 Fuzzy free dipoal hull odel under poibility and credibility eaure 297 (( ) ( + )) rp rp v x + R ( δ) x + π 0, j =,, n, u, v 0, r =,, ; i =,, ; j =,, n. u y y L δ y y and ( ) ( ) ( ) ( ) ur j u y L ( δ) y v x + R ( δ) x 0, j =,, n,under NDRS u y + R ( δ) y v x L ( δ) x 0, j =,, n,under NIRS ( ) ( ) u y L ( δ) y v x + R ( δ) x 0; j =,, n, under CRS y + R ( δ) y v x L ( δ) x 0; j =,, n. We preent the following definition to define the efficiency of a DMU: Definition 0: A DMU i called poibiliticly δ-efficient if the objective function of odel (4), π *, i greater than or equal to one at the poibility level δ; otherwie, it i called poibiliticly δ-inefficient. Theore: The h-poibility efficiency core i a non-increaing function of the poibility level h. Proof: Since L(x) and R(x) are continuou non-increaing function, then we have a follow: R (δ 2 ) R (δ ), L (δ 2 ) L (δ ) Then we ut have a follow: ( v xip + R δ2 xip ) v ( xip + R ( δ) xip ), j =,, n ( 2 ) ( v xip R δ xip v xip R δ xip ), j =,, n ( ( 2)( )) ( ( 2) ) rp rp (( ) ( )( )) rp rp u y y R δ y + y v x + R δ x + π ( ) u y y R δ y + y v x + R ( δ ) x + π 0, j =,, n Alo, iilar to the above contraint, the following can be verified for CRS, NIRS and NDRS technologie:

13 298 R.K. Shiraz et al. ( ) ( ) u y R ( δ) y v x + R ( δ) x 0, j =,, n, under NDRS ( ) ( ) u y + R ( δ) y v x R ( δ) x 0, j =,, n, under NIRS ( ) ( ) u y R ( δ) y v x + R ( δ) x 0; j =,, n, 2 ur ( j + ) ( ) = y R ( δ) y v x R ( δ) x 0; j,, n. Then, fro the above we have a follow: and v ( xip + R ( δ2 ) xip ), j =,, n, ( ( 2 ) v xip L δ xip ), j =,, n, ( ( 2 )( + )) v ( x + R ( δ2 ) x ) + π 0, j =,, n u y y R δ y y rp rp under CRS. 2 ( u y R δ2 y ) v x + R ( δ2) x 0, j =,, n, under NDRS 2 ( u y + R δ2 y ) v x R ( δ2) x 0, j =,, n, under NIRS 2 ( u y R δ2 y ) v x + R ( δ2) x 0; j =,, n, 2 u ( y 2 ) ( 2) + R δ y v x R δ x 0; j =,, n, under CRS Let ( u, v ) be optial olution of odel (4) for poibility level δ. Baed upon the above dicuion, ( u, v ) i a feaible olution of (4) for any poibility level δ 2 uch that δ δ 2. Conequently we ut have π π. 2 δ δ

14 Fuzzy free dipoal hull odel under poibility and credibility eaure 299 Propoition : If DMU p i poibiliticδ-efficient under poibility level δ =, then for all threhold level δ uch that δ < the DMU p i efficient. Proof: Thi can obviouly be obtained fro Theore 2. Propoition 2: If DMU p under threhold level δ = 0 i inefficient, then for all poibility level δ uch that δ > 0 the DMU p i inefficient. Proof: Thi can obviouly be obtained fro Theore The fuzzy FDH odel with credibility eaure In thi ection, we introduce the credibility approach to olve the fuzzy FDH odel and deduce it equivalent crip odel. In thi cae, the expected value operator i ued to convert the chance contraint into deterinitic contraint. Siilar to the expected value operator for a rando variable in probability theory, the expected value operator for a fuzzy variable uing the credibility eaure i defined by Liu (2002) a follow: Definition (Liu, 2002): Let ξ be a fuzzy variable. Then the expected value of ξ i + 0 defined by E( ξ) = Cr{ ξ rdr } Crξ { rdr } 0 provided that at leat one of the two integral i finite. Theore 3: Let λ = (,,, ) LR be a left and right fuzzy variable where and 2 are the ean value and and are the (non-negative) left and right pread. Then the expected value of fuzzy variable λ i a follow: E[ λ ] = ( ) L() t dt R() t dt Proof: Baed on the definition of the credibility eaure we have the following relationhip: { } ( { } { }) Cr λ r = Po λ r + Nec λ r 2 ( ) Cr λ r, r r L, r 2 =, r 2 r R, r + 2 0, r > + (5)

15 300 R.K. Shiraz et al. ( ) Cr λ r 0, r r L, r 2 =, r 2 r R, r 2 +, r > + Obviouly, it follow fro (6) that { } { } E( λ ) = Cr λ r dr Cr λ r dr 0 Cr{ λ rdr } = 0 and accordingly: 2 2 r dr + r = dr + L dr R dr (6) By aking the following variable change, r r = t, = t we can obtain: + = E( λ) Ltdt () Rtdt () Propoition 3: In particular, let A = ( abcd,,, ) and B = ( abc,, ) be trapezoidal and triangular fuzzy variable, repectively, then: E[ A ] = ( a+ b+ c+ d)and E[ B ] = ( a+ 2 b+ c). 4 4 Proof: Let A = ( abcd,,, ). For a trapezoidal fuzzy nuber we have R(x) = L(x) = x, and b a =, d c =. By uing Theore 3, we have following: E( A + ) = () () 2 2 L t dt+ 0 2 R t dt = ( tdt ) ( tdt ) = = ( 2 ( + ) + ) = ( a+ b+ c+ d). 4 4 The credibility approach ue the expected value of fuzzy variable to tranfor the fuzzy FDH odel into the credibility prograing FDH odel. Siilar to the expected value approach in tochatic prograing where rando variable are replaced by their expected value, in the credibility prograing FDH (CP-FDH) odel fuzzy variable are replaced by expected value, which are derived by uing credibility eaure. In thi way, the fuzzy FDH (FFDH) odel i tranfored into the following credibility prograing-fdh odel:

16 Fuzzy free dipoal hull odel under poibility and credibility eaure 30 ax π t.. E v x ip = E[], j =,, n, E u ( y y rp ) v x + π 0, j =,, n, u, v 0, r =,, ; i =,, ; j =,, n. (7) and E u y v x 0, j =,, n; under NIRS, E u y v x 0, j =,, n; under NDRS, E u y v x = 0, j =,, n; under CRS. Now, we ubtitute the following expected value of fuzzy variable to obtain a deterinitic FDH odel with variou return to cale auption: and ( y + y y y ) ( ) ( ) rp rp E u ( y yrp ) u = r, 2 y + yrp L() t dt+ y + yrp R() t dt 0 0 = ( ip ip ) ip ip E v x v x x x L() t dt x R() t dt. By applying the following expected value of the Fuzzy variable, we can contruct the following deterinitic LP odel to evaluate the efficiency of each DMU in an uncertain environent: ax π t.. ( ip ip ) ip ip v x + x x L() t dt+ x R() t dt = 2, j =,, n, 0 0 ( + rp rp ) ( + rp ) + ( + rp ) u y y y y y y L() t dt y y R() t dt 0 0 ( + ) = = = v x x x L() t dt x R() t dt + 2 π 0, j =,, n, u, v 0, r,, ; i,, ; j,, n. (8)

17 302 R.K. Shiraz et al. For NIRS, NDRS, and CRS we have repectively: ( + ) + ( ) ( + ) + u y y y L() t dt y R() t dt 0 0 v x + x x L() t dt+ x R() t dt 0, j =,, n, undernirs 0 0 u y y y L() t dt y R() t dt 0 0 v ( x x ) x ( + ) + u y y y L() t dt y R() t dt 0 0 ( ) Ltdt () + x Rtdt () 0, j=,, n, underndrs v x + x x L() t dt+ x R() t dt = 0, j =,, n, undercrs. 0 0 We hould note that in the above odel, the DMU p under aeent i aid to be efficient if the correponding optial olution i equal to unity; otherwie, DMU p i inefficient. 5 Nuerical exaple In thi ection, we ue a hypothetical exaple to exaine the applicability of the propoed odel. Conider five DMU with two fuzzy triangular input and two fuzzy triangular output a reported in Table. Thi data i denoted by (,, ) where i the centre value and and are the left and right tail, repectively. Table Fuzzy input and fuzzy output DMU Input Input 2 Output Output 2 (0.2, 4, 0.5) (0.2, 5., 0.2) (0.2, 2.6, 0.2) (0.3, 4., 0.3) 2 (0., 5.9, 0.) (0., 5.5, 0.) (0., 2.2, 0.) (0.2, 3.5, 0.2) 3 (0.2, 4.9, 0.5) (0., 2.6, 0.4) (0.2, 3.2, 0.5) (0.5, 5., 0.8) 4 (0.4, 8., 0.7) (0., 5.3, 0.) (0., 4.9, 0.4) (0.2, 5.7, 0.2) 5 (0.3, 6.5, 0.6) (0.2, 4., 0.5) (0.4, 6., 0.7) (0.6, 7.4, 0.9) Six different poibility (threhold) level of δ = 0, δ = 0.2, δ = 0.5, δ = 0.7, δ = 0.9 and δ = are conidered to copare the reult fro the FDH odel with variou return to cale auption, including CRS, NIRS, NDRS, and VRS. The coputational reult of the deterinitic equivalent of the efficiency odel (), for δ = 0, δ = 0.2, δ = 0.5, δ = 0.7, δ = 0.9 and δ = are preented in Table 2, 3, 4 and 5 for the CRS, NIRS, NDRS, and VRS cae, repectively.

18 Fuzzy free dipoal hull odel under poibility and credibility eaure 303 Table 2 FDH odel reult under the CRS auption DMU δ = 0 δ = 0.2 δ = 0.5 δ = 0.7 δ = 0.9 δ =.0 Credibility approach (CRS) Table 3 FDH odel reult under the NIRS auption DMU δ = 0 δ = 0.2 δ = 0.5 δ = 0.7 δ = 0.9 δ =.0 Credibility approach (NIRS) Table 4 FDH odel reult under the NDRS auption DMU δ = 0 δ = 0.2 δ = 0.5 δ = 0.7 δ = 0.9 δ =.0 Credibility approach (NDRS) Table 5 FDH odel reult under the VRS auption DMU δ = 0 δ = 0.2 δ = 0.5 δ = 0.7 δ = 0.9 δ =.0 Credibility approach (VRS) A hown in thee table, the DMU have a higher efficiency core under δ = 0 copared with other probability level. We have confired Propoition 2. In addition, under all given poibility level, DMU 3 and DMU 5 perfor better than DMU, 2, and 4 according to the fuzzy poibility FDH odel. Furtherore, DMU 2 and DMU 4 are inefficient under all given poibility level. Thi reult how that the efficiency i non-increaing (non-decreaing) in poibility level δ and therefore we have confired Theore 2.

19 304 R.K. Shiraz et al. We apply the fuzzy expected value odel (7) to calculate the efficiency eaure of the DMU a reported in the lat colun of Table 2, Table 3, Table 4, and Table 5, for the CRS, NIRS, NDRS and VRS cae, repectively. According to Table, DMU 5 in the CRS and NIRS cae, and DMU, 3 and 5 in the NDRS and VRS cae are identified a the efficient unit with a unity core. In uary, the propoed credibility approach (fuzzy expected value) odel i a rather traightforward forulation for eauring the efficiency of a group of DMU a well a providing adequate dicriinatory power in the preence of the fuzzy input and fuzzy output. 6 Concluion and future reearch direction Fuzzy DEA i a tool for coparing the perforance of a et of activitie or organiation in uncertain environent. The conventional FDH i deterinitic and aue that the input and the output are eaured preciely. However, the oberved value of the input and output data in real-world proble can potentially be fuzzy in nature. In thi paper, the concept of chance-contrained prograing wa ued to develop FDH odel with variou return to cale auption, including VRS, NIRS, NDRS, and CRS, for DMU with fuzzy data. We propoed two fuzzy FDH odel with repect to the poibility and the expected value (credibility approach) contraint. Finally, we ued a nuerical exaple to how the feaibility and the richne of the obtained olution, ince only real application can reveal the true value of the fraework. We plan to apply the propoed odel to a real-life elaborate cae tudy in the near future. Acknowledgeent The author would like to thank the anonyou reviewer and the editor for their inightful coent and uggetion. Reference Agrell, P.J. and Tind, J. (200) A dual approach to nonconvex frontier odel, Journal of Productivity Analyi, Vol. 6, No. 2, pp Carlon, C. and Korhonen, P. (986) A paraetric approach to fuzzy linear prograing, Fuzzy Set and Syte, Vol. 20, No., pp Charne, A., Cooper, W.W. and Rhode, E.L. (978) Meauring the efficiency of deciion aking unit, European Journal of Operational Reearch, Vol. 2, No. 6, pp Deprin, D., Siar, L. and Tulken, H. (984) Meauring labor efficiency in pot office, in March, M., Petieu, P. and Tulken, H. (Ed.): The Perforance of Public Enterprie: Concept and Meaureent, pp , Elevier, North Holland, Aterda. Duboi, D. and Prade, H. (978) Operation on fuzzy nuber, International Journal of Syte Science, Vol. 9, No. 6, pp Duboi, D. and Prade, H. (980) Fuzzy Set and Fuzzy Logic: Theory and Application, Acadeic Pre, New York. Duboi, D. and Prade, H. (988) Poibility Theory, Plenu, New York.

20 Fuzzy free dipoal hull odel under poibility and credibility eaure 305 Farnooh, R., Khanjani, R. and Chaji, A. (20) Stochatic FDH odel with variou return to cale auption in data envelopent analyi, Journal of Advanced Reearch in Applied Matheatic, Vol. 3, No., pp Guo, P. and Tanaka, H. (200) Fuzzy DEA: a perceptual evaluation ethod, Fuzzy Set and Syte, Vol. 9, No., pp Jahanhahloo, G.R., Kazei Matin, R. and Vencheh, A.H. (2004) On FDH effciency analyi with interval data, Applied Matheatic and Coputation, Vol. 59, No., pp Kao, C. and Liu, S.T. (2000a) Fuzzy efficiency eaure in data envelopent analyi, Fuzzy Set and Syte, Vol. 3, No. 3, pp Kao, C. and Liu, S.T. (2000b) Data envelopent analyi with iing data: an application to univerity librarie in Taiwan, Journal of the Operational Reearch Society, Vol. 5, No. 8, pp Kao, C. and Liu, S.T. (2003) A atheatical prograing approach to fuzzy efficiency ranking, International Journal of Production Econoic, Vol. 86, No. 2, pp Kao, C. and Liu, S.T. (2005) Data envelopent analyi with iprecie data: an application of Taiwan achinery fir, International Journal of Uncertainty, Fuzzine and Knowledge Baed Syte, Vol. 3, No. 2, pp Kerten, K. and Vanden Eeckaut, P. (999) Etiating return to cale uing non-paraetric deterinitic technologie: a new ethod baed on goodne-of-fit, European Journal of Operational Reearch, Vol. 3, No., pp Klir, G.J. (999) On fuzzy-et interpretation of poibility theory, Fuzzy Set and Syte, Vol. 08, No. 3, pp Leleu, H. (2006) A linear prograing fraework for free dipoal hull technologie and cot function: prial and dull odel, European Journal of Operational Reearch, Vol. 68, No. 2, pp León, T., Liern, V., Ruiz, J.L. and Sirvent, I. (2003) A fuzzy atheatical prograing approach to the aeent of efficiency with DEA odel, Fuzzy Set and Syte, Vol. 39, No. 2, pp Lertworairikul, S., Shu-Cherng, F., Joine, J.A. and Nuttle, H.L.W. (2003) Fuzzy data envelopent analyi (DEA): a poibility approach, Fuzzy Set and Syte, Vol. 39, No. 2, pp Liu, S.T. (2008) A fuzzy DEA/AR approach to the election of flexible anufacturing yte, Coputer & Indutrial Engineering, Vol. 54, No., pp Liu, S.T. and Chuang, M. (2009) Fuzzy efficiency eaure in fuzzy DEA/AR with application to univerity librarie, Expert Syte with Application, Vol. 36, No. 2P, pp Liu, Y.K. and Liu, B. (2002) Expected value of fuzzy variable and fuzzy expected value odel, IEEE Tranaction on Fuzzy Syte, Vol. 0, No. 4, pp Majid Zerafat Angiz, L., Erouznejad, A. and Mutafa, A. (202) Fuzzy data envelopent analyi: a dicrete approach, Expert Syte with Application, Vol. 39, No. 3, pp Pei-Huang, L. (2006) Multiple criteria ranking by fuzzy data envelopent analyi, WSEAS Tranaction on Coputer, Vol. 5, No. 5, pp Pendharkar, P. (202) Fuzzy claification uing the data envelopent analyi, Knowledge-baed Syte, July, Vol. 3, pp Sakawa, M. (993) Fuzzy Set and Interactive Multiobjective Optiization, Plenu Pre, New York. Sengupta, J.K. (992a). A fuzzy yte approach in data envelopent analyi, Coputer and Matheatic with Application, Vol. 24, No. 8, pp Sengupta, J.K. (992b) Meauring efficiency by a fuzzy tatitical approach, Fuzzy Set and Syte, Vol. 46, No., pp

21 306 R.K. Shiraz et al. Trianti, K. and Girod, O.A. (998) A atheatical prograing approach for eauring technical efficiency in a fuzzy environent, Journal of Productivity Analyi, Vol. 0, No., pp Tulken, H. (993) On FDH efficiency: oe ethodological iue and application to retail banking, court, and urban tranit, Journal of Productivity Analyi, Vol. 4, No., pp Tulken, H. and Vanden Eeckaut, P. (995a) Non-paraetric efficiency, progre and regre eaure for panel data: ethodological apect, European Journal of Operation Reearch, Vol. 80, No. 3, pp Tulken, H. and Vanden Eeckaut, P. (995b) Non-frontier eaure of effciency, progre and regre for tie erie data, International Journal of Production Econoic, Vol. 39, No. 2 3, pp Wang, Y.M. and Chin, K.S. (20) Fuzzy data envelopent analyi: a fuzzy expected value approach, Expert Syte with Application, Vol. 38, No. 9, pp Zadeh, L.A. (965) Fuzzy et, Inforation and Control, Vol. 8, No. 3, pp Zadeh, L.A. (978) Fuzzy et a a bai for a theory of poibility, Fuzzy Set and Syte, Vol., No., pp Zhou, Z., Lui, S., Ma, C., Liu, D. and Liu, W. (202) Fuzzy data envelopent analyi odel with aurance region: a note, Expert Syte with Application, Vol. 39, No. 2, pp Zierann, H.J. (996) Fuzzy Set Theory and It Application, 2nd ed., Kluwer Acadeic Publiher, Dordrecht, The Netherland.