DEA models with production trade-offs and weight restrictions

Transcription

1 Lughbrugh University Institutinal Repsitry DEA mdels with prductin trade-ffs and weight restrictins This item was submitted t Lughbrugh University's Institutinal Repsitry by the/an authr. Citatin: PODINOVSKI, V.V., 205. DEA mdels with prductin tradeffs and weight restrictins. IN: Zhu, J. (ed.). Data Envelpment Analysis: A Handbk f Mdels and Methds. New Yrk, U.S.A.: Springer US, pp Additinal Infrmatin: The final publicatin is available at Springer via Metadata Recrd: Versin: Accepted fr publicatin Publisher: c Springer Rights: This wrk is made available accrding t the cnditins f the Creative Cmmns Attributin-NnCmmercial-NDerivatives 4.0 Internatinal (CC BY-NC-ND 4.0) licence. Full details f this licence are available at: Please cite the published versin.

2 DEA Mdels with Prductin Trade-ffs and Weight Restrictins Victr V. Pdinvski Warwick Business Schl, University f Warwick, Cventry CV4 7AL, United Kingdm v.pdinvski@warwick.ac.uk Abstract There is a large literature n the use f weight restrictins in multiplier DEA mdels. In this chapter we prvide an alternative view f this subject frm the perspective f dual envelpment DEA mdels in which weight restrictins can be interpreted as prductin trade-ffs. The ntin f prductin trade-ffs allws us t state assumptins that certain simultaneus changes t the inputs and utputs are technlgically pssible in the prductin prcess. The incrpratin f prductin trade-ffs in the envelpment DEA mdel, r the crrespnding weight restrictins in the multiplier mdel, leads t a meaningful expansin f the mdel f prductin technlgy. The efficiency measures in DEA mdels with prductin trade-ffs retain their traditinal meaning as the ultimate and technlgically realistic imprvement factrs. This vercmes ne f the knwn drawbacks f weight restrictins assessed using ther methds. In this chapter we discuss the assessment f prductin trade-ffs, prvide the crrespnding theretical develpments and suggest cmputatinal methds suitable fr the slutin f the resulting DEA mdels. Keywrds: Data envelpment analysis, prductin trade-ffs, weight restrictins

3 Intrductin The cnventinal variable and cnstant returns-t-scale (VRS and CRS) DEA mdels can each be stated as tw mutually dual linear prgrams: as an envelpment r multiplier mdel (Charnes et al. 978; Banker et al. 984). The envelpment mdel is based n an explicit representatin f the prductin technlgy. The efficiency f decisin making units (DMUs) in this mdel is btained by their input r utput radial prjectin n the bundary f the technlgy. The dual multiplier mdel is stated in terms f variable vectrs f input and utput weights. This mdel assesses the efficiency f DMUs in terms f the rati f their aggregated weighted utputs t aggregated weighted inputs, in relatin t similar ratis calculated fr all bserved DMUs. One cmmn mdificatin f the multiplier mdel is based n the use f weight restrictins the incrpratin amng its cnstraints f additinal inequalities n the input and utput weights (Thanassulis et al. 2008; Cper et al. 20a). Weight restrictins are attractive because f their apparent managerial meaning and als because their use can significantly imprve the efficiency discriminatin f DEA mdels (Allen et al. 997; Thanassulis et al. 2004). A well-knwn drawback f weight restrictins arises frm the fact that their use in the multiplier mdel implicitly changes the mdel f prductin technlgy in the envelpment frm (Allen et al. 997). Specifically, weight restrictins enlarge the mdel f technlgy and generally shift the efficient frntier t a mre demanding level, as illustrated by Rll et al. (99). An bvius prblem with this is that the efficient prjectins f inefficient DMUs lcated n the expanded frntier may nt be prducible (technlgically realistic). Furthermre, the traditinal meaning if efficiency as the ultimate and technlgically feasible imprvement factr generally becmes unsubstantiated (Pdinvski 2004a; Førsund 203). The purpse f this chapter is t describe an apprach t the cnstructin f weight restrictins that by definitin des nt have the abve drawback. The idea is t cnsider the dual frms f weight restrictins induced in the envelpment mdels. These are additinal terms that are simultaneusly added t, r subtracted frm, the inputs and utputs f the units in the prductin technlgy. Fllwing Pdinvski (2004a), we refer t these terms as prductin trade-ffs. Weights restrictins and prductin trade-ffs are mathematically equivalent. Frm the practical pint f view they may, hwever, be regarded as different tls. While the terminlgy f weight restrictins is a natural language fr the elicitatin and cmmunicatin 2

4 f value judgements, the ntin f prductin trade-ffs makes us think in terms f prductin technlgy and pssible substitutins between its inputs and utputs. Prductin trade-ffs d nt generally fllw frm the data instead, they are additinal assumptins that we (r experts) are willing t make abut the prductin technlgy: that a certain simultaneus change (substitutin) f inputs and utputs is technlgically pssible, at all units. In this respect prductin trade-ffs shuld nt be cnfused with marginal rates f transfrmatin and substitutin between the inputs and utputs. The latter represent the slpes f the supprting hyperplanes t the technlgy and are generally different at different bundary units. Changing the inputs and utputs f a bundary unit in the prprtins based n the marginal rates (calculated at this unit) wuld keep the resulting unit n the supprting hyperplane this des nt mean that the resulting unit is prducible. In ther wrds, marginal rates represent the mvements (changes t inputs and utputs) that are tangent t the technlgy and are nt suppsed t result in prducible units. In cntrast, prductin tradeffs represent mvements that are nt necessarily tangent t the bundary f the true technlgy (that we are attempting t mdel), but are assumed t keep the resulting units technlgically pssible. The use f prductin trade-ffs fr the cnstructin f weight restrictins has been illustrated in different cntexts. These include the assessment f efficiency f university departments (Pdinvski 2007a), secndary schls (Khalili et al. 200), primary health care prviders (Amad and Sants 2009), primary diabetes care prviders (Amad and Dysn 2009), electricity distributrs (Sants et al. 20) and agricultural farms (Atici 202). The fllwing are a few examples f prductin trade-ffs emplyed in the abve studies.. Primary health care prvisin: the hspital utputs shuld nt deterirate if the number f nurses is reduced by and the number f dctrs is increased by (Amad and Sants 2009). This crrespnds t the weight restrictin stating that the weight attached t the number f dctrs is at least as large as the weight attached t the number f nurses. 2. Electricity distributin: a distributin utility shuld be able t increase the delivery f electricity by at least 40 KWh per Eur f increase f perating expenses the latter is chsen as a representative measure fr all distributin csts (Sants et al. 20). This implies that the weight attached t perating expenses (in Eurs) is greater than r equal t 40 times the weight attached t the number f KWh delivered. 3. Agricultural farms: the resurces required fr the prductin f tnne f wheat are sufficient fr the prductin f at least 0.75 tnnes f barley, at any farm in the given 3

5 regin (Atici 202). This implies that the weight attached t wheat is greater than r equal t 0.75 times the weight attached t barley. Prductin trade-ffs have exactly the same effect n the mdel f technlgy as weight restrictins: the technlgy expands but, in cntrast with the latter case, in a cntrlled way that we explicitly assume t be technlgically pssible. Because the expanded technlgy and, therefre, its efficient frntier are realistic in the prductin sense, this further implies that the radial targets are prducible and the efficiency measures retain their cnventinal technlgical meaning as pssible imprvement factrs. The use f prductin trade-ffs vercmes the knwn drawbacks f weight restrictins nt because they are different: as nted, bth are equivalent cncepts. The advantage f trade-ffs is that their assessment explicitly refers t the technlgy and requires ur judgement t be stated in the language f pssible changes t inputs and utputs. The assessed trade-ffs can be incrprated either in the envelpment mdel (which currently requires the use f a general linear ptimiser), r as equivalent weight restrictins in the multiplier mdel (which can be perfrmed in mst current DEA slvers). In the latter case, the weight restrictins d nt have the abve knwn general drawbacks because they are cnstructed by transfrmatin f prductin trade-ffs. We call this methd the trade-ff apprach t the cnstructin f weight restrictins. It is wrth mentining that several earlier studies came clse t the ntin f prductin trade-ffs. Charnes et al. (989), Rll et al. (99) and Halme and Krhnen (2000) shw that the incrpratin f weight restrictins in multiplier mdels induce dual terms that change the technlgy but d nt explre this relatin as a basis fr the assessment f weight restrictins that have a prductin meaning. The assessment f weight restrictins in sme earlier applicatins f DEA can als be viewed as being implicitly based n (r cnsistent with) the idea f prductin trade-ffs. Dysn and Thanassulis (988) cnsider a DEA mdel with a single input. In this study the lwer bund n each utput weight is related t the minimum amunt f the input required per unit f the utput. This is essentially a statement f a prductin trade-ff, althugh in a specific DEA mdel that cannt be easily generalised t the case f multiple utputs. In the assessment f bank branch perfrmance, Schaffnit et al. (997) and Ck and Zhu (2008) incrprate limits n the ratis f the weights f different transactin and maintenance activities. Such limits are based n the lwer and upper bunds n the amunts f time that such activities require and effectively express prductin trade-ffs between the activities. 4

6 2 Prductin Trade-ffs Fllwing Pdinvski (2004a), in this sectin we intrduce prductin trade-ffs as the dual frms f weight restrictins. It is als straightfrward t intrduce prductin trade-ffs independently and establish their dual relatinship t weight restrictins afterwards. We prefer the frmer apprach because it builds up n the already well-established cncept f weight restrictins in the DEA literature. Cnsider technlgy with m inputs and s utputs. The m s + + elements f are DMUs stated as the pairs ( XY, ), where X and Y are the input and utput vectrs, respectively. Let J = {,..., n} be the set f bserved DMUs. Each bserved DMU can als be stated as ( X, Y ), where j J. Dente ( Xο, Yο ) the unit in whse efficiency is being assessed. j j In rder fr the DEA mdels t be well-defined and avid the cnsideratin f special cases, we make the fllwing standard data assumptin: at least ne input and ne utput f each bserved DMU is strictly psitive. We als assume that every utput r =,..., s is strictly psitive fr at least ne bserved DMU j psitive fr at least ne bserved DMU j 2 Let m v + and J. J, and every input i =,..., m is strictly s u + be, respectively, the vectrs f input and utput weights used in the multiplier DEA mdels. Cnsider the fllwing K hmgeneus weight restrictins: u Q v P 0, t =,..., K, () t t where m Pt and s Qt are sme cnstant vectrs, fr all t, and symbl dentes transpsitin. Cmpnents f vectrs P t and Q t can be psitive, negative r zer. If, fr sme t, bth Pt 0 and Qt 0, the crrespnding weight restrictin t in () is called linked and is ften referred t as Assurance Regin II (Thmpsn et al. 990). Otherwise, the weight restrictin is nt linked and is termed Assurance Regin I, r plyhedral cne rati (Charnes et al. 989, 990). Suppse we wish t assess the efficiency f sme DMU ( Xο, Yο ) using the multiplier mdel with weight restrictins (). T be specific, we cnsider the case f CRS first, and cmment n the case f VRS afterwards. 5

7 The input radial efficiency f DMU ( Xο, Yο ) is btained as the ptimal value θ in the fllwing multiplier mdel that incrprates weight restrictins (): Mdel CRS : θ = max uy, (2) subject t v X =, uy j u Q t vx v P t j 0, j =,..., n, 0, t =,..., K, uv, 0. (In mdel (2) and belw, the vectr inequalities and mean that the crrespnding inequality is true fr each cmpnent.) Nte that, althugh mdel (2) maximises the aggregated utput uy f DMU ( Xο, Yο ), its dual envelpment frm (4) presented belw prjects the latter unit n the bundary f the technlgy by the radial cntractin f the input vectr X. This explains why mdel (2) and its dual are cnventinally referred t as input-minimisatin, r inputriented, mdels (Cper et al. 20b). Similarly, the utput radial efficiency f DMU ( Xο, Yο ) is equal t the inverse /η f the ptimal value η in the fllwing utput-maximisatin (r utput-riented) multiplier mdel: Mdel 2 CRS : η = min v X, (3) subject t uy=, uy j u Q t vx v P t j 0, j =,..., n, 0, t =,..., K, uv, 0. The ntin f prductin trade-ffs and their relatin t weight restrictins becmes apparent when we cnsider the envelpment mdels dual t (2) and (3). Using vectrs = (,..., ) and π = ( π,..., π K ), the dual t (2) can be stated as fllws: λ λ λ n 6

8 Mdel CRS : θ = min θ, (4) subject t n K λ Y + π Q Y ο, j j t t j= t= Mdel n K λ X + π P θx ο, j j t t j= t= λ 0, π 0, θ sign free. Similarly, the dual t (3) is the envelpment mdel 2 CRS : η = max η, (5) subject t n K λ Y + π Q ηy ο, j j t t j= t= n K λ X + π P X ο, j j t t j= t= λ 0, π 0, η sign free. In bth envelpment mdels (4) and (5) the first grup f terms n the left-hand side f their cnstraints defines a cmpsite unit ( Xλ, Yλ ) in the cnventinal CRS technlgy. This unit is further mdified by the pairs f vectrs used in prprtins π t 0 ( PQ, ), t =,..., K, (6) t t. In particular, vectr P t represents changes t the inputs, and vectr Q shws changes t the utputs. Therefre, each pair ( PQ, ) in (6) can be referred t t as a prductin trade-ff. It is clear that fr sme weight restrictins () the crrespnding prductin trade-ffs (6) may nt represent a technlgically pssible substitutin between the inputs and utputs. In this case the unit btained n the left-hand side f mdels (4) and (5) is generally nt prducible. An bvius way t vercme this prblem is t cnstruct technlgically realistic trade-ffs (6) in the first place. The efficiency f DMU ( Xο, Yο ) can then be assessed by slving either the envelpment mdels (4) and (5) r their dual multiplier mdels (2) and (3). In the latter case, the trade-ffs (6) shuld be cnverted t weight restrictins (). The abve prcess describes the trade-ff apprach t the cnstructin f weight restrictins. Its idea is that the weight restrictins () are assessed in the dual envelpment t t 7

9 space where they take n the frm f prductin trade-ffs (6). The latter are essentially additinal prductin assumptins based n ur understanding f the technlgy. Examples illustrating the trade-ff apprach are discussed in Sectin 3 belw. In the case f VRS, the dual relatinship between weight restrictins () and prductin trade-ffs (6) is the same as abve. The VRS analgues f the CRS envelpment mdels are the prgrams (4) and (5) with the additinal nrmalising cnditin n λ j =. (7) j= Belw we dente t the resulting envelpment VRS mdels as The crrespnding dual multiplier mdels are referred t as VRS and VRS and multiplier mdels utilise an additinal sign free variable u 0 dual t equality (7). 2 VRS, respectively. 2 VRS. Bth VRS Remark. The case f nn-hmgeneus weight restrictins is cnsidered in Pdinvski (2004a; 2005). Such restrictins have a nn-zer cnstant n the right-hand side f inequalities (), an example f which is abslute weight bunds (Dysn and Thanassulis 988). Nn-hmgeneus weight restrictins can als be related t prductin trade-ffs in the envelpment mdel, but frmula (6) is n lnger valid. The exact trade-ff induced by a nn-hmgeneus weight restrictin depends n the DMU ( X, Y ) under the assessment and the rientatin (input minimisatin r utput maximisatin) f the mdel. This cmplicates the assessment f nn-hmgeneus weight restrictins and makes them less attractive in practical applicatins. A further difficulty arising in DEA mdels with nn-hmgeneus weight restrictins is that the managerial meaning f the resulting efficiency btained via the multiplier DEA mdel may be unclear. In particular, the ptimal input and utput weights in the resulting mdels d nt generally represent the assessed DMU in the best light cmpared t the ther DMUs (Pdinvski and Athanasspuls 998; Pdinvski 999; 2004b). 3 Illustrative Example Belw we cnsider an example that illustrates the use f prductin trade-ffs in the assessment f efficiency f academic departments frm different universities using a hypthetical data set. The departments are assumed t be frm the same academic area (e.g., ecnmics). The chice f inputs and utputs in this example is the same as in Pdinvski (2007a) but the data set is different. 8

10 Table shws seven hypthetical university departments dented D, D2,, D7. The tw inputs include full academic staff and research staff. The three utputs include undergraduate students, master (pstgraduate) students and academic publicatins. T be specific, we cnsider the case f utput radial efficiency. Using the tw cnventinal CRS and VRS utput-maximisatin DEA mdels, we btain the efficiency scres as shwn in the secnd left clumns in Tables 2 and 3 titled CRS and VRS, respectively. It is nt surprising that, given the small set f bserved DMUs, the efficiency discriminatin is lw: in the case f CRS nly tw departments are inefficient, and in the case f VRS nly ne is inefficient. Table 4 shws the ptimal input and utput weights btained in the standard CRS mdel. The weights u, u 2 and u 3 crrespnd t the three utputs: undergraduate students, master students and publicatins, respectively. The weights v and v 2 crrespnd t the tw inputs: academic and research staff, respectively. Althugh ptimal weights are generally nt unique, the weights in Table 4 are cnsistent with the knwn drawback f cnventinal DEA mdels: the cmplete flexibility f weights ften results in zer weights attached t sme f the inputs and utputs. These represent the areas in which the DMU under the assessment is relatively weak (Thanassulis et al. 987; Dysn and Thanassulis 988). Fr example, department D4 has a relatively lw number f students per member f staff but the highest number f publicatins per staff. This is reflected in the ptimal weights attached t these utputs: bth undergraduate and master students have a zer weight attached t them. This implies that the DEA mdel used fr the assessment f department D4 effectively ignres the first tw utputs. Exactly the same efficiency scre fr department D4 is btained if we remve the tw types f student frm mdel specificatin and assess the efficiency f D4 based n the tw inputs and publicatins nly. Let us shw that bth the CRS and VRS DEA mdels can be imprved using simple prductin trade-ffs. 3.. Undergraduate and Master Students We start by cmparing the resurces (academic staff) that are used by the departments fr the teaching f undergraduate and master students. Assumptin. The teaching f ne undergraduate student des nt require mre resurces (academic staff time) than the teaching f ne master student. 9

11 shuld accept: We can restate the abve assumptin as the fllwing trade-ff that all departments 0 P = 0, Q =. (8) 0 The meaning f the abve trade-ff is straightfrward: it is pssible t replace ne master student (the value in the secnd cmpnent f vectr Q ) by ne undergraduate student (the value in the first cmpnent f vectr Q ). Fr this replacement, n change f the inputs (resurces) is needed: vectr P is a zer vectr. There shuld als be n change t the third utput (publicatins): the third cmpnent f vectr Q is zer. Prductin trade-ff (8) can be restated as a weight restrictin using frmula (): u u2 0. (9) This inequality implies that the weight attached t master students cannt be less than the weight attached t undergraduate students. The same weight restrictin may pssibly be btained by a value judgement but it is the riginal prductin trade-ff (8) that makes this weight restrictin meaningful in the prductin sense. Assumptin 2. The teaching f a master student may require mre resurces than an undergraduate student, hwever, by n mre than a factr f 3. Nte that the abve assumptin is nt a precise measure f the relative amunt f resurces required by the tw types f utput, and it shuld nt be fr tw reasns. First, the estimates f this rati may vary depending n the methdlgy used fr its calculatin even fr ne particular department. Secnd, even if the precise rati were pssible t assess, this wuld mst likely vary between the departments. Because f these uncertainties, Assumptin 2 is suppsed t be a safe cnservative estimate (an upper bund f different pssible estimates) that all departments shuld agree n. We state Assumptin 2 as the fllwing prductin trade-ff: 3 0 P2 = 0, Q2 =. (0) 0 The abve trade-ff means that n extra resurces shuld be claimed ( P 2 is a zer vectr) and there shuld be n detriment t the publicatins if the number f undergraduate 0

12 students is reduced by 3 and the number f master students is increased by. Using frmula (), prductin trade-ff (0) is restated as the weight restrictin 3u + u 0. () 2 Accrding t (), the weight attached t master students cannt be mre than 3 times larger than the weight attached t undergraduate students. Nte that the factr 3 des nt reflect the perceived imprtance f master students cmpared t undergraduates, as bth utputs may be deemed equally imprtant fr the departments r the decisin maker wh is assessing their efficiency. The factr 3 is btained as (the upper bund n) the rati f the resurces that these tw utputs require. This shuld be acceptable t all departments Research Staff and Publicatins Cnsider the rle f research staff in prducing academic publicatins. Because the rate f publicatins may vary between different departments and individual researchers, the fllwing tw assumptins are intended t be sufficiently cnservative. Assumptin 3. Each researcher shuld be able t publish at least ne paper in tw years. The abve statement can be stated as the fllwing prductin trade-ff: 0 P 0 3 =, Q3 = 0. (2) 0.5 This trade-ff implies that if the number f researchers is increased by, it shuld be pssible t increase the number f papers by 0.5 per year. Equivalently, using frmula (), the abve trade-ff translates t the fllwing linked weight restrictin: 0.5u v Assumptin 4. N department can justify a reductin f the number f papers by mre than 6 per year by referring t a lss f ne research staff. The number 6 in the abve statement is purely speculative and is simply used as an illustratin f a reasnably high research utput. In real applicatins this can be revised either way. Assumptin 4 is stated as the fllwing prductin trade-ff: 0 P 0 4 =, Q4 = 0. (3) 6 Equivalently, Assumptin 4 can be stated as the fllwing weight restrictin:

13 6u + v Prductin trade-ffs (2) and (3) effectively specify the lwer and upper bunds n the number f papers per average researcher at any department. Any publicatin rate belw the lwer bund f 0.5 is treated as evidence f inefficiency. A publicatin rate abve the upper bund f 6 is regarded as unrealistically high Academic Staff and Students There are different ways in which the link between academic staff and their utputs (students and publicatins) can be expressed. Belw we cnsider tw statements that link ne input and tw utputs simultaneusly in a single trade-ff. The idea f these tw assumptins is based n the cmmn use f student-t-staff ratis at academic departments and the expectatin f certain publicatin rates. Assumptin 5. One full academic pst is a sufficient resurce fr the number f undergraduate students at the department t increase by 0 and the number f publicatins t increase by 0.5. This assumptin is stated as the fllwing prductin trade-ff: P 0 5 = 0, Q5 = 0. (4) 0.5 It further translates t the linked weight restrictin: 0u + 0.5u v 0. (5) 3 Assumptin 6. A lss f ne academic pst shuld nt lead t a reductin f mre than 20 undergraduate students and 5 publicatins per year. This assumptin is represented by the fllwing trade-ff 20 P6 = 0, Q6 = 0, (6) 5 and the fllwing equivalent weight restrictin: 20u 5u + v 0. (7) Students and Publicatins Three university departments in ur data set, D, D4 and D6, can be regarded as researchintensive. They have a mderate teaching-t-staff rati and a relatively high publicatin rate. 2

14 Departments D3 and D7 are fcused primarily n the teaching. They have a high student-tstaff rati and a lw number f publicatins. Overall, this suggests that the departments in Table can be viewed as having different specialisatins. Highly specialised DMUs are ften shwn as efficient by DEA mdels. This is because the peer grups f units t which specialised units can be cmpared have t shw a similar specialisatin, which is a limiting factr. Belw we vercme the abve prblem by relating the prductin f students and publicatins by means f prductin trade-ffs. The latter are based n the evaluatin f the resurces (staff time) that are needed fr the generatin f the tw utputs. Assumptin 7. The reductin f the number f undergraduate students by 20 releases the academic staff time sufficient t write ne academic paper. As a justificatin f the abve statement, we can think f an academic member f staff being n a ne-year study leave. This invlves n teaching lad and an expectatin f several research utputs. The reductin f undergraduate students by 20 can be apprximately equated t ne year f staff time, and the publicatin f just ne paper is a cnservative estimate f the publicatin utput achievable within ne year. This assumptin is stated as the fllwing prductin trade-ff: 20 0 P7 = 0, Q7 = 0. (8) It further translates t the weight restrictin: 20u + u 0. 3 Assumptin 8. The reductin f the number f publicatins by 5 releases the academic staff time sufficient t increase the number f undergraduate students by 20. This assumptin is stated as the fllwing trade-ff: 0 P 20 8 = 0, Q8 = 0. (9) 5 It further translates t the weight restrictin: 20u 0.5u 0. 3 Taken tgether, trade-ffs (8) and (9) put the bunds n the rati between the resurces (staff time) required t teach undergraduate students and publish papers. Namely, the teaching f 20 undergraduate students may, depending n the department, equate t the 3

15 writing f between and 5 papers. If the number f students is reduced by 20, any department shuld be able t cmpensate fr this by increasing the number f publicatins by at least paper per year. If the number f students is increased by 20 (and the staff number is kept cnstant), this may be used t justify the reductin f publicatins by n mre than 5 papers per year Cmputatinal Results Tables 2 and 3 shw the utput radial efficiency f all departments in the CRS and VRS DEA mdels with different sets f prductin trade-ffs. We btained these results using a cmmn cmmercial slver. Obviusly, slving the envelpment and crrespnding multiplier mdels led t the same efficiency scres. As nted abve, in these tw tables, the clumns titled CRS and VRS crrespnd t the standard DEA mdels withut prductin trade-ffs. Mdels CRS k and VRS k, where k =,...,8, incrprate all prductin trade-ffs ( PQ, ), t =,..., k stated abve. Fr example, mdels CRS and VRS incrprate the single trade-ff ( PQ, ) as stated in (8). Mdels CRS 3 and VRS 3 incrprate three trade-ffs ( PQ, ), ( P2, Q 2) and ( P3, Q 3) t t. Mdels CRS 8 and VRS 8 incrprate all eight prductin trade-ffs. Bth tables allw us t bserve the gradual imprvement f efficiency discriminatin as additinal trade-ffs are prgressively incrprated. The final clumns CRS 8 and VRS 8 shw a significant imprvement ver the cnventinal CRS and VRS mdels. Table 5 shws the ptimal input and utput weights in mdel CRS 8. These weights were btained by slving the dual multiplier mdel with the eight weight restrictins equivalent t the prductin trade-ffs. In cmparisn t Table 4, all ptimal weights in Table 5 are strictly psitive. In this respect it shuld be nted that in practical applicatins f prductin trade-ffs the aim f making all ptimal weights strictly psitive may be a gal that is hard t achieve. The incrpratin f realistic prductin trade-ffs (r weight restrictins based n them) is a wrthwhile imprvement t the DEA mdel, even if this des nt cmpletely eliminate all zer weights in the ptimal slutin. 4. Graphical Illustratins T illustrate the effect f prductin trade-ffs n the technlgy, cnsider the fllwing tw examples. Bth are cncerned with the assessment f efficiency f university departments. Nte that these departments are different frm thse in Table. 4

16 Example. Let units A, B and C shwn in Figure be bserved departments. These departments are assumed t have the same level f a single input (staff) which is nt depicted, and different levels f tw utputs: undergraduate and master students. Because the input is equal, the shaded area represents bth the VRS and CRS technlgy induced by the three units. Mre precisely, the shaded area is the sectin f either technlgy fr the given level f input. Fr simplicity, we still refer t this sectin as the technlgy. The efficient frntier f this technlgy is the line segment AC. Department B is lcated n the bundary f technlgy but is dminated by A. It is therefre nly weakly efficient. The utput radial efficiency f all three departments is equal t. Cnsider prductin trade-ff (8). (We ignre the publicatins and research staff that are nt present in this example.) By the assumptin made, this trade-ff can be applied t any department. Fr example, starting at A, we can increase the number f its undergraduate students by and simultaneusly reduce the number f master students by. This prcedure can be repeated multiple times. Increasing the number f undergraduate students f department A by 00 and reducing the number f master students als by 00, we arrive at the hypthetical department D. Cntinuing this prcess, we induce the straight line AW. We have shwn that the line AW cnsists f prducible units and shuld therefre be regarded as part f the technlgy. Using the free dispsability f utputs, we shuld als add the nnnegative area belw AW t the technlgy. Nte that, if we start at any ther unit, e.g., at B r C, the applicatin f trade-ff (8) des nt add any further new pints t the technlgy. The use f trade-ff (8) allws us t add new units in the scenari in which the number f undergraduate students is increased. T cnsider the reductin f this input, we need t refer t prductin trade-ff (0). Starting at pint A and using the same lgic as abve, we mve away frm A t pint U. All pints n the line AU are prducible because we are replacing 3 undergraduate students by master student in this prcess, as in trade-ff (0). This adds the line AU t the technlgy, alng with the dminated nnnegative regin belw it. Overall, the specificatin f tw trade-ffs (8) and (0) results in the expansin f the technlgy frm the shaded area in Figure t the area belw the brken line UAW, and the latter is the new efficient frntier. Department A remains efficient, while departments B and C are n lnger efficient and are prjected n the units E and F, respectively. Nte that, because f the assumptins abut prductin trade-ffs (8) and (0), bth target units E and F are technlgically feasible. Therefre, the utput radial efficiency f the units B and C 5

17 retains its traditinal technlgical meaning. Namely, fr each unit the inverse f its utput radial efficiency is the ultimate imprvement factr by which bth f its utputs can be imprved. Example 2. In this example we illustrate the effect f linked prductin trade-ffs n the prductin technlgy. Fr simplicity we cnsider the case f VRS with a single input (academic staff) and a single utput (undergraduate students). The shaded area in Figure 2 crrespnds t the VRS technlgy induced by tw departments A and B. Bth departments are efficient in this technlgy. Cnsider the fllwing variants f linked prductin trade-ffs (4) and (6) adapted t ur example: P = ( ), ( ) 5 Q 5 = 0, (20) P = ( ), ( ) 6 Q 6 = 20. (2) We use the same lgic as in Example. Starting frm unit A and applying trade-ff (20) in different prprtins, we add the ray AW t the technlgy. Similarly, the applicatin f trade-ff (2) t unit A induces the line AK. Using free dispsability f input and utput, the VRS technlgy expands t the nnnegative area belw the brken line KAW, and the latter is its new efficient frntier. Nte that unit B is n lnger efficient in the expanded technlgy. Its utput radial efficiency is assessed by its prjectin n the unit E. Because the latter unit is prducible accrding t the stated trade-ff assumptins, it is a technlgically feasible efficient target fr department B. 5 CRS and VRS Technlgy with Prductin Trade-ffs Abve we defined prductin trade-ffs as the dual frms f weight restrictins. Their use in the example invlving university departments resulted in a meaningful expansin f the CRS and VRS technlgy and led t a significant imprvement f efficiency discriminatin. The missing link in the abve develpment is the definitin f technlgy with prductin trade-ffs. Belw we address this gap using the aximatic apprach t the definitin f technlgy pineered by Banker et al. (984). The main definitins and results f this sectin are based n the results f Pdinvski (2004a). 5. Aximatic Definitins The first three axims are the standard prductin assumptins that define the cnventinal VRS technlgy VRS. Adding the furth axim defines the CRS technlgy CRS. 6

18 Axim (Feasibility f bserved data). ( X, Y ), fr any j J. Axim 2 (Cnvexity). Technlgy is a cnvex set. Axim 3 (Free dispsability). If ( XY, ), Y Y 0 and X X, then ( X, Y ). Axim 4 (Prprtinality). If ( XY, ) and α 0, then ( αx, αy). j j The fllwing axim states that each f the prductin trade-ffs ( PQ, ) in (6) can be applied t any unit in technlgy, and any number f times (in any prprtin) π t 0 as lng as the resulting unit has nnnegative inputs and utputs. Axim 5 (Feasibility f prductin trade-ffs). Let ( XY, ). Then, fr each trade-ff ( PQ, ) in (6) and fr any π 0, the unit t t prvided X 0 and Y 0. t ( XY, ) = ( X+ π PY, + π Q), t t t t The next, and last, axim states that the prductin technlgy shuld be a clsed set. This is a standard prperty f prductin technlgies (Shephard 974, Färe et al. 985) that is ften autmatically satisfied and needs nt t be stated this is true in the cases f CRS, VRS and free dispsal hull technlgy f Deprins et al. (984). Hwever, as shwn by an example in Pdinvski (2004a), this is nt s fr technlgies that incrprate prductin trade-ffs as stated in Axim 5. Therefre, the fllwing axim needs t be explicitly stated. Axim 6 (Clsedness). Technlgy is a clsed set. The fllwing definitin is based n the minimum extraplatin principle intrduced t DEA by Banker et al. (984). Definitin. The CRS technlgy with trade-ffs (6) is the intersectin f all technlgies that satisfy Axims 6. It is straightfrward t verify that technlgy satisfies all Axims 6. Fr example, Axim 2 is satisfied because the intersectin f cnvex sets is a cnvex set. Definitin implies that is the smallest technlgy that satisfies all Axims 6. This means that it cntains nly thse DMUs that are required t satisfy the axims and n ther arbitrary units. The abve definitin is nt cnstructive, and its equivalent peratinal statement is given by the fllwing therem. t t 7

19 Therem (Pdinvski 2004a). Technlgy is the set f all units (, ) that can be stated in the frm n Y = λy + π Q e K j j t t j= t= m s XY + +, (22) n X = λ X + π P + d K, (23) j j t t j= t= n K where λ = ( λ,..., λ n) +, π = ( π,..., π K ) +, s e + and m d +. Therem prvides a meaningful interpretatin t the envelpment mdels (4) and (5). It shws that the radial imprvement f the input and, respectively, utput vectrs f the unit ( X, Y ) is perfrmed within the technlgy. Nte, hwever, that this interpretatin is crrect nly if the imprved unit has nnnegative input and utput vectrs, as required by Therem. This requirement is autmatically satisfied in mdel (5) because the utput-imprvement factr η is maximised. In mdel (4) the input-imprvement factr θ is minimised and may in sme cases becme negative. It may appear that we need t add the cnditin θ 0 t the cnstraints f mdel (4) this wuld remedy the prblem and guarantee that the minimisatin f θ is perfrmed in technlgy. While this is pssible, there are tw reasns why this may nt be a gd idea. First, adding the cnditin θ 0 t the cnstraints f mdel (4) wuld invalidate its duality with the multiplier mdel (2). The secnd and, perhaps, mre imprtant cnsideratin is that the feasibility f negative values f θ in mdel (4) indicates an incnsistency within the trade-ffs (6) r, equivalently, weight restrictins (). Allwing θ t take n negative values in the envelpment mdels make them self-testing fr errrs in the cnstructin f trade-ffs (r weight restrictins). We cnsider this issue in detail in the next sectin. m s Generally thugh, the nnnegativity cnditins (, ) are imprtant in the XY + + statement f technlgy and shuld nt be mitted unless prved redundant in a particular DEA mdel. This is discussed further in Sectin 7 (see Remark 2) in relatin t the additive DEA mdel based n the abve technlgy. In the case f VRS, we fllw the same lgic as abve and give the fllwing definitin. Definitin 2. The VRS technlgy with trade-ffs (6) is the intersectin f all technlgies that satisfy Axims 3, 5 and 6. 8

20 As in the abve case, it is straightfrward t verify that technlgy satisfies Axims 3, 5 and 6 and is, therefre, the smallest technlgy that satisfies them. Therem 2 (Pdinvski 2004a). Technlgy is the set f all units (, ) m s XY + + that can be stated in the frm (22) and (23), subject t the additinal nrmalising equality (7) and the same nnnegativity cnditins n vectrs λ, π, e and d as in Therem. The duality f weight restrictins and prductin trade-ffs allws us t give a psitive answer t the lng-standing questin f whether the use f weight restrictins in VRS DEA mdels is theretically sund (Thanassulis and Allen 998). The cunterargument is that in CRS mdels the marginal rates f transfrmatin and substitutin between inputs and utputs (that define the slpes f facets n the bundary f the technlgy) are invariant with respect t the scaling (r the size) f the unit, while in the VRS technlgy this is nt s. The main cncern is then that the weight restrictins that specify bunds n the marginal rates wuld be inapprpriate in the VRS technlgy because such rates change with the scale f peratins. This argument is weakened by the fact that the marginal rates in the CRS technlgy are still generally different at any tw units, unless ne is a scaled variant f the ther. The abve prblem des nt arise if we interpret weight restrictins as the dual frms f prductin trade-ffs and assess the latter in the first place. Indeed, if prductin trade-ffs (6) are assumed technlgically feasible in the CRS technlgy (in the sense f Axim 5), then they must be technlgically feasible in the VRS technlgy because the latter is a subset f. Therefre, any prductin trade-ffs (r weight restrictins based n them) that are deemed realistic and apprpriate in the CRS mdel, are als acceptable and can be used in the VRS mdel. 5.2 Sme Prperties f CRS and VRS Technlgies with Trade-ffs Belw we establish tw prperties f technlgies and. Therem 3. Technlgies and are plyhedral sets. In particular, is a plyhedral cne. Prf f Therem 3. The set P f all slutins { XY,, λπ,, ed, } t the set f linear equatins (22), (23) and inequalities XY,, λπ,, ed, 0 is a plyhedral set in 2( m s) + + n+ K. Technlgy in Definitin is the prjectin f P n its input and utput dimensins X and Y. 9

21 By the knwn prjectin lemma (see, e.g., Jnes et al. 2008, Lemma 3.), plyhedral set. Because is cnsidered in a similar way. is a satisfies Axim 4, it is a cne. The case f technlgy The secnd prperty is smewhat mre subtle. Withut prductin trade-ffs, the cnventinal CRS technlgy is the cne extensin f the VRS technlgy. This means that any unit ( XY, ) in the CRS technlgy is btained by the scaling f sme unit ( XY, ) in the VRS technlgy by sme factr α 0. This result is generally incrrect fr the CRS and VRS technlgies that incrprate prductin trade-ffs (althugh it is almst crrect in the sense defined belw). Example 3. Cnsider the CRS and VRS technlgies with a single input and single utput induced by the single bserved unit A = (2,). Suppse we specified the linked trade-ff: ( PQ, ) = (, 2). Figure 3 shws the resulting VRS technlgy as the shaded area belw the brken line GAF. Nte that the ray AF is btained by the applicatin f trade-ff ( PQ, ) t the unit A fllwing the same lgic as in Example 2. Furthermre, the CRS technlgy is the cne under the ray OE: the ray OE is btained by the applicatin f trade-ff ( PQ, ) t the zer unit the latter is included in the riginal CRS technlgy. This implies that, fr example, unit B = (, 2) is in technlgy. (As an alternative argument, unit B satisfies the cnditins f Therem with λ = 0 and π =.) It is, hwever, straightfrward t shw that there exists n unit ( XY, ) and α 0 such that B= α( XY, ). Example 3 shws that technlgy is generally nt the cne extensin f. Belw we prve that frmally, dente the cne extensin f as is the clsed cne extensin f. T state this m s {(, ) (, ), α 0: (, ) = α(, ) } cne = XY XY XY XY. Dente cl( cne ) the clsure f the set cne (intersectin f all clsed sets cntaining cne ). Therem 4. Technlgy is the clsed cne induced by : = cl( cne ). 20

22 Prf f Therem 4. By Therem 2, any ( XY, ) satisfies (22), (23) and (7) with sme vectrs λ, π, e and d. Fr any α 0, α( XY, ) satisfies (22) and (23) with the vectrs αλ, απ, αe and αd. By Therem, ( XY, ). Therefre, cne, and cl( cne ) cl =. (The last equality is true because satisfies Axim 6.) Cnversely, let ( XY, ). Then ( XY, ) satisfies (22) and (23) with sme vectrs λ, π, e and d. Let λ = λ. Tw cases arise. n j= j Case. Assume that λ > 0. Define. Then ( XY, ) VRS because it TO ( XY, ) = (/ λ )( XY, ) satisfies (22), (23) and (7) with λ = λ / λ, π = π / λ, e e λ = / and = /. Because d d λ ( XY, ) = α( XY, ) where α = λ, we have ( X, Y ) cne cl( cne ). Case 2. Assume that λ = 0. Therefre, λ = 0. (This is the case fr unit B in Example 3.) Cnsider the sequence f units ( X, Y ), k =,2,..., defined as fllws: k k n ( Xk, Yk) = ( X j, Yj) + kxy (, ) j= n. (24) Because bth terms n the right-hand side f (24) are nnnegative, each unit ( X, Y ) is nnnegative. It is straightfrward t verify that ( X, Y ) satisfies cnditins (22), (23) and (7) with the vectr λ k whse cmpnents are ( λ ) = /, j =,..., n, and vectrs πk = kπ ', ek = ke and dk ( Y ) k k k j n = kd. Therefre, ( Xk, Yk), fr all k =,2,... Define the sequence f units ( X, Y ) = ( / k) ( X, Y ). Obviusly, we have k k k k X k, k cne, fr all k. Nte that ( XY, ) is the limit unit f the sequence f units ( Xk, Yk). Indeed, based n (24), n ( X k, Y k) = ( X j, Yj) + ( XY, ) ( XY, ). k k j n + = Therefre ( X, Y ) cl( cne ). Because ( XY, ) is an arbitrary unit in, in bth cases and 2 we have cl( cne ). Taking int accunt the inverse embedding btained in the first part f the prf, we have = cl( cne ). k k 2

23 technlgy Therem 4 states that the CRS technlgy is btained frm the VRS by the scaling f its units by all factrs α 0, and subsequently adding all limit pints (units) t the resulting set. 6 Weight Restrictins and the Infeasibility Prblem It is well-knwn that the use f weight restrictins in multiplier mdels (2) and (3), and in their VRS analgues, may result in their infeasibility (see, e.g., Allen et al. 997, Pedraja- Chaparr et al. 997). A similar prblem may ccur when prductin trade-ffs are incrprated in envelpment DEA mdels. By duality, if a multiplier mdel with weight restrictins is infeasible, its dual envelpment mdel (which is always feasible) must have an unbunded bjective functin. The unbundness f the bjective functin η in the utput-maximisatin CRS mdel (5) and its VRS analgue indicates that the incrpratin f weight restrictins (prductin trade-ffs) has created an unlimited prductin f the utput vectr Y. (Because η Y can be taken t infinity while keeping the input vectr X cnstant.) This is incnsistent with the established prperties f prductin technlgies (Shephard 974, Färe et al. 985) and indicates that an errr has ccurred in the cnstructin f weight restrictins r trade-ffs. The unbundness f the bjective functin θ in the input-minimisatin mdel (4) r its VRS analgue implies that θ = 0 is feasible in the mdel. Cnsequently, the technlgy allws free prductin f the utput vectr Y frm the zer vectr f inputs θ X = 0X. This is an equally prblematic situatin that indicates that weight restrictins shuld be recnsidered. In the authr s experience based n teaching DEA t a large class f undergraduate students fr many years, wh were asked t use weight restrictins in their wrk, the abve infeasibility prblems are nt unusual. These are mre likely t happen if the mdel incrprates a relatively large number f weight restrictins f cmplex structure: thse that invlve several input and utput weights in ne inequality, as in (5) and (7). The use f trade-ffs fr the assessment f weight restrictins facilitates and ften encurages the frmulatin f cmplex weight restrictins. Fr example, weight restrictins (5) and (7) that have a clear meaning as stated in Assumptins 5 and 6 are unlikely t be stated using value judgements, because it may nt even be clear what they mean in value terms. 22

24 Pdinvski and Buzdine-Chameeva (203) shw that free and unlimited prductin f utput vectrs may ccur even if all multiplier mdels are feasible and all efficiency scres appear plausible. In such cases, the technlgy is mdelled incrrectly and the efficiency scres are als incrrect. One cannt therefre rely n the fact that the efficiency scres appear unprblematic there may still be an undetected underlying prblem with weight restrictins that invalidates the results f analysis and needs crrecting. Belw we utline the results presented in Pdinvski and Buzdine-Chameeva (203). These include a descriptin f the infeasibility (and unbundness) prblem caused by weight restrictins and the frms it can take, depending n the assumptin f returns t scale (VRS r CRS) and the rientatin f the mdel (input minimisatin r utput maximisatin). This leads t the frmulatin f analytical and cmputatinal tests that give us a cnclusive answer as t whether there is a prblem with weight restrictins. 6. Definitins and Examples We start with the fllwing tw definitins. Let, Y 0, be a vectr f utputs. s Y + Definitin 3. Technlgy allws free prductin f vectr Y if (0, Y ). Definitin 4. Technlgy allws unlimited prductin f vectr Y if there exists a vectr f inputs X such that ( X, αy ) fr all α 0. Pdinvski and Buzdine-Chameeva (203) prve that the abve tw ntins are equivalent in any cne technlgy, e.g., in technlgy : the existence f free prductin implies the existence f unlimited prductin, and vice versa. In a nn-cne technlgy, e.g., in, the tw ntins are generally different. Furthermre, in any cnvex technlgy (e.g., in and ), the specificatin f vectr X in Definitin 4 is unimprtant: if vectr Y can be prduced in an unlimited quantity α frm the input vectr X, then it can be prduced in an unlimited quantity frm the input vectr X f any ther unit ( XY, ) in the technlgy. It is straightfrward t verify that, under the nnnegativity assumptins made abut the bserved DMUs, cnventinal CRS and VRS prductin technlgies d nt allw free r unlimited prductin f utput vectrs, but the incrpratin f weight restrictins (prductin trade-ffs) may create it. The fllwing tw examples demnstrate this effect. Example 4. Suppse we made a mistake in the assessment f prductin trade-ffs (8) and (0), and stated them as fllws: 23

25 0 P 4 = 0, Q =, (25) P 2 = 0, Q = 2 2. (26) 0 It is easy t see that the abve trade-ffs induce unlimited prductin f the tw utputs (undergraduate and master students) in the VRS and CRS technlgy. Figure 4 is a mdificatin f Figure t this case. Starting frm unit A and applying trade-ff (25) 00 times, we substitute 00 master students by 400 undergraduate students. This creates pint E n the graph. Subsequently applying trade-ff (26) 00 times, we substitute 300 undergraduate students by 200 master students. The resulting unit A has 00 mre f bth types f student cmpared t the riginal unit A, and achieves this withut any extra input. We can cntinue this prcess and generate a further sequence f units A 2, A 3,, taking the prductin f utputs t infinity. (The lightly shaded area in Figure 4 shws the regin f units dminated by A 3. By free dispsability f utput, this regin is als included in the technlgy. As the sequence f units A t, t =,2,..., tends t infinity, the crrespnding dminated area cvers the whle nnnegative rthant.) Example 5. Cnsider the VRS technlgy as in Figure 2. Assume we replaced the prductin trade-ff (2) by the fllwing trade-ff: Q = 0. (27) P = ( ), ( ) Figure 5 shws the effect f trade-ff (27) n the VRS technlgy. Starting at unit A and cnsecutively applying this trade-ff, we generate the line AK which, tgether with the regin belw it, shuld be added t the technlgy. Nte that unit K has a zer input and a strictly psitive utput. This means that the expanded technlgy allws free prductin and indicates that trade-ff (27) shuld be recnsidered. Nte that the abve prblem cannt be bserved by the efficiency calculatins: the utput radial efficiency f departments A and B in this example is equal t and 0.5, respectively, and is nt suspicius. Hwever, because the slpe f the efficient bundary KA is incrrect, the calculated efficiencies are als incrrect. 24

26 6.2 Theretical Results Belw we give a cmplete characterisatin f prblematic utcmes in the CRS and VRS DEA mdels with weight restrictins (prductin trade-ffs) that are caused by free r unlimited prductin f vectr Y in the crrespnding technlgy. If any f such utcmes are bserved in practical cmputatins, this implies that an errr has ccurred in the assessment f weight restrictins (r, equivalently, prductin trade-ffs), and these need t be recnsidered. The first therem deals with the case f CRS. Therem 5 (Pdinvski and Buzdine-Chameeva 203). Let ( X, Y ) CRS TO and let X 0 and Y 0. (Fr example, ( X, Y ) may be an bserved unit.) Then the fllwing three statements are equivalent: (a) There exists free and unlimited prductin f utput vectr Y in technlgy. (b) The CRS input-minimisatin envelpment mdel CRS is unbunded r has a finite ptimal value θ = 0 value θ = 0, respectively.. Its dual multiplier mdel (c) The CRS utput-maximisatin envelpment mdel mdel 2 CRS is infeasible. CRS is infeasible r has an ptimal 2 CRS is unbunded. Its dual multiplier The next result deals with the case f VRS. Because in this technlgy the ntins f free and unlimited prductin are generally nt equivalent, these are cnsidered separately. Therem 6 (Pdinvski and Buzdine-Chameeva 203). Let ( X, Y ) and let X 0 and Y 0. (Fr example, ( X, Y ) may be an bserved unit.) Then the fllwing statements are true: (a) There exists free prductin f utput vectr Y in technlgy if and nly if the VRS input-minimisatin envelpment mdel VRS is either unbunded r has a finite ptimal value θ 0. Its dual multiplier mdel finite ptimal value θ 0. VRS is, respectively, infeasible r has a (b) There exists unlimited prductin f utput vectr Y in technlgy if and nly if the VRS utput-maximisatin multiplier mdel 2 VRS is unbunded. Its dual multiplier mdel 2 VRS is infeasible. 25

27 One f the differences between the cases f CRS and VRS highlighted by Therems 5 and 6 is that free prductin in the VRS technlgy may result in a finite negative value f the input efficiency θ. Fr example, cnsider unit G = (50,50) in the VRS technlgy in Figure 5. The input radial prjectin f G is H = ( 5,50). Slving the envelpment mdel VRS prduces the finite value θ = 5 / 50 = 0. and illustrates part (a) f Therem 6. The abve tw therems d nt slve the prblem f identifying prblematic weight restrictins (trade-ffs) cmpletely: even if n prblematic utcmes ccur with the assessment f all bserved units ( X, Y ), this guarantees nly that there is n free r j j unlimited prductin f the utput vectrs Y j f bserved units. This des nt hwever guarantee that there is n free r unlimited prductin f ther utput vectrs in the technlgy. Fr example, in the case f VRS technlgy in Figure 5, Therem 6 wuld nt identify any prblem when the input r utput radial efficiency f bth bserved units A and B is assessed. Pdinvski and Buzdine-Chameeva (203) suggest tw appraches, analytical and cmputatinal, that allw us t examine if the incrpratin f prductin trade-ffs (weight restrictins) has induced free r unlimited prductin in the technlgy. This task is simplified by the fllwing statement. Therem 7 (Pdinvski and Buzdine-Chameeva 203). The existence f free (and therefre unlimited prductin) f the utput vectr Y in technlgy is equivalent t the existence f either free r unlimited prductin f vectr Y (but nt necessarily bth) in technlgy. Accrding t Therem 7, if there is a prblem with free r unlimited prductin in either CRS r VRS technlgy, then there is a similar prblem in the ther. Because the ntins f free and unlimited prductin are equivalent in the CRS technlgy, and als because the chice f vectr X is unimprtant fr the latter ntin, it suffices t test fr the existence f unlimited prductin with the input vectr X f an arbitrary unit ( XY, ) in the CRS technlgy. Pdinvski and Buzdine-Chameeva (203) cnsider tw cases. The simpler case arises if weight restrictins () are nt linked. In this case the testing is reduced t verifying a simple algebraic cnditin. If weight restrictins () include linked restrictins, the testing is perfrmed by slving specially cnstructed linear prgrams. Belw we utline the tw cases. 26

28 6.3 Free Prductin with Nt Linked Trade-ffs The mst straightfrward case arises if the weight restrictins are nt linked. Then () can be restated as fllws: uq t 0 v P t 0,, 2 t =,..., K, (28) t =,..., K. (29) Therem 8 (Pdinvski and Buzdine-Chameeva 203). Technlgy des nt allw free (and unlimited) prductin if and nly if bth f the fllwing tw cnditins are satisfied: (a) there exists a strictly psitive vectr u > 0 that satisfies (28); (b) there exists a nnnegative vectr v 0 that satisfies (29) such that ( v ) X > 0 hlds fr all bserved units j =,..., n. (If either grup f weight restrictins (28) r (29) is missing, then the crrespnding cnditin (a) r (b) is remved frm the abve statement.) Nte that the vectrs u and v d nt need t satisfy the cnditins f mdels (2) r (3) all that is required is that such vectrs satisfy (28) and (29). In practical applicatins all inputs f all bserved DMUs j =,..., n are usually strictly psitive. In this case cnditin (b) f Therem 8 is equivalent t the simpler cnditin: there exists a nnzer vectr v 0 that satisfies (29). If sme f the inputs f bserved DMUs are equal t zer, the abve simplified cnditin des nt apply. Hwever, t prve that there is n free prductin, a simpler sufficient cnditin may be used. (Obviusly, if it is nt satisfied, this des nt mean that there is free prductin we need t use Therem 8 fr a definitive answer.) j Crllary. If there exist strictly psitive vectrs u > 0 and v > 0 that satisfy (28) and (29), then technlgy des nt allw free (and unlimited) prductin. As an illustratin, refer t Example in which we used the trade-ffs between undergraduate and master students as stated in (8) and (0). The resulting technlgy was illustrated in Figure. The crrespnding weight restrictins (9) and () are simultaneusly satisfied, fr example, by strictly psitive weights u = u 2 =. This means that cnditin (a) f Therem 8 is true. Because there are n weight restrictins invlving input weights, cnditin (b) f Therem 8 shuld be ignred. By Therem 8 r its Crllary, the tw 27

29 trade-ffs (8) and (0) d nt cause free r unlimited prductin in either CRS r VRS technlgy, which is cnsistent with Figure. Let us illustrate hw Therem 8 can be used t detect free prductin when it exists, even if all efficiency scres appear unprblematic. Example 6. In Example 4 we shwed hw the use f trade-ffs (25) and (26) resulted in the unlimited prductin f tw utputs (undergraduate and master students). If we use the same tw trade-ffs with the data set in Table, they induce unlimited prductin f the tw utputs in the same way but the prblem is nt bserved frm the efficiency calculatins and becmes hidden. Table 6 shws the efficiency scres (in %) in the CRS and VRS DEA mdels fr the departments as in Table. Bth the CRS and VRS mdels incrprate nly tw prductin trade-ffs (25) and (26). (These mdels are btained frm the mdels CRS 2 and VRS 2 discussed abve in which the gd trade-ffs (8) and (0) are replaced by the prblematic trade-ffs (25) and (26).) Nte that the results f cmputatins in Table 6 d nt appear prblematic the nly exceptin may be the unusually lw efficiency f department D2 in bth mdels. In such cases it is easy t miss the underlying prblem. T see if there is a prblem we use Therem 8 and restate prductin trade-ffs (25) and (26) as the weight restrictins 4u u 0, (30) 2 3u + 2u 0. (3) 2 It is straightfrward t shw that the abve inequalities cannt be satisfied by strictly psitive weights u and u 2. Indeed, adding the tw inequalities (30) and (3), we btain u u2 0 +, which des nt allw a strictly psitive slutin vectr. By Therem 8, prductin trade-ffs (25) and (26) induce free (and unlimited) prductin in the CRS technlgy, and the CRS efficiency scres are, althugh plausible, bviusly meaningless. By Therem 7, the efficiency scres in the VRS mdel are als incrrect. 6.4 Free Prductin with Linked Trade-ffs In the general case f linked weight restrictins () Pdinvski and Buzdine-Chameeva (203) develp tw cmputatinal prcedures t test if there is free (and unlimited) prductin in the CRS technlgy. Belw we describe ne f them. 28

30 The idea f this methd is simple and based n the fllwing fact: technlgy allws an unlimited prductin f a vectr Y if and nly if it allws an unlimited prductin f each f its individual psitive utputs, prvided all the ther individual utputs are taken equal t zer. (The nly if part f this statement is bvius. The if part fllws frm the fllwing. Suppse the technlgy allws the prductin f each individual utput ( Y ), r =,..., s, in any prprtin α 0, frm the input vectr X. Then the simple average f all s such units, each prducing the single utput α ( Y ) r, is the unit ( X,( α / sy ) ) CRS TO r. Because s is cnstant and α is arbitrarily large, technlgy allws an unlimited prductin f vectr Y.) The abve suggests that we can test fr unlimited prductin as fllws. First, we select any (e.g., bserved) unit ( X, Y ) CRS TO such that all cmpnents f vectr Y are strictly psitive: ( ) 0 Y >, fr all r =,..., s. If n such bserved unit exists, we can always r take the simple average f all bserved units. Because each utput r is strictly psitive fr at least ne bserved unit j, the average f all bserved units will have a strictly psitive utput vectr. Define s artificial utput vectrs U, r =,..., s, as fllws. Each f these vectrs has nly ne psitive cmpnent: U = Y (( ),0,...,0) r,, U = ( 0,...,0,( Y ) ) s s Cnsider s DMUs in the frm ( X, U ), where r =,..., s. Each f such units is dminated by the riginal unit ( X, Y ) and therefre ( X, U ) r r.. We can nw expand the set f bserved DMUs J by incrprating the abve s artificial units. Because the latter units are dminated, the technlgy remains unchanged. We nw slve s utput-maximisatin multiplier mdels, ne fr each unit ( X, U ρ ), ρ =,..., s. (We use index ρ t differentiate frm r in the same frmulatin.) η = min v X, (32) subject t uu ρ =, uy j vx j 0, j =,..., n, uu r v X 0, r =,..., s, 29

31 u Q t v P t 0, t =,..., K, uv, 0. Therem 9 (Pdinvski and Buzdine-Chameeva 203). Technlgy allws free (and unlimited) prductin if and nly if there exists a mdel (32) is infeasible. ρ =,..., s such that the multiplier Obviusly, instead f mdel (32), we can slve its dual envelpment mdel. In this case the infeasibility f mdel (32) is equivalent t the unbundness f the envelpment mdel. Als nte that the cnstraints uu v X 0 r in mdel (32) are redundant and can in principle be remved because, as discussed, units ( X, U ) are dminated. Frm the practical pint f view, hwever, it may be beneficial t keep mdel (32) as stated, because in this case it can be slved by standard DEA slvers. Example 7. As an illustratin, cnsider the university departments in Table and the eight trade-ffs discussed in Sectin 3. Because sme f these trade-ffs are linked, we use the methd based n prgram (32) t verify that the cmbinatin f this particular data set and trade-ffs des nt induce free r unlimited prductin. As the starting pint, let us chse department D as the unit ( X, Y ). (Alternatively, we can chse any department frm D t D6 fr this purpse, but nt D7 because its secnd utput is zer.) Fllwing the abve prcedure, define three artificial units with the vectr f inputs X = (92,5) as in department D, and the fllwing different utput vectrs: U = (800,0,0), U 2 = (0,200,0), U 3 = (0,0,90) We nw add the three units ( X, U ), ( X, U 2) and ( X, U 3) t the set f departments D D7. Because all three additinal departments are dminated by D, the technlgy des nt change. Finally, we assess the utput radial efficiency f the three additinal departments in the CRS multiplier mdel (32). The crrespnding three ptimal values f prgram (32) are finite and equal, respectively, t 3.632, 6.04 and (The utput radial efficiency f the three artificial units is, respectively, , and The utput radial efficiency f departments D D7, if calculated simultaneusly by the sftware, is the same as withut the additinal three units.) By Therem 9, the CRS (and cnsequently VRS) technlgy based n the data set in Table and the eight trade-ffs des nt allw free r unlimited prductin. r. 30

32 7 Slving DEA Mdels with Prductin Trade-ffs Cnventinal CRS and VRS DEA mdels (withut weight restrictins) are usually slved using either a tw-stage cmputatinal prcedure r an analgus single-stage methd utilizing a nn-archimedean ε (in practice taken equal t a very small psitive number). These methds are summarized in Thanassulis et al. (2008) and Cper et al. (20b). In many applicatins f DEA nly the radial efficiency f the DMUs is f interest, and the first stage f the tw-stage methd suffices fr this purpse. It identifies the radial prjectin f the assessed DMU n the bundary f the VRS r CRS technlgy and prduces the DMU s radial input r utput efficiency. Because the radial prjectin f an inefficient DMU may be nly weakly efficient, the identificatin f its efficient target (in the Paret sense) requires the secnd ptimisatin stage in which the sum f input and utput slacks is maximised. Perfrming the secnd stage identifies the efficient target f the DMU and the reference set f its efficient peers. The latter are the bserved DMUs j that have a crrespnding multiplier λ j > 0 in the ptimal slutin t the secnd-stage linear prgram. Pdinvski (2007b) shws that the applicatin f the standard secnd stage t DEA mdels with weight restrictins (r prductin trade-ffs) may result in a target unit with meaningless negative values f sme inputs. (This is unrelated t the issue f incnsistent weight restrictins discussed in the previus sectin.) In the suggested crrected prcedure, the cnventinal secnd stage is split int tw new stages, and the cmplete slutin methd becmes a three-stage prcedure. Depending n the purpse f a DEA study, nly the first, tw first r all three cmputatinal stages may need t be perfrmed. Belw we utline these three stages. We assume that the weight restrictins (prductin trade-ffs) have already been checked using the methds described in the previus sectin, and that the underlying VRS r CRS technlgy des nt allw free r unlimited prductin f nn-zer utput vectrs. Stage (Assessing the radial efficiency). This task is straightfrward and requires the slutin f the apprpriate CRS r VRS envelpment mdel, r their dual multiplier frms, as stated in Sectin 2. Stage 2 (Identifying efficient targets). An efficient target f DMU ( X, Y ) is btained by slving the specially cnstructed additive DEA mdel frmulated in Sectin 7.2 belw. Stage 3 (Identifying reference sets f efficient peer units). This stage is required because, even if the multiplier λ j is strictly psitive in an ptimal slutin t the mdel used at 3

33 Stage 2, the crrespnding bserved DMU j may be inefficient. An example f this is given in Pdinvski (2007b). The linear prgram slved at Stage 3 is presented in Sectin Stage : Assessing the radial efficiency Mst applicatins f DEA are cncerned nly with the input r utput radial efficiency f the units. In such applicatins this stage is the nly ne that needs perfrming. Depending n the assumptin f CRS r VRS and the rientatin f the mdel (input minimisatin r utput maximisatin), the radial efficiency f DMU ( X, Y ) is assessed by slving the crrespnding envelpment (r multiplier) mdel stated in Sectin 2. This stage als identifies the radial prjectin (target) unit ( X, Y ) f the DMU ( X, Y ). In the case f input minimisatin, ( X, Y ) ( θ X, Y) =, where θ is the input radial efficiency f DMU ( X, Y ). In the case f utput maximisatin, ( X, Y ) ( X, η Y) =, where η is the inverse utput radial efficiency f DMU ( X, Y ). (The value η is the ptimal value in the crrespnding envelpment and multiplier mdels that is inverse t the utput efficiency measure.) 7.2 Stage 2: Identifying Efficient Targets As in the case f cnventinal CRS and VRS DEA mdels, this stage shuld be perfrmed nly if we need t identify efficient targets f individual DMUs. In particular, the cmputatins at this stage d nt alter the radial efficiency assessed at Stage. The need f the secnd stage arises because the radial target ( X, Y ) assessed at Stage may be a weakly efficient unit and nt efficient in the Paret sense. The cnventinal secnd ptimisatin stage aims at maximising the sum f input and utput slacks that imprve the unit ( X, Y ). The same idea is applicable t DEA mdels with weight restrictins (prductin trade-ffs), but an additinal care has t be taken f the nnnegativity f inputs in the resulting efficient unit (which is autmatically maintained in the standard mdels withut weight restrictins). The fllwing prgram identifies pssible individual imprvements t the inputs and utputs f the unit ( X, Y ): s m s max εr + δi r= i= =, (33) subject t X δ Y + ε, (, ) 32

34 where the frm s ε +,, and technlgy is either r. m δ + T be specific, cnsider the case f CRS. Based n Therem, prgram (33) takes n s m s max εr + δi r= i= =, (34.) subject t n K λjyj + πtqt e= Y + e j= t=, (34.2) n K λjx j + πtpt + d = X d j= t=, (34.3) Y X + ε 0, (34.4) δ 0, (34.5) λπ,, ed,, ed, 0. (34.6) Nte that prgram (34) can be simplified. First, at any f its ptimal slutins the vectr e must be a zer vectr. Indeed, if we assume the cnverse ( e 0 and e 0 ) then redefining e = 0 and e = e + e keeps (34.2) true and imprves the bjective functin (34.), which is impssible due t the assumed ptimality f the current slutin. Therefre, vectr e in prgram (34) can be assumed zer and remved frm the frmulatin. Secnd, cnditin (34.4) is redundant because bth vectrs The resulting mdel is as fllws: s m s max εr + δi r= i= Y and ε are nnnegative. =, (35.) subject t n K λjyj + πtqt = Y + ε j= t=, (35.2) n K λjx j + πtpt + d = X d j= t=, (35.3) X δ 0, (35.4) λπ,, d, εd, 0. (35.5) Mdel (3) is the same as mdel (6) stated in Pdinvski (2007b). In the latter mdel the abve cnditin (35.4) is replaced by an equivalent requirement that the expressin n the left-hand side f equality (35.3) is nnnegative. 33

35 As already stated, we assume that technlgy des nt allw free and unlimited prductin. Therefre the bjective functin (35.) is bunded abve, and there exists an ptimal slutin t prgram (35) that we dente λ, π, d, ε, d. (36) This defines the efficient target f DMU ( X, Y ) as ( X, Y ) ( X δ, Y ε ) = +. (37) By the cnditins f mdel (35), ( X, Y ). Therem 0 (Pdinvski 2007b). DMU ( X, Y ) in (37) is efficient in technlgy. Obviusly, if all ptimal slacks in (35), and hence the ptimal value zer, the efficient target ( X, Y ) cincides with the radial target DMU ( X, Y ) is efficient if and nly if ( X, Y ) = ( X, Y ). σ, are equal t ( X, Y ). In particular, In the case f VRS, mdel (35) requires an additinal nrmalising cnditin (7). The same frmula (37) defines the efficient target ( X, Y ) in this case. Nte that the inequality (35.4) in mdel (35) guarantees that the maximisatin f the sum f cmpnent slacks (35.) is perfrmed within the technlgy by requiring that inputs remain nnnegative. As shwn by example in Pdinvski (2007b), the simple maximisatin f the sum f slacks withut cnditin (35.4) (in this case d culd be assumed t be a zer vectr) may result in negative values f sme f the inputs. Remark 2. Mdel (35) is an additive CRS DEA mdel based n technlgy assesses the efficiency f the unit. It ( X, Y ) by maximising the sum f cmpnent slacks ε r and δ i, prvided the resulting unit remains within the technlgy (and, in particular, des nt have negative inputs). In the case f VRS, we need t add the nrmalising cnditin (7) t the cnstraints f mdel (35). Mdel (35) and its VRS variant becme standard additive DEA mdels (Charnes et al. 985) in the absence f trade-ffs (6). Indeed, in this case the trade-ff terms n the lefthand side f cnditins (35.2) and (35.3) are mitted. Furthermre, the maximisatin f the sum f slack variables in (35) implies that at ptimality d = 0, and therefre vectr d can be remved frm the frmulatin. Finally, the nnnegativity cnditin (35.4) is redundant because, in the absence f trade-ffs, it fllws frm (35.3). 34

36 Like cnventinal additive DEA mdels, mdel (35) and its VRS variant can be used independently fr the assessment f efficiency f any unit ( X, Y ), withut the need t perfrm the first (radial prjectin) ptimisatin stage. 7.3 Stage 3: Identifying Reference Sets f Efficient Peer Units In cnventinal DEA mdels withut weight restrictins (prductin trade-ffs), the reference set f efficient peers cnsists f the bserved DMUs j such that λ j > 0 in an ptimal slutin t the secnd-stage ptimisatin mdel. In a DEA mdel with weight restrictins, an bserved DMU j with a strictly psitive value λ j in the ptimal slutin (36) may be inefficient an example f this is given in Pdinvski (2007b). As prved, in this case there exists an alternative ptimal slutin t prgram (35) that results in the same efficient target ( X, Y ) and fr which the cnditin λ > 0 implies that the bserved unit j is efficient, fr all j. Identifying such an ptimal slutin t (35) requires slving anther linear prgram. As with Stage 2, the cmputatins f Stage 3 shuld be perfrmed nly if needed. These cmputatins d nt affect the radial efficiency, radial targets and efficient targets already btained at Stages and 2. Fllwing Pdinvski (2007b), efficient peers f DMU ( X, Y ) crrespnding t the efficient target ( X, Y ) can be btained by maximising the sum f cmpnents f vectr d as the secndary gal in prgram (35), while keeping vectrs ε and δ at their ptimum level as in (36). In this case, by (37), the cnstant vectrs Y + ε and j X δ n the right-hand side f cnditins (35.2) and (35.3) can be replaced by Y and X, respectively. The resulting mdel takes n the frm: D m = max d, (38.) i= i subject t n K λ Y + π Q = Y, (38.2) j j t t j= t= n K λ X + π P + d = X, (38.3) j j t t j= t= λπ,, d 0. (38.4) Nte that the inequality (35.4) n lnger cntains decisin variables (because the vectr δ = δ is kept cnstant) and is mitted as redundant in prgram (38). 35

37 Because the bjective functin f prgram (38) is bunded abve, there exists an ptimal slutin λπ,,d t this prgram. Taken tgether with the cnstant vectrs ε and δ, slutin λπ,, d, ε, d (39) is an ptimal slutin t prgram (35). If the ptimal slutin (36) t prgram (35) is unique, then (39) is the same as (36). Otherwise, (39) is an ptimal slutin t (35) that additinally maximises the sum f cmpnents f vectr d as in (38.). Therem (Pdinvski 2007b). If λ j > 0 then DMU j is efficient in technlgy CRS TO and, cnsequently, in the smaller standard CRS technlgy CRS. An alternative mdel t (38) is btained in Pdinvski (2000). It has the same bjective (38.) as abve maximised ver the set f cnstraints (35.2) (35.5), with the additinal cnditin and keeping vectrs ε and δ variable. s m εr + δi = s, (40) r= i= The difference between mdel (38) and the latter mdel is that, by slving the frmer, we identify the reference sets fr DMU ( X, Y ) that are used in the cmpsitin f its specific efficient target ( X, Y ) which is fixed. In the latter apprach, the efficient target is nt fixed. The mdel based n cnditin (40) generally has alternative ptima λ, π, d, ε, d, each identifying a generally different efficient target ( X, Y ) and the crrespnding reference set f efficient peers j. The abve results extend t the case f VRS with bvius mdificatins. As nted, in the case f VRS mdel (35) incrprates the additinal nrmalising equality (7). The latter shuld als be incrprated in (38). Let ˆ λπ, ˆ,dˆ (4) be an ptimal slutin t prgram (38) with the cnditin (7). Therem 2 (Pdinvski 2007b). If ˆ λ j > 0 then DMU j is efficient in technlgy and, cnsequently, in the smaller standard VRS technlgy VRS. 36

38 The abve therem implies the existence f at least ne efficient DMU j J in any technlgy (under the assumptin that there is n free r unlimited prductin, as stated befrehand). Crllary 2. In any technlgy, there exists at least ne efficient bserved DMU. Prf f Crllary 2. Because f cnditin (7), in slutin (4) there exists a j such that ˆ λ j > 0. By Therem 2, DMU j is efficient in technlgy. Nte that Crllary 2 des nt uncnditinally extend t the case f CRS. Accrding t Therem stated in Charnes et al. (990), there exists at least ne efficient bserved DMU in the CRS technlgy, under the cnditin that weight restrictins () are nt linked. The fllwing example shws that the same statement is generally nt true in the case f linked weight restrictins. Example 8. Cnsider CRS technlgy discussed in Example 3 and illustrated in Figure 3. The nly bserved unit A = (2,) is inefficient in the CRS technlgy induced by itself and the single linked prductin trade-ff ( PQ, ) = (, 2). (The latter is equivalent t the linked weight restrictin 2u v 0.) Therefre, there are n efficient bserved units in technlgy in Figure 3. Furthermre, the utput radial efficiency f A is equal t Its unique efficient target is (2,4) it is cnstructed entirely frm the abve trade-ff ( PQ, ) applied 4 times t the rigin, with n cntributin frm the unit A itself. Therefre, unit A has n efficient peers amng bserved units, and the efficient target is cmpsed entirely frm the prductin trade-ff. Finally nte that A is efficient in the VRS technlgy in Figure 3, which is cnsistent with Crllary 2. 8 Cnclusin In this chapter we presented the ntin f prductin trade-ffs as the dual frms f weight restrictins. We explred varius theretical, methdlgical and cmputatinal issues arising frm the applicatin f prductin trade-ffs in DEA mdels. Althugh prductin trade-ffs are mathematically equivalent t weight restrictins, the assessment f the frmer is cnducted in the language f pssible changes t the inputs and utputs in the technlgy. In cntrast, the assessment f weight restrictins ften invlves value judgements that are mre managerial in nature and nt directly related t the technlgical pssibilities. 37

39 Based n the results f this chapter, the fllwing standard wrkflw can be suggested fr the practical implementatin f prductin trade-ffs and weight restrictins. This cnsists f three steps that may need t be repeated iteratively as the mdel is being mdified by the incrpratin f additinal trade-ffs. ) Cnstructin f prductin trade-ffs and weight restrictins. As illustrated in Sectin 3, prductin trade-ffs shuld represent realistic assumptins abut the technlgy. In practice, we shuld be certain that all bserved DMUs wuld be willing t accept the simultaneus changes stated by the trade-ffs. 2) Verificatin that the trade-ffs (r weight restrictins) d nt generate free r unlimited prductin fr the given set f bserved DMUs. As discussed in Sectin 6, this stage is imprtant because, if there is free r unlimited prductin in the technlgy, the results f the next stage may be incnsistent and puzzling. Alternatively, such results may appear unprblematic but still be errneus. This stage requires either the checking f simple inequalities r, in the case f linked weight restrictins, the use f standard DEA sftware with the extended set f bserved DMUs. 3) Cmputatin f efficiency, efficient targets and efficient peers. There are three stages in the cmputatinal prcedure described in Sectin 7. In many practical applicatins nly the first stage wuld be needed and may be perfrmed using standard DEA sftware. The implementatin f Stages 2 and 3 wuld currently require the use f general linear slvers. The use f prductin trade-ffs in DEA mdels, r the use f weight restrictins btained frm such trade-ffs, is interesting fr a number f reasns. First, prductin trade-ffs allw us t specify additinal infrmatin abut the technlgy that is nt therwise captured by the bserved data and standard prductin assumptins. This leads t a meaningful extensin f the cnventinal CRS r VRS prductin technlgy and results in a better-infrmed mdel f the prductin prcess. Furthermre, this generally imprves the efficiency discriminatin f the mdel in a technlgically meaningful way. Secnd, the use f prductin trade-ffs r weight restrictins based n them des nt have the well-knwn drawback f weight restrictins assessed by ther methds. The use f the latter generally leads t an uncntrlled expansin f the mdel f technlgy. In particular, the value judgements used in the cnstructin f weight restrictins cannt generally explain the technlgical meaning f the expanded technlgy and its new efficient 38

40 frntier. As a result, the radial and efficient targets f inefficient units may nt be prducible. The meaning f radial efficiency as the ultimate and technlgically feasible imprvement factr is n lnger preserved. In cntrast, the assessment f prductin trade-ffs explicitly takes int accunt the meaning f the resulting expansin f the technlgy. The use f such trade-ffs r weight restrictins based n them preserves the traditinal meaning f efficiency. Third, because the use f prductin trade-ffs results in a meaningful mdel f prductin technlgy, the well-established ntins f prductivity analysis such as returns t scale, prductivity change, and ther can be extended t it in a straightfrward fashin. In particular, the frmer can be explred by the generic methd f reference technlgies develped by Färe et al. (985) and further explred by Pdinvski (2004c). The Malmquist prductivity index in mdels with prductin trade-ffs was discussed in Alirezaee and Afsharian (200). Furth, the clear technlgical meaning f prductin trade-ffs allws us t make relatively cmplex statements invlving several inputs and utputs in a single trade-ff r weight restrictin. Examples f such statements were prductin trade-ffs (4) and (6) and the crrespnding weight restrictins (5) and (7). An advantage f such cmplex prductin trade-ffs is that they generally add mre pints t the mdel f prductin technlgy than simple statements, and therefre cntribute t better efficiency discriminatin. It is unlikely that weight restrictins (5) and (7) culd be btained using value judgements. Fifth, prductin trade-ffs can be used in DEA mdels that d nt have dual multiplier frms. An example f this is the FDH technlgy. Sixth, the interpretatin f weight restrictins as the dual frms f prductin tradeffs allws us t clarify and reslve sme theretical, methdlgical and cmputatinal issues arising in the cntext f weight restrictins. Fr example, as discussed, the interpretatin f weight restrictins in terms f prductin trade-ffs gives a psitive answer t the lng-standing questin f applicability f weight restrictins in the VRS technlgy. In Sectin 6 we shwed that the ntin f trade-ffs is instrumental in understanding the infeasibility and related prblems in DEA mdels with weight restrictins. 39

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44 Master students U B E A D C F W Undergraduates Figure Prductin trade-ffs expanding the technlgy in utput dimensins 43

45 900 Undergraduate students W A L E K B F Academic staff Figure 2 Linked prductin trade-ffs expanding the VRS technlgy 44

46 4 Output E F 3 2 B A G Input Figure 3 The VRS (dark grey) and CRS (light grey) technlgies induced by unit A and prductin trade-ff ( PQ, ) = (, 2) 45