DEA Models for Parallel Systems: Game-Theoretic Approaches

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1 Asia-Pacific Jurnal f Operatinal Research Vl. 32, N. 2 (2015) (22 pages) c Wrld Scientific Publishing C. & Operatinal Research Sciety f Singapre DOI: /S DEA Mdels fr Parallel Systems: Game-Theretic Appraches Juan Du Schl f Ecnmics and Management Tngji University Shanghai , P. R. China dujuan@tngji.edu.cn Je Zhu Internatinal Center fr Auditing and Evaluatin Nanjing Audit University Nanjing , P. R. China and Rbert A. Fisie Schl f Business Wrcester Plytechnic Institute Wrcester, MA 01609, USA jzhu@wpi.edu Wade D. Ck Schulich Schl f Business Yrk University, Trnt Ontari, Canada M3J 1P3 wck@schulich.yrku.ca Jiazhen Hu Schl f Ecnmics and Management Tngji University Shanghai , P. R. China hujiazhen@163.cm Received 11 May 2013 Accepted 3 March 2014 Published 13 Octber 2014 In many settings, systems are cmpsed f a grup f independent sub-units. Each sub-unit prduces the same set f utputs by cnsuming the same set f inputs. Cnventinal data envelpment analysis (DEA) views such a system as a black-bx, and uses the sum f the respective inputs and utputs f all relevant cmpnent units t calculate the system efficiency. Varius DEA-based mdels have been develped fr decmpsing the verall efficiency. This paper further investigates this kind f structure by using Crrespnding authr

2 J. Du et al. the cperative (r centralized) and nn-cperative (Stackelberg r leader fllwer) game thery cncepts. We shw that the existing DEA appraches can be viewed as a centralized mdel that ptimizes the efficiency scres f all sub-units jintly. The prpsed leader fllwer mdel will be useful when the pririty sequence is available fr sub-units. Cnsider, fr example, the evaluatin f relative efficiencies f a set f manufacturing facilities where multiple wrk shifts are perating. Management may wish t determine nt nly the verall plant efficiency, but as well, the perfrmance f each shift in sme pririty sequence. The relatinship between the system efficiency and cmpnent efficiencies is als explred. Our appraches are demnstrated with an example whse data set invlves the natinal frests f Taiwan. Keywrds: Data envelpment analysis (DEA); parallel systems; efficiency; centralized mdel; leader fllwer mdel. 1. Intrductin Data envelpment analysis (DEA) is an effective apprach fr measuring the relative efficiency f peer decisin making units (DMUs) that prduce the same set f utputs by cnsuming the same set f inputs. In cnventinal DEA mdels, DMUs are seen as black-bxes in the sense that the internal structure f DMUs is ignred. In recent years, a number f studies have lked at DMUs with netwrk structures. Fr example, Seifrd and Zhu (1999) use a series-cnnected tw-stage prcess t measure the prfitability and marketability f US cmmercial banks. T further address ptential cnflicts caused by intermediate measures existing between the tw stages, Ka and Hwang (2008), and Liang et al. (2008) prpse varius DEAbased methds t measure efficiency fr the verall prcess and fr each individual stage. Färe and Grsskpf (2000) develp a netwrk mdel t evaluate the verall netwrk DMU efficiency. One particular type f netwrk structure is a parallel system where a prductin system r a DMU can be cmpsed f a set f independent sub-units, with each cnsuming the same set f inputs t prduce the same set f utputs as is true f the entire system (Ka, 2009). A typical example prvided by Ka (2009) is a firm with several independently-perating plants. Each f the firm s inputs and utputs can be btained by summing up thse f all its plants respectively. In mdeling this kind f parallel structure, Beasley (1995), Ka (2009), Ka and Hwang (2010), and Castelli et al. (2010) prpse their wn versins f DEA mdels. All these mdels are essentially equivalent. Beasley (1995) develps a jint DEA maximizatin mdel fr determining teaching and research efficiencies fr university departments devted t the same discipline. Mrever, special situatins are cnsidered where certain resurces (general and equipment expenditure) are shared by sub-units (teaching and research activities). Ka (2009) further investigates the relatinship between the inefficiency f cmpnent units and the inefficiency f the entire parallel system, and prpses a parallel DEA mdel t calculate the verall and cmpnent efficiencies. Instead f maximizing efficiency, Ka s (2009) parallel mdel minimizes the inefficiency slack f a DMU, and decmpses the inefficiency slack int its prductin sub-units

3 DEA Mdels fr Parallel Systems: Game-Theretic Appraches Castelli et al. (2004) investigate single-level and tw-level hierarchical structures where each DMU is cmpsed f cnsecutive stages f parallel sub-units. Fr the tw-level situatin, they intrduce tw kinds f balancing cnstraints (virtual weight balancing cnstraints and flw balancing cnstraints), and accrdingly set up tw different DEA mdels. In particular, they prve that the maximum relative efficiency f a DMU is btained when it is cmpared with all the existing sub-units. Hwever, in their wrk, the peratin f each cmpnent unit is treated independently, and the relatinship amng these cmpnents is nt cnsidered. Castelli et al. (2010) prpsed an elementary DEA mdel which is further extended int shared flw, multi-level, and netwrk mdels. In rder t address bth series and parallel structures in netwrk systems, Ka and Hwang (2010) develp a relatinal mdel, based n which the system and cmpnent efficiencies can be directly determined. In terms f inefficiency, they shw that the system inefficiency is a weighted average f prcess/cmpnent inefficiencies. Hwever, the DEA-based mdels in the abve literature, including the jint maximizatin mdel (Beasley, 1995), parallel mdel (Ka, 2009), parallel relatinal mdel (Ka and Hwang, 2010), and elementary mdel (Castelli et al., 2010), may have alternative ptimal slutins, thus calculating prcess efficiencies directly frm its ptimality can lead t nn-unique efficiency decmpsitins. Thus an apprach fr testing fr unique efficiency decmpsitin is required. In the current paper, the parallel structure is further studied and extended using the game-theretic perspectives presented in Liang et al. (2008). In Liang et al. (2008), tw game appraches are prpsed fr series-cnnected tw-stage netwrk prcesses. One is a centralized mdel based upn the cncept f a cperative game where the verall efficiency is maximized in an effrt t ptimize bth stages efficiency scres jintly. The verall efficiency is defined as the weighted arithmetic average r gemetric average f stage efficiencies. The ther is a nn-cperative game mdel which assumes that ne stage is viewed as the leader with actin pririty t ptimize its efficiency first, and then the efficiency f the fllwer stage is calculated subject t maintaining the first-derived leader s efficiency. Based upn the Liang et al. (2008) cncept, we shw that the existing DEA appraches can be viewed as centralized mdels that ptimize the verall efficiency f a DMU subject t the unity restrictins n each cmpnent sub-unit and n the entire system. We then prpse a leader fllwer apprach that assumes that ne first decides n the relative imprtance f the sub-units, and then the ptimal relative efficiencies f the sub-units are derived in sequential and pririty rder. Such a cncept f evaluating bth verall and sub-unit efficiencies can be readily applicable in many industrial and service sectr settings. Cnsider, fr example, the prblem f evaluating the relative efficiencies f a set f manufacturing facilities where multiple wrk shifts are perating. Management may wish t determine nt nly the verall efficiency f the plant, but may, as well, wish t determine the perfrmance f each f the multiple shifts, and in sme pririty sequence

4 J. Du et al. In Ka s (2009) apprach, the verall efficiency f a DMU under evaluatin is measured in terms f inefficiency in frms f slacks. As a result, the verall slack is a sum f sub-unit slacks, and implies a frm f additive efficiency decmpsitin. We shuld pint ut, hwever, that these slacks in Ka s (2009) apprach are neither the standard DEA slacks in the secnd stage calculatin f the envelpment DEA mdel, nr the slacks in slacks-based mdels (Tne, 2001). We pint ut that when efficiency is measured in terms f rati efficiency as in the standard DEA mdel, such additive decmpsitin des nt arise and thus is nt inherent in the newly develped mdels. The remainder f the paper is rganized as fllws. The centralized and leader fllwer appraches fr measuring system and prcess efficiencies are develped in Secs. 2 and 3, respectively. The mathematical relatinship between the system and cmpnent efficiencies is als investigated. In Sec. 4, a real-wrld applicatin in frest prductin in Taiwan is used t illustrate bth appraches and cmpare results. Sectin 5 presents cncluding remarks. 2. The Centralized Apprach Suppse that there are n DMUs, with unit is dented by DMU ( =1,...,n), andwiththeith input and rth utput f DMU dented by x i (i =1,...,m)and y r (r =1,...,s), respectively. The cnventinal efficiency scre, E fr DMU is calculated by slving the fllwing CCR mdel (1) (Charneset al., 1978): s E =Max u ry r m v, ix i s u ry rj m v ix ij 1, j =1,...,n (1) u r,v i ε, r =1,...,s,,...,m. Nw, suppse that each DMU has a parallel internal structure as shwn in Fig. 1. Fr each DMU,therearek sub-units perating independently, dented by DMU p (p =1,...,k), and each f them utilizes the same m inputs t prduce the same s utputs (in varying amunts), dented by x p i (i =1,...,m)and yr p (r =1,...,s), respectively. The sum x i f all x p i ver p is the amunt f input i available t DMU,andy r,thesumfallyr p ver p is the amunt f utput r available t DMU. We first view this parallel system frm a centralized and cperative perspective, and determine a cmmn set f ptimal weights t maximize the verall efficiency fr each DMU. In a cperative sense, all cmpnent sub-units are suppsed t agree n the abslute imprtance f the entire system perfrmance. They cperate first t achieve the ptimal verall efficiency, after which the cmpnent efficiency is btained fr each sub-unit at the premise f maintaining the ptimal system efficiency

5 DEA Mdels fr Parallel Systems: Game-Theretic Appraches Fig. 1. A parallel prductin system with k sub-units. Therefre, fr DMU, we first ptimize its verall efficiency subject t the usual unity cnstraints in the frm f unity restrictins nt nly n the DMUs, but als n each sub-unit in each DMU. Because f the centralized assumptin, DMU has the central cntrl ver its sub-units. Thus the weight attached t each input/utput is assumed t be unified in bth system and cmpnent levels. The centralized mdel is given as fllws: s θ =Max u ry r m v, ix i s u ry rj m v ix ij 1, j =1,...,n (2) s u ry p rj m v ix p ij 1, p =1,...,k, j =1,...,n u r,v i ε, r =1,...,s,,...,m. Since x ij = k p=1 xp ij, y rj = k p=1 yp rj, it is bvius that the first set f cnstraints in mdel (2) is redundant. Using the Charnes Cper transfrmatin (Charnes and Cper, 1962), we get the fllwing linear prgram (3) equivalentt fractinal prgram (2): θ =Max µ r y r m µ r y p rj ω i x p ij 0, ω i x i =1 µ r,ω i ε, r =1,...,s,,...,m. p =1,...,k, j =1,...,n (3)

6 J. Du et al. In mdel (3), let µ r,r =1,...,s and ωi,i =1,...,m represent an ptimal slutin fr µ r and ω i. Then the verall efficiency scre fr the whle system is calculated as θ = s µ r y r, and the efficiency scre fr its sub-unit p, dented by θ p,p=1,...,k, can be cmputed as s θ p = µ r yp r m, p =1,...,k. ω i xp i One can find that this centralized mdel (3) is the same with the jint maximizatin mdel in Beasley (1995), the parallel relatinal mdel in Ka and Hwang (2010), and the elementary mdel in Castelli et al. (2010). If we measure the efficiency in (3) in terms f inefficiency as in Ka (2009), namely letting s p 0(p =1,...,k)represent the slacks fr DMU s inequality cnstraints in mdel (3), then the centralized mdel (3) is als equivalent t the parallel mdel prpsed by Ka (2009). Hwever, althugh the verall system efficiency is btained via similar mdels frm existing literature, we determine the efficiency decmpsitin amng all sub-units in a very different way as fllws. Nte that ur centralized mdel (3), r ther similar mdels frm existing studies, may have alternative ptimal slutins, which can lead t nn-unique efficiency decmpsitins fr sub-units. This can be demnstrated by a simple numerical example presented in Table 1. Suppse that there are tw DMUs cnsuming tw inputs t prduce tw utputs. Each DMU has tw parallel sub-units with the same types f inputs and utputs. We evaluate DMU 1 via centralized mdel (3) and btain tw ptimal slutins, which lead t tw different efficiency decmpsitins fr sub-units 1 and 2. Fr ne ptimal slutin, the verall efficiency fr DMU 1 is θ1 =0.5, and the efficiency scres fr sub-units 1 and 2 are θ1 1 =1andθ1 2 =0.3333, respectively. The crrespnding results frm the ther ptimal slutin are θ1 =0.5, θ1 1 =0.6667, and θ1 2 =0.4444, respectively. The abve results indicate that the standard apprach f calculating sub-unit efficiencies directly frm the ptimal slutins t related mdels, may lead t different cmpnent efficiency cmbinatins. Therefre, t test fr uniqueness, we fllw Ka and Hwang s (2008), r Liang et al. (2008) apprach t find a set f multipliers which satisfies the fllwing cnditins: (i) prduce the largest scre fr sub-unit with the first pririty, while maintaining the verall efficiency scre θ Table 1. Numerical example. DMUs Input1 Input2 Output1 Output2 DMU Sub-unit Sub-unit DMU Sub-unit Sub-unit

7 DEA Mdels fr Parallel Systems: Game-Theretic Appraches calculated frm mdel (3); (ii) prduce the largest scre fr the sub-unit with the qth (q = 2,...,k) pririty while maintaining the ptimal efficiencies fr the verall system and fr sub-units in the first (q 1) pririty psitins. Thrugh the abve prcess, we btain a set f efficiency scres fr all sub-units. Fr example, assume that sub-unit p 1 is given pre-emptive pririty. The fllwing mdel (4) determines its efficiency scre while maintaining the verall efficiency scre at θ. θ p1 =Max µ p1 µ p1 r y p1 r r yp rj m ω p1 i x p1 i =1 µ p1 r y r θ ω p1 i x p ij 0, p =1,...,k, j =1,...,n ω p1 i x i =0 µ p1 r,ωp1 i ε, r =1,...,s,,...,m. Fr sub-unit with the qth (q =2,...,k) pririty, the fllwing mdel (5) ptimizes its efficiency scre while maintaining the ptimal efficiencies fr the entire system and sub-units with the first t the (q 1)th pririty. θ pq =Max µ pq µ pq r y pq r r y p rj m ω pq i x pq i =1 µ pq r y r θ ω pq i x p ij 0, p =1,...,k, j =1,...,n µ pq r y pt r θ pt ω pq i x i =0 ω pq i x pt i =0, t =1,...,q 1 µ pq r,ω pq i ε, r =1,...,s,,...,m. Therefre, an efficiency decmpsitin is btained fr all cmpnent units f DMU as (θ 1,θ 2,...,θ k ). It is very likely that when k is greater than a certain number, there is nly ne unique ptimal slutin t mdel (5). Let represent the ptimal efficiency scre fr sub-unit p calculated frm mdel (4) when sub-unit p itself is given the first pririty, p =1,...,k.Ifˆθ p = θ p fr all p =1,...,k, then the ˆθ p (4) (5)

8 J. Du et al. Table 2. Efficiency results frm centralized apprach. DMUs Efficiency Sub-unit 1 Sub-unit 2 takes pririty takes pririty DMU Sub-unit Sub-unit DMU Sub-unit Sub-unit efficiency scre fr each sub-unit is uniquely determined by centralized mdel (3), indicating that a unique efficiency decmpsitin fr DMU is btained. Here, θ p is defined abve and cmputed directly frm ne ptimal slutin f mdel (3). T demnstrate the abve idea, we revisit the numerical example in Table 1. The efficiency results are reprted in Table 2. The secnd clumn lists efficiency scres fr DMU 1, 2 and their tw sub-units when sub-unit 1 is given the first pririty. The third clumn lists the crrespnding results when sub-unit 2 is given the first pririty t realize its ptimal efficiency. Frm the results in Table 2, we nte that a unique efficiency decmpsitin fr DMU 2 can be directly determined by centralized mdel (3). Hwever, fr DMU 1, the efficiency decmpsitin fr sub-units varies when different sub-units are given first pririty. Nte that in Ka s (2009) parallel mdel, the verall slack in the bjective functin is a sum f sub-units slacks in their cnstraints. As a result, Ka (2009) btains a frm f additive efficiency decmpsitin in terms f the slacks. If we use rati efficiency measures as in mdel (2) r(3), such decmpsitin is nt available. We pint ut, hwever, that the slacks in Ka s (2009) mdel are neither the standard DEA slacks in the secnd stage calculatin f the envelpment DEA mdel, nr the slacks in slacks-based mdels (Tne, 2001). As a result, these slacks d nt cmpletely indicate the levels f perfrmance inefficiency. One still needs t cnvert such slacks int rati efficiency measures t get the magnitude f inefficiency. Next we explre the mathematical relatinship between the verall efficiency θ and cmpnent efficiency scres θp (p =1,...,k) calculated frm mdels (4) and (5). Fr a sub-unit with the kth pririty, let {θ p k ; µ p k r,ω p k i,r = 1,...,s,i = 1,...,m} represent an ptimal slutin t mdel (5); then it is true that and µ p k r yr pt = θpt µ p k r y r = θ ω p k i x pt i ω p k i x i (6), t =1,...,k. (7)

9 DEA Mdels fr Parallel Systems: Game-Theretic Appraches Summing ver t n bth sides f Eq. (7), we have k µ p k r y r = θ pt ω p k i x pt i, (8) θ ω p k i x i = θ = θ = t=1 k θ pt t=1 k ( m ωp k i x pt i m t=1 ωp k i x i k t=1 ω p k i x pt i, (9) w pt θ pt, where w pt = ) θ pt, (10) m ωp k i x pt i m ωp k i x i. (11) Since 0 <w pt < 1and k t=1 w p t =1,Eq.(11) demnstrates that the verall efficiency is a weighted average f all cmpnent efficiencies. The weight attached t each cmpnent efficiency is the prprtin f ttal aggregated resurces devted t each sub-unit, reflecting the relative size f a sub-unit, and furthermre the relative imprtance r cntributin f each sub-unit s perfrmancecmparedt the system s verall perfrmance. It is als easily derived frm Eq. (11) that min p=1,...,k {θp } θ max p=1,...,k {θp }. This implies that in ur centralized apprach, althugh the verall efficiency is ptimized first, it still lies within the efficiency range f the sub-units. 3. The Leader Fllwer Apprach In this sectin, we view the parallel system shwn in Fig. 1 frm a nn-cperative game perspective, and suppse that each sub-unit intends t make its efficiency scre as high as pssible, given the current input and utput levels. Thus, we adpt the idea frm the Stackelberg game cncept where the leader firm/party mves first, and then the prcess cntinues fr each fllwer dwn thrugh the hierarchical structure. In that sense, the Stackelberg mdel is als referred t as leader fllwer mdel. In a parallel structure, if we assume that the actin sequence fr all sub-units under DMU is (p 1,p 2,...,p k ), then sub-unit p 1 is the leader and mves first, subunit p 2 is fllwer 1 and mves after sub-unit p 1 s actin, sub-unit p 3 is fllwer 2 and mves after sub-unit p 1 and p 2 s actin. This prcess cntinues until sub-unit p k (fllwer(k 1)) mves after sub-unit p 1 thrugh p k 1 s actin. In ther wrds, each subrdinate player executes his plicies after, and with the full knwledge f, his superir players. N matter what tactics will be taken by the fllwers, the best strategy fr leader p 1 is t ptimize its efficiency, and the best strategy fr fllwer (t 1), p t,t =1,...,k is t ptimize its efficiency subject t the requirement that the efficiencies fr leader and fllwer 1 thrugh (t 2) all remain unchanged. It implies that the decisin f any ne player will impact its subsequent players set f feasible chices

10 J. Du et al. The abve prcedure can be realized in a mathematical manner as fllws: (1) Fr leader sub-unit p 1, its efficiency scre is determined via the fllwing mdel (12), the result f which is the cnventinal CCR efficiency when all sub-units are treated as independent DMUs and cmpared tgether. e p1 e p2 =Max µ p1 µ p1 r yp1 r r yp rj m ω p1 i x p1 i =1 ω p1 i x p ij 0, p =1,...,k, j =1,...,n µ p1 r,ωp1 i ε, r =1,...,s,,...,m. (2) Based upn the leader s efficiency result, the efficiency fr fllwer 1 (sub-unit p 2 ) is derived by slving the fllwing linear mdel (13). =Max µ p2 r yr p2 e p k µ p2 r y p rj m ω p2 i x p2 i =1 µ p2 r y p1 r e p1 ω p2 i x p ij 0, p =1,...,k, j =1,...,n ω p2 i x p1 i =0 µ p2 r,ωp2 i ε, r =1,...,s,,...,m. (3) Based upn leader s and all the previus fllwers efficiency results, the efficiency fr fllwer (k 1) (sub-unit p k ) is derived frm the slutin t: =Max µ p k r y p k r µ p k r y p rj m ω p k i x p k i =1 µ p k r ypt r ept ω p k i x p ij 0, p =1,...,k, j =1,...,n ω p k i x pt i =0, t =1,...,k 1 µ p k r,ωp k i ε, r =1,...,s,,...,m (12) (13) (14)

11 DEA Mdels fr Parallel Systems: Game-Theretic Appraches (4) After the efficiency scres fr all sub-units are btained as e p,p=1,...,k,we calculate the verall efficiency fr DMU as e =Max µ r y r m µ r y p rj ω i x p ij 0, ω i x i =1 µ r y p r ep ω i x p i =0, p =1,...,k, j =1,...,n p =1,...,k (15) µ r,ω i ε, r =1,...,s,,...,m. Nte that in the nn-cperative (leader fllwer) mdel prpsed by Liang et al. (2008), the verall efficiency scre is directly determined as the prduct f tw individual stages scres. This is due t the special relatinship amng inputs, intermediate measures and utputs frm their series-cnnected tw-stage structures. Hwever, in ur parallel case, there is n similar relatinship between the system and cmpnent efficiencies. Therefre, nce all cmpnent efficiency scres are determined frm (1) t(3) in the abve prcedure, we need t further slve an additinal mdel (15) in rder t evaluate the verall efficiency fr the parallel system. The last set f cnstraints requires that the calculated efficiency scres fr sub-units remain unchanged. It is wrth nting that in the case f leader fllwer apprach, althugh the verall efficiency is ptimized last, after all sub-units btain their respective ptimal scres in a sequential manner, it des nt necessarily mean that the entire system will be surpassed by any sub-unit in terms f efficiency measurement. Rather, the verall efficiency e lies between the lwest and highest f sub-unit efficiency scres. Als, if we let {e ; µ r,ω i,r =1,...,s,i =1,...,m} be an ptimal slutin t mdel (15), we have e = = = µ ry r, k p=1 p=1 e p ωi x p i, ( k m ) ωi x p i e p. (16)

12 J. Du et al. Equatin (16) indicates that the verall efficiency is a weighted average f all cmpnent efficiencies. The weight attached t sub-unit p s efficiency is its aggregated inputs, reflecting t sme extent the relative imprtance r cntributin f sub-unit p s perfrmance. T demnstrate ur leader fllwer apprach, we cnsider the numerical example in Table 1. We first assume that sub-unit 1 is the leader, and sub-unit 2 is the fllwer. The crrespnding efficiency results are reprted in the secnd clumn f Table 3. The third clumn shws the efficiency results when sub-unit 2 is assumed t be leader. Cmparing the tw grups f efficiency scres, we ntice that DMU 1 has different system and sub-unit efficiencies when different sub-units act as leader. Hwever, fr DMU 2, its verall and cmpnent efficiencies remain the same n matter which sub-unit is the leader. Finally, we can als develp the leader fllwer apprach based upn inefficiency r slacks. (1) Fr leader sub-unit p 1, its efficiency scre is determined by mdel (12) which can be cnverted int the fllwing linear prgram: e p1 =1 s p1 =1 Min s p1 µ p1 r y p1 r ω p1 i x p1 i + sp1 =0 µ p1 r yp rj m ω p1 i x p1 i =1 ω p1 i x p ij 0, p =1,...,k, j =1,...,n (17) µ p1 r,ωp1 i ε, r =1,...,s,,...,m s p1 0. Table 3. Efficiency results frm leader fllwer mdel. DMUs Efficiency Sub-unit 1 Sub-unit 2 is leader is leader DMU Sub-unit Sub-unit DMU Sub-unit Sub-unit

13 DEA Mdels fr Parallel Systems: Game-Theretic Appraches (2) Based upn leader s efficiency result, the efficiency fr fllwer 1 (sub-unit p 2 ) is calculated by slving the fllwing linear mdel (18). e p2 =1 s p2 =1 Min s p2 µ p2 r yp2 r m ω p2 i x p2 i + sp2 =0 µ p2 r y p rj m ω p2 i x p ij 0, p =1,...,k, j =1,...,n ω p2 i x p2 i =1 (18) µ p2 r yp1 r (1 sp1 ) ω p2 i x p1 i =0 µ p2 r,ωp2 i ε, r =1,...,s,,...,m s p2 0. (3) Based upn the leader s and all the previus fllwers efficiency results, the efficiency fr fllwer (k 1) (sub-unit p k )iscalculatedby e p k =1 s p k µ p k r yp k µ p k r =1 Min s p k r m y p rj m ω p k i x p k i =1 µ p k r ypt r (1 spt ω p k i x p k i + sp k =0 ω p k i x p ij 0, p =1,...,k, j =1,...,n ) µ p k r,ωp k i ε, r =1,...,s,,...,m s p k 0. ω p k i x pt i =0, t =1,...,k 1 (4) After the ptimal slacks fr all sub-units are btained as s p,p =1,...,k,we calculate the verall efficiency fr DMU as e =1 s =1 Min s (19)

14 J. Du et al. µ r y r ω i x i + s =0 m µ r y p rj ω i x p ij 0, ω i x i =1 µ r y p r (1 s p ) p =1,...,k, j =1,...,n ω i x p i =0, µ r,ω i ε, r =1,...,s,,...,m p =1,...,k s 0. The mathematical relatinship between the verall ptimal slack s and cmpnent ptimal slacks s p (p =1,...,k) is expressed as s = k p=1 ( m ω i xp i )sp, where ωi is the ptimal value fr ω i (i =1,...,m)inmdel(20). This implies that similar t efficiency, the verall inefficiency is als a weighted average f all cmpnent inefficiencies, which was previusly mentined in Ka and Hwang (2010). (20) 4. Frest Prductin in Taiwan We here revisit the frest prductin example in Ka (2009) t illustrate the cperative and nn-cperative game appraches prpsed in this paper, and further cmpare ur efficiency results with Ka s (2009) results. As pinted ut by Ka (2009), the frest prductin system is a typical parallel prductin system, where each district has several sub-districts, referred t as wrking circles (WCs), perating independently. In Taiwan, the frestlands are divided int eight districts, and each is further divided int fur r five WCs. A WC is the basic unit in frest management, but it is nt regarded as a s-called independent unit because it des nt pssess an administratr. Only a district is viewed as an independent unit in this frest prductin system. The data set is presented in Table 4, which was previusly used by Ka (1998, 2000, 2009). There are fur inputs, including land in thusands f hectares, labr in persns, expenditures each year in ten-thusand New Taiwan dllars, and initial stcks befre the evaluatin perid in 10,000 cubic meters. Three utputs are taken int accunt, including timber prductin each year in cubic meters, sil cnservatin in 10,000 cubic meters, and recreatin each year in thusands f visits. Fr a detailed explanatin regarding the frest prductin system and the data set, refer t Ka (1998, 2000, 2009)

15 DEA Mdels fr Parallel Systems: Game-Theretic Appraches Table 4. Data fr Taiwan frest system. Wrking circles Inputs Outputs Land Labr Expenditures Initial stcks Timber Sil cns. Recreatin (1000 hectares) (persns) (10000 NTD) (10000 m 3 ) (m 3 ) (10000 m 3 ) (1,000 visits) Ltung District , , , (1) Taipei (2) Tai-ping-shan (3) Cha-chi (4) Nan-au (5) H-ping Hsinchu District , , , (6) Guay-shan (7) Ta-chi (8) Chu-tung , , , (9) Ta-hu , Tungshi District , , , (10) Shan-chi , (11) An-ma-shan (12) Li-yang , (13) Li-shan , , Nantu District , , , , (14) Tai-chung , (15) Tan-ta , (16) Pu-li , (17) Shui-li , (18) Chu-shan

16 J. Du et al. Table 4. (Cntinued) Wrking circles Inputs Outputs Land Labr Expenditures Initial stcks Timber Sil cns. Recreatin (1000 hectares) (persns) (10000 NTD) (10000 m 3 ) (m 3 ) (10000 m 3 ) (1,000 visits) Chiayi District , , , (19) A-li-shan (20) Fan-chi-hu , (21) Ta-pu (22) Tai-nan Pingtung District , , , , (23) Chih-shan , (24) Cha-chu , (25) Liu-guay , (26) Heng-chun Taitung District , , , (27) Kuan-shan , , , (28) Chi-ben (29) Ta-wu (30) Chan-kng Hualien District , , , , (31) Shin-chan , , (32) Nan-hua (33) Wan-yng (34) Yu-li , , ,

17 DEA Mdels fr Parallel Systems: Game-Theretic Appraches Table 5. Efficiency results fr Taiwan frest system. Wrking circles Centralized Leader fllwer Parallel CCR WC CCR district mdel mdel mdel efficiency efficiency Ltung District (1) Taipei (2) Tai-ping-shan (3) Cha-chi (4) Nan-au (5) H-ping Hsinchu District (6) Guay-shan (7) Ta-chi (8) Chu-tung (9) Ta-hu Tungshi District (10) Shan-chi (11) An-ma-shan (12) Li-yang (13) Li-shan Nantu District (14) Tai-chung (15) Tan-ta (16) Pu-li (17) Shui-li (18) Chu-shan Chiayi District (19) A-li-shan (20) Fan-chi-hu (21) Ta-pu (22) Tai-nan Pingtung District (23) Chih-shan (24) Cha-chu (25) Liu-guay (26) Heng-chun Taitung District (27) Kuan-shan (28) Chi-ben (29) Ta-wu (30) Chan-kng Hualien District (31) Shin-chan (32) Nan-hua (33) Wan-yng (34) Yu-li Table 5 reprts the results fr districts and WC btained frm ur centralized apprach and leader fllwer apprach in clumns 2 and 3, respectively. Within each district, withut any additinal infrmatin available t us, the pririty rder fr bth appraches is assumed t be set accrding t the manner in which they

18 J. Du et al. are numbered, that is frm the WC with the lwest number t the ne with the highest number. The results frm Ka s (2009) parallel mdel are shwn in clumn 4. As expected, the efficiency results frm ur centralized apprach are exactly the same as thse frm the parallel mdel. Furthermre, t test the uniqueness fr each district, we assume the pre-emptive pririty and slve mdel (4) fr each WC. All f the resulting ptimal efficiency scres are the same with ur centralized apprach. This implies that a unique efficiency decmpsitin is btained fr all WC under each district. Als, nte that fr bth f ur appraches, the verall efficiency fr each district lies within the efficiency range f its subrdinate WCs, and can be demnstrated t be a weighted average f related WC efficiency scres. If we treat the 34 WCs as independent DMUs, we can calculate their efficiency scres by slving the cnventinal CCR mdel (1). The results are listed in the fifth clumn f Table 5, and generally speaking, they are quite cnsistent with thse calculated frm the centralized apprach (r Ka s (2009) parallel mdel), and with thse frm the leader fllwer apprach. Since the CCR WC efficiency is the best scre that each WC can pssibly achieve, these CCR efficiencies are greater than r equal t the crrespnding efficiency results btained frm ur centralized r leader fllwer apprach. This is especially true fr ur leader fllwer apprach, where the first WC under each district is given the first pririty in efficiency ptimizatin. Thus, its leader fllwer efficiency result is equal t its CCR WC efficiency. Cmparing the efficiency results slved frm the centralized mdel with the CCR WC efficiency scres, we find that there are 9 ut f 34 WCs whse CCR efficiencies are greater than their centralized efficiencies by mre than 0.1. The largest difference ccurs at the 14th WC Tai-chung under Nantu district, where the efficiency scre is 1 versus This indicates that sacrificing the efficiency scre fr Tai-chung by (= ), culd realize a higher efficiency fr Nantu, the district it belngs t. If WC Tai-chung is allwed t be efficient with the scre 1, then the highest pssible efficiency scre fr Nantu district becmes , a drp by (= ) cmpared with its riginal centralized efficiency. We als ntice that 12 ut f 34 WCs have equal centralized efficiencies t their respective CCR efficiencies, 2/3 f which have an efficient scre f unity. Similarly, we cmpare the efficiency results btained frm the leader fllwer mdel with the CCR WC efficiencies, and find that 12 ut f 34 WCs with a CCR efficiency greater than the leader fllwer efficiency by mre than 0.1. The largest difference ccurs at the 26th WC Heng-chun under Pingtung district, which is 1 versus This is because Heng-chun is the last fllwer and is given the least pririty in efficiency ptimizatin amng all fur WCs belnging t Pingtung district. In ther wrds, its efficiency is sacrificed t realize higher efficiency scres fr the remaining three WCs frm the same district, all f which have pririty ver Heng-chun in ptimizing efficiency. Als, nte that 11 ut f 34 WCs have leader fllwer efficiencies equal t their respective CCR efficiencies, 8 f which d s because they are the leader in

19 DEA Mdels fr Parallel Systems: Game-Theretic Appraches the actin sequence within each DMU/district, and have the pririty t receive the best pssible efficiency scres. The last clumn f Table 5 shws the cnventinal CCR efficiency scres fr eight districts withut cnsidering any WC. As pinted ut by Ka (2009), ignring the requirement that each sub-unit shuld have an aggregated utput that is smaller than its aggregated input will lead t a higher efficiency measure fr every district. The efficiency results btained either frm the centralized r frm the leader fllwer apprach, shw that nne f the eight districts is efficient. Hwever, in terms f cnventinal CCR efficiency, nly tw districts (Ltung and Nantu) are inefficient, and the remaining six are efficient. This implies that bth f ur centralized apprach and leader fllwer apprach have a strnger discriminatin pwer in perfrmance evaluatin than the cnventinal DEA mdel. Als cmparing the cnstraints frm leader fllwer mdel (15) with thse frm centralized mdel (3), we find that the cnstraints f the leader fllwer verall-efficiency mdel are much strnger than thse f the centralized mdel. Therefre, the verall efficiency scres calculated frm the frmer are smaller than thse calculated frm the latter. We pint ut again that ur applicatin f the leader fllwer mdel in this particular prblem setting is illustrative nly, and acknwledge that in the absence f additinal infrmatin, ne may nt be able t easily decide upn which sub-unit r WC in a particular district shuld be treated as the leader r fllwer. Specifically, if nne f the WCs can legitimately be treated as leader, then the leader fllwer mdel may nt be apprpriate fr capturing sub-unit level efficiency. 5. Cnclusins In this paper, we first examine the existing DEA appraches n DMUs that have a parallel internal netwrk structures with independently-perating sub-units. We shw that the existing DEA appraches can be viewed as a DEA mdel adpting the cncept f cperative (r centralized) game thery. Next we further develp a DEA apprach based upn nn-cperative (Stackelberg/leader fllwer) game thery. The tw appraches study the same prblem frm different perspectives. The centralized mdel suppses all cmpnent sub-units agree n the abslute imprtance f the verall efficiency fr the entire system. The verall efficiency is ptimized first, after which an efficiency-decmpsitin is btained fr each sub-unit. On the cntrary, in the leader fllwer apprach, a pririty is placed n the sub-units. Each sub-unit determines its best pssible scre with the restrictin that it must fllw thse leader sub-units that cme befre it in the ptimizing sequence. The verall efficiency fr the DMU is calculated last subject t that all sub-units maintain their respective efficiency scres. The decisin n which f the tw appraches is mre preferable depends n the specific real-wrld applicatin. Additinal infrmatin frm specific empirical study supprts decisin-makers t make the chice n mdels and t decide the pririty

20 J. Du et al. rder fr cmpnent units. If additinal infrmatin indicates that ne cmpnent is f vital imprtance during a prductin prcess, then the leader fllwer apprach may be mre apprpriate fr perfrmance evaluatin. But if decisin-makers pay much mre attentin t the verall system rather than any individual sub-unit, r nne f the sub-units can legitimately be treated as leader, then the centralized apprach is a better chice fr efficiency analysis. We emphasize that the key assumptin made in this paper and in the wrk f Ka (2009), is that the parallel sub-units all prduce exactly the same utputs using the same inputs (albeit in differing amunts). In many sub-unit situatins, hwever, this prperty f identical utput/input factrs may nt hld. Cnsider, fr example, the case f an rganizatin where different business units perate within the DMU different wards in hspitals, service versus sales cmpnents in banks, different prductin lines in a factry, and s n. In future wrk, the authrs prpse t extend the develpment herein t accmmdate settings where nn-hmgenus parallel sub-units perate. Acknwledgment Dr. Juan Du thanks the supprt by the Natinal Natural Science Fundatin f China (Grant N ), and Chen Guang prject by Shanghai Municipal Educatin Cmmissin and Shanghai Educatin Develpment Fundatin (Grant N. 13CG19). This paper is partially funded by the Pririty Academic Prgram Develpment f Jiangsu Higher Educatin Institutins. References Beasley, JE (1995). Determining teaching and research efficiencies. Jurnal f the Operatinal Research Sciety, 46, Castelli, L, R Pesenti and W Ukvich (2004). DEA-like mdels fr the efficiency evaluatin f hierarchically structured units. Eurpean Jurnal f Operatinal Research, 154(2), Castelli, L, R Presenti and W Ukvich (2010). A classificatin f DEA mdels when the internal structure f the decisin making units is cnsidered. Annals f Operatins Research, 173, Charnes, A and WW Cper (1962). Prgramming with linear fractinal functinals. Naval Research Lgistics Quarterly, 15, Charnes, A, WW Cper and E Rhdes (1978). Measuring the efficiency f decisin making units. Eurpean Jurnal f Operatinal Research, 2, Färe, R and S Grsskpf (2000). Netwrk DEA. Sci-ecnmic Planning Sciences, 34, Ka, C (1998). Measuring the efficiency f frest districts with multiple wrking circles. Jurnal f Operatinal Research Sciety, 49, Ka, C (2000). Shrt-run and lng-run efficiency measures fr multi-plant firms. Annals f Operatins Research, 97, Ka, C (2009). Efficiency measurement fr parallel prductin systems. Eurpean Jurnal f Operatinal Research, 196,

21 DEA Mdels fr Parallel Systems: Game-Theretic Appraches Ka, C and SN Hwang (2008). Efficiency decmpsitin in tw-stage data envelpment analysis: An applicatin t nn-life insurance cmpanies in Taiwan. Eurpean Jurnal f Operatinal Research, 185(1), Ka, C and SN Hwang (2010). Efficiency measurement fr netwrk systems: IT impact n firm perfrmance. Decisin Supprt Systems, 48, Liang, L, WD Ck and J Zhu (2008). DEA mdels fr tw-stage prcesses: Game apprach and efficiency decmpsitin. Naval Research Lgistics, 55, Seifrd, LM and J Zhu (1999). Prfitability and marketability f the tp 55 US cmmercial banks. Management Science, 45(9), Tne, K (2001). A slacks-based measure f efficiency in data envelpment analysis. Eurpean Jurnal f Operatinal Research, 130, Bigraphy Juan Du is Assistant Prfessr at the Schl f Ecnmics and Management, Tngji University, Shanghai, China. She received PhD in Management Science and Engineering frm the University f Science and Technlgy f China (USTC) in late Her research interests fcus n data envelpment analysis (DEA), decisin analysis, multi-criteria decisin mdeling, and applicatins in health care and banking. She has articles published in peer-reviewed jurnals, such as Eurpean Jurnal f Operatinal Research, Annals f Operatins Research, OMEGA, Decisin Supprt Systems, and thers. Je Zhu is Prfessr f Operatins and Industrial Engineering, Schl f Business at Wrcester Plytechnic Institute, Wrcester, MA, USA. He is als a visiting Distinguished Prfessr at Nanjing Audit University, Nanjing, China. His research interests include issues f prductivity and benchmarking, and applicatins f Data Envelpment Analysis (DEA). He has published ver 100 articles in peer-reviewed jurnals such as Management Science, Operatins Research, IIE Transactins, Naval Research Lgistics, Eurpean Jurnal f Operatinal Research, Jurnal f Operatinal Research Sciety, Annals f Operatins Research, Cmputer and Operatins Research, OMEGA, Internatinal Jurnal f Prductin Ecnmics, Sci-Ecnmic Planning Sciences, Jurnal f Prductivity Analysis, INFOR, and thers. He has published and c-edited seven bks n perfrmance evaluatin and benchmarking using DEA. He is an Area Editr f OMEGA and an Assciate Editr f INFOR. He is als a member f Cmputers and Operatins Research Editrial Bard. He is the Assciate Series Editr fr the Springer Internatinal Series in Operatins Research and Management Sciences. He is recgnized as ne f the tp 10 authrs in DEA with respect t h-index and g-index. Wade D. Ck is Grdn Charltn Shaw Prfessr f Management Science, and Prfessr f Operatins Management and Infrmatin Systems, Schulich Schl f Business, Yrk University, Trnt, Canada. His research cvers peratins management and infrmatin systems, aiming at develping decisin supprt tls fr mdeling perfrmance and identifying best practice in rganizatins. He has mre than

22 J. Du et al. 140 articles published in refereed jurnals such as Management Science, Operatins Research, IIE Transactins, Naval Research Lgistics, Eurpean Jurnal f Operatinal Research, Jurnal f Operatinal Research Sciety, Annals f Operatins Research, Cmputer and Operatins Research, OMEGA, Jurnal f Prductivity Analysis, INFOR, and thers. He is an Assciate Editr f Operatins Research, and Eurpean Jurnal f Operatinal Research Editrial Bard Member. Jiazhen Hu is Prfessr at the Schl f Ecnmics and Management, Tngji University, Shanghai, China. His research interests include lgistics and supply chain management, management infrmatin system (MIS), enterprise peratins management. He has published mre than 30 articles in peer-reviewed academic jurnals, c-edited tw bks n supply chain management, and been apprved six natinal sftware cpyrights in China. He is Chair Prfessr f DHL, Germany