DEA Models for Two-Stage Processes: Game Approach and Efficiency Decomposition

Transcription

1 DEA Mdels fr Tw-Stage Prcesses: Game Apprach and Efficiency Decmpsitin Liang Liang, 1 Wade D. Ck, 2 Je Zhu 3 1 Schl f Business, University f Science and Technlgy f China, He Fei, An Hui Prvince, Peple s Republic f China Schulich Schl f Business, Yrk University, Trnt, Ontari, Canada M3J 1P3 3 Department f Management, Wrcester Plytechnic Institute, Wrcester, Massachusetts Received 29 Octber 2006; revised 25 May 2008; accepted 1 June 2008 DOI /nav Published nline 25 August 2008 in Wiley InterScience ( Abstract: Data envelpment analysis (DEA) is a methd fr measuring the efficiency f peer decisin making units (DMUs). This tl has been utilized by a number f authrs t examine tw-stage prcesses, where all the utputs frm the first stage are the nly inputs t the secnd stage. The current article examines and extends these mdels using game thery cncepts. The resulting mdels are linear, and imply an efficiency decmpsitin where the verall efficiency f the tw-stage prcess is a prduct f the efficiencies f the tw individual stages. When there is nly ne intermediate measure cnnecting the tw stages, bth the nncperative and centralized mdels yield the same results as applying the standard DEA mdel t the tw stages separately. As a result, the efficiency decmpsitin is unique. While the nncperative apprach yields a unique efficiency decmpsitin under multiple intermediate measures, the centralized apprach is likely t yield multiple decmpsitins. Mdels are develped t test whether the efficiency decmpsitin arising frm the centralized apprach is unique. The relatins amng the nncperative, centralized, and standard DEA appraches are investigated. Tw real wrld data sets and a randmly generated data set are used t demnstrate the mdels and verify ur findings Wiley Peridicals, Inc. Naval Research Lgistics 55: , 2008 Keywrds: data envelpment analysis (DEA); efficiency; game; intermediate measure; cperative; nn-cperative 1. INTRODUCTION Data envelpment analysis (DEA), intrduced by Charnes et al [2], is an apprach fr identifying best practices f peer decisin making units (DMUs), in the presence f multiple inputs and utputs. In many cases, DMUs may als have intermediate measures. Fr example, Seifrd and Zhu [8] use a tw-stage prcess t measure the prfitability and marketability f US cmmercial banks. In their study, prfitability is measured using labr and assets as inputs, and the utputs are prfits and revenue. In the secnd stage fr marketability, the prfits and revenue are then used as inputs, while market value, returns, and earnings per share are used as utputs. Chilingerian and Sherman [5] describe anther twstage prcess in measuring physician care. Their first stage is a manager-cntrlled prcess with inputs including registered nurses, medical supplies, and capital and fixed csts. These inputs generate the utputs r intermediate measures Crrespndence t: J. Zhu (jzhu@wpi.edu) (inputs t the secnd stage), including patient days, quality f treatment, drug dispensed, amng thers. The utputs f the secnd (physician cntrlled) stage include research grants, quality f patients, and quantity f individuals trained, by specialty. Seifrd and Zhu [8] use the standard DEA apprach, which des nt address ptential cnflicts between the tw stages arising frm the intermediate measures. Fr example, the secnd stage may have t reduce its inputs (intermediate measures) t achieve an efficient status. Such an actin wuld, hwever, imply a reductin in the first stage utputs, thereby reducing the efficiency f that stage. T address that cnflict issue, Chen and Zhu [3] and Chen et al. [4] present a linear DEA type mdel where the intermediate measures are set as decisin variables. Hwever, their individual stage efficiency scres d nt prvide infrmatin n the verall perfrmance and best-practice f the tw-stage prcess. The current study seeks alternative ways t (i) address the cnflict between the tw stages caused by the intermediate measures, and (ii) prvide efficiency scres fr bth 2008 Wiley Peridicals, Inc.

2 644 Naval Research Lgistics, Vl. 55 (2008) individual stages and the verall prcess. We mdel the tw-stage prcesses via cncepts adpted frm nncperative and cperative games. Fr example, suppse a DMU cnsists f a manufacturer and a retailer. In such a setting, traditinally the manufacturer hlds manipulative pwer and acts as a leader, and the retailer is treated as a fllwer in mdeling nncperative supply chains [6]. In a similar manner, ur nncperative apprach assumes that ne f the stages is the leader that seeks t maximize its DEA efficiency. Then the efficiency f the ther stage (the fllwer) is calculated subject t the leader-stage maintaining its DEA efficiency. The leader stage can be viewed as being mre imprtant than the ther stage(s) in imprving its efficiency. In a mre cperative envirnment, the manufacturer and retailer may wish t wrk tgether in determining price, rder quantity, and ther factrs t achieve maximum savings and/r prfit fr the manufacturer-retailer chain. Our specific apprach herein assumes that initially bth stages efficiency scres are maximized simultaneusly, while determining a set f ptimal (cmmn) weights assigned t the intermediate measures. It is pinted ut that this apprach is nt specifically in line with cnventinal cperative game thery lgic, where players wuld jintly decide upn a multiplier space that is acceptable. We refer t ur apprach as centralized, in that it is the cmbined stages that are f interest (see, e.g., Cachn [3]). We then apply a secnd rder mdel (see Appendix) t arrive at a cperative efficiency decmpsitin that is fair t bth players. In this latter sense ur cmbined centralized/cperative apprach is in the spirit f cperative games. It is shwn that bth the nn-cperative and centralized appraches yield an efficiency decmpsitin, where the verall efficiency f the tw-stage prcess is a prduct f thse f the tw individual stages. Nte that such an efficiency decmpsitin is nt available in the standard DEA apprach f Seifrd and Zhu [8], and the multi-stage appraches f Chen and Zhu [3]. The current study further shws that when there is nly ne intermediate measure, bth the nncperative and centralized appraches yield the same results, and unique efficiency decmpsitin ccurs, as is the case in applying the standard DEA mdel t each stage separately. Althugh the nncperative apprach yields a unique efficiency decmpsitin under multiple intermediate measures, the centralized apprach may yield multiple efficiency decmpsitins. Mdels are develped t test whether the centralized apprach t efficiency decmpsitin is unique. The rest f the article is rganized as fllws. Sectin 2 presents the generic tw-stage prcess. We then present in Sectins 3 and 4 ur nncperative (r leader-fllwer) mdel, and the centralized mdel. It is shwn hw t test fr uniqueness f efficiency decmpsitin. Sectin 5 discusses the relatins amng the standard DEA mdel and Figure 1. Tw-stage prcess. the nncperative and centralized appraches. The issue f unique efficiency decmpsitin is als studied. In Sectin 6 ur mdels are then applied t three data sets t verify ur findings. One data set is frm Wang et al. [9], and has nly ne intermediate measure. The mdels are then applied t the data set f Seifrd and Zhu [8] with tw intermediate measures. It is shwn that the efficiency decmpsitin is unique. Finally, t further examine differences that can ccur between utcmes frm the nncperative and centralized appraches, a randmly generated data set is examined. Cnclusins fllw in Sectin TWO-STAGE PROCESSES Cnsider a generic tw-stage prcess as shwn in Fig. 1, fr each f a set f n DMUs. Using the ntins in Chen and Zhu [3], we assume each DMU j (j = 1, 2,..., n) has m inputs x ij, (i = 1, 2,..., m) t the first stage, and D utputs z dj, (d = 1, 2,..., D) frm that stage. These D utputs then becme the inputs t the secnd stage and will be referred t as intermediate measures. The utputs frm the secnd stage are y rj, (r = 1, 2,..., s). Fr DMU j we dente the efficiency fr the first stage as ej 1 and the secnd as ej 2. On the basis f the radial (CRS) DEA mdel f Charnes et al. [2], we define D ej 1 = w dz dj m v and ej 2 ix = u ry rj D ij w (1) dz dj where v i, w d, w d, and u r are unknwn nn-negative weights. It is nted that w d can be set equal t w d, and in many if nt mst situatins this wuld be an apprpriate curse f actin. In the case examined herein we make the assumptin that the wrth r value accrded t the intermediate variables is the same regardless f whether they are being viewed as inputs r utputs. Clearly, ne can apply tw separate DEA analyses t the tw stages as in Seifrd and Zhu [8]. One criticism f such an apprach is the inherent cnflict that arises between these tw analyses. Fr example, suppse the first stage is DEA efficient and the secnd stage is nt. When the secnd stage imprves its perfrmance (by reducing the inputs z dj via an inputriented DEA mdel), the reduced z dj may render the first Naval Research Lgistics DOI /nav

3 Liang, Ck, and Zhu: DEA Mdels fr Tw-Stage Prcesses 645 stage inefficient. This indicates a need fr a DEA apprach that prvides fr crdinatin between the tw stages. Befre presenting ur mdels, it is useful t pint ut that given the individual efficiency measures ej 1 and e2 j,it is reasnable t define the efficiency f the verall tw-stage prcess either as 1 2 (e1 j + e2 j ) r e1 j e2 j. If the input-riented DEA mdel is used, then we shuld have ej 1 1 and e2 j 1. The abve definitin ensures that the tw-stage prcess is efficient if and nly if ej 1 = e2 j = 1. Finally, if we define e j = u r y rj m v ix ij as the tw-stage verall efficiency, ur mdels imply e j = ej 1 e2 j at ptimality. 3. NONCOOPERATIVE MODEL One frm f a nncperative game is characterized by the leader-fllwer assumptin. (The term nncperative game is used t characterize either leader-fllwer situatins, r nrmal frm/ simultaneus game situatins. In the game thery literature, the leader-fllwer paradigm is als referred t as the Stackelberg mdel, brrwed frm the ntin f Stackelberg games). Fr example, cnsider a case f nncperative advertising between a manufacturer (leader) and a retailer (fllwer). The manufacturer, if assumed t be the leader, determines its ptimal brand name investment and lcal advertising allwance, based n an estimatin f the lcal advertising that will be undertaken by the retailer t maximize its prfit. The retailer, as a fllwer n the ther hand, based n the infrmatin frm the manufacturer, determines the ptimal lcal advertising cst, t maximize its prfit [7]. In a similar manner, if we assume that the first stage is the leader, then the first stage perfrmance is mre imprtant, and the efficiency f the secnd stage (fllwer) is cmputed, subject t the requirement that the leader s efficiency stays fixed. Adpting the cnventin that the first stage is the leader, and the secnd stage, the fllwer, we calculate the efficiency fr the first stage, using the CCR mdel [2]. That is, we slve fr a specific DMU the liner prgramming mdel e 1 = Max w d z d w d z dj v i x i = 1 v i x ij 0 w d 0, d = 1, 2,..., D; v i 0, i = 1, 2,..., m. (2) Nte that since mdel (2) is the standard (CCR) DEA mdel, then e 1 is the regular DEA efficiency scre. Once we btain the efficiency fr the first stage, the secnd stage will nly cnsider thse variables w d that maintain e 1 = e1. Or, in ther wrds, the secnd stage nw treats D w dz dj as the single input subject t the restrictin that the efficiency scre f the first stage remains at e 1. The mdel fr cmputing e 2, the secnd stage s efficiency, can be expressed as e 2 = Max U ry r Q D w dz d U ry rj Q D w 1 dz dj w d z dj v i x ij 0 v i x i = 1 w d z d = e 1 U r, Q, w d, v i 0, r = 1, 2,..., s; d = 1, 2,..., D; i = 1, 2,..., m (3) Nte that in mdel (3), the efficiency f the first stage is set equal t e 1. Let u r = U r, r = 1, 2,..., s. Mdel (3) is Q then equivalent t the fllwing linear mdel e 2 ( ) = Max u r y r /e 1 u r y rj w d z dj 0 w d z dj v i x i = 1 v i x ij 0 w d z d = e 1 w d 0, d = 1, 2,..., D; v i 0, i = 1, 2,..., m; u r 0, r = 1, 2,..., s (4) In a similar manner, if we assume the secnd stage t be the leader, we first calculate the regular DEA efficiency (e 2) fr that stage, using the apprpriate CCR mdel. Then, ne slves the first stage (fllwer) mdel, with the restrictin Naval Research Lgistics DOI /nav

4 646 Naval Research Lgistics, Vl. 55 (2008) that the secnd stage scre, having already been determined, cannt be decreased frm that value. We finally nte that in (4), e 1 e2 = u r y r at ptimality, with m v i x i = 1. That is, e 1 s e2 = u r y r m v i x. i (A similar result is true, if the leader/fllwer definitin is reversed). This indicates that ur nn-cperative apprach implies an efficiency decmpsitin fr the tw-stage DEA analysis. That is, the verall efficiency is equal t the prduct f the efficiencies f individual stages. Further, nte that in the first-stage leader case, e 1 and e 2 are ptimal values t linear prgrams. Therefre, such an efficiency decmpsitin is unique. The same is true f the decmpsitin fllwing frm the secnd-stage leader case. It is pinted ut, hwever, that these tw decmpsitins may nt be the same. 4. CENTRALIZED MODEL An alternative apprach t measuring the efficiency f the tw stage prcess is t view them frm a centralized perspective, and determine a set f ptimal weights n the intermediate factrs that maximizes the aggregate r glbal efficiency scre (as wuld be true where the manufacturer and retailer jintly determine the price, rder quantity, etc. t achieve maximum prfit [7]). In ther wrds, the centralized apprach is characterized by letting w d = w d in (1), and the efficiencies f bth stages are evaluated simultaneusly. 1 Generally, the mdel fr maximizing the average f e 1 and e 2 is a nn-linear prgram. We nte, hwever, that because f the assumptin f w d = w d in (1), e 1 e2 becmes s u r y r m v ix i. Therefre, instead f maximizing the average f e 1 and e2, we have e centralized = Max e 1 e2 = u ry r m v ix i e 1 j 1 and e2 j 1 and w d = w d. (5) Mdel (5) can be cnverted int the fllwing linear prgram e centralized = Max u r y r u r y rj w d z dj 0 w d z dj v i x ij 0 1 Nte that in the end, a cmmn set f weights is assigned t bth stages in ur nn-cperative game apprach. Hwever, in that apprach, e 1 and e2 are nt ptimized simultaneusly. v i x i = 1 w d 0, d = 1, 2,..., D; v i, i = 1, 2,..., m; u r, r = 1, 2,..., s (6) Mdel (6) gives the verall efficiency f the tw-stage prcess. Assume the abve mdel (6) yields a unique slutin. We then btain the efficiencies fr the first and secnd stages, namely D e 1,Centralized = w d z d m v i x i = wd z d and = u r y r D w d z. (7) d If we dente the ptimal value t mdel (6) as e centralized, then we have e centralized = e 1,Centralized e 2,Centralized. Nte that ptimal multipliers frm mdel (6) may nt be unique, meaning that e 1,Centralized and e 2,Centralized may nt be unique. T test fr uniqueness, we can first determine the maximum achievable value f e 1,Centralized e 1+ = Max w d z d u r y r = e centralized w d z dj via v i x ij 0 u r y rj w d z dj 0 v i x i = 1 w d 0, d = 1, 2,..., D; v i 0, i = 1, 2,..., m; u r 0, r = 1, 2,..., s (8) It then fllws that the minimum f e 2 = ecentralized. e 1+ The maximum f is given by, which we dente by e 2+, can be calculated in a manner similar t the abve, and the minimum f e 1,Centralized is then calculated as e 1 k = e centralized /e 2+. Nte that e 1 = e 1+ if and nly f e 2 = e 2+. Nte als that if e 1 = e 1+ r e 2 = e 2+, then e1,centralized and e 2,Centralized are uniquely determined via mdel (6). If e 1 = e 1+ r e 2 = e 2+, then presumably sme flexibility exists in setting values fr e 1,Centralized and e 2,Centralized. A legitimate reasn fr taking advantage f such flexibility is ne f cperatin and Naval Research Lgistics DOI /nav

5 Liang, Ck, and Zhu: DEA Mdels fr Tw-Stage Prcesses 647 fairness. That is, in the spirit f cperative games, nce the ptimal value fr the centralized scre is determined, it is reasnable t search fr a decmpsitin that is as fair as pssible t bth parties. The Appendix prvides a prcedure t btain an alternative decmpsitin f e 1,Centralized and. 5. RELATIONS AND UNIQUE EFFICIENCY DECOMPOSITION In this sectin, we discuss the relatinships amng the abve develped nncperative and centralized mdels, and the standard DEA apprach. We shw that under the cnditin f ne intermediate measure, the nncperative, centralized and regular DEA appraches yield the same results. Let θ 1 and θ 2 be the (CCR) efficiency scres fr the tw stages. That is, fr a specific DMU, (i) θ 1 is the DEA efficiency based upn inputs f x i and utputs f z d and (ii) θ 2 is the DEA efficiency based upn inputs f z d and utputs f y r. We first cnsider a special case f ne intermediate measure. We have THEOREM 1: If there is nly ne intermediate measure, then e 1 = θ 1 and e2 = θ 2 regardless f the assumptin f whether the first stage is a leader r fllwer, where e 1 and are btained via ur nncperative apprach. e 2 PROOF: Suppse the first stage is a leader. Recall that mdel (1) is the standard DEA mdel fr the first stage. Therefre, e 1 = θ 1.Wenextprvee2 = θ 2. Cnsider mdel (3), which nw becmes U ry r e 2 = max Qwz s U ry rj 1 Qwz j wz j v i x ij 0 v i x i = 1 wz = e 1 U r, Q, w, v i 0, r = 1, 2,..., s; i = 1, 2,..., m (9) Nte that w = e1 z. Letting Q = Q(e 1 /z ), mdel (9) is equivalent t U ry r e 2 = max Q z s U ry rj 1 Q z j ( e 1 /z ) zj v i x i = 1 v i x ij 0 U r, Q, v i 0, r = 1, 2,..., s; i = 1, 2,..., m (10) Nte that values f v i d nt affect the ptimal value t mdel (10), indicating that (e 1 /z )z j m v ix ij 0 and m v ix i = 1 are redundant. As a result, mdel (10) becmes the standard DEA mdel fr the secnd stage, hence e 2 = θ 2. Similarly, it can be shwn that the therem is true when the first stage is a fllwer. Thus, Therem 1 indicates that when there is nly ne intermediate measure, the nn-cperative apprach yields the same result as applying the standard DEA mdel t each stage. Under the cnditin f multiple intermediate measures, we nte that the feasible regin f mdel (6) cntains the feasible regin f mdel (4). Thus, the ptimal value t mdel (6) must be greater than r equal t e 1 (4). This can be summarized as e2 THEOREM 2: Fr a specific DMU, e centralized where e centralized and e 2 apprach. arising frm mdel e 1 e2, is the ptimal value t mdel (6), and e 1 are btained via the nncperative (leader-fllwer) In the presence f a single intermediate measure, Therem 1 shws that e 1 and e 2 are respectively their DEA efficiency scres, hence are the maximum achievable efficiencies. Therefre, based upn Therems 1 and 2, we must have THEOREM 3: In the presence f a single intermediate measure, e centralized = θ 1 θ 2, with θ 1 = e1,centralized and θ 2 = e2,centralized, where θ 1 and θ 2 are the (CCR) efficiency scres fr the tw stages, respectively, and e 1,Centralized and are defined in (7). When there is a single intermediate measure, Therem 3 indicates that 1. nn-cperative and centralized appraches yield the same result as applying the standard DEA mdel t each stage, and 2. the efficiency decmpsitin under the mdel (6) is unique. We finally nte the fllwing is true with respect t the relatins between the nn-cperative and centralized appraches. Naval Research Lgistics DOI /nav

6 648 Naval Research Lgistics, Vl. 55 (2008) Table 1. IT data set. Fixed assets IT budget N. f emplyees Depsits Prfit Fractin f lans DMU ($ billin) ($ billin) (thusand) ($ billin) ($ billin) recvered THEOREM 4: 1. e 1,Centralized e0 1 when the secnd stage is the leader, 2. e 2,Centralized e 2 when the first stage is the leader. 3. θ 2 (= e2 ) e2,centralized, and θ 1 (= e1 ) e 1,Centralized always hld, regardless f which stage is the leader. 6. APPLICATION In this sectin, we cnsider three data sets. The first data set has a single intermediate measure that was first used in Wang et al. [9], and then in Chen and Zhu [3] in examining the IT impact n prductivity. The secnd data set cnsists f 30 tp US cmmercial banks and has tw intermediate measures [8]. The final data set is randmly generated using the RAND() functin in Excel Infrmatin Technlgy Table 1 presents the data set, which cnsists f 27 bservatins n firms in the banking industry. The inputs fr the first stage are fixed assets, numbers f emplyees, and IT investment. The intermediate measure is the depsits generated. The secnd stage utputs are prfits and the fractin f lans recvered. 2 Since there is nly ne intermediate measure, bth the nncperative (whichever is the leader), and centralized results are identical, with a unique efficiency decmpsitin. Table 2 reprts the results. In this case, the scres in clumns 2 and 3 are als the DEA efficiencies fr stage 1 and stage 2, respectively. Clumn 4 displays the centralized scre btained in (6), which is equal t the prduct f the related scres in clumns 2 and 3. These results verify Therems 1 and 3. The last clumn under the heading θ reprts the DEA efficiency when the intermediate measures are ignred, i.e., θ is a DEA scre with inputs being fixed assets, IT budget and emplyees, and utputs being prfit and fractin f lans recvered. It can be seen that θ = 1 fr DMU4, while bth stages are (DEA) inefficient. This pints t the fallacy f applying the DEA mdel directly, and ignring the intermediate measures. 2 Fr detailed discussin n the data, the reader is referred t Wang et al. [9]. Naval Research Lgistics DOI /nav

7 Liang, Ck, and Zhu: DEA Mdels fr Tw-Stage Prcesses 649 Table 2. Results fr IT data. DMU e 1 e 2 e 1 e2 θ Tp US Cmmercial Banks Seifrd and Zhu [8] examine the perfrmance f the US cmmercial banks in 1995 via a tw-stage prductin prcess defined in terms f prfitability and marketability. The inputs t the first stage are numbers f emplyees, assets ($millins) and equity ($millin). The intermediate measures are prfit ($millins) and revenue ($millins). Outputs frm the secnd stage are market value ($millins), earnings per share ($) and returns t the investrs (%). Table 3 displays the data fr the tp 30 banks, and Table 4 reprts the results f the applicatin f mdel (6). The last clumn shws, fr each DMU, the DEA scre fr the verall prcess when emplyees, assets and equity are used as the inputs and prfit and revenue are used as the utputs. Cnceptually, such a DEA scre is similar t the e 1 e2 in the nn-cperative apprach r ecentralized in the centralized apprach f mdel (6). Hwever, the relatin θ = θ 1 θ 2 des nt always hld. This can be seen by using the scres reprted in clumns 2 and clumn 6. Clumn 2 represents the DEA scres fr the first stage and clumn 6 represents the DEA scres fr the secnd stage. T test the uniqueness f ur efficiency decmpsitin under the centralized apprach, we als calculate e 1+ (mdel (8)) and e 2+. Our results indicate that e1 = e 1+ and e 2 = e 2+ fr all the DMUs. Therefre, e 1,Centralized and defined in (7), are uniquely determined via mdel (6) in ur case. Finally nte that the results in Table 4 als verify ur Therems 2 and 4. Nte als that e centralized = e 1 e2 hlds fr all the banks, where e 1 and e 2 represent the efficiency scres fr the tw stages when the first stage is treated as the leader. This may indicate that the first stage r the prfitability stage is mre imprtant Randmly Generated Data Set One is tempted t cnclude frm the previus example that there might be a direct cnnectin between the centralized ptimal scre and that arising frm the results f treating stage 1 as the leader. T gain further insights, a randmly generated set f data was created as displayed in Table 5. The utcmes frm the varius analyses appear in Table 6. Amng the 27 DMUs, 15 shw centralized scres that exceed the crrespnding aggregate scres in bth the stage 1 leader and stage 2 leader cases. Fr the remaining 12 DMUs, the fllwing are the utcmes: 1. Fr three f the DMUs (8, 10, and 12), the stage 1 leader scres and decmpsitin match thse f the centralized analysis, but differ frm the stage 2 leader results; 2. Fr six f the DMUs (1, 3, 7, 19, 20, and 27), the stage 2 leader results match thse f the centralized analysis, but differ frm thse f the stage 1 leader; 3. Fr the three DMUs 14, 16, and 23, all three analyses prduce the same results. This latter set f utcmes appears t pint t the general unpredictability f any cnnectin between the results frm the centralized apprach and thse frm the tw appraches invlving a leader and a fllwer. 7. CONCLUSIONS In many DEA situatins, DMUs may take the frm f multiple stages with intermediate measures. It has been recgnized that the existing DEA appraches, including the standard DEA mdels, d nt apprpriately address such multi-stage structures. This paper presents alternative ways t address the cnflict between stages caused by the intermediate measures, and at the same time prvide efficiency scres fr bth individual stages and the verall prcess. Our nncperative and centralized appraches shw that the verall efficiency f the tw-stage prcess is the prduct f efficiencies f the tw stages. Naval Research Lgistics DOI /nav

8 650 Naval Research Lgistics, Vl. 55 (2008) Table 3. US cmmercial bank. Bank Emplyees Assets Equity Revenue Prfits Market value Earnings Returns 1 Citicrp 85, , , , 690 3, BankAmerica Crp. 95, , , , 386 2, NatinsBank Crp. 58, , , , 298 1, Chemical Banking Crp. 39, , , , 884 1, J.P. Mrgan & C. 15, , , , 838 1, Chase Manhattan Crp. 33, , 173 9, , 336 1, First Chicag NBD Crp. 35, , 002 8, , 681 1, First Unin Crp. 44, , Banc One Crp. 46, , Bankers Trust New Yrk Crp. 14, , 000 5, 000 8, Fleet Financial Grup 30, Nrwest Crp. 45, PNC Bank Crp. 26, , 404 5, KeyCrp 28, , Bank f Bstn Crp. 17, , 397 3, Wells Farg & C. 19, , 316 4, 055 5, 409 1, Bank f New Yrk C. 15, , 685 5, 223 5, First Interstate Bancrp 27, , 071 4, , Melln Bank Crp. 24, , 129 4, 106 4, Wachvia Crp. 15, SunTrust Banks 19, Barnett Banks 20, , Natinal City Crp. 20, , 199 2, First Bank System 13, , 874 2, , Cmerica 13, , Batmen s Bancshares 17, , U.S. Bancrp 14, , CreStates Financial Crp. 13, , Republic New Yrk Crp. 4, , MBNA 11, Bank θ 1 Table 4. US cmmercial bank results. Stage 1 as the Leader Stage 2 as the Leader Centralized DEA (= e1 ) e2 e 1 e2 e 1O θ 2 (= e2o ) e 1O e 2O e 1,Centralized e Centralized Naval Research Lgistics DOI /nav θ

9 Liang, Ck, and Zhu: DEA Mdels fr Tw-Stage Prcesses 651 Table 4. (cntinued) Bank θ 1 Stage 1 as the Leader Stage 2 as the Leader Centralized DEA (= e1 ) e2 e 1 e2 e 1O θ 2 (= e2o ) e 1O e 2O e 1,Centralized e Centralized θ Table 5. Randmly generated data set. a DMU x1 x2 x3 z1 z2 z3 y1 y a This data set was generated using the RAND() functin in Excel. Table 6. Results fr randmly generated data set. DMU θ 1 (= e1 ) Stage 1 as the Leader Stage 2 as the Leader Centralized e2 e 1 e2 e 1O θ 2 (= e2o ) e 1O e 2O e 1,Centralized e Centralized Naval Research Lgistics DOI /nav

10 652 Naval Research Lgistics, Vl. 55 (2008) Table 6. (cntinued) DMU θ 1 (= e1 ) Stage 1 as the Leader Stage 2 as the Leader Centralized e2 e 1 e2 e 1O θ 2 (= e2o ) e 1O e 2O e 1,Centralized e Centralized APPENDIX In case f multiple ptimal slutins that lead t nnunique f e 1 and e2 in the centralized apprach, we develp the fllwing prcedure t achieve a fair and alternative distributin f e 1 and e2 between the tw stages. Suppse there exists a λ, such that [ λe 1 + (1 λ)e 1+ ] [ λe 2 + (1 λ)e 2+ ] = e = e Centralized (11) Let a = e 1 e2 +e 1+ e2+ 2e, b = 2(e e 1+ e2+ ) and c = e1+ e2+ e, then (11) becmes aλ 2 + bλ + c = 0. On the basis f the related slutin λ, we can btain a fair distributin f e 1 = λ e 1 + (1 λ )e 1+ and e 2 = λ e 2 + (1 λ )e 2+. We next need t test whether there exist a set f weights that are related t the abve efficiency distributin. If nt, we then find a set f weights and efficiency distributin that is clse t the abve distributin. Cnsider the fllwing mdel Max D W d z d0 m V i x i0 which is equivalent t Max w d z d u r y r = e Centralized [ λ e 2 + (1 λ )e 2+ ] D w d z d e Centralized 0 w d z dj v i x ij 0 u r y rj w d z dj 0 v i x i = 1 w d 0, d = 1, 2,..., D; v i 0, i = 1, 2,..., m; u r 0, r = 1, 2,..., s (13) U r y r0 m V i x i0 = e Centralized U r y r0 D λ e 2 + (1 λ )e 2+ W d z d0 Let the ptimal slutin be w d, v i, u r. Then e1 e 2 = E / D w d z d. Nw, cnsider U r y r Max D W d z d = D w d z d, and D W d z dj m 1, j = 1, 2,..., n V i x ij U r y rj D 1, j = 1, 2,..., n W p z pj W d 0, d = 1, 2,..., D; V i 0, i = 1, 2,..., m; U r 0, r = 1, 2,..., s (12) U r y r m V i x i D W d z d m V i x i D W d z dj m V i x ij = e Centralized [ λ e 1 + (1 λ )e 1+ ] 1, j = 1, 2,..., n Naval Research Lgistics DOI /nav

11 Liang, Ck, and Zhu: DEA Mdels fr Tw-Stage Prcesses 653 which is equivalent t U r y rj D 1, j = 1, 2,..., n W d z dj W d 0, d = 1, 2,..., D; V i 0, i = 1, 2,..., m; U r 0, r = 1, 2,..., s, (14) Max u r y r u r y r e Centralized v i x i = 0 [ λ e 1 + (1 λ )e 1+ ] m v i x i 1 0 u r y rj w d z dj 0 w d z dj v i x ij 0 w d z d = 1 w d 0, d = 1, 2,..., D; v i 0, i = 1, 2,..., m; u r 0, r = 1, 2,..., s (15) Let the ptimal slutin be w d, v i, u r, then e1 = e Centralized / s u r y r and e 2 = u r y r. Let ( D d = w d z d [ λ e 1 + (1 λ )e 1+ ] )2 ( + e Centralized / w d z d [ λ e 2 + (1 λ )e 2+ ] )2 ( d = e Centralized / u r y r [ λ e 1 + (1 λ )e 1+ ] ) 2 ( + u r y r [ λ e 2 + (1 λ )e 2+ ] ) 2 If d >d, then e 1 = ecentralized / u r y r and e 2 = u r y r; D If d >d, the e 1 = w d z d and e 2 = ecentralized / w d z d; D If d = d, then e 1 = w d z d and e 2 = ecentralized / e 1 = ecentralized / w d z d, r u r y r and e 2 = ACKNOWLEDGMENTS u r y r. The authrs thank tw annymus referees and the Assciate Editr fr their cnstructive cmments n an earlier versin f this paper. Prfessr Liang thanks the supprt by the NSFC f China (Grant N ). REFERENCES [1] G.P. Cachn, Supply chain crdinatin with cntracts, Handbk in peratins research and management science: Supply chain management, S. Graves and T. de Kk (Editrs), Elsevier, Nrth Hlland, [2] A. Charnes, W.W. Cper, and E. Rhdes, Measuring the efficiency f decisin making units, Eur J Oper Res 2 (1978), [3] Y. Chen and J. Zhu, Measuring infrmatin technlgy s indirect impact n firm perfrmance, Infr Technl Manage J 5 (2004), [4] Y. Chen, L. Liang, F. Yang, and J. Zhu, Evaluatin f infrmatin technlgy investment: A data envelpment analysis apprach, Cmput Oper Res 33 (2006), [5] J. Chilingerian and H.D. Sherman, Health care applicatins: Frm hspitals t physician, frm prductive efficiency t quality frntiers, Handbk n data envelpment analysis, W.W. Cper, L.M. Seifrd and J. Zhu (Editrs), Springer, Bstn, [6] J.F. Gaski, The thery f pwer and cnflict in channels f distributin, J Market 15 (1984), [7] Z.M. Huang and S.X. Li, C-p advertising mdels in a manufacturing-retailing supply chain: A game thery apprach, Eur J Oper Res 135 (2001), [8] L.M. Seifrd and J. Zhu, Prfitability and marketability f the tp 55 US cmmercial banks, Management Sci 45 (1999), [9] C.H. Wang, R. Gpal, and S. Zints, Use f data envelpment analysis in assessing infrmatin technlgy impact n firm perfrmance, Ann Oper Res 73 (1997), Naval Research Lgistics DOI /nav