AN EXTENDED AGGREGATED RATIO ANALYSIS IN DEA

J Syst Sci Syst Eng (Jun 2011) 20(2): 249-256ISSN: 1004-3756 (Paper) 1861-9576 (Online)DOI: 10.1007/s11518-011-5162-1CNII -2983/NTechnical NoteAN EXTENDED AGGREGATED RATIO ANALYSIS IN DEAMalin SONG'Jie Wu2Yumei WANG3'School of Statistics and Applied Mathematics, Anhui University of Finance & Economics, Bengbu, 233030, Chinasongmartin@163.com (网)School ofManagement, University of Science & Technology ofChina, Hefei, 230026, Chinajacky012@mail.ustc.edu.cn'School of Satistics and Applied Mathematics, Anhui University of Finance & Economics, Bengbu, 233030, Chinaacwangyumei@163.comAbstractData Envelopment Analysis (DEA) and Ratio Analysis (RA) are two widely used methods formeasuring units' productivity and any other criteria that could be assessed based on the available inputand output variables. A number of researchers have studied DEA and RA and noted the positive andnegative differences between them. Aggregated ratio analysis (ARA) model, which provide animportant linkage between DEA and RA theory, is. equivalent to the CCR DEA model, and thisequivalence property offers a great deal of opportunities for DEA to be interpreted and applied indifferent ways. This paper extends the results of ARA model and proposes an extended aggregatedratio analysis (EARA) model, similar as the development ftom CCR model to BCC model in DEAcontext. The proposed model can offer an insight into the characteristic of returns to scale, playing thecorresponding role as BCC model does. The numerical example is revisited in the paper and the resultsare compared.Keywords: Data Envelopment Analysis (DEA), aggregated ratio analysis, efficiency evaluation,returs to scale1. Introductionvariables. Although the two methods share theData Envelopment Analysis (DEA) andsame goal, their approach is different in manyRatio Analysis (RA) are two methods foraspects. In brief, DEA is a nonparametricbenchmarking production units' productivity,method that generates a single relative efficiencyproftability or any other criteria that could bescore, while considering multiple inputs andassessed based on the available input and outputoutputs simultaneously. RA, in contrast, uses theThis work is fnancial support by National Natural Science59), Ministry ofEducation Foundation of Humanities and Social Sciences of P.R.中国煤化工aton of NauralScience for Colleges and Universities in Anhui, China (KJ2011AMYHC N M H Gdation of Anbui,China (AHSK07-08D25, AHSKF09-10D116, AHSK09-10D14).◎Systems Engineering Society ofChina and Springer-Verlag Berlin Heidelberg 2011Song et al: An Extended Aggregated Ratio Analysis in DEA250J Syst Sci Syst Engratio of a single output to a single input andalso prove that the ARA model is equivalent togenerates a relative efficiency score by dividingthe DEA CCR model by Charmes et al. (1978),the aforesaid ratio by the corresponding "bestthat is, to any given DMU, the efficiency ofperformer's”ratio, on this specific ratioaggregated ratio model is the same as thedefinition. DEA and individual ratios agreeefficiency of CCR model, and the best DMU inweakly on the performance of the unit and it isaggregated ratio model is on the DEA frontier.because DEA reflects the overall efficiencyHowever, the ARA model in Wu et al. (2005)based on a“ratio” of the weighted sum ofcan't help to detect the RTS (returns to scale)relevant outputs over inputs and the RA is atype as DEA BCC model does. Thus it isfactor specific efficiency.necessary to advance an extended aggregatedA number of researchers have studied DEAratio analysis (EARA) model which can play theand RA and noted the positive and negativesimilar role as BCC model while maintainingdifferences between them, such as Chen et al.the advantages of ARA model. In this paper, we(2009), Thanassoulis et al. (1996), Cronje (2002),propose the EARA (extended aggregated ratioFeroz et al. (2003), Bowlin (2004) and so on.analysis) model, similar as the development thatBesides, there are some studies which attemptedthe CCR model evolves into the BCC model in0 combine or relate the DEA and RA. ForDEA context. This extended model can offer anexample, Chen & Ali (2002) characterized theinsight into the characteristic of returns to scale.inherent relationships between the DEA frontierThe rest of this paper unfolds as follows. SectionDMUs and output-input ratios. They showed2 presents the extended aggregated ratio analysisthat top-ranked performance by ratio was a DEAmodel. The returns to scale characterizations arefrontier point and DEA subsumed the premise ofillustrated in Section 3. A numerical illustrationthe RA. Being motivated by the study of Chen &is revisited in Section 4 and concluding remarksAli (2002), Wu et al. (2005) proposed anare in Section 5.aggregated ratio analysis model in DEA. Thisratio model has been proven to be equivalent to2. The Extended Aggregated Ratiothe CCR model. Some other related studies canAnalysis (EARA) Modelbe referred to Chen & McGinnis (2007), DespicWe assume that there are n DMUs to beet al. (2007), Emrouznejad & Amin (2009),evaluated. Each DMU consumes m differentGonzalez- Bravo (2007), Hollingsworth & Smithinputs to produce s different outputs. Let the(2003), Holl6 & Nagy (2006) and so on.inpuand output vectors of DMUIn the current paper, we will consider thebe x,+=(x2..xmy)" >0,y =(Yyj,Y2j，study of Wu et al. (2005) among the variousrelated studies of DEA and RA abovementioned...y).. >0, +.... respetively.As said before, Wu et al. (2005) present an ARAThen themathematicalprogramming(aggregated ratio analysis) model where theproblem of the BCC“ratio-form”DEA model inaggregated sum of ratios of all outputs to allBanl中国煤化工inputs is regarded as a performance index. TheyYHCNMHGSong et l: An ExtendedAggregated Ratio Analysis in DEAJSyst Sci Syst Eng251=它u4ryro -uo)/analysis framework, here we introduce thei=following extended aggregated ratio analysis(EARA) model for DMUg:s.t.(_ny-u)/2xySl,网ρ° =maxp= SpWo -只Poj=...n u,2E, y2E,el旨Xxoi= .,..n. r = .,...(1). S1, j=... .and the BCC envelopment fom is:θ° = minθpr2ε，l=1,2...mxs.(4)Zqy s0x01-=.2.m.Definition 2 DMUo is EARA efficient if thereexists an optimal solution in Model (4){211.,..mxs} and Po with pi>0川2>12...such that p' =1.From the model (4) and similar to the(2)discussions of Wu et al. (2005), we can find that=1a DMUo is EARA efficient if and only if it isDefinition 1 DMU is DEA eficient if and onlyDEA eficient.if there exists aroptimal solution(u',v' )= (w ,....v".. of (1) with3. Returns to Scale Characterizationsui' >0 and v >0, and an optimal solutionIn the previous section, we have established(@",N)of(2)suchthat 9 =h' =1. .the equivalent relationship between the EARAFor any given DMU;, the output-input ratiomodel and the BCC DEA model. In this section,vector w; is defined aswe now continue to extend our analysis to theproblem of returns to scale (increasing,w;={wy+1.=...mxs}decreasing, or constant) discussed in Banker etal. (1984). We shall examine returns to scale:1+1...m; =...xylocally at a point, say (xo,yo), on the efficientproduction surface, and relate it to the sign ofWu et al. (2005) proposed the followingthe intercept termPo in the fractionalaggregated ratio analysis (ARA) model:programming problem (4) for this purpose.p' =maxp= 2 PrWoAs shown in Banker et al. (1984), toascertainwhetherincreasing, constant2prmyS1, j+=..n.decreasing returns to scale are present at(x0,yo) we let Zo.8={(x,y)|(x,)=pr≥ε，1= .2...mnxs.(3)(1+ 8(x,y,)}, be a point in the neighborhoodTo judge the RTS (returns to scales)of中国煤化工e sitbly smallsituations for decision making units in ratiosAssufYHCNMH Gilitysetbe T=Song et al: An Extended Aggregated Ratio Analysis in DEA252J Syst Sci Syst Eng{[x,)|y 20 can be produced fromx20}，thenwe can say that=(1-1+7网x0(7)(a) Increasing returms to scale are present ifAccording to the discussion of retums toand only if there exists 8° >0 such that (a1)scale above, let Zo. g∈T, δ° >δ≥0 andZo,8∈T for δ >δ20 and (a2) Zo,8ET forZo,8ET，δ<0, |δkδ, since-8" <δ<0.(b) Decreasing returns to scale are present ifs1,j=,.,n, soifand only if there exists δ" > 0 such that (c1)Zo.8ET for δ >δ>0 and (c2) Zo.8∈TZo,8∈T, then (x,y)=((1+0)x,(1+ )yo)for -8* <δ≤0.∈T and丛_罗PosI, ie.(C) Constant retums to scale are present ifand only if there exists δ >0 such that (b1)(-5它一p:≤0.Zo.,s∈T forall δ such that |8|<δ , or (b2)1+ 8Zo.,8ET forall δ such that 0<8kδ .If Zo,8ET , thenIn fact, the EARA model (4) is equivalent to(x,y)= (1+ 8)xo,(1+ 8)yo)&T , andthe following model:之立p:其-之B>1,ie.p =mxp=stp。迎之旦.同间xy问x网i=lxi0 = x;os.t.之立p其立BsI, j-=..(1-;1+δ它一p6>0.Asaresult, Zo.8∈T, 8 >δ≥0 andPr2E,1i=..m; r=...s,Po is free. (5Zo,8ET, δ<0, |8Kδ", i.e.Obviously, if (xo,yo) is efficient under the-)p。≤0, 8 >δ20 andEARA model, p* =1,i.e.1-43公品pi;x_二P-1=0.(6)(1-1+8它二>0，δ<0, 18K8",90之”xio 台x= x;opi<0, |8K|δ"|. On the other side, |8| isNow, for the point (x,y)=((1+ 8)xq,(1+8)yo)e Zo,8, we havesuitably small, so there mustbe 8 >0 to makethe inequality |δ|<\8° | be true.p:些_S Po_1As a whole, DMU(xo,yo) is increasing之ix E[x;0returns to scale if and onlyif P' <0.(1+)yo .Similarly, we can discuss the situation of之之:台台(1+8)Xxio 台(1+)xioconstant and decreasing returms to scale, and we. Po--1have the following corollary.=l i问l(1+0)x0Corollary 1 The following conditions identify=号Po_号_Pohe中国煤化工for the EARA台xo台(+8)x0modeYHCNMHGSong et al: An Extended Aggregated Ratio Analysis in DEA」Syst Sci Syst Eng253(2-1) Increasing returmns to scale prevail atNow, we can qualify the three corollaries as(x,y) if and only if po<0 for all optimalfollows:solutions;Corollary 2 The following conditions identify(2-2) Decreasing returns to scale prevail atthe situation for returns to scale for the EARA(x0,y) if and only if po>0 for all optimalmodel given in (5) when the multiple optimalsolutions exist,(2-3) Constant retums to scale prevail at(1-1) Increasing returms to scale prevail at(xq,yo) if there is an optimal solution satisfying(x,yo) ifand onlyif p'<0;p'o=0.(1-2) Decreasing returns to scale prevail atOne practical problem with the criterion in(xg,Yo) ifandonlyif po >0;Corollary 1 above is that there may exist(1-3) In other cases, constant retums to scaleatermative optimal solutions for the EARA DEA'prevail at (xp,y).model where Po exceeds zero in some optimalsolution but falls short of zero in another optimal4. An Ilustrated Examplesolution for the same problem. Because theA data set ilustrated in Wu et al. (2005),solution algorithm terminates whenever anwhich was also used in Ali et al. (1995) andoptimal solution is reached, the decision aboutChen & Ali (2002), is revisited in this paper, inreturns to scale then becomes dependent onwhich there are 11 DMUs to be evaluated. Eachwhich particular optimal solution was reached.of them consumes two inputs 莉and 2 toTo implement the revised criterion in practice,produce two outputs )1 and y2. The data of inputs,we need the following two step procedure:outputs and output-input ratios can be seen inStep 1: Solve the EARA model (5), supposeWu, Liang, Huang and Li (2005).that the optimal value of the objective functionis p'.4.1 Efficiency AnalysisStep 2: Solve the following model toTable 1 reports the solution to model (4)determine the upper bound of Po.where we use single ratio data set, ie.p=maxPo2 and 兰. EARA result by our model isFiOslightly different from that by the BCC model立E.--5151-1.because our computation is done by the Excelsoftware, which has a system error.Pr2E, i=.,m,;.=..s,Po is free. (8)Similar as the BCC DEA model, in Table 1, themultipliers in our EARA model might also beTo determine the lower bound of p'o, wezero. For example, three multipliers, p1, p2 andcanchangetheobjectivefunction tcp3， are designed to be zero by DMU3 asPo = min Po. Obviously, for any Po, we haveindic中国煤化工f Table 1. Thept≥po≥Pr.zero:fYHCNMHGursfromDMU6Song et a山; An Extended Aggregated Ratio Analysis in DEA254J Syst Sci Syst Engto DMU11. To deal with the zero multiplierand the result of po is listed in the fith columnproblem, techniques in DEA can be utilized here.of Table 2. we can find that the values of pt ,For example, cone ratio and assurance regionspio and po forDMU3, DMU6, DMU7,can be adopted to restrict the multipliers.DMU8, DMU9, DMU10 and DMU11 are theAnyway, this is beyond this research and will besame, which means that the solution of model (5)left for further consideration.for these DMUs is unique, and we can determineIn the last line of Table 1, the values of p0the returms to scale result shown in the last linefor each DMU are ilustrated, which can indicateof Table 2 for these DMUs by Corollary 1. Asthe characteristic of returas to scale according tofor other DMUs, we can determine the result ofCorollary 1. While there may exist altemativereturns to scale according to Corollary 2, whichoptimal solutions for the EARA model, weis also shown in the last line of Table 2.cannot determine the retums to scale resultBesides, we apply the BCC model to checksimply according to the value of p0 here.the returns to scale states for those eleven DMUsagain. The results show the remarkable4.2 Returns to Scale Analysisconsistency with those in the rightmost columnAs discussed above, to apply the two-stepin Table 2. In other words, the numericalprocedure based on the discussion in Section 3,example approves the validity of our proposedwe first solve the EARA model (5) and themodel.optimal value of the objective function ρ' isillustrated in the second line of the Table 2.5. Concluding RemarksAfter the optimal values of the objectiveThis paper extends the ARA model which isfunction ρ° in the EARA model (5) areequivalent to the CCR DEA model and proposesdetermined, we can solve the model (8) toan extended aggregated ratio analysis EARA)determine the upper bound of Po which ismodel that is similarly extended ftom CCR toshown in the third column of Table 2. ToBCC model in DEA. We also revisit thdetermine the lower bound of Po ，we cannumerical example in previous studies using ourchange the objective function to po = min Po,EARA model and DEA BCC model, and theTable 1 EARAmodel in (4) with a single ratio data setDMUEARA score_ppp:p4pODMU 10.0380.11640.32050.01287.0787DMU220.1194.57530.45041.91332636.9DMU30.7291(00.1184-6.3158DMU40.68534.8605 .21.732.1442127.6DMU 51.4779.02150.20620.2017871.17DMU 60.03150.3242-1.0858DMU 70.3918-17.143DMU 80.4326DMU90.5211DMU 100.51350.2865中国煤化工15.96DMU 110.4387_MYHCNMHGSong et al: An Extended Aggegated Ratio Analysis in DEAJ Syst Sci Syst Eng255Table 2 Results of retums to scalep_PoPoRTS resultDMU130.5087.0787-17.143ConstantDMU22.5313E62636.9-13.333DMU30.7291-6.3158-6.3158.IncreasingDMU47.5311E82127.6-1.5162.DMU52.467E8-8.2759DMU60.9409-1.0858-1.0858.DMU70.3918DMU80.4326DMU90.5211- 6.3158DMU100.513515.96DecreasingDMU11.0.4387results ftom EARA model and DEA BCC modelevaluation in data envelopment analysis.show the remarkable consistency between theEuropean Jourmal of Operational Research,two models. Similar as the advantage of ARA80 (3): 462 473relative to CCR, we can also conclude that[2] Banker, R.D, Charmes, A. & Cooper, W.W.EARA can be used in more scenarios than BCC(1984). Some models forestimatingespecially in cases that some inputs aretechnical and scale inefficiencies in dataindependent to some outputs.envelopment analysis. Management Science,Future research perhaps appears in the30 (9): 1078-1092research on the difference of EARA technique [3] Bowlin, W.F. (2004). Financial analysis ofand BCC approach in some special scenarioscivil reserve air fleet participants using dataincuding that mentioned above. As a fact, CCR,envelopment analysis. European Jourmal ofBCC and other models build up the frameworkOperational Research, 154 (3): 691 709of DEA. Similarly, ARA, EARA and future[4] Charmnes, A, Cooper, W.W. & Rhodes, E.achievement in the area will build up the(1978). Measuring the eficiency of decisionframework of ratio analysis, which may bemaking units. EuropeanJournal ofanother research direction in future. Obviously,Operational Research, 2 (6): 429 444the realistic application of the proposed model[5] Chen, w.C. & McGinnis, LF. (2007).herein is also an interesting future research.Reconciling ratio analysis and DEA asperformance asssment tools. EuropeanAcknowledgmentJourmal of Operational Research, 178 (1):We would like to thank referees for their help277-291to improve the quality of the paper.[6] Chen, Y. & Ali, AI. (2002). Output -inputReferencesratio analysis and DEA frontier. European[1] Ali, AII, Lerme, C.S. & Seiford, .L.M.中国煤化工rch, 142 (3):(1995). Components of efficiency:HCNMHGSong et al: An Extended Aggregated Ratio Analysis in DEA256」Syst Sci Syst Eng[7] Chen, Y, Li, K.W, Xu, H.Y. & Liu, S.F.efficiency in the enlarged European Union.(2009). A DEA-TOPSIS method forMNB working papersmultiple criteriadecision analysis in[15]Thanassoulis, E, Boussofiane, A. & Dyson,emergency management. Jourmal of SystemsR. (1996). A comparison of dataScience and Systems Engineering, 18 (4):envelopment analysis and ratio analysis as489- 507tools for performance assessment. Omega,[8] Cronje, JJL. (2002). Data EnvelopmentIntermational Journal of ManagementAnalysis as a measure for technicalScience, 24 (3): 229-244efficiency measurement in banking - a [16]Wu, D., Liang, L., Huang, Z.M. & Li, S.research framework. Southern African(2005). Aggregated ratio analysis in DEA.Business Review, 6 (2): 32-41InternationalJourmalofInformation[9] Despic, O, Despic, M. & Paradi, J.C.Technology and Decision Making, 4 (3):(2007). DEA-R: ratio-based comparative369-384efficiency model, its mathematical relationto DEA and its use in applications. JoumalMalin Song is an Associate Professor in Schoolof Productivity Analysis, 28 (1): 33-44of Statstics and Applied Mathematics, Anhui[10]Emrouznejad, A. & Amin, G.R. (2009).University of Finance and Economics. He is aDEA models for ratio data: ConvexityStanding Director of Institute of Industrialconsideration.Applied MathematicalEconomics, Anhui, China, and Research FellowModeling, 33 (1): 486 498in Economic Development Research Center,[11]Feroz, E.H, Kim, S. & Raab, R.L. (2003).Anbui University of Finance and Economics.Financial statement analysis: a dataHis major field of study includes Environmentalenvelopment analysis approach. Jourmal ofEconomics and System Modeling and Analysis.the Operational Research Society, 54 (1):48-58Jie Wu is a Lecturer in School of Management,[12]Gonzalez-Bravo,M.I.(2007).University of Science & Technology of China.Prior-ratio-analysis procedure to improveHis major field of study includes Managementdata envelopment analysis for performanceDecision and System Modeling and Analysis.measurement. Journal of the OperationalResearch Society, 58 (9): 1214-1222Yumei Wang is a Professor in School of[13]Hllingsworth, B. & Smith, P. (2003). UseStatistics and Applied Mathematics, Anhuiof ratios in data envelopment analysis.University of Finance and Economics. HerApplied Economics Letters, 10 (11):major field of study includes System Modeling733-735and Analysis.[14]Holl6，D. & Nagy, M. (2006). Bank中国煤化工MHCNMHG