1、Int. J. Production Economics 112C3TungIn this research we employ data envelopment analysis (DEA) to measure the Malmquist productivity of semiconductorDEA is a multiple inputoutput efficient techni-efficiency scores for individual units as theirFa re et al. (1992, 1994a) developed the DEA-nology fro
2、ntier and technical efficiency, are alsofurther examined in this research. By the technologyARTICLE IN PRESSfrontier shift (FS), the development or decline of allDMUs is able to measure. Technical efficiency0925-5273/$-see front matter r 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.ijpe.2007
3、.03.015C3Corresponding author. Tel./fax: +88635731739.E-mail address: fliumail.nctu.edu.tw (F.-H.F. Liu).que that measures the relative efficiency of decision-making units (DMUs) using a linear programmingbased model. The technique is non-parametricbecause it requires no assumption about the weights
4、of the underlying production function. DEA wasoriginally proposed by Charnes et al. (1978) and thismodel is commonly referred to as a CCR model.The DEA frontier DMUs are those with maximumoutput levels for given input levels or with minimuminput levels for given output levels. DEA providesbased Malm
5、quist productivity index by CCRmodel. The DEA-based Malmquist productivity isa combined index that can be extended to measurethe productivity change of DMUs over time. It hasbeen applied in many ways, as described in Fa re etal. (1994b), Grifell-Tatje and Lovell (1996), Fulgi-niti and Perrin (1997),
6、 Lo thgren and Tambour(1999), Herrero and Pascoe (2004), Wei (2006) andothers. The two components embedded in Malm-quist productivity, measuring the changes in tech-of technical change, the measurement of the frontier forward shift, and the measurement of the frontier backward shift of acompany over
7、 two consecutive periods. This approach not only reveals patterns of productivity change and presents a newinterpretation along with the managerial implication of each Malmquist component, but also identifies the strategy shiftsof individual companies based upon isoquant changes. Therefore, one can
8、judge with greater accuracy whether or not suchstrategy shifts are favorable and promising. We use slacks-based measurement (SBM) and Super-SBM models to obtainmore accurate measurements. Comparison is made between the results from SBM/Super-SBM and CCR models.r 2007 Elsevier B.V. All rights reserve
9、d.Keywords: Data envelopment analysis; Malmquist productivity; Super-SBM1. Introduction technical efficiency measure, with a score of oneassigned to the frontier (efficient) units.packaging and testing firms in Taiwan from 2000 to 2003. Malmquist productivity has three components: the measurementDEA
10、 Malmquist productivitysemiconductorFuh-Hwa Franklin LiuDepartment of Industrial Engineering Abstract(2008) 367379measure: Taiwanesecompanies, Peng-hsiang WangUniversity, Hsin Chu, Taiwan 300, Republic of China (Taiwan)accepted 20 March not greater than the optimal CCR y0, and (II) ADMU (xi0,yr0) i
11、s CCR-efficient, if only if DMU0isARTICLE IN PRESSF.-H.F. Liu, p.-h. Wang / Int. J. Production Economics 112 (2008) 367379368change (TEC) is used to measure the change intechnical efficiency. It is also a measure of howmuch closer to the frontier the company (DMU)iswhen crossing the two consecutive
12、times. We defineTEC and Malmquist productivity as R3and R4,respectively, in Section 4.1 for the performancemeasurement.Chen and Ali (2004) applied the DEA Malmquistproductivity measure to the computer industries bythe CCR model to assess the four distance functionsof Malmquist productivity. Moreover
13、, they discov-ered more information about the two componentsthat obscure in the Malmquist productivity index.We define them as R1and R2in Section 3 for theperformance measurement in this research andaccount for the attributes. Their approach not onlyreveals patterns of productivity change and presen
14、tsa new interpretation along with the managerialimplication of each component, but also identifiesthe strategy shifts of individual DMUsinaparticular time period. They determined whethersuch strategy shifts were favorable and improving.However, the ratio efficiency y0*by the CCRmodel is not able to
15、take account of slacks. Forinstance, the optimal solution y0* 1 might be withpositive slacks. In the DEA Malmquist productiv-ity, the DMU0is regarded as efficient but actually, itshould be regarded as inefficient. Therefore, it isimportant to observe both the ratio efficiency andthe slacks. Some att
16、empts have been made to unifyy0*and slacks into a scalar measure.Charnes et al. (1985) developed the additivemodel of DEA, which deals directly with inputexcess and output shortfalls. But this model has noscalar measure (ratio efficiency) per se. Thus,although this model can discriminate betweeneffi
17、cient and inefficient DMUs by the existence ofslacks, it has no means of gauging the depth ofinefficiency, similar to y0*in the CCR model.Tone (2001) developed a slacks-based measure(SBM) of efficiency in DEA, which takes account ofscalar measure and slacks. Further, Tone (2002)developed a SBM of su
18、per efficiency (Super-SBM)in DEA for discriminating between efficient DMUs.Super efficiency measures the degree of superioritythat efficient DMU0possesses against other DMUs.To extend the investigation on influence fromslacks to Malmquist productivity index, Chen(2003) proposed a non-radial Malmquis
19、t productiv-ity index, which is able to eliminate possibleinefficiency represented by the non-zero slacks tomeasure the productivity change of three ChineseSBM-efficient. Moreover, because the CCR score isa radical measure and takes no account of slacks,the particular DMU0may have an efficiency scor
20、ey0* 1 although it has a shortfall sC3rX0, but aninefficiency score r0*p1 for SBM measure when thefactor is taken into account. In this case, we canreduce the misleading result with the SBM measure.On the other hand, the SBM score r0* 1 guaran-tees the particular DMU has the more preciseefficiency s
21、core. Tone (2004) discusses the differ-ences between the slack-based and radial-basedmajor industries. Instead, we employ the SBM andSuper-SBM models in this research. In addition toTEC (R3) and Malmquist productivity (R4) whichexisted in the traditional Malmquist productivitymeasurement, we also in
22、vestigate the twocomponentsR1and R2proposed by Chen andAli (2004) to interpret a more detailed manage-ment implication. The next section reviews how theDEA-based Malmquist productivity index works.We also present the Malmquist productivityapproach.2. DEA Malmquist productivity indexFa re et al. (199
23、2) construct the DEA-basedMalmquist productivity index as the geometricmean of the two Malmquist productivity indicesof Caves et al. (1982): one measures the change inefficiency and the other measures the change in thefrontier technology. The frontier technology, deter-mined by the efficient frontie
24、r, is estimated usingDEA for a set of DMUs.There are n DMUs under comparison for theirperformance. Let xijand yrjdenote the value of theith input (i 1,y,m) and the rth output(r 1,y,s)ofDMUj(j 1,y,n), respectively.The slack variables for the ith input and the rthoutput are, respectively, represented
25、by siC0and sr+,which indicate the input excess and output shortfall,respectively. The variable ljdenotes the weight ofDMUjwhile assessing the performance y0of theobject DMU0.Instead of a radial-based model, we now use theSBM model and explain the reason for thesubstitution. A notation with * in supe
26、rscriptindicates it is the optimal solution. We must firstknow two proved theorems: (I) The optimal SBM r0*approaches in depth.ARTICLE IN PRESSF.-H.F. Liu, p.-h. Wang / Int. J. Production Economics 112 (2008) 367379 369Let Daxb0; yb0 denote the relative efficiencyof a particular DMU0in period b agai
27、nst theperformance of those DMUs in period a. There arefour possible pairs (a,b) for analysis of theMalmquist productivities, (t,t), (t+1, t), (t,t+1)and (t+1,t+1). Hence, there are four distances tobe measured, Dtxt0; yt0, Dt1xt0; yt0, Dtxt10; yt10,and Dt1xt10; yt10, and they are denoted as theeffi
28、ciency score rC310, rC320, rC330and rC340, respectively.Let xi0tand yr0tdenote DMU0s ith input and rthoutput, respectively, in time period t. Employing theSBM model introduced in Tone (2001), the follow-ing model (M1) is used to measure the relativeefficiencies of DMU0for (a,b) equal to (t,t)or(t+1,
29、t+1).rC3q0 Min Daxb0; yb0k C01mXmi1SC0i=xbi0;q 1 and 4.S:t: k 1sXsr1Sr=ybr01;kxbi0Xnj1xbijklj SC0i; i 1;2; .; m, (M1)kybr0Xnj1ybrjkljC0 Sr; r 1;2; .; s,ljX0; j 1;2; .; n; kX0; SC0iX0,i 1;2; .; m; SrX0; r 1;2; .; s.The optimal solutions lC3j, k*, SC0C3i, SC3r, rC3q0areobtained. Further, the excess an
30、d the shortfall canbe obtained indirectly: sC0C3i SC0C3i=kC3; sC3rSC3r=kC3. For instance, rC310is the relative efficiencyscore. The values xbi0 xbi0C0 sC0C3i, i 1m, andybr0 ybr0 sC3r, r 1s are its projection points onthe efficient frontier constructed by the DMUsperformed in period a.If rC3q0 1, we
31、employ the Super-SBM modelintroduced in Tone (2002) to measure the distanceof DMU0to the frontier that is constructed by theother DMUs. The following model (M2) is used tocompute the distance pC3q0. Its projection point on thefrontier is obtained (Xb0; Yb0)whereXb0xbi0; i bb b bC3 C3 b12m and Y0yr0;
32、 r 12s. xi0 xi0=t ; yr0ybC3r0=tC3.pC3q0 Min1mXmi1exbi0xbi0; q 1 and 4:S:t: 1 1sXsr1eybr0ybr0;exbi0XXnj1;a0xaijLaj; i 1;2; .; m,eybr0pXnj1;a0yarjLaj; r 1;2; .; s,exbi0Xtxbi0; i 1;2; .; m,0peybr0ptybr0; r 1;2; .; s,LajX0; j 1;2; .; n; t40. M2The mixed period measures, (a,b) (t+1, t),which is defined a
33、s rC320for each DMU0, is computedas the optimal value to the following SBM model(M3). In particular, the object DMU0is alsoincluded in the production possibility set. Themodel is also used for the second mixed periodmeasures rC330where (a,b) (t,t+1).rC3q0 Min Daxb0; yb0k C01mXmi1SC0i=xbi0;q 2and3.S:
34、t: k 1sXsr1Sr=ybr01;kxbi0Xnj1xbijklj xbi0kln1 SC0i,i 1;2; .; m,kybr0Xnj1ybrjklj ybr0kln1C0 Sr,r 1;2; .; s,ljX0; j 1;2; .;n 1; kX0;SC0iX0; i 1;2; .; m;SrX0; r 1;2; .; s. M3If rC3q0 1, employ the following Super-SBM model(M4) to measure the super-efficiency score pC3q0.pC3q0 Min1mXmi1exbi0xbi0; q 2 an
35、d 3:S:t: 1 1sXseybr0yb;r1 r00 0 0 0as Oa1=OAt11and Ob1=OAt11, respectively. Thus,ARTICLE IN PRESSF.-H.F. Liu, p.-h. Wang / Int. J. Production Economics 112 (2008) 367379370exbi0XXnj1xaijLaj; i 1;2; .; m,eybr0pXnj1yarjLaj; r 1;2; .; s,exbi0Xtxbi0; i 1;2; .; m,0peybr0ptybr0; r 1;2; .; s, M4LajX0; j 1;
36、2; .; n; t40.The efficiency score rC3q0is replaced by the value pC3q0.Therefore rC310, rC320, rC330, and rC340fall into one of thethree ranges: 41, 1, or o1. The Malmquistproductivity index (Fa re et al., 1992) measures theproductivity change of a particular DMU0in periodt and (t+1):Mt10Dtxt10; yt10
37、Dtxt0; yt0Dt1xt10; yt10Dt1xt0; yt0“#1=2. (1)When Mt1041, this signifies a productivity gain;when Mt10o1, this signifies a productivity loss; andwhen Mt10 1, there is no change in productivity.The above measure is actually the geometricmean of two Malmquist productivity indices:technical efficiency c
38、hange (TEC0) and frontier shift(FS0)(Caves et al., 1982; Fa re et al. 1992).Mt10Dtxt10; yt10Dtxt0; yt0Dt1xt10; yt10Dt1xt0; yt0“#1=2 TEC0C3 FS0, 2TEC0Dt1xt10; yt10Dtxt0; yt0 R3, (3)FS0Dtxt10; yt10Dt1xt10; yt10Dtxt0; yt0Dt1xt0; yt0“#1=2R1C3 R21=2. 4TEC0is used to measure the change in technicalefficie
39、ncy; on the other hand, it is also a measure ofhow much closer to the boundary the company is inperiod (t+1) compared with period t.IfTEC0is1.0, the particular DMU0(maybe a company) hasthe same distance in periods (t+1) and t from therespective efficient boundaries. If TEC0is over 1.0,the company ha
40、s moved closer to the period (t+1)boundary than it was to the period t boundary; theconverse is the case if the TEC0is under 1.0. As forFS0, it is used to measure the technology frontierDtxt10; yt10=Dt1xt10; yt10Oa1=Ob1. Simi-larly, drawing a line connects the origin and pointA1t. The line intersect
41、s with the t-frontier and (t+1)-frontier at points g1and d1, respectively. Tables 1and 2 depict the models employed to measure thetwo distances. The signs of R1and R2in the lastshift between time periods t and (t+1). Fa re et al.(1992, 1994a) point out that a value of FS0less than1.0 indicates negat
42、ive shift of frontier or technicalregress; FS0greater than 1.0 indicates positive shiftof frontier or technical progress; FS0equal to 1.0indicates no shift in technology frontier.3. Insights from the Malmquist productivityapproachChen and Ali (2004) further analyzed the proper-ties of two ratios of
43、FS0, Dtxt10; yt10=Dt1xt10; yt10 and Dtxt0; yt0=Dt1xt0; yt0, thebackward and forward frontier shifts, respectively.They are the performance of DMU0in periods(t+1) and t against the frontiers of period t and(t+1).As depicted in Fig. 1, a companys performancein period t could be the six possible locati
44、ons,A1tA6t. The oblique line that connects the originand the intersection of the two frontiers is thetradeoff on the strategy changes. A1t,A2t,andA3tlocate on the upper part and inside the t-frontier,between the two frontiers, and outside the (t+1)-frontier, respectively. The distances of A2tand A3t
45、tothe t- and (t+1)-frontiers, respectively, are themeasurement of super-efficiencies. Similarly, A4t,A5t,and A6tlocate on the lower part and inside the(t+1)-frontier, between the two frontiers, andoutside the t-frontier, respectively. The distancesof A6tand A5tto the t-and(t+1)-frontiers,respectivel
46、y, are the measurement of super-efficien-cies. It is noticeable that the locations of the sixpoints At11At16have similar occasions.For convenience of illustration, we temporarilyemploy a radial model such as CCR to express theefficiency measurement of each point by the ratio ofdistances; for instanc
47、e, by drawing a line thatconnects the origin and point At11. The lineintersects with the t-frontier and (t+1)-frontier atpoints a1and b1, respectively. The ratio ofDtxt1; yt1 to Dt1xt1; yt1 could be expressedcolumns are visible from Fig. 1.ARTICLE IN PRESSF.-H.F. Liu, p.-h. Wang / Int. J. Production
48、 Economics 112 (2008) 367379 371In Fig. 1, a downward frontier shift (towards theorigin) from period t to (t+1) represents a positiveshift. The converse situation (away from the origin)represents a negative shift. For a company, fromperiod t to (t+1), the four possible frontier shiftsare as follows in (a)(d). The 36 possible movementsare depicted in Table 3.Input 1Input 2 Fig. 1. FrontierTable 1The computation of ratio R1t+1Dtxt10; yt10 Dt1xt10; yt10A1t+1Use M3 (rC330o1) Use M1 (rC340oA2t+1Use M4 (pC33041) Use M1 (rC340oA3t+1Use M4 (pC33041) Us
