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Super-efficiency based on a modified directional distance function
Omega 41 (2013) 621–625
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Super-efficiency based on a modified directional distance function Yao Chen a, Juan Du b,n, Jiazhen Huo b a b
Manning School of Business, University of Massachusetts at Lowell, Lowell, MA 01845, USA School of Economics and Management, Tongji University, 1239 Siping Road, Shanghai 200092, PR China
a r t i c l e i n f o
abstract
Article history: Received 23 March 2012 Accepted 23 June 2012 Processed by B. Lev Available online 4 July 2012
The problem of infeasibility arises in conventional radial super-efficiency data envelopment analysis (DEA) models under variable returns to scale (VRS). To tackle this issue, a Nerlove–Luenberger (N–L) measure of super-efficiency is developed based on a directional distance function. Although this N–L super-efficiency model does not suffer infeasibility problem as in the conventional radial superefficiency DEA models, it can produce an infeasible solution in two special situations. The current paper proposes to modify the directional distance function by selecting proper feasible reference bundles so that the resulting N–L measure of super-efficiency is always feasible. As a result, our modified VRS super-efficiency model successfully addresses the infeasibility issues occurring either in conventional VRS models or the N–L super-efficiency model. Numerical examples are used to demonstrate our approach and compare results obtained from various super-efficiency measures. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Data envelopment analysis (DEA) Directional distance function Efficiency Super-efficiency
1. Introduction Data envelopment analysis (DEA) is a method for measuring relative efficiency of peer decision making units (DMUs). In recent years, DEA has been applied to various settings, such as performance evaluations in Olympic Games [1], estimating the importance of objectives in agricultural economics [2], regional R & D investment evaluations in China [3], and bankruptcy assessment for corporations [4]. In an effort to differentiate the performance of efficient DMUs, Andersen and Petersen [5] develop a superefficiency model based upon the constant returns to scale (CRS) model [6]. However, when the concept of super-efficiency is applied to the variable returns to scale (VRS) model [7], the resulting model must be infeasible for certain DMUs [8]. Infeasibility restricts a wider use of super-efficiency DEA. Recent years have seen several studies addressing the infeasibility issue and the development of new super-efficiency models. For example, Lovell and Rouse [9] modify the conventional radial super-efficiency model by scaling up the concerning input vector (in an input-oriented case), or by scaling down the concerning output vector (in an output-oriented case). Also under the VRS assumption, Chen [10,11] replaces inefficient observations by their respective efficient projections, and performs super-efficiency analysis with this revised data set. Cook et al. [12] show that for infeasibility cases, one needs to adjust both the input and output levels to move an efficient DMU under evaluation onto the
n
Corresponding author. Tel.: þ86 18201700780. E-mail addresses: [email protected] (Y. Chen), [email protected], [email protected] (J. Du), [email protected] (J. Huo). 0305-0483/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.omega.2012.06.006
frontier formed by the remaining DMUs. They develop a twostage process to address the infeasibility issue. Lee et al. [13] develop an alternative two-stage process to addressing infeasibility issue in the conventional VRS super-efficiency models. On the other hand, based on the directional distance function [14], Ray [15] develops a procedure to obtain Nerlove–Luenberger (N–L) measure of super-efficiency in a single model to adjust both input and output levels. As a result, this N–L super-efficiency model does not pose a similar infeasibility problem in the conventional VRS super-efficiency models. However, Ray [15] points out that the N–L super-efficiency model fails in two special situations. First, no feasible solution exists if the zero input value is present in a DMU under evaluation and all other DMUs in the reference set are positive-valued in that input. Second, when an N–L super-efficiency score is greater than 2, the model will yield an efficient projection involving negative output quantities. In fact, zero data are problematic in any super-efficiency models. For example, Lee and Zhu [16] show that either the conventional VRS super-efficiency or the two-stage super-efficiency procedure in [12] will become infeasible when zero data are present. Therefore, it is necessary to address the two issues presented in [15]. The current paper shows that we can choose a proper reference input–output bundle in the directional distance function (DDF) [14], and modify Ray’s DDF-based VRS super-efficiency model [15]. The new super-efficiency model successfully addresses the infeasibility issues occurring either in conventional VRS models or the N–L super-efficiency model. The remainder of this paper is organized as follows. Section 2 introduces the DDF and its previous applications in N–L efficiency and super-efficiency assessments. Section 3 proposes a modified DDF, based on which a new VRS DEA model is developed for
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Y. Chen et al. / Omega 41 (2013) 621–625
super-efficiency measurement. This new super-efficiency model addresses the infeasibility issues occurring either in conventional VRS models or the N–L super-efficiency model. Section 4 illustrates the new approach by using the data set from [8]. Section 5 concludes with a summary of our contributions.
Assume that there are n DMUs producing the same set of outputs at the cost of the same set of inputs. Unit j is represented by DMUj ðj ¼ 1,:::,nÞ, whose ith input and rth output are denoted by xij ði ¼ 1,:::,mÞ and yrj ðr ¼ 1,:::,sÞ, respectively. Then under the standard assumptions of convexity and free disposability of inputs and outputs, the production possibility set (PPS) formed from the above set of n DMUs is represented by 8 n n < X X T ¼ ðxi ,yr Þxi Z lj xij ,i ¼ 1,:::,m; yr r lj yrj ,r ¼ 1,:::,s; : j¼1 j¼1 9 n = X lj ¼ 1, lj Z 0,j ¼ 1,:::,n ð1Þ ; j¼1
Consider an input–output bundle xio ,yro and a reference input–output bundle ðg x ,g y Þ. Then based on the PPS (1), the directional distance function (DDF) is defined as [14] ð2Þ
The reference bundle ðg x ,g y Þ can be chosen in an arbitrary way, which makes the DDF varies with reference to any specific DMU. Chambers et al. [14] select ðxio ,yro Þ for ð g x ,g y Þ, and obtain the standard DDF as Dðxio ,yro Þ ¼ max b : ðð1bÞxio ,ð1 þ bÞyro Þ A T
The related VRS model for calculating the N–L super-efficiency of DMUk is developed as [15] maxbk
2. Directional distance function and super-efficiency
Dðxio ,yro ; g x ,g y Þ ¼ max b : ðxio þ bg x ,yro þ bg y Þ A T
Then the directional distance function for DMUk concerning the new PPS (5) is Dk ðxik ,yrk Þ ¼ maxbk : ð1bk Þxik ,ð1 þ bk Þyrk A T k ð6Þ
ð3Þ
In DDF (3), each input is decreased and each output is increased simultaneously by the same proportion b. The DDF-based VRS model for efficiency with respect to PPS (1) is [14]
s:t:
n X
lj xij rð1bk Þxik ,i ¼ 1,:::,m
j¼1 jak n X
lj yrj Zð1 þ bk Þyrk ,r ¼ 1,:::,s
j¼1 j ak n X
lj ¼ 1, lj Z0,j ¼ 1,:::,n,j ak
ð7Þ
j¼1 j ak
A negative optimal value of bk indicates the same proportion to be scaled down for the output bundle while to be scaled up for the input bundle in order to get an attainable input–output mix in PPS (5). Smaller bk indicates that DMU is more N–L super-efficient. In general, model (7) is always feasible. However, Ray [15] points out two exceptions. One is that the unit under evaluation DMUk has at least one input io at the zero level, and all other DMUs in the reference set are positive-valued in that input, i.e., xio k ¼ 0 while xio j 4 0,j ¼ 1,:::,n,j a k. In such a case, DDF-based super-efficiency model (7) becomes infeasible because its first set of constraints cannot be satisfied. The other case is one where for some input io , there is n n P P lj xio j for all lj s combinations satisfying lj ¼ 1 and 2xio k o j¼1
j¼1
jak
j ak
lj Z 0,j ¼ 1,:::,n,j a k. Thus bk is restricted to a value lower than 1. This will result in a reference point with negative output values.
max b s:t:
n X
3. Modified DDF-based super-efficiency
lj xij r ð1bÞxio ,i ¼ 1,:::,m
j¼1 n X
lj yrj Z ð1 þ bÞyro ,r ¼ 1,:::,s
j¼1 n X
lj ¼ 1, lj Z 0,j ¼ 1,:::,n
ð4Þ
j¼1
As pointed out in [15], the optimal value of b is the Nerlove– Luenberger (N–L) measure of technical inefficiency for the evaluated DMU, whose efficiency can be calculated as ð1bÞ. Efficiency scores can be used to rank those technically inefficient DMUs, but fail to differentiate efficient DMUs. Thus super-efficiency, which implies the possible capability of a DMU in reducing its outputs or increasing its inputs without becoming inefficient, is also applied into directional distance function. For DMUk xik ,yrk , the PPS for super-efficiency is modified as 8 > > > < n n X X T k ¼ ðxi ,yr Þxi Z lj xij ,i ¼ 1,:::,m; yr r lj yrj ,r ¼ 1,:::,s; > > j¼1 j¼1 > : j ak
9 > > > = n X lj ¼ 1, lj Z 0,j ¼ 1,:::,n,j ak > > j¼1 > ; j ak
In this section, we tackle the above two infeasibility problems in N–L super-efficiency model (7). Note that these infeasibility issues are caused by the choice of ðg x ,g y Þ, which results in the same changing proportion b taken by each input (decreased) and output (increased) simultaneously. We can choose a different ð g x ,g y Þ so that the above infeasibility cases will not occur. In other words, we can consider a reference input–output bundle different from the conventional selection in [14], by using ðaxio 1, byro þ1Þ for ðg x ,g y Þ, and obtain a new directional distance function as Dðxio ,yro Þ ¼ max b : ð1baÞxio b,ð1þ bbÞyro þ b A T ð8Þ Here both a and b are pre-determined positive parameters. We will develop a procedure to decide for such parameters to address the two infeasibility issues. For DMUk , the super-efficiency model based upon (8) can be written as maxbk n X s:t: lj xij r ð1bk aÞxik bk ,i ¼ 1,:::,m
ð9:1Þ
j¼1 jak
jak
ð5Þ
n X
j¼1 j ak
lj yrj Z ð1 þ bk bÞyrk þ bk ,r ¼ 1,:::,s
ð9:2Þ
Y. Chen et al. / Omega 41 (2013) 621–625
n X
lj ¼ 1, lj Z 0,j ¼ 1,:::,n,j ak
ð9:3Þ
623
Therefore, if the following new inequality (13) is satisfied, the previous inequality (12) or (10) holds.
j¼1 j ak
Note that an appropriate negative value for bk will eliminate the infeasibility problem in model (9), which is caused by the special situation that DMUk is the only unit with zero value for some input (the first exception mentioned in [15]). Next we will focus on the second exception by preventing directional output targets 1 þ bk b yrk þ bk from taking negative values. Note that constraint (9.1) and (9.2) can be re-written as 8 n X > > > ðaxik þ1Þbk rxik lj xij ,i ¼ 1,:::,m > > > > j¼1 > > > < j ak n X > > > lj yrj yrk ,r ¼ 1,:::,s ðbyrk þ1Þbk r > > > > j¼1 > > > : jak
n X
lj yrj yrk
j¼1 yrk byrk þ 1
r bk r
j ak
,r ¼ 1,:::,s
byrk þ 1
r ¼ 1,:::,sj ¼ 1,:::,n
Z
1 yrj
i ¼ 1,:::,m
max xij min xij þ 1
j ¼ 1,:::,n
a min
xik
To ensure that bk has a feasible solution,
yrk byrk þ 1
lj xij
j¼1 j ak
r
axik þ 1
must be satisfied for any combination of i and r, i.e., xik
r ¼ 1,:::,s j ¼ 1,:::,n
j ¼ 1,:::,n
required to be strictly greater than zero, or equivalently, a min min xij þ1 1 i ¼ 1,:::,m j ¼ 1,:::,n 4 max max r ¼ 1,:::,sj ¼ 1,:::,n yrj max max xij min xij þ 1
i ¼ 1,:::,m
j ¼ 1,:::,n
Determination of parameters a and b: Case I: completely positive inputs We first suppose that all input values are positive for all DMUs, namely min min xij 40, and then have i ¼ 1,:::,m j ¼ 1,:::,n a4 max max ð1=yrj Þ max max xij min xij
yrk
max
ð10Þ
axik þ 1
yrk
¼ min
r ¼ 1,:::,s byrk þ 1
1
¼ min
¼
r ¼ 1,:::,s b þ 1 yrk
1 b þ max
1 r ¼ 1,:::,s yrk
ð11Þ Then inequality (10) becomes xik 1 b þ max
1 r ¼ 1,:::,s yrk
Z min
i ¼ 1,:::,m
n P
n P
lj xij
j¼1 j ak
axik þ1
¼ max
! axik þ 1 r max
i ¼ 1,:::,m
r max
i ¼ 1,:::,m
!
S lj xij xik =
j¼1 j ak
i ¼ 1,:::,m
! maxj ¼ 1,:::,n xij þ 1 min xij = a min xij þ 1 j ¼ 1,:::,n
j ¼ 1,:::,n
max xij min xij þ 1 = min a min xij þ 1 :
j ¼ 1,:::,n
ð12Þ
axik þ1 n
Since S lj xij r max xij , we have max j ¼ 1,:::,n
j ¼ 1,:::,n
min xij .
Thus
we
j ¼ 1,:::,n
obtain
the
i ¼ 1,:::,m
j ¼ 1,:::,n
j ¼ 1,:::,n
ranges
for
parameters a and b, which are a4
! max max xij min xij þ1 1 = max max ð1=yrj Þ i ¼ 1,:::,m
j ¼ 1,:::,n
j ¼ 1,:::,n
0 ob r a min
min xij ,
i ¼ 1,:::,m j ¼ 1,:::,n
!
max xij min xij þ1 max
j ¼ 1,:::,n
j ¼ 1,:::,n
min xij þ 1 =
i ¼ 1,:::,m j ¼ 1,:::,n
!
max
r ¼ 1,:::,s j ¼ 1,:::,n
max
i ¼ 1,:::,m
1=yrj . Any value com-
bination taken from the above ranges is a reasonable candidate for parameters of our DDF (8). Any specific choice of a and b depends on the practical factors and preferences of each decision maker. Further, if integer-restricted parameters are expected or required by the decision maker, it is easy to find an integervalued a, denoted by a, large enough to be greater than
! max max xij min xij þ1 1 = max max ð1=yrj Þ r ¼ 1,:::,s j ¼ 1,:::,n
lj xij xik
j¼1 j ak
i ¼ 1,:::,m
n
j¼1 j ak
i ¼ 1,:::,m
i ¼ 1,:::,m j ¼ 1,:::,n
min
In the following we deduct the mathematical relationship between parameters a and b from inequality (10). For the left-hand side of (10), we get r ¼ 1,:::,s byrk þ 1
þ 1 1 = min
lj xij
j¼ 1
jak
yrk r min r ¼ 1,:::,s byrk þ1 i ¼ 1,:::,m max
ð14Þ
j ¼ 1,:::,n
r ¼ 1,:::,s j ¼ 1,:::,n
Pn
ð13Þ
Then we obtain b r1= maxi ¼ 1,:::,m max xij min xij þ1 = j ¼ 1,:::,n j ¼ 1,:::,n max max 1=yrj , or equivalently a min min xij þ1 r ¼ 1,:::,s j ¼ 1,:::,n i ¼ 1,:::,m j ¼ 1,:::,n br a min min xij þ 1 = max max xij min xij þ 1 i ¼ 1,:::,m j ¼ 1,:::,n i ¼ 1,:::,m j ¼ 1,:::,n j ¼ 1,:::,n max max 1=yrj , from inequality (13). To make parar ¼ 1,:::,s j ¼ 1,:::,n meter b restricted to positive value, a min min xij þ1 = i ¼ 1,:::,m j ¼ 1,:::,n is maxi ¼ 1,:::,m max xij min xij þ 1 max max 1=yrj
r ¼ 1,:::,sj ¼ 1,:::,n
n P
j ¼ 1,:::,n
min xij þ1
i ¼ 1,:::,mj ¼ 1,:::,n
j ¼ 1,:::,n
Therefore, to obtain non-negative destinations of directional movement in outputs, b needs to be selected to ensure that 1 þ bk b yrk þ bk Z 0,r ¼ 1,:::,s, or equivalently, byrk þ 1 bk Z yrk , r ¼ 1,:::,s. Since ðaxik þ1Þ and byrk þ 1 are positive values, we obtain 8 n X > > xik lj xij > > > > j¼1 > > > > j ak > > < bk r ,i ¼ 1,:::,m axik þ 1 > > > > > > > > > > > > :
1 bþ max max
max
i ¼ 1,:::,m
j ¼ 1,:::,n
j ¼ 1,:::,n
min min xij , and simultaneously makes a min min xij i ¼ 1,:::,m j ¼ 1,:::,n i ¼ 1,:::,m j ¼ 1,:::,n ! ! þ 1 = max max xij min xij þ1 max max y1 4 1. i ¼ 1,:::,m
j ¼ 1,:::,n
r ¼ 1,:::,s j ¼ 1,:::,n
j ¼ 1,:::,n
rj
Thus a could be any positive integer greater than
max max ð1=yrj Þ þ 1 max max xij min xij þ 1 1 = r ¼ 1,:::,s j ¼ 1,:::,n
min
i ¼ 1,:::,m
j ¼ 1,:::,n
j ¼ 1,:::,n
min xij . One possible selection for a is the smallest one,
i ¼ 1,:::,mj ¼ 1,:::,n
while one candidate for b could be the greatest positive ! a min min xij þ1 = max max integer less than i ¼ 1,:::,m j ¼ 1,:::,n
i ¼ 1,:::,m
j ¼ 1,:::,n
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Y. Chen et al. / Omega 41 (2013) 621–625
xij min xij þ 1 max max 1=yrj . This integer choice of r ¼ 1,:::,sj ¼ 1,:::,n j ¼ 1,:::,n a, b will be used in the current study for calculations in the illustration. The above procedure for determining a and b is based upon the assumption that min min xij 4 0. If min min xij ¼ 0, the i ¼ 1,:::,m j ¼ 1,:::,n
i ¼ 1,:::,m j ¼ 1,:::,n
above parameter choices are no longer valid, and we need to develop an alternative approach for determining a and b. Case II: any zero input From the right-hand side of inequality (12), we uehave 1 ! C BB n B B = 1axik þ1 r max maxi ¼ 1,:::,m @@ S lj xij xik C A j¼1 i ¼ 1,:::,m 00
! max xij min xij =
j ¼ 1,:::,n
j ¼ 1,:::,n
j ak
1 C a min xij þ 1 C max xij min xij = min A r i ¼max j ¼ 1,:::,n 1,:::,m j ¼ 1,:::,n j ¼ 1,:::,n i ¼ 1,:::,m
a min xij
þ1 .
j ¼ 1,:::,n
Then inequality (12) or (10) will be satisfied if the following inequality (15) holds: max max xij min xij 1 i ¼ 1,:::,m j ¼ 1,:::,n j ¼ 1,:::,n ð15Þ Z a min min xij þ 1 b þ max max y1 r ¼ 1,:::,s j ¼ 1,:::,n
i ¼ 1,:::,m j ¼ 1,:::,n
rj
Since at least one DMU has zero input(s), and it is impossible for all DMUs to be at zero level in a same input, then at least for some i, it is true that max xij min xij 4 0, or j ¼ 1,:::,n j ¼ 1,:::,n max xij min xij 4 0. Then we obtain further max i ¼ 1,:::,m j ¼ 1,:::,n j ¼ 1,:::,n ! ! b r a min
min xij þ1 =
i ¼ 1,:::,m j ¼ 1,:::,n
max
j ¼ 1,:::,n
the
1=yrj
i ¼ 1,:::,m
max xij min xij max
j ¼ 1,:::,n
from inequality (15). Since
range for !
b
is
simplified
to
j ¼ 1,:::,n
r ¼ 1,:::,s
min
min xij ¼ 0, i ¼ 1,:::,mj ¼ 1,:::,n max xij b r 1= max i ¼ 1,:::,m
r ¼ 1,:::,s j ¼ 1,:::,n
positive integer depending on a specific decision maker. Note that the VRS DEA model is unit-invariant. Therefore, to obtain positive integer values for parameters a and b, all input values xij can be proportionally scaled down to different values denoted as xij , which satisfy max xij min xij max max 1=yrj 4 1. Thus 1= max i ¼ 1,:::,m
j ¼ 1,:::,n
j ¼ 1,:::,n
r ¼ 1,:::,sj ¼ 1,:::,n
for the numerical examples analyzed in this paper, we let a ¼ 1, and select the greatest positive integer less than max xij min xij max max 1=yrj for b. 1= max i ¼ 1,:::,m
j ¼ 1,:::,n
j ¼ 1,:::,n
max
max
r ¼ 1,:::,s j ¼ 1,:::,n
1=yrj o 0. We need to make some data transfor-
mations in scale. All input values in the original data set are scaled down by 0.01. Based upon the new data set, which are presented in columns 5–7 of Table 1, we have 1= max i ¼ 1,:::,m max xij min xij Þ max max 1=yrj ¼ 9:5. Thus we select j ¼ 1,:::,n
r ¼ 1,:::,s j ¼ 1,:::,n
j ¼ 1,:::,n
1 for a, and 9 for b in our DDF-based super-efficiency model (9), and obtain the related super-efficiency results 1bk in the last column of Table 1. Based on our newly-proposed DDF (8) with parameters a, b , the VRS super-efficiency model (9) avoids the infeasibility issue occurring either in conventional VRS models or N–L super efficiency model (7). By selecting legitimate parameters a, b , and choosing an appropriate value of bk (negative if superefficient), it always yields a feasible solution for the super efficiency score, which is calculated as 1bk . This is true even
j ¼ 1,:::,n
max 1=yrj , while parameter a can be any
min xij max j ¼ 1,:::,n
max
than half of the value in that input of any other DMUs. i.e., 2x11 ox1j ,j ¼ 2,3,4,5, and thus for all non-negative lj s (j ¼ 2,3,4,5) P combinations satisfying we have 2x11 o j ¼ 2,3,4,5 lj ¼ 1, P l x . For DMU 3, its input 2 is zero, while the rest j ¼ 2,3,4,5 j 1j four DMUs all have positive values for that input. Thus all possible convex combinations of input 2 of DMUs 1, 2, 4, 5 are strictly greater than zero, making the constraint for input 2 in N–L super-efficiency model (7) impossible to be satisfied. Both of these cases make the approach in [15] infeasible. The last column of Table 1 reports the super-efficiency scores obtained from our newly-proposed approach. To select parameters a and b, we note that from the original data in Table 1, 1= max max xij min xij i ¼ 1,:::,m j ¼ 1,:::,n j ¼ 1,:::,n !
r ¼ 1,:::,sj ¼ 1,:::,n
Table 2 Numerical example II [8]. DMU
Input 1x1
Input 2x2
Input 3x3
Output 1y1
Output 2y2
1 2 3 4 5 6 7 8 9 10
182 74 160 183 133 106 109 240 276 191
237 82 195 150 155 120 110 243 188 117
468 148 400 339 329 138 188 806 574 466
5008 1857 4041 2779 3506 1306 1515 7763 4577 3322
5303 2336 5001 2418 3602 956 2282 9601 6493 4233
Consider a simple numerical example listed in columns 2–4 of Table 1 where we have five DMUs with two inputs and a single output. For DMU 1, its value in input 1, x11 ¼ 2, is strictly less Table 3 Alternative measures of VRS super-efficiency. Table 1 Numerical example I. DMU Original
1 2 3 4 5
New
Input 1x1
Input 2x2
Output 1y1
Input 1x1
Input 2x2
Output 1y1
2 8 12 8 5
4 3 0 4 5
2 4 6 5 5
0.02 0.08 0.12 0.08 0.05
0.04 0.03 0 0.04 0.05
2 4 6 5 5
Modified DDFbased superefficiency
1.0294 0.9923 1.0323 0.9935 1.0152
DMU
Standard radial
L–R
N–L
DDF-based model (9)
1 2 3 4 5 6 7 8 9 10
1.0626 1.5277 0.9765 0.7354 0.9752 1.0725 0.7852 infeasible 0.9246 1.0602
1.0626 1.5277 0.9765 0.7354 0.9752 1.0725 0.7852 6 0.9246 1.0602
1.0285 1.4430 0.9889 0.8566 0.9881 1.0725 0.8863 1.3836 0.9581 1.0334
1.00559 1.05113 0.99789 0.97561 0.99776 1.00724 0.97871 1.38354 0.99302 1.00557
Y. Chen et al. / Omega 41 (2013) 621–625
when the DMU under evaluation is the only unit in the reference set with zero inputs.
625
super-efficiency measure, however, provides a quite different rank for super-efficient units, which is DMU 2, 8, 6, 10, 1, 3, 5, 9, 7, 4 from high to low.
4. Comparisons 5. Conclusions We use the data set presented in Table 2 to illustrate different results obtained from various VRS super-efficiency models. This data set is previously studied in [8] and [15]. Columns 2–5 in Table 3 report the super-efficiency measures obtained from the standard input–oriented VRS model [5], the input-oriented L–R method [9], the N–L measure [15], and our modified DDF-based model (9), respectively. The column identified as ‘‘Standard radial’’ presents the efficiency and super-efficiency scores obtained from the conventional input-oriented VRS super-efficiency model. DMUs 3, 4, 5, 7, 9 are inefficient units, while DMUs 1, 2, 6, 8, 10 are efficient. Among these five efficient units, four (DMUs 1, 2, 6, 10) have super-efficiency greater than one. DMU 8 does not have a feasible solution to the conventional VRS super-efficiency problem. Column ‘‘L–R’’ shows the input-oriented super-efficiency obtained from the Lovell–Rouse (L–R) method [9] by setting the scale factor a ¼ 6. Note that the L–R super-efficiency measure equals the scale factor 6 for DMU 8. Except for this infeasible DMU 8, all scores obtained from the conventional model are identical to those obtained via the L–R approach. Column ‘‘N–L’’ displays the N–L measure of super-efficiency [15] 1bk for all ten DMUs. The last column of Table 3 shows the results obtained from our DDF-based VRS super-efficiency model (9). A detailed procedure for parameter selection and model calculation is presented as follows: First, we need to determine a reasonable choice for parameters a and b in model (9). For the data set presented in Table 2,
max
max
r ¼ 1,:::,s j ¼ 1,:::,n
1 yrj
þ1 max max xij min xij þ 1 1 = i ¼ 1,:::,m
j ¼ 1,:::,n
Conventional radial VRS super-efficiency models become infeasible in certain situations. Although the Nerlove–Luenberger (N–L) measure of super-efficiency [15] avoids such infeasibility issues under VRS condition, it would fail in two special cases. This paper modifies the DDF-based super-efficiency in [15] so that the newlyproposed approach is always feasible. We point out that the values of parameters a and b depend on the specific data set in order to make all DMUs feasible in our DDF-based model. This does not indicate that inefficient DMUs will affect the efficiency scores. Note that the parameter ranges are calculated based on extreme values of related input/output measures. Therefore, the values influencing the choice regions of parameters are the maximum or minimum input/output values from the whole data set, and are not relevant to the specific group of inefficient DMUs.
Acknowledgments The authors are grateful for the constructive comments and suggestions from two anonymous reviewers on an earlier version of this paper. Dr. Juan Du would like to thank the support by the National Natural Science Foundation of China (Grant no. 71101108) and China Postdoctoral Science Foundation (Grant no. 20110490696). Professor Huo would like to thank the support by the National Natural Science Foundation of China (Grant no. 70832005).
j ¼ 1,:::,n
min
min xij is calculated as 9.0365. Thus we choose a ¼ 10, and a min min xij þ 1 = max max xij min xij i ¼ 1,:::,m j ¼ 1,:::,n i ¼ 1,:::,m j ¼ 1,:::,n j ¼ 1,:::,n þ1Þ max max 1=yrj is then computed as 1.1066. The only
References
i ¼ 1,:::,m j ¼ 1,:::,n
r ¼ 1,:::,s j ¼ 1,:::,n
candidate for b is 1. Thus our VRS super-efficiency model (9) based on DDF (8) is presented as maxbk n X s:t: lj xij rð110bk Þxik bk ,i ¼ 1,:::,m j¼1 j ak n X
lj yrj Z ð1þ bk Þyrk þ bk ,r ¼ 1,:::,s
j¼1 jak n X
ð16Þ
lj ¼ 1
j¼1 jak
lj Z0,j ¼ 1,:::,n,j a k Similar to the N–L super-efficiency, for super-efficient DMUs 1, 2, 6, 8, 10, their optimal values to model (9) are negative, making their super-efficiency measures all exceed unity. Our DDF-based super-efficiency can fully differentiate all ten units. Comparing the ranking results from various super-efficiency measures, we find that three approaches (including standard radial model, L–R method, and our DDF-based model) lead to exactly the same rank, which is DMU 8, 2, 6, 1, 10, 3, 5, 9, 7, 4 from high to low. The N–L
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Contents lists available at SciVerse ScienceDirect
Omega journal homepage: www.elsevier.com/locate/omega
Super-efficiency based on a modified directional distance function Yao Chen a, Juan Du b,n, Jiazhen Huo b a b
Manning School of Business, University of Massachusetts at Lowell, Lowell, MA 01845, USA School of Economics and Management, Tongji University, 1239 Siping Road, Shanghai 200092, PR China
a r t i c l e i n f o
abstract
Article history: Received 23 March 2012 Accepted 23 June 2012 Processed by B. Lev Available online 4 July 2012
The problem of infeasibility arises in conventional radial super-efficiency data envelopment analysis (DEA) models under variable returns to scale (VRS). To tackle this issue, a Nerlove–Luenberger (N–L) measure of super-efficiency is developed based on a directional distance function. Although this N–L super-efficiency model does not suffer infeasibility problem as in the conventional radial superefficiency DEA models, it can produce an infeasible solution in two special situations. The current paper proposes to modify the directional distance function by selecting proper feasible reference bundles so that the resulting N–L measure of super-efficiency is always feasible. As a result, our modified VRS super-efficiency model successfully addresses the infeasibility issues occurring either in conventional VRS models or the N–L super-efficiency model. Numerical examples are used to demonstrate our approach and compare results obtained from various super-efficiency measures. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Data envelopment analysis (DEA) Directional distance function Efficiency Super-efficiency
1. Introduction Data envelopment analysis (DEA) is a method for measuring relative efficiency of peer decision making units (DMUs). In recent years, DEA has been applied to various settings, such as performance evaluations in Olympic Games [1], estimating the importance of objectives in agricultural economics [2], regional R & D investment evaluations in China [3], and bankruptcy assessment for corporations [4]. In an effort to differentiate the performance of efficient DMUs, Andersen and Petersen [5] develop a superefficiency model based upon the constant returns to scale (CRS) model [6]. However, when the concept of super-efficiency is applied to the variable returns to scale (VRS) model [7], the resulting model must be infeasible for certain DMUs [8]. Infeasibility restricts a wider use of super-efficiency DEA. Recent years have seen several studies addressing the infeasibility issue and the development of new super-efficiency models. For example, Lovell and Rouse [9] modify the conventional radial super-efficiency model by scaling up the concerning input vector (in an input-oriented case), or by scaling down the concerning output vector (in an output-oriented case). Also under the VRS assumption, Chen [10,11] replaces inefficient observations by their respective efficient projections, and performs super-efficiency analysis with this revised data set. Cook et al. [12] show that for infeasibility cases, one needs to adjust both the input and output levels to move an efficient DMU under evaluation onto the
n
Corresponding author. Tel.: þ86 18201700780. E-mail addresses: [email protected] (Y. Chen), [email protected], [email protected] (J. Du), [email protected] (J. Huo). 0305-0483/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.omega.2012.06.006
frontier formed by the remaining DMUs. They develop a twostage process to address the infeasibility issue. Lee et al. [13] develop an alternative two-stage process to addressing infeasibility issue in the conventional VRS super-efficiency models. On the other hand, based on the directional distance function [14], Ray [15] develops a procedure to obtain Nerlove–Luenberger (N–L) measure of super-efficiency in a single model to adjust both input and output levels. As a result, this N–L super-efficiency model does not pose a similar infeasibility problem in the conventional VRS super-efficiency models. However, Ray [15] points out that the N–L super-efficiency model fails in two special situations. First, no feasible solution exists if the zero input value is present in a DMU under evaluation and all other DMUs in the reference set are positive-valued in that input. Second, when an N–L super-efficiency score is greater than 2, the model will yield an efficient projection involving negative output quantities. In fact, zero data are problematic in any super-efficiency models. For example, Lee and Zhu [16] show that either the conventional VRS super-efficiency or the two-stage super-efficiency procedure in [12] will become infeasible when zero data are present. Therefore, it is necessary to address the two issues presented in [15]. The current paper shows that we can choose a proper reference input–output bundle in the directional distance function (DDF) [14], and modify Ray’s DDF-based VRS super-efficiency model [15]. The new super-efficiency model successfully addresses the infeasibility issues occurring either in conventional VRS models or the N–L super-efficiency model. The remainder of this paper is organized as follows. Section 2 introduces the DDF and its previous applications in N–L efficiency and super-efficiency assessments. Section 3 proposes a modified DDF, based on which a new VRS DEA model is developed for
622
Y. Chen et al. / Omega 41 (2013) 621–625
super-efficiency measurement. This new super-efficiency model addresses the infeasibility issues occurring either in conventional VRS models or the N–L super-efficiency model. Section 4 illustrates the new approach by using the data set from [8]. Section 5 concludes with a summary of our contributions.
Assume that there are n DMUs producing the same set of outputs at the cost of the same set of inputs. Unit j is represented by DMUj ðj ¼ 1,:::,nÞ, whose ith input and rth output are denoted by xij ði ¼ 1,:::,mÞ and yrj ðr ¼ 1,:::,sÞ, respectively. Then under the standard assumptions of convexity and free disposability of inputs and outputs, the production possibility set (PPS) formed from the above set of n DMUs is represented by 8 n n < X X T ¼ ðxi ,yr Þxi Z lj xij ,i ¼ 1,:::,m; yr r lj yrj ,r ¼ 1,:::,s; : j¼1 j¼1 9 n = X lj ¼ 1, lj Z 0,j ¼ 1,:::,n ð1Þ ; j¼1
Consider an input–output bundle xio ,yro and a reference input–output bundle ðg x ,g y Þ. Then based on the PPS (1), the directional distance function (DDF) is defined as [14] ð2Þ
The reference bundle ðg x ,g y Þ can be chosen in an arbitrary way, which makes the DDF varies with reference to any specific DMU. Chambers et al. [14] select ðxio ,yro Þ for ð g x ,g y Þ, and obtain the standard DDF as Dðxio ,yro Þ ¼ max b : ðð1bÞxio ,ð1 þ bÞyro Þ A T
The related VRS model for calculating the N–L super-efficiency of DMUk is developed as [15] maxbk
2. Directional distance function and super-efficiency
Dðxio ,yro ; g x ,g y Þ ¼ max b : ðxio þ bg x ,yro þ bg y Þ A T
Then the directional distance function for DMUk concerning the new PPS (5) is Dk ðxik ,yrk Þ ¼ maxbk : ð1bk Þxik ,ð1 þ bk Þyrk A T k ð6Þ
ð3Þ
In DDF (3), each input is decreased and each output is increased simultaneously by the same proportion b. The DDF-based VRS model for efficiency with respect to PPS (1) is [14]
s:t:
n X
lj xij rð1bk Þxik ,i ¼ 1,:::,m
j¼1 jak n X
lj yrj Zð1 þ bk Þyrk ,r ¼ 1,:::,s
j¼1 j ak n X
lj ¼ 1, lj Z0,j ¼ 1,:::,n,j ak
ð7Þ
j¼1 j ak
A negative optimal value of bk indicates the same proportion to be scaled down for the output bundle while to be scaled up for the input bundle in order to get an attainable input–output mix in PPS (5). Smaller bk indicates that DMU is more N–L super-efficient. In general, model (7) is always feasible. However, Ray [15] points out two exceptions. One is that the unit under evaluation DMUk has at least one input io at the zero level, and all other DMUs in the reference set are positive-valued in that input, i.e., xio k ¼ 0 while xio j 4 0,j ¼ 1,:::,n,j a k. In such a case, DDF-based super-efficiency model (7) becomes infeasible because its first set of constraints cannot be satisfied. The other case is one where for some input io , there is n n P P lj xio j for all lj s combinations satisfying lj ¼ 1 and 2xio k o j¼1
j¼1
jak
j ak
lj Z 0,j ¼ 1,:::,n,j a k. Thus bk is restricted to a value lower than 1. This will result in a reference point with negative output values.
max b s:t:
n X
3. Modified DDF-based super-efficiency
lj xij r ð1bÞxio ,i ¼ 1,:::,m
j¼1 n X
lj yrj Z ð1 þ bÞyro ,r ¼ 1,:::,s
j¼1 n X
lj ¼ 1, lj Z 0,j ¼ 1,:::,n
ð4Þ
j¼1
As pointed out in [15], the optimal value of b is the Nerlove– Luenberger (N–L) measure of technical inefficiency for the evaluated DMU, whose efficiency can be calculated as ð1bÞ. Efficiency scores can be used to rank those technically inefficient DMUs, but fail to differentiate efficient DMUs. Thus super-efficiency, which implies the possible capability of a DMU in reducing its outputs or increasing its inputs without becoming inefficient, is also applied into directional distance function. For DMUk xik ,yrk , the PPS for super-efficiency is modified as 8 > > > < n n X X T k ¼ ðxi ,yr Þxi Z lj xij ,i ¼ 1,:::,m; yr r lj yrj ,r ¼ 1,:::,s; > > j¼1 j¼1 > : j ak
9 > > > = n X lj ¼ 1, lj Z 0,j ¼ 1,:::,n,j ak > > j¼1 > ; j ak
In this section, we tackle the above two infeasibility problems in N–L super-efficiency model (7). Note that these infeasibility issues are caused by the choice of ðg x ,g y Þ, which results in the same changing proportion b taken by each input (decreased) and output (increased) simultaneously. We can choose a different ð g x ,g y Þ so that the above infeasibility cases will not occur. In other words, we can consider a reference input–output bundle different from the conventional selection in [14], by using ðaxio 1, byro þ1Þ for ðg x ,g y Þ, and obtain a new directional distance function as Dðxio ,yro Þ ¼ max b : ð1baÞxio b,ð1þ bbÞyro þ b A T ð8Þ Here both a and b are pre-determined positive parameters. We will develop a procedure to decide for such parameters to address the two infeasibility issues. For DMUk , the super-efficiency model based upon (8) can be written as maxbk n X s:t: lj xij r ð1bk aÞxik bk ,i ¼ 1,:::,m
ð9:1Þ
j¼1 jak
jak
ð5Þ
n X
j¼1 j ak
lj yrj Z ð1 þ bk bÞyrk þ bk ,r ¼ 1,:::,s
ð9:2Þ
Y. Chen et al. / Omega 41 (2013) 621–625
n X
lj ¼ 1, lj Z 0,j ¼ 1,:::,n,j ak
ð9:3Þ
623
Therefore, if the following new inequality (13) is satisfied, the previous inequality (12) or (10) holds.
j¼1 j ak
Note that an appropriate negative value for bk will eliminate the infeasibility problem in model (9), which is caused by the special situation that DMUk is the only unit with zero value for some input (the first exception mentioned in [15]). Next we will focus on the second exception by preventing directional output targets 1 þ bk b yrk þ bk from taking negative values. Note that constraint (9.1) and (9.2) can be re-written as 8 n X > > > ðaxik þ1Þbk rxik lj xij ,i ¼ 1,:::,m > > > > j¼1 > > > < j ak n X > > > lj yrj yrk ,r ¼ 1,:::,s ðbyrk þ1Þbk r > > > > j¼1 > > > : jak
n X
lj yrj yrk
j¼1 yrk byrk þ 1
r bk r
j ak
,r ¼ 1,:::,s
byrk þ 1
r ¼ 1,:::,sj ¼ 1,:::,n
Z
1 yrj
i ¼ 1,:::,m
max xij min xij þ 1
j ¼ 1,:::,n
a min
xik
To ensure that bk has a feasible solution,
yrk byrk þ 1
lj xij
j¼1 j ak
r
axik þ 1
must be satisfied for any combination of i and r, i.e., xik
r ¼ 1,:::,s j ¼ 1,:::,n
j ¼ 1,:::,n
required to be strictly greater than zero, or equivalently, a min min xij þ1 1 i ¼ 1,:::,m j ¼ 1,:::,n 4 max max r ¼ 1,:::,sj ¼ 1,:::,n yrj max max xij min xij þ 1
i ¼ 1,:::,m
j ¼ 1,:::,n
Determination of parameters a and b: Case I: completely positive inputs We first suppose that all input values are positive for all DMUs, namely min min xij 40, and then have i ¼ 1,:::,m j ¼ 1,:::,n a4 max max ð1=yrj Þ max max xij min xij
yrk
max
ð10Þ
axik þ 1
yrk
¼ min
r ¼ 1,:::,s byrk þ 1
1
¼ min
¼
r ¼ 1,:::,s b þ 1 yrk
1 b þ max
1 r ¼ 1,:::,s yrk
ð11Þ Then inequality (10) becomes xik 1 b þ max
1 r ¼ 1,:::,s yrk
Z min
i ¼ 1,:::,m
n P
n P
lj xij
j¼1 j ak
axik þ1
¼ max
! axik þ 1 r max
i ¼ 1,:::,m
r max
i ¼ 1,:::,m
!
S lj xij xik =
j¼1 j ak
i ¼ 1,:::,m
! maxj ¼ 1,:::,n xij þ 1 min xij = a min xij þ 1 j ¼ 1,:::,n
j ¼ 1,:::,n
max xij min xij þ 1 = min a min xij þ 1 :
j ¼ 1,:::,n
ð12Þ
axik þ1 n
Since S lj xij r max xij , we have max j ¼ 1,:::,n
j ¼ 1,:::,n
min xij .
Thus
we
j ¼ 1,:::,n
obtain
the
i ¼ 1,:::,m
j ¼ 1,:::,n
j ¼ 1,:::,n
ranges
for
parameters a and b, which are a4
! max max xij min xij þ1 1 = max max ð1=yrj Þ i ¼ 1,:::,m
j ¼ 1,:::,n
j ¼ 1,:::,n
0 ob r a min
min xij ,
i ¼ 1,:::,m j ¼ 1,:::,n
!
max xij min xij þ1 max
j ¼ 1,:::,n
j ¼ 1,:::,n
min xij þ 1 =
i ¼ 1,:::,m j ¼ 1,:::,n
!
max
r ¼ 1,:::,s j ¼ 1,:::,n
max
i ¼ 1,:::,m
1=yrj . Any value com-
bination taken from the above ranges is a reasonable candidate for parameters of our DDF (8). Any specific choice of a and b depends on the practical factors and preferences of each decision maker. Further, if integer-restricted parameters are expected or required by the decision maker, it is easy to find an integervalued a, denoted by a, large enough to be greater than
! max max xij min xij þ1 1 = max max ð1=yrj Þ r ¼ 1,:::,s j ¼ 1,:::,n
lj xij xik
j¼1 j ak
i ¼ 1,:::,m
n
j¼1 j ak
i ¼ 1,:::,m
i ¼ 1,:::,m j ¼ 1,:::,n
min
In the following we deduct the mathematical relationship between parameters a and b from inequality (10). For the left-hand side of (10), we get r ¼ 1,:::,s byrk þ 1
þ 1 1 = min
lj xij
j¼ 1
jak
yrk r min r ¼ 1,:::,s byrk þ1 i ¼ 1,:::,m max
ð14Þ
j ¼ 1,:::,n
r ¼ 1,:::,s j ¼ 1,:::,n
Pn
ð13Þ
Then we obtain b r1= maxi ¼ 1,:::,m max xij min xij þ1 = j ¼ 1,:::,n j ¼ 1,:::,n max max 1=yrj , or equivalently a min min xij þ1 r ¼ 1,:::,s j ¼ 1,:::,n i ¼ 1,:::,m j ¼ 1,:::,n br a min min xij þ 1 = max max xij min xij þ 1 i ¼ 1,:::,m j ¼ 1,:::,n i ¼ 1,:::,m j ¼ 1,:::,n j ¼ 1,:::,n max max 1=yrj , from inequality (13). To make parar ¼ 1,:::,s j ¼ 1,:::,n meter b restricted to positive value, a min min xij þ1 = i ¼ 1,:::,m j ¼ 1,:::,n is maxi ¼ 1,:::,m max xij min xij þ 1 max max 1=yrj
r ¼ 1,:::,sj ¼ 1,:::,n
n P
j ¼ 1,:::,n
min xij þ1
i ¼ 1,:::,mj ¼ 1,:::,n
j ¼ 1,:::,n
Therefore, to obtain non-negative destinations of directional movement in outputs, b needs to be selected to ensure that 1 þ bk b yrk þ bk Z 0,r ¼ 1,:::,s, or equivalently, byrk þ 1 bk Z yrk , r ¼ 1,:::,s. Since ðaxik þ1Þ and byrk þ 1 are positive values, we obtain 8 n X > > xik lj xij > > > > j¼1 > > > > j ak > > < bk r ,i ¼ 1,:::,m axik þ 1 > > > > > > > > > > > > :
1 bþ max max
max
i ¼ 1,:::,m
j ¼ 1,:::,n
j ¼ 1,:::,n
min min xij , and simultaneously makes a min min xij i ¼ 1,:::,m j ¼ 1,:::,n i ¼ 1,:::,m j ¼ 1,:::,n ! ! þ 1 = max max xij min xij þ1 max max y1 4 1. i ¼ 1,:::,m
j ¼ 1,:::,n
r ¼ 1,:::,s j ¼ 1,:::,n
j ¼ 1,:::,n
rj
Thus a could be any positive integer greater than
max max ð1=yrj Þ þ 1 max max xij min xij þ 1 1 = r ¼ 1,:::,s j ¼ 1,:::,n
min
i ¼ 1,:::,m
j ¼ 1,:::,n
j ¼ 1,:::,n
min xij . One possible selection for a is the smallest one,
i ¼ 1,:::,mj ¼ 1,:::,n
while one candidate for b could be the greatest positive ! a min min xij þ1 = max max integer less than i ¼ 1,:::,m j ¼ 1,:::,n
i ¼ 1,:::,m
j ¼ 1,:::,n
624
Y. Chen et al. / Omega 41 (2013) 621–625
xij min xij þ 1 max max 1=yrj . This integer choice of r ¼ 1,:::,sj ¼ 1,:::,n j ¼ 1,:::,n a, b will be used in the current study for calculations in the illustration. The above procedure for determining a and b is based upon the assumption that min min xij 4 0. If min min xij ¼ 0, the i ¼ 1,:::,m j ¼ 1,:::,n
i ¼ 1,:::,m j ¼ 1,:::,n
above parameter choices are no longer valid, and we need to develop an alternative approach for determining a and b. Case II: any zero input From the right-hand side of inequality (12), we uehave 1 ! C BB n B B = 1axik þ1 r max maxi ¼ 1,:::,m @@ S lj xij xik C A j¼1 i ¼ 1,:::,m 00
! max xij min xij =
j ¼ 1,:::,n
j ¼ 1,:::,n
j ak
1 C a min xij þ 1 C max xij min xij = min A r i ¼max j ¼ 1,:::,n 1,:::,m j ¼ 1,:::,n j ¼ 1,:::,n i ¼ 1,:::,m
a min xij
þ1 .
j ¼ 1,:::,n
Then inequality (12) or (10) will be satisfied if the following inequality (15) holds: max max xij min xij 1 i ¼ 1,:::,m j ¼ 1,:::,n j ¼ 1,:::,n ð15Þ Z a min min xij þ 1 b þ max max y1 r ¼ 1,:::,s j ¼ 1,:::,n
i ¼ 1,:::,m j ¼ 1,:::,n
rj
Since at least one DMU has zero input(s), and it is impossible for all DMUs to be at zero level in a same input, then at least for some i, it is true that max xij min xij 4 0, or j ¼ 1,:::,n j ¼ 1,:::,n max xij min xij 4 0. Then we obtain further max i ¼ 1,:::,m j ¼ 1,:::,n j ¼ 1,:::,n ! ! b r a min
min xij þ1 =
i ¼ 1,:::,m j ¼ 1,:::,n
max
j ¼ 1,:::,n
the
1=yrj
i ¼ 1,:::,m
max xij min xij max
j ¼ 1,:::,n
from inequality (15). Since
range for !
b
is
simplified
to
j ¼ 1,:::,n
r ¼ 1,:::,s
min
min xij ¼ 0, i ¼ 1,:::,mj ¼ 1,:::,n max xij b r 1= max i ¼ 1,:::,m
r ¼ 1,:::,s j ¼ 1,:::,n
positive integer depending on a specific decision maker. Note that the VRS DEA model is unit-invariant. Therefore, to obtain positive integer values for parameters a and b, all input values xij can be proportionally scaled down to different values denoted as xij , which satisfy max xij min xij max max 1=yrj 4 1. Thus 1= max i ¼ 1,:::,m
j ¼ 1,:::,n
j ¼ 1,:::,n
r ¼ 1,:::,sj ¼ 1,:::,n
for the numerical examples analyzed in this paper, we let a ¼ 1, and select the greatest positive integer less than max xij min xij max max 1=yrj for b. 1= max i ¼ 1,:::,m
j ¼ 1,:::,n
j ¼ 1,:::,n
max
max
r ¼ 1,:::,s j ¼ 1,:::,n
1=yrj o 0. We need to make some data transfor-
mations in scale. All input values in the original data set are scaled down by 0.01. Based upon the new data set, which are presented in columns 5–7 of Table 1, we have 1= max i ¼ 1,:::,m max xij min xij Þ max max 1=yrj ¼ 9:5. Thus we select j ¼ 1,:::,n
r ¼ 1,:::,s j ¼ 1,:::,n
j ¼ 1,:::,n
1 for a, and 9 for b in our DDF-based super-efficiency model (9), and obtain the related super-efficiency results 1bk in the last column of Table 1. Based on our newly-proposed DDF (8) with parameters a, b , the VRS super-efficiency model (9) avoids the infeasibility issue occurring either in conventional VRS models or N–L super efficiency model (7). By selecting legitimate parameters a, b , and choosing an appropriate value of bk (negative if superefficient), it always yields a feasible solution for the super efficiency score, which is calculated as 1bk . This is true even
j ¼ 1,:::,n
max 1=yrj , while parameter a can be any
min xij max j ¼ 1,:::,n
max
than half of the value in that input of any other DMUs. i.e., 2x11 ox1j ,j ¼ 2,3,4,5, and thus for all non-negative lj s (j ¼ 2,3,4,5) P combinations satisfying we have 2x11 o j ¼ 2,3,4,5 lj ¼ 1, P l x . For DMU 3, its input 2 is zero, while the rest j ¼ 2,3,4,5 j 1j four DMUs all have positive values for that input. Thus all possible convex combinations of input 2 of DMUs 1, 2, 4, 5 are strictly greater than zero, making the constraint for input 2 in N–L super-efficiency model (7) impossible to be satisfied. Both of these cases make the approach in [15] infeasible. The last column of Table 1 reports the super-efficiency scores obtained from our newly-proposed approach. To select parameters a and b, we note that from the original data in Table 1, 1= max max xij min xij i ¼ 1,:::,m j ¼ 1,:::,n j ¼ 1,:::,n !
r ¼ 1,:::,sj ¼ 1,:::,n
Table 2 Numerical example II [8]. DMU
Input 1x1
Input 2x2
Input 3x3
Output 1y1
Output 2y2
1 2 3 4 5 6 7 8 9 10
182 74 160 183 133 106 109 240 276 191
237 82 195 150 155 120 110 243 188 117
468 148 400 339 329 138 188 806 574 466
5008 1857 4041 2779 3506 1306 1515 7763 4577 3322
5303 2336 5001 2418 3602 956 2282 9601 6493 4233
Consider a simple numerical example listed in columns 2–4 of Table 1 where we have five DMUs with two inputs and a single output. For DMU 1, its value in input 1, x11 ¼ 2, is strictly less Table 3 Alternative measures of VRS super-efficiency. Table 1 Numerical example I. DMU Original
1 2 3 4 5
New
Input 1x1
Input 2x2
Output 1y1
Input 1x1
Input 2x2
Output 1y1
2 8 12 8 5
4 3 0 4 5
2 4 6 5 5
0.02 0.08 0.12 0.08 0.05
0.04 0.03 0 0.04 0.05
2 4 6 5 5
Modified DDFbased superefficiency
1.0294 0.9923 1.0323 0.9935 1.0152
DMU
Standard radial
L–R
N–L
DDF-based model (9)
1 2 3 4 5 6 7 8 9 10
1.0626 1.5277 0.9765 0.7354 0.9752 1.0725 0.7852 infeasible 0.9246 1.0602
1.0626 1.5277 0.9765 0.7354 0.9752 1.0725 0.7852 6 0.9246 1.0602
1.0285 1.4430 0.9889 0.8566 0.9881 1.0725 0.8863 1.3836 0.9581 1.0334
1.00559 1.05113 0.99789 0.97561 0.99776 1.00724 0.97871 1.38354 0.99302 1.00557
Y. Chen et al. / Omega 41 (2013) 621–625
when the DMU under evaluation is the only unit in the reference set with zero inputs.
625
super-efficiency measure, however, provides a quite different rank for super-efficient units, which is DMU 2, 8, 6, 10, 1, 3, 5, 9, 7, 4 from high to low.
4. Comparisons 5. Conclusions We use the data set presented in Table 2 to illustrate different results obtained from various VRS super-efficiency models. This data set is previously studied in [8] and [15]. Columns 2–5 in Table 3 report the super-efficiency measures obtained from the standard input–oriented VRS model [5], the input-oriented L–R method [9], the N–L measure [15], and our modified DDF-based model (9), respectively. The column identified as ‘‘Standard radial’’ presents the efficiency and super-efficiency scores obtained from the conventional input-oriented VRS super-efficiency model. DMUs 3, 4, 5, 7, 9 are inefficient units, while DMUs 1, 2, 6, 8, 10 are efficient. Among these five efficient units, four (DMUs 1, 2, 6, 10) have super-efficiency greater than one. DMU 8 does not have a feasible solution to the conventional VRS super-efficiency problem. Column ‘‘L–R’’ shows the input-oriented super-efficiency obtained from the Lovell–Rouse (L–R) method [9] by setting the scale factor a ¼ 6. Note that the L–R super-efficiency measure equals the scale factor 6 for DMU 8. Except for this infeasible DMU 8, all scores obtained from the conventional model are identical to those obtained via the L–R approach. Column ‘‘N–L’’ displays the N–L measure of super-efficiency [15] 1bk for all ten DMUs. The last column of Table 3 shows the results obtained from our DDF-based VRS super-efficiency model (9). A detailed procedure for parameter selection and model calculation is presented as follows: First, we need to determine a reasonable choice for parameters a and b in model (9). For the data set presented in Table 2,
max
max
r ¼ 1,:::,s j ¼ 1,:::,n
1 yrj
þ1 max max xij min xij þ 1 1 = i ¼ 1,:::,m
j ¼ 1,:::,n
Conventional radial VRS super-efficiency models become infeasible in certain situations. Although the Nerlove–Luenberger (N–L) measure of super-efficiency [15] avoids such infeasibility issues under VRS condition, it would fail in two special cases. This paper modifies the DDF-based super-efficiency in [15] so that the newlyproposed approach is always feasible. We point out that the values of parameters a and b depend on the specific data set in order to make all DMUs feasible in our DDF-based model. This does not indicate that inefficient DMUs will affect the efficiency scores. Note that the parameter ranges are calculated based on extreme values of related input/output measures. Therefore, the values influencing the choice regions of parameters are the maximum or minimum input/output values from the whole data set, and are not relevant to the specific group of inefficient DMUs.
Acknowledgments The authors are grateful for the constructive comments and suggestions from two anonymous reviewers on an earlier version of this paper. Dr. Juan Du would like to thank the support by the National Natural Science Foundation of China (Grant no. 71101108) and China Postdoctoral Science Foundation (Grant no. 20110490696). Professor Huo would like to thank the support by the National Natural Science Foundation of China (Grant no. 70832005).
j ¼ 1,:::,n
min
min xij is calculated as 9.0365. Thus we choose a ¼ 10, and a min min xij þ 1 = max max xij min xij i ¼ 1,:::,m j ¼ 1,:::,n i ¼ 1,:::,m j ¼ 1,:::,n j ¼ 1,:::,n þ1Þ max max 1=yrj is then computed as 1.1066. The only
References
i ¼ 1,:::,m j ¼ 1,:::,n
r ¼ 1,:::,s j ¼ 1,:::,n
candidate for b is 1. Thus our VRS super-efficiency model (9) based on DDF (8) is presented as maxbk n X s:t: lj xij rð110bk Þxik bk ,i ¼ 1,:::,m j¼1 j ak n X
lj yrj Z ð1þ bk Þyrk þ bk ,r ¼ 1,:::,s
j¼1 jak n X
ð16Þ
lj ¼ 1
j¼1 jak
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