Efficiency analysis to incorporate interval-scale data

European Journal of Operational Research 207 (2010) 1116–1121

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Interfaces with Other Disciplines

Efficiency analysis to incorporate interval-scale data Akram Dehnokhalaji a,b, Pekka J. Korhonen a,*, Murat Köksalan a,c, Nasim Nasrabadi a,b, Jyrki Wallenius a a

Aalto University, School of Economics, P.O. Box 21220, 00076 Aalto, Finland Tarbiat Moallem University, Department of Mathematics, Tehran, Iran c Middle East Technical University, Department of Industrial Engineering, Ankara, Turkey b

a r t i c l e

i n f o

Article history: Received 4 April 2009 Accepted 17 March 2010 Available online 31 March 2010 Keywords: Data Envelopment Analysis Interval-scale Research evaluation

a b s t r a c t We develop an approach to efficiency analysis to enable us to incorporate interval-scale data in addition to ratio-scale data. Our approach introduces a measure of inefficiency and identifies efficient units as is done in Data Envelopment Analysis. The basic idea in our approach is to find the ‘‘best” hyperplane separating the units that are better and worse than each unit. ‘‘Best” is defined in such a way that the number of not-better units is maximal. The efficiency measure is defined as a proportion of not-better units to all units. The results are invariant under a strictly increasing linear re-scaling of any input- or output-variables. Thus zeroes or negative values do not cause problems for the analysis. The approach is used to analyze the data of the research evaluation exercise recently carried out at the University of Joensuu, Finland. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Since Charnes et al. (1978, 1979) developed the Data Envelopment Analysis (DEA) as a method for evaluating the relative (technical) efficiency of different Decision Making Units (DMUs) essentially performing the same task, it has become a very popular method. DEA is value-free in the sense that the efficiency evaluation is based on the available data without taking into account the decision maker’s (DM) preferences. The units are assumed to operate under similar conditions. Each of the units uses multiple inputs to produce multiple outputs. Based on information about existing input/output-values of the units and some preliminary assumptions, a so-called production possibility set (PPS) is specified. The production possibility set consists of all possible input/output combinations. It can be continuous or discrete. A specific part of the production possibility set is called an efficient frontier (surface). If a DMU lies on the efficient frontier, it is referred to as an efficient unit, otherwise it is considered inefficient. DEA also provides efficiency scores and a reference set for inefficient DMUs. The efficiency scores represent a degree of efficiency of the DMUs. The reference set for inefficient units consists of a set of efficient units and determines a virtual target unit on the efficient frontier. In DEA the target unit is found by projecting an inefficient DMU radially1 to the efficient surface. To check the efficiency * Corresponding author. E-mail address: Pekka.Korhonen@hse.fi (P.J. Korhonen). 1 The term ‘‘radial” in the traditional DEA-literature means that the purpose is to reach the efficient frontier by proportionally increasing the values of the current outputs or decreasing the values of the current inputs. 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.03.039

of a unit and to find the reference set and the efficiency score for inefficient units requires solving an LP-model. An implicit assumption in basic DEA models is that the inputand output-variables are measured on a ratio scale with positive or negative values. When any input-/output-variable has both negative and positive values, the basic DEA models do not work, because the projection vector cannot have positive and negative values in the same model. In case the values of the variables presented on an interval-scale are positive (or negative, but not mixed), we can technically use DEA to solve the problem, but the efficiency scores are generally meaningless, because they are not invariant under the translation of the variables. Translation invariant means that ‘‘an affine displacement of data does not alter the efficient frontier and the classification of DMUs as efficient or inefficient” (Ali and Seiford, 1990). They found the BCC model (Banker et al., 1984) and the additive model (Charnes et al., 1985) translation invariant, but generally the inefficiency scores in BCC-models obtained for the inefficient DMUs will be different (Ali and Seiford, 1990). However, the inefficiency scores remain invariant if the translation is made to input (output)-variables in the output (input)-oriented BCC model (Lovell and Pastor, 1995; Pastor, 1996). Halme et al. (2002) studied properties of the problem where the interval-scale variable is defined as the difference of two positive or negative ratio-scale variables. They proposed the decomposition of such a variable back to one input and one output variable both measured on a ratio scale. If the conventional DEA model is applied to this kind of transformed data, it is possible to classify an inefficient unit as efficient. We propose a novel approach to efficiency analysis. Our approach is inspired by DEA. One of the main advantages of our

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approach over DEA is that we can handle interval-scale data, whereas DEA cannot. The modified model is based on the idea of finding a hyperplane separating the units that are better and worse than each unit under consideration. The hyperplane is defined in such a way that the number of not-better units is maximal. The approach maintains the same optimal solution under re-scaling (using a linear transformation) of each input- and output-variable. The efficiency ‘‘score” is defined as the proportion of not-better units to all units, and it is independent of re-scaling. The efficient units generated by our model can be used as benchmarks (reference units). We illustrate the model with a small application. The basic idea for the modification is adapted from the article by Köksalan et al. (2010), in which the authors proposed a method to sort units into classes. Cherchye et al. (2000) proposed a similar idea, where the production possibility set is reduced until the unit under consideration becomes efficient. The paper is organized in five sections. In the next section, we present some theoretical considerations. We develop the model in Section 3 and apply it to a resource evaluation problem in Section 4. We conclude the paper in Section 5. 2. Theoretical considerations 2.1. Preliminary considerations Consider a production technology, where m inputs are needed p to produce p outputs. Let x 2 Rm þ and y 2 Rþ denote the inputs and outputs, respectively. We define the production possibility set (PPS) as follows:

P ¼ fðy; xÞ j y can be produced from xg

2 Rpþm þ ;

ð2:1Þ

which consists of all feasible inputs and outputs. As usual, we assume more is better in the outputs and less is better in the inputs. Efficient and weakly efficient points in set P are defined as follows: Definition 1. A point (y*,x*) 2 P is efficient in set P iff (if and only if) there does not exist another (y, x) 2 P such that y P y*, x 6 x* and (y, x) – (y*, x*). Definition 2. A point (y*, x*) 2 P is weakly efficient in set P iff there does not exist another (y, x) 2 P such that y > y* and x < x*. We next show that the efficient set in P is invariant under the transformation of the variables of the form: x ? ax + b, where x refers to an input or output variable and a > 0. Lemma 1. A point (u*,v*) 2 PL = {(u,v)ju = diag(ay)y + by and v = diag(ax)x + bx,ay > 0,ax > 0,(y,x) 2 P} is (weakly) efficient iff the corresponding (y*,x*) 2 P is (weakly) efficient, where u* = diag(ay)y* + by and v* = diag(ax)x* + bx.2 Proof. Assume (y*, x*) 2 P is efficient in P, but (u*, v*) 2 PL is not efficient in PL. Then there exists another (u, v) 2 PL such that u P u*, v 6 v* and (u, v)–(u*, v*). Hence diag (ay)y + by P diag (ay)y* + by and diag (ax)x + bx 6 diag (ax)x* + bx, from which it follows that y P y*, x 6 x* and (y, x)–(y*, x*). This is a contradiction with the assumption that (y*, x*) 2 P is efficient in P. Because we may write y = diag (ay)1u  by and x = diag (ax)1v  bx, we can prove as above that if (u*,v*) 2 PL is efficient in PL, then the corresponding (y*, x*)2P is efficient as well. For weakly efficient points the proof is similar. h

2

Notation diag(a) refers to a diagonal matrix with the vector a on the diagonal.

Table 1 Unified Formulation for BCC-models (fk j k 2 Rnþ and 1Tk = 1}). Envelopment model

Multiplier model

T

max h ¼ r þ e1 s s:t: Uk  rw  s ¼ u0

min h ¼ qT u0 þ g ð2:2aÞ

T

1 k¼1 k; s P 0

s:t:

 qT U þ g1 P 0

ð2:2bÞ

T

q w¼1 q P e1 g is free

(e > 0 ‘‘Non-Archimedean”)

The set consisting of all efficient points is called an efficient set and the set consisting of all weakly efficient points is called a weakly efficient set. In case the (weakly) efficient set also includes interior points, the term frontier is often used to refer to the set. 2.2. Basic data envelopment models Assume we have n decision making units (DMU), each consuming m inputs, and producing p outputs, where m and p are positive integers. This information can be presented by two matrices. Let X be an (m  n) – matrix and Y be a (p  n) – matrix containing observed input and output measures for the DMUs, respectively. In DEA it is traditionally assumed that the matrices X and Y consist of only non-negative elements3. We denote by xj (the jth column of X) the vector of inputs consumed by DMUj, and by xij the quantity of input i consumed by DMUj. A similar notation is used for outputs. We further assume that xj–0 and yj–0 for j 2 J = {1, . . . , n}, and for simplicity we assume that there are no duplicates in the data set. Furthermore, we denote 1 = [1, . . . , 1]T and refer by ei to the ith unit vector in Rn . In DEA, the existing data set is used to define the (practical) production possibility set as set P = {(y, x)jx P Xk, y 6 Yk, k 2 K}, where the definition of K depends on the assumptions made about the form of set P. In this paper, there is no need to emphasize the different roles   y of inputs and outputs. Therefore, we simply denote u ¼ x   Y and U ¼ . We call u an input/output-vector and use notaX   y0 to refer to the output and input values of the unit tion u0 ¼ x0 under consideration. The basic BCC-model (Banker et al., 1984) in a general form (Joro et al., 1998) is provided in Table 1.     0 y0 Vector w ¼ for for input-oriented models and w ¼ x0 0 output-oriented models. We may also deal with so-called   y0 . If some of the combined models by assuming that w ¼ x0 input- or output-variables are nondiscretionary, then we set the corresponding wi = 0. It can be shown that DMU0 is efficient if and only if the optimal value of the above model is equal to zero. The inefficiency score of DMU0 is equal to h*, where h* is the optimal objective value of model (2.2). 3. Development of the model 3.1. Motivation and illustration Our aim is to modify the basic DEA models in such a way that they are capable of dealing with interval-scale data in addition to 3 An implicit assumption is that the input- and output-variables are given on a ratio-scale.

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Table 2 The Data for an Illustrative Example. DMU

A

B

C

D

E

F

G

H

I

J

Output 1 Output 2 Input

1 3 1

2 1 1

2 4 1

3 6 1

5 3.5 1

5.5 2 1

7 5.5 1

6.5 1 1

8 2.5 1

5 5.75 1

ratio-scale data. Our approach is based on the idea of finding a hyperplane including the DMU under consideration,4 which classifies DMUs into two sets. In other words, we will compute the weights for the input- and output- variables such that the number of ‘‘better” units is minimal. The main difference between the traditional DEA models and our approach is that we do not measure the distance from the efficient frontier5 to DMU0. Instead, we find the minimum number of units we have to omit from the set of DMUs to make DMU0 efficient. This formulation enables us to deal with negative values in the variables without any problems, and the efficiency measure is invariant as well under the transformation of the form: x ? ax + b, where x refers to an input or output variable and a > 0. This is not generally true in DEA; however, the additive model is also translation invariant (Ali and Seiford, 1990). Our approach consists of many DEA features, such as the concept of efficiency, efficiency score, and benchmarking units (reference units). Moreover, each unit is presented in the most optimistic light. The basic idea is illustrated with the following example consisting of 10 DMUs (A, B, . . . , and J), each producing two outputs using the same amount of a single input. We may assume that the output-variables are measured on an interval-scale. The data are given in Table 2 and described in output space in Fig. 1. Let E be DMU0. Hyperplane H0 identifies the four units D, G, I and J to be excluded from the PPS in order to make E efficient. In a similar manner, hyperplane H1 identifies units G, I and J, and so on. Our goal is to choose the ‘‘best” hyperplane in such a way that the number of excluded units is minimal or equivalently the number of remaining units is maximal. Hence, hyperplanes such as H0 are not used as the basis of our efficiency measure. However, alternative hyperplanes may exist having the desired property that can be identified as ‘‘best”. Actually the only important aspect for us is the number of DMUs to be separated, not the hyperplane to be chosen. Note that if a unit is efficient, there is no need to exclude any units from the PPS. Köksalan et al. (2010) proposed a model in a different context, which they used to sort units into classes. We modified their model to serve our purposes of generalizing DEA to handle interval-scale data. Note that not all units that need to be excluded to make E efficient are necessarily efficient. At least one of them has to be efficient, though. We propose that all or a subjectively-selected subset of the efficient units that need to be removed may be used as benchmarks or targets (see for example Korhonen et al., 2003). In traditional DEA, the efficiency score is defined in relation to units on the efficient frontier solely. In our approach, the efficiency score may depend on inefficient units as well. For example, let us consider unit F in Fig. 1. In traditional DEA, units G and I are used to compute the efficiency score for F and the elimination of, for example, H has no impact on F’s efficiency score. In our approach, in addition to efficient units G and I, the inefficient unit H is also 4 In the sequel, we will use the notation DMU0 to refer to the unit under consideration. 5 Measuring the distance from the efficient frontier is a basic principle in DEA. In fact, even if the variables are in the interval-scale, models (2.2 a, b) can be used to identify efficient units as long as a projection vector w has a property w P 0, w – 0 (see, e.g., Korhonen and Luptacik, 2004). However, because a linear transformation is allowed to interval-scale variables, the traditional DEA-efficiency score is not useful, because it is not invariant to linear transformations.

needed to define the efficiency score. Thus, eliminating H has an impact on F’s efficiency score, unlike in traditional DEA. Some of the drawbacks of the efficiency scores in traditional DEA are further discussed in Köksalan and Tuncer (2009). 3.2. The model Our aim is to construct a model with which we can find, for each unit, the maximal subset of n units in which DMU0 is efficient. The optimal solution of model (2.2b) for the efficient units is zero. Thus we include this requirement into the model as a constraint. However, if the unit is not efficient, some of the inequalities - qTU + g1 P 0 are not fulfilled. To minimize the number of violated inequalities, we use a big M technique. Note that each inequality is associated with one unit. Violating an inequality constraint implies that we have to omit the corresponding unit to make DMU0 efficient. Thus we formulate and solve for each unit the following zero-one integer program:

Min x0 ¼ 1T z s:t: qT u0 þ g ¼ 0 qT U þ g1 þ Mz P 0ðM >> 0Þ

qT w ¼ 1 q P e1 ðwhere e > 0 is Non-ArchimedeanÞ g is free zj 2 f0; 1g;

ð2:3Þ

j ¼ 1; . . . ; n:

Notice that zj’s are zero-one variables in the above model. When zj is set to one, the corresponding constraint becomes redundant. It means that DMUj is excluded from PPS. In fact, the set of zj’s, with optimal values of zero, determines the maximal subset of the production possibility set in which DMU0 is efficient. Note that the model’s structure is very close to the multiplier BCC model in DEA. The role of e and g in model (2.3) is exactly the same as in the traditional DEA-model. The value e > 0 guarantees that weakly efficient units are not diagnosed efficient. The value of g determines the returns to scale property. For instance, if we restrict g = 0, then we have a constant returns to scale model. In defining vector w appropriately, we may deal with input-oriented, output-oriented, or combined DEA models (Joro et al., 1998). The only requirement for w is that w P 0 and w – 0. 3.3. Some theoretical results Theorem 1. Let (q*, g*, z*) be an optimal solution for (2.3) and J0 ¼ fj j zj ¼ 0; j 2 Jg, where J = {1, . . . , n}. Then J0 is the maximal subset of J for which DMU0 is efficient. Proof. The feasibility of (q*, g*, z*) implies that-q*Tu0 + g* = 0, q *Tw = 1, and -q*TU + g*1 P 0 for all j 2 J0. Because -q*Tu0 + g* = 0, it is the optimal value of the objective function of model (2.2b). Only considering the units for which j 2 J0, the constraints -q*TU + g*1 P 0 in addition to the other identical constraints of (2.3) make the solution (q*, g*, 0) feasible to (2.2b), implying that DMU0 is efficient in J0. To show the number of elements in J0 is maximal, assume DMU0 is efficient in J1 # J such that jJ1j > jJ0j. When model (2.2b) is solved for DMU0, its optimal solution is a feasible solution for (2.3) with the optimal objective value less than x0 , which is a contradiction. h Definition 3. The efficiency score of DMU0 is defined as e0 = |J0|/n, where |J0| denotes the cardinality of J0.

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Fig. 1. Illustrating a separating hyperplane H0 and the optimal hyperplane H1 for unit E.

Note that the minimum increment in the efficiency scores is 1/n and the possible number of different scores is n.

2. Because xj = A1yj  A1b and A1 = diag (1/aj), we may prove as above that k* P k.

Lemma 2. Consider the set of points X ¼ fxj 2 Rp j j 2 Jg where J = {1, . . . , n}, select a point from the set and refer to this point by index ‘‘o”. Let k be the solution of the following problem:

Hence, k = k*.

h

k ¼ fmaximal number of points for which wT xo P wT xj ; j 2 Jg;

Theorem 2. The efficiency and the efficiency score are invariant under a strictly increasing linear re-scaling of the variables.

s:t: 1T w ¼ 1;

Proof. The result follows from Lemma 2. h

w P 0:

ð2:4Þ

Then k is also the solution of problem 2.4, in which we have applied a non-singular linear re-scaling to the variables: yj = Axj + b, where A = diag (ai) is a diagonal matrix with elements a1 > 0, . . . , ap > 0 on the diagonal. Proof. Let k* be the solution of the problem: 

k ¼ fmaximal number of points for which

v T yo P v T yj ;

j 2 Jg;

T

v ¼ 1; v P 0:

s:t: 1

1. We prove that k P k*. For each j 2 J, for which vTyo P vTyj, we have vT(A xo + b) P vT(Axj + b)  vTA xo + vTb P vTA xj + vTb  vTA xo P vTA xj  wTxo P wTxj, when we define wk = vkak/ (Ri viai) P 0,k = 1, . . . , p,w – 0. Because 1Tw = 1 and w P 0, hence k P k*.

3.4. Example By using the data set from Table 2, we solve an output-oriented BCC version with model (2.3) for each DMU. The results are reported in Table 3. For each row A through J, in columns A through J, number zero indicates units that need not be removed from the PPS to guarantee the efficiency of the unit in the corresponding row. The column next to last represents the efficiency score of the units, based on our model (2.3). To discuss the problems associated with traditional DEA models, we also report the efficiency scores by using the output-oriented BCC-model (2.2) in the last column of Table 3. Both models diagnose efficient units correctly. On the other hand, the efficiency scores for inefficient units will differ. In the numerical example, the scores for each unit are close to each other. Note that if we subtract say two from each output value, our efficiency scores (Definition 3) will remain the same. However, we

Table 3 Example Results.

A B C D E F G H I J P

A

B

C

D

E

F

G

H

I

J

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 2

1 1 1 0 0 0 0 0 0 0 3

1 1 0 0 0 0 0 0 0 0 2

0 1 0 0 0 0 0 0 0 0 1

1 1 1 0 1 1 0 1 0 0 6

0 1 0 0 0 1 0 0 0 0 2

0 1 0 0 1 1 0 1 0 0 4

1 1 1 0 1 0 0 0 0 0 4

P

Efficiency score (2.3)

Efficiency score (2.2)

5 8 3 0 3 3 0 2 0 0

0.5 0.2 0.7 1 0.7 0.7 1 0.8 1 1

0.500 0.264 0.667 1 0.698 0.698 1 0.812 1 1

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are not even able to calculate such scores for the traditional DEAmodel. Note that the column sums tell how many times the unit belongs to the reference set of other units. In this problem, the efficiency scores are quite close to each other, but this is not necessarily true in general. We can easily create an example where this does not hold. For example, adding a constant, say 10, to each output variable changes the efficiency scores of the DEA model drastically, whereas our results stay the same as our approach is invariant to a linear transformation.

Table 4 Research Assessment Exercise Data Set at the University of Joensuu. #

Department

O1

O2

O3

O4

O5

O6

O7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Geography History Sociology Social Policy The Karelian Institute Economics and Business Law Biosciences Chemistry Computer & Statistics Mathematics Physics Humanities Translation Studies Finnish Language Cultural Research Theology Orthodox Theology Special Education Adult Education Applied Education Teacher Education Psychology Forest Engineer Forest Planning Management of Forest

5 5 5 4 6 3.5 5 5 7 6 5 7 5 5 5 5 4 4 4 4 4 4 4 5 5 6

6 6 6 5 6 4.5 6 6 7 5 6 7 6 6 6 6 6 4 4 5 4 3 6 5 6 7

5 4 6 3 5 3 5 5 6 5 6 6.5 5 5 5 5 5 4 2 1 1 1 5 5 5 7

4 4 0 0 4 2 5.5 5 7 4 5 7 5 6 5 4 5 3 3 2 2 4 4 4 4 5

6 5 4 3 5 3 5 6 6 4 6 6 5 5 4 6 6 4 3 1 2 2 2 5 6 7

5 5 3 4 6 4 6 6 7 3 5 7 5 5 6 6 6 4 5 3 4 3 5 5 6 7

6 4 3 3 6 3 5 7 6 4 6 6 4 4 4 5 6 3 3 1 3 2 2 4 6 7

4. Application International panels assessed the quality of research of different departments of the University of Joensuu, Finland, for the period of 2000–2006. The data have been taken from the evaluation reports, which are available at .. The panels were provided with the background information about the departments including their available resources (see Table 4). The data set consists of 26 departments. Each was assessed with seven criteria by using a scale where one represents ‘‘Poor” and seven represents ‘‘Excellent”. These criteria were used as outputs in our study. Identical inputs were assumed. We assume that the panels took into account the resources (inputs) in evaluating the departments. The scale can be regarded as a proxy to the value scale, which is an interval-scale. We cannot say that a score of two (‘‘Fair”) implies that it is two times better than ‘‘Poor”. The purpose of the scale is that equal value differences correspond to, roughly, the same preference differences. That is, it is assumed that the participants judge the differences between each category roughly equal. Without loss of information, we may also use the scale 3, . . . , 3 in our approach without any effect on our results. However, we cannot assume that a unit with a value of 7.0 on a criterion is twice ‘‘better” than another unit with a value 3.5 on the same criterion.

O1: Research Infrastructure. O2: Standard of Research at National Level. O3: Standard of Research at International Level. O4: Research Strategy. O5: International Activities. O6: National Co-operation. O7: International Co-operation. 7 = Excellent, 6 = Very good, 5 = Good, 4 = Average, 3 = Somewhat below average,2 = Fair, 1 = Poor. The 0-values for Sociology and Social Policy departments on criterion 1 ‘‘Research Strategy” indicate that these departments are new and did not produce any output in this criterion.

Table 5 Application Results. No

The optimal values of z’s 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 P

2

3

4

1 1

1

1

1

1

5

6

7

8

9

1

1

1 1

1

1

1

1

1

1

1 1 1 1 1 1 1

1

1 1 1 1 1 1 1

1 1 1 1 1

10

7

1

0

10

1

11

12

1 1 1 1

1 1 1 1 1 1 1

1

1 1 1 1 1 1 1

1

11

1

1

1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1

16

22

1 1 1 1 1 1 1

12

14

15

16

17

1

1

1

1

1

1

1

1

1

1

1

1

18

19

20

21

22

23

24

1 1 1

25

26

1 1

1 1 1 1 1 1 1 1

1

1

1

1

1 1 1

1 1 1

13

1

1 1 1

1 1 1 1 1 1 1

4

13

1 1 1 1 1 1 1 1 1 1 1 1 1 23

1 1 1

1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1

10

11

8

10

8

1

1 1 1 1 1

1 1 1 1 1 1 1

3

7

12

1 1

1

1

2

0

0

0

1 1 1 1 1 1 1 1 1 1 1 22

P

eff 2.3

6 12 4 18 3 18 3 1 1 4 3 0 7 2 5 5 4 16 16 18 20 18 14 12 4 0

.769 .538 .846 .308 .885 .308 .885 .962 .962 .846 .885 1.000 .731 .923 .808 .808 .846 .385 .385 .308 .231 .308 .462 .538 .846 1.000

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The optimal solution of model (2.3) is reported in Table 5. In this case, the output-orientation was a natural choice6. Departments 12 (Physics) and 26 (Management of Forest) both received an efficiency score of one. The next ‘‘best” units were departments eight (Biosciences) and nine (Chemistry). The weakest departments were related to education (19–22). Table 5 also provides information about the reference units for inefficient units. For instance, we can see that department 12 (Physics) belongs to the reference set of all inefficient units. In fact, if desired, we may introduce a new efficiency/productivity measure for units by reporting, how many times the department belongs to the reference set of other (inefficient) units. Note that the interpretation of efficiency scores differs from that of the traditional DEA, in which the scores tell how much we have to proportionally improve the output values (in this case) to reach the efficient frontier. In our approach, we do not assume the existence of the efficient frontier. Instead we operate with a finite set consisting of given units. The efficiency measurement is based on this set. In the spirit of DEA, we try to make each unit as efficient as possible. We search for the hyperplane separating the units into two sets (better and worse) in such a way that the number of worse units is maximal. The approach is quite suitable for comparing the research output of university departments in case no other information is available. It is easy to recognize ‘‘good” and ‘‘poor” units and for each inefficient unit we may identify the corresponding reference set consisting of possible benchmarking units. 5. Conclusions and discussion In this paper, we developed an approach that can measure the efficiency of units, even if some variables are measured on an interval-scale and some on a ratio-scale. A positive linear transformation is allowed for interval-scale input- and output-variables, implying that our efficiency measure is invariant under such a transformation. Although the basic idea is adopted from the traditional DEA, our approach alleviates the shortcoming of the original DEA that fails to produce meaningful results when the data has both positive and negative values or is interval-scale. The data in our application technically enables us to use DEA, because all values are positive. However, the underlying unknown scale is not a ratio-scale making the use of the traditional DEA inappropriate. Our approach has desirable properties. It can, for example, be used to rank/sort Decision Making Units into classes, providing an improved ranking scheme for DEA. Many of the ranking

6 To study the robustness of the results with respect to the choice of e, we made calculations by using e = 102 and e = 101. We noticed that the choice of e had no impact on the results.

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schemes do not perform very well (see, for example, the survey by Adler and Golany (2002)). Furthermore, it is easy to incorporate weight restrictions into our model. Such weight restrictions are commonly used in DEA to incorporate DM’s preferences into the analysis. Acknowledgment The research was supported by the Academy of Finland (Grant No. #121980). The authors thank the reviewers for useful comments. References Adler, N., Golany, B., 2002. Review of ranking methods in the data envelopment analysis context. European Journal of Operational Research 140, 249–265. Ali, A.I., Seiford, L.M., 1990. Translation invariance in data envelopment analysis. Operations Research Letters 9, 403–405. Banker, R.D., Charnes, A., Cooper, W.W., 1984. Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science 30, 1078–1092. Charnes, A., Cooper, W.W., Rhodes, E., 1978. Measuring efficiency of decision making units. European Journal of Operational Research 2, 429–444. Charnes, A., Cooper, W.W., Rhodes, E., 1979. Short Communication: Measuring efficiency of decision making units. European Journal of Operational Research 3, 339. Charnes, A., Cooper, W.W., Golany, B., Seiford, L.M., Stutz, J., 1985. Foundations of data envelopment analysis for Pareto–Koopmans efficient empirical production functions. Journal of Econometrics 30, 91–107. Cherchye, L., Kuosmanen, T., Post, G.T. (2000). New tools for dealing with errors-invariables in DEA, Catholic University of Leuven, Center for Economic Studies, Discussion Paper Series DPS 00.06. Halme, M., Joro, T., Koivu, M., 2002. Dealing with interval-scale data in data envelopment analysis. European Journal of Operational Research 137, 22–27. Joro, T., Korhonen, P., Wallenius, J., 1998. Structural comparison of data envelopment analysis and multiple objective linear programming. Management Science 44, 962–970. Korhonen, P., Luptacik, M., 2004. Eco-Efficiency analysis of power plants: An extension of data envelopment analysis. European Journal of Operational Research 154, 437–446. Korhonen, P., Stenfors, S., Syrjänen, M., 2003. Multiple objective approach as an alternative to radial projection in DEA. Journal of Productivity Analysis 20, 305– 321. Köksalan, M., Tuncer, C., 2009. A DEA-based approach to ranking multi-criteria alternatives. International Journal of Information Technology and Decision Making 8, 29–54. Köksalan, M., Büyükbasßaran, T., Özpeynirci, Ö., Wallenius, J., 2010. A flexible approach to ranking with an application to MBA programs. European Journal of Operational Research 201, 470–476. Lovell, C.A.K., Pastor, J.T., 1995. Unit invariant and translation invariant DEA models. Operations Research Letters 18, 147–151. Pastor, J.T., 1996. Translation invariance in data envelopment analysis: A generalization. Annals of Operations Research 66, 93–102.