A sequence of targets toward a common best practice frontier in DEA
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Abstract
Original data envelopment analysis models treat decisionmaking units as independent entities. This feature of data envelopment analysis results in significant diversity in input and output weights, which is irrelevant and problematic from the managerial point of view. In this regard, several methodologies have been developed to measure the efficiency scores based on common weights. Specifically, Ruiz and Sirvant (Omega 65:1–9, 2016) formulated an aggregated DEA model to minimize the gap between actual performances and best practices and identify a common best practice frontier. Their model is capable of determining target units for all units under evaluation, simultaneously, with the property that all of them are located on a common best practice frontier. However, in practice it is difficult for some units to achieve that specified target in a single step. Consequently, developing a methodology for assisting units to reach their corresponding targets, through a path of intermediate improving targets, is useful. This problem is investigated in this paper, and we propose a stepwise target setting approach which provides a path of intermediate targets for each unit. We study efficient and inefficient units separately and provide two distinct models for each category, although both of them are intrinsically similar. A simple numerical example and an application are also provided to illustrate our approach.
Keywords
DEA Target setting Common benchmarking Reference hyperplane Sequential targetsIntroduction
Data envelopment analysis (DEA) is a nonparametric linear programmingbased technique first developed by Charnes et al. (1978) for evaluating the performance of homogeneous decisionmaking units (DMUs) having multiple inputs and multiple outputs. Since then DEA has been implemented in various applications such as service quality evaluation (Najafi et al. 2015), evaluation of academic performance (Monfared and Safi 2013) and facility location problem (Razi 2018). In DEA, efficiency score of each unit is generally defined as the ratio of the weighted sum of outputs to the weighted sum of inputs. The corresponding input and output weights for each unit are then calculated by solving individual linear programming problems that maximize this ratio provided that the corresponding ratio for all the units does not exceed one. Units with efficiency score of one are submitted as efficient and lay on the efficient frontier of the technology, whereas a score less than one signals inefficiency. Zamani and Borzouei (2016) investigated the stability region for preserving this classification in the presence of variable returns to scale. On the other hand, one of the main functions of DEA is benchmarking. It means that for each inefficient unit, DEA determines a (virtual) target unit lying on the frontier that monitors the best levels of inputs and outputs for that unit to perform efficiently. The first research on this subject was developed by Frei and Harker (1999), who have investigated the issue of benchmarking based on projecting inefficient units onto the strongly efficient frontier of DEA, considering Euclidean distance. After that, some researchers generalized the idea of using Euclidean distance in order to define efficiency measure and also to project units onto the strongly efficient frontier, e.g., see Baek and Lee (2009), Amirteimoori and Kordrostami (2010), and Aparicio and Pastor (2014a, b). The issue of least distance to the frontier has been used in other applications of DEA, such as ranking units (e.g., see Ziari 2016; Aghayi and Tavana 2018). The concept of benchmarking based on least distance has also been developed by Pastor and Aparicio (2010), Ando et al. (2012, 2017), Aparicio and Pastor (2013, 2014a, b) and Aparicio et al. (2014). On the other hand, some authors like Cherchye and Van Puyenbroeck (2001) and Silva Portela et al. (2003) have developed some benchmarking approaches based in the issue of similarity and closeness. Their motivation was that a benchmark which is closer to the inefficient unit is easier to be reached. Other measures of efficiency, like the modified Russell measure and the slackbased measure, have been also utilized for benchmarking, e.g., see Gonzalez and Alvarez (2001) and Aparicio et al. (2007). However, benchmarking is considered as an important field of research in DEA and can be viewed from different perspectives such as neural network as discussed in Shokrollahpour et al. (2016), artificial units in Didehkhani et al. (2018) or Fuzzy DeNovo programming by Sarah and KhaliliDamghani (2018). For a complete review of other benchmarking models, the reader is referred to Aparicio (2016), Aparicio et al. (2017a, b).
An important feature of DEA in evaluating units is that it behaves naturally optimistic, in the sense that it primarily treats units as individual and independent entities, as the corresponding optimal weights for each unit are calculated by autonomous programs. Therefore, it would be likely to happen that the input and output weights differ considerably across all units, which is incompatible and irrelevant. Especially, flexibility in choosing each unit’s optimal weights will surely reduce the discrimination power of DEA in distinguishing efficient and inefficient units. This query usually causes critical arguments among researchers. In this regard, there are a variety of means to overcome the aforementioned difficulty. For an early review of methods on improving the power of discrimination in DEA see AnguloMeza and Lins (2002). One basic approach suggested by Li and Reeves (1999) is the multiple criteria DEA (MCDEA) technique. They formulated DEA in the framework of multicriteria decision making (MCDM) and tried to evaluate units applying MCDM techniques. Recently, Chaves et al. (2016) studied the main properties of the model proposed by Li and Reeves (1999). On the other hand, Bal and Orkcu (2007) formulated a goal programming problem for weight dispersion in DEA. Their approach was later improved by Bal et al. (2010). Furthermore, Ghasemi et al. (2014) suggested an approach for improving the discrimination power in MCDEA. This study was recently supplemented by Rubem et al. (2017) who introduced a weighted goal programming formulation to solve the MCDEA problem.
Another basic approach to improve the discrimination power of DEA is to determine a common set of weights for all units under assessment. This approach was first proposed by Roll et al. (1991). Actually, a common set of weights (CSWs) are basically considered as coefficients of a supporting hyperplane of the DEA technology at some efficient units. However, the question of how to determine such a hyperplane has been seen in different perspectives. One approach is to minimize the difference between the DEA efficiency scores and those obtained from the associated CSWs. This method was investigated by Despotis (2002) and Kao and Hung (2005). Other formulations have been also proposed, such as maximizing the sum of the efficiency ratios of all the units by Ganley and Cubbin (1992), maximizing the minimum of the efficiency ratios by Troutt (1997), or introducing ideal and antiideal virtual units by KhaliliDamghani and Fadaei (2018).
Additionally, Ruiz and Sirvant (2016) formulated a model of CSW in the framework of benchmarking, i.e., they formulated a program to globally minimize a weighted \(L_1\)distance of all the units to their corresponding benchmarks which are located on a common hyperplane of the DEA technology. In this regard, they suggested that only a facet of the DEA technology should be considered as the best practice frontier. Then, the coefficients of this hyperplane are implemented to calculate efficiency scores. Therefore, it is reasonable to establish benchmarks on this frontier. However, it should be noted that applying this method for target setting may involve deterioration of some of the observed input and output levels. In other words, in the common benchmarking approach the dominance criteria do not prevail necessarily. Moreover, this approach provides benchmark activities for all units under evaluation, simultaneously. Even for efficient units which are not located on the common best practice frontier, a different benchmark unit may be assigned, located on the underlying hyperplane. The inquiry that is highlighted here is that as all the benchmarks lay on a common supporting hyperplane of the technology, it is likely happen that the determined best practice efficient target may be very far from the corresponding unit. Therefore, it would be quite impossible for that unit to achieve its (final) target in one single step, because largescale input and output adjustments are quite problematic and very demanding for an inefficient unit. This issue is also of great importance even if the final target is obtained via conventional DEA model for individual units separately. In this regard, some researchers investigated the problem of stepwise target setting. This issue includes developing a set of intermediate targets which constitute a path toward the strongly efficient frontier on which the final target is located. The methods of stepwise target setting can be categorized into two types. The first category involves the approaches in which all intermediate targets and final target are chosen from among the set of observed (or existing) units. In contrast, all the approaches that allow all virtual units to be selected as intermediate and final target belong to the second category. In this regard, Seiford and Zhu (2003) were the first who developed a model for type one stepwise benchmarking. Their model has been formulated based on context dependent DEA and layering units into successive layers. After that, most of the existing models in this category adapted the idea of this idea, e.g., Lim et al. (2011) established a path of intermediate benchmarks by clustering units and layering them. Lozano and Villa (2005a, b) investigated the concept of efficiency improvement in DEA, which leads to a stepwise target setting model. Lozano and Villa (2005a) proposed a model which is based on bounded adjustments of inputs and outputs, such that only a limited portion of the corresponding input and output levels is allowed for reduction and expansion at each step, respectively. Also, their model guarantees that each of intermediate targets dominates the previous one. Finally, the algorithm terminates when an efficient target is reached. Also, Suzuki and Nijkamp (2011) developed a stepwise projection DEA model for public transportation in Japan. Khodakarami et al. (2014) formulated a twostage DEA model which has been applied on industrial Parks. Additionally, Fang (2015) developed a centralized resource allocation based on gradual efficiency improvement. For a complete review on the stepwise target setting models see Lozano and CalzadaInfante (2017). Moreover, Nasrabadi et al. (2018) investigated the concept of stepwise benchmarking in the presence of interval scale data in DEA.
The above discussion clarifies that the enquiry to investigate the problem of sequential benchmarking in case that all final benchmarks are located on a common hyperplane, is of great importance in DEA. In this study, we aim to investigate this subject in the framework of the CSW proposed by Ruiz and Sirvant (2016). Also, we incorporate the idea of bounded adjustments of Lozano and Villa (2005a, b) in our proposed method. Recalling that in the approach of Ruiz and Sirvant (2016) even efficient units that do not lay on the common best practice frontier are also assigned with a benchmark, a qualified model for determining a sequence of benchmarks should be applicable for both efficient and inefficient units. Therefore, we enquire about these two cases separately and establish different formulations for each one, although both models are basically similar.
This paper unfolds as follows. In “A common benchmarking model in DEA” section, we present some preliminary considerations of DEA and review the common benchmarking model developed by Ruiz and Sirvant (2016). Our proposed approach for sequential benchmarking is illustrated in “Developing a benchmark path toward the common best practice frontier” section, followed by a simple numerical example in “Numerical example” section. Then, “Application” section includes an empirical application and finally “Conclusion” section provides discussion and concluding remarks.
A common benchmarking model in DEA
In this section, we briefly present the common benchmarking model developed by Ruiz and Sirvant (2016). Consider a set of n decisionmaking units (DMUs), each consuming m inputs to produce s outputs. For \(j=1,\ldots ,n\), we denote unit j by the activity vector \((X_j,Y_j)\), where \(X_j\) and \(Y_j\) are input and output vectors of unit j, respectively. It is assumed that \(X_j=(x_{1j},\ldots ,x_{mj})^t \in {\mathcal {R}}^m_{\ge 0}\), \(X_j \ne 0\) and \(Y_j=(y_{1j},\ldots ,y_{sj})^t \in {\mathcal {R}}^s_{\ge 0}\), \(Y_j \ne 0\).
The production possibility set consisting all feasible input–output vectors (X, Y) is generally defined as \(T=\{(X,Y)  X\; {\text{ can }}\; {\text{ produce }}\;Y \}\). Each member of T is called a (virtual) activity. We call each unit j an observed activity or an observed unit, for \(j=1,\ldots ,n\).^{1} An activity \((\bar{X},\bar{Y})\in T\) is said to be nondominated or efficient in set T iff there exists no other \((X,Y)\in T\) such that \(X \le \bar{X}\) and \(Y \ge \bar{Y}\). The set of all observed extreme efficient units is denoted by E.
The CSW model of Ruiz and Sirvant (2016)
 1.
\(H^*=\{ (X,Y)\ \ U^* Y + V^* X + \gamma ^* =0 \}\) is the common best practice frontier of the technology which is used as the reference hyperplane in target setting.^{2}
 2.
The set \(RG=\{ k\ \ \lambda _k^{j*} >0\;{\text{ for }}\;{\text{ some }}\;j, 1\le j\le n \}\) provides the reference group of units which lay on the reference hyperplane \(H^*\).
 3.For unit j which is not in the reference group, the coordinates of the corresponding target on \(H^*\) is given by$$\begin{aligned} (X_j^*, Y_j^*)=\sum _{k \in E} {\lambda _k^{j*} (X_k, Y_k)}. \end{aligned}$$(3)
Note that model (2) can be easily transformed to a zero–one linear program by using transformations \(x=x^+ + x^\) and \(x=x^+  x^\) for \(x \in {\mathcal {R}}\).
Difficulties in common benchmarking
Data set of the numerical example
Unit  x  y  \(\mathbf x ^*\)  \(\mathbf y ^*\) 

A  1  2  4  7.5 
B  2  5  4  7.5 
C  4  7.5  4  7.5 
D  7  9  7  9 
E  12  10  7  9 
F  1.5  2.5  4  7.5 
G  3.5  5.5  4  7.5 
H  6.5  8  6.5  8.75 
I  11  9.5  7  9 
J  4  4  4  7.5 
K  5.5  4.5  5.5  8.25 
L  8.5  6.5  7  9 
M  10  8  7  9 
N  1.5  3.5  4  7.5 
P  5  8  5  8 
It is clear that all units in RG, i.e., units \(\mathbf{C }\), \(\mathbf{D }\) and \(\mathbf{P }\) coincide their corresponding targets. This also happens for the efficient nonextreme unit \(\mathbf{P }\) which is located on \(H^*\). Moreover, the common benchmark for efficient units \(\mathbf{A }\), \(\mathbf{B }\) and \(\mathbf{N }\) is unit \(\mathbf{C }\) and for unit \(\mathbf{D }\) is unit \(\mathbf{E }\). Additionally, the corresponding benchmark for the other inefficient units is a virtual (or observed) activity located on the intersection of \(H^*\) and \(T_v\).
Now, consider unit \(\mathbf{A }\). We observe that although it is efficient, but it has a different benchmark, i.e., unit \(\mathbf{C }\), which is approximately far from it. Therefore, it might be disappointing for unit \(\mathbf{A }\) to just consider such a far unit as its benchmark, since it is roughly impossible to reach it in a single move. Moreover, there exists a similar story for, e.g., the inefficient unit \(\mathbf{I }\).
This setting motivates us to address the above mentioned problem, i.e., to help each unit to achieve the common best practice frontier gradually. We follow a procedure to set up a path of intermediate benchmarks for each unit until we reach \(H^*\). The procedure is developed in next section.
Developing a benchmark path toward the common best practice frontier
In order to develop a procedure for establishing a benchmark path, efficient and inefficient units are investigated separately. Recalling that \(H^*\) is the reference hyperplane characterized by model (2) and assuming that unit “o” is under assessment, with \(o \notin RG\), we aim to establish a sequence of (improving) intermediate benchmarks for this unit that originates from it and converges to \(H^*\) in sequential steps. Let \((Targett)\) be the model which calculates the tth (intermediate) target from the preceding one. Then, we present the general scheme of our benchmarking algorithm as the following 3step procedure:

Step 1 Set \(t=0\), \((X_o^t,Y_o^t)=(X_o,Y_o)\).
 Step 2 Set \(t=t+1\) and solve \((Targett)\) for unit “o” to obtain$$\begin{aligned} \left(X_o^{t},Y_o^{t}\right)=\left(X_o^{t1}  S^{t*},Y_o^{t1} + S^{+t*}\right). \end{aligned}$$

Step 3 If \((X_o^{t},Y_o^{t}) \in H^*\), stop. Otherwise go to Step 2.
Now, the main question is that how to formulate \((Targett)\) for each unit. The answer is that the structure of \((Targett)\) depends on the efficiency status of unit “o” under consideration. We first try to formulate \((Targett)\) for an efficient unit, and then, we go to the case of an inefficient one.
A sequence of targets for an efficient unit
 1.
As unit “o” is efficient, all intermediate benchmarks are expected to be efficient, too. This means that the benchmark path must go through the efficient frontier until it reaches \(H^*\), which is an especial part of the efficient frontier. This implies that the efficiency status of all intermediate benchmarks is not allowed to be deteriorated along the path. Nevertheless, the benchmark path would inevitably involve deterioration in some of the observed input and/or output levels in return to improving the others.
 2.
The path is monotonically convergent to \(H^*\), i.e., the direction of the path is toward the common best practice frontier. This means that each intermediate benchmark is closer to the final target than the previous one. We verify this property in our approach by checking whether each individual input/output of unit “o” has improvement in its final target or not. If an individual input (output) has been improved (deteriorated) in the final target, we imply that this property should be satisfied for all intermediate benchmarks. Therefore, the sign of each input/output slack in intermediate targets is determined according to the sign of the corresponding slack in the final target.
 3.
As the path is expected to move toward \(H^*\) gradually, it is rational to approve bounded adjustments in input and output levels at each step. Hence, like Lozano and Villa (2005a, b) we set upper and lower bounds for input and output adjustments. These bounds are defined as a predetermined portion of their current levels. Note that the corresponding bounds are determined by the decision maker, based on some managerial limitations and they may differ for individual inputs and outputs at each step.
Theorem 1
Proof
The above three cases prove the first aforementioned statement. By a similar discussion on outputs, one can prove the second inequality, easily. \(\square\)
As we wish to establish a benchmark path converging to the common best practice frontier, we run model (4) until we reach the common frontier \(H^*\), i.e., we have \(U^* Y_o^{t} + V^* X_o^{t} + \gamma ^* =0\), where \((U^*,V^*,\gamma ^*)\) is an optimal solution of model (2).
A sequence of targets for an inefficient unit
 1.As unit “o” is inefficient, we aim to find its sequential benchmarks such that each intermediate benchmark is better that the previous one. Although in common benchmarking model (2) the domination criteria does not necessarily prevail, it is possible to determine intermediate benchmarks at each step in a way that each target is better than the previous one, in the sense of its weighted \(L_1\)distance to the efficient frontier. Hence, we consider the following weighed additive model of Cooper et al. (2011) evaluating (virtual) activity \((\bar{X},\bar{Y})\in T\), which minimizes the weighted \(L_1\)distance of this unit to the efficient frontier, as follows:where \(({\mathbf {w}}^{\mathbf {x}},{\mathbf {w}}^{\mathbf {y}})\in {\mathcal {R}}^{m+s}\) is a nonnegative weight vector. Our aim is to determine \((X_o^{t},Y_o^{t})\) from the previous (intermediate) benchmark \((X_o^{t1},Y_o^{t1})\) with the property that$${\text{ADD}}(\bar{X},\bar{Y}) =\begin{array}{lll}\min & \quad U \bar{Y} + V \bar{X}+\gamma &\\ {\hbox{s.t.}} &\quad U Y_j + V X_j+\gamma \ge 0, &\quad j\in E \\ &{} \quad U \ge {\mathbf {w}}^{\mathbf {y}},\quad V \ge {\mathbf {w}}^{\mathbf {x}}, \gamma \;{\text{free}}, \\ \end{array}$$(6)In other words, we wish to move along an improving direction in our benchmarking procedure. The following theorem provides us to find a sufficient condition for such a direction.$$\begin{aligned} {\text{ADD}}(X_o^{t},Y_o^{t}) \le {\text{ADD}}(X_o^{t1},Y_o^{t1}). \end{aligned}$$(7)
Theorem 2
Assume that \((U_o^*,V_o^*,\gamma _o^*)\) is an optimal solution of the weighted additive model (6) evaluating unit “o”. Then for each vector \((S^,S^+)\in {\mathcal {R}}^{m+s}\) such that \(U_o^* S^+ + V_o^* S^ \ge 0\), we have \({\text{ADD}}(X_o  S^,Y_o + S^+) \le {\text{ADD}}(X_o,Y_o)\).
Proof
The proof is straightforward if we apply the additive model (6) for evaluating \((X_o  S^,Y_o + S^+)\). By assumption, it can be verified that \((U_o^*,V_o^*,\gamma _o^*)\) is a feasible solution for model (6) evaluating \((X_o  S^,Y_o + S^+)\), which implies that \({\text{ADD}}(X_o  S^,Y_o + S^+) \le {\text{ADD}}(X_o,Y_o)\), and the proof is complete. \(\square\)
 2.
It should be kept in mind that a desirable benchmark path should go monotonically toward the best practice frontier. So, as before, the sign of input and output slacks is determined according to improvement or deterioration in the corresponding input and outputs of the final target. This will be observed in the proposed model similar to model (4).
 3.
The final consideration deals with bounded adjustments for inputs and outputs at each step. This issue can be observed in the proposed model similar to model (4).
Computational complexity
Both models (4) and (8) that have been developed for sequential target setting, are nonlinear due to existence of absolute value in their constraints, as well as their objective functions. Nevertheless, this issue can be resolved by introducing two nonnegative variables \(x^+\) and \(x^\), such that \(x=x^+  x^\) and \(x=x^+ + x^\), but it increases the computational complexity of the models, which is not of interest and should be avoided. In this regard, we propose another approach for linearizing these models which is more practical. Without loss of generality, we illustrate this technique for model (4).
Numerical example
Sequential target setting in the numerical example
Step  0  1  2  3  4  5  6  7 

A  (1,2)  (1.2,2.6)  (1.46,3.38)  (1.80,4.40)  (2.33,5.42)  (3.04,6.30)  (3.95,7.44)  (4,7.5) 
B  (2,5)  (2.6,5.75)  (3.38,6.725)  (4,7.5)  
C  (4,7.5)  Itself  
D  (7,9)  Itself  
E  (12,10)  (8.4,9.28)  (7,9)  
F  (1.5,2.5)  (1.75,3.25)  (2.08,4.23)  (2.70,5.49)  (3.51,6.88)  (4,7.5)  
G  (3.5,5.5)  (4,7.15)  (4,7.5)  
H  (6.5,8)  (6.5,8.75)  
I  (11,9.5)  (7.7,9)  (7,9)  
J  (4,4)  (4,5.2)  (4,6.76)  (4,7.5)  
K  (5.5,4.5)  (5.5,5.85)  (5.5,7.61)  (5.5,8.25)  
L  (8.5,6.5)  (7,8.45)  (7,9)  
M  (10,8)  (7,9)  
N  (1.5,3.5)  (1.85,4.55)  (2.41,5.51)  (3.13,6.41)  (4,7.5)  
P  (5,8)  (5,8) 
Application
Data set of 28 international airlines
Airline  Input1  Input2  Input3  Input4  Output1  Output2  

1  NIPPON  12,222  860  2008  6074  35,261  614 
2  CATHAY  12,214  456  1492  4174  23,388  1580 
3  GARUDA  10,428  304  3171  3305  14,074  539 
4  JAL  21,430  1351  2536  17,932  57,290  3781 
5  MALAYSIA  15,156  279  1246  2258  12,891  599 
6  QANTAS  17,997  393  1474  4784  28,991  1330 
7  SAUDIA  24,708  235  806  6819  18,969  760 
8  SINGAPORE  10,864  523  1512  4479  32,404  1902 
9  AUSTRIA  4067  62  241  587  2943  65 
10  BRITISH  51,802  1294  4276  12,161  67,364  2618 
11  FINNAIR  8630  185  303  1482  9925  157 
12  IBERIA  30,140  499  1238  3771  23,312  845 
13  LUFTHANSA  45,514  1078  3314  9004  50,989  5346 
14  SAS  22,180  377  1234  3119  20,799  619 
15  SWISSAIR  19,985  392  964  2929  20,092  1375 
16  PORTUGAL  10,520  121  831  1117  8961  234 
17  AIR CANADA  22,766  626  1197  4829  27,676  998 
18  AM. WEST  11,914  309  611  2124  18,378  169 
19  AMERICAN  80,627  2381  5149  18,624  133,796  1838 
20  CANADIAN  16,613  513  1051  3358  24,372  625 
21  CONTINENTAL  35,661  1285  2835  9960  69,050  1090 
22  DELTA  61,675  1997  3972  14,063  96,540  1300 
23  EASTERN  21,350  580  1498  4459  29,050  245 
24  NORTHWEST  42,989  1762  3678  13,698  85,744  2513 
25  PANAM  28,638  991  2193  7131  54,054  1382 
26  TWA  35,783  1118  2389  8704  62,345  1119 
27  UNITED  73,902  2246  5678  18,204  131,905  2326 
28  USAIR  53,557  1252  3030  8952  59,001  392 
Sequential benchmarking for efficient units: actual inputs/outputs and target path
Airline  Input1  Input2  Input3  Input4  Output1  Output2  ADD  

4  JAL  21,430  1351  2536  17,932  57,290  3781  0 
6  QANTAS  17,997  393  1474  4784  28,991  1330  0 
17,997  510.9  1288.9  4784  30,926.1  1330  0  
17,997  561.9  1288.9  4784  32,289.3  1330  0  
7  SAUDIA  24,708  235  806  6819  18,969  760  0 
17,295.6  305.5  806  4773.3  19,854.9  760  0  
15,906.3  397.2  806  4709.4  21,445.6  760  0  
15,906.3  464.7  806  4709.4  22,823.6  760  0  
8  SINGAPORE  10,864  523  1512  4479  32,404  1902  0 
11  FINNAIR  8630  185  303  1482  9925  157  0 
13  LUFTHANSA  45,514  1078  3314  9004  50,989  5346  0 
31,859.8  1143.5  3314  9817.1  66,285.7  4168.8  0  
23,821.3  1146.2  3314  9817.1  71,016.6  4168.8  0  
23,811.7  1146.3  3314  9817.1  71,023.1  4168.8  0  
15  SWISSAIR  19,985  392  964  2929  20,092  1375  0 
13,989.5  392  1110.5  3347.1  23,229.5  1375  0  
9792.7  392  1110.5  3347.1  23,959.0  1375  0  
8612.3  392  1110.5  3347.1  24,136.7  1375  0  
16  PORTUGAL  10,520  121  831  1117  8961  234  0 
7669.0  134.6  581.7  1117  8961  234  0  
6642.4  157.7  407.2  1314.2  8961  234  0  
6642.4  161.7  308.3  1314.2  8961  234  0  
18  AM. WEST  11,914  309  611  2124  18,378  169  0 
Sequential benchmarking for inefficient units: actual inputs/outputs and target path
Airline  Input1  Input2  Input3  Input4  Output1  Output2  ADD  

1  NIPPON  12,222  860  2008  6074  35,261  614  1.821 
15,888.6  661.4  1464.9  6074  35,261  798.2  0.598  
20,082.7  661.4  1283.9  6074  35,261  899.3  0  
2  CATHAY  12,214  456  1492  4174  23,388  1580  0.392 
12,176.2  492.3  1328.6  4174  29,872.7  1580  0  
3  GARUDA  10,428  304  3171  3305  14,074  539  1.879 
10,428  315.4  2219.7  3258.7  15,355.2  539  1.407  
10,428  315.4  1553.8  3258.7  15,355.2  539  1.095  
10,428  315.4  1087.7  3258.7  15,355.2  539  0.878  
10,428  315.4  761.4  3258.7  15,355.2  539  0.725  
10,428  315.4  549.8  3258.7  15,355.2  539  0  
5  MALAYSIA  15,156  279  1246  2258  12,891  599  0.952 
10,609.2  279  872.2  2306.9  16,047.5  599  0.540  
9653.1  279  619.8  2306.9  16,047.5  599  0  
9  AUSTRIA  4067  62  241  587  2943  65  0.282 
3388.2  73.2  168.7  587  3825.9  65  0.184  
3388.2  73.2  121.2  587  3935.9  65  0  
10  BRITISH  51,802  1294  4276  12,161  67,364  2618  3.226 
51,802  1419.9  2993.2  12,161  78,613.2  2618  0.438  
51v802  1419.9  2894.4  12,161  78,613.2  2618  0  
12  IBERIA  30,140  499  1238  3771  23,312  845  1.475 
21,098  499  1006.7  4081  28,026.6  845  0.06  
19,365.5  499  1006.7  4081  28,026.9  845  0  
14  SAS  22,180  377  1234  3119  20,799  619  1.2 
15,526  382  863.8  3119  21,374.8  619  0.384  
15,086.1  382  758.3  3119  21,374.8  619  0  
17  AIR CANADA  22,766  626  1197  4829  27,676  998  0.98 
22,766  591.1  1197  4829  33,252.2  998  0  
19  AMERICAN  80,627  2381  5149  18,624  133,796  1838  2.359 
82,346.6  2357.9  4624.7  18,624  133,796  2139.7  0  
20  CANADIAN  16,613  513  1051  3358  24,372  625  0.756 
16,613  480.8  1051  3519  28,792.5  625  0  
21  CONTINENTAL  35,661  1285  2835  9960  69,050  1090  1.635 
42,013.8  1228.8  2406.2  9960  69,050  1204.5  0  
22  DELTA  61,675  1997  3972  14,063  96,540  1300  3.158 
61,675  1772.5  3475.3  14,063  100,405.2  1630.7  0  
23  EASTERN  21,350  580  1498  4459  29,050  245  1.791 
21,350  580  1142.4  4272.4  33,734.1  318.5  0.077  
21,350  580  1142.4  4272.4  33,734.1  414.1  0.003  
21,350  580  1142.4  4272.4  33,734.1  417.7  0  
24  NORTHWEST  42,989  1762  3678  13,698  85,744  2513  2.026 
46,167.4  1550.1  3241.5  13,698  85,744  2513  0  
25  PANAM  28,638  991  2193  7131  54,054  1382  0.694 
29,683.5  917.9  2024.7  7131  54,054  1382  0  
26  TWA  35,783  1118  2389  8704  62,345  1119  0.813 
37,585.1  1095.8  2188.8  8704  62,345  1119  0  
27  UNITED  73,902  2246  5678  18,204  131,905  2326  3.684 
79,686.9  2310.4  4624.5  18,204  131,905  2326  0  
28  USAIR  53,557  1252  3030  8952  59,001  392  4.142 
50,735.5  1252  2372.3  8952  72,690.3  509.6  0.206  
50,735.5  1252  2372.3  8952  72,690.3  662.5  0.088  
50,735.5  1252  2372.3  8952  72,690.3  776.6  0 
We first observe that Table 4 does not provide target path airlines in RG, because for these units targets and actual inputs/outputs coincide. However, for the efficient units not in RG, a path of sequential targets toward the reference hyperplane is reported. We observe that all efficient units reach their corresponding final target in two or three steps. For QANTAS and SAUDIA, the targets are less demanding, but the other efficient airlines, i.e., LUFTHANSA, SWISSAIR and PORTUGAL, need considerable adjustments to reach their corresponding target. The maximum adjustment for these three is due to their first input which needs approximately 50% decrease from their actual levels. Also, note that there would be no change in the third input of LUFTHANSA, the second input and second output of SWISSAIR and the second output of PORTUGAL. On the other hand, as the results show the efficiency status of the efficient units do not deteriorate along the path, i.e., for efficient airlines, all intermediate (and final) targets have an optimal value of zero in model (6).
On the other hand, turning to inefficient airlines in Table 5, we observe different behaviors. Among these airlines, CATHAY has the longest path including 5 benchmarks toward its final target. This is due to the level of its third input which needs a decrement of approximately 84% from its actual level (3171 to 549.8). Meanwhile, the level of its first input and second output remains unchanged along the path. Then, the second longest path among the inefficient airlines is of length 3 and belong to EASTERN and USAIR, both consisting of an adjustment of nearly twice in the second output. Although, three inefficient airlines reach their corresponding final target in two steps, the majority of ten inefficient airlines achieve the reference hyperplane in just one step. Moreover, the last column in Table 5 presents the optimal value of the weighted additive model (6) for each (virtual) unit. Observing these values, which constitute a descending sequence for each inefficient unit with a final value of zero, the improving characteristic of the obtained sequential benchmarks is confirmed.
Conclusion
One of the main features of DEA is that it can be used for benchmarking, which is an important issue in management and economics. In practice, in a production technology the decision maker (DM) usually aims to first evaluate the efficiency status of decisionmaking units and classify them into efficient and inefficient categories, and then to determine a benchmark feasible and efficient activity for each inefficient unit. Benchmarking can assist inefficient units to improve their performance in comparison to best practices of others. On the other hand, applying an aggregated DEAbased model which finds a common set of weights to evaluating the efficiency score of all units, simultaneously, it is possible to determine a common supporting hyperplane of the technology as the best practice frontier. Thus, one could evaluate the units by means of the coefficients of this frontier and also to establish targets for all of them, on this common best practice frontier. However, in practice these targets may be difficult to reach in a single step, and therefore, it is required to propose a gradual improvement strategy to achieve the best practices. This research developed an approach of establishing a gradual improving path of targets for each unit which is not located on the best practice frontier. To apply the proposed method, the common set of weights (CSW) model of Ruiz and Sirvant (2016) is solved in order to find the final target for all units, simultaneously. Then for each unit, a path of targets is found which originates from that unit and proceeds gradually to the final efficient target which has been already determined. The obtained path is an improving one, in the sense that if the underlying unit is an efficient one, then all of the intermediate targets are efficient as well, and for inefficient units each of the intermediate targets is closer to the final target than the previous one, and also has a better performance than the previous one, in the framework of the weighted additive model of Cooper et al. (2011). Also, in the proposed model only a limited amount of adjustment is allowed. The portion of the current levels of inputs and outputs that is allowed for adjustment is determined by the DM based on his/her managerial points of view and practical limitations. Then, this feature guarantees that the path is roughly acceptable by the DM, as it is more practical and understandable.
This approach can be extended along different lines. One can investigate this issue for a free disposal hull (FDH) technology. Furthermore, the question of establishing a target path which in convergent to the final target in a predetermined number of steps might be interesting. Another possibility is to develop a procedure of stepwise target setting in the framework of other common weight methodologies, such as models proposed in Despotis (2002) or Kao and Hung (2005).
Footnotes
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