A Review on Data Envelopment Analysis (DEA)
Chuen Tse Kuah, Kuan Yew Wong, Farzad Behrouzi Faculty of Mechanical Engineering, Universiti Teknologi Malaysia,
81310 UTM Skudai, Malaysia.
e-mails: [email protected], [email protected], [email protected]
Abstract—This paper is a review paper focusing on the methodological development of Data Envelopment Analysis (DEA), a multi-factor performance measurement and improvement tool. Since its introduction in 1978, vast studies have been done on DEA, causing significant growth in its methodology and applications in the real world. The purpose of this paper is to provide a general introduction to DEA. The basic DEA models and some important methodological extensions of DEA, such as multilevel DEA models, stochastic DEA models, and fuzzy DEA models, are discussed in the paper. In addition, some current and future research trends are highlighted.
Keywords-DEA; performance improvement
I. INTRODUCTION
Since Data Envelopment Analysis (DEA) was first introduced by Charnes, Cooper, and Rhodes in 1978 [1], the simple yet powerful method has been vastly developed and used to assess the relative efficiencies of multiple-input multiple-output decision making units (DMUs). The popularity of DEA is due to its ability to measure relative efficiencies of multiple-input and multiple-output DMUs without prior weights on the inputs and outputs. To date, DEA is still widely researched and is being applied in many areas and domains. For example, agriculture industry [2], banking industry [3], information technology and information system [4], education [5], airline [6], computer industry [7], power plant [8], sport [9], stock market [10], government [11], supply chain [12], and many more.
In the current paper, we present various fundamental DEA models and some important methodological extensions of DEA that have enhanced its effectiveness. In addition, we also present some current research trends of DEA. However, readers should be noted that the coverage of this paper is not meant to be complete as the volume of literature is immense. Therefore, for those who are interested in a thorough discussion on the various topics of DEA, please refer to [13,14,15,16].
Section II gives a general review on some basic DEA models. Section III examines a number of extensions of DEA model. Section IV discusses several multilevel DEA models. A discussion on stochastic and fuzzy DEA models is presented in Section V, followed by conclusions in section VI.
II. BASIC DEAMODELS
This section describes some primary DEA models which are important in the development of DEA methodology. A. CCR Model
CCR DEA model was the first DEA model, which was proposed in [1]. It was named after the three authors (Charnes, Cooper, and Rhodes). In DEA, the organization under study is called a DMU. A DMU is an entity responsible for converting input(s) into output(s) and whose performances are to be evaluated. Consider there are n DMUs: DMU1, DMU2, …, and DMUn. Each DMUj, (j = 1, 2, …, n) uses m inputs xij (i = 1, …, m) and generates s outputs yrj (r = 1, …, s). Let the input weights vi (i = 1, …, m) and the output weights ur (r = 1, …, s) as variables. Let the DMUj to be evaluated on any trial be designated as DMU0 (0 = 1, 2, …, n). The efficiency of each DMU0, e0, is thus found by solving the linear programming below, which is known as multiplier form in DEA,
(1)
The model is run n times in identifying the relative efficiency scores of all the DMUs. Each DMU selects a set of input weights vi and output weights ur that maximize its efficiency score. Generally, a DMU is efficient if it obtains the maximum score of 1; else, a DMU is inefficient.
For every inefficient DMU, CCR identifies a set of corresponding efficient DMUs that can be utilized as benchmarks for improvement. The benchmarks can be obtained from the dual problem of (1). By duality, the model can be written as envelopment form, as shown in (2). Readers can refer to [17] on how to convert a multiplier form linear programming into envelopment form.
(2)
Problem (2) has an optimal solution when θ = 1, λ = 1, and λ ≠ 0. The DMUs with the optimal solution are referred to as efficient DMUs. For inefficient DMUs, CCR projects them onto the efficient frontier (the segment formed by efficient DMUs) by reducing each input by the proportionality factor θ, which is obtained from the envelopment model above, while maintaining the output levels. Another way of projecting the inefficient DMUs to the efficient frontier is by increasing the outputs by a proportionality factor 1/θ.
Despite its simplicity, CCR has some limitations. First, an inefficient DMU and its benchmarking DMUs may not be similar in their operations. The reason, as pointed out in [18], is due to the composite DMU does not exist in reality. To surmount this problem, performance-based clustering methods have been used by researchers to cluster similar DMUs into groups. The efficient DMUs in a cluster are utilized to benchmark other inefficient DMUs in a particular cluster [19].
The second limitation of CCR model is that it assumes constant return to scale (CRS), which may not be true for some applications. To address this issue, researchers have implemented variable return to scale (VRS) into the original DEA model. The basic VRS model is known as BCC (Banker, Charnes, Cooper) model [20], which will be explained in the following section.
B. BCC Model
In BCC model [20], VRS is assumed and the efficient frontier is formed by the convex hull of the existing DMUs. The envelopment form of BCC is,
(3)
and other constraints in (2). Note that BCC differs from CCR in that it has the additional convexity constraint, . A DMU0 is BCC-efficient if it has an optimal solution of θ = 1, λ = 1, and λ ≠ 0. The difference between CRS and VRS is illustrated with a single input-single output example in Fig. 1 and Fig. 2, which show the efficient frontiers of the models. In CCR, due to the CRS assumption, only DMU C is efficient. While for BCC, due to the VRS assumption, the efficient frontier is spanned by DMUs A, C, D, and F; these DMUs are BCC-efficient. Generally, a DMU is BCC-efficient if it is CCR-efficient but it is not necessary that a BCC-efficient DMU is CCR-efficient.
Referring to Fig. 2, that line segment connecting point A and point C, forms the increasing return to scale segment of the efficient frontier; constant return to scale occurring at point C; while all points to the right of point C is experiencing decreasing return to scale. For more detailed discussion on return to scale classification in DEA, readers can refer to [21].
Figure 1. Efficient frontier of CCR model
Figure 2. Efficient frontier of BCC model III. EXTENSIONS OF BASIC DEAMODELS A. Additive Model
There are two types of measures in DEA, viz. radial and non-radial. Both CCR and BCC are radial models. Radial means the inputs (or outputs) are reduced (increased) proportionally while maintaining the outputs (or inputs). Additive model is a non-radial DEA model, which moves an inefficient DMU to the efficient frontier by reducing inputs and increasing outputs concurrently.
Additive model, which is also known as Pareto-Koopmans model, was first introduced in [22]. Both CCR and BCC models can be modified to be additive models. Problem (4) shown below is an additive model arose from BCC model, (4)
where and are the slacks, specifically, is the input excesses and is the output shortfalls. A DMU is additive-efficient if all the slacks equal to zero at its optimum solution. Note that the efficient frontier generated by this additive model is exactly the same with the one generated by
(3) due to the convexity constraint . Therefore, a DMU is additive-efficient if and only if it is BCC-efficient.
An additive-inefficient DMU can be projected to the efficient frontier by adjusting the inputs and outputs using the slacks, as shown below,
! (5)
! (6)
A weakness of the additive model was it does not provide an actual measure of efficiency. To address this issue, a measure of efficiency for additive model was proposed [22],
" #$ % % & (7)
Dividing and by and render these slacks units invariant. While # controls the overall scale. Charnes et al. [22] suggested # to be '$ &. " provides an actual measure of efficiency for additive model. However, notice that the optimum value of " is zero. This is not consistent with the efficiency measures as in CCR and BCC models. B. Slacks-Based Model (SBM)
To overcome the weakness in the additive model, a variant of additive model was proposed in [23,24], namely, slacks-based model (SBM) presenting a measure of efficiency of, ( ) * + ,- -.'/-0 ) * 1 ,2 3'4 20 2 (8)
subjecting to the same constraints in (4). This fractional programming problem can be transformed into linear programming as shown in [24]. The optimum solution of SBM is 1, which can only be achieved when all the slacks equal to zero. This is consistent with CCR and BCC models. C. Hybrid Measures
Difference between radial approach and non-radial approach in DEA exists in the characterization of input or output items. Radial inputs and outputs change proportionally; while non-radial inputs and outputs change non-proportionally. CCR and BCC are two radial models, while additive model and SBM are two examples of non-radial models. These models can be applied to evaluate a group of DMUs’ efficiencies based on the characterization of their inputs and outputs. If all inputs and outputs are radial, CCR or BCC may be used; if all are non-radial, then additive model or SBM may be utilized.
However, there are situations where the inputs and/or outputs are a mixture of radial and non-radial. These differences should be reflected in the measurement of efficiency. Cooper et al. [14] therefore integrated radial and non-radial models in a unified framework and presented a hybrid measure of efficiency, which includes both radial and non-radial in the model.
D. Russel Measure
Another DEA model that is important in the methodology development of DEA is Russel Measure (RM), which was first formulated by Färe and Lovell [25]. RM aggregates both output and input efficiencies in the framework of a radial measure and treatments between input and output orientation are not required. RM, despite its usefulness, is too complicated in computation wise because it was formulated as a nonlinear programming problem [26]. RM was then revisited by a number of researchers. For instance, RM was extended into an improved model named as Enhanced Russell Graph Measure (EGRM) [26]. EGRM has reduced the computational complication of RM, but as a trade off, it can only give approximate RM scores.
A recent development of RM is published in [27]. By utilizing second-order cone programming (SOCP), the researchers are able to solve RM directly, without depending on EGRM. The proposed SOCP approach can provide an exact RM score. Their research is a major breakthrough in RM measurement. Interested readers are recommended to refer to their research paper.
E. Free Disposal Hull (FDH)
Free Disposal Hull (FDH) model in [28] is another model which has received a considerable amount of researchers’ attention. Instead of the convexity assumption that most DEA models use, FDH employs the assumption that the efficient frontier is constructed only by observed DMUs. The idea of FDH is to ensure that efficiency measurements are the results of actual observed performances. The concept is illustrated in Fig. 3. As can be seen, the boundary of the set and its connection represents the "hull" that envelops all the production possibilities from the actual observation.
The basic FDH model is an easy method to use, in fact, it can be extended from the CCR or BCC model with an additional constraint 5 67. For more discussion on the FDH model, one can refer to [14,28,29,30].
F. Cross Efficiency Model
Cross efficiency model, which was initiated by Sexton, Silkman, and Hogan [31], allows a DMU to be evaluated by not just with its own optimal weights, but also all the other DMUs’ optimum weights. In cross efficiency model, first the optimal weights for the inputs and outputs of each DMU are computed using a typical DEA procedure. Next, each DMU is to be evaluated using the sets of weights, obtaining efficiency scores for each DMU. Lastly, cross efficiency score for a DMU is the average value of its efficiency scores. The DMUs can then be ranked according to their cross efficiency scores. Further discussion on cross efficiency model can be found in [19]. Numerous applications and improved models of cross efficiency can be found in the literature, for example, [32,33].
G. Super Efficiency Model
In the super-efficiency models, the DMU under evaluation is excluded from the reference set. In other words, the super-efficiency score of a DMU is based on a reference set constructed from all other DMUs except itself.
Figure 3. Efficient frontier of FDH model
The concept of super-efficiency was proposed in [34]. There are troubles with these super-efficiency measures, for examples, the super-efficiency measures are lacking of units invariance and there are non-solution possibilities in VRS super-efficiency model [35].
Not much effort was given to resolve the non-solution problem. Only until recently, an alternative approach to resolve this non-solution problem in super-efficiency model has been proposed in [36].
Super-efficiency models have been widely applied in different application, for instances, ranking of efficient DMUs [34], determining efficiency regions [37], detecting outliers [38], and detecting the extreme efficient DMUs [39].
IV. DISCUSSIONS ON MULTILEVEL AND MULTISTAGE MODELS
The models presented thus far are single level models. However, there are applications where the efficiency has to be viewed in a multilevel or multistage perspective. The four models that will be discussed in this section are among the various multilevel and multistage DEA models.
A. Network DEA Models
One of the drawbacks of the traditional DEA models is the neglect of intermediate activities. A simple example is evaluating efficiencies of companies which consist of multiple divisions in each company. Each division utilizes its own input resources for producing its own outputs and there are linking activities between the divisions. In traditional DEA models, the companies are evaluated without considering the continuity of links between divisions. Some DEA models employ multiple steps, using intermediate products as outputs in one division and as inputs in another division, to evaluate the companies and their divisions. This, however, cannot deal with the linking activities in one single step. Therefore, network DEA models were developed to deal with this issue.
Färe and Grosskopf [40] originated the studies on network DEA. The researchers investigated the so-called “black box” of DEA. Their research was then improved by other researchers. Examples of extensions and applications of network DEA are, evaluating organizations with complex internal structures in [41], evaluating pharmacies productivity and customer satisfaction in [42], and a slacks-based network DEA model, which can decompose the
overall efficiency into divisional efficiencies, has been proposed in [43]. There is an increasing number of current studies being done on network DEA, thus, it can be a direction for future work.
B. Multi-component DEA Models
While the multilevel models which have been discussed previously can be termed as serial processes, some application areas involve multiple components or functions that can be termed as parallel processes [16]. It is sometimes needed to obtain a measure of performance for the DMUs and for the components or functions in the DMUs at the same time. The concept of partial efficiency measures was first introduced in [44]. The situation is further complicated when there are shared resources between the components or functions. The DEA structure has to be modified to be able to split those resources among the various components or functions. Cook and Green [45] proposed the use of a multi-component DEA model that can identify the overall performance of a DMU and isolate particular components which the company can improve on. In addition, the model can be utilized to identify the core business of a company. C. Hierarchical DEA Models
Some situations in multilevel efficiency measurement might involve hierarchical structures or groupings. Hierarchical DEA concept was introduced to evaluate DMUs that fall naturally into hierarchies and groupings. Hierarchical DEA enables one to view and evaluate efficiencies of DMUs at various levels. The efficiency scores of DMUs might be adjusted based on the level or group they fall into.
An application of hierarchical model to evaluate power plant efficiencies was demonstrated in [46]. Each power plant is made up of several operating units. By utilizing hierarchical DEA model, the researchers were able to evaluate both the efficiency of each power unit relative to other power units, and the efficiency of the power plants. In their research, the hierarchical efficiency was viewed as a multi-stage process. Cook and Green [47] revised and improved the model. This improved hierarchical efficiency measurement can be viewed at all levels simultaneously. As pointed out by Cook and Green, the application of DEA principles to hierarchical structures is an important and potential area for research.
D. Supply Chain DEA Models
Supply chain is a relatively new and fast emerging area in which multilevel DEA models are being applied. A number of DEA-based approaches have been used to evaluate supply chain efficiency. Wong and Wong [12], for instance, have provided a realistic framework to study supply chain performances. Several DEA models and approaches to measure the overall efficiency of supply chain, as well as the efficiencies of the members in the supply chain were presented in [48].
V. DISCUSSIONS ON STOCHASTIC AND FUZZY DEA As pointed out in [49], traditional DEA models in general share one common limitation - they do not allow stochastic variations in input-output data. The assumption that the data are always measured with precision is very questionable in applications that are highly dynamic. Two primary methods that are used to overcome this weakness of DEA are stochastic DEA and fuzzy DEA.
A. Stochastic DEA Models
Researchers have incorporated stochastic input and output variations into DEA models. Particularly, stochastic DEA models allow the existence of data errors and provide probabilistic results. Examples of applications are as follow, stochastic DEA and chance constrained programming approaches were utilized to evaluate technical efficiencies in [50]; Khodabakhshi and Asgharian [51] demonstrated the use of stochastic DEA on input relaxation measure of efficiency; stochastic data were incorporated to estimate the most productive scale size in DEA [49]; and Cooper et al. [52] employed chance constrained programming approaches to handle congestion issues in stochastic DEA.
B. Fuzzy DEA Models
Application of fuzzy mathematical programming approaches is another alternative to deal with data variations in DEA. For instances, [53] demonstrated a fuzzy mathematical programming approach to assess efficiencies with DEA models; a method to represent the efficiency of a DMU as a fuzzy efficiency was described in [54]; DEA model was incorporated with fuzzy random inputs and outputs in [55]; and [56] presented an integrated fuzzy DEA, fuzzy C-means and computer simulation for optimization of operator allocation in cellular manufacturing systems.
Increasing studies are being done in both the stochastic and fuzzy DEA models to deal with the variations in data, congestion, ranking, and variable coefficients in DEA. However, it should be noted that stochastic and fuzzy are not always favorable in DEA applications. For situation where the data are known to be inaccurate, and the decision makers want a rough estimate on the efficiencies of the DMUs, stochastic or fuzzy DEA models might be favorable. Else if the data is highly accurate, those deterministic models should be utilized instead of stochastic or fuzzy DEA models.
VI. CONCLUSIONS AND FUTURE TREND
This paper is intended to provide a general review of several fundamental DEA models and various important methodological extensions of DEA. Readers must be noted that the applications of DEA are vast and this paper is not intended to cover all the models and issues. For instances, there are significant studies in the areas of sensitivity analysis in DEA, modeling of different types of variables, multiplier restrictions in DEA, resource constrained DEA, and interfaces of DEA with other disciplines.
Several current and future research trends in DEA have been highlighted in the text, namely, network DEA models, hierarchical DEA models, application of DEA in supply chain efficiency measurement, stochastic DEA models, and
lastly, fuzzy DEA models. All of these could be the directions for future research.
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