Dynamic DEA: A slacks-based measure approach☆
Introduction
Measurement of intertemporal efficiency change has long been a subject of concern in DEA. The window analysis by Klopp [15] was the first approach for this purpose (see also Charnes et al. [4]). Based on Malmquist [16], Färe et al. [9] developed the Malmquist index in the DEA framework. This model can decompose the intertemporal efficiency change into catch-up and innovation (Frontier-shift) effects.
However, these models do not account for the effect of carry-over activities between two consecutive terms. For each term these models have inputs and outputs but the connecting activities between terms are not accounted explicitly.
The dynamic DEA model proposed by Färe and Grosskopf [8] is the first innovative scheme for dealing formally with these inter-connecting activities. See also Bogetoft et al. [2], Chen [5], Kao [14], Nemoto and Goto [17], [18], Park and Park [19], Sueyoshi and Sekitani [21] and Chang et al. [3] for further references. The dynamic DEA models have close connections with the network DEA models. For the latter subject, refer to Färe and Grosskopf [8], Tone and Tsutsui [24] and Avkiran [1] among others.
In this paper, we extend their model within the slacks-based measure framework proposed by Tone [22] and Pastor et al. [20]. Hence, our model is non-radial and can deal with inputs/outputs individually, enabling us to obtain non-uniform input/output factor efficiencies, contrary to the radial approaches that assume proportional changes in inputs/outputs and provide only one common uniform input/output factor efficiency. We can put weights to input/output items according to their importance. Furthermore, we categorize the carry-overs, called links, into four types, i.e. desirable (good), undesirable (bad), discretionary (free) and non-discretionary (fixed), reflecting actual characteristics of carry-over activities. As for orientations of the model, we have three types: input-, output- and non-oriented.
Thus, this paper has the following novel features. (1) As a dynamic model, we can compare a long range performance of companies. (2) Adoption of non-radial SBM models enables us to deal with inputs and outputs individually, and hence their non-proportional changes are allowed. (3) Carry-over activities (links) are categorized into four types, i.e. desirable (good), undesirable (bad), discretionary (free) and non-discretionary (fixed), and hence we can cope with demands of researchers and practitioners correctly and properly. (4) We developed three orientations for every model: input-, output- and non-oriented models. Thus, in accordance with the research purpose, we can choose appropriate models for evaluation. If input (output) side efficiency is the main target, we can choose input-(output-) oriented models. If both input and output efficiencies are to be evaluated concurrently, we can apply non-oriented models.
This paper can be positioned as an extension of the network SBM in Tone and Tsutsui [24] to the dynamic structure.
The rest of this paper unfolds as follows. In Section 2, we describe our dynamic model (DSBM) and discuss links and their characteristics. The production possibility set and models are presented in Section 3. Objective functions and efficiency of DSBM along with projections are presented in Section 4. We extend our models in Section 5 including the treatment of the initial condition. In Section 6 we exhibit an illustrative example, where we also compare our results with the traditional separation model which deals with time series data term by term. Since uniqueness of the optimal solution is an important matter for identifying term efficiencies and for projecting inefficient DMUs to efficiency status, we report an experiment on this subject using a real world dataset in Section 7. We conclude this paper in the last section.
Section snippets
Dynamic structure
Traditional DEA models deal usually with efficiency of input resources vs. output products of associated decision making units (DMUs) within cross sectional data. We extend them to dynamic situations as exhibited in Fig. 1. We observe n DMUs over T terms. At each term t, each DMU has its respective inputs and outputs along with the carry-over (link) to the next term t+1. We assume that we have a panel data through terms 1 to T. So, we look at the concerned enterprises as a continuum between the
Production possibility set and models
We deal with n DMUs (j=1,…,n) over T terms (t=1,…,T). At each term, DMUs have common m inputs , p non-discretionary (fixed) inputs , s outputs and r non-discretionary (fixed) outputs . Let , , and denote the observed (discretionary) input, non-discretionary input, (discretionary) output and non-discretionary output values of DMU j at term t, respectively. We symbolize the four category links as zgood,
Objective functions and efficiency
We evaluate the overall efficiency of DMUo (o=1,…, n) taking (, , , , , ) as variables, in the following three orientations, i.e. input-, output- and non-orientations. The input-oriented models deal mainly with reduction of input-related factors while producing at least the observed output-related factor levels. In our DSBM models, we maximize relative slacks in inputs and undesirable (bad) links. In the output-oriented models, we attempt to maximize
Extensions of models
We discuss two extensions of the fore-mentioned model.
An illustrative example
We generated a sample dataset as exhibited in Table 1.
In this dataset, we assume one input, one output, and one link for four years (T1–T4) regarding eight DMUs (A to H). In other words, they can be respectively labor inputs, products and capital assets which are carried over to the next period. Input amounts are increasing during the period for all DMUs, while we assumed four settings for output amounts, i.e. stable (A and B), increasing (C and D), decreasing (E and F) and volatile (G and H).
An experiment on the uniqueness issue of optimal solutions
As we mentioned in Section 4.1, non-radial dynamic SBM model may result in multiple optimum solutions for term efficiencies. In this section we examine uniqueness of optimal solutions using a real world data set.
Concluding remarks
In this paper, we have proposed a slacks-based dynamic DEA (DSBM) model. As an SBM model, DSBM is non-radial and can deal with inefficiency of inputs, outputs and links individually as contrast to the radial models which assume proportional changes of inputs or outputs. Furthermore, we categorized carry-over activities into four types, i.e. good, bad, free and fixed. As the model orientation, we proposed three types, i.e. input-, output- and non-oriented, and offered a program for evaluating
Acknowledgments
We are grateful to two anonymous referees for their valuable comments and suggestions by which our first draft has been substantially improved.
References (24)
Opening the black box of efficiency analysis: an illustration with UAE banks
Omega
(2009)- et al.
Using Malmquist indexes to measure changes in the productivity and efficiency of US accounting firms before and after the Sarbanes—Oxley Act
Omega
(2009) - et al.
model with new efficiency measures to incorporate the dynamic effect in production networks
European Journal of Operational Research
(2009) - et al.
Benchmarking and regulation: international electricity experience
Utilities Policy
(2000) - et al.
Dynamic data envelopment analysis modeling intertemporal behavior of a firm in the presence of productive inefficiencies
Economic Letters
(1999) - et al.
Measurement of multiperiod aggregative efficiency
European Journal of Operational Research
(2009) - et al.
An enhanced DEA Russell, graph efficiency measure
European Journal of Operational Research
(1999) - et al.
Returns to scale in dynamic DEA
European Journal of Operational Research
(2005) A slacks-based measure of efficiency in data envelopment analysis
European Journal of Operational Research
(2001)- et al.
Network DEA: a slacks-based measure approach
European Journal of Operational Research
(2009)
A developmental study of data envelopment analysis in measuring the efficiency of maintenance units in the US Air Forces
Annals of Operations Research
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Research supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science.