This paper proposes a new efficiency benchmarking methodology that is capable of incorporating probability while still preserving the advantages of a distribution-free and nonparametric modeling technique. This new technique developed in this paper will be known as the DEA-Chebyshev model. The foundation of DEA-Chebyshev model is based on the model pioneered by Charnes, Cooper, and Rhodes in 1978 known as Data Envelopment Analysis (DEA). The combination of normal DEA with DEA-Chebyshev frontier (DCF) can successfully provide a good framework for evaluation based on quantitative data and qualitative intellectual management knowledge. The simulated dataset was tested on DEA-Chebyshev model. It has been statistically shown that this model is effective in predicting a new frontier, whereby DEA efficient units can be further differentiated and ranked. It is an improvement over other methods, as it is easily applied, practical, not computationally intensive, and easy to implement.
There has been a substantial amount of research conducted in the area of stochastic evaluation of efficiency, such as the stochastic frontier approach (SFA) [
This paper proposes to develop a new efficiency benchmarking methodology that is capable of incorporating probability while still preserving the advantages of a function-free and nonparametric modeling technique. This new technique developed in this paper will be known as the DEA-Chebyshev model. The objectives are to first distinguish amongst top performers and second to define a probable feasible target for the empirically efficient units (as they are found from the usual DEA models) with respect to the DEA-Chebyshev frontier (DCF). This can be achieved by incorporating management's expertise (qualitative component) along with the available data (quantitative component) to infer this new frontier. The foundation of DEA-Chebyshev model is based on the model pioneered by Charnes et al. in 1978 [ It is nonparametric and does not require It has the ability to simultaneously handle multiple inputs and outputs without making prior judgments of their relative importance (i.e., function-free) It can provide a single measurement of performance based upon multiple inputs and outputs.
DEA ensures that the production units being evaluated will only be compared with others from the same “cultural” environment, provided, of course, that they operate under the same environmental conditions.
The rest of the paper is organized as follows. Section
This section provides the applicable literature on past and present researches relating to stochastic models and weight-restricted models designed for performance measurements. They show the relevance of well-known methodologies used for estimating efficiency scores and constructing the approximated frontier in order to account, as well as possible, for noise which can have diverse effects on efficiency evaluation of human performance-dependent entities.
Aigner et al. [
Stochastic DEA is a DEA method that attempts to account for and filter out noise by incorporating stochastic variations of inputs and outputs while still maintaining the advantages of DEA [
Chance-constrained programming was first developed by Charnes and Cooper [
All these CCP formulations have considered normal distributions for the probability of staying within the constraints. This method is effective when qualitative data is not available. However, expert opinion from management cannot be discounted with regard to data dispersion from the expected or correct values. Unfortunately, the current CCP is strictly a quantitative analysis based on empirical data and whose variations are said to be of a predefined distributional form.
In an “unrestricted” DEA model, the weights are assigned to each DMU such that it would appear as favourable as possible, which is an inherent characteristic of DEA. Hence, there is a concern when largely different weights may be assigned to the same inputs and outputs in the LP solutions for different DMUs. This motivated the development of weight-restricted models such as the “assurance region” (AR) [
The motivation behind weight-restricted models is to redefine the DEA frontier so as to make it as
Weight restriction models deal directly with the model’s inconsistencies in a practical sense using qualitative information, whereas stochastic models deal with data discrepancies and inconsistencies using quantitative approaches to infer to the degree of data disparity. Although the motivations of these two methods are similar, the underlying objectives for their developments are not the same. Both are valid extensions of the normal DEA model in attempting to correct the frontier.
The Assurance Region (AR) model was developed by Thompson et al. [
The cone-ratio (CR) method was developed by Charnes et al. [
Before we begin to make modifications to incorporate probability into the basic DEA model, it is crucial that the types of errors are identified, which are sources of data disparity. These can be segregated into 2 categories; systematic and nonsystematic errors. Nonsystematic errors are typically defined to be statistical noise, which are random normal
The design of the new DEA model is intended to take into account the possibility of data disparity that affect productivity analysis while preserving the advantages that DEA offers in order to estimate the true level of efficiency. Due to data disparity, normal DEA results may contain two components of the error term. The first refers to statistical noise which follows a normal distribution, while the second refers to the technical inefficiency which is said to follow a truncated normal or a half-normal distribution. This can be achieved by relaxing the LP constraints to allow for these variations which may provide a better approximation of the level of efficiency.
The following general linear programming model illustrates the mathematical form of systematic and nonsystematic errors as defined previously. Variation in the variable (
Four scenarios are illustrated later which describes sources of data disparity. The notations are as follows:
The following are equations defining the relationship between the observed and true or expected values for both inputs and outputs in a productivity analysis such as SFA where measurement errors and/or random noise and inefficiencies are a concern in parametric estimations:
The following four scenarios illustrate the impact of different errors and were constructed using the notations given previously. These scenarios follow the definition by Tomlinson [
The term “measurement error” does not simply imply that data had been misread or collected erroneously. According to Tomlinson [
Deterministic methods such as DEA are not designed to handle cases in which, due to uncertainty, constraints may be violated although infrequently. Various methods have been employed to transform the basic DEA approach to include stochastic components. Two of the more popular methods are chance-constraint programming (CCP) and stochastic DEA. An extensive literature survey has revealed that CCP DEA has always assumed a normal distribution. The objective of this research is to redefine the probabilities employed in CCP productivity analysis, which would accommodate problems emanating from various scenarios where errors are independent but convoluted without assuming any distributional form. The independent and convoluted properties of the error terms make it difficult to distinguish between them, and hence, a distribution-free approach will be employed.
The advantage of using CCP is that it maintains the nonparametric form of DEA. It allows modeling of multiple inputs and outputs with ease. There is no ambiguity in defining a distribution or the interpretation of the results as had been demonstrated in the Normal-Gamma parametric SFA model [
The focus of this paper is on the further development of DEA coupled with CCP. The benefit of applying CCP to DEA is such that the multidimensional and nonparametric form of DEA is maintained. To drop the
The CCP formulation shown in (
The first constraint in (
Using the same mathematical formulation shown in (
The advantages of using DEA-Chebyshev model as an efficiency evaluation tool are that it provides an approximation of performance given that random errors and inefficiencies do exist, and these deviations are considered, either through expert opinion or through data inference. Nevertheless, the results should always be subject to management scrutiny. This method also provides for ranking efficient DMUs.
In a simplified explanation, the Chebyshev theorem states that the fraction of the dataset lying within
DEA-Chebyshev model developed in this paper will not be restricted to any one distribution but instead will assume an unknown distribution. A distribution-free approach will be used to represent the stochastic nature of the data. This approach is applied to the basic DEA model using chance-constraint programming. This distribution-free method is known as the Chebyshev inequality. It states that
Let a random variable
Other methods considered to define the probabilities for DEA-Chebyshev model were the distribution-free linear constraint set (or linear approximation), the unit sphere method, and the quantile method. These methods were tested to determine which of them would provide the best estimate of the
The distribution-free approaches tested were the Chebyshev extended lemma (
Allen et al. have proven in their paper [
Since, we do not want to place an
A
Two general assumptions have been made when constructing the model. First, nuisance parameters (including confounding variables) will affect efficiency scores causing them to differ from the true performance level if they are not accounted for in the productivity analysis. Second, variations in the observed variables can arise from both statistical noise and measurement errors and are convoluted.
In the simulation to follow, as an extension to the general assumptions mentioned previously, we will assume that variations in outputs are negligible and will average out to zero [
The assumption regarding the disparity between the observed and expected inputs is to illustrate the input-oriented DEA-Chebyshev model. In input-oriented models, the outputs are not adjusted for efficiency, but the inputs are based on the weights applied to those DMUs that are efficient. This assumption regarding errors can be reversed between inputs and outputs depending on expert opinions and the objective of the analysis (i.e., input versus output-oriented models).
As an extension of Land et al. [
Theoretically, the DEA, algorithm allows the evaluation of models containing strictly outputs with no inputs and vice versa. In doing so, it neglects the fact that inputs are crucial for the production of outputs. However, the properties of a production process are such that they must contain inputs in order to produce outputs. Let the
An input-oriented BCC model will be used to illustrate this work. Here,
Since it is difficult to establish a specific form of distribution with empirical data due to the convolution of different types of errors, a distribution-free approach is taken. In this case, the Chebyshev one-sided inequality [
The value of
The deterministic LP formulation for DEA-Chebyshev model can be written in the following mathematical form:
The value of
It may not be sufficient to develop a model that technically sounds with appropriate theoretical proofs. We cannot discount the fact that management expertise can play an important role in defining the corrected frontier nor should we cause the user to reject the model. Hence, DEA-Chebyshev model is designed to incorporate management input, which can become a crucial factor in the modeling process. One of the major advantages of this model is its flexibility as compared to models that require a distributional form. It can provide crucial information to management based upon their expertise and experience in their own field of specialization thereby redefining the efficient frontier.
In DEA-Chebyshev model,
The value for
Hence, for
When deviation due to errors is negligible, then %
Unlike the straightforward method in which DEA scores are calculated, DEA-Chebyshev model efficiency scores are slightly more complicated to obtain. There are five stages to the determination of the best efficiency rating for a DMU.
An efficient DMU with a smaller standard deviation implies a smaller confidence region in which the EFF resides, hence, this particular DMU is considered to be more robustly efficient since it is closer to the EFF. It can be conjectured that for DEA efficient DMUs, When In general, the mean efficiency score in DEA-Chebyshev model is such that
Five data sets, each containing 15 DMUs in a two-input one-output scenario, were generated in order to illustrate the approximation of the EFF using the DEA-Chebyshev model. This will demonstrate the proximity of the DCF to the EFF. A comparison is drawn between the results provided by the DCF, DEA, and the CCP input-oriented VRS models as compared against the EFF.
The first data set shown in Table
Control group: the error-free production units.
DMU | Output | Input 1 | Input 2 |
---|---|---|---|
1 | 12.55 | 2 | 12 |
2 | 10.43 | 3 | 8 |
3 | 9.68 | 4 | 6 |
4 | 9.53 | 5 | 4.8 |
5 | 9.68 | 6 | 4 |
6 | 10.01 | 7 | 3.43 |
7 | 10.43 | 8 | 3 |
8 | 11.45 | 10 | 2.4 |
9 | 11.99 | 11 | 2.18 |
10 | 12.55 | 12 | 2 |
11 | 13.12 | 13 | 1.85 |
12 | 14.25 | 15 | 1.6 |
13 | 15.36 | 17 | 1.41 |
14 | 16.46 | 19 | 1.26 |
15 | 16.99 | 20 | 1.2 |
The experimental groups are generated from the control group with the error components. Their outputs are the same as the control groups and are held deterministic, while inputs are stochastic containing confounded measurement errors distributed as half-normal nonzero inefficiency
Four experimental groups with variations and inefficiencies introduced to both inputs while keeping outputs constant.
DMU | Output | Experimental Grp 1 | Experimental Grp 2 | Experimental Grp 3 | Experimental Grp 4 | ||||
---|---|---|---|---|---|---|---|---|---|
Input 1 | Input 2 | Input 1 | Input 2 | Input 1 | Input 2 | Input 1 | Input 2 | ||
1 | 12.55 | 3.16 | 12.5 | 2.34 | 12.85 | 2.91 | 12.6 | 2.68 | 13.92 |
2 | 10.43 | 3.69 | 9.08 | 1.6 | 10.07 | 2.34 | 8.23 | 3.32 | 8.34 |
3 | 9.68 | 4.88 | 8.41 | 3.58 | 5.97 | 6.1 | 6.43 | 4.25 | 6.53 |
4 | 9.53 | 5.27 | 5.31 | 7.28 | 9.43 | 7.84 | 3.96 | 6.44 | 4.25 |
5 | 9.68 | 8.39 | 7.43 | 6.98 | 5.9 | 7.64 | 2.96 | 9.93 | 3.55 |
6 | 10.01 | 9.17 | 3.8 | 7.04 | 5.57 | 9.6 | 4.01 | 10.46 | 4.98 |
7 | 10.43 | 10.92 | 3.11 | 9.6 | 3.26 | 7.71 | 2.9 | 6.29 | 2.95 |
8 | 11.45 | 13.14 | 3.95 | 11.41 | 1.88 | 10.38 | 3.14 | 11.71 | 3.05 |
9 | 11.99 | 9.33 | 2.85 | 11.53 | 4.75 | 13.88 | 0.59 | 13.25 | 2.47 |
10 | 12.55 | 10.38 | 7.43 | 13.94 | 2.46 | 12.55 | 4.44 | 12.19 | 3.73 |
11 | 13.12 | 12.67 | 1.69 | 12.46 | 4.79 | 13.53 | 1.1 | 13.24 | 1.1 |
12 | 14.25 | 17.59 | 4.8 | 15.71 | 2.09 | 16.57 | 2.27 | 14.14 | 2.08 |
13 | 15.36 | 17.35 | 4.23 | 17.33 | 4.44 | 15.35 | 1.38 | 15.47 | 2.25 |
14 | 16.46 | 19.13 | 1.4 | 20.33 | 3.49 | 19.11 | 0.06 | 18.67 | 0.57 |
15 | 16.99 | 19.98 | 2.51 | 19.31 | 4.85 | 20.57 | 1.21 | 19.32 | 2.59 |
The DEA results were calculated using ProDEA, while CCP and DEA-Chebyshev model results were calculated using MathCad. The CCP LP formulation follows that from [
Table
DEA and CCP efficiency evaluation for simulation 1.
DEA |
|
CCP (U) |
CCP (L) |
Average |
CCP |
|
|
|
|
---|---|---|---|---|---|---|---|---|---|
DMU1 |
|
1.674 | 0.795 | 1.52 | 1.158 |
|
1.834 (8) | 2.124 (6) | 1.979 |
DMU2 |
|
1.58 | 0.762 | 1.259 | 1.011 |
|
1.31 (3) | 1.18 (5) | 1.245 |
DMU3 |
|
0 | 0.694 | 1.074 | 0.884 |
|
0 | 0.323 | 0.162 |
DMU4 |
|
2.56 | 0.69 | 1.277 | 0.984 |
|
3.458 (7) | 1.785 (8) | 2.621 |
DMU5 |
|
0 | 0.481 | 0.852 | 0.666 |
|
0 | 0 | 0 |
DMU6 |
|
0 | 0.706 | 1.089 | 0.898 |
|
0 | 0.5463 | 0.273 |
DMU7 |
|
0 | 0.653 | 1.094 | 0.873 |
|
0 | 0.5199 | 0.26 |
DMU8 |
|
0 | 0.538 | 0.885 | 0.711 |
|
0 | 0 | 0 |
DMU9 |
|
4.778 | 0.777 | 1.238 | 1.008 |
|
4.876 (10) | 2.415 (9) | 3.645 |
DMU10 |
|
0 | 0.665 | 0.894 | 0.779 |
|
0 | 0 | 0 |
DMU11 |
|
1.105 | 0.82 | 1.593 | 1.206 |
|
0.0996 (3) | 2.37 (9) | 1.235 |
DMU12 |
|
0 | 0.666 | 0.819 | 0.743 |
|
0 | 0 | 0 |
DMU13 |
|
0 | 0.772 | 0.962 | 0.867 |
|
0 | 0 | 0 |
DMU14 |
|
2.302 | 0.912 | 2.154 | 1.533 |
|
1.532 (4) | 2.134 (6) | 1.833 |
DMU15 |
|
1 | 0.924 | 2.906 | 1.915 |
|
1.892 (2) | 1.601 (5) | 1.747 |
DEA and CCP efficiency evaluation for simulation 2.
DEA |
|
CCP (U) |
CCP (L) |
Average |
CCP |
|
|
|
|
---|---|---|---|---|---|---|---|---|---|
DMU1 |
|
1.222 | 0.803 | 1.702 | 1.252 |
|
1.61 (6) | 1.449 (5) | 1.53 |
DMU2 |
|
1 | 0.759 | 1.924 | 1.341 |
|
0.875 (2) | 1.117 (6) | 0.996 |
DMU3 |
|
4.377 | 0.699 | 1.329 | 1.014 |
|
4.205 (8) | 2.998 (7) | 3.602 |
DMU4 |
|
0 | 0.425 | 0.764 | 0.595 |
|
0 | 0 | 0 |
DMU5 |
|
0 | 0.615 | 1.012 | 0.814 |
|
0 | 0.0678 | 0.034 |
DMU6 |
|
0 | 0.639 | 1.038 | 0.839 |
|
0 | 0.3006 | 0.15 |
DMU7 |
|
0 | 0.73 | 1.164 | 0.947 |
|
0 | 0.7558 | 0.378 |
DMU8 |
|
2.872 | 0.78 | 1.629 | 1.204 |
|
3.263 (10) | 2.305 (10) | 2.784 |
DMU9 |
|
0 | 0.727 | 0.963 | 0.845 |
|
0 | 0 | 0 |
DMU10 |
|
0 | 0.779 | 1.243 | 1.011 |
|
0 | 0.7534 | 0.377 |
DMU11 |
|
0 | 0.789 | 1.026 | 0.907 |
|
0 | 0.0958 | 0.048 |
DMU12 |
|
3.074 | 0.847 | 1.64 | 1.243 |
|
2.603 (6) | 2.132 (8) | 2.367 |
DMU13 |
|
0 | 0.856 | 1.033 | 0.944 |
|
0 | 0.1427 | 0.071 |
DMU14 |
|
1 | 0.888 | 1.439 | 1.163 |
|
0.259 (2) | 1.264 (4) | 0.761 |
DMU15 |
|
1.455 | 0.922 | 1.514 | 1.218 |
|
2.186 (5) | 1.62 (5) | 1.903 |
DEA and CCP efficiency evaluation for simulation 3: if the data contains small nonsystematic errors, the DEA model outperforms the CCP. CCP works well under conditions where inefficiency has not been partially offset by noise.
DEA |
|
CCP (U) |
CCP (L) |
Average |
CCP |
|
|
|
|
---|---|---|---|---|---|---|---|---|---|
DMU1 |
|
1 | 0.794 | 1.566 | 1.18 |
|
1.136 (4) | 1.283 (2) | 1.20945 |
DMU2 |
|
1.901 | 0.731 | 1.603 | 1.167 |
|
3.148 (11) | 1.305 (4) | 2.22655 |
DMU3 |
|
0 | 0.659 | 1.003 | 0.831 |
|
0 | 0 | 0 |
DMU4 |
|
0 | 0.67 | 1.079 | 0.874 |
|
0 | 0 | 0 |
DMU5 |
|
2.986 | 0.728 | 1.235 | 0.982 |
|
0 (0) | 2.137 (7) | 1.0685 |
DMU6 |
|
0 | 0.571 | 0.954 | 0.762 |
|
0 | 0 | 0 |
DMU7 |
|
2.704 | 0.725 | 1.24 | 0.982 |
|
5.681 (10) | 2.598 (7) | 4.13975 |
DMU8 |
|
0 | 0.705 | 1.028 | 0.867 |
|
0 | 0 | 0 |
DMU9 |
|
1 | 0.791 | 2.408 | 1.599 |
|
0 (0) | 1.963 (10) | 0.98141 |
DMU10 |
|
0 | 0.664 | 0.88 | 0.772 |
|
0 | 0 | 0 |
DMU11 |
|
1 | 0.799 | 1.298 | 1.048 |
|
0 | 0.6928 | 0.3464 |
DMU12 |
|
0 | 0.674 | 0.926 | 0.8 |
|
0 | 0 | 0 |
DMU13 |
|
2.409 | 0.893 | 1.161 | 1.027 |
|
2.634 (8) | 0.451 (3) | 1.54245 |
DMU14 |
|
1 | 0.936 | 29.92 | 15.43 |
|
0.585 (2) | 3.528 (6) | 2.05655 |
DMU15 |
|
1 | 0.926 | 2.77 | 1.848 |
|
1.816 (2) | 1.041 (2) | 1.42865 |
DEA and CCP efficiency evaluation for simulation 4.
DEA |
|
CCP (U) |
CCP (L) |
Average |
CCP |
|
|
|
|
---|---|---|---|---|---|---|---|---|---|
DMU1 |
|
1.036 | 0.797 | 1.613 | 1.205 |
|
1.182 (7) | 1.383 (3) | 1.283 |
DMU2 |
|
1 | 0.726 | 1.294 | 1.01 |
|
1.954 (6) | 0.911 (3) | 1.432 |
DMU3 |
|
1.255 | 0.773 | 1.207 | 0.99 |
|
0 (0) | 0.715 (4) | 0.358 |
DMU4 |
|
0 | 0.667 | 1.129 | 0.898 |
|
0 | 0.939 | 0.469 |
DMU5 |
|
0 | 0.462 | 0.99 | 0.726 |
|
0 | 0 | 0 |
DMU6 |
|
0 | 0.428 | 0.815 | 0.622 |
|
0 | 0 | 0 |
DMU7 |
|
5.52 | 0.712 | 1.367 | 1.039 |
|
7.079 (13) | 3.819 (10) | 5.449 |
DMU8 |
|
0 | 0.57 | 0.981 | 0.775 |
|
0 | 0 | 0 |
DMU9 |
|
0 | 0.601 | 1.013 | 0.807 |
|
0 | 0.018 | 0.009 |
DMU10 |
|
0 | 0.696 | 0.929 | 0.812 |
|
0 | 0 | 0 |
DMU11 |
|
2.009 | 0.797 | 1.781 | 1.289 |
|
0 (0) | 2.338 (9) | 1.169 |
DMU12 |
|
0 | 0.829 | 1.079 | 0.954 |
|
0 | 0.518 | 0.259 |
DMU13 |
|
1.87 | 0.935 | 1.098 | 1.017 |
|
1.455 (3) | 0.506 (3) | 0.981 |
DMU14 |
|
1.31 | 0.912 | 3.899 | 2.406 |
|
1.303 (5) | 2.734 (7) | 2.018 |
DMU15 |
|
1 | 0.922 | 2.743 | 1.832 |
|
2.028 (3) | 1.119 (2) | 1.573 |
In this simulation, because we do expect data collected to be reasonably reliable, a less conservative model would be a better choice. Conservative models tend to provide results with greater standard deviation and therefore produce an estimate with less accuracy. The four simulations were designed to test CCP, DEA, and DEA-Chebyshev model to determine the accuracy of the results obtained in comparison to the EFF. The results for DEA, CCP, and DCF for all four simulations using the values of
When the degree of deviation from observed performance levels is available, the results generated using DEA-Chebyshev model are generally a more precise approximation of the EFF compared to CCP, which assumes the normal distribution. From the simulations, it has been shown that the alpha values based on the deviation from the observed level of performance consistently produce the best approximations. The estimated degree of deviation due to inefficiency from the observed level of performance is formulated as follows:
Qualitative information: determining the value for
Simulation 1 |
|
| |
Simulation 2 |
|
| |
Simulation 3 |
|
| |
Simulation 4 |
|
Note that in the simulations, the correction factor is set to
DEA-Chebyshev model efficiency analysis from simulation 1 at
|
|
|
|
|
St. dev |
| |
---|---|---|---|---|---|---|---|
DMU1 | 0.786 | 1.548 |
|
|
|
|
|
DMU2 | 0.751 | 1.272 |
|
|
|
|
|
DMU3 | 0.683 | 1.082 | 0 | 0.357 | 0.179 | 0.282 | 0.883 |
DMU4 | 0.673 | 1.287 | 3.491 (7) | 1.72 (8) | 2.605 (0.02) | 0.434 | 0.98 |
DMU5 | 0.47 | 0.858 | 0 | 0 | 0 | 0.275 | 0.664 |
DMU6 | 0.696 | 1.096 | 0 | 0.591 | 0.295 | 0.283 | 0.896 |
DMU7 | 0.643 | 1.104 | 0 | 0.547 | 0.274 | 0.326 | 0.874 |
DMU8 | 0.531 | 0.892 | 0 | 0 | 0 | 0.255 | 0.712 |
DMU9 | 0.767 | 1.249 |
|
|
|
|
|
DMU10 | 0.659 | 0.898 | 0 | 0 | 0 | 0.169 | 0.779 |
DMU11 | 0.813 | 1.628 | 0.101 (3) | 2.328 (9) | 1.214 (0.006) | 0.577 | 0.906 |
DMU12 | 0.662 | 0.822 | 0 | 0 | 0 | 0.113 | 0.742 |
DMU13 | 0.768 | 0.965 | 0 | 0 | 0 | 0.14 | 0.867 |
DMU14 | 0.906 | 2.225 |
|
|
|
|
|
DMU15 | 0.92 | 3.232 |
|
|
|
|
|
DEA-Chebyshev model efficiency analysis from simulation 2 at
|
|
|
|
|
St. dev |
| |
---|---|---|---|---|---|---|---|
DMU1 | 0.793 | 1.739 |
|
|
|
|
|
DMU2 | 0.748 | 1.964 | 0.809 (1) | 1.178 (7) | 0.994 (0.25) | 0.86 | 0.874 |
DMU3 | 0.684 | 1.342 |
|
|
|
|
|
DMU4 | 0.417 | 0.771 | 0 | 0 | 0 | 0.25 | 0.594 |
DMU5 | 0.604 | 1.02 | 0 | 0.115 | 0.058 | 0.294 | 0.812 |
DMU6 | 0.628 | 1.047 | 0 | 0.337 | 0.169 | 0.296 | 0.837 |
DMU7 | 0.719 | 1.174 | 0 | 0.764 | 0.382 | 0.322 | 0.947 |
DMU8 | 0.769 | 1.657 |
|
|
|
|
|
DMU9 | 0.719 | 0.967 | 0 | 0 | 0 | 0.176 | 0.843 |
DMU10 | 0.78 | 1.264 | 0 | 0.794 | 0.397 | 0.342 | 0.89 |
DMU11 | 0.782 | 1.03 | 0 | 0.115 | 0.057 | 0.175 | 0.906 |
DMU12 | 0.84 | 1.664 |
|
|
|
|
|
DMU13 | 0.852 | 1.037 | 0 | 0.152 | 0.076 | 0.131 | 0.944 |
DMU14 | 0.887 | 1.46 | 0.646 (1) | 1.273 (4) | 0.959 (0.08) | 0.405 | 0.943 |
DMU15 | 0.918 | 1.556 |
|
|
|
|
|
DEA-Chebyshev model efficiency analysis from simulation 3 at
|
|
|
|
|
St. dev |
| |
---|---|---|---|---|---|---|---|
DMU1 | 0.794 | 1.566 |
|
|
|
|
|
DMU2 | 0.731 | 1.603 |
|
|
|
|
|
DMU3 | 0.659 | 1.003 | 0 | 0 | 0 | 0.213 | 0.833 |
DMU4 | 0.67 | 1.079 | 0 | 0.461 | 0.23 | 0.255 | 0.877 |
DMU5 | 0.728 | 1.235 | 0.377 (1) | 2.111 (8) | 1.244 (0.06) | 0.3195 | 0.985 |
DMU6 | 0.571 | 0.954 | 0 | 0 | 0 | 0.24 | 0.765 |
DMU7 | 0.725 | 1.24 |
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DMU8 | 0.705 | 1.028 | 0 | 0.027 | 0.014 | 0.204 | 0.87 |
DMU9 | 0.791 | 2.408 | 0.805 (1) | 1.061 (8) | 0.933 (0.005) | 0.9715 | 0.905 |
DMU10 | 0.664 | 0.88 | 0 | 0 | 0 | 0.1347 | 0.774 |
DMU11 | 0.799 | 1.298 | 0 | 1.077 | 0.538 | 0.2745 | 0.921 |
DMU12 | 0.674 | 0.926 | 0 | 0 | 0 | 0.157 | 0.803 |
DMU13 | 0.893 | 1.161 |
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DMU14 | 0.936 | 29.92 | 1 (1) | 2.718 (6) | 1.859 (0.04) | 17.971 | 1 |
DMU15 | 0.926 | 2.77 |
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DEA-Chebyshev model efficiency analysis from simulation 4 at
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DMU2 | 0.729 | 1.291 | 1.951 (6) | 0.918 (3) | 1.4347 (0.05) | 0.398 | 1 |
DMU3 | 0.776 | 1.204 | 0 (0) | 0.719 (4) | 0.359 (0.1) | 0.303 | 0.99 |
DMU4 | 0.67 | 1.126 | 0 | 0.92 | 0.46 | 0.322 | 0.898 |
DMU5 | 0.464 | 0.987 | 0 | 0 | 0 | 0.37 | 0.726 |
DMU6 | 0.43 | 0.813 | 0 | 0 | 0 | 0.271 | 0.622 |
DMU7 | 0.716 | 1.363 |
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DMU8 | 0.572 | 0.979 | 0 | 0 | 0 | 0.288 | 0.775 |
DMU9 | 0.603 | 1.01 | 0 | 0.015 | 0.007 | 0.288 | 0.807 |
DMU10 | 0.697 | 0.928 | 0 | 0 | 0 | 0.163 | 0.812 |
DMU11 | 0.799 | 1.767 | 0 (0) | 2.327 (9) | 1.164 (0.002) | 0.685 | 0.9 |
DMU12 | 0.831 | 1.077 | 0 | 0.512 | 0.256 | 0.174 | 0.954 |
DMU13 | 0.884 | 1.097 | 2.217 (4) | 0.514 (3) | 1.366 (0.06) | 0.15 | 0.991 |
DMU14 | 0.913 | 3.862 |
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DMU15 | 0.923 | 2.682 |
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Note: In Tables
The Tables
All the efficiency evaluation tools will be measured against the control group to determine which of these would provide the best approximation method. Both CCP and DEA-Chebyshev model efficiency scores are defined in the same manner. The upper and lower bounds of the frontier determine the region where the EFF may likely be and is approximated by the DCF efficiency score,
Using the results obtained in Step II, the four simulated experimental groups are adjusted using their respectively efficiency scores. The virtual DMUs are the DMUs from the four experimental groups in which their inputs have been reduced according to their efficiency scores from Step II, according to the contraction factor,
In this step, in order to test the hypothesis, the 12 data sets of virtual DMUs are each aggregated with the control group, forming a sample size of 30 DMUs per simulation. “DMU#” denotes the control group (or “sample one”) and “V.DMU#” denotes the efficient virtual units derived from the experimental group (or “sample two”) using the efficiency scores generated by DEA, CCP, and DEA-Chebyshev model, respectively. There are 12 data sets in total: three for each of the simulations (three input contraction factors per DMU, from DEA, CCP (normal), and DEA-Chebyshev model). The inputs for the virtual DMUs calculated from each of these three methodologies for the same experimental group will be different. The sample size of 30 DMUs in each of the 12 sets is a result of combining the 15 error-free DMUs with the 15 virtual DMUs. These 30 DMUs are then evaluated using ProDEA (software). It is logical to use DEA for our final analysis to scrutinize the different methods since this is a deterministic method, which would work perfectly in an error-free situation. The DEA results for the 4 simulations are given in Table
Deterministic efficiency results for all four simulations with an aggregate of 30 DMUs; 15 from the control group and another 15 virtual units calculated according to CCP and DEA, respectively.
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DMU1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
DMU2 | 1 | 1 | 1 | 0.986 | 0.946 | 0.942 | 0.962 | 0.962 | 0.962 | 1 | 0.937 | 1 |
DMU3 | 1 | 1 | 1 | 0.96 | 0.96 | 0.96 | 1 | 1 | 1 | 1 | 0.981 | 1 |
DMU4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.989 | 0.977 | 0.989 |
DMU5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.945 | 0.94 | 0.945 |
DMU6 | 1 | 1 | 1 | 1 | 1 | 1 | 0.991 | 0.987 | 0.988 | 0.888 | 0.888 | 0.888 |
DMU7 | 1 | 1 | 1 | 1 | 1 | 1 | 0.965 | 0.965 | 0.965 | 0.901 | 0.885 | 0.885 |
DMU8 | 1 | 0.965 | 0.963 | 1 | 1 | 1 | 0.971 | 0.933 | 0.943 | 0.914 | 0.872 | 0.872 |
DMU9 | 0.991 | 0.937 | 0.935 | 1 | 1 | 1 | 0.968 | 0.917 | 0.931 | 0.918 | 0.863 | 0.863 |
DMU10 | 0.978 | 0.906 | 0.903 | 1 | 1 | 1 | 0.962 | 0.901 | 0.919 | 0.926 | 0.87 | 0.871 |
DMU11 | 0.966 | 0.882 | 0.878 | 1 | 1 | 1 | 0.954 | 0.893 | 0.913 | 0.932 | 0.876 | 0.877 |
DMU12 | 0.985 | 0.931 | 0.929 | 1 | 1 | 1 | 0.934 | 0.903 | 0.911 | 0.939 | 0.906 | 0.914 |
DMU13 | 0.996 | 0.966 | 0.965 | 1 | 1 | 1 | 0.914 | 0.909 | 0.912 | 0.949 | 0.932 | 0.939 |
DMU14 | 1 | 0.991 | 0.991 | 1 | 1 | 1 | 0.973 | 0.957 | 0.973 | 0.967 | 0.967 | 0.967 |
DMU15 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
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V.DMU1 | 0.889 | 0.885 | 0.884 | 0.921 | 0.921 | 0.921 | 0.898 | 0.999 | 0.898 | 0.841 | 0.84 | 0.84 |
V.DMU2 | 0.86 | 0.86 | 0.86 | 1 | 1 | 1 | 1 | 0.987 | 0.993 | 0.938 | 1 | 0.938 |
V.DMU3 | 0.864 | 0.872 | 0.873 | 1 | 1 | 1 | 1 | 1 | 1 | 0.931 | 0.92 | 0.941 |
V.DMU4 | 0.929 | 0.944 | 0.948 | 0.976 | 0.972 | 0.974 | 1 | 0.971 | 0.986 | 0.982 | 0.979 | 0.984 |
V.DMU5 | 0.926 | 0.943 | 0.946 | 0.934 | 0.943 | 0.945 | 1 | 1 | 1 | 1 | 1 | 1 |
V.DMU6 | 0.915 | 0.927 | 0.928 | 0.955 | 0.966 | 0.968 | 1 | 0.998 | 1 | 0.999 | 0.963 | 0.964 |
V.DMU7 | 0.959 | 0.947 | 0.946 | 0.926 | 0.926 | 0.927 | 1 | 1 | 1 | 1 | 1 | 1 |
V.DMU8 | 0.977 | 0.954 | 0.951 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.929 | 0.93 |
V.DMU9 | 1 | 0.99 | 0.987 | 0.956 | 0.946 | 0.946 | 0.989 | 0.989 | 0.989 | 1 | 0.898 | 0.899 |
V.DMU10 | 0.959 | 0.938 | 0.936 | 0.933 | 0.959 | 0.958 | 1 | 1 | 1 | 0.989 | 0.958 | 0.959 |
V.DMU11 | 1 | 1 | 1 | 0.939 | 0.94 | 0.94 | 0.938 | 0.954 | 0.952 | 1 | 1 | 1 |
V.DMU12 | 0.977 | 0.953 | 0.952 | 0.933 | 0.932 | 0.932 | 0.972 | 0.989 | 0.988 | 0.996 | 0.976 | 0.987 |
V.DMU13 | 0.971 | 0.975 | 0.975 | 0.903 | 0.899 | 0.899 | 0.998 | 1 | 1 | 0.992 | 1 | 0.995 |
V.DMU14 | 0.986 | 0.98 | 0.979 | 0.872 | 0.924 | 0.924 | 0.99 | 1 | 1 | 1 | 1 | 1 |
V.DMU15 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
In order to determine if the frontiers created by these models are substantially different from that of the control group (or the error-free units), the rank-sum-test and statistical hypothesis test for mean differences were used.
The DEA-Chebyshev model is scrutinized using several statistical methods, which show that there is a strong relationship between the DCF and the EFF. All the statistical tools used to test the DCF against the EFF have produced consistent conclusions that the
Hypothesis tests for mean differences of efficiency scores. Sample 1 is denoted as the “Control group” and sample 2 is denoted as the “Virtual group”.
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CCP efficiency evaluation |
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The
There can be more than one way of ranking efficient units. In the simplest (or naïve) case, empirically efficient DMUs can be ranked according to the score
Table
“Naïve” ranking of empirically efficient DMUs in order of declining levels of efficiency. Values in bold correspond to DEA efficient units with a score of “1”.
Rank |
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Simulation 1 |
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DMU6 | 0.896 | |
DMU3 | 0.883 | |
DMU7 | 0.874 | |
DMU13 | 0.867 | |
DMU10 | 0.779 | |
DMU12 | 0.742 | |
DMU8 | 0.712 | |
DMU5 | 0.664 | |
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Simulation 2 | DMU10 | 1.022 |
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DMU7 | 0.947 | |
DMU13 | 0.944 | |
DMU11 | 0.906 | |
DMU9 | 0.843 | |
DMU6 | 0.837 | |
DMU5 | 0.812 | |
DMU4 | 0.594 | |
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Simulation 3 |
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DMU4 | 0.877 | |
DMU8 | 0.87 | |
DMU3 | 0.833 | |
DMU12 | 0.803 | |
DMU10 | 0.774 | |
DMU6 | 0.765 | |
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Simulation 4 |
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DMU12 | 0.954 | |
DMU4 | 0.898 | |
DMU10 | 0.812 | |
DMU9 | 0.807 | |
DMU8 | 0.775 | |
DMU5 | 0.726 | |
DMU6 | 0.622 |
This method of ranking is naïve because it ignores the standard deviation, which indicates the robustness of a DMU's efficiency score to the possible errors and the
The ranking in the order of robustness of a DMU begins with the efficiency score defined as
Ranking from the most efficient down, those DMUs which have a DEA-Chebyshev model score of
Ranking of efficient DMUs according to robustness based on their standard deviations. The DMUs in bold denote the empirically efficient DMUs.
Simulation 1 | Simulation 2 | Simulation 3 | Simulation 4 | ||||||||
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0.98 | 0.43388 | DMU7 | 0.947 | 0.3218 |
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0.906 | 0.57657 | DMU13 | 0.944 | 0.13124 |
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DMU6 | 0.896 | 0.28298 |
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DMU3 | 0.883 | 0.28164 | DMU11 | 0.906 | 0.17515 |
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0.905 | 0.97149 | DMU12 | 0.969 | 0.17444 |
DMU7 | 0.874 | 0.32605 | DMU10 | 0.89 | 0.34217 | DMU4 | 0.877 | 0.25512 | DMU4 | 0.899 | 0.32244 |
DMU13 | 0.867 | 0.13958 |
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0.874 | 0.85998 | DMU8 | 0.87 | 0.20386 | DMU10 | 0.818 | 0.16313 |
DMU10 | 0.779 | 0.16935 | DMU9 | 0.843 | 0.17572 | DMU3 | 0.833 | 0.21305 | DMU9 | 0.774 | 0.28765 |
DMU12 | 0.742 | 0.11335 | DMU6 | 0.837 | 0.29614 | DMU12 | 0.803 | 0.15726 | DMU8 | 0.754 | 0.28786 |
DMU8 | 0.712 | 0.25534 | DMU5 | 0.812 | 0.2938 | DMU10 | 0.774 | 0.1347 | DMU5 | 0.747 | 0.3701 |
DMU5 | 0.664 | 0.2745 | DMU4 | 0.594 | 0.25039 | DMU6 | 0.765 | 0.24013 | DMU6 | 0.6 | 0.27103 |
Additional analyses were conducted by taking the observed DMUs in each simulation and evaluating them against the EFF, DEA, CCP, and DEA-Chebyshev model results. If DCF is a good approximation of the EFF, then the efficiency scores for the observed DMUs should not be substantially different from the efficiency scores generated by the EFF. This also holds true for CCP.
The efficiency scores of the observed DMUs from the experimental groups determined by the EFF (to be denoted as “exp.grp+EFF”) will provide a benchmark for evaluating the DEA frontier (“exp.grp+DEA”), CCP (normal) frontier (“exp.grp+CCP”), and the corrected frontier (“exp.grp+DCF”). A comparison is drawn between the efficiency scores of the experimental groups generated by the four frontiers.
The hypothesis is that the mean of the efficiency scores for the 15 observed units in the “exp.grp+EFF” group and the “exp.grp+DCF” group should be approximately the same (i.e., the difference is not statistically significant). From Table
Statistical analysis for frontier comparisons. Observed DMUs are evaluated against the 3 different frontiers to determine their efficiency scores which are calculated using the normal DEA model and to determine if the efficiency scores for each group are substantially different when comparing EFF to DEA, EFF to DCF, and EFF to CCP.
Exp.grp+ |
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Table
Outliers have a tendency to exhibit large standard deviations, which is translated to large confidence limits. Consequently, the reason for establishing DCF and CCP scores is to reduce the likelihood of a virtual unit from becoming an outlier. Also, the results generated by the stochastic models (as opposed to deterministic ones) such as the DCF and CCP can be greatly affected because the efficiency scores are generally not restricted to 1.00. In reality, outliers are not always easily detected. If the data set contains some outliers, the stochastic models may not perform well. DMU14 in Simulation 3 is an example of this problem. It can be solved by either removing the outliers or by imposing weight restrictions. However, weight restrictions are not within the scope of this paper.
Traditional methods of performance analysis are no longer sufficient in a fast paced constantly evolving environment. Observing past data alone is not adequate for future projections. The DEA-Chebyshev model is designed to bridge the difference between conventional performance measurements and new techniques to incorporate relevance into such measures. This algorithm not only provides a multidimensional evaluation technique, but it has successfully incorporated a new element into an existing deterministic technique (DEA). This is known as the
The combination of normal DEA with DCF can successfully provide a good framework for evaluation based on quantitative data and qualitative intellectual knowledge of management. When no errors are expected, then standard DEA models will suffice. DCF is designed such that in the absence of errors, the model will revert to a DEA model. This occurs when the
The simulated dataset was tested on DEA-Chebyshev model. It has been statistically proven that this model is an effective tool with excellent accuracy to detect or predict the EFF frontier as a new efficiency benchmarking technique. It is an improvement over other methods, easily applied, practical, not computationally intensive, and easy to implement. The results have been promising thus far. The future work includes using a real data application to illustrate the usefulness of DEA-Chebyshev model.
Note that semi-positive is defined to be the nonpositive characteristics of all data where at least one component in every input and output sector is positive; mathematically, Let Each pair of input Inefficiency: For any semi-positive PPS where Convexity: if Ray unboundedness: if a PPS Any semi-positive linear combination of PPS in
Therefore, satisfying
Definition of
As theoreticians have shown and used in the past the characteristic of errors in efficiency analysis, the simulations generated will incorporate those elements. These characteristics are statistical noise: inefficiency:
Data variability is caused by statistical noise, measurement errors, and inefficiency. There errors can arise from either exogenous or endogenous variables such as poor management, economic growth, and environmental and sociological contributions.
The corrected frontier is defined such that the production possibility space will always be greater than that of the DEA spaces (see if DMUs on the DEA frontier are a subset of those on the DCF frontier; some DEA efficient units will appear inefficient in DCF, in which case, the frontier is shifted away from the PPS (i.e., expansion of the PPS).
Although we do not have formal proof of the convergence of the corrected frontier to the EFF, due to the convergence of the DEA estimator to the EFF,