Category: Learning modules »

Slacks based measure or SBM analysis in DEA

Slacks based measure (SBM) is a non-radial model to solve the problem in the “additive model” developed by Charnes, Cooper, & Rhodes in 1978. This model can discriminate between efficient and inefficient Decision-Making Units (DMU). However it does not provide an efficiency measure, so a decision maker cannot interpret the performance of the DMU. Therefore “super SBM” model was introduced to determine and rank efficient DMU. The efficiency scores are dimensionless and fall between 0 and 1, thus allowing for the comparison of different DMU in terms of efficiency.

Why SBM model?

SBM model uses “slacks” to show excess input and shortfalls in the output then it directly deals with them by maximizing these slacks. SBM analysis provides an efficiency score which is units-invariant and a monotone function of input slacks and output slacks. To rank efficiency, super SBM was introduced by Tone & Tsutsui (2001b). It is appropriate for evaluating efficiencies when inputs and outputs may change non-proportionally.

Super SBM model can be expanded by the super efficiency to the DEA models. Super efficiency rate refers to the distance between the inputs and outputs of both units. The distance is shown in variable ρ. Assuming that there exist a set of ‘n’  DMU producing the same set of outputs which consume the same set of inputs. Input and output matrix is matrix (X, Y), where (Input) X= (xij) ϵ Rmxn and (Output) Y= (yij) ϵ Rsxn.

Equations of the SBM model

λ is a nonnegative vector in Rn. The vector S ϵ Rm and S+ ϵ Rs, shows an excess input and a short falling output, respectively.  The equation for the SBM model is as follows:

Figure 1: Equation for SBM analysis model

Figure 1: Equation for SBM model

Suppose (ρ*, λ*,s-*, s+*) is the optimal condition for SBM analysis and (x0 , y0) is SBM efficient of DMU. When ρ* = 1, s-*=0 and s+*=0 (or there is no excess input and a short falling output). A super-efficiency model helps in ranking DMU. Below is the formula for super SBM analysis:

Figure 2: Equation for SBM model

Figure 2: Equation for super SBM model

Challenges of super SBM model

One of the major challenges in super SBM model is that it involves solving a complex equation. This becomes more tedious when there are large units of input and outputs. It takes a lot of effort and time to estimate the efficiency of different decision-making units manually. Therefore, it is optimal to use statistical packages like DEA solver, max DEA, and others. They are more popular than manual calculations and other measures of evaluating efficiency.

Interpretation of super SBM analysis results

This module shows the following process in determining efficiency for 10 DMU and ranks them on the basis efficiency by using Super SBM model. It uses MaxDEA software to assess efficiency. MaxDEA 7 basic is available free of cost, however, the ‘pro’ version is a paid software.

Input and output dataset for 10 DMU

OUTPUT

INPUT

DMU

Exports percentage of GDP

Export

Import

tariff

Exchange rates

1 68.12 3,069,559.00 2.27 30.73
2 19.99 3,488,123.37 1.70 1.04
3 29.67 3,696,265.19 5.02 0.64
4 100.63 4,138,413.19 1.35 3.67
5 45.40 4,736,995.93 3.79 0.75
6 75.63 4,984,467.88 1.72 3.15
7 53.88 6,682,944.91 3.96 1,094.85
8 24.50 13,177,694.49 3.04 6.20
9 15.92 13,544,244.95 3.54 97.60
10 13.64 23,869,948.68 2.65 1.00

By using maxDEA software and applying super SBM model, efficiency score for all DMU can be determined  and ranked. DEA-SBM model is non-radial and non-oriented and can deal with inputs and outputs individually. The purpose is to minimize the input and output slacks, resulting in this fractional program.

Efficiency scores and ranking from super SBM model

DMU

Score

Rank

1 0.248082 8
2 0.541691 6
3 0.651521 5
4 1 1
5 1 1
6 0.796988 4
7 0.151898 10
8 0.359254 7
9 0.227246 9
10 1 1

Determining the efficiency score of DMU

The results show that DMU4, DMU5, and DMU6 are the most efficient decision-making units achieving highest efficiency score of 1. The efficiency score of all other DMUs is less than 1 in DEA-SBM models. It reflects these DMUs have input excesses and output shortfalls. This analysis can be used in yearly data. The mean efficiency score for specific DMU over a period of time and year time efficiency can be evaluated too.

References

How to perform unit root test?

Unit root indicates a stochastic trend in the time series. Sometimes it is known as “random walk with drift”. A time series dataset will show a systematic unpredictable pattern if it has the unit root. If a time series dataset has the unit root, the regression result will be unreasonable and provide a spurious result (in which there is large r-squared value even if the data is uncorrelated) and errant behavior (in which t-rations will not follow at- distributions). Therefore it is important to perform unit root test. Read more »

How to perform Johansen cointegration test?

If a series is nonstationary in time series without a constant mean and constant variance, the regression results will be spurious. But regression results can be reliable when a linear combination of non-stationary series (dependent and independent) removes the stochastic trend and produces stationary residuals. Therefore, it is implied that variables are co-integrated. Co-integrated also assumes that there is the occurrence of stochastic non-stationary series, underlying two or more process (p). Read more »

Importance of Granger causality test

Granger causality is a method to examine the causality between two variables in a time series. “Causality” is related to cause and effect notion, although it is not exactly the same. It is a statistical concept which is based on the prediction. If X variable’s Granger causes Y, then past values of X should contain information that helps in predicting Y. Read more »

Understanding random operating curves or ROC analysis

Previous articles in this module on logistic regression and discriminant analysis explained how to know the classification of a group of observations based on some selected variables. In results, the articles predicted a binary classification (in the case of logistic regression) and classified the observations (like student hired or not hired). Receiver Operating Curve (ROC) is an extension of such classifications. Performance of binary classifier system in the case of ROC analysis can be tested. Read more »

Demographic data representation in Nvivo

The previous article explained how to generate queries within the data processed. This article explains the succeeding step of data visualization of results generated for the case research. Data visualization is a technique for data representation in the form of tables, charts and diagrams. This article explains representation of demographic information of the participants (e.g. teachers) based on nodes in Nvivo. Read more »

Data analysis by generating Nvivo coding query

The next step after processing data through coding and creating memos and classifications is analysis. Nvivo coding query eases the understanding of nodes and their interconnections. Given the vast array of nodes generated, researchers find it difficult to connect two nodes. Therefore examining elements and checking if such connection is possible is also challenging. Read more »

Creating and managing Nvivo memo

Memos are notes which can be linked on a project item in Nvivo. They are used to record and maintain elements of a project. They are a part of ‘Sources’ folder and are saved in ‘Memos’ folder. A memo can be made on the entire project or a single part of it. For example, memo can be created for recording notes about a particular interview. Read more »

We are looking for candidates who have completed their master's degree or Ph.D. Click here to know more about our vacancies.