Variable returns to scale in DEA and summary of efficiency and slacks

By Avishek Majumder & Priya Chetty on October 10, 2017

Variable returns to scale (VRS) is a type of frontier scale used in data envelopment analysis (DEA). It helps to estimate efficiencies whether an increase or decrease in input or outputs does not result in a proportional change in the outputs or inputs respectively (Cooper, Seiford, & Zhu, 2011). This method includes both increasing and decreasing returns to scale. Hence, VRS may exhibit increasing, constant and decreasing returns to scale when working in Data Envelopment Analysis Program (DEAP).

Difference between constant returns to scale (CRS) and variable returns to scale (VRS) in DEA

No proportional change for input and output variables (Reddy, 2015). Proportional change for input and output variables.
Based on increasing or decreasing returns to scale (Tsai & Mar Molinero, 2002). Based on constant input or output variable.
Based on model described by Banker, Chames and Cooper and so BCC model or VRS frontier (Benicio & De Mello, 2015; Reddy, 2015). Based on Charnes, Cooper & Rhodes model of DEA and so CCR model or CRS frontier.
In DEAP, VRS frontier model shows technical efficiency difference between VRS and CRS (Kao & Liu, 2011). In DEAP, it shows only one technical efficiency (constant).
Interpretations complicated. Better in making interpretations.
Used only when specifically it is required to check for increasing or decreasing returns (Kao & Liu, 2011). Most commonly used.

Differences between VRS and CRS

Efficiency summary of case banks

Before discussing the interpretations and findings, look into the previous article, where the procedure to extract, apply and execution of data to perform the analysis is discussed. However, this article will only include interpretations and findings and how VRS differs from CRS.

Summary of efficiencies in VRS DEA
Summary of efficiencies

The first image above shows efficiency summary for input oriented VRS DEA and the second image shows summary for output oriented VRS DEA. The above image shows the difference in technical efficiency between CRS and VRS frontier. In contrast to CRS model, the current VRS model showed four banks as efficient, namely 7,8,9 and 10 while CRS showed only two efficient banks.

Furthermore, the scale efficiency is the “unit where the size of operations is optimal so that any modifications on its size will render the unit less efficient” (Kao & Liu, 2011, p.225). Scale efficiency can be achieved by dividing the total efficiency by the technical efficiency. Moreover, scale efficiency shows whether the returns are increasing or decreasing by identifying the banks by  increasing returns to scale and decreasing returns to scale (Banker, Emrouznejad, Lopes, & Almeida, 2012). Similarly, for input oriented VRS DEA, except for bank 7 and 8, all other banks have increasing returns to scale. 7th and 8th banks with scale efficiency presented neither irs or drs. Again only 1st bank shows drs, while 4, 6, 7 and 8 show constant returns to scale.

Interpretation for efficiency summary

Increasing returns to scale occurs when the output increases by a larger proportion than the increase in inputs during the production process (Banker et al., 2004, p.346).

Similarly, “if output increases by less than proportional change in inputs, there are decreasing returns to scale”. Thereby, it can be interpreted that the Total Profits and Loans of the bank has increased by a larger proportion of the given amount of Total Capitals and Deposits and thus they need to increase or decrease its Total Capitals and Deposits to become efficient (Benicio & De Mello, 2015). Similarly, in case of output-oriented, bank 1 has a decreasing return to scale, which means that the values of Total Profits and Loans are lesser than the given values of Total Capitals and Deposits.

Moreover, Kao & Liu, (2011) says that scale efficiency value is also given by:

Scale Efficiency = CRS TE/VRS TE; 
TE = Technical efficiency, 
CRS = constant returns to scale 
VRS = variable returns to scale
Graph showing the banks having technical efficiencies
Graph showing the banks having technical efficiencies

The graph represents both input and output oriented VRS DEA, banks 7, 8, 9 and 10 with technical efficiency of 1. The chart shows both inefficient banks and efficient banks. It also shows the degree of improvement made so as to become efficient. Thus, their returns to scale must improve by effective inputs and outputs so as to become efficient. For example, bank 1 from output oriented VRS DEA, needs to improve its finances so as to achieve a rise of 68.4% to become efficient.

Summary of slacks

Moreover, the findings from the summary of slacks can further be interpreted the same way as it was interpreted for the CRS DEA model. As here, for image A, bank 2 have increasing returns to scale, so bank 2 has scale returns discrepancy of 114575.911 for Deposits. Similarly, from image B, bank 2 shows IRS thus, bank 2 has scale returns discrepancy of 986837.387.

The values differ in VRS DEA, from what was found from the evaluations in CRS DEA. The values found in variable returns to scale DEA are better than CRS. However, it is mainly because the variable in VRS DEA shows which organization has increased outputs or lowered outputs. Thereby defining increased and decreased returns to scale. However, CRSDEA is simple and easy to interpret. Moreover, a majority of the efficiency tests of benchmarking based on CRS frontier (Coelli, 2008).

Interpretations of slacks

Summary of slacks in mrs dea
Summary of slacks


Input oriented slacks in the original values
Input oriented slacks in the original values


Output oriented slacks in the original values
Output oriented slacks in the original values

From the above graph, it is evident that bank 5 (UBI bank) has discrepancies in its values. The Total capitals and Loans which on correcting will become efficient. It also means that this bank will follow the financial trend of another bank (shown in summary of peers) so as to become efficient. Similarly, in figure 3 bank 1 (SBI) also has discrepancies in its Total capital and Deposits and will have to refer to an efficient bank’s finances to become efficient.


  • Banker, R., Emrouznejad, A., Lopes, A. L. M., & Almeida, M. R. de. (2012). Data Envelopment Analysis: Theory and Applications. 10th International Conference on DEA, 1, 1–305.
  • Benicio, J., & De Mello, J. C. S. (2015). Productivity analysis and variable returns of scale: DEA efficiency frontier interpretation. Procedia Computer Science, 55(Itqm), 341–349.
  • Coelli, T. J. (2008). A Guide to DEAP Version 2.1: A Data Envelopment Analysis (Computer) Program. CEPA Working Papers, 1–50. Retrieved from
  • Cooper, W. W., Seiford, L. M., & Zhu, J. (2011). Handbook on Data Envelopment Analysis. In Chapter 1:Data Envelopment Analysis (pp. 1–39).
  • Dr. G. Thirupati Reddy. (2015). Comparison and Correlation Coefficient between CRS and VRS models of OC Mines. International Journal of Ethics in Engineering & Management Education, 2(1), 2348–4748.
  • Kao, C., & Liu, S.-T. (2011). Scale Efficiency Measurement in Data Envelopment Analysis with Interval Data: A Two-Level Programming Approach. Journal of CENTRUM Cathedra: The Business and Economics Research Journal, 4(2), 224–235.


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