# Data Envelopment Analysis

Dateneinhüllanalyse (DEA) and data envelopment analysis are terms for a technique for efficiency analysis in the field of Operations Research , which in economics has been widely used. It serves as a comparative measurement of the efficiency of organizational units or decision-making units .

## General description of the DEA

The DEA is traced back to Charnes, Cooper and Rhodes, although there are also earlier uses (Brockhoff 1970). It represents a technique for measuring the relative efficiency of so-called Decision Making Units (DMUs). A decision making unit can be any object that is determined by inputs (e.g. costs, workload in hours) and outputs (e.g. Turnover, quality level) can be characterized. Decision units can e.g. B. Universities, hospitals, bank branches, branches of a trading group or plants of an automobile manufacturer.

All decision units in a group of decision units have the same inputs and outputs. So that the application of the DEA delivers a meaningful result, only decision units that are similar should be taken into account in an application. It should z. B. no hospitals can be compared to universities. With the help of the DEA, the relative efficiency of the decision-making units is measured, since the decision-making units within the group serve as a benchmark.

The DEA enables the user to consider multiple inputs and outputs. These factors are often not comparable with one another (e.g. the turnover measured in money and the level of quality). Therefore, the inputs and the outputs are multiplied with weightings. A special feature of DEA compared to other efficiency analysis techniques is that the weightings of the inputs and outputs are determined within the model. The user does not have to specify this.

To assess the efficiency of the decision units, an efficiency value is calculated for each decision unit. Based on the observed inputs and outputs of a DMU, ​​this efficiency or inefficiency value measures the distance to the efficient edge (data envelope). This efficient edge is formed from the group of decision units that are taken into account in the respective DEA application. From the efficiency value of a decision-making unit, improvement potentials for its management can be derived directly.

## Mathematical classification

When applying the DEA to a group of decision units, an optimization problem must be solved for each decision unit. In its basic form, a DEA model is a problem of quotient programming. Because the efficiency value of a decision unit is a quotient whose numerator contains the sum of the weighted outputs and whose denominator contains the sum of the weighted inputs.

${\ displaystyle E_ {i} = {\ frac {\ sum _ {j = 1} ^ {J} w_ {ji} \ cdot y_ {ji}} {\ sum _ {k = 1} ^ {K} v_ { ki} \ cdot x_ {ki}}}}$ ${\ displaystyle \ forall i = 1, ..., I}$ ${\ displaystyle E_ {i}}$ : Efficiency values : Outputs : Inputs : Output weights : Input weights
${\ displaystyle y_ {ji}}$ ${\ displaystyle x_ {ki}}$ ${\ displaystyle w_ {ji}}$ ${\ displaystyle v_ {ki}}$ Solving a quotient programming problem is not easy because the objective function is not linear. Therefore the problem is converted into a linear programming problem with the help of the so-called Charnes-Cooper transformation.

Each DEA ​​model can be represented in the envelope form and in the multiplier form. A model in the envelope form can be converted into the multiplier form and vice versa by means of a primal-dual transformation .

## Historical development

Charnes, Cooper and Rhodes developed the basic DEA model in 1978. It was later called the CCR model after the initial letters of its developers. This model assumes constant returns to scale . In 1984, Banker, Charnes, and Cooper introduced the variable returns to scale BCC model.

Window analysis is another development. In this, the efficiency of a decision-making unit is compared with one another in different periods. This allows statements to be made about the efficiency development of decision-making units. In addition, DEA models have emerged that calculate with fuzzy numbers by making use of fuzzy logic approaches .

Data Envelopment Analysis has been used in a wide variety of areas of business since the 1960s. It is traditionally used to assess organizational units, i.e. departments or branches. More recent scientific papers also refer to the applicability to individuals. The Data Envelopment Analysis enables employees to be assessed fairly based on many factors.

Because of the high proportion of efficient decision units in both the CCR and the BCC model, Sexton et al. Cross-efficiency analysis proposed, in which a second optimization criterion is used in addition to the original DEA model. Depending on the implementation, each decision unit evaluates all other decision units with its optimal factor weights. The average of all these external evaluations then results in the cross-efficiency of the decision-making units and usually achieves a clear ranking and unique factor weightings.

## literature

• M. Afsharian (2019): A Metafrontier-based Yardstick Competition Mechanism for Incentivising Units in Centrally Managed Multi-group Organizations . Annals of Operations Research.
• M. Afsharian (2017): Metafrontier Efficiency Analysis with Convex and Non-convex Metatechnologies by Stochastic Nonparametric Envelopment of Data . Economics Letters, 160, 1-3.
• H. Ahn (2014): Data Envelopment Analysis - More than Benchmarking . Controller Magazin 39, Vol, pp. 63-65.
• RD Banker, A. Charnes, WW Cooper (1984): Some Models for Estimating Technical and Scale Inefficiency in Data Envelopment Analysis . Management Science, Vol. 30, No. 9, pp. 1078-1092.
• K. Brockhoff (1970): On the Quantification of MArginal Productivity of Industrial Research by Estimating a Production <Function for a Single Firm . German Economic Review, vol. 8, pp. 202-229.
• A. Charnes, W. Cooper, E. Rhodes (1978): Measuring the efficiency of decision making units . European Journal of Operational Research, Vol. 2, No. 6, pp. 429-444.
• U. Cantner , H. Hanusch, H. (1998): Efficiency Analysis Using Data Envelopment Analysis . Economics studies, Volume 27, Issue 5, pp. 228–237.
• H. Dyckhoff, K. Allen (1999): Theoretical justification of an efficiency analysis using Data Envelopment Analysis (DEA) . Schmalenbach's Journal for Business Research, Volume 51, Issue 5, pp. 411–436.
• WW Cooper, LM Seiford, K. Tone (2000): Data Envelopment Analysis . Boston / Dordrecht / London 2000.
• S. Hülsmann, ML Peters: Data Envelopment Analysis in the Banking Industry - Theory and Practical Application . Saarbrücken 2007. ISBN 978-3-8364-0109-8
• M. Richter, E. Borsch (2017): Efficiency Measurement Using Data Envelopment Analysis - Production Theory Basics and the Example of Student Learning at Universities . WiSt - Business Studies, vol. 46, no. 5, 2017, pp. 20–26.
• H. Scheel (2000): Efficiency measures of the Data Envelopment Analysis . Wiesbaden 2000.
• A. Kleine (2002): DEA efficiency, decision making and production theory basics of data envelope analysis . Wiesbaden 2002.
• TJ Coelli (2006): An Introduction to Efficiency and Productivity Analysis . Springer 2006
• WW Cooper (2006): Introduction to Data Envelopment Analysis and Its Uses . Springer 2006.
• T.Wenk (2006): Performance Measurement Systems and their use as a management system . Shaker 2006. ISBN 3-8322-4901-X

## Individual evidence

1. See: Boles, JS, Donthu, N. and Lohtia, R., 1995. Salesperson Evaluation Using Relative Performance Efficiency: The Application of Data Envelopment Analysis. Journal of Personal Selling & Sales Management, 15 (3), pp. 31–49 and “How are they doing? The Application of Data Envelopment Analysis to Develop a Strategic Performance Appraisal on Individual Level ", M. Koch-Rogge, G. Westermann, Ch. Wilbert, R. Willis in Ron Sanchez, Aimé Heene (ed.) A Focused Issue on Building New Competences in Dynamic Environments (Research in Competence-Based Management, Volume 7) Emerald Group Publishing Limited, pp.193-218, September 2014. See also the article with examples and a comparison of DEA, Balanced Scorecard and Simple Ranking: “Objective and fair employee appraisal: doctoral candidate develops new procedure ” .
2. Thomas R. Sexton, Richard H. Silkman, Andrew J. Hogan: Data envelopment analysis: Critique and extension . In: New Directions for Program Evaluation . No. 32 , 1986, pp. 73-105 .