# Data Envelopment Analysis

Data Envelopment Analysis

Data Envelopment Analysis (DEA) is a nonparametric method in operations research and economics for the estimation of production frontiers. It is used to empirically measure productive efficiency of decision making units (or DMUs). There are also parametric approaches which are used for the estimation of production frontiers (see Lovell & Schmidt 1988 for an early survey).

History

In microeconomic production theory a firm's input and output combinations are depicted using a production function. Using such a function one can show the maximum output which can be achieved with any possible combination of inputs, that is, one can construct a production technology frontier. (Seiford & Thrall 1990). Some 30 years ago DEA analysis (and frontier techniques in general) set out to answer the question of how to use this principle in empirical applications while overcoming the problem that for actual firms (or other DMUs) one can never observe all the possible input-output combinations.

Building on the ideas of Farrell (1957), the seminal work "Measuring the efficiency of decision making units" by Charnes, Cooper & Rhodes (1978) applies linear programming to estimate an empirical production technology frontier for the first time. Since then, there have been a large number of books and journal articles written on DEA or applying DEA on various sets of problems. Other than comparing efficiency across DMUs within an organization, DEA has also been used to compare efficiency across firms. There are several types of DEA with the most basic being CCR based on Charnes, Cooper & Rhoades, however there are also DEA which address varying returns to scale, either CRS (constant returns to scale) or VRS (variable). The main developments of DEA in the 1970s and 1980s are documented by Seiford & Thrall (1990).

Techniques

Data Envelopment Analysis (DEA) is a Linear Programming methodology to measure the efficiency of multiple Decision Making Units (DMUs) when the production process presents a structure of multiple inputs and outputs.

Some of the benefits of DEA are:
* no need to explicitly specify a mathematical form for the production function
* proven to be useful in uncovering relationships that remain hidden for other methodologies
* capable of handling multiple inputs and outputs
* capable of being used with any input-output measurement
* the sources of inefficiency can be analysed and quantified for every evaluated unit

In the DEA methodology, formally developed by Charnes, Cooper and Rhodes (1978), efficiency is defined as a weighted sum of outputs to a weighted sum of inputs, where the weights structure is calculated by means of mathematical programming and constant returns to scale (CRS) are assumed. In 1984, Banker, Charnes and Cooper developed a model with variable returns to scale (VRS).

Assume that we have the following data:

* Unit 1 produces 100 pieces of items per day, and the inputs are 10 dollars of materials and 2 labour-hours
* Unit 2 produces 80 pieces of items per day, and the inputs are 8 dollars of materials and 4 labour-hours
* Unit 3 produces 120 pieces of items per day, and the inputs are 12 dollars of materials and 1.5 labour-hours

To calculate the efficiency of unit 1, we define the objective function as

* maximize Efficiency = (u1 * 100) / (v1 * 10 + v2 * 2)

which is subject to all efficiency of other units (efficiency cannot larger than 1):

* subject to the efficiency of unit 1: (u1 * 100) / (v1 * 10 + v2 * 2) <= 1
* subject to the efficiency of unit 2: (u1 * 80) / (v1 * 8 + v2 * 4) <= 1
* subject to the efficiency of unit 3: (u1 * 120) / (v1 * 12 + v2 * 1.5) <= 1

and non-negativity:

* all u and v >= 0.

But since linear programming cannot handle fraction, we need to transform the formulation, such that we limit the denominator of the objective function and only allow the linear programming to maximize the numerator.

So the new formulation would be:

* maximize Efficiency = u1 * 100
* subject to the efficiency of unit 1: (u1 * 100) - (v1 * 10 + v2 * 2) <= 0
* subject to the efficiency of unit 2: (u1 * 80) - (v1 * 8 + v2 * 4) <= 0
* subject to the efficiency of unit 3: (u1 * 120) - (v1 * 12 + v2 * 1.5) <= 0
* subject to v1 * 10 + v2 * 2 = 1
* all u and v >= 0.

Inefficiency measuring with DEA

Data Envelopment Analysis (DEA) has been recognized as a valuable analytical research instrument and a practical decision support tool. DEA has been credited for not requiring a complete specification for the functional form of the production frontier nor the distribution of inefficient deviations from the frontier. Rather, DEA requires general production and distribution assumptions only. However, if those assumptions are too weak, inefficiency levels may be systematically underestimated in small samples. In addition, erroneous assumptions may cause inconsistency with a bias over the frontier. Therefore, the ability to alter, test and select production assumptions is essential in conducting DEA-based research. However, the DEA models currently available offer a limited variety of alternative production assumptions only.

References

* Banker, R.D., R.F. Charnes, & W.W. Cooper (1984) "Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis, "Management Science" vol. 30, pp. 1078-1092.

* Charnes, A., W. Cooper, & E., Rhodes (1978) "Measuring the efficiency of decision-making units," "European Journal of Operational Research" vol. 2, pp. 429-444.

* Emrouznejad, A. (2001) "An Extensive Bibliography of Data Envelopment Analysis (DEA), Volume I: Working Papers.", Business School, University of Warwick. [http://www.deazone.com/bibliography/index.htm]

* Farrell, J.M. (1957) "The Measurement of Productive Efficiency," "Journal of the Royal Statistical Society" vol. 120, pp. 253-281.

* Lovell, C.A.L., & P. Schmidt (1988) "A Comparison of Alternative Approaches to the Measurement of Productive Efficiency, in Dogramaci, A., & R. Färe (eds.) "Applications of Modern Production Theory: Efficiency and Productivity", Kluwer: Boston.

* Ramanathan, R. (2003) "An Introduction to Data Envelopment Analysis: A tool for Performance Measurement," Sage Publishing.

* Seiford, L.M., & R.M. Thrall (1990) "Recent Developments in DEA: The Mathematical Programming Approach to Frontier Analysis," "Journal of Econometrics" vol. 46: pp. 7-38.

* [http://www.deazone.com DEA Zone] , A comprehensive website on Data Envelopment Analysis

* OR Notes by J E Beasley [http://people.brunel.ac.uk/~mastjjb/jeb/or/dea.html DEA]

* "DEA" [http://finance.kiev.bz Finance.kiev.bz]
* [http://www.deaos.com DEA Online Software] , Solve your DEA models with a professional software
* [http://www.ewepa.org/index.php] ,European Workshop on Efficiency and Productivity Analysis
* [http://www.euro-online.org/display.php?file=wg_info.php&wgid=17&title=EWG-WS-EPA-WS-Efficiency-WS-and-WS-productivity-WS-analysis&parent=303] ,Group on Efficiency and Productivity Analysis that is active within EURO
* [http://www.springerlink.com/content/100296/] ,Journal of Productivity Analysis, Kluwer Publishers

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