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[clarification needed]. It is used to empirically measure productive efficiency of decision making units (or DMUs). Non-parametric approaches have the benefit of not assuming a particular functional form/shape for the frontier, however they do not provide a general relationship (equation) relating output and input. There are also parametric approaches which are used for the estimation of production frontiers (see Lovell & Schmidt 1988 for an early survey). These require that the shape of the frontier be guessed beforehand by specifying a particular function relating output to input. One can also combine the relative strengths from each of these approaches in a hybrid method (Tofallis, 2001) where the frontier units are first identified by DEA and then a smooth surface is fitted to these. This allows a best-practice relationship between multiple outputs and multiple inputs to be estimated.

"The framework has been adapted from multi-input, multi-output production functions and applied in many industries. DEA develops a function whose form is determined by the most efficient producers. This method differs from the Ordinary Least Squares (OLS) statistical technique that bases comparisons relative to an average producer. Like Stochastic Frontier Analysis (SFA), DEA identifies a "frontier" on which the relative performance of all utilities in the sample can be compared: DEA benchmarks firms only against the best producers. It can be characterized as an extreme point method that assumes that if a firm can produce a certain level of output utilizing specific input levels, another firm of equal scale should be capable of doing the same. The most efficient producers can form a 'composite producer', allowing the computation of an efficient solution for every level of input or output. Where there is no actual corresponding firm, 'virtual producers' are identified to make comparisons" (Berg 2010)

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 (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. In Germany, the procedure was used earlier to estimate the marginal productivity of R&D and other factors of production (Brockhoff 1970). 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 & Rhodes, 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.

"DEA has been used for both production and cost data. Utilizing the selected variables, such as unit cost and output, DEA software searches for the points with the lowest unit cost for any given output, connecting those points to form the efficiency frontier. Any company not on the frontier is considered inefficient. A numerical coefficient is given to each firm, defining its relative efficiency. Different variables that could be used to establish the efficiency frontier are: number of employees, service quality, environmental safety, and fuel consumption. An early survey of studies of electricity distribution companies identified more than thirty DEA analyses—indicating widespread application of this technique to that network industry. (Jamasb, T. J., Pollitt, M. G. (2001). A number of studies using this technique have been published for water utilities. The main advantage to this method is its ability to accommodate a multiplicity of inputs and outputs. It is also useful because it takes into consideration returns to scale in calculating efficiency, allowing for the concept of increasing or decreasing efficiency based on size and output levels. A drawback of this technique is that model specification and inclusion/exclusion of variables can affect the results." (Berg 2009)

Some of the advantages 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

Some of the disadvantages of DEA are:

• results are sensitive to the selection of inputs and outputs (Berg 2010).
• you cannot test for the best specification (Berg 2010).
• the number of efficient firms on the frontier tends to increase with the number of inputs and output variables (Berg 2010).

Sample Applications and Example

Sample Applications

DEA is commonly applied in the electric utilities sector. For instance a government authority can choose Data Envelope Analysis as their measuring tool to design an individualized regulatory rate for each firm based on their comparative efficiency. The input components would include man-hours, losses, capital (lines and transformers only), and goods and services. The output variables would include number of customers, energy delivered, length of lines, and degree of coastal exposure. (Berg 2010)

DEA is also regularly used to assess the efficiency of public and not-for-profit organizations, e.g. hospitals (Kuntz, Scholtes & Vera 2007; Kuntz & Vera 2007; Vera & Kuntz 2007) or police forces (Thanassoulis 1995; Sun 2002).

Example

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 be larger than 1):

• subject to the efficiency of unit 1: (u1 × 100) / (v1 × 10 + v2 × 2) ≤ 1
• subject to the efficiency of unit 2: (u2 * 80) / (v1 * 8 + v2 * 4) ≤ 1
• subject to the efficiency of unit 3: (u3 * 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: (u2 * 80) - (v1 * 8 + v2 * 4) ≤ 0
• subject to the efficiency of unit 3: (u3 * 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

• Coelli, T.J., D.P. Rao, C.J. O'Donnell, and G.E. Battese, An Introduction to Efficiency and Productivity Analysis, Springer, 2005.
• 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.
• Berg, S. (2010). "Water Utility Benchmarking: Measurement, Methodology, and Performance Incentives." International Water Association.
• Brockhoff, K. (1970). "On the Quatification of the MArginal Productivity of Industrial Research by Estimating a Production Function for a Single Firm", German Economic Review vol. 8, pp. 202-229.
• Emrouznejad A., Barnett R. Parker, Gabriel Tavares (2008) Evaluation of research in efficiency and productivity: A survey and analysis of the first 30 years of scholarly literature in DEA, Socio-Economic Planning Sciences, 42(3):151-157.[1]
• Kuntz L., Vera A. (2007): Modular organization and hospital performance. Health Services Management Research 20(1), 48–58.
• Kuntz, L., S. Scholtes, A. Vera (2007), Incorporating Efficiency in Hospital Capacity Planning in Germany, European Journal of Health Economics 8(3), 213-223.
• 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.
• Sun, S. (2002), Measuring the relative efficiency of police precincts using data envelopment analysis, Socio-Economic Planning Sciences, 36(1), 51-71.
• Thanassoulis, E. (1995): Assessing police forces in England and Wales using data envelopment analysis, European Journal of Operational Research, 87(3), 641-657.
• Vera, A. & Kuntz, L. (2007), Process-based organization design and hospital efficiency, Health Care Management Review, 32(1), 55-65.

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