Data envelopment analysis

Building on the ideas of Farrell (1957),[5] the seminal work "Measuring the efficiency of decision making units" by Charnes, Cooper & Rhodes (1978)[1] 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. Since then, there have been a large number of books and journal articles written on DEA or applying DEA on various sets of problems.

Starting at the CCR model by Charnes, Cooper and Rhodes,[1] many extensions to DEA have been proposed in the literature. They range from adapting implicit model assumption such as input and output orientation, distinguishing technical and allocative efficiency,[6] adding limited disposability[7] of inputs/outputs or varying returns-to-scale[8] to techniques that utilize DEA results and extend them for more sophisticated analyses, such as stochastic DEA[9] or cross-efficiency analysis.[10]

In a one-input, one-output scenario, efficiency is merely the ratio of output over input that can be produced and comparing several entities/DMUs based on it is trivial. However, when adding more inputs or outputs the efficiency computation becomes more complex. Charnes, Cooper, and Rhodes (1978)[1] in their basic DEA model (CCR) define the objective function to find ${\displaystyle DMU_{j}'s}$ efficiency ${\displaystyle (\theta _{j})}$ as:

${\displaystyle \max \quad \theta _{j}={\frac {\sum \limits _{m=1}^{M}y_{m}^{j}u_{m}^{j}}{\sum \limits _{n=1}^{N}x_{n}^{j}v_{n}^{j}}},}$

where the ${\displaystyle DMU_{j}'s}$ known ${\displaystyle M}$ outputs ${\displaystyle y_{1}^{j},...,y_{m}^{j}}$ are multiplied by their respective weights ${\displaystyle u_{1}^{j},...,u_{m}^{j}}$ and divided by the ${\displaystyle N}$ inputs ${\displaystyle x_{1}^{j},...,x_{n}^{j}}$ multiplied by their respective weights ${\displaystyle v_{1}^{j},...,v_{n}^{j}}$.

The efficiency score ${\displaystyle \theta _{j}}$ is sought to be maximized, under the constraints that using those weights on each ${\displaystyle DMU_{k}\quad k=1,...,K}$, no efficiency score exceeds one:

${\displaystyle {\frac {\sum \limits _{m=1}^{M}y_{m}^{k}u_{m}^{j}}{\sum \limits _{n=1}^{N}x_{n}^{k}v_{n}^{j}}}\leq 1\qquad k=1,...,K,}$

and all inputs, outputs and weights have to be non-negative. To allow for linear optimization, one typically constrains either the sum of outputs or sum of inputs to equal a fixed value (typically 1. See later for an example).

Because this optimization problem's dimensionality is equal to the sum of its inputs and outputs, selecting the smallest number of inputs/outputs that collectively, accurately capture the process one attempts to characterize is crucial. Because the production frontier envelopment is done empirically, several guidelines exist on the minimum required number of DMUs for good discriminatory power of the analysis, given homogeneity of the sample. This minimum number of DMUs varies between twice the sum of inputs and outputs (${\displaystyle 2(M+N)}$) and twice the product of inputs and outputs (${\displaystyle 2MN}$).

Some advantages of DEA approach are:

• no need to explicitly specify a mathematical form for the production function
• capable of handling multiple inputs and outputs
• capable of being used with any input-output measurement, although ordinal variables remain tricky
• the sources of inefficiency can be analysed and quantified for every evaluated unit
• using the dual of the optimization problem identifies which DMUs is evaluating itself against which other DMUs

Some of the disadvantages of DEA are:

• results are sensitive to the selection of inputs and outputs
• high efficiency values can be obtained by being truly efficient or having a niche combination of inputs/outputs
• the number of efficient firms on the frontier increases with the number of inputs and output variables
• a DMU's efficiency scores may be obtained by using non-unique combinations of weights on the input and/or output factors

Assume that we have the following data:

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

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

• ${\displaystyle MaxEfficiency:(100u_{1})/(10v_{1}+2v_{2})}$

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

• Efficiency of unit 1: ${\displaystyle (100u_{1})/(10v_{1}+2v_{2})\leq 1}$
• Efficiency of unit 2: ${\textstyle (80u_{1})/(8v_{1}+4v_{2})\leq 1}$
• Efficiency of unit 3: ${\displaystyle (120u_{1})/(12v_{1}+1.5v_{2})\leq 1}$

and non-negativity:

• ${\displaystyle u,v\geq 0}$

A fraction with decision variables in the numerator and denominator is nonlinear. Since we are using a linear programming technique, we need to linearize the formulation, such that the denominator of the objective function is constant (in this case 1), then maximize the numerator.

The new formulation would be:

• OF
  • ${\displaystyle MaxEfficiency:100u_{1}}$
• ST
  • Efficiency of unit 1: ${\displaystyle 100u_{1}-(10v_{1}+2v_{2})\leq 0}$
  • Efficiency of unit 2: ${\textstyle 80u_{1}-(8v_{1}+4v_{2})\leq 0}$
  • Efficiency of unit 3: ${\displaystyle 120u_{1}-(12v_{1}+1.5v_{2})\leq 0}$
  • Denominator of nonlinear OF: ${\displaystyle 10v_{1}+2v_{2}=1}$
  • Non-negativity: ${\displaystyle u,v\geq 0}$

A desire to Improve upon DEA, by reducing its disadvantages or strengthening its advantages has been a major cause for many discoveries in the recent literature. The currently most often DEA-based method to obtain unique efficiency rankings is called cross-efficiency. Originally developed by Sexton et al. in 1986,[10] it found widespread application ever since Doyle and Green's 1994 publication.[11] Cross-efficiency is based on the original DEA results, but implements a secondary objective where each DMU peer-appraises all other DMU's with its own factor weights. The average of these peer-appraisal scores is then used to calculate a DMU's cross-efficiency score. This approach avoids DEA's disadvantages of having multiple efficient DMUs and potentially non-unique weights.[12] Another approach to remedy some of DEA's drawbacks is Stochastic DEA,[9] which synthesizes DEA and SFA.[13]

• Charnes A., W. W. Cooper and E. Rhodes (1978). “Measuring the Efficiency of Decision Making Units.” EJOR 2: 429-444.
• Banker R.D. (1984). "Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis". Management Science. 30 (9): 1078–1092. doi:10.1287/mnsc.30.9.1078.
• Brockhoff K. (1970). "On the Quantification of the Marginal Productivity of Industrial Research by Estimating a Production Function for a Single Firm". German Economic Review. 8: 202–229.
• Charnes A. (1978). "Measuring the efficiency of decision-making units" (PDF). European Journal of Operational Research. 2 (6): 429–444. doi:10.1016/0377-2217(78)90138-8.
• Cook, W.D., Tone, K., and Zhu, J., Data envelopment analysis: Prior to choosing a model, OMEGA, 2014, Vol. 44, 1-4.
• Farrell M.J. (1957). "The Measurement of Productive Efficiency". Journal of the Royal Statistical Society. 120 (3): 253–281. doi:10.2307/2343100. JSTOR 2343100.
• 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.
• Sickles, R., & Zelenyuk, V. (2019). Measurement of Productivity and Efficiency: Theory and Practice. Cambridge: Cambridge University Press. doi:10.1017/9781139565981 https://assets.cambridge.org/97811070/36161/frontmatter/9781107036161_frontmatter.pdf
• Sun S (2002). "Measuring the relative efficiency of police precincts using data envelopment analysis". Socio-Economic Planning Sciences. 36 (1): 51–71. doi:10.1016/s0038-0121(01)00010-6.
• Thanassoulis E (1995). "Assessing police forces in England and Wales using data envelopment analysis". European Journal of Operational Research. 87 (3): 641–657. doi:10.1016/0377-2217(95)00236-7.
• Tofallis C (2001). "Combining two approaches to efficiency assessment". Journal of the Operational Research Society. 52 (11): 1225–1231. doi:10.1057/palgrave.jors.2601231. hdl:2299/917. S2CID 15258094. SSRN 1353122.

• OR Notes by J E Beasley DEA
• , Journal of Productivity Analysis, Kluwer Publishers