Data envelopment analysis

Data envelopment analysis (DEA) is a nonparametric method in operations research and economics for the estimation of production frontiers.^[1] DEA has been applied in a large range of fields including international banking, economic sustainability, police department operations, and logistical applications^[2][3][4] Additionally, DEA has been used to assess the performance of natural language processing models, and it has found other applications within machine learning.^[5][6][7]

DEA is used to empirically measure productive efficiency of decision-making units (DMUs). Although DEA has a strong link to production theory in economics, the method is also used for benchmarking in operations management, whereby a set of measures is selected to benchmark the performance of manufacturing and service operations.^[8] In benchmarking, the efficient DMUs, as defined by DEA, may not necessarily form a “production frontier”, but rather lead to a “best-practice frontier.”^{[1][9]: 243–285}

In contrast to parametric methods that require the ex-ante specification of a production- or cost-function, non-parametric approaches compare feasible input and output combinations based on the available data only.^[10] DEA, one of the most commonly used non-parametric methods, owes its name to its enveloping property of the dataset's efficient DMUs, where the empirically observed, most efficient DMUs constitute the production frontier against which all DMUs are compared. DEA's popularity stems from its relative lack of assumptions, the ability to benchmark multi-dimensional inputs and outputs as well as its computational ease owing to it being expressable as a linear program, despite its task to calculate efficiency ratios.^[11]

Building on the ideas of Farrell,^[12] the 1978 work "Measuring the efficiency of decision-making units" by Charnes, Cooper & Rhodes^[1] applied linear programming to estimate, for the first time, an empirical, production-technology frontier. In Germany, the procedure had earlier been used 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 about applying DEA to various sets of problems.

Starting with the CCR model, named after Charnes, Cooper, and Rhodes,^[1] many extensions to DEA have been proposed in the literature. They range from adapting implicit model assumptions such as input and output orientation, distinguishing technical and allocative efficiency,^[13] adding limited disposability^[14] of inputs/outputs or varying returns-to-scale^[15] to techniques that utilize DEA results and extend them for more sophisticated analyses, such as stochastic DEA^[16] or cross-efficiency analysis.^[17]

In a one-input, one-output scenario, efficiency is merely the ratio of output over input that can be produced, while 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 (the 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 the 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. And 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 the 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 DMU 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: ${\displaystyle (80u_{1})/(8v_{1}+4v_{2})\leq 1}$
• 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 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,^[17] it found widespread application ever since Doyle and Green's 1994 publication.^[18] 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.^[19] Another approach to remedy some of DEA's drawbacks is Stochastic DEA,^[16] which synthesizes DEA and Stochastic Frontier Analysis (SFA).^[20]

• Charnes, Abraham; Cooper, William Wager; Rhodes, E. (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. Retrieved 27 January 2022.
• Charnes, Abraham; Cooper, William; Lewin, Arie; Seiford, Lawrence (1995). Data Envelopment Analysis: Theory, Methodology, and Applications. Springer Science & Business Media. ISBN 9780792394808.
• Mahmoudi, Amin; Abbasi, Mehdi; Deng, Xiaopeng (2021). "Evaluating the Performance of the Suppliers Using Hybrid DEA-OPA Model: A Sustainable Development Perspective". Group Decision and Negotiation. 31 (2): 335–362. doi:10.1007/s10726-021-09770-x. ISSN 0926-2644. S2CID 254498857.
• Banker, R. D.; Charnes, A.; Cooper, William Wager (September 1984). "Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis" (PDF). Management Science. 30 (9): 1078–1092. doi:10.1287/mnsc.30.9.1078. S2CID 51901687. Retrieved 27 January 2022.
• 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.
• Banker, R. D.; Charnes, A.; Cooper, William Wager (September 1984). "Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis" (PDF). Management Science. 30 (9): 1078–1092. doi:10.1287/mnsc.30.9.1078. S2CID 51901687. Retrieved 27 January 2022.
• Cook, Wade D.; Hababou, Moez; Tuenter, Hans J. H. (November 2000). "Multicomponent Efficiency Measurement and Shared Inputs in Data Envelopment Analysis: An Application to Sales and Service Performance in Bank Branches". Journal of Productivity Analysis. 14 (3): 209–224. doi:10.1023/A:1026598803764. JSTOR 41781515.

• Cook, Wade D.; Tone, Kaoru; Zhu, Joe (April 2014). "Data envelopment analysis: Prior to choosing a model". Omega. 44 (C): 1–4. doi:10.1016/j.omega.2013.09.004.

• Cooper, William Wager; Seiford, Lawrence; Zhu, Joe (2000). "A unified additive model approach for evaluating inefficiency and congestion with associated measures in DEA". Socio-Economic Planning Sciences. 34 (1): 1–25. doi:10.1016/S0038-0121(99)00010-5.
• Cooper, William Wager; Seiford, Lawrence; Zhu, Joe (2000). "A unified additive model approach for evaluating inefficiency and congestion with associated measures in DEA". Socio-Economic Planning Sciences. 34 (1): 1–25. doi:10.1016/S0038-0121(99)00010-5.
• Cooper, William Wager; Seiford, Lawrence M.; Tone, Kaoru (2007). Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-Solver Software (2 ed.). Springer Publishing.
• Cooper, William Wager; Seiford, Lawrence M.; Zhu, Joe, eds. (2011). Handbook on Data Envelopment Analysis. International Series in Operations Research & Management Science. Vol. 164 (2 ed.). Springer Publishing. ISBN 978-1441961501.
• Dyson, R. G.; Allen, R.; Camanho, A. S.; Podinovski, V. V.; Sarrico, C. S.; Shale, E. A. (2001-07-16). "Pitfalls and protocols in DEA". European Journal of Operational Research. Data Envelopment Analysis. 132 (2): 245–259. doi:10.1016/S0377-2217(00)00149-1.
• Doyle, John; Green, Rodney (1994). "Efficiency and Cross-efficiency in DEA: Derivations, Meanings and Uses". Journal of the Operational Research Society. 45 (5): 567–578. doi:10.1057/jors.1994.84. ISSN 0160-5682. S2CID 122161456.
• Emrouznejad, Ali; Banker, Rajiv; Ray, Subhash; Chen, Lei (2016). "Recent Applications of Data Envelopment Analysis". Proceedings of the 14th International Conference on Data Envelopment Analysis. {{cite journal}}: Cite journal requires |journal= (help)
• Farrell, Michael James (1957). "The Measurement of Productive Efficiency". Journal of the Royal Statistical Society. 120 (3): 253–290. doi:10.2307/2343100. JSTOR 2343100.
• Fried, Harold O.; Lovell, C. A. Knox; Schmidt, Shelton S. (2008). The Measurement of Productive Efficiency and Productivity Growth. Oxford University Press. ISBN 978-0-19-804050-7.
• Guerrero, Nadia; Aparicio, Juan; Valero-Carreras, Daniel (2022). "Combining data envelopment analysis and machine learning". Mathematics. 10 (6): 909. doi:10.3390/math10060909.
• 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.
• Olesen, Ole B.; Petersen, Niels Christian (2016). "Stochastic Data Envelopment Analysis—A review". European Journal of Operational Research. 251 (1): 2–21. doi:10.1016/j.ejor.2015.07.058. ISSN 0377-2217.
• Ramanathan, R. (2003). An Introduction to Data Envelopment Analysis: A tool for Performance Measurement. N.Delhi: SAGE Publishing.
• Sexton, Thomas R. (1986). "Data envelopment analysis: Critique and extension". New Directions for Program Evaluation. 1986 (32): 73–105. doi:10.1002/ev.1441.
• Sickles, Robin; Zelenyuk, Valentin (2019). Measurement of Productivity and Efficiency - Theory and Practice (PDF). Cambridge University Press. ISBN 978-1-107-68765-3. Retrieved 27 January 2022.
• Thanassoulis, Emmanuel (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.
• Zhou, Zachary; Zachariah, Alisha; Conathan, Devin; Kline, Jeffery (2022). "Assessing Resource-Performance Trade-off of Natural Language Models using Data Envelopment Analysis". Proceedings of the 3rd Workshop on Evaluation and Comparison of NLP Systems. Association for Computational Linguistics: 11–20. arXiv:2211.01486.
• Koronakos, Gregory; Sotiropoulos, Dionysios (2020). "A Neural Network approach for Non-parametric Performance Assessment". 2020 11th International Conference on Information, Intelligence, Systems and Applications (IISA. IEEE. pp. 1–8. doi:10.1109/IISA50023.2020.9284346. ISBN 978-1-6654-2228-4. S2CID 228097834.

• Sun, Shinn (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. Retrieved 27 January 2022.
• Tofallis, Chris (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. Retrieved 27 January 2022.

• Data Envelopment Analysis official website
• Journal of Productivity Analysis official website