Data envelopment analysis: Meaning, Criticisms & Real-World Uses

What Is Data Envelopment Analysis?

Data Envelopment Analysis (DEA) is a non-parametric method used to measure the relative efficiency of a set of comparable entities, often referred to as Decision Making Units (DMUs). Falling under the broader category of Performance Measurement within quantitative analysis, DEA assesses how effectively these DMUs convert multiple inputs into multiple outputs. Unlike traditional statistical methods, Data Envelopment Analysis does not require a pre-specified functional form relating inputs to outputs, making it flexible for various applications. It constructs a "frontier" of best practices based on the observed data, against which all other DMUs are evaluated. Those DMUs lying on the frontier are considered efficient, while others are deemed inefficient relative to the best performers.

History and Origin

The foundational work for Data Envelopment Analysis emerged from the concept of Farrell efficiency measures, first introduced by Michael J. Farrell in 1957. Farrell's work laid the groundwork for the non-parametric approach to productivity and efficiency measurement. The formalization of Data Envelopment Analysis as a distinct methodology, however, is attributed to Abraham Charnes, William W. Cooper, and Edwardo Rhodes (CCR) in their seminal 1978 paper, "Measuring the Efficiency of Decision Making Units." This paper introduced the first DEA model, often referred to as the CCR model, which was capable of handling multiple inputs and multiple outputs for evaluating organizational efficiency. The path from Farrell's conceptual insights to the widely recognized CCR model is explored in detail in an academic paper on its origins.⁵ Subsequent developments, such as the Banker, Charnes, and Cooper (BCC) model in 1984, extended DEA to account for returns to scale that are not necessarily constant, further broadening its applicability.

Key Takeaways

Data Envelopment Analysis (DEA) is a non-parametric method for evaluating the relative efficiency of comparable entities.
It identifies a "best practice" frontier based on observed input-output data.
DMUs are assigned an efficiency score between 0 and 1, with 1 indicating full efficiency relative to the frontier.
DEA does not require a pre-defined functional relationship between inputs and outputs.
It can identify sources and magnitudes of inefficiency, indicating areas for improvement.

Formula and Calculation

At its core, Data Envelopment Analysis calculates an efficiency score for each DMU by solving a series of Linear Programming problems. For a given DMU, the objective is to maximize a ratio of weighted outputs to weighted inputs, subject to the constraint that the same ratio for all other DMUs is less than or equal to one. The mathematical representation for the efficiency score ((h_o)) of a specific DMU (o) (often called the "target DMU") is:

\begin{aligned} \text{maximize} \quad & h_o = \frac{\sum_{r=1}^{s} u_r y_{ro}}{\sum_{i=1}^{m} v_i x_{io}} \\ \text{subject to} \quad & \frac{\sum_{r=1}^{s} u_r y_{rj}}{\sum_{i=1}^{m} v_i x_{ij}} \le 1 \quad \text{for } j=1, \ldots, n \\ & u_r, v_i \ge \epsilon \quad \text{for all } r, i \end{aligned}

Where:

(y_{ro}) = amount of output (r) produced by DMU (o)
(x_{io}) = amount of input (i) used by DMU (o)
(u_r) = weight assigned to output (r)
(v_i) = weight assigned to input (i)
(s) = number of outputs
(m) = number of inputs
(n) = total number of DMUs
(\epsilon) = a small non-Archimedean constant (ensures all weights are positive)

This fractional programming problem is typically converted into an equivalent linear programming problem for easier optimization and solution using standard software. The solution provides the optimal weights (u_r) and (v_i) that maximize the efficiency of the target DMU relative to its peers.

Interpreting the Data Envelopment Analysis

The result of a Data Envelopment Analysis is an efficiency score for each DMU, typically ranging from 0 to 1 (or 0% to 100%). A score of 1 (or 100%) indicates that the DMU is operating on the efficient frontier, meaning it is considered efficient relative to the other DMUs in the sample. This DMU is deemed a "best practice" unit and serves as a benchmarking reference point for less efficient units.

DMUs with scores less than 1 are considered inefficient. The score quantifies their relative inefficiency; for instance, a DMU with an efficiency score of 0.80 suggests it is 80% as efficient as its peer group and could potentially produce the same outputs using 20% fewer inputs, or produce 25% more outputs with the same inputs, by adopting the best practices of efficient DMUs. Data Envelopment Analysis also identifies specific "peer" DMUs on the frontier that collectively form the benchmark for an inefficient unit, providing actionable insights for resource allocation and improvement.

Hypothetical Example

Consider a hypothetical scenario involving three small regional banks, Bank A, Bank B, and Bank C. Each bank uses "Number of Employees" as an input and generates "Number of Loans Processed" and "Total Deposits" as outputs.

Bank A: 100 employees, 500 loans, $100M deposits
Bank B: 120 employees, 650 loans, $110M deposits
Bank C: 90 employees, 480 loans, $95M deposits

A Data Envelopment Analysis would evaluate each bank individually:

For Bank A: The DEA model would try to find weights for employees, loans, and deposits that make Bank A look as efficient as possible, while ensuring that no other bank achieves an efficiency score greater than 1 with those same weights.
For Bank B: The same process is repeated. Bank B, despite having more employees, might appear highly efficient if its output levels (loans and deposits) are proportionally higher, pushing it closer to or onto the efficiency frontier.
For Bank C: The model calculates its efficiency relative to A and B. If Bank A can produce similar outputs with fewer employees, or more outputs with the same employees, it might serve as a peer for Bank C, indicating that Bank C is less efficient.

Let's assume the DEA calculation results in:

Bank A: Efficiency Score = 1.00
Bank B: Efficiency Score = 1.00
Bank C: Efficiency Score = 0.85

In this example, Bank A and Bank B are on the efficiency frontier, representing best practices. Bank C is 85% as efficient as its peers. This suggests Bank C could improve its financial performance by examining the operational strategies of Bank A and Bank B to achieve similar outputs with less input or greater outputs with the same input.

Practical Applications

Data Envelopment Analysis is a versatile tool applied across various sectors for assessing efficiency and productivity. In public services, it helps evaluate the performance of schools, hospitals, police departments, and libraries by comparing their output (e.g., student test scores, patient outcomes) against inputs (e.g., teaching staff, medical equipment). For instance, DEA has been used to assess healthcare systems' efficiency in transforming inputs into health outcomes.⁴

In the private sector, DEA assists in analyzing the efficiency of branches within a bank, retail stores in a chain, or manufacturing plants in a corporation. It can be particularly useful in investment management to compare the efficiency of different fund managers or portfolio strategies based on various risk-adjusted returns (outputs) and management fees or capital employed (inputs). Researchers have also utilized DEA to conduct an analysis of efficiency in car companies.³ DEA's ability to handle multiple inputs and outputs simultaneously, without imposing rigid functional forms, makes it valuable for complex organizational assessments where traditional ratios might fall short.

Limitations and Criticisms

While Data Envelopment Analysis offers significant advantages, it also has limitations. A primary criticism is its sensitivity to outliers and measurement errors; since DEA constructs a frontier based on observed data, any inaccuracies or unusual data points can significantly distort the efficiency scores. Unlike econometrics approaches like Stochastic Frontier Analysis, DEA is deterministic, meaning it does not account for statistical noise or random variations in the data. All deviations from the frontier are attributed to inefficiency.

Another limitation is that DEA provides only relative efficiency scores. A DMU might be 100% efficient within the sample, but still inefficient when compared to an unobserved ideal or a larger population. Furthermore, selecting the appropriate inputs and outputs is crucial, as misspecification can lead to misleading results. If a DMU appears highly efficient by manipulating which inputs and outputs are chosen, the analysis loses validity. Academic discussions, such as an academic discussion regarding its critical evaluation,² highlight the ongoing scrutiny and refinements within the DEA methodology, particularly concerning its interpretation of Pareto-Koopmans efficiency and handling of varying conditions. Moreover, DEA's non-parametric nature means it does not yield parameters for a production function that can be easily generalized beyond the observed sample. Applying DEA requires careful consideration of data quality and the context of the DMUs being analyzed, along with potential implications for risk management if decisions are made solely based on its output.

Data Envelopment Analysis vs. Regression Analysis

Data Envelopment Analysis and Regression Analysis are both analytical techniques used to understand relationships between variables, but they differ fundamentally in their approach to measuring performance and efficiency.

Feature	Data Envelopment Analysis (DEA)	Regression Analysis
Methodology	Non-parametric, mathematical programming (e.g., Linear Programming)	Parametric, statistical modeling (e.g., Ordinary Least Squares)
Objective	Measures relative efficiency against a "best practice" frontier	Identifies average relationships and trends
Functional Form	No pre-specified functional form required	Requires a pre-specified functional form (e.g., linear, logarithmic)
Error Handling	Assumes all deviations from frontier are inefficiency; no noise	Accounts for statistical noise/random errors
Output Interpretation	Efficiency scores (0-1); identifies peer benchmarks	Coefficients indicate impact of independent variables on dependent variable
Outlier Sensitivity	Highly sensitive to outliers, which define the frontier	Less sensitive; outliers can be identified and may influence fit

While Data Envelopment Analysis aims to identify best performers and quantify relative inefficiencies by constructing an empirical frontier, Regression Analysis seeks to model the average relationship between inputs and outputs, often with the goal of prediction or understanding the typical behavior of a system. A comprehensive text on Data Envelopment Analysis highlights how various DEA models serve as non-parametric alternatives to traditional econometric models for efficiency measurement.¹

FAQs

Can Data Envelopment Analysis compare any group of entities?

No, Data Envelopment Analysis is designed for comparing homogeneous entities or Decision Making Units (DMUs) that perform similar tasks and operate under similar conditions, using the same types of inputs to produce the same types of outputs. Comparing a hospital to a manufacturing plant, for instance, would be inappropriate.

What does an efficiency score of less than 1 mean in DEA?

An efficiency score less than 1 indicates that a DMU is operating below the efficiency frontier, meaning it is not achieving the maximum possible outputs given its inputs, or it is not using the minimum possible inputs to achieve its outputs, relative to its peers. It implies there is room for improvement by adopting best practices observed among the efficient units.

Is Data Envelopment Analysis suitable for small datasets?

DEA's performance can be affected by the size of the dataset. If the number of DMUs is small relative to the number of inputs and outputs, many DMUs might appear efficient, making it difficult to differentiate performance. A general guideline is that the number of DMUs should be at least three times the sum of the number of inputs and outputs to yield meaningful results for benchmarking purposes.