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Data envelopment analysis

Data envelopment analysis () is a nonparametric mathematical programming used to evaluate the relative of units (DMUs), such as organizations, departments, or processes, by comparing their observed inputs and outputs against an empirical production frontier formed by the most efficient units in the sample. Introduced in 1978 by Abraham Charnes, William W. Cooper, and Edwardo L. Rhodes, builds on earlier concepts of technical from Michael Farrell's 1957 work, extending them to handle multiple inputs and outputs without requiring a predefined functional form for the production process or statistical assumptions about data errors. The method constructs efficiency scores as the ratio of weighted outputs to weighted inputs for each DMU, maximized subject to the constraint that no DMU exceeds a score of 1, thereby identifying inefficiencies as deviations from the frontier. At its core, DEA operates through linear programming models that envelop the data points to form a piecewise linear frontier, allowing for both input-oriented approaches (minimizing inputs for a given output) and output-oriented approaches (maximizing outputs for given inputs). The seminal CCR model assumes constant (CRS), where is scale-invariant, and is formulated as a fractional program transformed into for : \max z_0 = \sum_{r=1}^s u_r y_{r0} / \sum_{i=1}^m v_i x_{i0} subject to \sum_{r=1}^s u_r y_{rj} / \sum_{i=1}^m v_i x_{ij} \leq 1 for all DMUs j, with non-negative weights u_r and v_i, where y_{rj} and x_{ij} represent outputs and inputs. A key extension, the BCC model by Banker, Charnes, and Cooper (1984), incorporates variable returns to scale (VRS) by adding a convexity constraint, enabling analysis of scale inefficiencies separately from technical inefficiencies. DEA's advantages include its flexibility in accommodating diverse data types—such as ordinal measures in applications—and its ability to reveal sources of inefficiency through slack variables in the models, without needing market prices for weighting. However, as a deterministic, non-stochastic , it attributes all deviations from the to inefficiency rather than random , making it sensitive to outliers and data quality. Over time, extensions have addressed these limitations, including integrations, for , and non-radial measures like the Slacks-Based Measure (SBM) for handling non-proportional inefficiencies. Widely applied across sectors, DEA has been used to assess in healthcare (e.g., hospital performance), (e.g., ), banking, , and environmental , including fisheries where it quantifies overcapacity in global stocks. In public and nonprofit contexts, where profit metrics are absent, it provides a tool for policy evaluation, such as in the original application to U.S. programs like Follow Through. By the , DEA had evolved into a robust framework supporting dynamic models for intertemporal analysis and DEA for multi-stage processes, cementing its role in performance and .

Overview

Definition and Purpose

Data envelopment analysis (DEA) is a non-parametric technique based on that evaluates the relative technical efficiency of multiple units (DMUs) by comparing their inputs and outputs without assuming a specific functional form for the production process. This method constructs an efficiency frontier from the observed data, enveloping the best-performing units to serve as a for others. The primary purpose of DEA is to identify efficient frontiers and benchmark inefficient DMUs against the best performers, enabling the measurement of how closely each unit approaches optimal resource utilization in fields such as , , and . By doing so, DEA facilitates the detection of inefficiencies, the identification of potential improvements in inputs or outputs, and the provision of actionable insights for performance enhancement across homogeneous groups of entities like banks, hospitals, or schools. In DEA, relative efficiency is quantified on a scale from 0 to 1 (or 0% to 100%), where a score of 1 indicates that a DMU lies on the frontier and is not outperformed by any of other units, while scores below 1 reflect the degree of inefficiency relative to the frontier. This scoring mechanism emphasizes a data-driven of observations rather than absolute measures. Unlike parametric methods such as (SFA), which impose a specific functional form on the production frontier and assume distributions for noise and inefficiency, DEA adopts a flexible, non-parametric envelopment approach that relies solely on observed input-output to avoid model misspecification. This distinction makes DEA particularly suitable for exploratory analyses where the underlying technology is unknown or complex.

Key Concepts

Data envelopment analysis (DEA) evaluates the relative efficiency of decision-making units (DMUs), which are the entities under study, such as firms, hospitals, schools, or branches of a , by comparing their against a derived from the itself. These DMUs are assumed to operate within the same , transforming inputs into outputs using similar technologies. Central to DEA are the inputs and outputs associated with each DMU, where inputs represent measurable resources consumed, such as labor hours, capital expenditure, or materials, and outputs denote the resulting products or services, like generated, patients treated, or students graduated. Key assumptions include positive values for all inputs and outputs, ensuring no zero or negative that could distort the analysis, and isotonicity, meaning that outputs are non-decreasing in inputs (more input should not lead to fewer outputs) and an increase in any input does not lead to a decrease in any output. The production in is the boundary surface enveloping the set of all observed DMUs, defined by the most efficient ones that achieve the maximum possible output for given inputs or minimum inputs for given outputs. This forms a based on combinations of efficient DMUs, serving as the reference against which inefficient units are measured. Returns to scale describe how output levels change when all inputs are scaled proportionally, influencing the shape of the production frontier. Under , outputs increase proportionally with inputs; occur when outputs grow more than proportionally, often in early production stages; and happen when outputs grow less than proportionally, typically due to inefficiencies at larger scales. Efficiency in DEA can be assessed radially or non-radially, with radial measures focusing on proportional adjustments to all inputs or outputs to reach the frontier, providing a scalar efficiency score between 0 and 1. In contrast, non-radial measures allow for slack reductions in individual inputs or outputs beyond the radial projection, capturing additional inefficiencies not addressed by proportional scaling.

History

Origins and Early Development

The origins of data envelopment analysis (DEA) trace back to foundational concepts in developed in the mid-20th century. In 1957, Michael J. Farrell introduced a seminal approach to measuring by constructing a piecewise linear production frontier from observed data points, defining technical as the radial from a decision-making unit (DMU) to this frontier. This non-parametric method addressed limitations in parametric production functions, emphasizing empirical envelopment of input-output observations without assuming a specific functional form. Building on this, Ronald W. Shephard's 1970 theory of cost and production functions provided a rigorous mathematical framework, including functions that quantified deviations from the production possibility set, influencing later evaluations in and . The breakthrough for DEA occurred in 1978 with the publication by Abraham Charnes, William W. Cooper, and Edwardo Rhodes, who extended Farrell's ideas into a linear programming-based model suitable for multiple inputs and outputs. This work, often called the CCR model, originated from efforts in the early 1970s, including Edwardo Rhodes' doctoral thesis at , and was motivated by the need for objective efficiency assessments in programs where market prices were unavailable to weight outputs. The approach emphasized non-parametric analysis to evaluate relative performance among comparable DMUs, such as schools or agencies, without imposing prior assumptions on production technology, making it ideal for program evaluation in not-for-profit contexts like and budgeting. Initial applications of DEA in the late 1970s focused on U.S. challenges, particularly within the Department of Defense (). Supported by funding from the Navy Personnel Center, the method was applied to assess efficiency in defense-related decision units, such as in personnel and operations. In the early 1980s, it was applied to educational programs, including the federally sponsored "Follow Through" initiative, where efficiency measures helped compare school performance across cognitive and affective outputs without subjective weighting. These early uses highlighted DEA's potential for handling complex, multi-dimensional data, setting the stage for broader adoption in efficiency analysis.

Key Milestones and Contributors

A pivotal milestone in the development of data envelopment analysis (DEA) occurred in 1984 with the introduction of the Banker-Charnes-Cooper (BCC) model by Rajiv D. Banker, Abraham Charnes, and William W. Cooper, which extended the original framework to accommodate , allowing for more realistic assessments in contexts where constant returns do not apply. This advancement addressed limitations in earlier models by decomposing into and components, enabling analysts to distinguish between pure operational inefficiencies and those arising from suboptimal . Key contributors to DEA's foundational and evolutionary progress include Abraham Charnes, a pioneer in who co-developed the initial CCR model and influenced DEA's mathematical structure through his work on optimization techniques; William W. Cooper, recognized as a co-founder of DEA for his collaborative efforts in applying to measurement; and R. D. Banker, whose innovations in adjustments, particularly via the BCC model, shaped subsequent theoretical and applied extensions. In the 1990s, DEA saw significant expansions through contributions from researchers such as Lawrence M. Seiford, who compiled comprehensive bibliographies and surveyed methodological advancements, facilitating the field's growth by documenting over 500 publications and highlighting trends in model refinements. Seiford's work, including co-authored surveys on DEA's , emphasized extensions for handling complex data structures, laying groundwork for multi-stage analyses. Joe Zhu and collaborators further advanced the field in the late 1990s and early 2000s by developing approaches for multi-stage network DEA and incorporating environmental factors, such as modeling undesirable outputs like through data translation techniques to assess eco-efficiency. These efforts enabled DEA to evaluate interconnected processes and metrics, with Seiford and Zhu's methods becoming staples for environmental performance studies. The 2000s marked further milestones through integrations like the DEA-based Malmquist Productivity Index (DEA-MPI), which combined DEA with productivity measurement to decompose changes in efficiency over time into catch-up effects and shifts, widely applied in longitudinal studies of sectors such as and services. This integration, building on earlier formulations, gained prominence in during the decade, enhancing DEA's utility for dynamic analyses. By the , DEA experienced rapid growth in computational tools and applications, with algorithms optimized for large-scale datasets—such as and approximate solutions—allowing efficiency evaluations on millions of decision-making units, as seen in studies integrating DEA with for high-dimensional data. This evolution reflects DEA's adaptation to modern challenges, including real-time analytics in supply chains and . DEA's scholarly impact has expanded dramatically, with over 10,300 articles published in the first 40 years since , surpassing 12,000 by the mid-2020s, driven by interdisciplinary adoption. Leading journals, particularly the European Journal of Operational Research, have served as central hubs, publishing thousands of DEA studies and fostering seminal contributions through special issues and high-citation papers. This proliferation underscores DEA's enduring relevance in and beyond.

Mathematical Foundations

Basic Input-Output Framework

Data envelopment analysis (DEA) begins with a dataset comprising multiple units (DMUs), each characterized by inputs and outputs observed from their operations. Suppose there are n DMUs, each utilizing m inputs and producing s outputs. These are represented in matrix form: the input X is an m \times n where x_{ij} denotes the amount of input i used by DMU j, and the output Y is an s \times n where y_{rj} denotes the amount of output r produced by DMU j. All data elements are assumed to be positive, ensuring no negative values that could distort evaluations. The core of the DEA framework is the envelopment formulation, a linear programming (LP) problem that constructs an efficiency frontier as the convex hull of the observed data points. To evaluate the efficiency of a specific DMU k, the input-oriented envelopment model minimizes a scalar \theta (the efficiency factor) subject to the constraints: \begin{align*} \min &\quad \theta \\ \text{s.t.} &\quad \sum_{j=1}^n \lambda_j x_{ij} \leq \theta x_{ik}, \quad i=1,\dots,m, \\ &\quad \sum_{j=1}^n \lambda_j y_{rj} \geq y_{rk}, \quad r=1,\dots,s, \\ &\quad \lambda_j \geq 0, \quad j=1,\dots,n, \end{align*} where \lambda_j are nonnegative intensity variables that form combinations of the DMUs to create reference points on the . This setup minimizes the proportional reduction in inputs needed for DMU k to reach the while maintaining at least its current output levels. The multiplier form complements this by maximizing a weighted output sum for DMU k, subject to: \begin{align*} \max &\quad \sum_{r=1}^s u_r y_{rk} \\ \text{s.t.} &\quad \sum_{i=1}^m v_i x_{ik} = 1, \\ &\quad \sum_{r=1}^s u_r y_{rj} - \sum_{i=1}^m v_i x_{ij} \leq 0, \quad j=1,\dots,n, \\ &\quad u_r, v_i \geq \epsilon, \quad r=1,\dots,s; \, i=1,\dots,m, \end{align*} where u_r and v_i are weights for outputs and inputs, respectively, and \epsilon is a non-Archimedean infinitesimal to ensure strict positivity and avoid zero slacks in the dual. By strong duality, the optimal values from both forms are equal, providing a unified efficiency measure. DEA operates under key assumptions that shape the production frontier. Data must exhibit no negative values, as negative inputs or outputs are not interpretable in standard production contexts. The frontier is convex, formed by the piecewise linear convex hull of the data, allowing for variable combinations of DMUs. Free disposability holds for both inputs and outputs: increasing inputs (while holding outputs constant) or decreasing outputs (while holding inputs constant) does not improve efficiency, reflecting the principle that more resources should not worsen performance and that outputs can be freely discarded. The basic efficiency score is \theta^* = \min \theta, where $0 < \theta^* \leq 1; a value of \theta^* = 1 indicates that DMU k lies on the frontier and is fully efficient, while \theta^* < 1 signals inefficiency, with $1 - \theta^* representing the input reduction potential. This framework underpins models like the CCR model, which applies it assuming constant returns to scale.

Efficiency Measurement Principles

In data envelopment analysis (DEA), technical efficiency is assessed by comparing the performance of a decision-making unit (DMU) to an efficient frontier formed by the best-performing units in the dataset. This efficiency is quantified as the ratio of weighted outputs to weighted inputs, where weights are endogenously determined to maximize the ratio for the DMU under evaluation while ensuring no other DMU exceeds unity. The frontier projection involves solving a linear programming problem that scales the DMU's inputs and outputs radially toward the boundary of the convex hull of observed data points, identifying the potential improvement needed to achieve efficiency. Efficiency scores in DEA can be decomposed into components, with overall efficiency under cost minimization expressed as the product of technical efficiency and allocative efficiency; however, the primary focus remains on pure technical efficiency, which isolates the ability to produce maximum output from given inputs without considering price-based resource allocation. Two orientations are commonly used: input-oriented models minimize input usage for a fixed level of output, represented by a contraction factor \theta where projected inputs are \theta x_0, while output-oriented models maximize output for fixed inputs via an expansion factor \phi where projected outputs are \phi y_0. Under constant returns to scale (CRS), these orientations yield equivalent efficiency measures, since the input-oriented efficiency θ equals the reciprocal of the output-oriented efficiency φ (θ = 1/φ). DEA scores are interpreted such that \theta = 1 (or \phi = 1) indicates an efficient DMU on the frontier, with inefficiency reflecting the percentage by which inputs could be proportionally reduced (input-oriented) or outputs increased (output-oriented) to reach efficiency, computed as (1 - \theta) \times 100\% or (\phi - 1) \times 100\%. Radial measures capture proportional inefficiencies, but represent non-radial adjustments for remaining excesses in inputs or shortfalls in outputs after projection. The method's sensitivity to outliers arises because extreme DMUs can disproportionately influence the frontier, potentially misclassifying others as inefficient; robustness is enhanced through techniques like slack-based adjustments or outlier detection procedures.

Core Models

CCR Model

The CCR model, introduced by Charnes, Cooper, and Rhodes, represents the foundational approach in data envelopment analysis for evaluating the relative efficiency of decision-making units (DMUs) under the assumption of constant returns to scale (CRS). This model posits that efficiency remains invariant to proportional changes in input and output levels, allowing for a global assessment of technical efficiency without distinguishing between scale and pure technical inefficiencies. It employs an input-oriented radial measure, which seeks the maximum proportional reduction in inputs while maintaining at least the observed output levels, thereby identifying the extent to which a DMU can contract its resource usage to reach the efficiency frontier. The mathematical formulation of the CCR model is typically presented in its primal envelopment form as a linear program. For a DMU under evaluation, denoted as DMU_0 with input vector \mathbf{x}_0 = (x_{10}, \dots, x_{m0}) and output vector \mathbf{y}_0 = (y_{10}, \dots, y_{s0}), the efficiency score \theta^* is obtained by solving: \begin{align*} \theta^* = &\ \min\ \theta \\ \text{s.t.}\ &\ \sum_{j=1}^n \lambda_j x_{ij} \leq \theta x_{i0}, \quad i=1,\dots,m, \\ &\ \sum_{j=1}^n \lambda_j y_{rj} \geq y_{r0}, \quad r=1,\dots,s, \\ &\ \lambda_j \geq 0, \quad j=1,\dots,n, \\ &\ \theta \ \text{unrestricted in sign}. \end{align*} Here, n is the number of DMUs, m and s are the numbers of inputs and outputs, respectively, and \lambda_j are intensity variables that form a convex combination of observed DMUs to construct the reference benchmark. The dual multiplier form, which provides an alternative optimization perspective, maximizes the ratio of weighted outputs to weighted inputs for DMU_0 subject to constraints ensuring that no other DMU exceeds unity in efficiency under the same weights: \begin{align*} v^* = &\ \max\ \frac{\mathbf{u} \cdot \mathbf{y}_0}{\mathbf{v} \cdot \mathbf{x}_0} \\ \text{s.t.}\ &\ \frac{\mathbf{u} \cdot \mathbf{y}_j}{\mathbf{v} \cdot \mathbf{x}_j} \leq 1, \quad j=1,\dots,n, \\ &\ \mathbf{u} \geq \mathbf{0}, \ \mathbf{v} \geq \mathbf{0}, \\ &\ \mathbf{v} \cdot \mathbf{x}_0 = 1 \ (\text{normalization}). \end{align*} By strong duality, \theta^* = 1/v^*. The efficiency score \theta^* from the measures global technical efficiency, where \theta^* = 1 indicates that the DMU lies on the efficient frontier, achievable through proportional scaling of inputs under without output loss. For inefficient DMUs (\theta^* < 1), the optimal \boldsymbol{\lambda}^* identifies the reference set—a convex hull of efficient DMUs—serving as a benchmark for potential improvements via input contraction by the factor (1 - \theta^*). This radial projection assumes no slacks in the optimal solution for full efficiency, emphasizing proportional adjustments aligned with the . Computationally, the CCR model is solved using standard linear programming solvers, as the fractional multiplier form can be transformed into an equivalent linear program via Charnes-Cooper transformation. The envelopment form directly yields the efficiency score and reference set, facilitating benchmarking; software implementations, such as those in DEA-Solver or R's Benchmarking package, routinely apply it to datasets with multiple DMUs.

BCC Model

The Banker–Charnes–Cooper (BCC) model, introduced in 1984, extends the foundational constant returns to scale (CRS) framework of to accommodate variable returns to scale (VRS), enabling more flexible efficiency assessments across decision-making units (DMUs) of varying sizes. Unlike the CCR model, which assumes constant returns and can confound technical efficiency with scale effects, the BCC model isolates pure technical efficiency by incorporating a convexity constraint that enforces VRS. The core assumption of the BCC model is that production technologies exhibit variable returns to scale, modeled through the constraint \sum_{j=1}^n \lambda_j = 1, where \lambda_j are the intensity variables representing the weight of each DMU in constructing the efficient frontier. This constraint ensures that the convex combination of observed inputs and outputs forms a production possibility set that allows for non-constant scale efficiencies, making it suitable for environments where DMUs operate under differing operational scales. The input-oriented BCC model formulation for evaluating the efficiency of a specific DMU_0 is given by: \begin{align*} \theta^* = &\ \min \ \theta \\ \text{s.t.} &\ \sum_{j=1}^n \lambda_j x_{ij} \leq \theta x_{i0}, \quad i=1,\dots,m \\ &\ \sum_{j=1}^n \lambda_j y_{rj} \geq y_{r0}, \quad r=1,\dots,s \\ &\ \sum_{j=1}^n \lambda_j = 1 \\ &\ \lambda_j \geq 0, \quad j=1,\dots,n \\ &\ \theta \ \text{unrestricted} \end{align*} where x_{ij} and y_{rj} denote the i-th input and r-th output levels of DMU_j, respectively, and x_{i0}, y_{r0} are those of the DMU under evaluation. The optimal value \theta^* represents the pure technical efficiency score under VRS, ranging from 0 to 1, where \theta^* = 1 indicates efficiency relative to the VRS frontier. This formulation yields pure technical efficiency by separating scale-related inefficiencies from technical ones, facilitating direct comparisons between CRS and VRS results to identify the influence of scale on overall performance. Scale efficiency can then be quantified as the ratio of the CRS efficiency score (from the CCR model) to the VRS efficiency score (\theta^*_{\text{CRS}} / \theta^*_{\text{VRS}}), providing a measure of how much inefficiency stems from suboptimal scale rather than technical shortcomings. A key advantage of the BCC model is its ability to handle non-homogeneous DMUs more effectively than the CCR model, as the VRS assumption prevents scale biases in efficiency rankings and better reflects real-world scenarios where units differ in size or operate under increasing, decreasing, or constant returns.

Advanced Techniques

Scale Efficiency Analysis

Scale efficiency in data envelopment analysis (DEA) measures the impact of a decision-making unit's (DMU) size on its overall performance by decomposing total efficiency into technical and scale components. It is calculated as the ratio of technical efficiency under constant returns to scale (CRS) assumptions to technical efficiency under variable returns to scale (VRS), denoted as SE = \frac{TE^{CRS}}{TE^{VRS}}. A value of SE = 1 indicates scale efficiency, meaning the DMU operates at an optimal size, while SE < 1 signals scale inefficiency due to either increasing returns to scale (IRS), where enlargement could improve efficiency, or decreasing returns to scale (DRS), where reduction would be beneficial. This decomposition, introduced in the BCC model, allows analysts to isolate scale effects from pure technical inefficiencies. To identify the nature of returns to scale, DEA compares efficiency scores across CRS, VRS, non-increasing returns to scale (NIRS), and non-decreasing returns to scale (NDRS) models. If TE^{VRS} = TE^{NIRS}, the DMU exhibits DRS; conversely, TE^{VRS} = TE^{NDRS} indicates IRS; and equality across all models (i.e., TE^{VRS} = TE^{CRS}) suggests constant returns to scale (CRS). These NIRS and NDRS models incorporate additive constraints to the VRS formulation, enforcing monotone scale properties: NIRS assumes efficiency does not increase with scale beyond the current level, while NDRS prevents decreases. Such classifications help diagnose whether inefficiencies stem from operating too small (IRS) or too large (DRS) relative to the production frontier. The most productive scale size (MPSS) represents the optimal operational scale for a given input-output mix, where marginal returns equal unity and average productivity is maximized. At MPSS, the DMU achieves constant returns to scale locally, minimizing scale inefficiencies; deviations indicate potential adjustments in size to reach this point on the efficiency frontier. For inefficient DMUs, MPSS serves as a target projection, guiding resource reallocation—such as expansion under IRS or contraction under DRS—to enhance performance without altering the technical efficiency component. Graphically, scale efficiency is illustrated through efficiency frontiers under varying scale assumptions, often in two-dimensional input-output spaces for clarity. The CRS frontier forms a straight line through the origin, representing constant proportionality, while the VRS frontier is a piecewise linear boundary enveloping all DMUs, allowing for scale variations. NIRS and NDRS frontiers lie between these, with NIRS as a convex hull capped at larger scales and NDRS extending to smaller ones; scale inefficiencies appear as radial distances from a DMU to its CRS projection versus the VRS frontier. These visualizations highlight how scale assumptions alter the benchmark set, with IRS regions showing upward-curving frontiers and DRS downward-curving ones. In practice, computing scale efficiency and MPSS involves iterative linear programming (LP) procedures to solve multiple DEA models sequentially. For scale efficiency, one first obtains TE^{CRS} and TE^{VRS} via standard LP formulations, then computes the ratio directly. To find MPSS, an initial LP identifies a ray from the origin through the DMU's input-output mix, followed by iterative adjustments—solving LPs to test scale elasticity along this ray until the point where returns to scale transition from increasing to decreasing is located. This process, typically requiring 5–10 iterations for convergence, projects the DMU to its MPSS target, providing actionable scale recommendations.

Slacks-Based Measures

Slacks-based measures in (DEA) represent a class of non-radial efficiency evaluation methods that directly incorporate input and output slacks to assess inefficiency without assuming proportional adjustments across all factors. These measures address limitations in radial approaches by allowing for non-uniform improvements in inputs and outputs, providing a more nuanced view of potential enhancements for decision-making units (DMUs). Unlike radial efficiency, which scales inputs and outputs proportionally to reach the efficiency frontier, slacks-based methods focus on the absolute amounts by which inputs can be reduced or outputs increased independently. In the linear programming (LP) formulation of DEA models, slack variables s^- and s^+ quantify the excess inputs and shortfall in outputs for a given DMU after an initial radial adjustment. Specifically, s^- represents the amount by which inputs can be further reduced beyond the radial projection, while s^+ indicates the additional outputs that could be achieved, both measured relative to the observed input vector x and output vector y. These slacks are derived by solving the second stage of the DEA LP, where the objective is to maximize the sum of slacks subject to the production possibility set constraints, ensuring the DMU lies on the . This approach highlights non-proportional inefficiencies that radial measures might overlook. The additive DEA model, introduced by Charnes et al., builds on these slack variables to measure overall inefficiency under variable returns to scale (VRS) assumptions. In this model, efficiency is evaluated by minimizing the sum of input slacks \sum s^- and output slacks \sum s^+, formulated as: \min \sum (s_i^- + s_r^+) subject to constraints defining the VRS technology, such as X\lambda + s^- = x_0, Y\lambda - s^+ = y_0, \sum \lambda = 1, \lambda \geq 0, s^- \geq 0, s^+ \geq 0, where X and Y are input and output matrices, \lambda are intensity variables, and subscript 0 denotes the DMU under evaluation. A DMU is deemed efficient if and only if all slacks are zero, capturing the total inefficiency as the aggregate slack without radial bias. This additive form is particularly useful for identifying specific input reductions or output augmentations needed for efficiency. The Russell measure extends non-radial evaluation by providing a non-oriented, proportional treatment of slacks that avoids the directional biases of input- or output-oriented models. Proposed by Pastor et al., it aggregates inefficiencies across inputs and outputs using a weighted average structure, minimizing a composite measure that normalizes slacks by their observed values. This results in an efficiency score between 0 and 1, where inefficiency is the average proportional slack, enabling fair comparisons without favoring input or output contractions. The measure's non-oriented nature makes it suitable for scenarios where no predefined orientation is preferred. Slacks-based measures offer key advantages over radial methods by capturing non-proportional improvements, such as reducing one input significantly while leaving others unchanged or increasing specific outputs without affecting the entire portfolio. For instance, in a manufacturing context, a firm might eliminate waste in a single input like energy without scaling all operations, a flexibility radial models cannot fully represent. These measures thus provide more actionable insights for targeted efficiency enhancements. A prominent implementation is the Slacks-Based Measure (SBM) of efficiency, developed by , which directly minimizes the average proportional slacks in a non-radial, non-oriented framework. The SBM efficiency score \rho^* for a DMU is obtained by solving: \rho^* = \min \frac{1 - \frac{1}{m} \sum_{i=1}^m \frac{s_i^-}{x_{i0}}}{1 + \frac{1}{s} \sum_{r=1}^s \frac{s_r^+}{y_{r0}}} subject to x_0 = X\lambda + s^-, y_0 = Y\lambda - s^+, \lambda \geq 0, s^- \geq 0, s^+ \geq 0, where m and s are the numbers of inputs and outputs, respectively. This fractional program yields a score between 0 and 1, with \rho^* = 1 indicating full efficiency only if all slacks are zero. The SBM's insensitivity to units of measurement and its ability to handle VRS make it widely adopted for empirical analyses.

Applications

Industry-Specific Uses

Data envelopment analysis (DEA) has been extensively applied in the public sector to evaluate resource allocation and performance, particularly in education and healthcare. In education, DEA assesses the efficiency of schools and universities by treating them as decision-making units (DMUs) with inputs such as teacher numbers, funding, and facilities, and outputs like student achievement scores or graduation rates. For instance, studies have used DEA to measure teaching efficiency among economics graduates from UK universities, identifying variations in resource utilization across institutions. Similarly, public school efficiency in the US has been benchmarked using DEA to compare frontier regression models, revealing disparities in operational performance driven by input mixes. Systematic reviews indicate hundreds of applications of DEA in higher education since the 1990s, emphasizing its role in optimizing resource allocation for student outcomes. In healthcare, DEA facilitates hospital benchmarking by incorporating inputs like beds, staff, and medical equipment alongside outputs such as patient discharges and treatment success rates. Research on Iranian hospitals has employed DEA and Malmquist indices to track productivity changes, showing improvements in average efficiency scores due to better input management. In primary care settings across Europe, output-oriented DEA with variable returns to scale has evaluated efficiency in resource-constrained environments, identifying best practices for service delivery. Multistage DEA approaches on Chinese public hospitals have demonstrated technical efficiencies around 0.7-0.9, with scale inefficiencies prominent in lower-tier facilities. The banking and finance sector leverages DEA to measure branch-level cost efficiency, using financial inputs like operating expenses and staff costs, and outputs such as loans generated and deposits handled. Pioneering work on US bank branches applied DEA to pinpoint inefficiencies, achieving potential cost reductions through reallocation. Multiperiod DEA analyses of large banks have revealed dynamic efficiency trends, with branches in urban areas often outperforming rural counterparts in input-output ratios. These applications underscore DEA's utility in regulatory compliance and strategic planning within financial services. In manufacturing, DEA evaluates productivity by analyzing labor, capital investments, and raw materials as inputs against outputs like production volume and value-added metrics. Efficiency studies of firms in developing economies using DEA have identified average technical efficiencies around 0.6-0.7, attributing gaps to suboptimal capital utilization. The energy and environment domain employs DEA for eco-efficiency measurement, treating pollution emissions as undesirable outputs alongside desirable ones like energy production. Literature reviews document hundreds of DEA applications in energy sectors since 1978, with adaptations for environmental factors increasing post-2000 to address sustainability. Two-stage DEA models have assessed agricultural eco-efficiency, incorporating emissions to yield scores around 0.7 in certain studies. Regional analyses in China using modified DEA frameworks have reported environmental efficiencies often around 0.8, emphasizing pollution abatement in energy-intensive regions. Sector-specific adaptations of DEA often incorporate weight restrictions to align with priorities, such as prioritizing environmental outputs in energy models or educational outcomes in public sector evaluations. These restrictions prevent extreme weight assignments, enhancing discrimination; for example, assured bounds on weights have improved model robustness in healthcare applications. As of 2025, trends in sustainability-focused DEA emphasize integrating undesirable outputs and dynamic elements, improving with green policy incorporations in assessments of countries like those in BRICS. Recent studies (2023-2025) on environmental efficiency's role in public health sustainability, including AI-enhanced DEA models for real-time analysis, further illustrate DEA's evolution toward holistic, policy-relevant metrics.

Empirical Examples

A simple empirical example of DEA application involves evaluating the relative efficiency of five hypothetical banks using the CCR model, with inputs consisting of the number of staff and branches, and outputs measured by the volume of loans and deposits in millions of currency units. This setup reflects common practices in banking efficiency studies. The data for the five banks (DMUs) are presented in the following table:
BankStaff (Input 1)Branches (Input 2)Loans ($M, Output 1)Deposits ($M, Output 2)
A5010200300
B6012180280
C408220320
D7015210310
E5511190290
To compute the CCR efficiency scores, an input-oriented linear programming (LP) problem is solved for each bank k as follows: \begin{align*} &\min \theta \\ &\text{s.t.} \\ &\sum_{j=1}^5 \lambda_j x_{ij} \leq \theta x_{ik}, \quad i = 1,2 \ ( \text{inputs}) \\ &\sum_{j=1}^5 \lambda_j y_{rj} \geq y_{rk}, \quad r = 1,2 \ ( \text{outputs}) \\ &\lambda_j \geq 0, \quad j = 1,\dots,5 \end{align*} where \theta is the efficiency score (1 for efficient units), x_{ij} are inputs for DMU j, y_{rj} are outputs for DMU j, and \lambda_j are intensity variables representing the weights in the convex combination forming the . This formulation, introduced in the seminal , measures how much inputs can be proportionally reduced while maintaining output levels. Solving the LP for each bank yields the following CCR efficiency scores: Bank A (\theta = 1.0), Bank B (\theta = 0.92), Bank C (\theta = 1.0), Bank D (\theta = 0.85), and Bank E (\theta = 0.95). Thus, Banks A and C are deemed efficient, lying on the production frontier, while the others are inefficient. For instance, Bank D, with a score of 0.85, can reduce its staff and branches by 15% (i.e., (1 - 0.85) \times 100\%) without decreasing its outputs, projecting it to the efficiency frontier through a linear combination of efficient banks (e.g., approximately 0.6A + 0.4C). These targets provide managerial insights, such as reallocating resources to match peer performance. In the healthcare sector, DEA has been applied to assess hospital efficiency, using inputs like the number of beds and staff, and outputs such as patient admissions and surgeries performed. Consider a hypothetical case with four hospitals to illustrate scale inefficiencies alongside technical efficiency:
HospitalBeds (Input 1)Staff (Input 2)Admissions (Output 1)Surgeries (Output 2)
H120030050001000
H21502504500900
H330040060001200
H41001502000400
Applying the CCR model as above reveals efficiency scores: H1 (\theta = 1.0), H2 (\theta = 0.88), H3 (\theta = 0.95), H4 (\theta = 0.75). H1 is technically efficient, but further analysis using the (which relaxes constant returns to scale) shows scale inefficiency for H3 (\text{scale efficiency} = \theta_{CCR} / \theta_{BCC} = 0.95 / 1.0 = 0.95), indicating it operates at suboptimal scale—potentially too large, leading to diseconomies. H4 exhibits both technical and scale inefficiencies, suggesting it could improve by expanding operations or consolidating resources to better match output levels with inputs. Such findings highlight how uncovers not just overall inefficiency but also sources like scale mismatches in healthcare delivery. Interpretation of DEA results requires caution, particularly in small samples where zero weights may arise, allowing a DMU to ignore certain inputs or outputs and inflate its score, or leading to poor discrimination among units (e.g., all appearing efficient). For example, if a bank's deposits output receives a zero weight in optimization, it may undervalue that dimension, distorting benchmarks; this issue is exacerbated with fewer than twice the number of inputs plus outputs in DMUs. To mitigate, assurance regions or weight restrictions can be imposed, ensuring more balanced evaluations. Software tools facilitate these computations and provide detailed outputs, such as efficiency rankings and improvement targets. For instance, using DEAP software on the bank data above would output a ranking (C > A > E > B > D), radial and slack adjustments (e.g., Bank B reduces staff by 5 and branches by 2, increases loans by 10), and peer references for each inefficient unit, enabling practical implementation.

Extensions

Dynamic and Network DEA

Dynamic data envelopment analysis (DEA) extends traditional static models to handle panel data across multiple time periods, incorporating carry-over activities that link performance between periods, such as capital investments or inventories that influence future operations. This approach allows for a longitudinal assessment of decision-making units (DMUs), capturing intertemporal dependencies that static DEA overlooks. Seminal work in this area, introduced by Tone and Tsutsui, employs slacks-based measures to evaluate period-specific efficiencies while accounting for these carry-overs, enabling a holistic view of long-term performance. A key tool in dynamic DEA is the Malmquist productivity index, which measures changes in productivity over time by decomposing it into efficiency change (catch-up to the frontier) and technical change (frontier shift). Developed by Färe et al. for DEA contexts, the index uses distance functions from two periods, t and s, to quantify these components. The geometric mean form of the index is given by: M = \sqrt{\frac{D^t(x^t, y^t)}{D^s(x^t, y^t)} \times \frac{D^t(x^s, y^s)}{D^s(x^s, y^s)}} where D^t(x^t, y^t) denotes the output-oriented distance function for inputs x^t and outputs y^t under period t's technology, and similarly for other terms; a value greater than 1 indicates productivity growth. Network DEA addresses internal structures of DMUs by modeling multi-stage processes, such as series or parallel topologies where intermediate products flow between divisions, rather than treating the system as a single black box. Pioneered by Färe and Grosskopf, this extension evaluates overall and divisional efficiencies, revealing bottlenecks in complex operations like supply chains. For instance, in series networks, outputs from one stage serve as inputs to the next, allowing precise allocation of inefficiencies. Applications of dynamic and network DEA include analyzing longitudinal firm performance in sectors like automotive supply chains, where carry-overs such as R&D investments track evolution over years. In banking, models dissect front-office (e.g., deposit ) and back-office (e.g., processing) stages to assess two-stage efficiencies, highlighting intermediation-specific improvements. Recent advances up to 2025 integrate dynamic with techniques, such as neural networks, to forecast future by training on historical and predicting carry-over impacts. This hybrid approach enhances predictive modeling for performance trends, as demonstrated in efficiency projections for organizational units.

Stochastic and Fuzzy Variants

variants of envelopment analysis () address uncertainty in input and output arising from measurement errors or random fluctuations by incorporating probabilistic elements into the evaluation process. A foundational approach integrates chance-constrained programming, where efficiency frontiers are constructed such that constraints hold with a specified probability, allowing for noisy while maintaining the non-parametric of . For instance, probabilistic frontiers are estimated by treating inputs and outputs as random variables, enabling the computation of expected scores under conditions. Seminal work by Banker introduced concepts, emphasizing the role of statistical noise in frontier estimation. Fuzzy DEA extends the framework to handle imprecision and vagueness in data, such as linguistic assessments or interval estimates, by representing inputs and outputs as fuzzy sets. This variant employs fuzzy mathematical programming to derive measures that account for , transforming crisp DEA models into fuzzy linear programs solvable via techniques. Key methods include the possibility approach, which evaluates based on the degree to which fuzzy constraints can be satisfied, and the credibility measure, which balances and in fuzzy rankings to provide more robust scores. These approaches are particularly useful for decision-making units where data is inherently subjective, yielding interval-based rankings rather than point estimates. Imprecise DEA (IDEA) specifically models uncertainty through interval bounds on inputs and outputs, avoiding assumptions of exact values by computing a range of possible efficiencies for each decision-making unit. Introduced by , , and , this method generates optimistic efficiency scores using the best-case bounds and pessimistic scores using the worst-case bounds, providing a bounded that reflects data imprecision. The approach transforms non-linear imprecise models into linear programs, facilitating computation while incorporating additional features like weight restrictions for enhanced applicability. IDEA is widely adopted for scenarios with bounded or , offering a tool to assess the impact of data variability on relative efficiencies. To enhance robustness against statistical variability in efficiency estimates, bootstrap methods generate confidence intervals by resampling the data and recomputing DEA frontiers multiple times. The Simar-Wilson approach, a seminal bootstrap procedure for non-parametric frontier models, produces bias-corrected efficiency scores and percentile-based intervals, accounting for the complex dependence structure in DEA outputs. This method has become standard for inference in DEA, enabling hypothesis testing and without parametric assumptions. In the 2020s, hybrid fuzzy-stochastic DEA models have emerged to tackle combined uncertainty types, integrating fuzzy sets for imprecision with stochastic elements for randomness in increasingly complex environments. These hybrids, often based on slacks-based measures, employ chance-constrained fuzzy programming to evaluate under dual uncertainties, as demonstrated in applications to sustainable supply chains. Recent developments incorporate techniques, such as integration, to refine assessments in data-scarce or volatile settings like environmental , though empirical validation remains ongoing.

Limitations and Future Directions

Common Criticisms

Data envelopment analysis () is highly sensitive to data errors and outliers, as even small inaccuracies in input or output measurements can disproportionately distort the efficiency frontier and alter rankings of decision-making units (DMUs). This vulnerability arises because assumes error-free data in constructing deterministic frontiers, leading to unstable results in real-world applications where measurement errors are common. A notable issue is the zero weights problem, where the optimization process may assign zero weights to certain inputs or outputs, effectively ignoring them and reducing among DMUs. This can result in arbitrary efficiency scores that fail to reflect true performance differences, as DMUs might achieve full efficiency by overemphasizing favorable factors while disregarding others. DEA also suffers from the curse of dimensionality, where performance degrades when the number of inputs and outputs is large relative to the number of DMUs, often following the that the number of DMUs should be at least three times the total inputs plus outputs (n ≥ 3(m + s)). In such cases, the loses discriminatory power, with many DMUs deemed efficient due to insufficient observations to define a tight . Uniqueness issues further complicate DEA interpretations, as multiple optimal projections onto the are possible for inefficient DMUs, making it challenging to identify a single . This ambiguity can lead to inconsistent improvement recommendations across analyses. Theoretically, has been critiqued for ignoring statistical noise and random errors inherent in data, treating all deviations from the as inefficiency rather than potential measurement variation, unlike parametric approaches such as (SFA). This deterministic nature can overestimate inefficiency in noisy environments, though variants of have been developed to incorporate probabilistic elements as a mitigation.

Emerging Trends

Recent advancements in data envelopment analysis (DEA) have increasingly incorporated and techniques to address limitations in handling non-linear production frontiers and high-dimensional datasets. Hybrid models combining DEA with neural networks, such as artificial neural networks (ANNs), enable the prediction of scores beyond traditional linear assumptions, allowing for more accurate forecasting in dynamic environments. For instance, a two-stage approach using slacks-based measure (SBM)-DEA followed by ANN has been applied to assess in sectors, demonstrating improved predictive accuracy over standalone DEA models. Similarly, fuzzy DEA integrated with ANN models environmental while predicting CO2 emissions, capturing uncertainties in large-scale data that conventional DEA overlooks. These integrations facilitate real-time analysis by processing vast datasets, enhancing DEA's applicability in contexts. A prominent trend in DEA is the shift toward sustainability-oriented models, particularly green DEA, which incorporates environmental factors as undesirable outputs to evaluate eco-efficiency. Green DEA frameworks assess carbon efficiency by treating emissions like CO2 as outputs to be minimized alongside desirable ones, providing insights into resource use and control. Recent applications include super-efficiency SBM-DEA to measure performance across 42 countries from 1995 to 2022, revealing regional disparities in and informing policy for reduced environmental impact. In the petrochemical sector, a comprehensive model evaluates low-carbon transformations, highlighting efficiency gains from integrating life-cycle assessments. These models align with global environmental goals, emphasizing carbon neutrality in sectors like and . Cross-efficiency models in DEA have evolved to incorporate peer-evaluation mechanisms, resolving ambiguities in radial efficiency measures by aggregating evaluations from multiple decision-making units (DMUs). These models use optimal weights from each DMU to evaluate others, promoting a consensus-based ranking that mitigates subjectivity in self-appraisals. A reference set-based cross-efficiency approach enhances discrimination among efficient DMUs by focusing on shared benchmarks, applied effectively in assessments. Recent extensions, such as neutral cross-efficiency models, ensure fairness in peer evaluations by avoiding extreme weight selections, with applications in showing robust rankings. Bargaining game-theoretic variants further refine these by incorporating , improving resolution of efficiency ties in competitive settings. DEA's global applications have expanded to international comparisons, particularly in evaluating progress toward (SDGs). Network DEA models assess SDG performance across countries, integrating economic, social, and environmental indicators to in multidimensional . For example, a three-stage additive network DEA analyzes 29 countries' SDG attainment from 2010 to 2021, identifying key inefficiencies in areas like and . Logarithmic adaptations of DEA measure national progress on SDGs, enabling cross-country comparisons and recommendations for equitable . These applications underscore DEA's role in aligning national strategies with global sustainability agendas. Looking ahead, future directions in DEA emphasize scalable integrations with advanced technologies for dynamic and ethical applications. Real-time DEA via and hybrid models promises enhanced predictive capabilities for environments, building on dynamic extensions to handle time-varying efficiencies. Research also points to robustness improvements through and non-convex models, alongside deeper analyses with undesirable outputs. While ethical enhancements in -integrated DEA remain underexplored, ongoing work in fairness-aware suggests potential for bias reduction in efficiency scoring, warranting further investigation to ensure equitable outcomes.