Understanding Data Envelopment Analysis (DEA): Principles, Models, and Applications

Principles and Theory of Data Envelopment Analysis

A decision‑making unit (DMU) represents any economic system or production process that transforms a set of inputs into outputs, aiming to maximize benefit; DMUs can be universities, firms, countries, or the same unit at different times.

When evaluating DMUs, the basis is the input and output data, which provides a measure of relative effectiveness.

(1) Effectiveness is relative, based on mutual comparison.

(2) Each DMU’s effectiveness depends on the ratio of aggregated inputs to aggregated outputs.

Data Envelopment Analysis (DEA), introduced by Charnes, Cooper and others, uses the concept of relative efficiency to assess multiple‑input‑multiple‑output units via linear‑programming models, offering an objective, weight‑free evaluation.

(1) DEA treats input‑output weights as variables, selecting the most favorable weights for each DMU, thus avoiding subjective weighting.

(2) It does not require an explicit functional relationship between inputs and outputs.

The main advantage of DEA is that it derives optimal weights from actual data, eliminating many subjective factors and providing strong objectivity.

DEA is built on relative efficiency, convex analysis, and linear programming; since the first model in 1978, it has been refined and widely applied in sectors such as education, healthcare, and cultural services.

DEA Models and Procedure

Model Introduction

Evaluation of DMUs relies on a set of input indicators and a set of output indicators; inputs represent resources consumed, outputs represent results achieved.

The basic DEA model considers a DMU with input vector x and output vector y . For multiple DMUs, each has its own input and output vectors.

Because inputs and outputs have different roles, DEA aggregates them into a single composite input and output by assigning appropriate weights.

Weights are initially unknown and are determined by solving an optimization problem that maximizes each DMU’s efficiency score under the constraint that no DMU can exceed a score of one.

The resulting linear‑programming formulation (after the Charnes‑Cooper transformation) yields the optimal efficiency score for each DMU.

Dual Model

Linear programming duality provides an alternative representation that is often easier to interpret economically.

The dual model introduces slack variables to convert inequality constraints into equalities.

Theorem 1: Both the primal and dual linear programs have feasible solutions and thus optimal values; if the optimal values are equal, the solutions correspond.

Definition 1: A DMU is weakly DEA‑efficient if the optimal value of the primal equals one.

Definition 2: A DMU is DEA‑efficient if the primal optimal value equals one and all slack variables are zero.

Weak DEA efficiency indicates that a DMU meets the basic condition for efficiency, while DEA efficiency implies that all inputs and outputs contribute indispensably.

Theorem 2: (1) Weak DEA efficiency is equivalent to the primal optimal value being one. (2) DEA efficiency requires the primal optimal value to be one and all slack variables to be zero.

DEA also distinguishes scale returns:

(1) If a certain condition holds, the DMU exhibits constant returns to scale.

(2) If the condition does not hold but another inequality is satisfied, the DMU shows increasing returns to scale.

(3) If neither condition holds, the DMU exhibits decreasing returns to scale.

Testing DEA efficiency can be done via the primal or dual linear program; specialized models simplify this assessment and quantify the gap between inefficient and efficient DMUs.

Further extensions of the basic DEA model address various practical complexities, but they are beyond the scope of this overview.

Reference

Modern Comprehensive Evaluation Methods and Selected Cases, Du Dong, Pang Dahua, Wu Yan (eds.), 3rd edition, Tsinghua University Press, 2015.