Unlocking Efficiency: How Data Envelopment Analysis (DEA) Evaluates Decision‑Making Units

Thoughts and Principles of Data Envelopment Analysis

An economic system or production process can be viewed as a unit operating within a feasible range, consuming certain inputs to produce certain outputs. Such units, called Decision‑Making Units (DMUs), aim to maximize benefit. DMUs may represent universities, enterprises, or even countries, and groups of similar DMUs share goals, environments, and input‑output indicators.

Evaluation of DMUs relies on their input and output data to compare relative effectiveness. Effectiveness involves two aspects: (1) it is based on mutual comparison, thus relative; (2) it depends on the ratio of aggregated inputs to aggregated outputs.

Data Envelopment Analysis (DEA), introduced by A. Charnes and W. W. Cooper, uses the concept of relative efficiency to assess multiple‑input, multiple‑output units. DEA computes efficiency scores via mathematical programming, ranks DMUs, identifies inefficient ones, and suggests improvement directions. Its key strengths are handling many inputs/outputs and avoiding preset weight assumptions.

DEA treats input‑output weights as variables, choosing values most favorable to each DMU, thus eliminating subjective weight selection.

DEA does not require explicit functional relationships between inputs and outputs.

DEA’s most notable advantage is that weights are derived endogenously from data, ensuring objectivity.

Since its first model in 1978, DEA has been refined and widely applied in sectors such as education, healthcare, and cultural services, offering a unique advantage for evaluating multi‑input, multi‑output economic systems.

Model and Steps of DEA

Model Introduction

In social, economic, and management contexts, we often need to evaluate the relative efficiency of similar departments, enterprises, or time periods—collectively called DMUs. Evaluation uses a set of input indicators (resources consumed) and output indicators (results produced).

The basic DEA model considers each DMU with an input vector x and an output vector y . For a set of n DMUs, each has m input types and s output types. The model seeks weights for inputs and outputs that maximize each DMU’s efficiency score while ensuring that no DMU exceeds a score of 1.

Because the relationship between inputs and outputs is often unknown, DEA treats the weight vectors as variables and determines them through optimization.

Each DMU obtains an efficiency index θ . By adjusting weights, we can achieve the maximum possible θ for a given DMU, leading to the following linear‑programming formulation (Charnes‑Cooper transformation):

Subject to:

Weighted sum of inputs ≤ 1 for each DMU.

Weighted sum of outputs ≥ θ for the DMU under evaluation.

The solution provides the optimal weights and the efficiency score.

Dual Model

Linear programming has a powerful dual theory. The dual of the DEA model offers economic interpretation and simplifies analysis.

The dual formulation introduces slack variables for inputs and outputs, converting inequality constraints to equalities.

Theorem 1: Both the primal and dual linear programs have feasible solutions and thus optimal values.

Definition 1: If the optimal value of the primal equals 1, the DMU is weakly DEA‑efficient.

Definition 2: If the dual solution contains positive slack variables, the DMU is DEA‑efficient.

Weak DEA efficiency satisfies basic feasibility; DEA efficiency indicates that all inputs and outputs contribute indispensably to the unit’s performance.

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

Economic interpretation: DEA can distinguish technical efficiency (no excess inputs or deficient outputs) and scale efficiency (optimal size). The following conditions apply:

If there exist weights such that both input excess and output shortfall are zero, the DMU is technically and scale efficient.

If some slack variables are positive, the DMU is weakly efficient (either technical or scale inefficiency).

If neither condition holds, the DMU is inefficient.

Scale returns can also be inferred from the optimal values of the model.

Various extensions of the basic DEA model have been developed to address different practical situations, 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.