Unlock Efficiency with Data Envelopment Analysis (DEA) in Operations Research

When evaluating the efficiency of various entities such as firms, projects, or institutions, a simple output‑to‑input ratio is intuitive but limited, especially with multiple inputs and outputs; Data Envelopment Analysis (DEA) overcomes these limits by automatically assigning optimal weights, providing a more comprehensive and accurate efficiency assessment.

1. Basic Concept of DEA

Consider a set of decision‑making units (DMUs). Each DMU uses several inputs (e.g., labor, capital) to produce several outputs.

\(x_{ij}\) denotes the amount of the \(i\)‑th input for the \(j\)‑th DMU.

\(y_{rj}\) denotes the amount of the \(r\)-th output for the \(j\)‑th DMU.

The goal is to evaluate the relative efficiency of each DMU.

2. DEA Mathematical Model

2.1 Basic Model

To assess the efficiency of DMU \(j\), the following linear‑programming model can be used:

Objective: maximize the weighted sum of outputs. Constraints: the weighted sum of inputs for DMU \(j\) is normalized to 1, and for every DMU the weighted output minus weighted input must be ≤ 0. Weights must be non‑negative.

2.2 Production Frontier (PPF)

In DEA, the production possibility frontier (PPF) represents the set of best‑practice units that achieve the highest possible efficiency. Units on the frontier are considered efficient; those below it are inefficient.

The DEA model identifies which DMUs lie on the frontier and which lie below it, revealing potential improvement opportunities.

2.3 Interpretation of the Model

Objective function: maximizes the weighted output of a DMU; weights reflect the relative importance of each output.

Constraint 1: ensures the weighted sum of inputs for the evaluated DMU equals 1, providing a standardization for comparison.

Constraint 2: guarantees that no other DMU can achieve a weighted output‑minus‑input greater than zero, i.e., no other DMU can be more efficient than the one under evaluation.

Constraint 3: enforces non‑negative weights, preserving the model’s practical meaning.

The DEA model evaluates relative efficiency without requiring an explicit production function, making it suitable for multi‑input, multi‑output contexts.

3. DEA Applications

3.1 Case Study: Hospital Efficiency Assessment

Suppose we evaluate three hospitals. Inputs are the number of doctors and total hospital expenditure; the output is the annual number of patients cured.

Using DEA, we construct a linear‑programming model for each hospital to obtain optimal input and output weights and compute efficiency scores.

3.2 Python Implementation

<code>import pulp

def DEA_evaluation(input_data, output_data, inputs, outputs):
    """
    Evaluate the efficiency of decision‑making units using DEA.

    Parameters:
    - input_data: dict of DMU names to input values.
    - output_data: dict of DMU names to output values.
    - inputs: list of input names.
    - outputs: list of output names.

    Returns:
    A dict mapping DMU names to their efficiency scores.
    """
    n = len(input_data)
    efficiencies = {}
    for dm in input_data.keys():
        prob = pulp.LpProblem("DEA", pulp.LpMaximize)
        v = pulp.LpVariable.dicts("v", inputs, lowBound=0)
        u = pulp.LpVariable.dicts("u", outputs, lowBound=0)
        prob += pulp.lpSum([u[r] * output_data[dm][r] for r in outputs]), "Objective"
        prob += pulp.lpSum([v[i] * input_data[dm][i] for i in inputs]) == 1, "Standardization"
        for dmu in input_data.keys():
            prob += (pulp.lpSum([u[r] * output_data[dmu][r] for r in outputs])
                     - pulp.lpSum([v[i] * input_data[dmu][i] for i in inputs]) <= 0,
                     f"Efficiency_{dmu}")
        prob.solve()
        efficiencies[dm] = pulp.value(prob.objective)
    return efficiencies

# Example data
input_data = {
    'A': {'医生人数': 10, '医院开销': 5},
    'B': {'医生人数': 15, '医院开销': 7},
    'C': {'医生人数': 8,  '医院开销': 4}
}
output_data = {
    'A': {'病人治愈人数': 1000},
    'B': {'病人治愈人数': 1300},
    'C': {'病人治愈人数': 800}
}
inputs = ['医生人数', '医院开销']
outputs = ['病人治愈人数']

DEA_evaluation(input_data, output_data, inputs, outputs)
</code>

Result:

<code>{'A': 1.0, 'B': 0.9285714230000001, 'C': 1.0}</code>

3.3 Result Interpretation

Hospitals A and C lie on the production frontier with efficiency scores of 1, indicating best‑practice performance. Hospital B falls below the frontier with a score of 0.93, showing a 7 % efficiency gap and room for improvement.

DEA’s main limitation is that it measures only relative efficiency and is sensitive to outliers.

4. Variants of the DEA Model

4.1 Weighted DEA

Allows decision‑makers to assign different importance levels to inputs and outputs, reflecting real‑world priorities.

4.2 Network DEA

Divides a DMU into sub‑processes or stages, each with its own inputs and outputs, enabling deeper internal efficiency analysis.

4.3 Dynamic DEA

Extends the static model by evaluating efficiency over multiple time periods, useful for long‑term performance assessment.

4.4 DEA with Undesirable Outputs

Incorporates negative outputs such as pollution or waste, providing a more comprehensive efficiency picture.

4.5 Cross‑Efficiency DEA

Enables comparison of DMUs from different datasets on a common efficiency scale.

Data Envelopment Analysis is a powerful tool for understanding and comparing the efficiency of decision‑making units, helping to identify under‑performers and guide overall efficiency improvements.

References:

Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2(6), 429‑444.

Emrouznejad, A., & Yang, G. L. (2018). A survey and analysis of the first 40 years of scholarly literature in DEA: 1978–2016. Socio‑Economic Planning Sciences, 61, 4‑8.