Understanding Input‑ and Output‑Oriented DEA Models with Variable Returns to Scale

The efficiency frontier determined by the input‑oriented DEA model represents the case of variable returns to scale; the effective frontier identified by the model can be called the variable returns to scale frontier. Below is a brief introduction to returns to scale and the efficient frontier.

Consider the five decision‑making units (DMUs) shown in the figure, each with a single input and single output. The variable‑returns‑to‑scale efficient frontier consists of DMUs representing increasing returns to scale (IRS), constant returns to scale (CRS), and decreasing returns to scale (DRS). Model (2.2) is an input‑oriented DEA model with VRS; a DMU that lies inside the convex hull of the frontier points is inefficient and should reduce its input to reach the efficiency projection point. In an output‑oriented model, the projection point would involve increasing its output.

The output‑oriented DEA model can be described (when a specific DMU is the object of evaluation) similarly to the input‑oriented version. The boundaries identified by the output‑ and input‑oriented DEA models are identical; the output‑oriented model seeks coefficients that, while keeping the input level unchanged, increase the output, which yields the efficiency score. If the score equals 1, the current output cannot be proportionally increased and the DMU lies on the efficient frontier; if the score is less than 1, the same input level can generate more output. A feasible solution always exists, so the model has a solution. In the output‑oriented model, a higher efficiency score indicates greater inefficiency.

Before applying the output‑oriented model to evaluate the efficiency of 15 top wealth cities, we first consider a simple example shown below, with four DMUs, one input and two outputs, and all four DMUs share the same input level.

In the figure, one DMU is efficient. Applying the model to compute the efficiency of DMU 4 yields a set of constraints (s.t.) and a solution that shows DMU 4 is inefficient; its efficiency projection point requires increasing both outputs to reach the frontier.

Reference:

Data Envelopment Analysis: A Balanced Benchmark Method / (Ed.) Wade D. Cook, Joe Zhu; translated by Wu Huaqing, Beijing: Science Press, 2017.9