Evaluating University Efficiency with DEA and Virtual Decision Units

Suppose we need to evaluate the efficiency of four universities, each with two inputs and three outputs, as shown in the table below:

The two input indicators are: (1) annual public or private funding received; (2) total number of faculty. The three output indicators are: (1) average annual number of graduates; (2) average graduate salary; (3) total number of published academic papers.

We then construct virtual decision units (VDUs). For each university a distinct VDU is formed as a weighted combination of the existing decision units (the four universities). Let the unknown weights w₁, w₂, w₃, w₄ satisfy w_i ≥ 0 and Σ w_i = 1. The VDU’s two inputs are the weighted sums of the original inputs, and its three outputs are the weighted sums of the original outputs. By assigning different values to the four weights we obtain different VDUs.

Compared with the optimal VDU, we want to know whether a real decision unit can achieve the same or better performance by increasing outputs or decreasing inputs. Assuming the VDU’s inputs are less than or equal to the real unit’s inputs and its outputs are greater than or equal to the real unit’s outputs, the condition can be expressed mathematically for each decision unit.

Next, using an input‑oriented DEA model, we aim to reduce inputs while keeping output levels unchanged. For a given decision unit we formulate the following linear program:

Minimize θ subject to Σ λ_j x_{ij} ≤ θ x_{i0} for each input i, Σ λ_j y_{rj} ≥ y_{r0} for each output r, λ_j ≥ 0.

In these models the decision variables are the λ coefficients and the efficiency score θ; the objective function is θ itself. The models can be solved with standard DEA software.

Reference: “Modern Integrated Evaluation Methods and Selected Cases” by Du Dong, Pang Qinghua, and Wu Yan.