How Variable Returns to Scale DEA Reveals Input/Output Slack in a Two‑Stage Model

In variable‑returns‑to‑scale (VRS) input‑oriented DEA, efficiency scores only capture the proportion of input reduction and output increase. The following table illustrates this concept:

Applying the input‑oriented VRS DEA model to decision unit (DU) 4 yields an input slack, i.e., a possible reduction in the input (reaction time). This individual input reduction is called an input slack.

After solving the model, both input and output slacks exist. However, due to multiple optimal solutions, the model may incorrectly suggest that all slacks are zero, even when non‑zero slacks are feasible.

To determine possible non‑zero slack values, we employ the following linear‑programming model:

... (model formulation omitted for brevity) ...

Applying this model to DU 4 gives the optimal slack values.

Definition 1 (DEA full efficiency): A DMU is fully efficient if and only if conditions (1) and (2) hold.

Definition 2 (DEA weak efficiency): A DMU is weakly efficient if and only if conditions (1) and (2) hold for some inputs and outputs, and there exist non‑zero slacks or reductions.

In the accompanying figure, DUs 1, 2, and 3 are efficient, while DU 4 is weakly efficient. The two‑stage VRS input‑oriented DEA consists of:

Stage 1: Maximize input reduction to obtain the optimal solution.

Stage 2: Identify optimal slack variables to further move the DMU toward the efficiency frontier.

Weakly efficient DMUs generate multiple feasible solutions; if no weakly efficient units exist, the second stage becomes unnecessary, though this cannot be known a priori.

Similarly, the VRS output‑oriented DEA can be expressed as:

... (output‑oriented model formulation omitted) ...

Definition 3 (DEA efficiency for output‑oriented model): Full efficiency holds when conditions (1) and (2) are satisfied for all inputs and outputs; weak efficiency holds when they hold for some inputs/outputs with non‑zero slacks.

The DEA results can be interpreted as follows:

(1) If the conditions are satisfied, the DMU lies on the efficient frontier; otherwise, it is inefficient and can improve by reducing inputs or increasing outputs.

(2) The left‑hand side of the DEA model forms the “reference set,” while the right‑hand side represents the evaluated DMU. Non‑zero weights are benchmark coefficients that define virtual efficient units, guiding how to reduce inputs or increase outputs to achieve efficiency.

DEA full efficiency and relative efficiency are defined as:

Definition 4 (Full efficiency): A DMU is fully efficient if, without worsening any other input or output, no improvement is possible for any input or output.

Definition 5 (Relative efficiency): A DMU is relatively efficient if no other DMU can improve any input or output without worsening others.

Reference:

Data Envelopment Analysis: A Balanced Benchmarking Method, Wade D. Cook & Joe Zhu, translated by Wu Huaqing, Science Press, 2017.