Understanding Input‑Oriented DEA: Modeling Efficiency Frontiers

In the DEA model, there are two ways to characterize the efficient frontier: input‑oriented and output‑oriented. The following describes the input‑oriented approach, which minimizes inputs while keeping output levels unchanged.

Let DMU_j denote the j‑th decision‑making unit among n units, and let x_{ij} and y_{rj} represent the i‑th input and r‑th output of unit j, respectively. The weights λ_j are unknown, and n is the number of units. The left‑hand side of the inequalities forms the convex hull of all units’ inputs and outputs, while the right‑hand side corresponds to the unit under evaluation.

The model contains three groups of constraints, actually amounting to m input constraints and s output constraints, where each input and each output contributes a separate constraint. Thus the model can be expressed as:

min θ

subject to

∑_{j=1}^{n} λ_j x_{ij} ≤ θ x_{i0}, i = 1,…,m

∑_{j=1}^{n} λ_j y_{rj} ≥ y_{r0}, r = 1,…,s

λ_j ≥ 0, j = 1,…,n

Here θ is the efficiency score. Because θ is feasible, the optimal solution satisfies θ ≤ 1 . If θ = 1 , the evaluated unit lies on the efficient frontier; if θ < 1 , the unit is inside the frontier, meaning its inputs can be proportionally reduced while maintaining the same output level.

Before constructing the data model, consider an example with five decision units. Over one week each unit generates a profit of $1500 using different combinations of supply‑chain cost and response time.

The figure below shows the five units and the efficient frontier.

Units 1, 2, 3, and 4 lie on the frontier. Applying the model to evaluate unit 5 yields the following results:

Reference:

Data Envelopment Analysis: Balanced Scorecard Method / (Ed.) Wade D. Cook, Joe Zhu; translated by Wu Huaqing, Beijing: Science Press, September 2017.