Understanding DEA Models: CRS, VRS, and Efficiency Decomposition

DEA Model

Notation

Sets:

DMUs: set of decision‑making units.

Inputs: set of input variables.

Outputs: set of output variables.

Decision variables:

λ_i: dual decision variable for the i‑th DMU.

s_i⁻: input slack for the i‑th DMU.

v_j: weight of the j‑th input variable.

u_r: weight of the r‑th output variable.

Parameters:

x_{ij}: value of the j‑th input for DMU i.

y_{ir}: value of the r‑th output for DMU i.

CRS Model

Charnes, Cooper and Rhodes (CCR) introduced the constant‑returns‑to‑scale (CRS) DEA model in 1978, also called the CCR model. It can measure overall efficiency (OE) and identify the most productive scale (MPSS) for a specific firm.

Primal form

The primal form is a fractional programming problem.

This form contains:

Number of decision variables (unspecified).

Number of constraints (unspecified).

Dual form

Because the primal is a fractional (non‑linear) program, dual theory is used to transform it into a linear dual problem, simplifying computation.

The dual also contains the same numbers of decision variables and constraints.

VRS Model

Banker, Charnes and Cooper (BCC) proposed the variable‑returns‑to‑scale (VRS) DEA model in 1984, also known as the BCC model, which relaxes the constant‑returns assumption and can measure technical efficiency (TE) of a specific firm.

Primal form

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Dual form

(Details not provided in the source.)

How to evaluate efficiency?

Input‑oriented model

As illustrated, the point represents the current input state, and the optimal point represents the most efficient input level; the model seeks proportional reduction of inputs without changing outputs.

Output‑oriented model

The diagram shows the current input state and the optimal input state; the model aims to proportionally expand outputs while keeping inputs unchanged.

Decomposing overall efficiency

Components

Overall Efficiency (OE) is the optimal solution of the CRS model.

Technical Efficiency (TE) is the optimal solution of the VRS model.

Scale Efficiency (SE) is calculated as OE divided by TE; it reflects the effect of returns to scale on productivity.

Interpretation

This input‑oriented model illustrates how overall efficiency is composed of technical and scale components.

References

[1] Coelli, T. J., Rao, D. S. P., O'Donnell, C. J., & Battese, G. E. (2005). An introduction to efficiency and productivity analysis. Springer. [2] Data envelopment analysis. (2020, June 9). Wikipedia. [3] Lee, C. Y., & Johnson, A. L. (2012). Two‑dimensional efficiency decomposition to measure the demand effect in productivity analysis. European Journal of Operational Research, 216(3), 584‑593. [4] Returns to scale. (2020, April 16). Wikipedia. [5] Vörösmarty, G., & Dobos, I. (2020). A literature review of sustainable supplier evaluation with Data Envelopment Analysis. Journal of Cleaner Production.