Programming and Solving the CCR Dual Model in DEA Method Using MATLAB

Data Envelopment Analysis (DEA) is a non-parametric method for evaluating the relative efficiency of Decision Making Units (DMUs), with the CCR model being its most fundamental form. The corresponding dual model can directly solve for efficiency values through linear programming.

### Core Model Concept The objective of the CCR dual model is to calculate the efficiency score θ (0 ≤ θ ≤ 1) for a specific DMU, where θ=1 indicates that the DMU lies on the efficiency frontier. The model is implemented through the following steps: Variable Definition: For each DMU, define input variables (such as resource consumption) and output variables (such as product revenue). Constraint Construction: Ensure that the weighted outputs of all DMUs do not exceed their weighted inputs, with θ reflecting the compressible proportion of inputs for the current DMU. Linear Programming Solution: Transform the problem into a linear programming problem that minimizes θ, with constraints incorporating the input-output relationships of all DMUs.

### MATLAB Implementation Key Points Input-Output Matrices: Organize input and output data for each DMU into matrices X and Y, where each column represents a DMU. Linear Programming Function: Utilize MATLAB's `linprog` function with the following configuration: - Objective function: minimize θ - Constraints include: * Technical constraints (weighted outputs ≤ weighted inputs for other DMUs) * Non-negative weight constraints (to avoid meaningless negative weight solutions) Result Interpretation: The θ value represents the efficiency score, where values close to 1 indicate high efficiency, while lower values suggest the need for optimization in input or output structure.

### Extended Applications Returns to Scale Analysis: The model can be extended to the BCC model by making adjustments, enabling further analysis of scale efficiency. Sensitivity Testing: Add or remove DMUs/variables to observe changes in efficiency values and verify model stability.

This method is applicable to scenarios such as financial efficiency evaluation and resource allocation optimization. MATLAB's matrix computation capabilities enable efficient handling of large-scale DMU calculations through vectorized operations and optimized linear programming solvers.