Evaluating City Efficiency with DEA’s CCR and BCC Models

DEA Core Idea

DEA evaluates the relative efficiency of decision‑making units (DMUs) with multiple inputs and outputs by constructing a production frontier and comparing actual outputs to the maximal possible outputs under the same resource consumption.

CCR Model

The CCR model, proposed by Charnes, Cooper and Rhodes in 1978, assumes constant returns to scale and formulates a linear programming problem to maximize each DMU’s efficiency score, which is the ratio of weighted outputs to weighted inputs.

For a DMU, the objective is to maximize θ subject to Σλ_j x_ij ≤ x_i0 and Σλ_j y_rj ≥ θ y_r0, with λ_j ≥ 0.

Case Study: City Efficiency Evaluation

An example uses 15 U.S. cities as DMUs, with housing price, monthly rent, and bread price as inputs, and number of doctors per 1,000 people and household income as outputs. The CCR model is applied to compute each city’s efficiency score (θ).

Key steps:

Identify inputs and outputs for each city (e.g., Seattle: inputs – $586,000 housing price, $581 rent, $1.45 bread; outputs – 4.49 doctors per 1,000, $46,928 household income).

Formulate the CCR linear program to maximize Seattle’s efficiency.

Solve the linear program using a solver such as Python’s pulp library.

Interpret the results: a score of 1 indicates the city lies on the efficiency frontier; scores below 1 reveal potential for improvement.

The solved scores show Seattle, Denver, Philadelphia, Raleigh‑Durham, St. Louis, Washington, Baltimore, and Boston achieve a score of 1, while other cities have scores ranging from 0.781 to 0.982, indicating varying degrees of inefficiency.

BCC Model

The BCC model, introduced by Banker, Charnes and Cooper in 1984, extends the CCR model by adding a convexity constraint (Σλ_j = 1) to allow variable returns to scale, making the model suitable for environments where scale effects change.

In practice, CCR is used when constant returns to scale are assumed, whereas BCC is applied when scale efficiency varies.

DEA is widely used beyond city analysis, including banks, hospitals, and schools, to identify inefficiencies and guide resource‑allocation improvements.