Unlocking Rank Sum Ratio: A Step-by-Step Guide to Comprehensive Evaluation

Rank Sum Ratio (RSR) Method

The Rank Sum Ratio (RSR) comprehensive evaluation method creates a dimensionless statistic by converting raw data into ranks, allowing direct sorting of evaluation objects based on their RSR values.

Sample rank: For a sample of size $n$ drawn from a univariate population, order the observations from smallest to largest. The position of each observation in this ordered list is its rank, denoted $R_i$, and the collection of all $R_i$ forms the rank statistics.

For example, given a set of sample data, the ordered statistics are the sorted values, while the rank statistics are their corresponding positions.

Assume a comprehensive evaluation problem involves m evaluation objects and k indicators, with observed indicator values forming a data matrix.

2 Steps

2.1 Rank Assignment

Rank each column of the data matrix. For benefit (larger‑the‑better) indicators, rank from smallest to largest; for cost (smaller‑the‑better) indicators, rank from largest to smallest. When indicator values are equal, assign the average rank. The resulting rank matrix is denoted $R$.

2.2 Compute Rank Sum Ratio (RSR)

If all indicator weights are equal, compute RSR using the formula:

RSR = (sum of ranks for an object) / (maximum possible sum of ranks)

When indicator weights differ, compute a weighted RSR:

RSR = (∑ w_j * R_{ij}) / (∑ w_j * R_{max,j})

where $w_j$ is the weight of the $j$‑th indicator.

2.3 Rank Sorting by RSR

Sort the evaluation objects according to their RSR values; a larger RSR indicates a better evaluation result.

Reference

司守奎，孙玺菁. Python数学实验与建模.