# Aggregation of sustainability indicators by shifted geometric means and comparison with established rankings

## Description

This spreadsheet file contains data and calculations to supplement manuscript with tentative title Y. A. Phillis, V. S. Kouikoglou, E. Grigoroudis, and F. D. Kanellos, "On a mathematical theory of sustainability assessment". We start with n sustainability indicators relevant to a country whose sustainability is assessed. The following are common assumptions in most sustainability assessment approaches: a) each indicator i, i = 1, …, n, is normalized into a dimensionless number x_i on [0, 1], where 0 corresponds indicator values deemed unsustainable and 1 corresponds to completely sustainable values b) each indicator i is assigned a weight of importance w_i where w_i>0 and Σ w_i = 1. The shifted geometric mean of x_i is defined by GM_c = Π (c + x_i)^{w_i} - c where c is a positive parameter. For simplicity we choose c=1 and use GM_1 as an alternative national sustainability index. We compare some publicly available sustainability indices with GM_1 using the same indicator values and weights. The spreadsheet contains several sheets grouped by color. Each color corresponds to one of the following assessment approaches 1) Human Development Index (HDI) 2) Environmental Performance Index (EPI) 3) Two TOPSIS-based assessments using the EPI data (TOPSIS: Technique for Order of Preference by Similarity to Ideal Solution) 4) Sustainability Assessment by Fuzzy Evaluation (SAFE) Each sheet contains one of the following: i) Indicator and sustainability indices from sources mentioned in the "Steps to reproduce" section below. ii) Calculations involving GM (GM_1) or TOPSIS iii) Computation of correlation coefficients (Pearson r, Kendall tau) between the overall sustainability indices obtained by two models.

## Files

## Steps to reproduce

1) Human Development Index (HDI) Data, weights and HDI: https://hdr.undp.org/sites/default/files/data/2020/2020_Statistical_Annex_Table_1.xlsx 2) Environmental Performance Index (EPI) Data (some missing values) and EPI: https://epi.yale.edu/downloads/epi2022results05302022.csv Weights: https://epi.yale.edu/downloads/epi2022weights05302022.csv 3) TOPSIS method using EPI data with some missing values Standard TOPSIS (topsisEPI): Calculations based on https://en.wikipedia.org/wiki/TOPSIS Version 2 (topsis2EPI): Use of original EPI indicators (i.e., without applying TOPSIS normalization) and the L1-distance (sum of absolute deviations) instead of the standard L2-distance (square root of sum of squared deviations). 4) Sustainability Assessment by Fuzzy Evaluation (SAFE) Data: Grigoroudis, Evangelos; Kouikoglou, Vassilis; Phillis, Yannis (2020), “Data for: SAFE 2019: Updates and new sustainability findings worldwide”, Mendeley Data, V1, doi: 10.17632/n8rbrmnzf4.1 SAFE Index: Grigoroudis, E., Kouikoglou, V. S., & Phillis, Y. A. (2021). SAFE 2019: Updates and new sustainability findings worldwide. Ecological Indicators, 121, 107072. The above data sets are also used to calculate sustainability indices based on the GM_1 function and the TOPSIS model.