On A Mathematical Theory Of Sustainability Assessment

Yannis A. Phillis, Vassilis S. Kouikoglou, Evangelos Grigoroudis
School of Production Engineering and Management, Technical University of Crete, Chania 73100, Greece
Fotis D. Kanellos
School of Electrical and Computer Engineering, Technical University of Crete, Chania 73100, Greece

We compile a number of postulates to be satisfied by a function that mathematically assesses sustainability of some type: national, urban, energy, etc. These postulates lay the mathematical foundations of a sustainability theory and lead to a simple model based on shifted geometric means combining basic sustainability indicators into an overall index. The model has a number of desirable properties and generalizes the weighted arithmetic and weighted geometric means, which are commonly used aggregation functions in sustainability assessments. Numerical results demonstrate the closeness of the model to other established techniques of sustainability.

Sustainability assessment, mathematics of sustainability, sustainability indicators, data aggregation

Sustainability assessments of an entity, be it a nation, a city, an energy or a transportation system etc., rely on data to evaluate human welfare and environmental integrity. Each of these two fundamental components is a combination of more specialized indicators organized into various hierarchical levels. For example, the human dimension of sustainability encompasses socio-economic, technological and political aspects which are further elaborated using different groups of indicators. Most assessment models normalize indicators from their physical domains into a common interval representing a range from the lowest to the highest levels of sustainability. Normalized indicators of the same group are then combined into more composite indicators through aggregation functions which capture the relationships among and the relative importance of indicators belonging to the same dimension. Composite indicators are further aggregated by exploiting the hierarchy from bottom to top to eventually arrive at a single numerical value of overall sustainability.
Existing approaches to measure sustainability differ in scope, suite of indicators, and normalization and aggregation procedures. We review some commonly used definitions and assessment frameworks of national sustainability and sustainable development.
The Human Development Index (HDI) uses four indicators: life expectancy at birth, expected years of schooling for children, mean years of schooling for adults aged 25 years and older, and the logarithm of gross national income per capita. The indicators are first transformed into a scale from 0 to 1 by a linear interpolation between thresholds of unsustainable and sustainable values and then combined into an overall index using arithmetic and geometric means. HDI was first released in 1990 and is updated annually by the United Nations Development Programme (2022).
The Environmental Performance Index (EPI) measures the closeness of a country's performance to established environmental policy targets. Its latest version (Wolf et al., 2022) uses 40 indicators relevant to three policy objectives: climate change, environmental integrity, and ecosystem vitality. The indicators are first converted to dimensionless numbers from 0 to 100 and then successively aggregated using weighted arithmetic means into more composite indicators and the overall index. Missing data are imputed either via predictive models involving past or correlated variables or by averaging data of neighboring countries. For some indicators the imputed values include a penalty for failing to report information. EPI is successor to the Environmental Sustainability Index (ESI), which contained additional indicators to assess social and political aspects of sustainability (Esty et al., 2005).
Sustainability Assessment by Fuzzy Evaluation (SAFE) is a hierarchical fuzzy system whose first version appeared in 2001. Its most recent release (Grigoroudis et al., 2021) uses 69 time series of indicators grouped in various components of ecological and human sustainability. The Pressure-State-Response classification of OECD (1991) is used to describe each component. Pressure indicators assess the negative impacts on the corresponding component, state describes the prevailing conditions, and response indicators reflect the actions taken to improve the state. Each indicator time series is transformed into a single value which captures the latest trends, it is then normalized in [0, 1], and finally it is combined with other normalized indicators through a sequential fuzzy reasoning process to obtain the SAFE index.
A model of national environmental sustainability proposed by Liu (2007) uses the indicators of ESI. The Analytic Hierarchy Process (Saaty, 1980) is used to assign weights to and aggregate well-defined indicators such as water quality, while a fuzzy reasoning scheme similar to SAFE is used to aggregate composite indicators using subjective criteria and qualitative information.
The Sustainable Society Index (SSI) introduced by van de Kerk and Manuel (2008) is a combination of separate indices corresponding to three dimensions: Human, Environmental and Economic Wellbeing. Each dimension is assessed using five to nine indicators which are normalized and aggregated using geometric means. All variables are assigned a numerical value from 1 (weakest sustainability) to 10 (strongest sustainability). SSI is updated every second year since 2006.
All the above models provide country rankings. About one hundred other models of sustainability, sustainable development, and human well-being have been surveyed by Yang (2014). Recent comprehensive reviews on weighting and aggregation methods for constructing composite indicators can be found in Gan et al. (2017) and Greco et al. (2019).
The simplest aggregation functions are the arithmetic and geometric means and their weighted counterparts, also known as additive and multiplicative functions. These are the most commonly used models for sustainability assessments. However they have certain drawbacks that limit their applicability. For example, an additive function cannot be used to describe an indicator that is critically important for the sustainability of the whole entity since a decline of such an indicator below a certain unsustainability level can, by the additive property, be compensated for by an increase in sustainability levels of other indicators. Multiplicative functions have the opposite property, i.e., they treat all indicators as being critically important. Fuzzy logic and other universal function approximation tools overcome these limitations, sometimes at the price of increased computational complexity and need for sophisticated parameter tuning.
In the present paper we propose an extension to the geometric mean which avoids the drawbacks of the previous models. The new function satisfies a number of sustainability postulates and provides sustainability assessments and country rankings which have strongly positive correlations with some of the models reviewed above and variants thereof...read more in pdf

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