Introducing the comprehensive value function for sustainability full-spectrum assessment

Citation: Boix-Cots, D.; Pardo-Bosch, F.; Pujadas, P. Introducing the Comprehensive Value Function for Sustainability Full-Spectrum Assessment. Sustainability 2024,16, 2617. https://doi.org/10.3390/ su16072617 Academic Editors: Malgorzata Jasiulewicz-Kaczmarek and Nita Yodo Received: 31 January 2024 Revised: 19 March 2024 Accepted: 20 March 2024 Published: 22 March 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). sustainability Article Introducing the Comprehensive Value Function for Sustainability Full-Spectrum Assessment David Boix-Cots 1,* , Francesc Pardo-Bosch 2,3 and Pablo Pujadas 2,3 1Department of Civil and Environmental Engineering, Universitat Politècnica de Catalunya · BarcelonaTech (UPC), C/Jordi Girona 1-3, 08034 Barcelona, Spain 2Department of Project and Construction Engineering, Universitat Politècnica de Catalunya · BarcelonaTech (UPC), Av. Diagonal 647, 08028 Barcelona, Spain; francesc.par[email protected] (F.P.-B.); [email protected] (P.P.) 3Group of Construction Research and Innovation (GRIC), C/Colom, 11, Ed. TR5, 08222 Terrassa, Spain *Correspondence: [email protected] Abstract: This paper introduces the comprehensive value function, a novel framework designed to address multi-dimensional challenges in sustainability assessment within decision-making processes. Multi-criteria decision-making methods based on multi-attribute utility theory excel at integrating environmental, social, and economic factors but typically focus on positive and neutral impacts. This limitation often results in the oversight of critical negative consequences, thus restricting their effectiveness in all-encompassing sustainability evaluations. The proposed framework addresses this gap by quantifying the degree of satisfaction across a full spectrum of sustainability impacts and integrating negative outcomes essential for holistic assessments. The necessity of this model is highlighted by the shortcomings of current practices, where adverse impacts are frequently overlooked and existing frameworks fail to foresee the negative repercussions of various alternatives. By facilitating the creation of positive, negative, or piecewise functions, the CVF provides a comprehensive reflection of impacts, essential for well-informed sustainability decisions. Integrating this function into established decision-making models leads to a more balanced approach adept at navigating the intricate trade-offs inherent in sustainable development. Organised systematically, the paper presents the comprehensive value function, its application across various domains, and a concluding section synthesising the findings. Keywords: comprehensive value function; sustainability assessment; decision making; multiple criteria decision making; multi-attribute analysis 1. Introduction In modern decision-making, the integration of sustainability has become paramount, acknowledging the profound and long-lasting impacts of human activities on the environment, society, and economies [ 1 ]. However, its complexity, characterised by its multidimensional nature, poses significant challenges in its evaluation and the inclusion of all its variables in the decision-making process [ 2 ]. As the global community strives to integrate sustainability into analyses to achieve consideration of multiple sustainable development goals, the need for robust, nuanced, and holistic evaluation methods has never been more pressing. In addressing these complexities, multi-criteria decision-making (MCDM) methods have emerged as vital tools. They facilitate the consideration of diverse and often conflicting criteria, enabling decision-makers to navigate the intricate trade-offs inherent in sustainability issues [ 3 ]. The efficacy of MCDM methods lies in their ability to synthesise a range of environmental, social, and economic factors into a coherent decision-making framework, thus offering a more comprehensive understanding of the decision-making process. Sustainability 2024,16, 2617. https://doi.org/10.3390/su16072617 https://www.mdpi.com/journal/sustainability Sustainability 2024,16, 2617 2 of 14 Among the various MCDM methodologies, the MIVES (Spanish acronym, “Modelo Integrado de Valor para una Evaluación Sostenible”, in English: value model for the evaluation of sustainability) method has gained prominence. MIVES represents a significant advance in sustainability assessment, as it integrates a three-pillar sustainability requirement framework with the multi attribute utility theory, offering a structured and systematic approach to decision-making. The core of this method is the application of a value function [ 4 ], a single mathematical function that converts the qualitative and quantitative variables of the indicators, with their different units and scales, into a non-dimensional scale comprised between 0 and 1, which represents the minimum and the maximum degree of satisfaction of the decision-maker. Thus, every criterion and indicator can be aggregated to obtain a sustainability index. MIVES has been extensively utilised to comprehensively assess the sustainability indices of various alternatives in general building analysis, primarily due to the expertise of its developers in civil engineering and construction proceedings. These analyses include evaluations focused on different types of structures and materials. For example, in structural analyses, it has been applied to tunnels [ 5 – 9 ], bridges [ 10 , 11 ], and footbridges [ 12 , 13 ], industrial buildings [ 14 – 16 ] or skyscrapers [ 17 ], mass housing [ 18 ], and modular residential towers [ 19 ]. It has also been used in assessing temporary [ 20 – 23 ], and self-promoting housing [ 24 ], as well as in evaluating the sustainability of construction materials like wood [ 25 , 26 ], concrete [ 27 ], and prefabricated structures [ 28 ]. Furthermore, MIVES has been instrumental in evaluating multiple construction elements. These include bricks and adobe [29,30], columns [31], slabs [32], and various others such as pipes [33,34], earth retaining walls [ 35 ], trenches [ 36 , 37 ], structural concrete retention tanks [ 38 ], piles [ 39 , 40 ], interior panels [41], roofs [42–44] and facades [45,46]. The good performance of MIVES in sustainability analysis and its ability to specifically adapt to each problem have allowed it to be applied in many other fields. These applications include advancements in Spanish sustainability codes and software [ 47 – 49 ], educational and development assessments such as school construction technologies and active learning strategies, R&D technology selection and advancing project management innovations [50–54] , resource analyses for sustainable public fund allocation [ 55 – 63 ] or tourism [ 64 ], and promotion of industrial heritage assets [ 65 ]. Energy studies have explored sustainable power generation [ 66 ], biomass processing [ 67 ], and solar energy systems [ 68 ], while the machinery sector has seen MIVES applied to optimise designs and operational efficiencies in wind turbine systems [ 69 , 70 ], heat exchangers [ 71 ], and for enhancing waste utilisation in both educational facilities and civil engineering [ 72 , 73 ]. Even risk assessments have utilised MIVES to address climate impacts on urban development and to enhance safety in the financial sector [ 74 – 78 ], and a specific, multiple criteria group decision-making method, HIVES [ 79 ], has been developed to further refine this approach. This method enriches the decision-making process by incorporating diverse perspectives from multiple stakeholders, ensuring a more comprehensive and democratic evaluation of sustainability metrics [80]. However, as our understanding of sustainability deepens, it becomes increasingly clear that positive and neutral impacts are only part of the picture. The MIVES applications already presented, while varied and extensive, can only assess positive and neutral impacts [ 81 ], inadvertently marginalising the significance of negative consequences [ 82 ]. The limitations of such an approach become increasingly evident in the current socio-economic and environmental context, where negative outcomes can have profound implications [ 83 ]: from resource depletion and social inequity [ 84 ] to ecological degradation [ 85 ], each carrying the potential to substantially reverse sustainability gains. Recognising and addressing these negative impacts is essential for a holistic sustainability strategy. Negative outcomes, if ignored, can significantly undermine sustainability efforts and lead to unintended consequences [ 86 ]. Consequently, a more nuanced and inclusive model is necessary—one that anticipates and mitigates potential adverse effects—to ensure that sustainability efforts are not only well-intentioned but also truly effective and comprehensive. Sustainability 2024,16, 2617 3 of 14 Addressing this gap, this paper proposes the necessary evolution of the original value function methodology [ 4 ]: the comprehensive value function. The enhancement of the traditional value function approach is characterised by the inclusion of a full range of sustainability impacts, notably incorporating those that are negative. A more equitable and realistic framework for sustainability assessment is thus provided, one that acknowledges the complex interplay of gains and setbacks inherent in the pursuit of sustainable development. Designed to integrate with the existing MIVES model, the comprehensive value function expands its evaluative scope and strengthens its ability to navigate the intricacies of modern sustainability challenges. The paper is organised as follows: The Section 2presents the comprehensive value function. In Section 3various application examples are exhibited, showcasing the practical application and implications of the proposal. The Section 4gives a general conclusion. 2. The Comprehensive Value Function The comprehensive value function (CVF) represents an evolution of the original value function methodology: a construct designed to measure the degree of satisfaction derived from the values of assessed indicators or criteria, while considering the preferences of decision-makers. In contrast to the original proposal, which was designed primarily to represent preferences [ 4 ] and thus limited to positive or neutral values, the comprehensive value function addresses a broader spectrum. The real-world application of MIVES across various domains has further underscored the need for this broader spectrum approach with the CVF. In practice, certain indicators have been identified as having adverse impacts significant enough to be considered negative values. Additionally, in situations where the MIVES framework, established by one organisation, is utilised to assess alternatives offered by different entities, the possibility of encountering unanticipated negative impacts arises. This is particularly evident in cases where the alternative developers lack comprehensive control over their projects’ development. Such scenarios reinforce the necessity of a value function that is adept at accommodating negative scenarios, a crucial aspect for an all-encompassing sustainability assessment. To address this complexity, the evolved function integrates an ‘n’ factor, which can assume both positive and negative values, thus offering the possibility to generate a positive, a negative, or a piecewise positive–negative function. This integration allows the function to capture a full range of impacts—positive, neutral, and negative—offering a more holistic and realistic valuation of outcomes, crucial for effective sustainability decision-making. The development of the CVF consists of four distinct steps, as depicted in Figure 1. For each indicator under assessment, these steps are systematically applied in the following sequence: First, identifying the nature of the CVF by determining if it is positive, negative, or piecewise, which includes selecting its initial typology and tendency. Second, establishing the satisfaction points that delineate the CVF’s operational range. Third, deciding on the function’s shape—whether linear, concave, convex, or S-shaped. Lastly, formulating the mathematical expression that captures the essence of the value function. A description of each of these steps is presented in the sections below. Sustainability 2024, 16, x FOR PEER REVIEW 4 of 15 Figure 1. CVF development steps. 2.1. Defining the CVF’s Nature The formulation of the CVF commences with discerning its inherent nature, which is essential for shaping its subsequent structure. This crucial step involves examining the indicator in question and the satisfaction it imparts to the decision-maker, which determines the tendency and whether the CVF will manifest as a single function—exclusively positive or negative—or as a piecewise function, which may exhibit varying tendencies within its range. In scenarios where the indicator’s impact is unidirectional, the CVF will be single, meaning it is uniformly positive or negative across its range. For example, a positive increasing function could represent the percentage of green space in urban development, where a higher percentage directly increases satisfaction. A positive decreasing function might be applied to indicators such as energy consumption, where less energy use denotes higher satisfaction. Conversely, a negative increasing function could relate to the level of pollutants emitted, as higher emissions would decrease satisfaction. Finally, a negative decreasing function might be appropriate for an indicator like recovery time from a disaster, where longer recovery times result in increasingly negative satisfaction. Piecewise functions, however, introduce a more nuanced representation. A positive– negative piecewise function could be applied to an indicator like vehicle flow near a construction site: low traffic disruption (positive) is acceptable up to a certain limit, beyond which increased disruption leads to dissatisfaction (negative). Alternatively, a positive– positive piecewise function might describe the relationship between the built area of a development project and its utility; initial increases in the built area yield higher utility (the first positive phase), but after reaching optimal utility, further expansion might prioritise other factors such as community spaces (the second positive phase). Moreover, a negative–negative piecewise function could be relevant for an indicator like water consumption during construction phases, where initial reductions in consumption are more valued (the first negative phase), but subsequent reductions might offer diminishing satisfaction due to already achieved efficiencies (the second negative phase). 2.2. Establishing the Satisfaction Points In the CVF definition process, the establishment of satisfaction points is a critical step, defining the dominion within which the function operates. These satisfaction points, denoted as S min and S max , represent the extremities of the function’s range on the x-axis in absolute value, as shown in Figure 2. This figure illustrates general examples of potential single CVFs along with their corresponding satisfaction points. For a CVF that is single positive and increasing, S min and S max are typically set at satisfaction values of 0 and 1, respectively. Conversely, for a single positive and decreasing CVF, the satisfaction values are reversed, with S min at 1 and S max at 0. In cases of strictly negative impacts, for a single negative and increasing CVF, the satisfaction values range from −1 (S min ) to 0 (S max ). Conversely, for a single negative and decreasing CVF, S min is set at 0 and S max at −1. Moreover, for functions that must encapsulate both positive and negative impacts—a piecewise positive–negative function—the satisfaction points can extend from −1 for S min to 1 for S max . Figure 1. CVF development steps. Sustainability 2024,16, 2617 4 of 14 2.1. Defining the CVF’s Nature The formulation of the CVF commences with discerning its inherent nature, which is essential for shaping its subsequent structure. This crucial step involves examining the indicator in question and the satisfaction it imparts to the decision-maker, which determines the tendency and whether the CVF will manifest as a single function—exclusively positive or negative—or as a piecewise function, which may exhibit varying tendencies within its range. In scenarios where the indicator’s impact is unidirectional, the CVF will be single, meaning it is uniformly positive or negative across its range. For example, a positive increasing function could represent the percentage of green space in urban development, where a higher percentage directly increases satisfaction. A positive decreasing function might be applied to indicators such as energy consumption, where less energy use denotes higher satisfaction. Conversely, a negative increasing function could relate to the level of pollutants emitted, as higher emissions would decrease satisfaction. Finally, a negative decreasing function might be appropriate for an indicator like recovery time from a disaster, where longer recovery times result in increasingly negative satisfaction. Piecewise functions, however, introduce a more nuanced representation. A positive– negative piecewise function could be applied to an indicator like vehicle flow near a construction site: low traffic disruption (positive) is acceptable up to a certain limit, beyond which increased disruption leads to dissatisfaction (negative). Alternatively, a positive– positive piecewise function might describe the relationship between the built area of a development project and its utility; initial increases in the built area yield higher utility (the first positive phase), but after reaching optimal utility, further expansion might prioritise other factors such as community spaces (the second positive phase). Moreover, a negative– negative piecewise function could be relevant for an indicator like water consumption during construction phases, where initial reductions in consumption are more valued (the first negative phase), but subsequent reductions might offer diminishing satisfaction due to already achieved efficiencies (the second negative phase). 2.2. Establishing the Satisfaction Points In the CVF definition process, the establishment of satisfaction points is a critical step, defining the dominion within which the function operates. These satisfaction points, denoted as S min and S max , represent the extremities of the function’s range on the x-axis in absolute value, as shown in Figure 2. This figure illustrates general examples of potential single CVFs along with their corresponding satisfaction points. For a CVF that is single positive and increasing, S min and S max are typically set at satisfaction values of 0 and 1, respectively. Conversely, for a single positive and decreasing CVF, the satisfaction values are reversed, with S min at 1 and S max at 0. In cases of strictly negative impacts, for a single negative and increasing CVF, the satisfaction values range from − 1 (S min ) to 0 (S max ). Conversely, for a single negative and decreasing CVF, S min is set at 0 and S max at − 1. Moreover, for functions that must encapsulate both positive and negative impacts—a piecewise positive–negative function—the satisfaction points can extend from − 1 for S min to 1 for Smax. Sustainability 2024, 16, x FOR PEER REVIEW 5 of 15 Figure 2. Single CVF examples: (A) positive increasing, (B) positive decreasing, (C) negative increasing, and (D) negative decreasing. The setting of these satisfaction points is informed by four specific criteria: 1. Existing rules and regulations: The satisfaction points are often delineated by legal and regulatory frameworks, which define the minimum acceptable standards and maximum permissible limits for the indicators; 2. Experience of the experts: Expert judgement, rooted in knowledge from previous projects and empirical evidence, also plays a pivotal role. This expertise helps to shape the CVF by establishing satisfaction points that reflect practical realities and historical precedents; 3. Decision-maker’s preferences: The perspectives and preferences of decision-makers are integral to the CVF. These preferences can influence the determination of satisfaction points, aligning the function with the strategic objectives and priorities of the stakeholders involved; 4. Value produced by the considered alternatives: The actual performance of different alternatives against the indicators sets the empirical bounds of the CVF. This criterion ensures that the function is calibrated to real-world data, accommodating the breadth of outcomes that the assessed alternatives may exhibit. In application, the CVF’s satisfaction points are not static; they are subject to adjustment based on the evolving nature of the indicators and the context of their assessment. This dynamic calibration allows the CVF to maintain relevance and accuracy in diverse situations, reflecting the true scope of decision-makers’ satisfaction or dissatisfaction with the sustainability outcomes being assessed. If the values derived from certain alternatives extend beyond the predetermined xaxis domain, there are two possible courses of action. First, if these values transgress the absolute minimum or maximum constraints, particularly those mandated by regulatory standards, they may be omitted from consideration. Second, if the values exceed these constraints, they can be capped at the established limits. For functions representing positive impacts, values that surpass the established upper limit are capped at 1, while those falling below the established lower limit are adjusted to 0. For functions representing negative impacts, any value falling below the lower threshold is set to −1, and any value exceeding the upper threshold is limited to 0. When dealing with piecewise functions that combine both positive and negative impacts, the values are managed accordingly: positive values exceeding the upper limit are capped at 1, and negative values dipping below the lower limit are set to −1. 2.3. Deciding on the Function’s Shape Upon establishing the satisfaction points, the next step is to determine their shape by connecting these points through a specific function. The selection among the suggested shapes—concave, convex, linear, or S-shaped—depends on the decision-maker’s response to the indicators and their strategic goals. For positive functions, a concave curve is selected when it is crucial to quickly distance from the lower bound of satisfaction. This shape is effective when even small improvements from the least favourable condition are highly valued. In contrast, a convex Figure 2. Single CVF examples: (A) positive increasing, (B) positive decreasing, (C) negative increasing, and (D) negative decreasing. Sustainability 2024,16, 2617 5 of 14 The setting of these satisfaction points is informed by four specific criteria: 1. Existing rules and regulations: The satisfaction points are often delineated by legal and regulatory frameworks, which define the minimum acceptable standards and maximum permissible limits for the indicators; 2. Experience of the experts: Expert judgement, rooted in knowledge from previous projects and empirical evidence, also plays a pivotal role. This expertise helps to shape the CVF by establishing satisfaction points that reflect practical realities and historical precedents; 3. Decision-maker’s preferences: The perspectives and preferences of decision-makers are integral to the CVF. These preferences can influence the determination of satisfaction points, aligning the function with the strategic objectives and priorities of the stakeholders involved; 4. Value produced by the considered alternatives: The actual performance of different alternatives against the indicators sets the empirical bounds of the CVF. This criterion ensures that the function is calibrated to real-world data, accommodating the breadth of outcomes that the assessed alternatives may exhibit. In application, the CVF’s satisfaction points are not static; they are subject to adjustment based on the evolving nature of the indicators and the context of their assessment. This dynamic calibration allows the CVF to maintain relevance and accuracy in diverse situations, reflecting the true scope of decision-makers’ satisfaction or dissatisfaction with the sustainability outcomes being assessed. If the values derived from certain alternatives extend beyond the predetermined xaxis domain, there are two possible courses of action. First, if these values transgress the absolute minimum or maximum constraints, particularly those mandated by regulatory standards, they may be omitted from consideration. Second, if the values exceed these constraints, they can be capped at the established limits. For functions representing positive impacts, values that surpass the established upper limit are capped at 1, while those falling below the established lower limit are adjusted to 0. For functions representing negative impacts, any value falling below the lower threshold is set to − 1, and any value exceeding the upper threshold is limited to 0. When dealing with piecewise functions that combine both positive and negative impacts, the values are managed accordingly: positive values exceeding the upper limit are capped at 1, and negative values dipping below the lower limit are set to −1. 2.3. Deciding on the Function’s Shape Upon establishing the satisfaction points, the next step is to determine their shape by connecting these points through a specific function. The selection among the suggested shapes—concave, convex, linear, or S-shaped—depends on the decision-maker’s response to the indicators and their strategic goals. For positive functions, a concave curve is selected when it is crucial to quickly distance from the lower bound of satisfaction. This shape is effective when even small improvements from the least favourable condition are highly valued. In contrast, a convex shape is utilised when greater value is placed on advancements towards the upper bound of satisfaction, encouraging efforts that strive for the highest sustainability outcomes. Negative functions also utilise these shapes, but with an inverted perspective on satisfaction. A convex curve is chosen when initial reductions in the indicator lead to a rapid decrease in satisfaction. Conversely, a concave shape would indicate that as the indicator approaches the lower bound, any further decreases have a more pronounced effect on reducing satisfaction, highlighting the urgency of preventing conditions that lead to the worst sustainability outcomes. For both positive and negative functions, the linear shape denotes a direct and consistent relationship between changes in the indicator and shifts in satisfaction, without any acceleration or deceleration in the rate of change, and an S-shaped curve represents a situation where incremental changes near the middle of the range have the most significant Sustainability 2024,16, 2617 6 of 14 effect on satisfaction, with lesser impact near the established bounds. This function is particularly useful for differentiating between alternatives that cluster around the median value of an indicator. As highlighted, the CVF is capable of assuming various forms, each exemplified in Figure 3. This figure presents various typical configurations of the CVF, delineated for both positive and negative ranges. It is important to note that each configuration is adaptable and can be applied inversely across different quadrants. For example, a function characterised as increasing in the positive domain can be mirrored to represent an increasing function in the negative domain. Similarly, what is depicted as a decreasing function in the negative range can correspond to a decreasing function in the positive range. This flexibility is crucial for accurately modelling the multifaceted nature of sustainability indicators. Sustainability 2024, 16, x FOR PEER REVIEW 6 of 15 shape is utilised when greater value is placed on advancements towards the upper bound of satisfaction, encouraging efforts that strive for the highest sustainability outcomes. Negative functions also utilise these shapes, but with an inverted perspective on satisfaction. A convex curve is chosen when initial reductions in the indicator lead to a rapid decrease in satisfaction. Conversely, a concave shape would indicate that as the indicator approaches the lower bound, any further decreases have a more pronounced effect on reducing satisfaction, highlighting the urgency of preventing conditions that lead to the worst sustainability outcomes. For both positive and negative functions, the linear shape denotes a direct and consistent relationship between changes in the indicator and shifts in satisfaction, without any acceleration or deceleration in the rate of change, and an S-shaped curve represents a situation where incremental changes near the middle of the range have the most significant effect on satisfaction, with lesser impact near the established bounds. This function is particularly useful for differentiating between alternatives that cluster around the median value of an indicator. As highlighted, the CVF is capable of assuming various forms, each exemplified in Figure 3. This figure presents various typical configurations of the CVF, delineated for both positive and negative ranges. It is important to note that each configuration is adaptable and can be applied inversely across different quadrants. For example, a function characterised as increasing in the positive domain can be mirrored to represent an increasing function in the negative domain. Similarly, what is depicted as a decreasing function in the negative range can correspond to a decreasing function in the positive range. This flexibility is crucial for accurately modelling the multifaceted nature of sustainability indicators. Figure 3. Possible CVF shapes. 2.4. Formulating the Mathematical Expression The final step is to formulate the mathematical expression that will accurately represent the value of each sustainability indicator. This expression is derived from the established satisfaction points, the determined nature and shape of the CVF, and the parameters governing its behaviour, as contained in Equation (1). 𝑉 =𝑛∗𝐵∗󰇩1−𝑒·󰇡  󰇢  󰇪 (1) where V ind is the satisfaction index of the evaluated indicator; n is the positivity or negativity parameter; Figure 3. Possible CVF shapes. 2.4. Formulating the Mathematical Expression The final step is to formulate the mathematical expression that will accurately represent the value of each sustainability indicator. This expression is derived from the established satisfaction points, the determined nature and shape of the CVF, and the parameters governing its behaviour, as contained in Equation (1). Vind =n∗B∗1−e−K·(X−Smin C)P(1) where Vind is the satisfaction index of the evaluated indicator; nis the positivity or negativity parameter; Bis a parameter that allows the function to remain within 0 and 1, where 1 is assumed to be the highest satisfaction value. This parameter is determined by Equation (2); Smin is the minimum satisfaction value point on the x-axis; Smax is the x maximum satisfaction value point on the x-axis; Xis the indicator value that generates the value equal to Vind; Pdefines the shape of the curve. P= 1, the curve is linear; P< 1, the curve is concave for positive CVFs and convex for negative CVFs; P> 1, the curve is S-shaped or convex for positive CVFs and concave for negative CVFs; Cis a parameter that defines the inflexion x-value point for P> 1 curves; Kis a parameter that defines the y-value C point. Sustainability 2024,16, 2617 7 of 14 B=1 1−e−K·(Smax−Smin C)P(2) This mathematical expression encapsulates the essence of the CVF, translating the qualitative and quantitative aspects of sustainability indicators into a non-dimensional value that can be aggregated to compute an overall sustainability index. The flexibility of the CVF is showcased in its capacity to adapt to a wide array of indicators, reflecting the diverse impacts that various decisions have on sustainability goals. 3. Examples of Applications In this section, examples of the CVF are presented to illustrate its application, following the development steps in Figure 1. The examples of positive CVFs, already extensively covered in current literature (considering n= 1), are not included. Instead, the focus is on negative and piecewise functions, highlighting the CVF’s capability to address a wide array of sustainability indicators and impacts that are multifaceted or predominantly adverse. 3.1. Water Quality Index The first indicator example is the Water Quality Index, which encapsulates a multitude of parameters that reflect the health and cleanliness of water bodies, ranging from chemical composition to biological integrity, using a percentage value. This indicator is identified as negative, based on the understanding that lower water quality leads to increased dissatisfaction. The function’s initial typology is set as decreasing, reflecting the reality that as water quality worsens (measured by increasing values of pollutants or decreasing values of cleanliness), the level of satisfaction diminishes. The satisfaction points are established with reference to a standardised scale of water quality. S max is set at 100, representing the ideal or best possible water quality, where satisfaction is at its maximum. S min , conversely, is set at 0, aligning with the poorest water quality scenario, where satisfaction is at its lowest. This delineation of satisfaction points captures the full operational range of the CVF in the context of water quality assessment. The shape of the CVF for the Water Quality Index is determined to be S-shaped (see Figure 4) with S max = 100, S min = 0, C= 50, K= 1.5, P= 2, a choice motivated by the nature of the water quality’s impact on satisfaction: a first phase where small quality changes are accepted, a second phase with a rapid decrease in satisfaction as the water quality moves away from the ideal condition, and a third phase in which the decision maker already shows practically minimum values. Sustainability 2024, 16, x FOR PEER REVIEW 8 of 15 Figure 4. Water Quality Index indicator’s CVF. 3.2. Recycled Waste The second indicator example is a waste management indicator, focusing on the proportion of recycled waste. This indicator is crucial for monitoring compliance with EU recycling goals and for identifying areas where waste reduction efforts can be intensified. The indicator is designated as negative, considering that as the volume of non-recycled waste increases, especially beyond European Union (EU) targets, it leads to greater environmental impact and public dissatisfaction. These targets allow us to find out the S max and S min : the EU has state that each member state should recycle or prepare for reuse at least 55% of municipal waste. Therefore, for any administration (from local to national), the S max is set to 55% and the S min is set to 0% of recycled waste. The typology of the function is set as convex (see Figure 5) with S max = 55, S min = 0, C = 20, K = 2, and P = 1, intending to illustrate that small increases in non-recycled waste volumes may not significantly affect satisfaction initially, but as volumes approach and exceed regulatory limits, dissatisfaction declines rapidly. Figure 5. Recycled waste indicator’s CVF. 3.3. Land Degradation The third indicator example is land degradation, which reflects the loss of the Earth’s land surface’s biological or economic productivity and complexity. This provides a valuable tool for policymakers, land managers, and environmental conservationists to quantify the impact of land degradation and prioritise restoration and prevention strategies. Given the critical importance of land for agriculture, habitat, and overall ecological balance, this indicator is inherently negative, with increased degradation leading to higher dissatisfaction due to its adverse effects on food security, biodiversity, and climate regulation. Therefore, the CVF for this indicator is identified as a negative function, with the initial typology set as increasing to signify that as land degradation intensifies (measured by increasing rates of soil erosion or decreasing soil health), the level of dissatisfaction similarly rises. Satisfaction points for this indicator can be established in accordance with the EU Soil Strategy for 2030, which aims for a neutral soil degradation index (SDI) by the year 2030. However, since a universally standardised soil degradation index is not established by the −1.00 −0.80 −0.60 −0.40 −0.20 0.00 1009080706050403020100 Satisfaction Index Water Quality Index (%) −1.00 −0.80 −0.60 −0.40 −0.20 0.00 5549.54438.53327.52216.5115.50 Satisfaction Index Recycled waste (%) Figure 4. Water Quality Index indicator’s CVF. 3.2. Recycled Waste The second indicator example is a waste management indicator, focusing on the proportion of recycled waste. This indicator is crucial for monitoring compliance with EU recycling goals and for identifying areas where waste reduction efforts can be intensified. The indicator is designated as negative, considering that as the volume of non-recycled waste increases, especially beyond European Union (EU) targets, it leads to greater envi- Sustainability 2024,16, 2617 8 of 14 ronmental impact and public dissatisfaction. These targets allow us to find out the S max and S min : the EU has state that each member state should recycle or prepare for reuse at least 55% of municipal waste. Therefore, for any administration (from local to national), the S max is set to 55% and the S min is set to 0% of recycled waste. The typology of the function is set as convex (see Figure 5) with S max = 55, S min = 0, C= 20, K= 2, and P= 1, intending to illustrate that small increases in non-recycled waste volumes may not significantly affect satisfaction initially, but as volumes approach and exceed regulatory limits, dissatisfaction declines rapidly. Sustainability 2024, 16, x FOR PEER REVIEW 8 of 15 Figure 4. Water Quality Index indicator’s CVF. 3.2. Recycled Waste The second indicator example is a waste management indicator, focusing on the proportion of recycled waste. This indicator is crucial for monitoring compliance with EU recycling goals and for identifying areas where waste reduction efforts can be intensified. The indicator is designated as negative, considering that as the volume of non-recycled waste increases, especially beyond European Union (EU) targets, it leads to greater environmental impact and public dissatisfaction. These targets allow us to find out the S max and S min : the EU has state that each member state should recycle or prepare for reuse at least 55% of municipal waste. Therefore, for any administration (from local to national), the S max is set to 55% and the S min is set to 0% of recycled waste. The typology of the function is set as convex (see Figure 5) with S max = 55, S min = 0, C = 20, K = 2, and P = 1, intending to illustrate that small increases in non-recycled waste volumes may not significantly affect satisfaction initially, but as volumes approach and exceed regulatory limits, dissatisfaction declines rapidly. Figure 5. Recycled waste indicator’s CVF. 3.3. Land Degradation The third indicator example is land degradation, which reflects the loss of the Earth’s land surface’s biological or economic productivity and complexity. This provides a valuable tool for policymakers, land managers, and environmental conservationists to quantify the impact of land degradation and prioritise restoration and prevention strategies. Given the critical importance of land for agriculture, habitat, and overall ecological balance, this indicator is inherently negative, with increased degradation leading to higher dissatisfaction due to its adverse effects on food security, biodiversity, and climate regulation. Therefore, the CVF for this indicator is identified as a negative function, with the initial typology set as increasing to signify that as land degradation intensifies (measured by increasing rates of soil erosion or decreasing soil health), the level of dissatisfaction similarly rises. Satisfaction points for this indicator can be established in accordance with the EU Soil Strategy for 2030, which aims for a neutral soil degradation index (SDI) by the year 2030. However, since a universally standardised soil degradation index is not established by the −1.00 −0.80 −0.60 −0.40 −0.20 0.00 1009080706050403020100 Satisfaction Index Water Quality Index (%) −1.00 −0.80 −0.60 −0.40 −0.20 0.00 5549.54438.53327.52216.5115.50 Satisfaction Index Recycled waste (%) Figure 5. Recycled waste indicator’s CVF. 3.3. Land Degradation The third indicator example is land degradation, which reflects the loss of the Earth’s land surface’s biological or economic productivity and complexity. This provides a valuable tool for policymakers, land managers, and environmental conservationists to quantify the impact of land degradation and prioritise restoration and prevention strategies. Given the critical importance of land for agriculture, habitat, and overall ecological balance, this indicator is inherently negative, with increased degradation leading to higher dissatisfaction due to its adverse effects on food security, biodiversity, and climate regulation. Therefore, the CVF for this indicator is identified as a negative function, with the initial typology set as increasing to signify that as land degradation intensifies (measured by increasing rates of soil erosion or decreasing soil health), the level of dissatisfaction similarly rises. Satisfaction points for this indicator can be established in accordance with the EU Soil Strategy for 2030, which aims for a neutral soil degradation index (SDI) by the year 2030. However, since a universally standardised soil degradation index is not established by the scientific community, it necessitates the creation of a composite index. This composite index could be formulated as a weighted sum of multiple key components, each representing a critical aspect of soil health. These components might include soil pollutants (SP), biodiversity (BIO), organic matter or soil fertility (OM), land take or arable land (LT), and erosion rate (ER). Equation (3) illustrates an example of how these components can be integrated, with each component assigned a specific weight, denoted as α . These weights reflect the relative importance or impact of each component on the overall soil degradation assessment. SDI =αSP·SP +αBIO·BIO +αOM·OM +αLT·LT +αER·ER (3) Considering the proposed example, S max is set at the neutral point (SDI = 50%), where land degradation begins to have a noticeable impact on land productivity. S min is determined at a critical level of degradation (SDI = 100%). The shape of the CVF is chosen to be convex with S max = 50, S min = 100, C= 50, K= 5, P= 1 (see Figure 6), illustrating high levels of dissatisfaction with low percentage losses of SDI, arriving at its minimum at nearly 85%, a point where the land’s ability to recover is significantly compromised and the negative impacts on ecosystem services are severe. Sustainability 2024,16, 2617 9 of 14 Sustainability 2024, 16, x FOR PEER REVIEW 9 of 15 scientific community, it necessitates the creation of a composite index. This composite index could be formulated as a weighted sum of multiple key components, each representing a critical aspect of soil health. These components might include soil pollutants (SP), biodiversity (BIO), organic matter or soil fertility (OM), land take or arable land (LT), and erosion rate (ER). Equation (3) illustrates an example of how these components can be integrated, with each component assigned a specific weight, denoted as α. These weights reflect the relative importance or impact of each component on the overall soil degradation assessment. 𝑆𝐷𝐼=𝛼 ·𝑆𝑃𝛼 ·𝐵𝐼𝑂𝛼 ·𝑂𝑀𝛼 ·𝐿𝑇𝛼 ·𝐸𝑅 (3) Considering the proposed example, S max is set at the neutral point (SDI = 50%), where land degradation begins to have a noticeable impact on land productivity. S min is determined at a critical level of degradation (SDI = 100%). The shape of the CVF is chosen to be convex with S max = 50, S min = 100, C = 50, K = 5, P = 1 (see Figure 6), illustrating high levels of dissatisfaction with low percentage losses of SDI, arriving at its minimum at nearly 85%, a point where the land’s ability to recover is significantly compromised and the negative impacts on ecosystem services are severe. Figure 6. Land degradation indicator’s CVF. 3.4. Piecewise Functions In this subsection, attention is turned to one of the key features of the CVF: its capability to embody both positive and negative values through a piecewise function. This characteristic facilitates a nuanced assessment of indicators with dualistic aspects. The already-presented examples are adapted into piecewise functions by constructing a positive CVF and integrating it with the negative CVF. This holistic approach enables a thorough evaluation, recognising both the detrimental outcomes when performance falls below standards and the beneficial outcomes when benchmarks are met or surpassed. For the Water Quality Index, the adaptation involves establishing a point that signifies the lowest index value consumers deem acceptable. This point bifurcates the existing negative CVF into separate positive and negative components. As illustrated in the example, this critical juncture—acting as the S min for the positive CVF and the new S max for the negative CVF—is set at 75%. Consequently, the parameters of the negative CVF are adjusted to S max = 75, S min = 0, C = 100, K = 7, and P = 1. Simultaneously, the positive CVF is configured with S max = 100, S min = 75, C = 100, K = 7, and P = 1. This configuration culminates in the composite piecewise CVF presented in Figure 7. −1.00 −0.80 −0.60 −0.40 −0.20 0.00 50 55 60 65 70 75 80 85 90 95 100 Satisfaction Index SDI (%) Figure 6. Land degradation indicator’s CVF. 3.4. Piecewise Functions In this subsection, attention is turned to one of the key features of the CVF: its capability to embody both positive and negative values through a piecewise function. This characteristic facilitates a nuanced assessment of indicators with dualistic aspects. The already-presented examples are adapted into piecewise functions by constructing a positive CVF and integrating it with the negative CVF. This holistic approach enables a thorough evaluation, recognising both the detrimental outcomes when performance falls below standards and the beneficial outcomes when benchmarks are met or surpassed. For the Water Quality Index, the adaptation involves establishing a point that signifies the lowest index value consumers deem acceptable. This point bifurcates the existing negative CVF into separate positive and negative components. As illustrated in the example, this critical juncture—acting as the S min for the positive CVF and the new S max for the negative CVF—is set at 75%. Consequently, the parameters of the negative CVF are adjusted to S max = 75, S min = 0, C= 100, K= 7, and P= 1. Simultaneously, the positive CVF is configured with S max = 100, S min = 75, C= 100, K= 7, and P= 1. This configuration culminates in the composite piecewise CVF presented in Figure 7. Sustainability 2024, 16, x FOR PEER REVIEW 10 of 15 Figure 7. Water Quality Index indicator with a piecewise CVF. For the recycled waste indicator, appending a positive CVF enables the analysis of the entire range of recycled waste percentages. This dual-function approach not only highlights the negative implications of not meeting sustainable waste management targets but also recognises the benefits of surpassing them. Specifically, a positive CVF is established with S max = 100 and S min = 55, C = 20, K = 2, and P = 1. This structure, shown in Figure 8, allows for a balanced assessment, penalising underperformance while rewarding achievements beyond the set targets. Figure 8. Recycled waste indicator with a piecewise CVF. Similarly, for the land degradation indicator, a positive CVF function can be directly added to the existing negative CVF function. This composite model captures the entire spectrum of land use outcomes, quantifying not only the detrimental effects of surpassing certain land degradation thresholds but also highlighting the advantages of maintaining or improving land quality and incentivising sustainable land management and rehabilitation efforts. For instance, a positive CVF is structured with the parameters S max = 0, representing an ideal state of no land degradation, S min = 50, C = 50, K = 5, and P = 1. Consequently, the amalgamated piecewise function shown in Figure 9 provides a more comprehensive assessment of land sustainability. −1.00 −0.80 −0.60 −0.40 −0.20 0.00 0.20 0.40 0.60 0.80 1.00 0 102030405060708090100 Satisfaction index Water Quality Index (%) −1.00 −0.80 −0.60 −0.40 −0.20 0.00 0.20 0.40 0.60 0.80 1.00 0 112233445564738291100 Satisfaction Index Recycled waste (%) Figure 7. Water Quality Index indicator with a piecewise CVF. For the recycled waste indicator, appending a positive CVF enables the analysis of the entire range of recycled waste percentages. This dual-function approach not only highlights the negative implications of not meeting sustainable waste management targets but also recognises the benefits of surpassing them. Specifically, a positive CVF is established with S max = 100 and S min = 55, C= 20, K= 2, and P= 1. This structure, shown in Figure 8, allows for a balanced assessment, penalising underperformance while rewarding achievements beyond the set targets. Similarly, for the land degradation indicator, a positive CVF function can be directly added to the existing negative CVF function. This composite model captures the entire spectrum of land use outcomes, quantifying not only the detrimental effects of surpassing certain land degradation thresholds but also highlighting the advantages of maintaining or improving land quality and incentivising sustainable land management and rehabilitation efforts. For instance, a positive CVF is structured with the parameters S max = 0, representing an ideal state of no land degradation, S min = 50, C= 50, K= 5, and P= 1. Consequently,