Multiple Criteria Assessment of Insulating Materials with a Group Decision Framework Incorporating Outranking Preference Model and Characteristic Class Profiles
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Abstract
We present a group decision making framework for evaluating sustainability of the insulating materials. We tested thirteen materials on a model that was applied to retrofit a traditional rural building through roof’s insulation. To evaluate the materials from the socioeconomic and environmental viewpoints, we combined life cycle costing and assessment with an adaptive comfort evaluation. In this way, the performances of each coating material were measured in terms of an incurred reduction of costs and consumption of resources, maintenance of the cultural and historic significance of buildings, and a guaranteed indoor thermal comfort. The comprehensive assessment of the materials involved their assignment to one of the three preferenceordered sustainability classes. For this purpose, we used a multiple criteria decision analysis approach that accounted for preferences of a few tens of rural buildings’ owners. The proposed methodological framework incorporated an outrankingbased preference model to compare the insulating materials with the characteristic class profiles while using the weights derived from the revised Simos procedure. The initial sorting recommendation for each material was validated against the outcomes of robustness analysis that combined the preferences of individual stakeholders either at the output or at the input level. The analysis revealed that the most favorable materials in terms of their overall sustainability were glass wool, hemp fibres, kenaf fibres, polystyrene foam, polyurethane, and rock wool.
Keywords
Multiple criteria decision analysis Group decision Characteristic profiles ELECTRE TRIrC Insulating materials Sustainability1 Introduction
This paper presents a group decision framework for evaluating sustainability of the insulating materials to retrofit traditional rural buildings. The importance of this research derives from the previous studies on both retrofitting solutions tailored to traditional rural buildings as well as judging an overall desirability of coating materials (see, e.g., Krarti 2015; Fabbri et al. 2012; Ma et al. 2012; Yung and Chan 2012; MartínezMolina et al. 2016). These studies prove that energy efficiency and thermal comfort are crucial for the maintenance of historic buildings.
The context of the study is that of a typical farmhouse in central Italy. The incorporated building model derives from the analysis of over 800 farmhouses surveyed by the census of the scattered rural buildings of the municipality of Perugia (Umbria region). The high landscape values of traditional buildings and the legislation about their preservation prevent external alterations (Mazzarella 2015). Therefore, the most viable solutions are to intervene on the roof of these structures, increasing their thermal inertia with coating materials (Verbeeck and Hens 2005; Kumar and Suman 2013; Taylor et al. 2000).
We comprehensively evaluate the materials for the roof insulation by considering economic, social, and environmental viewpoints. For this purpose, we incorporate a life cycle costing (LCC) approach, a life cycle assessment (LCA), and a dynamic thermal simulation for the evaluation of energy savings and thermal comfort. As such, we aim at identifying the materials that guarantee the indoor thermal comfort, at the same time reducing the consumption of resources in their entire life cycle as well as maintaining cultural and historic significance of the buildings. In this perspective, we differentiate from the vast majority of previous studies concerning coating materials which incorporate a monodisciplinary approach (Copiello 2017).
To provide an overall sustainability assessment of coating materials, we incorporate Multiple Criteria Decision Analysis (MCDA). MCDA offers a diversity of approaches designed for providing the decision makers (DMs) with a recommendation concerning a set of alternatives evaluated in terms of multiple conflicting points of view. Few applications of MCDA methods for the evaluation of building materials, which are reported in the literature (Ginevicius et al. 2008) deal mainly with the environmental sustainability of materials (Papadopoulos and Giama 2007; Khoshnava et al. 2016). Some combinations of LCA and MCDA were considered by Santos et al. (2017) and Piombo et al. (2016). Applications which included both LCC and LCA for the definition of criteria to be used in MCDA are still rare (Piombo et al. 2016). Decision analysis methods used in the abovementioned studies involved different variants of AHP (Motuziene et al. 2016; Khoshnava et al. 2016), PROMETHEE II (Kumar et al. 2017), Weighted Sum, TOPSIS (Čuláková et al. 2013), VIKOR, and COPRAS (Ginevicius et al. 2008).

We formulate the considered problem in terms of multiple criteria sorting, thus aiming at assigning the materials to a set of predefined and ordered sustainability classes (categories) rather than at ordering them from the best to the worst;

We assess the insulating materials while taking into account preferences of multiple DMs (owners of rural houses), thus incorporating group decision making tools into the evaluation framework;

The adopted assignment procedure builds upon outrankingbased comparison of the insulating materials with the characteristic profiles composed of the perclass most representative performances on all criteria (Kadziński et al. 2015b);

The research results are validated against the outcomes of robustness analysis that takes into account all sets of weights compatible with either the ranking of criteria provided by each DM within the revised Simos (SRF) procedure (Figueira and Roy 2002) or a group compromise ranking of criteria that is constructed with an original procedure proposed in this paper.
2 Review of Multiple Criteria Sorting Group Decision Methods
The objective of the case study presented in this paper is to give an easily interpretable comprehensive assessment of the insulating materials’ sustainability. This is achieved by assigning them to a set of predefined and ordered decision classes based on their performances on multiple criteria (Kadziński et al. 2015b). While computing the sorting recommendation, we account for the preferences of a group of experts and stakeholders. This requires implementation of a group decision making framework.
As realworld situations often involve multiple stakeholders, some methods have been proposed to support groups in making collective sorting decisions (Daher and Almeida 2010). These approaches can be distinguished at different levels. In particular, they differ in terms of a preference model employed to represent preferences of the DMs. Furthermore, an underlying classification rule may involve analysis of a single preference model instance or all sets of parameters compatible with the DMs’ preference information. Moreover, sorting methods can be divided with respect to the level on which individual viewpoints are aggregated (Dias and Climaco 2000). Finally, some approaches account for the importance degrees of the involved DMs, while other methods assume that all DMs play the same role in the committee.
Among multiple criteria sorting group decision methods, outrankingbased approaches are prevailing. Most decision support systems in this stream incorporate Electre TRIB (Yu 1992; Roy 1996). For example, Dias and Climaco (2000) proposed an approach that admits each DM to specify imprecise constraints on the parameters of an outranking model, then exploits a set of compatible parameters using robust assignment rule, and finally aggregates individual perspectives in a disjunctive or conjunctive manner (thus, not accounting for the DMs’ powers). The former accepts an assignment if it is justified by at least one DM, whereas the latter confirms some classification only if it is consistent with the preferences of all DMs. In this way, a group may agree on some result even if its members do not share the same model parameters. This idea was extended by Damart et al. (2007) to an interactive preference disaggregation approach that accepts assignment examples provided by different DMs. The method incorporates robustness analysis by deriving for each DM the possible class assignments (confirmed by at least one compatible preference model instance) and guides the group on sorting exemplary alternatives by exhibiting the levels of consensus between the DMs. Analogously, Shen et al. (2016) developed an adaptive approach under intuitionistic fuzzy environment that allows to reach a classification with an acceptable individual and group consensus levels. Moreover, de Morais Bezerra et al. (2017) enriched Electre TRIB with the tools for visualizing the comparison of individual results and procedures for guiding the changes of model parameters for deriving a better consensus.
Furthermore, Jabeur and Martel (2007) proposed a framework, which derives a collective sorting decision at the output level from the individual nonrobust classifications by additionally accounting for the relative importance of group members. Then, Morais et al. (2014) used a stochastic variant of Electre TRIB, called SMAATRI, to consider uncertainty in criteria weights and to derive for each DM the shares of the relevant parameter vectors that assign a given alternative to a certain category. An overview of thus obtained individual results leads to a collective recommendation. Conversely, Cailloux et al. (2012) employed assignment examples provided by multiple DMs for reaching an agreement at the input level. In particular, they proposed some linear programming models for deriving a joint set of boundary class profiles and veto thresholds.
As far as outrankingbased sorting approaches incorporating a model typical for PROMETHEE are concerned, Nemery (2008) extended the FlowSort method to group decision making. His proposal derives an assignment for each alternative from its relative comparison (strength and weakness) against the boundary or central class profiles specified by each individual DM. A similar idea was implemented by Lolli et al. (2015) in FlowSortGDSS. The underlying procedure derives class assignments by comparing comprehensive (global) net flows of alternatives and reference profiles. The proposed sorting rules distinguish between scenarios in which analysis of the individual assignments leads to either univocal or nonunanimous recommendation. Although the viewpoints of different DMs are aggregated at the output level, the method defines some consistency conditions on the preference information (in particular, reference profiles) provided by the individual DMs.
The majority of existing valuebased approaches derive a sorting recommendation incorporating robustness analysis and not differentiating between the roles played by the DMs. In particular, the \(\hbox {UTADIS}^\mathrm{GMS}\)GROUP method (Greco et al. 2012) accounts for the assignment examples provided by each DM and derives collective results that concern two levels of certainty. The first level refers to the necessary and possible consequences of individual preference information, which is typical for Robust Ordinal Regression (ROR) (Greco et al. 2010; Kadziński et al. 2015b). The other level is related to the necessity or possibility of a support that a particular assignment is given in the set of DMs. This method was further adapted by Liu et al. (2015) to account for the uncertain evaluations represented with the evidential reasoning approach, to provide some measures on the agreement between the DMs, and to derive a collective univocal assignment.
Conversely, Kadziński et al. (2013) aimed at a joint representation of assignment examples provided by all DMs by a set of additive value functions and investigating the necessary and possible consequences of applying the latter on the set of alternatives. When there is no value function compatible with preferences of all DMs, some linear programming techniques can be used to remove a minimal subset of inconsistent assignment examples. A similar approach was proposed by Cai et al. (2012), though additionally accounting for the DMs’ priorities. The latter ones intervene in the selection of a representative value function and in resolving inconsistency in the provided assignment examples. These priorities are updated with the progressive preference elicitation process to reflect the preciseness, quantity and consistency of the example decisions supplied by each DM.
Finally, when it comes to using “if ...then ...” decision rules for representing preferences of the DMs, one proposed various extensions of the Dominancebased Rough Set Approach (DRSA) (Greco et al. 2001). These accept preference information in form of individual assignment examples. First, Greco et al. (2006) introduced some concepts (e.g., multiunion and megaunion) related to dominance with respect to minimal profiles of evaluations provided by different DMs. Then, Chen et al. (2012) proposed to aggregate the recommendations suggested by individual linguistic decision rules into an overall assignment be means of a Dempster–Shafer Theory. The crucial concepts incorporated in the DRSA sorting method proposed by Sun and Ma (2015) are a dominance relation on the set of multiple sorting decisions (each provided by an individual DM) and a multiagent conflict analysis framework. Furthermore, Chakhar and Saad (2012) and Chakhar et al. (2016) illustrated how to combine individual approximations of class unions and derive collective decision rules that permit classification of all alternatives in a way consistent with the judgments of all DMs. These approaches measure the contribution of each expert to the collective assignment in terms of the individual quality of classification. Finally, Kadziński et al. (2016) adapted the principle of ROR to a group decision framework with DRSA, thus considering all sets of rules compatible with the individual assignment examples and combining their indications only at the output level.
In this paper, we propose an outrankingbased group decision approach that incorporates Electre TRIrC. Thus, it derives the assignments by comparing alternatives with the characteristic class profiles rather than with the boundary profiles as in Electre TRIB. The basic procedure we use takes into account a single preference model instance (incorporating criteria weights derived from the SRF procedure) for each DM and aggregates the individual viewpoints at the output level. While still aggregating the preferences at the output level, we extend the basic framework to offer results of robustness analysis with multiple sets of parameters compatible with the DMs’ value systems. Additionally, we propose a new algorithm for constructing a group compromise ranking of criteria, hence offering aggregation of the individual viewpoints also at the input level. At all stages, we assume that the involved stakeholders have the same importance degrees. Moreover, instead of providing precise assignments, our framework offers acceptability indices indicating the support that is given to the assignment of each alternative to various classes by different DMs and/or preference model instances compatible with their preferences.
3 Multiple Criteria Decision Analysis Method for the Assessment of Insulating Materials
This section describes a threestage multiple criteria decision analysis method that has been used to evaluate the insulating materials while taking into account preferences of a group of stakeholders. Firstly, we discuss the Electre TRIrC method (Kadziński et al. 2015b) that has been employed to assign the materials to a set of predefined and ordered classes. It incorporates the SRF procedure to compute the criteria weights (Figueira and Roy 2002). The method has been extended to a group decision setting to derive for each material some group class acceptability indices, which indicate the proportion of stakeholders that accept an assignment of the material to a given class. Secondly, we have adapted Stochastic Multicriteria Acceptability Analysis (SMAA; Lahdelma and Salminen 2001; Tervonen and Figueira 2008; Tervonen et al. 2007) to the context of Electre TRIrC and SRF procedure. It has been used to conduct robustness analysis (Roy 2010) for the results obtained in the first part, i.e., to validate their certainty while avoiding the arbitrary choice of criteria weights, which is conducted by the SRF procedure. Thirdly, we have proposed an algorithm for constructing a group compromise ranking of criteria based on the orders provided by the individual DMs. This ranking of criteria has been used as an input for SMAA to offer yet another view on the stability of computed results.

\(A=\left\{ {a_1 ,a_2 ,\ldots ,a_n } \right\} \) is a set of alternatives (insulating materials);

\(G=\left\{ {g_1 ,g_2 ,\ldots ,g_m } \right\} \) is a family of evaluation criteria that represent relevant points of view on the quality of assessed alternatives;

\(g_j \left( a \right) \) is the performance of alternative a with respect to criterion \(g_j \), \(j=1,\ldots ,m\) (when presenting the method, without loss of generality, we assume that all criteria are of gain type, i.e., the greater the performance, the better);

\(C_1 ,C_2 ,\ldots ,C_p \) are the preference ordered classes to which alternatives should be assigned; we assume that \(C_h \) is preferred to \(C_{h1} \) for \(h=2,\ldots ,p\).
3.1 Assessment of Insulating Materials Within a Group Decision Framework Incorporating Electre TRIrC and the SRF Procedure
In this section, we present the Electre TRIrC method (Kadziński et al. 2015b) that is used to assign the materials to a set of predefined and ordered classes. The method derives for each material a possibly imprecise assignment by constructing and exploiting an outranking relation S (Figueira et al. 2013). This relation quantifies an outcome of the comparison between the materials and a set of characteristic class profiles (Rezaei et al. 2017). In what follows, we discuss the main steps of the incorporated approach.
Step 1 For each class \(C_h \), provide the most typical (representative) performances on all criteria \(g_j , j=1,\ldots ,m,\) thus specifying the characteristic profiles \(b_h \), \(h=1,\ldots ,p\) (Almeida Dias et al. 2010). Defining such profiles was found intuitive and manageable by the involved experts, which was the main reason for incorporating Electre TRIrC in the study. The set of all characteristic profiles is denoted by B.
Steps 2–7 are conducted separately for each Decision Maker (\(DM_k \), \(k=1,\ldots ,K)\) in \(\partial ^{\mathrm{K}}=\left\{ {DM_1 ,DM_2 ,\ldots ,DM_K } \right\} \).

Assign some importance rank \(l^{k}\left( j \right) \) to each criterion \(g_j \); this is attained by ordering the cards with criteria names from the least to the most important (the greater \(l^{k}\left( j \right) \), the greater \(w_j^k \); some criteria can be assigned the same rank, thus being judged indifferent);

Quantify a difference between importance coefficients of the successive groups of criteria judged as indifferent, \(L_s^k \) and \(L_{s+1}^k \), by inserting \(e_s^k \) white (empty) cards between them (the greater \(e_s^k \), the greater the difference between the weights assigned to the criteria contained in \(L_{s+1}^k \) and \(L_s^k )\);

Specify ratio \(Z^{k}\) between the importances of the most and the least significant criteria denoted by \(L_{v\left( k \right) }^k \) and \(L_1^k \).

a being preferred to \(b_h \) (\(aS^{k}b_h \wedge not\left( {b_h S^{k}a} \right) \Rightarrow a\succ _k b_h )\);

\(b_h \) being preferred to a (\(b_h S^{k}a\wedge not\left( {aS^{k}b_h } \right) \Rightarrow b_h \succ _k a)\);

a being indifferent with \(b_h \) (\(aS^{k}b_h \wedge b_h S^{k}a \Rightarrow a\sim _k b_h )\);

a being incomparable with \(b_h \) (\(not\left( {aS^{k}b_h } \right) \wedge not\left( {b_h S^{k}a} \right) \Rightarrow a?_k b_h )\).
3.2 Stochastic Multicriteria Acceptability Analysis with Electre TRIrC
The SRF procedure derives the precise weight values from the ranking of criteria, intensities of preference, and ratio between the most and the least important criteria provided by \(DM_{k}\) applying some arbitrary rule (Figueira and Roy 2002). However, there exist multiple weight vectors compatible with such incomplete preference information. Recently, many researchers have raised the robustness concern in view of the SRF procedure to quantify the impact of uncertainty in the selection of an arbitrary weight vector on the stability of computed recommendation. In particular, Siskos and Tsotsolas (2015) proposed a set of robustness rules for the SRF procedure to obtain tangible and adequately supported results. Then, Govindan et al. (2017) suggested to exploit the whole set of compatible weight vectors to construct the necessary and possible results being confirmed by, respectively, all or at least one compatible vector. Further, Corrente et al. (2017) adapted the stochastic analysis of recommendation with the SRF procedure to the context of Electre III. We follow the latter research direction and integrate Stochastic Multicriteria Acceptability Analysis (Lahdelma and Salminen 2001; Tervonen et al. 2007) to handle possibly imprecise weight values compatible with the ranking of criteria and to derive robust recommendation with Electre TRIrC.

\(\left[ {O1} \right] \) ensures that criteria ranked better by \(DM_k\) will be assigned greater weight;

\(\left[ {O2} \right] \) guarantees that criteria deemed indifferent by \(DM_k \) will be assigned equal weights;

\(\left[ {O3} \right] \) sets the ratio Z between weights of the most and the least significant criteria;

\(\left[ {O4} \right] \) respects the intensities of preference for different pairs of criteria that have been quantified with the number of inserted empty cards;

\(\left[ {O5} \right] \) normalizes the weights.
3.3 Selection of a Group Compromise Ranking of Criteria
In this section, we introduce a procedure for deriving a compromise complete ranking of criteria based on the rankings provided individually by each \(DM_k\) within the SRF procedure. The procedure builds on the algorithm that was introduced by Govindan et al. (2017) for constructing a utilitarian ranking of alternatives. Hence, we adopt an idea of minimizing a sum of of distances between the compromise ranking and all individual rankings.
Definition of distances \(\delta \left( R_{k^{\prime }}^{jl} ,R_{k^{\prime \prime }}^{jl} \right) \) between different pairwise relations
\(R_{k^{\prime }}^{jl} \Big \backslash R_{k^{\prime \prime }}^{jl} \)  \(g_j \succ _{k^{\prime \prime }} g_l \left( \succ _{k^{\prime \prime }}^{jl} \right) \)  \(g_j \prec _{k^{\prime \prime }} g_l \quad \left( \prec _{k^{\prime \prime }}^{jl} \right) \)  \(g_j \sim _{k^{\prime \prime }} g_l\left( \sim _{k^{\prime \prime }}^{jl} \right) \) 

\(g_j \succ _{k^{\prime }} g_l \left( \succ _{k^{\prime }}^{jl} \right) \)  0  2  1 
\(g_j \prec _{k^{\prime }} g_l \left( \prec _{k^{\prime }}^{jl} \right) \)  2  0  1 
\(g_j \sim _{k^{\prime \prime }} g_l \left( \sim _{k^{\prime }}^{jl} \right) \)  1  1  0 

\(p_\partial ^{jl} \) represents a weak preference of \(g_j \) over \(g_l \) in the compromise ranking (i.e., in case \(p_\partial ^{jl} =1\), then \(g_j \succ _\partial g_l \) or \(g_j \sim _\partial g_l )\); note that \(p_\partial ^{jl} \) and \(p_\partial ^{lj} \) can be used to instantiate one of the three relations \(\succ _\partial ^{jl} \), \(\sim _\partial ^{jl} \), or \(\prec _\partial ^{jl} \) for \(g_j \) and \(g_l \); that is, if \(p_\partial ^{jl} \hbox {}=1\) and \(p_\partial ^{lj} =0\), then \(g_j \succ _\partial g_l \); if \(p_\partial ^{jl} =0\) and \(p_\partial ^{lj} =1\), then \(g_j \prec _\partial g_l \); if \(p_\partial ^{jl} =1\) and \(p_\partial ^{lj} =1\), then \(g_j \sim _\partial g_l \);

\(i_\partial ^{jl} \) represents an indifference \(\sim _\partial \) between \(g_j \) and \(g_l \) (i.e., in case \(p_\partial ^{jl} =1\) and \(p_\partial ^{lj} =1\), then \(i_\partial ^{jl} =1\) and \(g_j \sim _\partial g_l \); see [R3]).
Once a group compromise ranking of criteria is constructed, we conduct robustness analysis with SMAA in the same way as described in the previous section for an individual DM. This leads us to deriving cumulative group compromise class stochastic acceptability indices \(CuCCSAI^{\partial ^{\mathrm{K}}}\left( {a,h} \right) \).
3.4 Decision Aiding with the Proposed Approach
Multiple criteria sorting decisions can be aided with the proposed group decision making framework through the process illustrated in Fig. 1. It starts with specifying the sets of alternatives, criteria, and ordered classes as well as the alternatives’ evaluations (performances) on the criteria.
Further, the method derives three types of results. These indicate a support that is given to the assignment of considered alternatives to different classes via the application of Electre TRIrC for different sets of weights and cutting levels compatible with the preferences of the involved experts. In two cases, the preferences of the individual stakeholders are aggregated only at the output level. Depending on whether these individual preferences are processed using the SRF procedure or the Monte Carlo simulation, the method computes, respectively, group class acceptability indices or cumulative group class stochastic acceptability indices. In the third case, the preferences are aggregated at the input level by constructing a group compromise ranking of criteria. Then, the method applies SMAA to derive cumulative group compromise class stochastic acceptability indices.
Finally, these three types of outcomes should be analyzed and combined into the recommended assignments. This is straightforward in case the support given to the assignment of alternatives to decision classes by different results is similar. In case of ambiguous indications by different procedures, the inconsistency needs to be raised by a decision analyst.
Obviously, it is not required to use all three types of procedures and respective results for each study. This may be useful when offering different viewpoints on the robustness of sorting recommendation is desired. Otherwise, one can employ just a single procedure for processing the experts’ preferences depending on whether they should be aggregated at the input or output level and whether the robustness analysis should be incorporated into a particular study.
4 Results of Multiple Criteria Assessment of Insulating Materials with the Outranking Preference Model and Characteristic Class Profiles
The study aims at evaluating overall sustainability of coating materials used in buildings retrofitting. We consider 13 materials listed in Table 2 (they are denoted by \(A=\left\{ {a_1 ,a_2 ,\ldots ,a_{13} } \right\} )\). All materials having a thickness of 15cm were placed internally on the roof of a model building typical for central Italy, and evaluated from the socioeconomic and environmental viewpoints. The six relevant criteria which have been used to assess the materials are: hour of discomfort (\(g_1 \); DH),\(\hbox {CO}_{2}\) avoidance (\(g_2\)); Net Present Value (\(g_3\); NPV), human health (\(g_4\)); ecosystem quality (\(g_5\)), and consumed resources (\(g_6\)). In what follows, we explain their meaning.

Human health (\(g_4 \); the less, the better) which is derived from the analysis of the following normalized impact categories: carcinogens, respiratory organics and inorganics, climate change, radiation, and ozone layer;

Ecosystem quality (\(g_5 \); the less, the better) which is made up by the following three normalized impact categories: ecotoxicity, acidification/eutrophication, and land use;

Resources (\(g_6 \); the less, the better) which aggregates two normalized impact categories: minerals and fossil fuels.
Performances of 13 insulating materials with respect to 6 criteria
Insulating material  a  \(g_1\)  \(g_2\)  \(g_3\)  \(g_4\)  \(g_5\)  \(g_6\) 

Performance unit  –  Hours  kg of \(\hbox {CO}_{2}\)  €  Points  Points  Points 
Autoclave aerated complete  \(a_{1}\)  4889.339  158.63  283.41  0.009703  0.000636  0.015876 
Corkslab  \(a_{2}\)  3974.451  178.49  282.01  0.022122  0.018376  0.040660 
Expanded perlite  \(a_{3}\)  3893.646  179.11  326.26  0.006451  0.000759  0.043280 
Fibreboard hard  \(a_{4}\)  3657.799  185.29  243.45  0.039111  0.014516  0.136345 
Glass wool  \(a_{5}\)  3681.898  187.35  316.92  0.010608  0.001307  0.033364 
Gypsum fibreboard  \(a_{6}\)  7051.231  103.24  135.88  0.047131  0.003916  0.070469 
Hemp fibres  \(a_{7}\)  3921.449  182.59  334.10  0.002336  0.003079  0.008207 
Kenaf fibres  \(a_{8}\)  3685.510  186.82  341.79  0.004760  0.015137  0.003079 
Mineralized wood  \(a_{9}\)  4392.808  167.63  245.45  0.042932  0.004548  0.083149 
Plywood  \(a_{10}\)  7636.502  87.58  71.26  0.095717  0.201332  0.126167 
Polystyrene foam  \(a_{11}\)  3750.482  187.13  322.02  0.002801  0.000217  0.016521 
Polyurethane  \(a_{12}\)  3357.309  194.18  330.35  0.013225  0.000564  0.043280 
Rock wool  \(a_{13}\)  3659.441  188.45  346.14  0.019183  0.000825  0.009846 
The performances of 13 insulating materials with respect to 6 criteria are provided in Table 2. For all materials but hemp fibres, Ecoinvent Database (Ecoinvent 2010) was used as a source of foreground and background data related to both production and assembly processes as well as to the transport, electricity and fuel consumption. Instead, for the hemp processes the underlying data was derived from Zampori et al. (2013).
The objective of the case study is to give an easily interpretable comprehensive assessment of the materials’ sustainability. This is achieved by assigning them to a set of three predefined and ordered classes: \(C_1\) (low sustainability), \(C_2\) (medium sustainability), and \(C_3\) (high sustainability).
The study involved elicitation of preferences from the two groups of stakeholders. On one hand, a characteristic profile \(b_h\) for each class \(C_h\), \(h=1,2,3\), has been collectively specified by the experts from the universitybased engineering team specialized in the materials and retrofitting of rural buildings. On the other hand, the preferences on the importance of individual criteria have been elicited individually from multiple stakeholders who were owners of rural buildings interested in a renovation of their houses for improving the energetic performance. Thus, they can be perceived as potential consumers of the insulating materials.
Performances of the characteristic profiles for three classes
Profile  \(g_1\)  \(g_2\)  \(g_3\)  \(g_4\)  \(g_5\)  \(g_6\) 

\(b_1 \)  7051.231  158.63  135.88  0.042932  0.015137  0.083149 
\(b_2 \)  4392.808  182.59  283.41  0.013225  0.003079  0.043280 
\(b_3 \)  3659.441  187.35  330.35  0.004760  0.000636  0.009846 
4.1 Results of Multiple Criteria Assessment of the Insulating Materials Within a Group Decision Framework Incorporating Electre TRIrC and the SRF Procedure
The weights representing the importance of individual criteria have been elicited from the rural buildings’ owner. In what follows, we call them stakeholders. Overall, we approached 63 owners by explaining them the characteristics of different materials, the interpretation of all criteria and their relation to different phases of the materials’ life cycle. Among them, 38 stakeholders (let us denote them by \(\partial ^{\mathrm{K}}=\left\{ {DM_1 ,DM_2 ,\ldots ,DM_{38}}\right\} )\) claimed to understand the meaning and role of different criteria, and expressing their willingness to provide preferences on the criteria importance.
The order of cards with criteria names (ranks \(l^{k}\left( j \right) )\), white cards \(e_s^k \), and ratio \(Z^{k}\) provided by the three selected DMs in the SRF procedure, the weights \(w_j^k \) derived from the SRF procedure, and the cutting level \(\lambda ^{k}\)
\(DM_1 (Z^{1}=10, \lambda ^{1}=0.714\))  \(DM_2 (Z^{2}=5, \lambda ^{2}=0.696)\)  \(\cdots \)  \(DM_{38} (Z^{38}=5, \lambda ^{38}=0.682\))  

\(g_j \)  \(l^{1}\left( j \right) \)  \(e_s^1 \)  \(w_j^1 \)  \(g_j \)  \(l^{2}\left( j \right) \)  \(e_s^2 \)  \(w_j^2 \)  \(\cdots \)  \(g_j \)  \(l^{38}\left( j \right) \)  \(e_s^{38} \)  \(w_j^{38} \) 
\(g_1 \)  1  0.024  \(g_3 \)  1  0.049  \(\cdots \)  \(g_1 , \quad g_3 \)  1  0.045  
1  \(g_1 \)  2  0.088  2  
\(g_3 \)  2  0.085  1  \(\cdots \)  \(g_2 , \quad g_4 , \quad g_5 , \quad g_6 \)  2  0.227  
2  \(g_4 \)  3  0.167  \(\cdots \)  
\(g_2 \)  3  0.177  \(g_2 \)  4  0.206  \(\cdots \)  
1  \(g_5 , \quad g_6 \)  5  0.245  \(\cdots \)  
\(g_4 , \quad g_5 , \quad g_6 \)  4  0.238  \(\cdots \) 
Credibility indices and class assignments obtained with ELECTRE TRIrC for four exemplary materials for \(DM_1 \) (cutting level \(\lambda ^{1}=0.714)\)
\(b_1 \)  \(b_2 \)  \(b_3 \)  \(\left[ {C_L^1 \left( a \right) ,C_R^1 \left( a \right) } \right] \)  \(b_1 \)  \(b_2 \)  \(b_3 \)  \(\left[ {C_L^1 \left( a \right) ,C_R^1 \left( a \right) } \right] \)  

\(a_{1}\)  \(\succ \)  \(\succ \)  \(\prec \)  \(\left[ {C_2 ,C_2 } \right] \)  \(a_{6}\)  ?  \(\prec \)  \(\prec \)  \(\left[ {C_1 ,C_1 } \right] \) 
\(\sigma ^{1}\left( {a_1 ,b_h } \right) \)  1.000  0.799  0.238  \(\sigma ^{1}\left( {a_6 ,b_h } \right) \)  0.585  0.000  0.000  
\(\sigma ^{1}\left( {b_h ,a_1 } \right) \)  0.177  0.286  1.000  \(\sigma ^{1}\left( {b_h ,a_6 } \right) \)  0.524  1.000  1.000  
\(a_{11}\)  \(\succ \)  \(\succ \)  ?  \(\left[ {C_3 ,C_3 } \right] \)  \(a_{12}\)  \(\succ \)  \(\succ \)  \(\prec \)  \(\left[ {C_3 ,C_3 } \right] \) 
\(\sigma ^{1}\left( {a_{11} ,b_h } \right) \)  1.000  1.000  0.476  \(\sigma ^{1}\left( {a_{12} ,b_h } \right) \)  1.000  1.000  0.524  
\(\sigma ^{1}\left( {b_h ,a_{11} } \right) \)  0.000  0.000  0.524  \(\sigma ^{1}\left( {b_h ,a_{12} } \right) \)  0.000  0.476  0.738 
Class assignments obtained with Electre TRIrC for all materials and different stakeholders
a  \(DM_1 \)  \(DM_2 \)  \(DM_3 \)  \(DM_4 \)  \(DM_5 \)  \(DM_6 \)  \(DM_7 \)  \(DM_8 \)  \(DM_9 \)  \(DM_{10}\)  \(\cdots \)  \(DM_{38} \) 

\(a_{1}\)  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  \(\cdots \)  [\(C_{2}\),\(C_{2}\)] 
\(a_{2}\)  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  \(\cdots \)  [\(C_{2}\),\(C_{2}\)] 
\(a_{3}\)  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  \(\cdots \)  [\(C_{2}\),\(C_{2}\)] 
\(a_{4}\)  [\(C_{1}\),\(C_{1}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{1}\),\(C_{2}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{2}\),\(C_{3}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{2}\),\(C_{3}\)]  [\(C_{2}\),\(C_{2}\)]  \(\cdots \)  [\(C_{2}\),\(C_{2}\)] 
\(a_{5}\)  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  \(\cdots \)  [\(C_{3}\),\(C_{3}\)] 
\(a_{6}\)  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  \(\cdots \)  [\(C_{1}\),\(C_{1}\)] 
\(a_{7}\)  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{2}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{2}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  \(\cdots \)  [\(C_{3}\),\(C_{3}\)] 
\(a_{8}\)  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  \(\cdots \)  [\(C_{3}\),\(C_{3}\)] 
\(a_{9}\)  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{1}\),\(C_{1}\)]  \(\cdots \)  [\(C_{1}\),\(C_{1}\)] 
\(a_{10}\)  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{1}\)]  \(\cdots \)  [\(C_{1}\),\(C_{1}\)] 
\(a_{11}\)  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  \(\cdots \)  [\(C_{3}\),\(C_{3}\)] 
\(a_{12}\)  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  \(\cdots \)  [\(C_{3}\),\(C_{3}\)] 
\(a_{13}\)  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{2}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  \(\cdots \)  [\(C_{3}\),\(C_{3}\)] 
Group class acceptability indices \(E^{\partial }\left( {a,h} \right) \)
\(h \quad \backslash \quad a\)  \(a_{1}\)  \(a_{2}\)  \(a_{3}\)  \(a_{4}\)  \(a_{5}\)  \(a_{6}\)  \(a_{7}\)  \(a_{8}\)  \(a_{9}\)  \(a_{10}\)  \(a_{11}\)  \(a_{12}\)  \(a_{13}\) 

1  0.00  0.16  0.00  0.34  0.00  1.00  0.00  0.00  0.76  1.00  0.00  0.00  0.00 
2  0.95  1.00  1.00  0.76  0.00  0.00  0.21  0.16  0.24  0.00  0.00  0.00  0.08 
3  0.08  0.00  0.00  0.24  1.00  0.00  0.97  1.00  0.00  0.00  1.00  1.00  1.00 
The analysis of \(E^{\partial }\left( {a,h} \right) \) leads to indicating the assignments which are necessary (in case \(E^{\partial }\left( {a,h} \right) =1)\), possible (if \(E^{\partial }\left( {a,h} \right) >0)\), and impossible (if \(E^{\partial }\left( {a,h} \right) =0)\) in terms of the support they are provided in the group of stakeholders. Additionally, these results clearly indicate the most and the least probable assignments. In particular, for each material we are able to indicate the class with the greatest support among all stakeholders. It is \(C_{1}\) for \(a_{6},a_{9}\) and \(a_{10}\), \(C_{2}\) for \(a_{1}, a_{2}, a_{3}\) and \(a_{4}\), or \(C_{3}\) for \(a_{5}, a_{7}, a_{8}\), \(a_{11}, a_{12}\), and \(a_{13}\). The support which is given to the assignment of the materials to other classes is significantly smaller. For clarity of presentation, in all tables exhibiting stochastic acceptability indices (Tables 7, 8, 9 and 11), the text in bold indicates the class with the greatest support for a given material.
4.2 Results of Stochastic Multicriteria Acceptability Analysis with Electre TRIrC
To validate the recommendation for insulating materials against the arbitrary choice of weights conducted with the SRF procedure, we applied SMAA. For each stakeholder, we considered a sample of 10000 uniformly distributed weight vectors compatible with the ranking of criteria (s)he provided within the SRF procedure.
Class range stochastic acceptability indices \(CRSAI^{1}\left( {a,\left[ {L,R} \right] } \right) \) and cumulative class stochastic acceptability indices \(CuCSAI^{1}\left( {a,h} \right) \) for all materials for \(DM_1 \)
CRSAIs  CuCSAIs  

a  [\(C_{1}\),\(C_{1}\)]  [\(C_{1}\),\(C_{2}\)]  [\(C_{2}\),\(C_{2}\)]  [\(C_{1}\),\(C_{3}\)]  [\(C_{2}\),\(C_{3}\)]  [\(C_{3}\),\(C_{3}\)]  \(C_{1}\)  \(C_{2}\)  \(C_{3}\) 
\(a_{1}\)  0.000  0.000  0.825  0.000  0.000  0.175  0.000  0.825  0.175 
\(a_{2}\)  0.000  0.175  0.825  0.000  0.000  0.000  0.175  1.000  0.000 
\(a_{3}\)  0.000  0.000  1.000  0.000  0.000  0.000  0.000  1.000  0.000 
\(a_{4}\)  0.717  0.000  0.283  0.000  0.000  0.000  0.717  0.283  0.000 
\(a_{5}\)  0.000  0.000  0.000  0.000  0.000  1.000  0.000  0.000  1.000 
\(a_{6}\)  1.000  0.000  0.000  0.000  0.000  0.000  1.000  0.000  0.000 
\(a_{7}\)  0.000  0.000  0.000  0.000  0.000  1.000  0.000  0.000  1.000 
\(a_{8}\)  0.000  0.000  0.000  0.000  0.175  0.825  0.000  0.175  1.000 
\(a_{9}\)  1.000  0.000  0.000  0.000  0.000  0.000  1.000  0.000  0.000 
\(a_{10}\)  1.000  0.000  0.000  0.000  0.000  0.000  1.000  0.000  0.000 
\(a_{11}\)  0.000  0.000  0.000  0.000  0.000  1.000  0.000  0.000  1.000 
\(a_{12}\)  0.000  0.000  0.000  0.000  0.175  0.825  0.000  0.175  1.000 
\(a_{13}\)  0.000  0.000  0.000  0.000  0.175  0.825  0.000  0.175  1.000 
When it comes to a group decision perspective, the cumulative group class stochastic acceptability indices \(CuCSAI^{\partial ^{\mathrm{K}}}\left( {a,h} \right) \) are presented in Table 9. Their values are very similar to the group class acceptability indices \(E^{\partial }\left( {a,h} \right) \) reported in the previous section. The main differences concern a slightly increased support given to the minority class for some alternatives (see, e.g., \(a_{1}\) to \(C_{3}\), or \(a_{2}\) to \(C_{1}\), \(a_{8}\), and \(a_{12}\) to \(C_{2})\).
Cumulative group class stochastic acceptability indices \(CuCSAI^{\partial ^{\mathrm{K}}}\left( {a,h} \right) \) for all materials
\(h\backslash a\)  \(a_{1}\)  \(a_{2}\)  \(a_{3}\)  \(a_{4}\)  \(a_{5}\)  \(a_{6}\)  \(a_{7}\)  \(a_{8}\)  \(a_{9}\)  \(a_{10}\)  \(a_{11}\)  \(a_{12}\)  \(a_{13}\) 

1  0.018  0.226  0.000  0.338  0.000  1.000  0.000  0.000  0.794  1.000  0.000  0.000  0.000 
2  0.897  0.995  1.000  0.760  0.000  0.000  0.188  0.222  0.206  0.000  0.000  0.063  0.107 
3  0.131  0.000  0.000  0.213  1.000  0.000  0.986  0.998  0.000  0.000  1.000  1.000  0.999 
4.3 Results of Stochastic Multicriteria Acceptability Analysis for a Group Compromise Ranking of Criteria
The numbers of stakeholders indicating a preference or indifference (in the round brackets) for all pairs of criteria
\(g_j \)  \(g_1 \)  \(g_2 \)  \(g_3 \)  \(g_4 \)  \(g_5 \)  \(g_6 \) 

\(g_1 \)  –  14 (2)  22 (8)  15 (2)  11 (2)  10 (2) 
\(g_2 \)  22 (2)  –  25 (2)  17 (9)  4 (16)  3 (16) 
\(g_3 \)  8 (8)  11 (2)  –  12 (1)  9 (2)  9 (1) 
\(g_4 \)  21 (2)  12 (9)  25 (1)  –  6 (10)  7 (10) 
\(g_5 \)  24 (2)  18 (16)  27 (2)  22 (10)  –  1 (31) 
\(g_6 \)  26 (2)  19 (16)  28 (1)  21 (10)  6 (31)  – 
Obviously, one needs to bear in mind that the compromise ranking of criteria minimizes the sum of distances between relations observed for all pairs of criteria in all individual rankings. In this perspective, it may not be considered representative by all individuals (see, e.g., \(DM_7, DM_9 \), \(DM_{12}, DM_{17}, DM_{19}, DM_{20},\) or \(DM_{36} )\) whose preferences are represented in the compromise ranking to a marginal degree (i.e., an overall distance between their ranking and the compromise one is substantial).
Such a compromise ranking of criteria has been used to simulate DMs’ joint preferences within SMAA. Consistently with the previous sections, the cutting level \(\lambda \) was assumed to be equal to the sum of weights of the three most significant criteria. The results of robustness analysis are materialized with the cumulative group compromise class stochastic acceptability indices \(CuCCSAI^{\partial ^{\mathrm{K}}}\left( {a,h} \right) \) (see Table 11).
Cumulative group compromise class stochastic acceptability indices \(CuCCSAI^{\partial ^{\mathrm{K}}}\left( {a,h} \right) \) for all materials
\(h\backslash a\)  \(a_{1}\)  \(a_{2}\)  \(a_{3}\)  \(a_{4}\)  \(a_{5}\)  \(a_{6}\)  \(a_{7}\)  \(a_{8}\)  \(a_{9}\)  \(a_{10}\)  \(a_{11}\)  \(a_{12}\)  \(a_{13}\) 

1  0.000  0.419  0.000  0.533  0.000  1.000  0.000  0.000  1.000  1.000  0.000  0.000  0.000 
2  0.581  1.000  1.000  0.467  0.000  0.000  0.000  0.419  0.000  0.000  0.000  0.000  0.000 
3  0.419  0.000  0.000  0.000  1.000  0.000  1.000  1.000  0.000  0.000  1.000  1.000  1.000 
4.4 Summary

Low (\(C_1 )\): gypsum fibreboard (\(a_{6})\), mineralized wood (\(a_{9})\) and plywood (\(a_{10})\);

Low (\(C_1 )\) or medium (\(C_2 )\): fibreboard hard (\(a_{4})\);

Medium (\(C_2 )\): autoclave aerated complete (\(a_{1})\), corkslab (\(a_{2})\), and expanded perlite (\(a_{3})\);

High (\(C_3 )\): glass wool (\(a_{5})\), hemp fibres (\(a_{7})\), kenaf fibres (\(a_{8})\), polystyrene foam (\(a_{11})\), polyurethane (\(a_{12})\), and rock wool (\(a_{13})\).

\(a_{10}\) is worse than \(b_{1}\) on all criteria, thus being assigned to the worst class \(C_{1}\); in the same spirit, \(a_{6}\) is worse than \(b_{1}\) on \(g_{2}\), \(g_{4}\), and \(g_{5 }\) (thus, on 3 out of 4 considered environmental criteria), and not better than \(b_{2}\) on any criterion, which makes \(C_{1}\) its most desired class;

\(a_{3}\) is better than \(b_{1}\) and worse than \(b_{3}\) on all criteria, which makes its performance vector typical for \(C_{2}\);

\(a_{12}\) and \(a_{13}\) are at least as good as \(b_{2}\) on all criteria and better than \(b_{3}\) on four criteria (\(g_1 \), \(g_2 \), \(g_3 \), \(g_5\) or \(g_1 \), \(g_2 \), \(g_3 \), \(g_6\), respectively (note that both scenarios include two accounted socioeconomic criteria, \(g_1 \) and \(g_3 ))\), which makes their assignment to \(C_{3}\) the most justified.
Subsets of criteria on which the materials attain at least as good performances as these of the characteristic profiles \(b_1 \), \(b_2 \), and \(b_3 \) of three decision classes
Insulating material  a  \(b_1 \)  \(b_2 \)  \(b_3 \) 

Autoclave aerated  \(a_{1}\)  \(g_1 , g_2 ,\, g_3 , g_4 , g_5 , g_6 \)  \(g_3 , g_4 , g_5 , g_6 \)  \(g_5 \) 
Corkslab  \(a_{2}\)  \(g_1 , g_2 , \,g_3 , g_4 , g_6 \)  \(g_1 , g_3 , g_6 \)  
Expanded perlite  \(a_{3}\)  \(g_1 , g_2 ,\, g_3 , g_4 , g_5 , g_6 \)  \(g_1 , g_3 , g_4 , g_5 , g_6 \)  
Fibreboard hard  \(a_{4}\)  \(g_1 , g_2 , \,g_3 , g_4 , g_5 \)  \(g_1 , g_2 ,\, g_3 \)  \(g_1 \) 
Glass wool  \(a_{5}\)  \(g_1 , g_2 , \,g_3 , g_4 , g_5 , g_6 \)  \(g_1 , g_2 , g_3 , \,g_4 , g_5 , g_6 \)  \(g_2 \) 
Gypsum fibre board  \(a_{6}\)  \(g_1 , g_3 , g_6\)  
Hemp fibres  \(a_{7}\)  \(g_1 , g_2 , g_3 , g_4 , g_5 , g_6 \)  \(g_1 , g_2 , g_3 , g_4 , g_5 , g_6 \)  \(g_3 , g_4 , g_6 \) 
Kenaf fibres  \(a_{8}\)  \(g_1 , g_2 , g_3 , \,g_4 , g_5 , g_6 \)  \(g_3 , g_4 , g_6 \)  \(g_3 , g_4 , g_6 \) 
Mineralized wood  \(a_{9}\)  \(g_1 , g_2 ,\, g_3 , g_4 , g_5 , g_6 \)  \(g_1 , g_3 \)  
Plywood  \(a_{10}\)  
Polystyrene foam  \(a_{11}\)  \(g_1 , g_2 , \,g_3 , g_4 , g_5 , g_6 \)  \(g_1 , g_2 , g_3 ,\, g_4 , g_5 , g_6 \)  \(g_4 , g_5 \) 
Polyurethane  \(a_{12}\)  \(g_1 , g_2 ,\, g_3 , g_4 , g_5 , g_6 \)  \(g_1 , g_2 , g_3 , \,g_4 , g_5 , g_6 \)  \(g_1 , g_2 , g_3 ,\,g_5 \) 
Rock wool  \(a_{13}\)  \(g_1 , g_2 , g_3 ,\, g_4 , g_5 , g_6 \)  \(g_1 , g_2 , g_3 ,\, g_4 , g_5 , g_6 \)  \(g_1 , g_2 ,g_3 ,\,g_6 \) 
5 Conclusions
We considered a multiple criteria problem of sustainability assessment of insulating materials. We combined Life Cycle Costing, Life Cycle Assessment, and adaptive comfort evaluation to derive performances of these materials on six socioeconomic and environmental criteria. The comprehensive assessment of the materials involved their assignment to three preferenceordered sustainability classes. The classification was performed with a group decision counterpart of the Electre TRIrC method that compares alternatives with the characteristic class profiles defined by the experts.
To derive a recommendation that would reflect viewpoints of a wide spectrum of potential customers, we accounted for the preference information of a few tens of rural buildings’ owners being interested in the roof’s insulation. The initial recommendation was derived by computing the proportion of stakeholders who accepted an assignment of a particular material to a given class. These results were subsequently validated against the outcomes of a twofold robustness analysis realized with the Monte Carlo simulation. The latter exploited the space of all criteria weights compatible with either each stakeholder’s preference information provided in the SRF procedure or collective ranking of criteria that was derived with an original algorithm proposed in this paper.
The threestage analysis revealed that the most sustainable materials were glass wool, hemp fibres, kenaf fibres, polystyrene foam, polyurethane, and rock wool. This was mainly due to their favorable performances quantified with the Net Present Value and Ecoindicators. On the contrary, gypsum fibreboard, mineralized wood and plywood were assessed as the least sustainable materials. This can be justified in terms of their poor performances on thermal comfort, human health, and ecosystem quality. Overall, the proposed method provided greater clarity for decision making and guaranteed credibility in the eyes of the traditional rural houses’ owners. Moreover, all research results—concerning both materials’ performances on the individual criteria and comprehensive sorting recommendation—were well perceived by the experts on insulating materials in Italy.
The proposed framework can be applied to other decision contexts than that of a typical farmhouse in central Italy. This would require, however, accounting for a comfort model as well as warm and cold periods suitable to a particular geographical context, specification of a relevant lifespan for the investment, and adapting life cycle assessment to the reality of a particular study.
From the methodological viewpoint, we envisage the following future developments. Firstly, we plan to extend the SRF procedure to a group decision context so that it tolerates intensities of preference for different pairs of criteria and accepts information on different roles (weights) of the decision makers. Secondly, we aim at extending the proposed group decision framework to methods dealing with choice and ranking problems. This would require elaboration of the algorithms for deriving a compromise recommendation that would appropriately combine results of robustness analysis computed individually for each stakeholder.
Notes
Acknowledgements
The work of Miłosz Kadziński and Grzegorz Miebs was supported by the Polish Ministry of Science and Higher Education under the Iuventus Plus program in 2016–2019 Grant Number IP2015 029674  0296/IP2/2016/74.
Supplementary material
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