1 Introduction
-
We formulate the considered problem in terms of multiple criteria sorting, thus aiming at assigning the materials to a set of pre-defined 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 outranking-based comparison of the insulating materials with the characteristic profiles composed of the per-class 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
3 Multiple Criteria Decision Analysis Method for the Assessment of Insulating Materials
-
\(A=\left\{ {a_1 ,a_2 ,\ldots ,a_n } \right\} \) 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 \(C_{h-1} \) for \(h=2,\ldots ,p\).
3.1 Assessment of Insulating Materials Within a Group Decision Framework Incorporating Electre TRI-rC and the SRF Procedure
-
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 Multi-criteria Acceptability Analysis with Electre TRI-rC
-
\(\left[ {O1} \right] \) 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
\(R_{k^{\prime }}^{jl} \Big \backslash R_{k^{\prime \prime }}^{jl} \)
|
\(g_j \succ _{k^{\prime \prime }} g_l \left( \succ _{k^{\prime \prime }}^{jl} \right) \)
|
\(g_j \prec _{k^{\prime \prime }} g_l \quad \left( \prec _{k^{\prime \prime }}^{jl} \right) \)
|
\(g_j \sim _{k^{\prime \prime }} g_l\left( \sim _{k^{\prime \prime }}^{jl} \right) \)
|
---|---|---|---|
\(g_j \succ _{k^{\prime }} g_l \left( \succ _{k^{\prime }}^{jl} \right) \)
| 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]).
3.4 Decision Aiding with the Proposed Approach
4 Results of Multiple Criteria Assessment of Insulating Materials with the Outranking Preference Model and Characteristic Class Profiles
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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.
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 |
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 TRI-rC and the SRF Procedure
\(DM_1 (Z^{1}=10, \lambda ^{1}=0.714\)) |
\(DM_2 (Z^{2}=5, \lambda ^{2}=0.696)\)
|
\(\cdots \)
|
\(DM_{38} (Z^{38}=5, \lambda ^{38}=0.682\)) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(g_j \)
|
\(l^{1}\left( j \right) \)
|
\(e_s^1 \)
|
\(w_j^1 \)
|
\(g_j \)
|
\(l^{2}\left( j \right) \)
|
\(e_s^2 \)
|
\(w_j^2 \)
|
\(\cdots \)
|
\(g_j \)
|
\(l^{38}\left( j \right) \)
|
\(e_s^{38} \)
|
\(w_j^{38} \)
|
\(g_1 \)
| 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 |
\(\cdots \)
|
\(b_1 \)
|
\(b_2 \)
|
\(b_3 \)
|
\(\left[ {C_L^1 \left( a \right) ,C_R^1 \left( a \right) } \right] \)
|
\(b_1 \)
|
\(b_2 \)
|
\(b_3 \)
|
\(\left[ {C_L^1 \left( a \right) ,C_R^1 \left( a \right) } \right] \)
| ||
---|---|---|---|---|---|---|---|---|---|
\(a_{1}\)
|
\(\succ \)
|
\(\succ \)
|
\(\prec \)
|
\(\left[ {C_2 ,C_2 } \right] \)
|
\(a_{6}\)
| ? |
\(\prec \)
|
\(\prec \)
|
\(\left[ {C_1 ,C_1 } \right] \)
|
\(\sigma ^{1}\left( {a_1 ,b_h } \right) \)
| 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 |
a
|
\(DM_1 \)
|
\(DM_2 \)
|
\(DM_3 \)
|
\(DM_4 \)
|
\(DM_5 \)
|
\(DM_6 \)
|
\(DM_7 \)
|
\(DM_8 \)
|
\(DM_9 \)
|
\(DM_{10}\)
|
\(\cdots \)
|
\(DM_{38} \)
|
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(a_{1}\)
| [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] |
\(\cdots \)
| [\(C_{2}\),\(C_{2}\)] |
\(a_{2}\)
| [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] |
\(\cdots \)
| [\(C_{2}\),\(C_{2}\)] |
\(a_{3}\)
| [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{2}\)] |
\(\cdots \)
| [\(C_{2}\),\(C_{2}\)] |
\(a_{4}\)
| [\(C_{1}\),\(C_{1}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{1}\),\(C_{2}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{2}\),\(C_{3}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{2}\),\(C_{3}\)] | [\(C_{2}\),\(C_{2}\)] |
\(\cdots \)
| [\(C_{2}\),\(C_{2}\)] |
\(a_{5}\)
| [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] |
\(\cdots \)
| [\(C_{3}\),\(C_{3}\)] |
\(a_{6}\)
| [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] |
\(\cdots \)
| [\(C_{1}\),\(C_{1}\)] |
\(a_{7}\)
| [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{2}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{2}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] |
\(\cdots \)
| [\(C_{3}\),\(C_{3}\)] |
\(a_{8}\)
| [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] |
\(\cdots \)
| [\(C_{3}\),\(C_{3}\)] |
\(a_{9}\)
| [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{2}\),\(C_{2}\)] | [\(C_{1}\),\(C_{1}\)] |
\(\cdots \)
| [\(C_{1}\),\(C_{1}\)] |
\(a_{10}\)
| [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] | [\(C_{1}\),\(C_{1}\)] |
\(\cdots \)
| [\(C_{1}\),\(C_{1}\)] |
\(a_{11}\)
| [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] |
\(\cdots \)
| [\(C_{3}\),\(C_{3}\)] |
\(a_{12}\)
| [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] |
\(\cdots \)
| [\(C_{3}\),\(C_{3}\)] |
\(a_{13}\)
| [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{2}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] | [\(C_{3}\),\(C_{3}\)] |
\(\cdots \)
| [\(C_{3}\),\(C_{3}\)] |
\(h \quad \backslash \quad a\)
|
\(a_{1}\)
|
\(a_{2}\)
|
\(a_{3}\)
|
\(a_{4}\)
|
\(a_{5}\)
|
\(a_{6}\)
|
\(a_{7}\)
|
\(a_{8}\)
|
\(a_{9}\)
|
\(a_{10}\)
|
\(a_{11}\)
|
\(a_{12}\)
|
\(a_{13}\)
|
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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
|
4.2 Results of Stochastic Multi-criteria Acceptability Analysis with Electre TRI-rC
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
|
\(h\backslash a\)
|
\(a_{1}\)
|
\(a_{2}\)
|
\(a_{3}\)
|
\(a_{4}\)
|
\(a_{5}\)
|
\(a_{6}\)
|
\(a_{7}\)
|
\(a_{8}\)
|
\(a_{9}\)
|
\(a_{10}\)
|
\(a_{11}\)
|
\(a_{12}\)
|
\(a_{13}\)
|
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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 Multi-criteria Acceptability Analysis for a Group Compromise Ranking 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) | – |
\(h\backslash a\)
|
\(a_{1}\)
|
\(a_{2}\)
|
\(a_{3}\)
|
\(a_{4}\)
|
\(a_{5}\)
|
\(a_{6}\)
|
\(a_{7}\)
|
\(a_{8}\)
|
\(a_{9}\)
|
\(a_{10}\)
|
\(a_{11}\)
|
\(a_{12}\)
|
\(a_{13}\)
|
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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 socio-economic criteria, \(g_1 \) and \(g_3 ))\), which makes their assignment to \(C_{3}\) the most justified.
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 |
\(a_{13}\)
|
\(g_1 , g_2 , g_3 ,\, g_4 , g_5 , g_6 \)
|
\(g_1 , g_2 , g_3 ,\, g_4 , g_5 , g_6 \)
|
\(g_1 , g_2 ,g_3 ,\,g_6 \)
|