1 Introduction
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in case alternative \(a\) is necessarily preferred to alternative \(b\), what are the minimal sets of pairwise comparisons provided by the DM that induce a necessary preference for a pair \((a,b)\) in consequence of using all compatible value functions;
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in case alternative \(a\) is not even possibly preferred to alternative \(b\), what are the maximal sets of the provided pairwise comparisons that admit a possible preference relation for a pair \((a,b)\) in consequence of using at least one compatible value function;
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in case alternative \(a\) is ranked at positions between \(P^*(a)\) and \(P_*(a)\) (\(P^*(a) \le P_*(a)\)) by all compatible value functions, what are the minimal sets of the provided pairwise comparisons that imply such a ranking interval, and what are the maximal sets of provided pairwise comparisons that admit \(a\) being ranked higher than \(P^*(a)\) or lower than \(P_*(a)\).
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in case alternative \(a\) is necessarily assigned to class \(C_h\), what are the minimal sets of assignment examples provided by the DM that induce such a necessary assignment;
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in case alternative \(a\) is possibly assigned to an interval of classes \([C_L, C_R]\), what are the minimal sets of the provided assignment examples that imply such a possible assignment, and what are the maximal sets of assignment examples that admit \(a\) being assigned to a class worse than \(C_L\) or better than \(C_R\).
2 Notation
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\(A=\{ a_1, a_2, \ldots , a_i, \ldots , a_n \}\)—a finite set of \(n\) alternatives;
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\(A^\mathrm{R} = \{ a^{*}, b^{*}, \ldots \}\)—a finite set of reference alternatives, on which the DM accepts to express holistic preferences; we assume that \(A^\mathrm{R} \subseteq A\);
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\(G = \{ g_1, g_2,\ldots , g_j, \ldots , g_m \}\)—a finite set of \(m\) evaluation criteria, \(g_j:~A\rightarrow \mathbb {R}\) for all \(j \in J=\{ 1,2, \ldots , m \}\); although real-coded they may have ordinal or cardinal scales;
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\(C_h, h=1, \ldots , p\)—pre-defined preference ordered classes such that \(C_{h+1}\) is preferred to \(C_h\), \(h=1, \ldots , p-1\); \(H=\{1, 2, \ldots , p\}\);
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\(X_j = \{ x_j \in \mathbb {R} : g_j(a_i)=x_j, a_i \in A \}\)—the set of all different evaluations on \(g_j\), \(j \in J\); without loss of generality, we assume that all criteria have an increasing direction of preference, i.e., the greater \(g_j(a_i)\), the more preferred alternative \(a_i\) on criterion \(g_j\), for all \(j \in J\) and \(a_i \in A\);
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\(x_j^1, x_j^2, \ldots , x_j^{n_j(A)}\)—the ordered values of \(X_j\), \(x_j^k < x_j^{k+1}, k=1, \ldots , n_j(A)-1\), where \(n_j(A) = |X_j|\) and \(n_j(A) \le n\); consequently, \(X =\prod _{j=1}^m X_j\) is the evaluation space.
3 Robust ordinal regression based on multi-attribute value theory
3.1 Robust ordinal regression for multiple criteria ranking
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Necessary weak preference relation, \(\succsim ^\mathrm{N}\), that holds for a pair of alternatives \((a,b) \in A \times A\), in case \(U(a) \ge U(b)\) for all compatible value functions;
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Possible weak preference relation, \(\succsim ^\mathrm{P}\), that holds for a pair of alternatives \((a,b) \in A \times A\), in case \(U(a) \ge U(b)\) for at least one compatible value function.
Country |
\(g_1\)
|
\(g_2\)
|
\(g_3\)
|
\(g_4\)
|
\(g_5\)
| |
---|---|---|---|---|---|---|
1 | Australia (AU) | 10.00 | 8.93 | 7.78 | 8.75 | 10.00 |
2 | China (CN) | 0.00 | 4.64 | 2.78 | 6.25 | 1.18 |
3 | Fiji (FJ) | 6.50 | 5.21 | 3.33 | 5.00 | 8.24 |
4 | Indonesia (ID) | 6.92 | 7.14 | 5.00 | 6.25 | 6.76 |
5 | Japan (JP) | 9.17 | 7.86 | 5.56 | 8.75 | 9.41 |
6 | Kazkhstan (KZ) | 2.67 | 2.14 | 3.33 | 4.38 | 5.59 |
7 | Kyrgyzstan (KG) | 5.75 | 1.86 | 2.78 | 5.00 | 5.00 |
8 | Laos (LA) | 0.00 | 3.21 | 1.11 | 5.00 | 1.18 |
9 | Malayasia (MY) | 6.08 | 5.71 | 4.44 | 7.50 | 6.18 |
10 | Mongolia (MN) | 9.17 | 6.08 | 3.89 | 5.63 | 8.24 |
11 | Myanmar (MM) | 0.00 | 1.79 | 0.56 | 5.63 | 0.88 |
12 | New Zealand (NZ) | 10.00 | 8.57 | 8.33 | 8.13 | 10.00 |
13 | North Korea (KP) | 0.83 | 2.50 | 0.56 | 1.25 | 0.00 |
14 | Papua New G. (PG) | 7.33 | 6.43 | 4.44 | 6.25 | 8.24 |
15 | Philippines (PH) | 9.17 | 5.36 | 5.00 | 3.75 | 9.12 |
16 | Singapore (SG) | 4.33 | 7.50 | 2.78 | 7.50 | 7.35 |
17 | South Korea (KR) | 9.58 | 7.14 | 7.22 | 7.50 | 7.94 |
18 | Taiwan (TW) | 9.58 | 7.50 | 6.67 | 5.63 | 9.71 |
19 | Tajikistan (TJ) | 1.83 | 0.79 | 2.22 | 6.28 | 1.18 |
20 | Thailand (TH) | 4.83 | 6.43 | 5.00 | 5.63 | 6.47 |
21 | Timor Leste (TL) | 7.00 | 5.57 | 5.00 | 6.25 | 8.24 |
22 | Turkmenistan (TM) | 0.00 | 0.79 | 2.78 | 5.00 | 0.59 |
23 | Uzbekistan (UZ) | 0.08 | 0.79 | 2.78 | 5.00 | 0.59 |
24 | Vietnam (VN) | 0.83 | 4.29 | 2.78 | 4.38 | 1.47 |
\(B^\mathrm{R}_1 = (\mathrm{JP} \succ \mathrm{KR})\)
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\(B^\mathrm{R}_2 = (\mathrm{FJ} \succ \mathrm{TH})\)
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\(B^\mathrm{R}_3 = (\mathrm{PH} \succ \mathrm{MN})\)
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\(B^\mathrm{R}_4 = (\mathrm{ID} \succ \mathrm{MY})\)
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\(B^\mathrm{R}_5 = (\mathrm{KG} \succ \mathrm{KZ})\)
|
Country |
\(P^*(a)-P_*(a)\)
|
\(P^*_D(a)-P^D_*(a)\)
| RPRs | RCORE |
---|---|---|---|---|
AU | 1–2 | 1–2 | – | – |
CN | 17–20 | 10–21 |
\(\{1,2,3\}\)
|
\(\{1,2,3\}\)
|
FJ | 10–13 | 9–16 |
\(\{2,5\}\)
|
\(\{2,5\}\)
|
ID | 8–12 | 5–13 |
\(\{2,3,4\}\), \(\{2,3,5\}\)
|
\(\{2,3\}\)
|
JP | 3–4 | 3–6 |
\(\{1\}\)
|
\(\{1\}\)
|
KZ | 16–16 | 14–23 |
\(\{2,3,5\}\)
|
\(\{2,3,5\}\)
|
KG | 13–15 | 13–21 |
\(\{2,5\}\)
|
\(\{2,5\}\)
|
LA | 19–23 | 15–24 |
\(\{1,2\}\), \(\{1,3\}\), \(\{2,3\}\), \(\{3,5\}\)
| – |
MY | 11–14 | 5–14 |
\(\{2,3,4\}\), \(\{2,3,5\}\)
|
\(\{2,3\}\)
|
MN | 6–9 | 4–14 |
\(\{2,3,5\}\)
|
\(\{2,3,5\}\)
|
MM | 20–24 | 15–24 |
\(\{3\}\), \(\{2,4\}\)
| – |
NZ | 1–2 | 1–2 | – | – |
KP | 19–24 | 18–24 |
\(\{2\}\), \(\{3\}\)
| – |
PG | 6–9 | 6–13 |
\(\{1,2\}\), \(\{1,3\}\)
|
\(\{1\}\)
|
PH | 5–8 | 5–23 |
\(\{2,3,5+\}\)
|
\(\{2,3,5\}\)
|
SG | 9–15 | 4–16 |
\(\{1,3,5\}\), \(\{2,3,5\}\)
|
\(\{3,5\}\)
|
KR | 4–7 | 3–10 |
\(\{1,5\}\)
|
\(\{1,5\}\)
|
TW | 3–5 | 3–12 |
\(\{3\}\)
|
\(\{3\}\)
|
TJ | 17–22 | 10–24 |
\(\{2,3\}\), \(\{1,3,5\}\)
|
\(\{3\}\)
|
TH | 12–15 | 7–15 |
\(\{2\}\)
|
\(\{2\}\)
|
TL | 6–11 | 4–13 |
\(\{2,3\}\), \(\{3,5\}\)
|
\(\{3\}\)
|
TM | 21–24 | 18–24 |
\(\{1,4\}\), \(\{2,3\}\), \(\{2,4\}\)
| – |
UZ | 20–23 | 18–23 |
\(\{1,3\}\), \(\{1,4\}\)
|
\(\{1\}\)
|
VN | 17–19 | 15–23 |
\(\{2,3\}\)
|
\(\{2,3\}\)
|
3.2 Robust ordinal regression for multiple criteria sorting
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The possible assignment \(C_\mathrm{P}(a)\) is defined as the set of indices of classes \(C_h\) for which there exists at least one compatible pair \((U,\mathbf t )\) assigning \(a\) to \(C_h\) (denoted by \(a \rightarrow ^\mathrm{P} C_h\));
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The necessary assignment \(C_\mathrm{N}(a)\) is defined as the set of indices of classes \(C_h\) for which all compatible pairs \((U,\mathbf t )\) assign \(a\) to \(C_h\) (denoted by \(a \rightarrow ^\mathrm{N} C_h\)).
\(A^\mathrm{R}_1 = (\mathrm{JP} \rightarrow C_4)\)
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\(A^\mathrm{R}_2 = (\mathrm{KR} \rightarrow C_4)\)
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\(A^\mathrm{R}_3 = (\mathrm{TW} \rightarrow C_3)\)
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\(A^\mathrm{R}_4 = (\mathrm{TH} \rightarrow C_3)\)
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\(A^\mathrm{R}_5 = (\mathrm{SG} \rightarrow C_3)\)
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\(A^\mathrm{R}_6 = (\mathrm{KZ} \rightarrow C_2)\)
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\(A^\mathrm{R}_7 = (\mathrm{KG} \rightarrow C_2)\)
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\(A^\mathrm{R}_8 = (\mathrm{CN} \rightarrow C_1)\)
|
Country |
\(C^\mathrm{P}_L-C^\mathrm{P}_R\)
|
\(C^\mathrm{N}\)
| APRs | ACORE |
---|---|---|---|---|
AU |
\(C4\)
|
\(C4\)
|
\(\{1\}\), \(\{2\}\)
| – |
CN |
\(C1\)
|
\(C1\)
|
\(\{8\}\)
|
\(\{8\}\)
|
FJ |
\(C2-C3\)
| – |
\(\{3,5,7\}\), \(\{3,6,7\}\)
|
\(\{3,7\}\)
|
ID |
\(C3\)
|
\(C3\)
|
\(\{3,4,5\}\), \(\{3,4,6,7,8\}\)
|
\(\{3,4\}\)
|
JP |
\(C4\)
|
\(C4\)
|
\(\{1\}\)
|
\(\{1\}\)
|
KZ |
\(C2\)
|
\(C2\)
|
\(\{6\}\)
|
\(\{6\}\)
|
KG |
\(C2\)
|
\(C2\)
|
\(\{7\}\)
|
\(\{7\}\)
|
LA |
\(C1\)
|
\(C1\)
|
\(\{8\}\)
|
\(\{8\}\)
|
MY |
\(C2-C4\)
| – |
\(\{6\}\), \(\{7\}\)
| – |
MN |
\(C2-C3\)
| – |
\(\{3,6\}\), \(\{3,7\}\)
|
\(\{3\}\)
|
MM |
\(C1\)
|
\(C1\)
|
\(\{8\}\)
|
\(\{8\}\)
|
NZ |
\(C4\)
|
\(C4\)
|
\(\{1\}\), \(\{2\}\)
| – |
KP |
\(C1\)
|
\(C1\)
|
\(\{1,3,4,6,8\}\), \(\{1,3,5,6,8\}\)
|
\(\{1,3,6,8\}\)
|
PG |
\(C2-C3\)
| – |
\(\{3,5,6\}\), \(\{3,5,7\}\), \(\{3,7,8\}\)
|
\(\{3\}\)
|
PH |
\(C2-C3\)
| – |
\(\{3,7,8\}\)
|
\(\{3,7,8\}\)
|
SG |
\(C3\)
|
\(C3\)
|
\(\{5\}\)
|
\(\{5\}\)
|
KR |
\(C4\)
|
\(C4\)
|
\(\{2\}\)
|
\(\{2\}\)
|
TW |
\(C3\)
|
\(C3\)
|
\(\{3\}\)
|
\(\{3\}\)
|
TJ |
\(C1-C2\)
| – |
\(\{6,7,8\}\)
|
\(\{6,7,8\}\)
|
TH |
\(C3\)
|
\(C3\)
|
\(\{4\}\)
|
\(\{4\}\)
|
TL |
\(C3\)
|
\(C3\)
|
\(\{3,4,5,6,8\}\), \(\{3,4,6,7,8\}\), \(\{3,4,5,7,8\}\)
|
\(\{3,4,8\}\)
|
TM |
\(C1\)
|
\(C1\)
|
\(\{8\}\)
|
\(\{8\}\)
|
UZ |
\(C1\)
|
\(C1\)
|
\(\{5,6,8\}\)
|
\(\{5,6,8\}\)
|
VN |
\(C1-C2\)
| – |
\(\{6,8\}\), \(\{7,8\}\)
|
\(\{8\}\)
|
4 Explaining recommendation in terms of decision maker’s preference information
4.1 Explaining recommendation of robust multiple criteria ranking
4.1.1 Necessary and possible preference relation
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Since \(E^{A^\mathrm{R}}_\mathrm{RANK}\) is feasible, IIS in \(E^\mathrm{N}(a,b)\) needs to contain constraint \(U(b) - U(a) \ge \varepsilon \). Thus, IIS contains constraints on monotonicity and normalization of the preference model and/or constraints corresponding to pairwise comparisons provided by the DM that are in conflict with \(U(b)>U(a)\), i.e., induce \(U(a) \ge U(b)\) for all compatible value functions.
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A preferential reduct for \(a \succsim ^\mathrm{N} b\) is equivalent to a set of pairwise comparisons whose corresponding constraints are included in an IIS in \(E^\mathrm{N}(a,b)\).
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\(\mathrm{PR(MN} \succsim ^\mathrm{N} \mathrm{SG}) = \{B^\mathrm{R}_3, B^\mathrm{R}_5\}\Rightarrow \) “the necessary preference of MN over SG is a consequence of the DM’s two pairwise comparisons: \(B^\mathrm{R}_3 = (\mathrm{PH} \succ \mathrm{MN})\) and \(B^\mathrm{R}_5 = (\mathrm{KG} \succ \mathrm{KZ})\)”;
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\(\mathrm{PR_1(TW} \succsim ^\mathrm{N} \mathrm{TL}) = \{B^\mathrm{R}_3\}\) and \(\mathrm{PR_2(TW} \succsim ^\mathrm{N} \mathrm{TL}) = \{B^\mathrm{R}_4\}\) \(\Rightarrow \) “the necessary preference of TW over TL is a consequence of the DM’s single pairwise comparison: either \(B^\mathrm{R}_3 = (\mathrm{PH} \succ \mathrm{MN})\) or \(B^\mathrm{R}_4 = (\mathrm{ID} \succ \mathrm{MY})\)”.
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Since \(E^{A^\mathrm{R}}_\mathrm{RANK}\) is feasible, constraint \(U(a) - U(b) \ge 0\) is in conflict with some constraints of \(E^{A^\mathrm{R}}_\mathrm{RANK}\), thus making \(E^\mathrm{P}(a,b)\) infeasible.
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Preferential construct for \(\mathrm{not}(a \succsim ^\mathrm{P} b)\) is equivalent to a set of pairwise comparisons whose corresponding constraints are included in a MFS together with constraint set \(E^{A^\mathrm{R}}_\mathrm{BASE}\) and constraint \(U(a) - U(b) \ge 0\).
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Identifying such a MFS is equivalent to finding a minimal set of pairwise comparisons whose corresponding constraints need to be removed from \(E^\mathrm{P}(a,b)\) so that the remaining constraint set is feasible.
4.1.2 Extreme ranking analysis
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\(\mathrm{RPR(TW)} = \{B^\mathrm{R}_3\}\), \(P^*(\mathrm{TW})=P^*_D(\mathrm{TW})=3\) and \(P_*(\mathrm{TW})=5 < P_*^D(\mathrm{TW})=12\) \(\Rightarrow \) “TW is not ranked at a position worse than \(5\) (i.e., between \(6\) and \(12\)) by any compatible value function, because the DM provided a pairwise comparison \(B^\mathrm{R}_3 = (\mathrm{PH} \succ \mathrm{MN})\)”;
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\(\mathrm{RPR(MN)} = \{B^\mathrm{R}_2, B^\mathrm{R}_3, B^\mathrm{R}_5\}\), \(P^*(\mathrm{MN})=6 > P^*_D(\mathrm{MN})=4\) and \(P_*(\mathrm{MN})=9 < P_*^D(\mathrm{MN})=14\) \(\Rightarrow \) “MN is not ranked at a position better than \(6\) (i.e., between \(4\) and \(5\)) nor worse than \(9\) (i.e., between \(10\) and \(14\)) by any compatible value function, because the DM provided pairwise comparisons \(B^\mathrm{R}_2 = (\mathrm{FJ} \succ \mathrm{TH})\), \(B^\mathrm{R}_3 = (\mathrm{PH} \succ \mathrm{MN})\), and \(B^\mathrm{R}_5 = (\mathrm{KG} \succ \mathrm{KZ})\)”.
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There is a single upper rank preferential construct \(\mathrm{URPC(TJ)} = \{B^\mathrm{R}_1, B^\mathrm{R}_2,\) \(B^\mathrm{R}_4, B^\mathrm{R}_5\}\), which means that “\(B^\mathrm{R}_3\) is the only pairwise comparison in \(B^\mathrm{R}\) preventing \(TJ\) from being ranked better than 17th”;
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There are two lower rank preferential constructs \(\mathrm{LRPC_1(TJ)} = \{B^\mathrm{R}_1,\) \(B^\mathrm{R}_3, B^\mathrm{R}_4\}\) and \(\mathrm{LRPC_2(TJ)} = \{B^\mathrm{R}_3, B^\mathrm{R}_4, B^\mathrm{R}_5\}\), which means that “either (\(B^\mathrm{R}_2\) and \(B^\mathrm{R}_5\)) or (\(B^\mathrm{R}_1\) and \(B^\mathrm{R}_2\)) need to be removed from the set of provided pairwise comparisons \(B^\mathrm{R}\) to allow TJ being ranked worse than 22nd”.
4.2 Explaining recommendation of robust multiple criteria sorting
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\(\mathrm{APR_1(AU)} = \{A^\mathrm{R}_1\}\) and \(\mathrm{APR_2(AU)} = \{A^\mathrm{R}_2\}\) \(\Rightarrow \) “the necessary assignment of AU to \(C_4\) is a consequence of the DM’s single assignment example, either \(A^\mathrm{R}_1 = (\mathrm{JP} \rightarrow C_4)\) or \(A^\mathrm{R}_2 = (\mathrm{KR} \rightarrow C_4)\)”;
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\(\mathrm{APR(SG)} = \{A^\mathrm{R}_3, A^\mathrm{R}_7, A^\mathrm{R}_8\}\) \(\Rightarrow \) “SG is neither assigned to a class better than \(C_3\) nor to a class worse than \(C_2\) by any compatible value function, because the DM provided the following assignment examples: \(A^\mathrm{R}_3 = (\mathrm{TW} \rightarrow C_3)\), \(A^\mathrm{R}_5 = (\mathrm{SG} \rightarrow C_3)\), and \(A^\mathrm{R}_8 = (\mathrm{CN} \rightarrow C_1)\)”.