UTADIS: UTilités Additives DIScriminantes (Additive Utility Sorting)

Overview

Additive-utility preference-disaggregation sorting via LP-learned class thresholds. Output typically class_label (higher value = preferred).

Strengths

• •Method-specific: Additive-utility preference-disaggregation sorting via LP-learned class thresholds

Limitations

• •Assumes: Pre-defined ordered classes (C_1 P C_2 ... P C_Q) exist and are meaningful
• •Assumes: Decision-maker's reference classification of training alternatives is available and reliable
• •Assumes: Additive utility model is acceptable (criteria preferentially independent)
• •Assumes: Marginal utilities can be approximated by piecewise-linear functions on α_i breakpoints

Assumptions to verify

• •Pre-defined ordered classes (C_1 P C_2 ... P C_Q) exist and are meaningful
• •Decision-maker's reference classification of training alternatives is available and reliable
• •Additive utility model is acceptable (criteria preferentially independent)
• •Marginal utilities can be approximated by piecewise-linear functions on α_i breakpoints

When not to use

• •No reference classification available → use UTA (ranking) or unsupervised clustering
• •Non-ordered nominal classes → use UTADIS extensions or different sorting method (e.g., DRSA, ELECTRE-TRI nominal)
• •Strongly interacting criteria → consider Choquet integral or ANP-based sorting

Edge cases

• •if U(a) ≥ u_1; C_k if u_k ≤ U(a) < u_{k-1} (k = 2,…,Q−1); C_Q if U(a) < u_{Q-1}. Alternatives with non-zero σ⁺ or σ⁻ are misclassified by the LP: their classification by the rule above differs from t

Common pitfalls

• •Hatalı: 'UTADIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Pre-defined ordered classes (C_1 P C_2 ... P C_Q) exist and are meaningful
• •Hatalı: 'UTADIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision-maker's reference classification of training alternatives is available and reliable
• •Hatalı: 'UTADIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Additive utility model is acceptable (criteria preferentially independent)
• •Hatalı: 'UTADIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Marginal utilities can be approximated by piecewise-linear functions on α_i breakpoints
• •Hatalı: UTADIS'yi 'No reference classification available → use UTA (ranking) or unsupervised clustering' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: UTADIS'yi 'Non-ordered nominal classes → use UTADIS extensions or different sorting method (e.g., DRSA, ELECTRE-TRI nominal)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: UTADIS'yi 'Strongly interacting criteria → consider Choquet integral or ANP-based sorting' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: For each criterion i, define the value range G_i = [g_{i*}, g_i^*] (worst-to-best, direction-encoded by endpoint choice) and partition into α_i − 1 equal sub-intervals via α_i breakpoints g_i^j (Eq.(2)). u_i(g_{i*}) = 0 is enforced; u_i(g_i^*) > 0 is to be learned by the LP. Formül: g_{i}^{j} = g_{i*} + \frac{j-1}{\alpha_{i}-1}\bigl(g_{i}^{*} - g_{i*}\bigr),\quad j=1,\ldots,\alpha_{i} Anchor: Ulucan-Atıcı 2009 §1 Eq.(2)
2. 2.Adım 2 (F2): Step 2: For each alternative a and criterion i, locate the sub-interval [g_i^j, g_i^{j+1}] containing g_i(a) and compute the marginal utility by linear interpolation (Eq.(3)). Apply the Siskos-Yannacopoulos 1985 substitution w_{ij} = u_i(g_i^{j+1}) − u_i(g_i^j) ≥ 0 (Eq.(5)) so that u_i(g_i^j) = Σ_{k=1}^{j-1} w_{ik} (Eq.(6)); the global utility then has the linear form (Eq.(7)). Formül: u_{i}\!\bigl[g_{i}(a)\bigr] = \sum_{k=1}^{j-1} w_{ik} + \frac{g_{i}(a)-g_{i}^{j}}{g_{i}^{j+1}-g_{i}^{j}}\,w_{ij};\;\; U(a) = \sum_{i=1}^{m} u_{i}\!\bigl[g_{i}(a)\bigr] Anchor: Ulucan-Atıcı 2009 §1 Eqs.(5)-(8); Siskos-Yannacopoulos 1985 substitution
3. 3.Adım 3 (F3): Step 3: Formulate the sorting LP (Eqs.(21)-(27)). Decision variables: w_{ij} (utility increments), u_k (Q−1 class thresholds with u_1 > u_2 > … > u_{Q−1}), σ⁺(a) (lower-bound classification slack), σ⁻(a) (upper-bound classification slack). Objective (Eq.(21)): min F = Σ_{a∈C_1} σ⁺(a) + … + Σ_{a∈C_k} [σ⁺(a) + σ⁻(a)] + … + Σ_{a∈C_Q} σ⁻(a). Constraints: for C_1 (best): U(a) − u_1 + σ⁺(a) ≥ 0 (Eq.(22)); for C_k middle (k=2,…,Q−1): U(a) − u_{k−1} − σ⁻(a) ≤ −δ AND U(a) − u_k + σ⁺(a) ≥ 0 (Eq.(23)); for C_Q (worst): U(a) − u_{Q−1} − σ⁻(a) ≤ −δ (Eq.(24)). Normalisation (Eq.(25)): Σ_i Σ_j w_{ij} = 1. Threshold ordering (Eq.(26)): u_{k−1} − u_k ≥ s, k=2,…,Q−1. Non-negativity (Eq.(27)): w_{ij}, σ⁺, σ⁻ ≥ 0; u_k unconstrained in sign (but typically ≥ 0). Formül: \min F = \sum_{a\in C_{1}}\sigma^{+}(a) + \sum_{k=2}^{Q-1}\sum_{a\in C_{k}}\!\bigl[\sigma^{+}(a)+\sigma^{-}(a)\bigr] + \sum_{a\in C_{Q}}\sigma^{-}(a)\;\text{s.t. Eqs.(22)-(27)} Anchor: Ulucan-Atıcı 2009 §1 Eqs.(21)-(27); Zopounidis-Doumpos 1999a
4. 4.Adım 4 (F4): Step 4: Class assignment per Eqs.(14)-(17): compute U(a) = Σ_i u_i[g_i(a)] for every alternative (Eq.(8)); then C_1 if U(a) ≥ u_1; C_k if u_k ≤ U(a) < u_{k-1} (k = 2,…,Q−1); C_Q if U(a) < u_{Q-1}. Alternatives with non-zero σ⁺ or σ⁻ are misclassified by the LP: their classification by the rule above differs from their reference class, and the σ-values quantify the violation magnitude. Formül: a\in C_{1}\Leftrightarrow U(a)\ge u_{1};\;\; a\in C_{k}\Leftrightarrow u_{k}\le U(a)<u_{k-1};\;\; a\in C_{Q}\Leftrightarrow U(a)<u_{Q-1} Anchor: Ulucan-Atıcı 2009 §1 Eqs.(14)-(17)

Commonly paired with

• •endogenous (UTADIS-derived) (primary)

How to cite