UTA: UTilités Additives (Additive Utility Assessment)

Overview

Regression-based additive utility elicitation (LP, single-error). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Regression-based additive utility elicitation (LP, single-error)

Limitations

• •Assumes: Decision-maker's holistic ranking of the reference set is available and reliable
• •Assumes: An additive utility model is acceptable (criteria preferentially independent)
• •Assumes: Marginal utilities can be reasonably approximated by piecewise-linear functions on α_i breakpoints

Assumptions to verify

• •Decision-maker's holistic ranking of the reference set is available and reliable
• •An additive utility model is acceptable (criteria preferentially independent)
• •Marginal utilities can be reasonably approximated by piecewise-linear functions on α_i breakpoints

When not to use

• •No holistic reference ranking is available → use ELECTRE/PROMETHEE (outranking) or weight-based MAVT
• •Strongly interacting criteria → consider Choquet integral or ANP
• •Pure sorting/classification problem → use UTADIS

Edge cases

• •if a_k ≻ a_{k+1}, = 0 if a_k ~ a_{k+1}; (ii) monotonicity (Eq.(14)): u_i(g_i^{j+1}) − u_i(g_i^j) ≥ s_i ∀i,∀j=1…α_i−1; (iii) normalisation (Eq.(8)): Σ_i u_i(g_i^*) = 1, u_i(g_{*,i}) = 0; (iv) non-negat
• •when F* = 0 (or near-zero).

Common pitfalls

• •Hatalı: 'UTA bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision-maker's holistic ranking of the reference set is available and reliable
• •Hatalı: 'UTA bu varsayımı kontrol etmeden uygulamak'. Doğrusu: An additive utility model is acceptable (criteria preferentially independent)
• •Hatalı: 'UTA bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Marginal utilities can be reasonably approximated by piecewise-linear functions on α_i breakpoints
• •Hatalı: UTA'yi 'No holistic reference ranking is available → use ELECTRE/PROMETHEE (outranking) or weight-based MAVT' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: UTA'yi 'Strongly interacting criteria → consider Choquet integral or ANP' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: UTA'yi 'Pure sorting/classification problem → use UTADIS' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: For each criterion i, define α_i breakpoints g_i^j (j=1,…,α_i) on the user-specified range [g_{*,i}, g_i^*] (worst-to-best, direction-encoded by which endpoint is g_i^*). The α_i − 1 segments are the support of the piecewise-linear marginal utility u_i. 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: Jacquet-Lagrèze & Siskos 1982, p.154, breakpoint definition
2. 2.Adım 2 (F2): Step 2: For each reference action a∈A′ and criterion i, find j s.t. g_i^j ≤ g_i(a) ≤ g_i^{j+1} and linearly interpolate the marginal utility u_i[g_i(a)]. The global value of a is u′[g(a)] = Σ_i u_i[g_i(a)] + σ(a), where σ(a) ≥ 0 is the single error variable (Eq.(8)). Formül: u_{i}\!\bigl[g_{i}(a)\bigr] = u_{i}(g_{i}^{j}) + \frac{g_{i}(a) - g_{i}^{j}}{g_{i}^{j+1} - g_{i}^{j}}\!\left[u_{i}(g_{i}^{j+1}) - u_{i}(g_{i}^{j})\right];\;\; u'\!\bigl[g(a)\bigr] = \sum_{i=1}^{n} u_{i}\!\bigl[g_{i}(a)\bigr] + \sigma(a) Anchor: Jacquet-Lagrèze & Siskos 1982, p.154 (interpolation) & Eq.(8) (global utility with single error)
3. 3.Adım 3 (F3): Step 3: Solve PL1: min F = Σ_{a∈A′} σ(a) subject to (i) preference constraints (Eq.(11)-(12)): for consecutive (a_k, a_{k+1}) in the reference ranking, u′[g(a_k)] − u′[g(a_{k+1})] ≥ δ if a_k ≻ a_{k+1}, = 0 if a_k ~ a_{k+1}; (ii) monotonicity (Eq.(14)): u_i(g_i^{j+1}) − u_i(g_i^j) ≥ s_i ∀i,∀j=1…α_i−1; (iii) normalisation (Eq.(8)): Σ_i u_i(g_i^*) = 1, u_i(g_{*,i}) = 0; (iv) non-negativity: u_i(g_i^j) ≥ 0, σ(a) ≥ 0. Optimal F = F*. The estimated optimal utility U*(g) is the solution of PL1. Formül: [\text{PL1}]\;\; \min F = \sum_{a\in A'}\sigma(a)\;\;\text{s.t. Eqs.(8),(11),(12),(14) and }u_{i}(g_{i}^{j})\geq 0,\;\sigma(a)\geq 0 Anchor: Jacquet-Lagrèze & Siskos 1982, Eqs.(8),(11),(12),(14),(15) p.155-157
4. 4.Adım 4 (F4): Step 4: Post-optimality (Eqs.(16)-(19)): augment PL1 with F ≤ F* + k(F*) (Eq.(16)-(17)) to form polyhedron ℘ (Eq.(18)); for i = 1,…,n solve the 2n LPs (Eq.(19)) [min] u_i(g_i^*) and [max] u_i(g_i^*) on ℘. The mean utility function ū(g) (mean of the 2n post-optimal solutions) is the recommended representative; both U*(g) and ū(g) yield rankings consistent with the reference ranking when F* = 0 (or near-zero). Formül: [\text{Eq.(18) }℘:\,F\leq F^{*}+k(F^{*})];\quad[\text{Eq.(19)}]\;\;[\min]\,u_{i}(g_{i}^{*})\;\text{and}\;[\max]\,u_{i}(g_{i}^{*})\;\text{on }℘,\;i=1,\ldots,n;\quad \bar{u}(g)=\tfrac{1}{2n}\sum_{\ell=1}^{2n}u^{(\ell)}(g) Anchor: Jacquet-Lagrèze & Siskos 1982, Eqs.(16)-(19) p.157-159

Commonly paired with

• •endogenous (UTA-derived) + UTA (primary)

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