UTA - UTilités Additives (Additive Utility Assessment)

Overview

UTA infers an additive utility function u(g) = Σ_i u_i(g_i) from the decision-maker's holistic weak order on a reference set A′, by minimising the sum of single errors F = Σ_{a∈A′} σ(a) (Eq.(15)). u_i are piecewise-linear on α_i breakpoints (paper p.154). Post-optimality (Eqs.(16)-(19)) explores the polyhedron F ≤ F* + k(F*) via 2n LPs (min/max u_i(g_i^*)) to give a mean utility ū(g) and implicit-weight intervals. If F* > 0 the reference ranking is inconsistent with additive utility; if persistent infeasibility arises in monotonicity, consider UTASTAR (Siskos & Yannacopoulos 1985, double-error variant).

Output

utility, higher is better

Data

Crisp, complete numeric plus reference ranking

Weights

Derived internally, no weight source needed

Size

2+ alternatives, 3-12 criteria works best

Used for

Alternative selection, Supplier evaluation, consumer choice modelling, preference learning

How it works

1. 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.

   $$g_i^j=g_{\ast ,i}+\frac{j-1}{{\alpha }_i-1}(g_i^{\ast }-g_{\ast ,i}),j=1,\dots ,{\alpha }_i$$

   Jacquet-Lagrèze & Siskos 1982, p.154, breakpoint definition

2. 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)).

   $$u_i[g_i(a)]=u_i(g_i^j)+\frac{g_i(a)-g_i^j}{g_i^{j+1}-g_i^j}\left[u_i(g_i^{j+1})-u_i(g_i^j)\right];u^{\prime }[g(a)]=\sum _{i=1}^n{u}_i[g_i(a)]+\sigma (a)$$

   Jacquet-Lagrèze & Siskos 1982, p.154 (interpolation) & Eq.(8) (global utility with single error)

3. 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.

   $$[\text{PL1}]\min F=\sum _{a\in A^{\prime }}\sigma (a)\text{s.t. Eqs.(8),(11),(12),(14) and }{u}_i(g_i^j)\ge 0,\sigma (a)\ge 0$$

   Jacquet-Lagrèze & Siskos 1982, Eqs.(8),(11),(12),(14),(15) p.155-157

4. 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).

   $$[\text{Eq.(18) }\wp :F\le F^{\ast }+k(F^{\ast })];[\text{Eq.(19)}][\min ]u_i(g_i^{\ast })\text{and}[\max ]u_i(g_i^{\ast })\text{on }\wp ,i=1,\dots ,n;\bar{u}(g)=\frac{1}{2n}\sum _{\ell =1}^{2n}{u}^{(\ell )}(g)$$

   Jacquet-Lagrèze & Siskos 1982, Eqs.(16)-(19) p.157-159

Look elsewhere when

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

Edge cases and pitfalls

• •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).

• F* > 0: reference ranking inconsistent with additive utility - inspect Δ(a_k, a_{k+1}) constraints and either accept the inconsistency (UTA quantifies it) or switch to UTASTAR (always-feasible double-error variant).

• Dict-format reference ranking {alt: rank}: must sort by rank value (1 = best) before solving - enumerating insertion order silently produces wrong LP.

• Cost criterion direction: paper convention encodes direction in (g_{*,i}, g_i^*) - for cost g_{*,i} > g_i^* (worst value larger) - so the same breakpoint formula and interpolation handle both max and min without explicit reversal. Reversing arrays a second time inside the interpolator is a bug.

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