UTA* - Additive utility disaggregation from reference ranking (revised UTA)

Overview

UTA* - Additive utility disaggregation from reference ranking (revised UTA)

Output

utility, higher is better

Data

Crisp, complete numeric matrix

Weights

Needs a weight source

Size

2+ alternatives, 3-12 criteria works best

Used for

Alternative selection, Supplier evaluation

How it works

1. 1

   Express global utility u'(g(a_k)) of each reference action a_k in terms of marginal utilities u_i(g_i^j), and reparameterize via non-negative step variables w_ij = u_i(g_i^{j+1}) − u_i(g_i^j) with u_i(g_i^1) = 0.

   $$u_i(g_i^1)=0\forall i;u_i(g_i^j)=\sum _{t=1}^{j-1}{w}_{it}\forall i,j=2,\dots ,{\alpha }_i-1;w_{ij}=u_i(g_i^{j+1})-u_i(g_i^j)\ge 0$$

   Siskos-Yannacopoulos 1985 [Greco-Ehrgott-Figueira 2016 Ch.9 §9.2.3 Eqs.(9.18)-(9.19)]

2. 2

   Introduce double error functions σ⁺(a_k), σ⁻(a_k) on each reference action and define the consecutive-pair difference Δ(a_k, a_{k+1}) = [u(g(a_k)) − σ⁺(a_k) + σ⁻(a_k)] − [u(g(a_{k+1})) − σ⁺(a_{k+1}) + σ⁻(a_{k+1})].

   $$\Delta (a_k,a_{k+1})=[u(\mathbf{g}(a_k))-{\sigma }^{+}(a_k)+{\sigma }^{-}(a_k)]-[u(\mathbf{g}(a_{k+1}))-{\sigma }^{+}(a_{k+1})+{\sigma }^{-}(a_{k+1})]$$

   Siskos-Yannacopoulos 1985 [book Ch.9 §9.2.3 Eq.(9.20)]

3. 3

   Solve the LP that minimises the total error z = Σ_k [σ⁺(a_k) + σ⁻(a_k)] subject to: Δ(a_k, a_{k+1}) ≥ δ if a_k ≻ a_{k+1}, Δ(a_k, a_{k+1}) = 0 if a_k ~ a_{k+1}; Σ_{i,j} w_ij = 1 (normalization); w_ij ≥ 0; σ⁺, σ⁻ ≥ 0. δ is a small positive number.

   $$[\min ]z=\sum _{k=1}^m[{\sigma }^{+}(a_k)+{\sigma }^{-}(a_k)]\text{s.t.}\Delta (a_k,a_{k+1})\ge \delta \text{if}{a}_k\succ a_{k+1},\Delta (a_k,a_{k+1})=0\text{if}{a}_k\sim a_{k+1},\sum _{i=1}^n\sum _{j=1}^{{\alpha }_i-1}{w}_{ij}=1,w_{ij}\ge 0,{\sigma }^{+}(a_k)\ge 0,{\sigma }^{-}(a_k)\ge 0$$

   Siskos-Yannacopoulos 1985 [book Ch.9 §9.2.3 Eq.(9.21)]

4. 4

   Post-optimality / stability analysis: test for multiple or near-optimal LP solutions on the polyhedron Σ[σ⁺ + σ⁻] ≤ z* + ε; if non-unique, compute the mean additive value function across criterion-wise max/min LPs over u_i(g_i*) = Σ_j w_ij.

   $$u_i(g_i^{\ast })=\sum _{j=1}^{{\alpha }_i-1}{w}_{ij}\forall i;\sum _{k=1}^m[{\sigma }^{+}(a_k)+{\sigma }^{-}(a_k)]\le z^{\ast }+\epsilon$$

   Siskos-Yannacopoulos 1985 [book Ch.9 §9.2.3 Eqs.(9.22)-(9.23)]

5. 5

   Apply the inferred additive value function u'(g(a)) = Σ_i u_i(g_i(a)) to ALL alternatives (not just the reference set) to produce the final ranking by descending u'.

   $$u^{\prime }(\mathbf{g}(a))=\sum _{i=1}^n{u}_i(g_i(a))\forall a\in A$$

   Siskos-Yannacopoulos 1985 [book Ch.9 §9.2.1 Eq.(9.6)]

Look elsewhere when

Assumptions to verify

• Criteria preferences are independent (no synergistic interactions)
• Compensation is acceptable: high score on one criterion can offset low on another
• Decision matrix is complete (no missing values)

Edge cases and pitfalls

• •ties u_i(g_i^j), and reparameterize via non-negative step variables w_ij = u_i(g_i^{j+1}) − u_i(g_i^j) with u_i(g_i^1) = 0.
• •if a_k ≻ a_{k+1}, Δ(a_k, a_{k+1}) = 0 if a_k ~ a_{k+1}; Σ_{i,j} w_ij = 1 (normalization); w_ij ≥ 0; σ⁺, σ⁻ ≥ 0. δ is a small positive number.
• •if non-unique, compute the mean additive value function across criterion-wise max/min LPs over u_i(g_i*) = Σ_j w_ij.

• Applying UTASTAR without verifying this assumption.

  Requirement: Criteria preferences are independent (no synergistic interactions)

• Applying UTASTAR without verifying this assumption.

  Requirement: Compensation is acceptable: high score on one criterion can offset low on another

• Applying UTASTAR without verifying this assumption.

  Requirement: Decision matrix is complete (no missing values)

• Using UTASTAR when: Criteria strongly correlated → consider DEMATEL/ANP for interdependence.

  An alternative method is recommended in this situation.

• Using UTASTAR when: Non-compensatory preferences → consider outranking (ELECTRE/PROMETHEE).

  An alternative method is recommended in this situation.

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