Ranking

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

Siskos, Y., Yannacopoulos, D. · 1985

Overview

Additive utility disaggregation: LP from ordinal reference judgements. Output typically utility (higher value = preferred).

Strengths

•Method-specific: Additive utility disaggregation: LP from ordinal reference judgements

Limitations

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

Method assistant

Grounded explanations: it explains the method, it does not compute.

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)

When not to use

•Criteria strongly correlated → consider DEMATEL/ANP for interdependence
•Non-compensatory preferences → consider outranking (ELECTRE/PROMETHEE)

Edge cases

•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.

Common pitfalls

•Hatalı: 'UTASTAR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criteria preferences are independent (no synergistic interactions)
•Hatalı: 'UTASTAR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Compensation is acceptable: high score on one criterion can offset low on another
•Hatalı: 'UTASTAR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix is complete (no missing values)
•Hatalı: UTASTAR'yi 'Criteria strongly correlated → consider DEMATEL/ANP for interdependence' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
•Hatalı: UTASTAR'yi 'Non-compensatory preferences → consider outranking (ELECTRE/PROMETHEE)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1.Adım 1 (F1): Step 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. Formül: u_{i}(g_{i}^{1}) = 0\ \forall i;\ u_{i}(g_{i}^{j}) = \sum_{t=1}^{j-1} w_{it}\ \forall i,\ j=2,\ldots,\alpha_i-1;\ w_{ij} = u_{i}(g_{i}^{j+1}) - u_{i}(g_{i}^{j}) \geq 0 Anchor: Siskos-Yannacopoulos 1985 [Greco-Ehrgott-Figueira 2016 Ch.9 §9.2.3 Eqs.(9.18)-(9.19)]
2.Adım 2 (F2): Step 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})]. Formül: \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})] Anchor: Siskos-Yannacopoulos 1985 [book Ch.9 §9.2.3 Eq.(9.20)]
3.Adım 3 (F3): Step 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. Formül: [\min] z = \sum_{k=1}^{m}[\sigma^{+}(a_k) + \sigma^{-}(a_k)] \quad \text{s.t.}\quad \Delta(a_k, a_{k+1}) \geq \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} \geq 0,\ \sigma^{+}(a_k) \geq 0,\ \sigma^{-}(a_k) \geq 0 Anchor: Siskos-Yannacopoulos 1985 [book Ch.9 §9.2.3 Eq.(9.21)]
4.Adım 4 (F4): Step 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. Formül: 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)] \leq z^{{\ast}} + \varepsilon Anchor: Siskos-Yannacopoulos 1985 [book Ch.9 §9.2.3 Eqs.(9.22)-(9.23)]
5.Adım 5 (F5): Step 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'. Formül: u'(\mathbf{g}(a)) = \sum_{i=1}^{n} u_{i}(g_{i}(a))\ \forall a \in A Anchor: Siskos-Yannacopoulos 1985 [book Ch.9 §9.2.1 Eq.(9.6)]

Commonly paired with

•AHP + UTASTAR (high)
•BWM + UTASTAR (high)
•ENTROPY + UTASTAR (high)
•CRITIC + UTASTAR (high)
•SWARA + UTASTAR (high)

How to cite

Siskos, Y.; Yannacopoulos, D. (1985). UTASTAR: An ordinal regression method for building additive value functions. Investigación Operativa.

BibTeX RIS APA (.txt)

Edge cases

•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.

Common pitfalls

•Hatalı: 'UTASTAR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criteria preferences are independent (no synergistic interactions)

•Hatalı: 'UTASTAR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Compensation is acceptable: high score on one criterion can offset low on another

•Hatalı: 'UTASTAR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix is complete (no missing values)

•Hatalı: UTASTAR'yi 'Criteria strongly correlated → consider DEMATEL/ANP for interdependence' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

•Hatalı: UTASTAR'yi 'Non-compensatory preferences → consider outranking (ELECTRE/PROMETHEE)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1.Adım 1 (F1): Step 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. Formül: u_{i}(g_{i}^{1}) = 0\ \forall i;\ u_{i}(g_{i}^{j}) = \sum_{t=1}^{j-1} w_{it}\ \forall i,\ j=2,\ldots,\alpha_i-1;\ w_{ij} = u_{i}(g_{i}^{j+1}) - u_{i}(g_{i}^{j}) \geq 0 Anchor: Siskos-Yannacopoulos 1985 [Greco-Ehrgott-Figueira 2016 Ch.9 §9.2.3 Eqs.(9.18)-(9.19)]

2.Adım 2 (F2): Step 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})]. Formül: \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})] Anchor: Siskos-Yannacopoulos 1985 [book Ch.9 §9.2.3 Eq.(9.20)]

3.Adım 3 (F3): Step 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. Formül: [\min] z = \sum_{k=1}^{m}[\sigma^{+}(a_k) + \sigma^{-}(a_k)] \quad \text{s.t.}\quad \Delta(a_k, a_{k+1}) \geq \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} \geq 0,\ \sigma^{+}(a_k) \geq 0,\ \sigma^{-}(a_k) \geq 0 Anchor: Siskos-Yannacopoulos 1985 [book Ch.9 §9.2.3 Eq.(9.21)]

4.Adım 4 (F4): Step 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. Formül: 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)] \leq z^{{\ast}} + \varepsilon Anchor: Siskos-Yannacopoulos 1985 [book Ch.9 §9.2.3 Eqs.(9.22)-(9.23)]

5.Adım 5 (F5): Step 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'. Formül: u'(\mathbf{g}(a)) = \sum_{i=1}^{n} u_{i}(g_{i}(a))\ \forall a \in A Anchor: Siskos-Yannacopoulos 1985 [book Ch.9 §9.2.1 Eq.(9.6)]