L2T-MULTIMOORA - 2-Tuple Linguistic MULTIMOORA (Balezentis & Balezentis 2011)

Overview

L2T-MULTIMOORA extends crisp MULTIMOORA (Brauers-Zavadskas 2010) to 2-tuple linguistic environments using the Herrera-Martínez 2000 (s, α) representation. The method computes three independent sub-rankings - Ratio System (arithmetic mean), Reference Point (Chebyshev max-deviation from per-criterion maxima), and Full Multiplicative Form (geometric mean) - all in 2-tuple algebra via Δ⁻¹/Δ symbolic translation. The Brauers-Zavadskas 2011 dominance theory then aggregates the three sub-rankings into a final ordinal rank. Absolute dominance (rank 1 in all three sub-rankings) signals a robust winner. The 'dominance_score' shown in I.primary_output_table_columns is the sum of sub-ranks (lower better) used as a tiebreak when dominance_count is tied.

Output

ordinal rank, lower is better

Data

Linguistic 2-Tuple, 2tuple complete or transformable

Size

2+ alternatives, 4-10 criteria works best

Used for

Supplier selection under linguistic uncertainty, Multi-criteria evaluation with mixed crisp/linguistic data, Sustainability assessment with qualitative criteria

How it works

1. 1

   Validate the 2-tuple decision matrix T = [t_ij] = [(s_ij, α_ij)] in BLTS S_{g+1}. Apply E-1, E-2, E-3 checks. Determine criterion directions (J1 = benefit, J2 = cost).

   $$T=[t_{ij}]=[(s_{ij},{\alpha }_{ij})],s_{ij}\in S_{g+1},{\alpha }_{ij}\in [-0.5,0.5)$$

   Balezentis 2011 §3.2 input definition; Herrera-Martínez 2000 Def. 3

2. 2

   Transform cost criteria via 2-tuple negation: u_ij = t_ij for j ∈ J1, u_ij = Neg(t_ij) for j ∈ J2. Negation: Neg(s_k, α) = Δ(g − Δ⁻¹(s_k, α)). Produces uniform-benefit matrix U.

   $$u_{ij}=\{\begin{array}{ll}{t}_{ij},& j\in J_1\\ \Delta (g-{\Delta }^{-1}(t_{ij})),& j\in J_2\end{array}$$

   Balezentis 2011 Eq.(20); Herrera-Martínez 2000 Def. 5 (negation)

3. 3

   Ratio System: arithmetic mean of Δ⁻¹(u_ij) per row, then Δ back to 2-tuple. y_i = Δ((1/n) Σ_j Δ⁻¹(u_ij)). Higher y_i → higher Ratio System rank.

   $$y_i=\Delta \left(\frac{1}{n}\sum _{j=1}^n{\Delta }^{-1}(u_{ij})\right),i=1,\dots ,m$$

   Balezentis 2011 Eq.(21); Herrera-Martínez 2000 Def. 6 (arithmetic mean of 2-tuples)

4. 4

   Reference Point: per-column maxima u*_j = max_i u_ij (via Δ⁻¹). For each row, take Chebyshev distance d_i = max_j |Δ⁻¹(u*_j) − Δ⁻¹(u_ij)|. Smaller d_i → higher Reference Point rank.

   $$u_j^{\ast }=\underset{i}{\max }{\Delta }^{-1}(u_{ij});d_i=\underset{j}{\max }|u_j^{\ast }-{\Delta }^{-1}(u_{ij})|;\underset{i}{\min }{d}_i\text{ is best}$$

   Balezentis 2011 Eq.(22); Brauers-Zavadskas 2006 Eq.(14) Tchebycheff metric

5. 5

   Full Multiplicative Form: geometric mean of Δ⁻¹(u_ij) per row, then Δ back. U_i = Δ((∏_j Δ⁻¹(u_ij))^{1/n}). Higher U_i → higher Full Multiplicative rank.

   $$U_i=\Delta \left({\left(\prod _{j=1}^n{\Delta }^{-1}(u_{ij})\right)}^{1/n}\right),i=1,\dots ,m$$

   Balezentis 2011 Eq.(23); Brauers-Zavadskas 2010 Eq.(15) Full Multiplicative Form

6. 6

   Dominance theory aggregation: compose final rank from three sub-rankings (ratio_rank, ref_rank, full_mult_rank). Alternative A dominates B iff A has a strictly better rank-number in at least 2 of the 3 sub-rankings. Final ordering: sort by (−dominance_count, sum_of_subranks) ascending. Absolute dominance corresponds to (1,1,1) sub-ranks.

   $${\text{rank}}_i^{\text{final}}\propto -|\{j\ne i:i{\succ }_{\text{2-of-3}}j\}|;\text{tiebreak by }\sum _k{r}_i^{(k)}\text{ (ratio + ref + full)}$$

   Balezentis 2011 §3.2 final paragraph; Brauers-Zavadskas 2011 dominance theory

Fits when / Look elsewhere when

Assumptions to verify

• All decision matrix entries are valid 2-tuples (s_i, α) in the chosen BLTS
• BLTS granularity g is uniform across all cells
• Criterion directions (max/min) are explicit
• Equal-weight criteria assumption is acceptable for the application

Edge cases and pitfalls

• •L2T-MULTIMOORA için kritik edge case'ler: (1) Linguistic term set cardinality g+1 ODD seçilmeli (3, 5, 7, 9; Miller 1956 7±2 cognitive limit); g+1 EVEN → 'neutral' kategori yok, semantic symmetric bozulur. (2) τ-mapping Eq.10: α_i = max_y min{μ_I(y), μ_s_i(y)} - interval/TFN'nin overlap'i en yüksek linguistic label'a atanır; multiple-label overlap durumunda paper 'max-min' kuralı bağlanır. (3) Dominance theory aggregation üç ranking'i birleştirir; tam çelişki (örn. A1 Ratio'da 1., Reference'da son, Multiplicative'da orta) → Brauers 2010 majority rule + circular reasoning kontrolü. (4) Cost kriterler N_C: Ratio System'de ÇIKARILIR Eq.7; Multiplicative'da PAYDA'ya konur Eq.9 - bu MOORA gelenek (TOPSIS gibi pre-normalize flip kullanılmaz). (5) S_g sınır durumlar: a_0=0, c_g=0 - extreme labels yarı-açık üçgenler, ortadaki labels tam üçgen Eq.2.

• BLTS choice affects results - if input data is on different linguistic scales (e.g., S5 for some criteria, S7 for others), they must all be transformed to a common BLTS before this manifest applies. Balezentis 2011 uses S7 as BLTS with χ-translation (Eqs.10-11) - that step is upstream of this manifest.

• Negation operator orientation: F2 uses Neg(s_k, α) = Δ(g − Δ⁻¹(s_k, α)) where g is BLTS granularity (= number of terms − 1). Mis-specifying g leads to incorrect cost-to-benefit conversion.

• Three sub-rankings can disagree: when no alternative exhibits absolute dominance, the dominance theory aggregation may yield a partial order. Implementation must convert partial order to total order via deterministic tiebreaks (sum_of_subranks then alternative_id).

• MULTIMOORA equal-weight assumption: the seminal method weights all criteria equally. Introducing explicit weights changes the method to weighted-MULTIMOORA-2T, which is a known but separate variant (not covered by this manifest's F.steps).

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