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

Overview

L2T-MULTIMOORA (MULTIMOORA-2T), Brauers-Zavadskas (2006/2010) MULTIMOORA yönteminin Baležentis ve Baležentis (2011, Transformations in Business and Economics) tarafından Herrera-Martinez (2000) 2-TUPLE LINGUISTIC representation ile genişletilmiş halidir. Paper'ın kilit özelliği HYBRID DATA fusion: tek bir karar matrisinde crisp sayı, interval sayı, triangular fuzzy number (TFN) ve linguistic label birlikte bulunabilir; tümü τ:[0,1] → F(S_{g+1}) mapping ile (s_i, α) 2-tuple'a dönüştürülür (Eq.10): α ∈ [-0.5, 0.5] symbolic translation korunur, yani bilgi kaybı YOKTUR. 2-tuple'ın gücü Δ ve Δ^{-1} ile sürekli domain elde etmesinden gelir: Δ(β) = (s_round(β), β - round(β)) Eq.3; Δ^{-1}(s_i, α) = i + α ∈ [0, g]. MULTIMOORA üç alt-yöntemi (Ratio System, Reference Point Min-Max-Tchebycheff, Full Multiplicative) sırasıyla 2-tuple arithmetic mean Eq.7, 2-tuple distance Eq.8 ve 2-tuple geometric mean Eq.9 ile yeniden tanımlanır; Dominance theory (Brauers 2010) üç ranking'i birleştirir. Linguistic varyantın MOORA-katmanlı tarafa kıyasla erdemi: granularity-of-uncertainty ile uzman 'medium-high' gibi sözel kategoriler arasında compensatory tradeoff yapabilir.

Strengths

• •Method-specific: Linguistic ratio/reference/multiplicative ensemble: 2-Tuple Linguistic Variable (2TL: (s_i, α))
• •Preserves linguistic_2tuple uncertainty through the pipeline rather than premature crispification at elicitation

Limitations

• •Rank reversal known on alternative-set changes (ref: inherited from crisp MULTIMOORA; cf. Brauers-Zavadskas 2011 dominance theory tackles tie-breaking but rank reversal under alternative addition still possible)
• •Assumes: All decision matrix entries are valid 2-tuples (s_i, α) in the chosen BLTS
• •Assumes: BLTS granularity g is uniform across all cells
• •Assumes: Criterion directions (max/min) are explicit
• •Assumes: Equal-weight criteria assumption is acceptable for the application

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

When not to use

• •Criterion importance is heterogeneous and must enter the algorithm: use weighted-MULTIMOORA-2T variant instead
• •Inputs are crisp without linguistic content: use base crisp MULTIMOORA directly
• •Only one sub-ranking is needed: pick the appropriate single-arm method (MOORA Ratio, MOORA Reference, or Full Multiplicative Form)

Edge cases

• •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.

Common pitfalls

• •Doktora öğrencisi tuzakları: (a) 2-tuple (s_i, α)'yi 'ordered pair' veya 'fuzzy interval' sanmak: bu Herrera-Martinez SYMBOLIC TRANSLATION mekanizmasıdır; α label'lar arasındaki SAPMA değil, β-i yuvarlama hatasıdır; (b) Δ^{-1}(s_i, α) = i+α formülünü 'continuous → discrete' kayıp olarak sanmak: aslında SÜREKLİ domain elde etmemizin tam aracıdır, kayıp ROUND adımındadır ama α telafi eder; (c) Hybrid data conversion'da τ-mapping yerine 'average linguistic match' önermek: paper kanonik MAX-MIN overlap (Gong 2007); (d) Dominance theory'yi 'Borda count' ile özdeşleştirmek: Dominance theory Brauers 2010'da MAJORITY + CIRCULAR REASONING kontrolü, Borda toplam-puan; (e) Cost kriter Multiplicative'da PAYDAYA konduğunu görmemek: N_C terimler U_i = (Π benefits)/(Π costs) ile çıkar; Ratio System'de SUBTRACTION ile.

Worked example

1. 1.Adım 1 (F1): Step 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). Formül: T = [t_{ij}] = [(s_{ij}, \alpha_{ij})],\ s_{ij} \in S_{g+1},\ \alpha_{ij} \in [-0.5, 0.5) Anchor: Balezentis 2011 §3.2 input definition; Herrera-Martínez 2000 Def. 3
2. 2.Adım 2 (F2): Step 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. Formül: u_{ij} = \begin{cases} t_{ij}, & j \in J_1 \\ \Delta(g - \Delta^{-1}(t_{ij})), & j \in J_2 \end{cases} Anchor: Balezentis 2011 Eq.(20); Herrera-Martínez 2000 Def. 5 (negation)
3. 3.Adım 3 (F3): Step 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. Formül: y_i = \Delta\!\left(\dfrac{1}{n}\sum_{j=1}^{n}\Delta^{-1}(u_{ij})\right),\ i = 1,\dots,m Anchor: Balezentis 2011 Eq.(21); Herrera-Martínez 2000 Def. 6 (arithmetic mean of 2-tuples)
4. 4.Adım 4 (F4): Step 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. Formül: u^{*}_{j} = \max_{i}\,\Delta^{-1}(u_{ij});\ \ d_i = \max_{j}\,\bigl|u^{*}_{j} - \Delta^{-1}(u_{ij})\bigr|;\ \ \min_i d_i \text{ is best} Anchor: Balezentis 2011 Eq.(22); Brauers-Zavadskas 2006 Eq.(14) Tchebycheff metric
5. 5.Adım 5 (F5): Step 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. Formül: U_i = \Delta\!\left(\left(\prod_{j=1}^{n} \Delta^{-1}(u_{ij})\right)^{1/n}\right),\ i = 1,\dots,m Anchor: Balezentis 2011 Eq.(23); Brauers-Zavadskas 2010 Eq.(15) Full Multiplicative Form
6. 6.Adım 6 (F6): Step 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. Formül: \text{rank}_{i}^{\text{final}} \propto -\bigl|\{j \neq i : i \succ_{\text{2-of-3}} j\}\bigr| ; \ \text{tiebreak by } \sum_{k} r^{(k)}_i \text{ (ratio + ref + full)} Anchor: Balezentis 2011 §3.2 final paragraph; Brauers-Zavadskas 2011 dominance theory

Commonly paired with

• •n_a + MULTIMOORA (uncommon)

How to cite