L2T-SAW - Linguistic extension of L2T-SAW

Overview

l2t-saw extends L2T-SAW to handle Linguistic uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using 2-Tuple Linguistic Variable (2TL: (s_i, α)) algebra. The final scores are defuzzified via Δ^{-1}(s_i, α) = i + α before ranking.

Output

utility, higher is better

Data

Linguistic 2-Tuple, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Linguistic 2-Tuple MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Construct the 2-tuple linguistic decision matrix x̃_ij = (s_ij, α_ij) on linguistic term set S = {s_0,…,s_q}; define criteria weights w_j summing to 1. Δ⁻¹(s_i,α) = i + α is the canonical defuzzification.

   $$\tilde{D}=[(s_{ij},{\alpha }_{ij}){]}_{m\times n};{\Delta }^{-1}(s_i,\alpha )=i+\alpha ;\sum _{j=1}^n{w}_j=1$$

   Report §3.6 Steps 1-2; Cid-López 2018 2-Tuple SAW

2. 2

   Step 3 - Weighted 2-tuple score Score_i = Δ(Σ_j w_j · Δ⁻¹(s_ij, α_ij)) for each alternative; aggregation kept in 2-tuple representation with no information loss.

   $${\text{Score}}_i=\Delta \left(\sum _{j=1}^n{w}_j\cdot {\Delta }^{-1}(s_{ij},{\alpha }_{ij})\right)$$

   Report §3.6 Formula 1 - weighted 2-tuple SAW score

3. 3

   Step 4 - Rank alternatives in descending order of the defuzzified Score_i (compared via Δ⁻¹).

   $$\text{rank}={\text{argsort}}_{\text{desc}}\left({\Delta }^{-1}({\text{Score}}_i)\right)$$

   Report §3.6 Step 4 - ranking

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Linguistic 2-Tuple numbers/tuples
• Underlying crisp method's compensation assumption holds in uncertain space
• All decision-maker(s) and experts use the same linguistic/uncertainty scale

Edge cases and pitfalls

• Value-space violation: ensure all entries satisfy 2TL: s_i ∈ S (linguistic term set), α ∈ [-0.5,0.5) before computation.

• Defuzzification method affects ranking: Δ^{-1}(s_i, α) = i + α is the canonical choice but alternatives exist.

Works with

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