L2T-COPRAS - Linguistic extension of L2T-COPRAS

Overview

l2t-copras extends L2T-COPRAS 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} with Δ⁻¹(s_i,α)=i+α; define criterion weights w_j and direction sets Ω_max (benefit) and Ω_min (cost).

   $$\tilde{D}=[(s_{ij},{\alpha }_{ij}){]}_{m\times n};{\Delta }^{-1}(s_i,\alpha )=i+\alpha ;{\Omega }_{max}\cup {\Omega }_{min}=\{1,\dots ,n\}$$

   Report §3.5 Step 1; Gai 2021 ITL-COPRAS

2. 2

   Weighted 2-tuple decision matrix d̃_ij = Δ(w_j · Δ⁻¹(s_ij,α_ij)) using 2-tuple weighted-aggregation arithmetic.

   $${\tilde{d}}_{ij}=\Delta \left(w_j\cdot {\Delta }^{-1}(s_{ij},{\alpha }_{ij})\right)$$

   Report §3.5 Step 1 - weighted normalised matrix

3. 3

   Sum of benefit P_i = Σ_{j ∈ Ω_max} Δ⁻¹(d̃_ij) and sum of cost R_i = Σ_{j ∈ Ω_min} Δ⁻¹(d̃_ij).

   $$P_i=\sum _{j\in {\Omega }_{max}}{\Delta }^{-1}({\tilde{d}}_{ij}),R_i=\sum _{j\in {\Omega }_{min}}{\Delta }^{-1}({\tilde{d}}_{ij})$$

   Report §3.5 Formulas 2-3 - sum of benefit/cost criteria

4. 4

   Relative significance Q_i = P_i + (Σ_k R_k)/(R_i · Σ_k 1/R_k) combining benefit and cost terms.

   $$Q_i=P_i+\frac{{\sum }_{k=1}^m{R}_k}{R_i\cdot {\sum }_{k=1}^m\frac{1}{R_k}}$$

   Report §3.5 Formula 4 - relative significance

5. 5

   Utility degree N_i = (Q_i / max_k Q_k) × 100% with descending ranking.

   $$N_i=\frac{Q_i}{{\max }_k{Q}_k}\times 100\%;\text{rank}={\text{argsort}}_{\text{desc}}(N_i)$$

   Report §3.5 - utility degree and 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

How to cite