PiF-SAW - Picture extension of SAW

Overview

pif-saw extends SAW to handle Picture uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Picture Fuzzy Set (PiFS: μ, η, ν; μ+η+ν ≤ 1) algebra. The final scores are defuzzified via score function S = μ − ν before ranking.

Output

utility, higher is better

Data

Picture Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Picture Fuzzy MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Construct picture fuzzy decision matrix D = (α_ij)_{m×n} where α_ij = ⟨μ_ij, η_ij, ν_ij⟩ with μ_ij + η_ij + ν_ij ≤ 1.

   $$D=({\alpha }_{i}j{)}_{m\times n};{\alpha }_{i}j=⟨{\mu }_{i}j,{\eta }_{i}j,{\nu }_{i}j⟩;{\mu }_{i}j+{\eta }_{i}j+{\nu }_{i}j\le 1$$

   Cuong 2013 Def.1; Garg 2017 Sec.4 Step 1

2. 2

   Cost-complement normalization: for cost (min) criteria r_ij = α_ij^c = ⟨ν_ij, η_ij, μ_ij⟩; for benefit (max) criteria r_ij = α_ij. Converts all criteria to benefit-type without altering μ+η+ν ≤ 1.

   $$r_{i}j={\alpha }_i{j}^c=⟨{\nu }_{i}j,{\eta }_{i}j,{\mu }_{i}j⟩if{c}_j\in cost;r_{i}j={\alpha }_{i}j=⟨{\mu }_{i}j,{\eta }_{i}j,{\nu }_{i}j⟩if{c}_j\in benefit$$

   Garg 2017 Eq.8 (Sec.4 Step 2)

3. 3

   PFWA aggregation per alternative with weight vector w: r_i = PFWA_w(r_i1, ..., r_in). μ_i = 1 - ∏(1-μ_ij)^w_j; η_i = ∏(η_ij)^w_j; ν_i = ∏(ν_ij)^w_j.

   $$r_i=PFW{A}_w(r_{i}1,...,r_{i}n)=⟨1-\prod _{j=1}^n(1-{\mu }_{i}j{)}^{w_j},\prod _{j=1}^n({\eta }_{i}j{)}^{w_j},\prod _{j=1}^n({\nu }_{i}j{)}^{w_j}⟩$$

   Garg 2017 Eq.4 (PFWA), Sec.4 Step 3

4. 4

   Defuzzify r_i using Garg score function S(α) = μ - η - ν.

   $$s_i=S(r_i)={\mu }_i-{\eta }_i-{\nu }_i,s_i\in [-1,1]$$

   Garg 2017 Eq.1 (Sec.4 Step 4)

5. 5

   Rank alternatives in descending order of s_i. Ties broken by accuracy H(α) = μ + η + ν (Garg 2017 Eq.2).

   $$ranking=argsor{t}_{d}esc(s_i);ontieuseH(\alpha )=\mu +\eta +\nu$$

   Garg 2017 Sec.4 Step 5

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Picture Fuzzy 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

• •Ties broken by accuracy H(α) = μ + η + ν (Garg 2017 Eq.2).

• Value-space violation: ensure all entries satisfy PiFS: μ (membership), η (neutral), ν (non-membership); μ+η+ν ≤ 1 before computation.

• Defuzzification method affects ranking: score function S = μ − ν is the canonical choice but alternatives exist.

Works with

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