PiF-TOPSIS - Picture extension of TOPSIS

Overview

PIF-TOPSIS follows Sindhu 2019: build the PF decision matrix R, extract PFPIS/PFNIS per-criterion using extrema rules (Eq.7-8), compute weighted absolute-difference similarity with a max-term (Eq.6), and rank by relative closeness CR_i = S⁺/(S⁺+S⁻). Weights may be DM-supplied (default) or derived from an LP model (Sindhu Eq.11 - optional F4/F5). Highest CR_i is the best alternative.

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 the picture-fuzzy decision matrix R = [⟨α_ij, γ_ij, β_ij⟩]_{m×n} from DM information. α = degree of acceptance, γ = degree of neutral, β = degree of rejection; α+γ+β ≤ 1.

   $$R=[⟨{\alpha }_{i}j,{\gamma }_{i}j,{\beta }_{i}j⟩{]}_{m\times n}$$

   Sindhu 2019 §PF-TOPSIS Step 1

2. 2

   Determine PFPIS ∇⁺_p and PFNIS ∇⁻_p per-criterion. For benefit criteria j ∈ J₁: PFPIS = ⟨max_i α_ij, min_i γ_ij, min_i β_ij⟩; PFNIS = ⟨min_i α_ij, min_i γ_ij, max_i β_ij⟩. For cost criteria j ∈ J₂: PFPIS = ⟨max_i α_ij, max_i γ_ij, min_i β_ij⟩; PFNIS = ⟨min_i α_ij, max_i γ_ij, max_i β_ij⟩.

   $${\nabla }_p^{+}=⟨ma{x}_i\alpha ,mi{n}_i\gamma ,mi{n}_i\beta ⟩ifj\in J_{1}else⟨ma{x}_i\alpha ,ma{x}_i\gamma ,mi{n}_i\beta ⟩;{\nabla }_p^{-}=⟨mi{n}_i\alpha ,mi{n}_i\gamma ,ma{x}_i\beta ⟩ifj\in J_{1}else⟨mi{n}_i\alpha ,ma{x}_i\gamma ,ma{x}_i\beta ⟩$$

   Sindhu 2019 Eq.(7)-(8)

3. 3

   Compute weighted similarity S⁺_p(A_i, ∇⁺_p) and S⁻_p(A_i, ∇⁻_p) via Eq.(6) for each alternative. S(A,B) = 1 − (1/3) Σ_j w_j · ([|Δα|+|Δγ|+|Δβ|] + max(|Δα|, |Δγ|, |Δβ|)).

   S_p(A, B) = 1 − (1/3) Σ_{j=1}^n w_j · ([|α_A_j − α_B_j| + |γ_A_j − γ_B_j| + |β_A_j − β_B_j|] + max(|α_A_j − α_B_j|, |γ_A_j − γ_B_j|, |β_A_j − β_B_j|))

   Sindhu 2019 Eq.(6),(9),(10)

4. 4

   Construct the LP objective Z = Σ_i (S⁺_{pi} − S⁻_{pi}) used in Sindhu 2019 when weights are unknown. If the DM supplies explicit weights (this manifest's default), F4 is skipped.

   $$Z={\Sigma }_i(S_{pi}^{+}(A_i,{\nabla }_p^{+})-S_{pi}^{-}(A_i,{\nabla }_p^{-}))subjectto\Sigma w_j=1,DM-suppliedlinearconstraintson{w}_j$$

   Sindhu 2019 Eq.(11)

5. 5

   Solve the LP from F4 to obtain w_j. Skipped when weights are provided exogenously.

   $$maximizeZ(w_1,...,w_n)s.t.linearconstraints;standardLP(Vanderbei2014reference[26])$$

   Sindhu 2019 Step 5; Vanderbei 2014

6. 6

   Recompute S⁺_p and S⁻_p with the final weight vector (identical formula to F3, just with finalised w_j).

   $$SameasF3withfinalised{w}_j$$

   Sindhu 2019 Step 6

7. 7

   Compute relative closeness CR_i = S⁺_{pi} / (S⁺_{pi} + S⁻_{pi}) for each alternative. Rank descending by CR_i; the best alternative has maximum CR.

   $$C{R}_i=S_{pi}^{+}/(S_{pi}^{+}+S_{pi}^{-});rankbyC{R}_{i}descending$$

   Sindhu 2019 Eq.(12)

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Picture Fuzzy Numbers (α + γ + β ≤ 1).
• Criterion weights either (a) DM-supplied with Σ w_j = 1, or (b) derivable from LP with DM-supplied linear constraints.
• Criterion directions (max/min) are explicit - PFPIS/PFNIS rules differ between J1 and J2.
• Underlying TOPSIS compensation assumption holds: distances/similarities aggregate linearly across criteria.

Edge cases and pitfalls

• •If the DM supplies explicit weights (this manifest's default), F4 is skipped.
• •when weights are provided exogenously.

• Value-space violation: ensure every entry satisfies α+γ+β ≤ 1 BEFORE building PFPIS/PFNIS - extrema across invalid PFNs lead to spurious ideals.

• PFNIS γ-component for J1: Sindhu Eq.(8) uses min γ (not max γ). Confusion with PFNIS γ = max γ would invert the neutral-degree contrast.

• max-term in similarity: omitting the max(|Δα|,|Δγ|,|Δβ|) component reduces Eq.(6) to a plain Hamming distance and changes the ranking. Both terms are part of Sindhu's distance, not optional.

• Cost-criterion direction (J2): PFPIS uses max γ (not min γ); PFNIS uses max γ as well. This is asymmetric to the benefit case and easy to get wrong.

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