PiF-TOPSIS: Picture extension of TOPSIS

Overview

Picture outranking/ranking: Picture Fuzzy Set (PiFS: μ, η, ν; μ+η+ν ≤ 1). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Picture outranking/ranking: Picture Fuzzy Set (PiFS: μ, η, ν; μ+η+ν ≤ 1)
• •Preserves picture uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

• •Rank reversal known on alternative-set changes (ref: Inherited from crisp TOPSIS (Hwang & Yoon 1981); Belton-Gear 1983; García-Cascales & Lamata 2012 specifically on TOPSIS rank-reversal.)
• •Assumes: Decision matrix entries are valid Picture Fuzzy Numbers (α + γ + β ≤ 1).
• •Assumes: Criterion weights either (a) DM-supplied with Σ w_j = 1, or (b) derivable from LP with DM-supplied linear constraints.
• •Assumes: Criterion directions (max/min) are explicit: PFPIS/PFNIS rules differ between J1 and J2.
• •Assumes: Underlying TOPSIS compensation assumption holds: distances/similarities aggregate linearly across criteria.

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.

When not to use

• •Crisp data sufficient: use base TOPSIS directly.
• •Linguistic terms: use Zeng 2019 (LPF-TOPSIS) or fuzzy linguistic methods instead.
• •Higher-rung membership constraint needed (μ + η + ν > 1 but μ^q + η^q + ν^q ≤ 1 for some q > 1): use Ali 2024 q-ROPFS-TOPSIS variant.

Edge cases

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

Common pitfalls

• •Hatalı: 'PIF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Picture Fuzzy Numbers (α + γ + β ≤ 1).
• •Hatalı: 'PIF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criterion weights either (a) DM-supplied with Σ w_j = 1, or (b) derivable from LP with DM-supplied linear constraints.
• •Hatalı: 'PIF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criterion directions (max/min) are explicit: PFPIS/PFNIS rules differ between J1 and J2.
• •Hatalı: 'PIF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying TOPSIS compensation assumption holds: distances/similarities aggregate linearly across criteria.
• •Hatalı: PIF-TOPSIS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PIF-TOPSIS'yi 'Linguistic terms' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PIF-TOPSIS'yi 'Higher-rung membership constraint needed (μ + η + ν > 1 but μ^q + η^q + ν^q ≤ 1 for some q > 1)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 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. Formül: R = [⟨α_ij, γ_ij, β_ij⟩]_{m×n} Anchor: Sindhu 2019 §PF-TOPSIS Step 1
2. 2.Adım 2 (F2): Step 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⟩. Formül: ∇⁺_p = {⟨max_i α, min_i γ, min_i β⟩ if j∈J₁ else ⟨max_i α, max_i γ, min_i β⟩}; ∇⁻_p = {⟨min_i α, min_i γ, max_i β⟩ if j∈J₁ else ⟨min_i α, max_i γ, max_i β⟩} Anchor: Sindhu 2019 Eq.(7)-(8)
3. 3.Adım 3 (F3): Step 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|)). Formül: 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|)) Anchor: Sindhu 2019 Eq.(6),(9),(10)
4. 4.Adım 4 (F4): Step 4 (optional): 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. Formül: Z = Σ_i (S⁺_{pi}(A_i, ∇⁺_p) − S⁻_{pi}(A_i, ∇⁻_p)) subject to Σ w_j = 1, DM-supplied linear constraints on w_j Anchor: Sindhu 2019 Eq.(11)
5. 5.Adım 5 (F5): Step 5 (optional): Solve the LP from F4 to obtain w_j. Skipped when weights are provided exogenously. Formül: maximize Z(w_1, ..., w_n) s.t. linear constraints; standard LP (Vanderbei 2014 reference [26]) Anchor: Sindhu 2019 Step 5; Vanderbei 2014
6. 6.Adım 6 (F6): Step 6: Recompute S⁺_p and S⁻_p with the final weight vector (identical formula to F3, just with finalised w_j). Formül: Same as F3 with finalised w_j Anchor: Sindhu 2019 Step 6
7. 7.Adım 7 (F7): Step 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. Formül: CR_i = S⁺_{pi} / (S⁺_{pi} + S⁻_{pi}); rank by CR_i descending Anchor: Sindhu 2019 Eq.(12)

Commonly paired with

• •n_a + PIF-TOPSIS (common)

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