PiF-TODIM: Picture extension of TODIM

Overview

Picture outranking/ranking: Picture Fuzzy Number (PiFN: ⟨μ, η, ν⟩; μ+η+ν ≤ 1) with prospect-theory loss aversion. Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Picture outranking/ranking: Picture Fuzzy Number (PiFN: ⟨μ, η, ν⟩; μ+η+ν ≤ 1) with prospect-theory loss aversion
• •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 TODIM (Gomes & Lima 1992); reference-attribute choice (max-weight criterion) can change ranking on dataset edits.)
• •Assumes: Decision matrix entries are valid Picture Fuzzy Numbers (μ + η + ν ≤ 1).
• •Assumes: Criterion weights are DM-supplied with Σ w_j = 1.
• •Assumes: Criterion directions (max/min) are explicit; cost columns get Cuong-complement.
• •Assumes: Loss-aversion factor θ is chosen meaningfully (default 2.5).

Assumptions to verify

• •Decision matrix entries are valid Picture Fuzzy Numbers (μ + η + ν ≤ 1).
• •Criterion weights are DM-supplied with Σ w_j = 1.
• •Criterion directions (max/min) are explicit; cost columns get Cuong-complement.
• •Loss-aversion factor θ is chosen meaningfully (default 2.5).

When not to use

• •Crisp data sufficient: use base TODIM directly.
• •DM is risk-neutral and prospect-theory weighting is unwanted: use PIF-TOPSIS or PIF-VIKOR.
• •Weights are unknown: PF-TODIM requires explicit weights to define the reference attribute.

Edge cases

• •tie, accuracy H(α) = μ + η + ν breaks the tie (Wei Def.5).

Common pitfalls

• •Hatalı: 'PIF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Picture Fuzzy Numbers (μ + η + ν ≤ 1).
• •Hatalı: 'PIF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criterion weights are DM-supplied with Σ w_j = 1.
• •Hatalı: 'PIF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criterion directions (max/min) are explicit; cost columns get Cuong-complement.
• •Hatalı: 'PIF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Loss-aversion factor θ is chosen meaningfully (default 2.5).
• •Hatalı: PIF-TODIM'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PIF-TODIM'yi 'DM is risk-neutral and prospect-theory weighting is unwanted' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: PIF-TODIM'yi 'Weights are unknown' 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 = [⟨μ_ij, η_ij, ν_ij⟩]_{m×n} from DM information. For cost criteria j (direction='min'), apply Cuong complement (μ, η, ν) → (ν, η, μ) so that subsequent score/distance computation is uniformly oriented toward 'higher score is better'. Formül: R = [⟨μ_ij, η_ij, ν_ij⟩]_{m×n}; if direction_j = 'min' then r_ij^complement = ⟨ν_ij, η_ij, μ_ij⟩ Anchor: Wei 2018 §3 (matrix); Cuong 2013 (complement)
2. 2.Adım 2 (F2): Step 2: Compute score S(r_ij) = (1 + μ_ij − ν_ij)/2 ∈ [0,1] for every entry. Score is used to compare two PFNs (Wei 2018 Def.3, via Abdellaoui et al. 2017). On score-tie, accuracy H(α) = μ + η + ν breaks the tie (Wei Def.5). Formül: S(α) = (1 + μ_α − ν_α) / 2; tie-breaker: H(α) = μ_α + η_α + ν_α Anchor: Wei 2018 Eq.(3)-(4); Abdellaoui et al. 2017; Wei 2017a
3. 3.Adım 3 (F3): Step 3: Identify reference attribute G_r = argmax_j w_j and compute relative weights w_jr = w_j / w_r ∈ (0, 1]. Formül: w_r = max_j w_j; w_jr = w_j / w_r, j = 1, ..., n Anchor: Wei 2018 Eq.(9)
4. 4.Adım 4 (F4): Step 4: For every pair of alternatives (A_i, A_t) and every criterion G_j, compute the normalised picture-fuzzy Hamming distance d(r_ij, r_tj) = (1/2)(|μ_ij−μ_tj| + |η_ij−η_tj| + |ν_ij−ν_tj|). Formül: d(α, β) = (1/2) (|μ_α − μ_β| + |η_α − η_β| + |ν_α − ν_β|) Anchor: Wei 2018 Eq.(11)
5. 5.Adım 5 (F5): Step 5: For each criterion G_j and each ordered pair (A_i, A_t), compute per-criterion dominance φ_j(A_i, A_t). Piecewise on score comparison: (a) S(r_ij) > S(r_tj) → φ_j = √(w_jr · d(r_ij, r_tj) / Σ_k w_kr) [gain]; (b) S(r_ij) = S(r_tj) → φ_j = 0; (c) S(r_ij) < S(r_tj) → φ_j = −(1/θ) · √(d(r_ij, r_tj) · Σ_k w_kr / w_jr) [loss, attenuated by θ]. Formül: φ_j(A_i, A_t) = { +√(w_jr·d(r_ij,r_tj)/Σ_k w_kr) if S(r_ij) > S(r_tj); 0 if S(r_ij)=S(r_tj); −(1/θ)·√(d(r_ij,r_tj)·Σ_k w_kr/w_jr) if S(r_ij)<S(r_tj) } Anchor: Wei 2018 Eq.(10)
6. 6.Adım 6 (F6): Step 6: Aggregate overall dominance δ(A_i, A_t) = Σ_{j=1..n} φ_j(A_i, A_t). Produces the m×m dominance matrix δ. Formül: δ(A_i, A_t) = Σ_{j=1}^n φ_j(A_i, A_t), i, t = 1, ..., m Anchor: Wei 2018 Eq.(13)
7. 7.Adım 7 (F7): Step 7: Compute global value δ(A_i) = (Σ_t δ(A_i, A_t) − min_k Σ_t δ(A_k, A_t)) / (max_k Σ_t δ(A_k, A_t) − min_k Σ_t δ(A_k, A_t)) ∈ [0, 1]. Rank alternatives in descending order of δ(A_i); best has δ = 1, worst has δ = 0. Formül: δ(A_i) = (Σ_t δ(A_i, A_t) − min_k Σ_t δ(A_k, A_t)) / (max_k Σ_t δ(A_k, A_t) − min_k Σ_t δ(A_k, A_t)); rank by δ(A_i) descending Anchor: Wei 2018 Eq.(15)

Commonly paired with

• •PIF-CIMAS + PIF-TODIM (common)
• •PIF-SWARA + PIF-TODIM (emerging)

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