PiF-TODIM - Picture extension of TODIM

Overview

PIF-TODIM follows Wei 2018: build PF decision matrix R, compute score S(α)=(1+μ-ν)/2 and Hamming distance d (Eq.3,11), identify reference attribute G_r=argmax w_j and relative weights w_jr=w_j/w_r (Eq.9), compute per-criterion dominance φ_j piecewise on score comparison with loss-aversion factor θ (Eq.10), aggregate δ(A_i, A_t)=Σ_j φ_j (Eq.13), and rank by min-max normalised global value δ(A_i) (Eq.15). Highest δ(A_i)=1 is best alternative; lowest δ(A_i)=0 is worst.

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 with risk-/loss-averse DMs, MAGDM under epistemic uncertainty, Technology / supplier selection with prospect-theory considerations

How it works

1. 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'.

   $$R=[⟨{\mu }_{i}j,{\eta }_{i}j,{\nu }_{i}j⟩{]}_{m\times n};ifdirectio{n}_j{=}^{\prime }mi{n}^{\prime }then{r}_i{j}^{c}omplement=⟨{\nu }_{i}j,{\eta }_{i}j,{\mu }_{i}j⟩$$

   Wei 2018 §3 (matrix); Cuong 2013 (complement)

2. 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).

   $$S(\alpha )=(1+{\mu }_{\alpha }-{\nu }_{\alpha })/2;tie-breaker:H(\alpha )={\mu }_{\alpha }+{\eta }_{\alpha }+{\nu }_{\alpha }$$

   Wei 2018 Eq.(3)-(4); Abdellaoui et al. 2017; Wei 2017a

3. 3

   Identify reference attribute G_r = argmax_j w_j and compute relative weights w_jr = w_j / w_r ∈ (0, 1].

   $$w_r=ma{x}_j{w}_j;w_{j}r=w_j/w_r,j=1,...,n$$

   Wei 2018 Eq.(9)

4. 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|).

   $$d(\alpha ,\beta )=(1/2)(|{\mu }_{\alpha }-{\mu }_{\beta }|+|{\eta }_{\alpha }-{\eta }_{\beta }|+|{\nu }_{\alpha }-{\nu }_{\beta }|)$$

   Wei 2018 Eq.(11)

5. 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].

   $${\phi }_j(A_i,A_t)=+\surd (w_{j}r\cdot d(r_{i}j,r_{t}j)/{\Sigma }_k{w}_{k}r)ifS(r_{i}j)>S(r_{t}j);0ifS(r_{i}j)=S(r_{t}j);-(1/\theta )\cdot \surd (d(r_{i}j,r_{t}j)\cdot {\Sigma }_k{w}_{k}r/w_{j}r)ifS(r_{i}j)<S(r_{t}j)$$

   Wei 2018 Eq.(10)

6. 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.

   $$\delta (A_i,A_t)={\Sigma }_{j=1}^n{\phi }_j(A_i,A_t),i,t=1,...,m$$

   Wei 2018 Eq.(13)

7. 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.

   $$\delta (A_i)=({\Sigma }_t\delta (A_i,A_t)-mi{n}_k{\Sigma }_t\delta (A_k,A_t))/(ma{x}_k{\Sigma }_t\delta (A_k,A_t)-mi{n}_k{\Sigma }_t\delta (A_k,A_t));rankby\delta (A_i)descending$$

   Wei 2018 Eq.(15)

Fits when / Look elsewhere when

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).

Edge cases and pitfalls

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

• Reference attribute choice: Wei 2018 uses G_r = argmax_j w_j. If multiple attributes share the maximum weight, ties must be broken deterministically (e.g., lowest index) - otherwise different choices yield different w_jr and different φ_j matrices.

• θ parameter: large θ amplifies loss aversion (negative φ_j contributions grow); small θ < 1 inverts amplification (losses are smaller than gains). Wei 2018 uses θ = 2.5 - a θ < 1 may yield ranking inconsistent with prospect-theory intent.

• Score-function dependence: PF-TODIM uses S(α)=(1+μ-ν)/2 to compare PFNs, NOT direct PFN-component comparison. Two PFNs with identical S but different (μ, η, ν) are treated as equal in φ_j (tie-broken by accuracy H).

• Cost-criterion handling: Wei 2018 §4.1 example uses all-benefit attributes. For mixed J1/J2, this manifest applies Cuong complement (μ, η, ν) → (ν, η, μ) per cost criterion in F1 before scoring. Alternative interpretations (inverting the score comparison sign per cost criterion) exist; verify with literature for the specific application.

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