PiF-VIKOR - Picture extension of VIKOR

Overview

pif-vikor extends VIKOR 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, lower 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

   Build the initial picture fuzzy decision matrix R_1 = [r_ij]_{m×n} where r_ij = ⟨μ_ij, η_ij, ν_ij⟩ with μ + η + ν ≤ 1. Direction (max/min) of each criterion is recorded.

   $$r_{i}j=⟨{\mu }_{i}j,{\eta }_{i}j,{\nu }_{i}j⟩,{\mu }_{i}j+{\eta }_{i}j+{\nu }_{i}j\le 1;Fan2023\S Phase2Step1.$$

   Fan, Cheng, Wu 2023, Phase 2 Step 1

2. 2

   Standardize R_1 → R_4 using picture fuzzy complement for cost criteria: r_ij ← ⟨ν_ij, η_ij, μ_ij⟩ if criterion j is cost; benefit criteria pass through unchanged.

   $$ifdirectio{n}_j{=}^{\prime }mi{n}^{\prime }:r_{i}j\leftarrow ({\nu }_{i}j,{\eta }_{i}j,{\mu }_{i}j);else{r}_{i}junchanged.Fan2023\S Phase2Step2(alsoTable3row\to Table4rowtransform).$$

   Fan, Cheng, Wu 2023, Phase 2 Step 2

3. 3

   Determine PIS r_j^+ and NIS r_j^- column-wise using Hamming distance to picture-fuzzy anchors (1,0,0) and (0,0,1). r_j^+ = arg min_i d_H(r_ij, (1,0,0)); r_j^- = arg min_i d_H(r_ij, (0,0,1)). Ties on r_j^+ broken by max d_H to (0,0,1). d_H(α,β) = ½(|μ_α−μ_β| + |η_α−η_β| + |ν_α−ν_β|).

   $$d_H(\alpha ,\beta )=½(|{\mu }_{\alpha }-{\mu }_{\beta }|+|{\eta }_{\alpha }-{\eta }_{\beta }|+|{\nu }_{\alpha }-{\nu }_{\beta }|);r_j^{+}=argmi{n}_i{d}_H(r_{i}j,(1,0,0));r_j^{-}=argmi{n}_i{d}_H(r_{i}j,(0,0,1)).Fan2023\S Phase2Step3(Tables5-6).$$

   Fan, Cheng, Wu 2023, Phase 2 Step 3

4. 4

   Compute group utility S_i and individual regret R_i. S_i = Σ_j w_j · d_H(r_ij, r_j^+) / d_H(r_j^-, r_j^+); R_i = max_j { w_j · d_H(r_ij, r_j^+) / d_H(r_j^-, r_j^+) }. Eqs.(15)-(16).

   $$S_i={\Sigma }_j{w}_j\cdot d_H(r_{i}j,r_j^{+})/d_H(r_j^{-},r_j^{+})-Eq.(15);R_i=ma{x}_j{w}_j\cdot d_H(r_{i}j,r_j^{+})/d_H(r_j^{-},r_j^{+})-Eq.(16).Fan2023.$$

   Fan, Cheng, Wu 2023, Eqs.(15)-(16)

5. 5

   Compute Q_i = v · (S_i − S*)/(S^- − S*) + (1 − v) · (R_i − R*)/(R^- − R*), where S* = min_i S_i, S^- = max_i S_i, R* = min_i R_i, R^- = max_i R_i, and v = compromise coefficient (default v = 0.5). Eq.(17).

   $$Q_i=v\cdot (S_i-S\ast )/(S^{-}-S\ast )+(1-v)\cdot (R_i-R\ast )/(R^{-}-R\ast );S\ast =min,S^{-}=max,R\ast =min,R^{-}=max,defaultv=\mathrm{0.5.}Fan2023Eq.(17).$$

   Fan, Cheng, Wu 2023, Eq.(17)

6. 6

   Sort alternatives in ascending order of Q_i to obtain ranking. Apply Opricovic compromise tests: T1 (Acceptable advantage) - Q(A^(2)) − Q(A^(1)) ≥ 1/(m−1); T2 (Acceptable stability) - A^(1) is also best in S or R ranking. If both hold, A^(1) is the unique compromise. If only T1 holds, compromise set = {A^(1), A^(2)}. If only T2 holds, compromise set = {A^(1), …, A^(L)} where Q(A^(L)) − Q(A^(1)) < 1/(m−1).

   $$ranking=argsor{t}_{a}sc(Q);T1:Q(A^{(}2))-Q(A^{(}1))\ge 1/(m-1);T2:A^{(}1)bestinSorR;compromis{e}_{s}etperOpricovicrules.Fan2023Phase2Step6(alsoOpricovic1998).$$

   Fan, Cheng, Wu 2023, Phase 2 Step 6; Opricovic 1998

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

• •if criterion j is cost; benefit criteria pass through unchanged.
• •Ties on r_j^+ broken by max d_H to (0,0,1). d_H(α,β) = ½(|μ_α−μ_β| + |η_α−η_β| + |ν_α−ν_β|).
• •default v = 0.5). Eq.(17).
• •If both hold, A^(1) is the unique compromise. If only T1 holds, compromise set = {A^(1), A^(2)}. If only T2 holds, compromise set = {A^(1), …, A^(L)} where Q(A^(L)) − Q(A^(1)) < 1/(m−1).

• 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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