IVIF-VIKOR - Interval-Valued Intuitionistic Fuzzy VIKOR (Park, Cho & Kwun 2011)

Overview

IVIF-VIKOR extends Opricovic 1998 compromise-ranking VIKOR to handle Interval-Valued Intuitionistic Fuzzy uncertainty (Atanassov-Gargov 1989). Each alternative is measured by two aggregations relative to the interval-valued intuitionistic PIS: S̃ (sum of weighted distance ratios - group utility) and R̃ (max of weighted distance ratios - worst-case regret). The VIKOR compromise index Q̃ blends these by the strategy weight v: v=0.5 (consensus), v>0.5 (voting by majority), v<0.5 (veto). Lower Q̃ = better. The best alternative is the compromise solution only if BOTH C1 (acceptable advantage Q̃_[2]-Q̃_[1] ≥ 1/(m-1)) AND C2 (stability - best in S̃ or R̃) hold.

Output

compromise index, lower is better

Data

Interval Intuitionistic Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Public transportation alternative selection, Hydropower system selection, Land-use restraint strategies, Compromise decision-making under conflicting criteria, MAGDM under epistemic uncertainty (with IIFHG group aggregation extension)

How it works

1. 1

   Determine the interval-valued intuitionistic PIS O* and NIS O- by componentwise max (for benefit) / min (for cost) over alternatives. For benefit criterion i ∈ J1: r̃_i+ = ⟨[max_j a_ij, max_j b_ij], [min_j c_ij, min_j d_ij]⟩; r̃_i- = ⟨[min_j a_ij, min_j b_ij], [max_j c_ij, max_j d_ij]⟩. For cost criterion i ∈ J2: swap max ↔ min on a,b and c,d. Park 2011 Eqs. 16-17. O- = {⟨u_i, (min_j r̃_ij | i∈J1), (max_j r̃_ij | i∈J2)⟩ | i=1,...,m}^T = (r̃_1-, r̃_2-, ..., r̃_m-)^T [Eq.(17)] For i ∈ J1 (benefit): r̃_i+ = ⟨[max_j a_ij, max_j b_ij], [min_j c_ij, min_j d_ij]⟩; r̃_i- = ⟨[min_j a_ij, min_j b_ij], [max_j c_ij, max_j d_ij]⟩ For i ∈ J2 (cost): r̃_i+ = ⟨[min_j a_ij, min_j b_ij], [max_j c_ij, max_j d_ij]⟩; r̃_i- = ⟨[max_j a_ij, max_j b_ij], [min_j c_ij, min_j d_ij]⟩

   $$O\ast ={⟨u_i,(ma{x}_j{\tilde{r}}_{i}j|i\in J1),(mi{n}_j{\tilde{r}}_{i}j|i\in J2)⟩|i=1,...,m}^T=({\tilde{r}}_1+,{\tilde{r}}_2+,...,{\tilde{r}}_m+{)}^T[Eq.(16)]$$

   Park-Cho-Kwun 2011 §4 p.241 Eqs.(16)-(17)

2. 2

   Compute the average score S̃_j (Eq. 18) and worst group score R̃_j (Eq. 21) for each alternative O_j using the Burillo-Bustince d1 distance (default). The ratio w_i·d1(r̃_i+, r̃_ij)/d1(r̃_i+, r̃_i-) measures the relative closeness of alternative j to the PIS under criterion i; S̃_j sums these over criteria (sum-aggregator), R̃_j takes the max (worst-case aggregator). Eqs. 19-20 / 22-23 give the d2 (modified Burillo-Bustince) and dH (Grzegorzewski) variants. S̃_j^{d1} = Σ_{i=1}^m w_i · d1(r̃_i+, r̃_ij) / d1(r̃_i+, r̃_i-) [Eq.(18)] R̃_j^{d1} = max_{1 ≤ i ≤ m} [ w_i · d1(r̃_i+, r̃_ij) / d1(r̃_i+, r̃_i-) ] [Eq.(21)] Alternative distance variants (selectable via parameter distance_measure): d2 modified Burillo-Bustince - Eqs.(19),(22) dH Grzegorzewski Hamming - Eqs.(20),(23)

   $$d1(⟨[a1,b1],[c1,d1]⟩,⟨[a2,b2],[c2,d2]⟩)=|a1-a2|+|b1-b2|+|c1-c2|+|d1-d2|$$

   Park-Cho-Kwun 2011 §4 p.241-242 Eqs.(18)-(23)

3. 3

   Compute the VIKOR compromise index Q̃_j blending the sum-aggregator S̃_j and the worst-case-aggregator R̃_j by the strategy weight v (default 0.5 = consensus). v > 0.5 = voting by majority (S̃ dominant), v < 0.5 = veto (R̃ dominant). where S̃* = min_j S̃_j, S̃- = max_j S̃_j [Eq.(25)] R̃* = min_j R̃_j, R̃- = max_j R̃_j [Eq.(26)] Default v = 0.5 (consensus). v > 0.5 = voting by majority; v < 0.5 = veto.

   $${\tilde{Q}}_j=v\cdot ({\tilde{S}}_j-\tilde{S}\ast )/(\tilde{S}--\tilde{S}\ast )+(1-v)\cdot ({\tilde{R}}_j-\tilde{R}\ast )/(\tilde{R}--\tilde{R}\ast )[Eq.(24)]$$

   Park-Cho-Kwun 2011 §4 p.243 Eqs.(24)-(26)

4. 4

   Rank alternatives by S̃, R̃ and Q̃ values in ascending order. Three ranking lists S̃_[·], R̃_[·], Q̃_[·] are obtained. The alternative O_{j1} with minimum Q̃ is proposed as compromise solution if both C1 (acceptable advantage) and C2 (stability) hold. Also compute ascending lists for S̃ and R̃. Compromise conditions (Park 2011 §4 p.243 from Opricovic 1998): C1 (acceptable advantage): Q̃_[2] - Q̃_[1] ≥ DQ where DQ = 1 / (m - 1) C2 (stability in decision-making process): O_{j1} is also ranked best in S̃_[·] or R̃_[·] If C1 and C2 hold → unique compromise solution = O_{j1}. If only C2 fails → compromise set = {O_{j1}, O_{j2}} where Q̃_{j2} = Q̃_[2]. If C1 fails → compromise set = {O_{j1}, O_{j2}, ..., O_{jk}} for the maximum k with Q̃_[k] - Q̃_[1] < DQ.

   $$Rank{O}_{j}byascending{\tilde{Q}}_j\to list{\tilde{Q}}_{[}1]\le {\tilde{Q}}_{[}2]\le ...\le {\tilde{Q}}_{[}m]$$

   Park-Cho-Kwun 2011 §4 p.243 (C1, C2) + Opricovic 1998 §2 Step 5

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid IVIFNs (b+d ≤ 1)
• Weights are positive and sum to 1
• Strategy weight v ∈ [0,1] (default 0.5 = consensus)
• PIS and NIS are distinct for every criterion (d(r̃_i+, r̃_i-) > 0)
• Decision-maker accepts compromise-ranking framing (vs. distance-to-ideal TOPSIS or prospect-theory TODIM)

Edge cases and pitfalls

• •default 0.5 = consensus). v > 0.5 = voting by majority (S̃ dominant), v < 0.5 = veto (R̃ dominant).
• •if both C1 (acceptable advantage) and C2 (stability) hold.

• Value-space violation: ensure all entries satisfy IVIFN constraints 0 ≤ a ≤ b ≤ 1, 0 ≤ c ≤ d ≤ 1, b+d ≤ 1 (Atanassov-Gargov 1989).

• v parameter semantics: v ∈ [0,1] is the strategy weight, NOT a probability. v > 0.5 emphasizes group utility (S̃); v < 0.5 emphasizes worst-case regret (R̃); v = 0.5 is consensus. Default 0.5.

• PIS/NIS coincidence: if for some criterion i d(r̃_i+, r̃_i-) = 0 (alternatives identical under that criterion), the ratio is undefined. Engine must skip that criterion contribution or emit E-5 error.

• Cost criteria handling: PIS/NIS definitions in Eqs. 16-17 swap max ↔ min for cost criteria; do not invert the IVIFN components themselves.

• C1/C2 compromise conditions: the best Q̃ alone does NOT guarantee a unique compromise solution. Always evaluate C1 (acceptable advantage Q̃_[2]-Q̃_[1] ≥ 1/(m-1)) and C2 (best Q̃ alternative is also best in S̃ or R̃). If either fails, report the compromise SET rather than a single winner.

• Distance metric choice affects rankings: Park 2011 §5 shows ranking varies between d1 (Burillo-Bustince), d2 (modified), and dH (Grzegorzewski) for the same problem. Default d1 (simplest, additive); document the choice.

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