Fuzzy VIKOR: Fuzzy extension of VIKOR

Overview

Fuzzy outranking/ranking: Triangular Fuzzy Number (TFN: l, m, u). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Fuzzy outranking/ranking: Triangular Fuzzy Number (TFN: l, m, u)
• •Preserves fuzzy_TFN 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 base; cf. Belton-Gear 1983, Wang-Luo 2009)
• •Assumes: Decision matrix entries are valid Fuzzy (Triangular) numbers/tuples
• •Assumes: Underlying crisp method's compensation assumption holds in uncertain space
• •Assumes: All decision-maker(s) and experts use the same linguistic/uncertainty scale

Assumptions to verify

• •Decision matrix entries are valid Fuzzy (Triangular) numbers/tuples
• •Underlying crisp method's compensation assumption holds in uncertain space
• •All decision-maker(s) and experts use the same linguistic/uncertainty scale

When not to use

• •Crisp data sufficient: use base VIKOR directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •If C1 fails, return the maximum prefix A^(1)..A^(M) for which Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}.

Common pitfalls

• •Hatalı: 'FUZZY-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Fuzzy (Triangular) numbers/tuples
• •Hatalı: 'FUZZY-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'FUZZY-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: FUZZY-VIKOR'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: FUZZY-VIKOR'yi 'Aggregation operator (PFWA/PFOWA/etc.) not specified' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Determine the best f_j* and worst f_j^- value of every criterion across alternatives, respecting benefit/cost direction. Formül: f_{j}^{*} = \begin{cases}\max_{i} x_{ij} & j \in J \\ \min_{i} x_{ij} & j \in J'\end{cases},\quad f_{j}^{-} = \begin{cases}\min_{i} x_{ij} & j \in J \\ \max_{i} x_{ij} & j \in J'\end{cases} Anchor: Opricovic & Tzeng 2004, §2 Eq.(3)
2. 2.Adım 2 (F2): Step 2: Compute the group utility S_i and the individual regret R_i. S aggregates weighted normalised regret (L_1-metric); R is the maximum weighted regret (L_∞-metric). Formül: \tilde{S}_i = \sum_{j=1}^n \tilde{w}_j \otimes \left( \frac{\tilde{f}_j^* \ominus \tilde{x}_{ij}}{\tilde{f}_j^* \ominus \tilde{f}_j^-} \right) Anchor: Opricovic & Tzeng 2004, §2 Eq.(4)
3. 3.Adım 3 (F3): Step 3: Compute the VIKOR index Q_i as a convex combination of normalised S and R, weighted by the compromise coefficient v. Formül: \tilde{Q}_i = v \frac{\tilde{S}_i \ominus \tilde{S}^*}{\tilde{S}^- \ominus \tilde{S}^*} \oplus (1-v) \frac{\tilde{R}_i \ominus \tilde{R}^*}{\tilde{R}^- \ominus \tilde{R}^*} Anchor: Opricovic & Tzeng 2004, §2 Eq.(5)
4. 4.Adım 4 (F4): Step 4: Propose A^(1) (the lowest-Q alternative) as compromise solution iff both C1 (acceptable advantage) and C2 (acceptable stability) hold. If C1 fails, return the maximum prefix A^(1)..A^(M) for which Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}. Formül: DQ = \dfrac{1}{m-1};\quad C1: Q(A^{(2)}) - Q(A^{(1)}) \ge DQ;\quad C2: A^{(1)} \text{ is best in } S \text{ or in } R Anchor: Opricovic & Tzeng 2004, §2 Eqs.(6)-(7)

Commonly paired with

• •FUZZY-AHP + FUZZY-VIKOR (common)

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