IVIF-ARAS - Aralık-Değerli Sezgisel Bulanık ARAS (Büyüközkan & Göçer 2018)

Formül adımları

Analiz motorunun yöntem bildirimindeki (manifest F.steps) adımlar; raporlardaki formüllerle aynı kaynaktır.

1. Step 10 — Build the IVIF decision matrix x̃_ij and prepend an optimal performance row x̃_0 (Eq. 18). For benefit criterion j ∈ J1: μ^L(x̃_0j) = max_i μ^L(x̃_ij), μ^U(x̃_0j) = max_i μ^U(x̃_ij), ν^L(x̃_0j) = min_i ν^L(x̃_ij), ν^U(x̃_0j) = min_i ν^U(x̃_ij). For cost criterion j ∈ J2: reverse (min on μ, max on ν). Büyüközkan-Göçer 2018 Eqs. 19-20.

   $$X̃=[x{̃}_{0}j;x{̃}_{i}j](m+1)\times nmatrixwithoptimalrowprepended[Eq.(18)]Forj\in J1(benefit):{\mu }^L(x{̃}_{0}j)=ma{x}_i{\mu }^L(x{̃}_{i}j),{\mu }^U(x{̃}_{0}j)=ma{x}_i{\mu }^U(x{̃}_{i}j){\nu }^L(x{̃}_{0}j)=mi{n}_i{\nu }^L(x{̃}_{i}j),{\nu }^U(x{̃}_{0}j)=mi{n}_i{\nu }^U(x{̃}_{i}j)[Eq.(19)]Forj\in J2(cost):{\mu }^L(x{̃}_{0}j)=mi{n}_i{\mu }^L(x{̃}_{i}j),{\mu }^U(x{̃}_{0}j)=mi{n}_i{\mu }^U(x{̃}_{i}j){\nu }^L(x{̃}_{0}j)=ma{x}_i{\nu }^L(x{̃}_{i}j),{\nu }^U(x{̃}_{0}j)=ma{x}_i{\nu }^U(x{̃}_{i}j)[Eq.(20)]$$
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   X̃ = [x̃_0j; x̃_ij] (m+1) × n matrix with optimal row prepended [Eq.(18)] For j ∈ J1 (benefit): μ^L(x̃_0j) = max_i μ^L(x̃_ij), μ^U(x̃_0j) = max_i μ^U(x̃_ij) ν^L(x̃_0j) = min_i ν^L(x̃_ij), ν^U(x̃_0j) = min_i ν^U(x̃_ij) [Eq.(19)] For j ∈ J2 (cost): μ^L(x̃_0j) = min_i μ^L(x̃_ij), μ^U(x̃_0j) = min_i μ^U(x̃_ij) ν^L(x̃_0j) = max_i ν^L(x̃_ij), ν^U(x̃_0j) = max_i ν^U(x̃_ij) [Eq.(20)]

2. Step 11 — Normalize each criterion column j across alternatives (including the optimal row x̃_0) using the Xu vector normalization (Eqs. 21-22). Each component of the IVIFN is divided by the L2 norm of its column.

   $$a_i{j}^L={\mu }_i{j}^L/sqrt({\Sigma }_{l=0}^m(({\mu }_l{j}^L{)}^2+({\mu }_l{j}^U{)}^2))[Eq.(21)]a_i{j}^U={\mu }_i{j}^U/sqrt({\Sigma }_{l=0}^m(({\mu }_l{j}^L{)}^2+({\mu }_l{j}^U{)}^2))b_i{j}^L={\nu }_i{j}^L/sqrt({\Sigma }_{l=0}^m(({\nu }_l{j}^L{)}^2+({\nu }_l{j}^U{)}^2))[Eq.(22)]b_i{j}^U={\nu }_i{j}^U/sqrt({\Sigma }_{l=0}^m(({\nu }_l{j}^L{)}^2+({\nu }_l{j}^U{)}^2))Result:r{̃}_{i}j=⟨[a_i{j}^L,a_i{j}^U],[b_i{j}^L,b_i{j}^U]⟩,i\in 0,1,...,m,j\in 1,...,n.$$
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   a_ij^L = μ_ij^L / sqrt(Σ_{l=0}^m ((μ_lj^L)^2 + (μ_lj^U)^2)) [Eq.(21)] a_ij^U = μ_ij^U / sqrt(Σ_{l=0}^m ((μ_lj^L)^2 + (μ_lj^U)^2)) b_ij^L = ν_ij^L / sqrt(Σ_{l=0}^m ((ν_lj^L)^2 + (ν_lj^U)^2)) [Eq.(22)] b_ij^U = ν_ij^U / sqrt(Σ_{l=0}^m ((ν_lj^L)^2 + (ν_lj^U)^2)) Result: r̃_ij = ⟨[a_ij^L, a_ij^U], [b_ij^L, b_ij^U]⟩, i ∈ {0,1,...,m}, j ∈ {1,...,n}.

3. Step 12 — Apply criterion weights via IVIF scalar multiplication (Eq. 10 with λ = w_j) to obtain the weighted-normalized matrix R̂ (Eq. 23). Step 13 — Sum the weighted-normalized IVIFN cells across criteria (IVIF sum Eq. 6) to get the optimality function P̃_i (Eq. 24). Defuzzify P̃_i → P_i using the score function (μ^L + μ^U + (1-ν^L) + (1-ν^U))/4 ∈ [0,1] (default), or paper Eq. 25 variant.

   $$Step12—Weighted-normalizedmatrix(IVIFscalarmultEq.10):\lambda ã=⟨[1-(1-{\mu }^L{)}^{\lambda },1-(1-{\mu }^U{)}^{\lambda }],[({\nu }^L{)}^{\lambda },({\nu }^U{)}^{\lambda }]⟩[Eq.(10)]R{̂}_{i}j=w_j{\otimes }_{\lambda }r{̃}_{i}j[Eq.(23)]Step13—Optimalityfunction(IVIFsumEq.6):ã\oplus b̃=⟨[{\mu }_a^L+{\mu }_b^L-{\mu }_a^L·{\mu }_b^L,{\mu }_a^U+{\mu }_b^U-{\mu }_a^U·{\mu }_b^U],[{\nu }_a^L·{\nu }_b^L,{\nu }_a^U·{\nu }_b^U]⟩[Eq.(6)]P{̃}_i={\oplus }_{j=1}^{n}r{̂}_{i}j[Eq.(24)]Defuzzification(default—Hwang-Lin-stylescore,[0,1]-bounded,monotonic,best=1):P_i=({\mu }^L(P{̃}_i)+{\mu }^U(P{̃}_i)+(1-{\nu }^L(P{̃}_i))+(1-{\nu }^U(P{̃}_i)))/4PaperEq.25variantavailableviaparameterdefuzzificatio{n}_{m}ethod{=}^{\prime }pape{r}_{e}q{25}^{\prime }.$$
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   Step 12 — Weighted-normalized matrix (IVIF scalar mult Eq. 10): λã = ⟨[1-(1-μ^L)^λ, 1-(1-μ^U)^λ], [(ν^L)^λ, (ν^U)^λ]⟩ [Eq.(10)] R̂_ij = w_j ⊗_λ r̃_ij [Eq.(23)] Step 13 — Optimality function (IVIF sum Eq. 6): ã ⊕ b̃ = ⟨[μ_a^L+μ_b^L - μ_a^L·μ_b^L, μ_a^U+μ_b^U - μ_a^U·μ_b^U], [ν_a^L·ν_b^L, ν_a^U·ν_b^U]⟩ [Eq.(6)] P̃_i = ⊕_{j=1}^n r̂_ij [Eq.(24)] Defuzzification (default — Hwang-Lin-style score, [0,1]-bounded, monotonic, best=1): P_i = (μ^L(P̃_i) + μ^U(P̃_i) + (1-ν^L(P̃_i)) + (1-ν^U(P̃_i))) / 4 Paper Eq. 25 variant available via parameter defuzzification_method='paper_eq25'.

4. Step 14 — Compute the utility degree Q_i = P_i / P_0 (Eq. 26), the ratio of each alternative's defuzzified optimality to that of the optimal reference row. Step 15 — Rank alternatives by descending Q_i; the alternative with the highest Q_i is the best.

   $$Q_i=P_i/P_0(i=1,...,m)[Eq.(26)]where{P}_{0}isthedefuzzifiedoptimalityoftheoptimalreferencerowx{̃}_0.Rank:Q_{[1]}\ge Q_{[2]}\ge ...\ge Q_{[m]}(descending).Bestalternative=A_{[1]},worst=A_{[m]}.$$
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   Q_i = P_i / P_0 (i = 1, ..., m) [Eq.(26)] where P_0 is the defuzzified optimality of the optimal reference row x̃_0. Rank: Q_{[1]} ≥ Q_{[2]} ≥ ... ≥ Q_{[m]} (descending). Best alternative = A_{[1]}, worst = A_{[m]}.

Sezgi

Utility-degree ranking MCDM under Interval-Valued Intuitionistic Fuzzy uncertainty (IVIFN: ⟨[μ⁻,μ⁺],[ν⁻,ν⁺]⟩; μ⁺+ν⁺ ≤ 1) - additive sum of weighted-normalized IVIF performance ratings benchmarked against an optimal reference row. Output typically utility_degree (higher value = preferred).

Sonucu okuma: IVIF-ARAS extends Zavadskas-Turskis 2010 Additive Ratio ASsessment (ARAS) to handle Interval-Valued Intuitionistic Fuzzy uncertainty (Atanassov-Gargov 1989). Each alternative's overall performance is summarized by an optimality function P̃_i (IVIF sum of weighted-normalized criterion ratings) and benchmarked against an optimal reference row x̃_0 (componentwise max of memberships and min of non-memberships for benefit criteria, reversed for cost). The utility degree Q_i = P_i/P_0 expresses how close each alternative is to the optimal benchmark; values lie in (0, 1] for non-degenerate inputs and the highest Q_i identifies the best alternative.

Varsayımlar

• Decision matrix entries are valid IVIFNs (μ⁺+ν⁺ ≤ 1)
• Weights are positive and sum to 1
• Defuzzification choice documented (score default vs paper_eq25)
• Optimal reference row x̃_0 has at least one non-trivial criterion
• Decision-maker accepts utility-degree framing (vs. compromise-ranking VIKOR or distance-to-ideal TOPSIS)

Ne zaman kullanılmaz

• Classical data sufficient - use base ARAS (Zavadskas-Turskis 2010) directly
• Triangular fuzzy data sufficient - use ARAS-F (Turskis-Zavadskas 2010)
• Grey numbers sufficient - use ARAS-G (Turskis-Zavadskas 2010)
• Single-valued IFS already provides enough granularity - use IF-ARAS
• Decision-maker wants prospect-theory loss aversion - use IVIF-TODIM instead
• Decision-maker wants compromise-ranking with stability conditions - use IVIF-VIKOR instead

Sınırlılıklar

• Rank reversal known on alternative-set changes (ref: Inherited from crisp ARAS (Zavadskas-Turskis 2010). Optimality function P_i depends on the optimal reference row x̃_0 which is constructed from the alternative set (Eqs. 19-20). Adding or removing alternatives can shift x̃_0 and the Xu-normalization denominator (Eqs. 21-22), changing every alternative's P_i and Q_i = P_i/P_0 simultaneously - known rank-reversal vulnerability of reference-based MCDM methods.)
• Assumes: Decision matrix entries are valid IVIFNs (μ⁺+ν⁺ ≤ 1)
• Assumes: Weights are positive and sum to 1
• Assumes: Defuzzification choice documented (score default vs paper_eq25)
• Assumes: Optimal reference row x̃_0 has at least one non-trivial criterion

Sık yapılan hatalar

• Değer-uzayı ihlali: tüm girişlerin IVIFN kısıtlarını sağladığından emin olun.
• Maliyet kriter işleme: maliyet kriterler için x̃_0 oluştururken max ↔ min yönünü tersine çevir.
• Xu normalizasyonu x̃_0'ı içerir: normalizasyon toplamı (Eqs. 21-22) m alternatife ek olarak optimal referans satırını da içermelidir.
• Durulaştırma polaritesi: skor-tabanlı durulaştırıcı en iyi IVIFN'yi 1'e, en kötüsünü 0'a eşler. Paper Eq. 25'in bilinen polarite sorunu vardır.
• Sıralama tersinmesi: ARAS alternatiften türetilen referans satır x̃_0 kullanır; alternatif eklenmesi/çıkarılması x̃_0 ve normalizasyon paydasını değiştirir.
• IVIF AHP ağırlık çıkarımı (Adımlar 1-9) F.steps kapsamı dışıdır. F.steps tek-DM çekirdeği ağırlıkları crisp simpleks olarak verilmiş varsayar.