IVIF-VIKOR - Aralık-Değerli Sezgisel Bulanık VIKOR (Park, Cho & Kwun 2011)

Formül adımları

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

Step 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.

$O * = {⟨ u_{i}, (m a x_{j} r ̃_{i} j | i \in J 1), (m i n_{j} r ̃_{i} j | i \in J 2) ⟩ | i = 1, . . ., m}^{T} = (r ̃_{1} +, r ̃_{2} +, . . ., r ̃_{m} +)^{T} [E q . (16)] O - = {⟨ u_{i}, (m i n_{j} r ̃_{i} j | i \in J 1), (m a x_{j} r ̃_{i} j | i \in J 2) ⟩ | i = 1, . . ., m}^{T} = (r ̃_{1} -, r ̃_{2} -, . . ., r ̃_{m} -)^{T} [E q . (17)] F o r i \in J 1 (b e n e f i t) : r ̃_{i} + = ⟨ [m a x_{j} a_{i} j, m a x_{j} b_{i} j], [m i n_{j} c_{i} j, m i n_{j} d_{i} j] ⟩; r ̃_{i} - = ⟨ [m i n_{j} a_{i} j, m i n_{j} b_{i} j], [m a x_{j} c_{i} j, m a x_{j} d_{i} j] ⟩ F o r i \in J 2 (c o s t) : r ̃_{i} + = ⟨ [m i n_{j} a_{i} j, m i n_{j} b_{i} j], [m a x_{j} c_{i} j, m a x_{j} d_{i} j] ⟩; r ̃_{i} - = ⟨ [m a x_{j} a_{i} j, m a x_{j} b_{i} j], [m i n_{j} c_{i} j, m i n_{j} d_{i} j] ⟩$

LaTeX
O* = {⟨u_i, (max_j r̃_ij | i∈J1), (min_j r̃_ij | i∈J2)⟩ | i=1,...,m}^T = (r̃_1+, r̃_2+, ..., r̃_m+)^T [Eq.(16)] 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]⟩
Step 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.

$d 1 (⟨ [a 1, b 1], [c 1, d 1] ⟩, ⟨ [a 2, b 2], [c 2, d 2] ⟩) = | a 1 - a 2 | + | b 1 - b 2 | + | c 1 - c 2 | + | d 1 - d 2 | S ̃_{j}^{d 1} = Σ_{i = 1}^{m} w_{i} \cdot d 1 (r ̃_{i} +, r ̃_{i} j) / d 1 (r ̃_{i} +, r ̃_{i} -) [E q . (18)] R ̃_{j}^{d 1} = m a x_{1 \leq i \leq m} [w_{i} \cdot d 1 (r ̃_{i} +, r ̃_{i} j) / d 1 (r ̃_{i} +, r ̃_{i} -)] [E q . (21)] A l t e r n a t i v e d i s t a n c e v a r i a n t s (s e l e c t a b l e v i a p a r a m e t e r d i s t a n c e_{m} e a s u r e) : d 2 m o d i f i e d B u r i l l o - B u s t i n c e — E q s . (19), (22) d H G r z e g o r z e w s k i H a m m i n g — E q s . (20), (23)$

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

$Q ̃_{j} = v \cdot (S ̃_{j} - S ̃ *) / (S ̃ - - S ̃ *) + (1 - v) \cdot (R ̃_{j} - R ̃ *) / (R ̃ - - R ̃ *) [E q . (24)] w h e r e S ̃ * = m i n_{j} S ̃_{j}, S ̃ - = m a x_{j} S ̃_{j} [E q . (25)] R ̃ * = m i n_{j} R ̃_{j}, R ̃ - = m a x_{j} R ̃_{j} [E q . (26)] D e f a u l t v = 0.5 (c o n s e n s u s) . v > 0.5 = v o t i n g b y m a j o r i t y; v < 0.5 = v e t o .$

LaTeX
Q̃_j = v · (S̃_j - S̃*) / (S̃- - S̃*) + (1 - v) · (R̃_j - R̃*) / (R̃- - R̃*) [Eq.(24)] 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.
Step 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.

$R a n k O_{j} b y a s c e n d i n g Q ̃_{j} \to l i s t Q ̃_{[} 1] \leq Q ̃_{[} 2] \leq . . . \leq Q ̃_{[} m] A l s o c o m p u t e a s c e n d i n g l i s t s f o r S ̃ a n d R ̃ . C o m p r o m i s e c o n d i t i o n s (P a r k 2011 § 4 p .243 f r o m O p r i c o v i c 1998) : C 1 (a c c e p t a b l e a d v a n t a g e) : Q ̃_{[} 2] - Q ̃_{[} 1] \geq D Q w h e r e D Q = 1 / (m - 1) C 2 (s t a b i l i t y i n d e c i s i o n - m a k i n g p r o c e s s) : O_{j 1} i s a l s o r a n k e d b e s t i n S ̃_{[} \cdot] o r R ̃_{[} \cdot] I f C 1 a n d C 2 h o l d \to u n i q u e c o m p r o m i s e s o l u t i o n = O_{j 1} . I f o n l y C 2 f a i l s \to c o m p r o m i s e s e t = O_{j 1}, O_{j 2} w h e r e Q ̃_{j 2} = Q ̃_{[} 2] . I f C 1 f a i l s \to c o m p r o m i s e s e t = O_{j 1}, O_{j 2}, . . ., O_{j k} f o r t h e m a x i m u m k w i t h Q ̃_{[} k] - Q ̃_{[} 1] < D Q .$

LaTeX
Rank O_j by ascending Q̃_j → list Q̃_[1] ≤ Q̃_[2] ≤ ... ≤ Q̃_[m] 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.

Sezgi

Compromise-ranking MCDM under Interval-Valued Intuitionistic Fuzzy uncertainty (IVIFN: ⟨[μ⁻,μ⁺],[ν⁻,ν⁺]⟩; μ⁺+ν⁺ ≤ 1) - Lp-metric distance to interval-valued intuitionistic PIS. Output typically compromise_index (lower value = preferred).

Sonucu okuma: 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.

Varsayımlar

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)

Ne zaman kullanılmaz

Classical data sufficient - use base VIKOR (Opricovic 1998) directly
Single-valued IFS already provides enough granularity - use IF-VIKOR (Devi 2011)
Decision-maker wants prospect-theory loss aversion - use IVIF-TODIM instead
Decision-maker wants strict distance-to-ideal ranking without compromise stability conditions - use IVIF-TOPSIS instead

Sınırlılıklar

Rank reversal known on alternative-set changes (ref: Inherited from crisp VIKOR (Opricovic 1998). Compromise solution depends on PIS/NIS bounds - alternative addition/removal can shift PIS/NIS and trigger rank reversal. Acceptable-advantage condition C1 (Q̃_{[2]}-Q̃_{[1]} ≥ 1/(m-1)) acts as a stability gate but does not eliminate reversal.)
Assumes: Decision matrix entries are valid IVIFNs (b+d ≤ 1)
Assumes: Weights are positive and sum to 1
Assumes: Strategy weight v ∈ [0,1] (default 0.5 = consensus)
Assumes: PIS and NIS are distinct for every criterion (d(r̃_i+, r̃_i-) > 0)

Sık yapılan hatalar

Değer-uzayı ihlali: tüm girişlerin IVIFN kısıtlarını sağladığından emin olun.
v parametresi anlam: v ∈ [0,1] strateji ağırlığı, olasılık DEĞİL. v > 0.5 grup yararını (S̃), v < 0.5 en kötü-durum pişmanlığını (R̃) öne çıkarır. Varsayılan 0.5.
PIS/NIS çakışması: bir kriter için d(r̃_i+, r̃_i-) = 0 ise oran tanımsız. Motor o kriter katkısını atlamalı veya E-5 hatası vermeli.
Maliyet kriter işleme: Eqs. 16-17'deki PIS/NIS tanımları maliyet kriterler için max ↔ min'i değiştirir; IVIFN bileşenlerini ters çevirmeyin.
C1/C2 uzlaşı koşulları: en iyi Q̃ tek başına eşsiz uzlaşı çözümünü garantilemez. Her zaman C1 ve C2'yi değerlendir. Biri başarısız ise tek kazanan yerine uzlaşı KÜMESİ raporla.
Uzaklık ölçüsü seçimi sıralamayı etkiler: Park 2011 §5 aynı problem için d1/d2/dH arasında sıralamanın değiştiğini gösterir. Varsayılan d1; seçimi belgele.

Hesap adımları ve dayanakları

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]⟩

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

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

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

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