IVIF-COPRAS - Aralık-Değerli Sezgisel Bulanık COPRAS (Davoudabadi-Mousavi-Mohagheghi-Vahdani 2019)

Formül adımları

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

Adım IV — Xu normalleştirme: her kriter j için, μ-bileşenleri Σ_i sqrt(a_ij²+b_ij²) ile, ν-bileşenleri Σ_i sqrt(c_ij²+d_ij²) ile normalleştir. IVIFN değer-uzayını korur.

$F o r e a c h c r i t e r i o n j = 1, . . ., n : d e n o m_{m} u_{j} = s q r t (Σ_{i = 1}^{m} (a_{i} j ² + b_{i} j ²)) [E q . (14) - (15)] d e n o m_{n} u_{j} = s q r t (Σ_{i = 1}^{m} (c_{i} j ² + d_{i} j ²)) [E q . (16) - (17)] T h e n f o r e a c h a l t e r n a t i v e i : ñ_{i} j = ⟨ [a_{i} j / d e n o m_{m} u_{j}, b_{i} j / d e n o m_{m} u_{j}], [c_{i} j / d e n o m_{n} u_{j}, d_{i} j / d e n o m_{n} u_{j}] ⟩ [E q . (18) - (19)] N o t a t i o n n o t e : D a v o u d a b a d i 2019 § 3 u s e s (μ^{L}, μ^{U}, ν^{L}, ν^{U}) = (a, b, c, d) f o r t h e I V I F N ⟨ [a, b], [c, d] ⟩ c o m p o n e n t s .$

LaTeX
For each criterion j = 1, ..., n: denom_mu_j = sqrt( Σ_{i=1}^{m} (a_ij² + b_ij²) ) [Eq. (14)-(15)] denom_nu_j = sqrt( Σ_{i=1}^{m} (c_ij² + d_ij²) ) [Eq. (16)-(17)] Then for each alternative i: ñ_ij = ⟨[ a_ij/denom_mu_j , b_ij/denom_mu_j ], [ c_ij/denom_nu_j , d_ij/denom_nu_j ]⟩ [Eq. (18)-(19)] Notation note: Davoudabadi 2019 §3 uses (μ^L, μ^U, ν^L, ν^U) = (a, b, c, d) for the IVIFN ⟨[a,b],[c,d]⟩ components.
Adım VII — Ağırlıklı normalleştirilmiş matris n̂_ij = w_j ⊙ ñ_ij (Eq. 3 IVIF skaler çarpım ile).

$n ̂_{i} j = w_{j} ⊙ ñ_{i} j = ⟨ [1 - (1 - a_{i} j)^{w_{j}}, 1 - (1 - b_{i} j)^{w_{j}}], [c_{i} j^{w_{j}}, d_{i} j^{w_{j}}] ⟩ [E q . (28) v i a E q . (3)]$

LaTeX
n̂_ij = w_j ⊙ ñ_ij = ⟨[ 1 - (1-a_ij)^{w_j} , 1 - (1-b_ij)^{w_j} ], [ c_ij^{w_j} , d_ij^{w_j} ]⟩ [Eq. (28) via Eq. (3)]
Adım IX — Fayda IVIF toplam B_i (Eq. 29) NB üzerinden, maliyet IVIF toplam C_i (Eq. 30) NC üzerinden. Her ikisi Eq. 1 IVIF ⊕ kullanır.

$I V I F \oplus o p e r a t i o n (E q . 1) : ⟨ [a, b], [c, d] ⟩ \oplus ⟨ [a^{'}, b^{'}], [c^{'}, d^{'}] ⟩ = ⟨ [a + a^{'} - a a^{'}, b + b^{'} - b b^{'}], [c c^{'}, d d^{'}] ⟩ B_{i} = \oplus_{j \in N B} n ̂_{i} j (m a x - d i r e c t i o n c r i t e r i a) [E q . (29)] C_{i} = \oplus_{j \in N C} n ̂_{i} j (m i n - d i r e c t i o n c r i t e r i a) [E q . (30)] w h e r e N B = j : c r i t e r i a_{t} y p e s [j] =^{'} m a x^{'} (p r o f i t), N C = j : c r i t e r i a_{t} y p e s [j] =^{'} m i n^{'} (c o s t) .$

LaTeX
IVIF ⊕ operation (Eq. 1): ⟨[a,b],[c,d]⟩ ⊕ ⟨[a',b'],[c',d']⟩ = ⟨[a+a'-aa', b+b'-bb'], [cc', dd']⟩ B_i = ⊕_{j ∈ NB} n̂_ij (max-direction criteria) [Eq. (29)] C_i = ⊕_{j ∈ NC} n̂_ij (min-direction criteria) [Eq. (30)] where NB = {j : criteria_types[j] = 'max'} (profit), NC = {j : criteria_types[j] = 'min'} (cost).
Adım X — Göreli ağırlık Q_i (Eq. 31). B_i ve C_i üzerinde Garg IVIF skor (GIS, Eq. 7). GIS(α; λ) = λ · (a+b)/2 + (1-λ) · (1 - (c+d)/2). Varsayılan λ = 0.6.

$G I S s c o r e f u n c t i o n (E q . 7, λ - p a r a m e t r i z e d) : G I S (⟨ [a, b], [c, d] ⟩; λ) = λ \cdot (a + b) / 2 + (1 - λ) \cdot (1 - (c + d) / 2) [E q . (7)] C o m p o u n d r e l a t i v e w e i g h t (E q . 31) : Q_{i} = G I S (B_{i}; λ) + (Σ_{k} G I S (C_{k}; λ)) / (G I S (C_{i}; λ) \cdot Σ_{k} (1 / G I S (C_{k}; λ))) [E q . (31)] N o t e : S u m_{k = 1 . . m} r u n s o v e r a l l m a l t e r n a t i v e s . T h e c o s t - s i d e f r a c t i o n i s t h e c l a s s i c a l Z a v a d s k a s - K a k l a u s k a s 1996 C O P R A S d e n o m i n a t o r t r a n s p o s e d i n t o I V I F s c o r e - s p a c e v i a G I S .$

LaTeX
GIS score function (Eq. 7, λ-parametrized): GIS(⟨[a,b],[c,d]⟩; λ) = λ · (a+b)/2 + (1-λ) · (1 - (c+d)/2) [Eq. (7)] Compound relative weight (Eq. 31): Q_i = GIS(B_i; λ) + ( Σ_k GIS(C_k; λ) ) / ( GIS(C_i; λ) · Σ_k (1/GIS(C_k; λ)) ) [Eq. (31)] Note: Sum_{k=1..m} runs over all m alternatives. The cost-side fraction is the classical Zavadskas-Kaklauskas 1996 COPRAS denominator transposed into IVIF score-space via GIS.
Adım XI — Fayda derecesi D_i (Eq. 32), maksimum Q_i'nin yüzdesi. En yüksek Q olan alternatif D = 100% alır. Azalan D'ye göre sırala.

$Q_{m} a x = m a x_{i = 1 . . m} Q_{i} D_{i} = (Q_{i} / Q_{m} a x) \times 100 R a n k i n g : d e s c e n d i n g D_{i} . B e s t a l t e r n a t i v e = a r g m a x_{i} D_{i} (= a r g m a x_{i} Q_{i}) .$

LaTeX
Q_max = max_{i=1..m} Q_i D_i = ( Q_i / Q_max ) × 100% [Eq. (32)] Ranking: descending D_i. Best alternative = argmax_i D_i (= argmax_i Q_i).

Sezgi

Compound-proportional ranking MCDM under Interval-Valued Intuitionistic Fuzzy uncertainty (IVIFN: ⟨[μ⁻,μ⁺],[ν⁻,ν⁺]⟩; μ⁺+ν⁺ ≤ 1) - Xu normalization + IVIF profit/cost sums + Garg score (GIS) compound Q_i + utility degree D_i (%). Output typically utility_degree_percent (higher value = preferred).

Sonucu okuma: IVIF-COPRAS extends Zavadskas-Kaklauskas 1996 crisp COPRAS (Complex Proportional Assessment) to handle Interval-Valued Intuitionistic Fuzzy uncertainty (Atanassov-Gargov 1989). Each alternative is summarized by two IVIF aggregates: B_i (profit IVIF sum over benefit criteria) and C_i (cost IVIF sum over cost criteria). These are projected to crisp scores via Garg's λ-parametrized score function GIS (Eq. 7). The compound relative weight Q_i adds GIS(B_i) to a cost-side fraction that rewards alternatives with LOW cost-side scores. Utility degree D_i = Q_i / Q_max × 100% gives the percentage-of-best ranking signal - higher D is better. The best alternative achieves D = 100%.

Varsayımlar

Decision matrix entries are valid IVIFNs (b+d ≤ 1)
Weights are non-negative and sum to 1 (paper Table 4 sums to 0.997 due to rounding - lenient O-3 tolerance 1e-3)
At least one profit criterion exists (NB ≠ ∅)
λ ∈ [0,1] (default 0.6 from Davoudabadi 2019 §4.1)
Decision-maker accepts compound-proportional-ranking framing (vs. distance-to-ideal TOPSIS or compromise VIKOR)

Ne zaman kullanılmaz

Classical data sufficient - use base COPRAS (Zavadskas-Kaklauskas 1996) directly
Single-valued IFS already provides enough granularity - use IF-COPRAS
Decision-maker wants prospect-theory loss aversion - use IVIF-TODIM instead
Decision-maker wants compromise-stability conditions (C1/C2) - use IVIF-VIKOR instead
All criteria are cost-direction (pure-all-cost configuration unsupported per Eq. 31)

Sınırlılıklar

Rank reversal known on alternative-set changes (ref: Inherited from crisp COPRAS (Zavadskas-Kaklauskas 1996; cf. Belton-Gear 1983, Wang-Luo 2009). The relative-weight Q_i formula depends on the cost-side denominator sum_GIS_C × (1/GIS_C_i) × Σ(1/GIS_C_k) which couples all alternatives; adding/removing an alternative re-scales this denominator and can shift the ranking.)
Assumes: Decision matrix entries are valid IVIFNs (b+d ≤ 1)
Assumes: Weights are non-negative and sum to 1 (paper Table 4 sums to 0.997 due to rounding - lenient O-3 tolerance 1e-3)
Assumes: At least one profit criterion exists (NB ≠ ∅)
Assumes: λ ∈ [0,1] (default 0.6 from Davoudabadi 2019 §4.1)

Sık yapılan hatalar

Değer-uzayı ihlali: tüm girişlerin IVIFN kısıtlarını sağladığından emin olun.
Tüm-maliyet konfigürasyonu desteklenmez: Davoudabadi 2019 Eq. 31 NB ≠ ∅ gerektirir (en az bir fayda kriteri).
λ parametresi anlam: λ ∈ [0,1] GIS üyelik-tamamlayıcı dengesi, olasılık DEĞİL. λ = 0 sadece üye-değil tamamlayıcı, λ = 1 sadece üyelik, λ = 0.5 simetrik, λ = 0.6 paper varsayılanı.
GIS payda sıfırı: herhangi bir alternatif için GIS(C_i; λ) = 0 ise (sadece dejenere IVIFN uçlarında olası), Eq. 31 maliyet-tarafı kesiri tanımsız.
Maliyet kriter işleme: maliyet ve fayda ayrımı Adım IX'da uygulanır (B/C toplam bölümü) - normalleştirmede DEĞİL. Maliyet sütunları için (μ,ν) ↔ (ν,μ) takasını Xu normalleştirmeden önce YAPMAYIN.
Sıra ters çevirme: COPRAS alternatif eklendiğinde/çıkarıldığında sıra ters çevirme gösterir.

Hesap adımları ve dayanakları

Step IV - Xu normalization: for each criterion j, normalize μ-components by Σ_i sqrt(a_ij²+b_ij²) and ν-components by Σ_i sqrt(c_ij²+d_ij²). This preserves IVIFN value-space (μ⁺+ν⁺ ≤ 1) under columnwise rescaling. Davoudabadi 2019 §3 Eqs. 14-19. denom_mu_j = sqrt( Σ_{i=1}^{m} (a_ij² + b_ij²) ) [Eq. (14)-(15)] denom_nu_j = sqrt( Σ_{i=1}^{m} (c_ij² + d_ij²) ) [Eq. (16)-(17)] Then for each alternative i: ñ_ij = ⟨[ a_ij/denom_mu_j , b_ij/denom_mu_j ], [ c_ij/denom_nu_j , d_ij/denom_nu_j ]⟩ [Eq. (18)-(19)] Notation note: Davoudabadi 2019 §3 uses (μ^L, μ^U, ν^L, ν^U) = (a, b, c, d) for the IVIFN ⟨[a,b],[c,d]⟩ components.

Dayanak: Davoudabadi-Mousavi-Mohagheghi-Vahdani 2019 §3 Step IV, Eqs.(14)-(19)
Step VII - Weighted normalized matrix n̂_ij = w_j ⊙ ñ_ij via Eq. 3 IVIF scalar multiplication. Davoudabadi 2019 §3 Eq. 28.

Dayanak: Davoudabadi-Mousavi-Mohagheghi-Vahdani 2019 §3 Step VII, Eq.(28) via Eq.(3)
Step IX - Profit IVIF sum B_i (Eq. 29) over benefit criteria j ∈ NB (max-direction), cost IVIF sum C_i (Eq. 30) over cost criteria j ∈ NC (min-direction). Both use Eq. 1 IVIF ⊕ operation. ⟨[a,b],[c,d]⟩ ⊕ ⟨[a',b'],[c',d']⟩ = ⟨[a+a'-aa', b+b'-bb'], [cc', dd']⟩ B_i = ⊕_{j ∈ NB} n̂_ij (max-direction criteria) [Eq. (29)] C_i = ⊕_{j ∈ NC} n̂_ij (min-direction criteria) [Eq. (30)] where NB = {j : criteria_types[j] = 'max'} (profit), NC = {j : criteria_types[j] = 'min'} (cost).

Dayanak: Davoudabadi-Mousavi-Mohagheghi-Vahdani 2019 §3 Step IX, Eqs.(29)-(30) via Eq.(1)
Step X - Relative weight Q_i (Eq. 31). Uses Garg's IVIF score function GIS (Eq. 7) on B_i and C_i. GIS(α; λ) = λ · (a+b)/2 + (1-λ) · (1 - (c+d)/2) ∈ [0,1] balances membership average and non-membership complement. Default λ = 0.6 (Davoudabadi 2019 §4.1). GIS(⟨[a,b],[c,d]⟩; λ) = λ · (a+b)/2 + (1-λ) · (1 - (c+d)/2) [Eq. (7)] Compound relative weight (Eq. 31): Q_i = GIS(B_i; λ) + ( Σ_k GIS(C_k; λ) ) / ( GIS(C_i; λ) · Σ_k (1/GIS(C_k; λ)) ) [Eq. (31)] Note: Sum_{k=1..m} runs over all m alternatives. The cost-side fraction is the classical Zavadskas-Kaklauskas 1996 COPRAS denominator transposed into IVIF score-space via GIS.

Dayanak: Davoudabadi-Mousavi-Mohagheghi-Vahdani 2019 §3 Step X, Eq.(31) using GIS Eq.(7)
Step XI - Utility degree D_i (Eq. 32) as percentage of the maximum Q_i across alternatives. The alternative with the highest Q achieves D = 100%. Rank by descending D (higher is better). D_i = ( Q_i / Q_max ) × 100% [Eq. (32)] Ranking: descending D_i. Best alternative = argmax_i D_i (= argmax_i Q_i).

Dayanak: Davoudabadi-Mousavi-Mohagheghi-Vahdani 2019 §3 Step XI, Eq.(32)