IF-MAUT - Sezgisel Bulanık Çok-Nitelikli Fayda Teorisi (IFWA-temelli toplamlı fayda)

Formül adımları

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

Adım 1 — Her hücre için IFN aksiyomlarını doğrula (μ+ν ≤ 1); kesin ağırlıkların n-simpleks üzerinde olduğunu kontrol et (Σω=1). IF karar matrisini R = (r_ij)_{m×n}, r_ij = (μ_ij, ν_ij) olarak kur.

$R = (r_{i j})_{m \times n}, r_{i j} = (μ_{i j}, ν_{i j}), μ_{i j} + ν_{i j} \leq 1, π_{i j} = 1 - μ_{i j} - ν_{i j}$

LaTeX
R = (r_{ij})_{m \times n},\ r_{ij} = (\mu_{ij}, \nu_{ij}),\ \mu_{ij} + \nu_{ij} \le 1,\ \pi_{ij} = 1 - \mu_{ij} - \nu_{ij}
Adım 2 — Her cost kriter j için (yön='min'), girişleri (μ_ij, ν_ij) → (ν_ij, μ_ij) ile tümle. Benefit kriterler değişmez. Bu, her kriteri benefit yönüne çevirerek IFWA'nın tercih anlamını korumasını sağlar.

${\hat{r}}_{i j} = {\begin{matrix} (μ_{i j}, ν_{i j}) & if j is benefit \\ (ν_{i j}, μ_{i j}) & if j is cost \end{matrix}$

LaTeX
\hat{r}_{ij} = \begin{cases} (\mu_{ij}, \nu_{ij}) & \text{if } j \text{ is benefit} \\ (\nu_{ij}, \mu_{ij}) & \text{if } j \text{ is cost} \end{cases}
Adım 3 — Her A_i alternatifi için, n IFN'lik satırı kesin kriter ağırlıkları ω ile Xu 2007 IFWA operatörü kullanarak tek bir fayda IFN'ine U_i'ye agrege et.

$U_{i} = {IFWA}_{ω} ({\hat{r}}_{i 1}, \dots, {\hat{r}}_{i n}) = (1 - \prod_{j = 1}^{n} (1 - μ_{i j})^{ω_{j}}, \prod_{j = 1}^{n} ν_{i j}^{ω_{j}})$

LaTeX
U_i = \text{IFWA}_{\omega}(\hat{r}_{i1}, \ldots, \hat{r}_{in}) = \left(1 - \prod_{j=1}^{n} (1 - \mu_{ij})^{\omega_j},\ \prod_{j=1}^{n} \nu_{ij}^{\omega_j}\right)
Adım 4 — Her alternatif için Chen-Tan 1994 skor fonksiyonunu hesapla S(U_i) = μ_{U_i} − ν_{U_i}.

$S (U_{i}) = μ_{U_{i}} - ν_{U_{i}} \in [- 1, 1]$

LaTeX
S(U_i) = \mu_{U_i} - \nu_{U_i} \in [-1, 1]
Adım 5 — Eşit skorlarda eşitlik kırma için Hong-Choi 2000 doğruluk fonksiyonu H(U_i) = μ_{U_i} + ν_{U_i}.

$H (U_{i}) = μ_{U_{i}} + ν_{U_{i}} \in [0, 1]$

LaTeX
H(U_i) = \mu_{U_i} + \nu_{U_i} \in [0, 1]
Adım 6 — (S, H) leksikografik ikilisine göre alternatifleri azalan olarak sırala (Xu 2007 Def 3.1): önce S(U_i)'ye göre, eşitlik halinde H(U_i)'ye göre.

$A_{i} ≻ A_{k} ⟺ S (U_{i}) > S (U_{k}) or (S (U_{i}) = S (U_{k}) \land H (U_{i}) > H (U_{k}))$

LaTeX
A_i \succ A_k \iff S(U_i) > S(U_k) \text{ or } (S(U_i) = S(U_k) \land H(U_i) > H(U_k))

Sezgi

Aggregation-operator-based additive ranking under Intuitionistic Fuzzy uncertainty (IFN: μ, ν; μ+ν ≤ 1). Output typically utility_score (higher value = preferred).

Sonucu okuma: IF-MAUT extends additive utility theory to the IF setting via the Xu 2007 IFWA aggregation operator. Each alternative's row of IFN ratings is aggregated into a single utility IFN U_i, then defuzzified by the Chen-Tan score S(U_i) = μ − ν for ranking. Cost criteria are complemented at Step F2 via (μ, ν) → (ν, μ). On ties, Hong-Choi accuracy H = μ + ν breaks them.

Varsayımlar

Decision matrix entries are valid IFNs (μ+ν ≤ 1)
Preferential independence: per-criterion utility is additively separable
Criterion weights are crisp and sum to 1 (Xu 2007 IFWA assumption)
Expert consensus on the IFN scale (e.g. mapping linguistic terms to (μ, ν) pairs)

Ne zaman kullanılmaz

Classical data sufficient - use base MAUT directly
Group decision-making with multiple DM weights needed - use IF-TOPSIS (Boran 2009) or extend with the outer IFWA loop
Strong interaction between criteria - additive utility assumption violated, consider Choquet integral-based IF-MADM
IF-SAW already used on the same problem - output is numerically identical, redundant

Sınırlılıklar

Assumes: Decision matrix entries are valid IFNs (μ+ν ≤ 1)
Assumes: Preferential independence: per-criterion utility is additively separable
Assumes: Criterion weights are crisp and sum to 1 (Xu 2007 IFWA assumption)
Assumes: Expert consensus on the IFN scale (e.g. mapping linguistic terms to (μ, ν) pairs)

Sık yapılan hatalar

Değer-uzayı ihlali: her giriş μ, ν ∈ [0,1] ve μ+ν ≤ 1 sağlamalı. Dilsel terimlerle elde edilen derecelendirmeler geçerli IFN'lere eşlenmelidir.
IFWA'yı (kesin ağırlıklar ω) IFWG / IFOWA / IFHA ile karıştırma: IF-MAUT yalnızca IFWA kullanır.
IF-SAW ile operasyonel örtüşme: kesin kriter ağırlıkları ve IFWA agregasyonu altında IF-MAUT ile IF-SAW aynı sayısal çıktıyı üretir.

Hesap adımları ve dayanakları

Validate IFN axioms (μ+ν ≤ 1) on every matrix entry; verify crisp weights on the n-simplex (Σω=1). Construct the IF decision matrix R = (r_ij)_{m×n}, r_ij = (μ_ij, ν_ij).

Dayanak: Atanassov 1986 (IFS axioms); Xu 2007 §II
For each cost criterion j (direction='min'), complement entries via (μ_ij, ν_ij) → (ν_ij, μ_ij). Benefit criteria are unchanged. This converts every criterion to benefit-direction so that IFWA preserves preference semantics.

Dayanak: Atanassov 1986 IFN complement; standard IF-MADM normalisation convention (e.g. Boran 2009 Step 5 cost branch)
For each alternative A_i, aggregate the row of n IFNs into a single utility IFN U_i using the Xu 2007 IFWA operator with crisp criterion weights ω. The IFWA closed form (Xu 2007 Eq. 12) gives U_i = (1 - ∏(1-μ_ij)^ω_j, ∏ ν_ij^ω_j).

Dayanak: Xu 2007 Def 3.3 + Theorem 3.4 (Eqs. 11-12 IFWA closed form)
Compute the Chen-Tan 1994 score S(U_i) = μ_{U_i} − ν_{U_i} for each alternative. S(U_i) ∈ [−1, 1] is the primary defuzzified utility used for ranking.

Dayanak: Chen & Tan 1994 score function; Xu 2007 Eq. (4)
Compute the Hong-Choi 2000 accuracy H(U_i) = μ_{U_i} + ν_{U_i} for tie-breaking on equal scores. H ∈ [0, 1]; higher H means lower hesitancy π = 1 − H.

Dayanak: Hong & Choi 2000 accuracy function; Xu 2007 Eq. (5)
Rank alternatives in descending order of the (S, H) lex pair per Xu 2007 Def 3.1: primary sort by S(U_i) descending; on equality of S, break ties by H(U_i) descending; if both S and H equal, the alternatives are equivalent under IFN comparison.

Dayanak: Xu 2007 Def 3.1 (score-then-accuracy comparison)