DHF-EDAS - EDAS yönteminin Çift Kararsız Bulanık (DHF) uzantısı

Formül adımları

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

1. Adım 1 (Ning §4) — Uzman başına p adet PDHFE karar matrisi Ʒ=(Ʒ_{ij}^k); uzman ağırlık vektörü θ (Σθ_k=1), öznel öznitelik ağırlık vektörü w (Σw_j=1) ve fayda/maliyet ayrımı. Pure DHFE = tekdüze τ_i=1/#ħ, υ_j=1/#λ.

   $$\Im =({\Im }_{ij}^k{)}_{m\times n},k=1,\dots ,p;{\Im }_{ij}^k=⟨{\hslash }_{ij}^k|{\tau }_{ij}^k,{\lambda }_{ij}^k|{\upsilon }_{ij}^k⟩;\sum _{k=1}^p{\theta }_k=1;\sum _{j=1}^n{w}_j=1$$
   LaTeX

   \Im = (\Im_{ij}^{k})_{m\times n},\quad k=1,\dots,p;\ \Im_{ij}^{k}=\langle \hbar_{ij}^{k}|\tau_{ij}^{k},\ \lambda_{ij}^{k}|\upsilon_{ij}^{k}\rangle;\ \sum_{k=1}^{p}\theta_{k}=1;\ \sum_{j=1}^{n}w_{j}=1

2. Adım 2 (Ning Eq.21) — p uzman PDHF matrisini PDHFWA operatörü (uzman ağırlıkları θ_k) ile birleşik karar matrisi D=(d_{ij})_{m×n}'ye birleştir.

   $$d_{ij}={⨁}_{k=1}^p{\theta }_k{\Im }_{ij}^k={\bigcup }_{{\gamma }_{ij}^k\in {\hslash }_{ij}^k,{\eta }_{ij}^k\in {\lambda }_{ij}^k}\{\left\{[1-{\prod }_{k=1}^p(1-{\gamma }_{ij}^k{)}^{{\theta }_k}]\right|{\prod }_{k=1}^p{p}_{{\gamma }_{ij}^k}\},\left\{{\prod }_{k=1}^p\right({\eta }_{ij}^k{)}^{{\theta }_k}\left|{\prod }_{k=1}^p{q}_{{\eta }_{ij}^k}\right\}\}$$
   LaTeX

   d_{ij} = \bigoplus_{k=1}^{p} \theta_{k}\,\Im_{ij}^{k} = \bigcup_{\gamma_{ij}^{k}\in \hbar_{ij}^{k},\ \eta_{ij}^{k}\in \lambda_{ij}^{k}} \left\{ \left\{\left[1-\prod_{k=1}^{p}\bigl(1-\gamma_{ij}^{k}\bigr)^{\theta_{k}}\right]\Big|\prod_{k=1}^{p} p_{\gamma_{ij}^{k}}\right\},\ \left\{\prod_{k=1}^{p}\bigl(\eta_{ij}^{k}\bigr)^{\theta_{k}}\Big|\prod_{k=1}^{p} q_{\eta_{ij}^{k}}\right\} \right\}

3. Adım 3 (Ning §4) — Toplu PDHF matrisini normalize matris N=(n_{ij})'ye dönüştür: fayda (j∈B) için n_{ij}=d_{ij}; maliyet (j∈C) için n_{ij}=d_{ij}^{C} (PDHFE tümleyeni: ħ↔λ, olasılık etiketleri korunur).

   $$n_{ij}=\{\begin{array}{cc}{d}_{ij}\hfill & j\in B(\text{benefit})\hfill \\ d_{ij}^C\hfill & j\in C(\text{cost})\hfill \end{array},d_{ij}^C=⟨{\lambda }_{ij}|{\upsilon }_{ij},{\hslash }_{ij}|{\tau }_{ij}⟩$$
   LaTeX

   n_{ij} = \begin{cases} d_{ij} & j\in B\ (\text{benefit}) \\ d_{ij}^{C} & j\in C\ (\text{cost}) \end{cases},\quad d_{ij}^{C} = \langle \lambda_{ij}|\upsilon_{ij},\ \hbar_{ij}|\tau_{ij}\rangle

4. Adım 4 (Ning Eq.11) — Skor matrisi S=(s_{D_α}(n_{ij})): s_{D_α}(Ʒ)=(1+s_ħ-s_λ)/2; s_ħ=Σ τ_i γ_i, s_λ=Σ υ_j η_j; Atanassov tereddüt bölünmesi α (Huang–Li 2013 Eq.8, varsayılan α=0.5).

   $$s_{D_{\alpha }}(\Im )={\displaystyle \frac{1+s_{\hslash }(\Im )-s_{\lambda }(\Im )}{2}},s_{\hslash }(\Im )=\sum _i{\tau }_i{\gamma }_i,s_{\lambda }(\Im )=\sum _j{\upsilon }_j{\eta }_j$$
   LaTeX

   s_{D_{\alpha}}(\Im) = \dfrac{1 + s_{\hbar}(\Im) - s_{\lambda}(\Im)}{2},\quad s_{\hbar}(\Im) = \sum_{i} \tau_{i}\gamma_{i},\ s_{\lambda}(\Im) = \sum_{j} \upsilon_{j}\eta_{j}

5. Adım 5 (Ning Eq.22) — Skor matrisi S'yi normalize skor matrisi Q'ya öznitelik bazında min–max normalize ile dönüştür: fayda için (s-min)/(max-min); maliyet için (max-s)/(max-min).

   $$s_{ij}=\{\begin{array}{cc}{\displaystyle \frac{s_{D_{\alpha }}(n_{ij})-{min}_i{s}_{D_{\alpha }}(n_{ij})}{{max}_i{s}_{D_{\alpha }}(n_{ij})-{min}_i{s}_{D_{\alpha }}(n_{ij})}}\hfill & s_{D_{\alpha }}(n_{ij})\in B\hfill \\ [\mathrm{6pt}]{\displaystyle \frac{{max}_i{s}_{D_{\alpha }}(n_{ij})-s_{D_{\alpha }}(n_{ij})}{{max}_i{s}_{D_{\alpha }}(n_{ij})-{min}_i{s}_{D_{\alpha }}(n_{ij})}}\hfill & s_{D_{\alpha }}(n_{ij})\in C\hfill \end{array}$$
   LaTeX

   s_{ij} = \begin{cases} \dfrac{s_{D_{\alpha}}(n_{ij}) - \min_{i} s_{D_{\alpha}}(n_{ij})}{\max_{i} s_{D_{\alpha}}(n_{ij}) - \min_{i} s_{D_{\alpha}}(n_{ij})} & s_{D_{\alpha}}(n_{ij})\in B \\[6pt] \dfrac{\max_{i} s_{D_{\alpha}}(n_{ij}) - s_{D_{\alpha}}(n_{ij})}{\max_{i} s_{D_{\alpha}}(n_{ij}) - \min_{i} s_{D_{\alpha}}(n_{ij})} & s_{D_{\alpha}}(n_{ij})\in C \end{cases}

6. Adım 6 (Ning Eq.13–20) — Birleşik öznitelik ağırlık vektörü ϖ; Lagrange göreli-entropi amacı Eq.(19) min F=Σ ϖ_j ln(ϖ_j/ω_j)+Σ ϖ_j ln(ϖ_j/η_j)+Σ ϖ_j ln(ϖ_j/w_j) s.t. Σ ϖ_j=1, ϖ_j≥0; ω=CRITIC nesnel ağırlık, η=PDHF entropi ağırlığı, w=öznel ağırlık; kapalı form Lagrange çözümü Eq.(20).

   $${\varpi }_j={\displaystyle \frac{\sqrt{{\omega }_j{\eta }_j{w}_j}}{{\sum }_{j=1}^n\sqrt{{\omega }_j{\eta }_j{w}_j}}},\text{s.t.}\sum _{j=1}^n{\varpi }_j=1,{\varpi }_j\ge 0$$
   LaTeX

   \varpi_{j} = \dfrac{\sqrt{\omega_{j}\,\eta_{j}\,w_{j}}}{\sum_{j=1}^{n}\sqrt{\omega_{j}\,\eta_{j}\,w_{j}}},\quad \text{s.t.}\ \sum_{j=1}^{n}\varpi_{j}=1,\ \varpi_{j}\geq 0

7. Adım 7 (Ning Eq.23) — Öznitelik başına ortalama normalize skor PDHFAV; Adım 5'teki Q'nun sütun ortalamasıdır.

   $$PDHFAV=[PDHFA{V}_j{]}_{1\times n},PDHFA{V}_j={\displaystyle \frac{{\sum }_{i=1}^m{s}_{ij}}{m}}$$
   LaTeX

   PDHFAV = [PDHFAV_{j}]_{1\times n},\quad PDHFAV_{j} = \dfrac{\sum_{i=1}^{m} s_{ij}}{m}

8. Adım 8 (Ning Eq.24–26) — Ortalamadan Pozitif/Negatif Sapma PDHFPDA/PDHFNDA; fayda için Eq.25, maliyet için Eq.26.

   $$\{\begin{array}{c}PDHFPD{A}_{ij}={\displaystyle \frac{max(0,s_{ij}-PDHFA{V}_j)}{PDHFA{V}_j}}\hfill \\ PDHFND{A}_{ij}={\displaystyle \frac{max(0,PDHFA{V}_j-s_{ij})}{PDHFA{V}_j}}\hfill \end{array}(j\in B);\{\begin{array}{c}PDHFPD{A}_{ij}={\displaystyle \frac{max(0,PDHFA{V}_j-s_{ij})}{PDHFA{V}_j}}\hfill \\ PDHFND{A}_{ij}={\displaystyle \frac{max(0,s_{ij}-PDHFA{V}_j)}{PDHFA{V}_j}}\hfill \end{array}(j\in C)$$
   LaTeX

   \begin{cases} PDHFPDA_{ij} = \dfrac{\max\bigl(0,\, s_{ij}-PDHFAV_{j}\bigr)}{PDHFAV_{j}} \\ PDHFNDA_{ij} = \dfrac{\max\bigl(0,\, PDHFAV_{j}-s_{ij}\bigr)}{PDHFAV_{j}} \end{cases}\ (j\in B);\quad \begin{cases} PDHFPDA_{ij} = \dfrac{\max\bigl(0,\, PDHFAV_{j}-s_{ij}\bigr)}{PDHFAV_{j}} \\ PDHFNDA_{ij} = \dfrac{\max\bigl(0,\, s_{ij}-PDHFAV_{j}\bigr)}{PDHFAV_{j}} \end{cases}\ (j\in C)

9. Adım 9 (Ning Eq.27) — Adım 6'daki birleşik ağırlık ϖ ile öznitelikler boyunca ağırlıklı toplamlar PDHFSP_i, PDHFSN_i.

   $$PDHFS{P}_i=\sum _{j=1}^n{\varpi }_{j}PDHFPD{A}_{ij},PDHFS{N}_i=\sum _{j=1}^n{\varpi }_{j}PDHFND{A}_{ij}$$
   LaTeX

   PDHFSP_{i} = \sum_{j=1}^{n} \varpi_{j}\, PDHFPDA_{ij},\quad PDHFSN_{i} = \sum_{j=1}^{n} \varpi_{j}\, PDHFNDA_{ij}

10. Adım 10 (Ning Eq.28) — PDHFSP/PDHFSN'yi maksimumlarına göre normalize et.

    $$PDHFNS{P}_i={\displaystyle \frac{PDHFS{P}_i}{{max}_{i}PDHFS{P}_i}},PDHFNS{N}_i=1-{\displaystyle \frac{PDHFS{N}_i}{{max}_{i}PDHFS{N}_i}}$$
    LaTeX

    PDHFNSP_{i} = \dfrac{PDHFSP_{i}}{\max_{i} PDHFSP_{i}},\quad PDHFNSN_{i} = 1 - \dfrac{PDHFSN_{i}}{\max_{i} PDHFSN_{i}}

11. Adım 11 (Ning Eq.29) — Nihai PDHF değerlendirme skoru PDHFAS_i=(PDHFNSP_i+PDHFNSN_i)/2, PDHFAS_i∈[0,1].

    $$PDHFA{S}_i=\frac{1}{2}(PDHFNS{P}_i+PDHFNS{N}_i),PDHFA{S}_i\in [0,1]$$
    LaTeX

    PDHFAS_{i} = \tfrac{1}{2}\bigl(PDHFNSP_{i} + PDHFNSN_{i}\bigr),\quad PDHFAS_{i}\in[0,1]

12. Adım 12 (Ning §4) — Alternatifleri PDHFAS_i'ye göre AZALAN sırada sırala; en büyük PDHFAS optimaldir.

    $$X^{*}=\mathrm{\backslash arg}{max}_{i}PDHFA{S}_i;\text{rank descending in }PDHFA{S}_i$$
    LaTeX

    X^{*} = \arg\max_{i} PDHFAS_{i};\quad \text{rank descending in } PDHFAS_{i}

Sezgi

Dual Hesitant outranking/ranking - Dual Hesitant Fuzzy Element (DHFE: h(x) membership set, g(x) non-membership set). Output typically utility (higher value = preferred).

Sonucu okuma: DHF-EDAS (Ning et al. 2023) is the PDHF-MAGDM extension of EDAS: p decision-makers each supply a PDHF (Probabilistic Dual Hesitant Fuzzy) matrix, aggregated by PDHFWA (Eq.21); cost columns are complemented (Step 3); the score function s_{D_α}(Ʒ)=(1+s_ħ-s_λ)/2 (Eq.11) reduces each cell to a crisp value; the score matrix is min-max normalised per attribute (Eq.22); the combined attribute weight ϖ (Eq.20) fuses CRITIC objective weight ω, PDHF entropy weight η, and decision-maker subjective weight w via Lagrange relative-entropy minimisation; the PDHFEDAS pipeline (PDHFAV→PDA/NDA→SP/SN→NSP/NSN→PDHFAS) yields the final appraisal score, ranked in DESCENDING order. Pure DHFE inputs are the special case τ_i=1/#ħ, υ_j=1/#λ (uniform probabilities); the algorithm collapses to dual-hesitant EDAS without code change. The Atanassov hesitation split α defaults to 0.5; α=1 maximises membership contribution, α=0 maximises non-membership.

Varsayımlar

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

Ne zaman kullanılmaz

• Classical data sufficient - use base EDAS directly (avoid unnecessary uncertainty layer)
• Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Sınırlılıklar

• 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 Dual Hesitant Fuzzy 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

Sık yapılan hatalar

• PDHFE vs DHFE karışıklığı: motor ikisini de kabul eder; pure DHFE girdileri otomatik olarak tekdüze olasılık etiketleri ile PDHFE'ye yükseltilir. Olasılıkları ikinci kez 1/#ħ ile çarpma.
• Skor fonksiyonu α: Eq.(11) gizli bir Atanassov tereddüt bölünmesi α taşır (Huang-Li 2013 Eq.8). α≠0.5 yakın eşitliklerde sıralamayı çevirebilir; α∈{0.25, 0.5, 0.75} duyarlılığını koş.
• Birleşik ağırlık stratejisi: varsayılan 'lagrange_combined' Eq.(20) CRITIC + PDHF entropi + öznel'i eşit Lagrange ağırlığı ile birleştirir. Kullanıcı tek kaynağa güveniyorsa weight_method'u 'subjective_only'/'objective_only' yap; Eq.(20) içinde bir bileşeni sessizce sıfırlama (Lagrange çözümü sıfır argümanda ıraksar).