D-TOPSIS: Modified TOPSIS based on D-Numbers (Deng Evidence Theory)

Overview

D-Number ranking: D:Ω→[0,1] with Σ_{B⊆Ω} D(B) ≤ 1; incomplete evidence allowed (Deng 2012). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: D-Number ranking: D:Ω→[0,1] with Σ_{B⊆Ω} D(B) ≤ 1; incomplete evidence allowed (Deng 2012)
• •Preserves d_number uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

• •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 D-Numbers 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

Assumptions to verify

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

When not to use

• •Crisp data sufficient: use base TOPSIS directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •if multiple experts, combine D-numbers via Dempster orthogonal sum D_1⊕D_2; convert each D-number to its expected value E(D_{ij}) = Σ_k v_k · (Σ_{x∈B_k} x)/|B_k| to obtain a crisp matrix x_{ij}.

Common pitfalls

• •Hatalı: 'D-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid D-Numbers numbers/tuples
• •Hatalı: 'D-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'D-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: D-TOPSIS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: D-TOPSIS'yi 'Aggregation operator (PFWA/PFOWA/etc.) not specified' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 0: Validate D-number decision matrix entries D_{ij}={(B_k,v_k)} satisfying Σ_k v_k ≤ 1 and B_k⊆Ω; if multiple experts, combine D-numbers via Dempster orthogonal sum D_1⊕D_2; convert each D-number to its expected value E(D_{ij}) = Σ_k v_k · (Σ_{x∈B_k} x)/|B_k| to obtain a crisp matrix x_{ij}. Formül: E(D_{ij}) = \sum_{k=1}^{n_{ij}} v_{k}\cdot\frac{\sum_{x\in B_{k}} x}{|B_{k}|},\quad D_{1}\oplus D_{2}(B) = \frac{\sum_{B_{i}\cap C_{j}=B} D_{1}(B_{i})D_{2}(C_{j})}{1-\sum_{B_{i}\cap C_{j}=\emptyset} D_{1}(B_{i})D_{2}(C_{j})} Anchor: Fei 2016 §3 Steps 1-3; Deng 2012
2. 2.Adım 2 (F2): Step 1: Vector normalisation on the D-number-derived crisp matrix: scale each column to unit Euclidean norm. Formül: r_{ij} = \dfrac{x_{ij}}{\sqrt{\sum_{i=1}^{m} x_{ij}^{2}}}, \quad i=1,\ldots,m;\ j=1,\ldots,n Anchor: Fei 2016 §3 Step 4; Hwang & Yoon 1981 Eq.(4.1)
3. 3.Adım 3 (F3): Step 2: Weighted normalised matrix: multiply each column by its weight. Formül: v_{ij} = w_{j} \cdot r_{ij} Anchor: Hwang & Yoon 1981, p.130 Eq.(4.2)
4. 4.Adım 4 (F4): Step 3: Determine the positive-ideal A⁺ and negative-ideal A⁻ solutions per criterion direction. Formül: A^{+} = \{v_{1}^{+}, \ldots, v_{n}^{+}\},\ v_{j}^{+} = \begin{cases} \max_{i} v_{ij} & j \in J \\ \min_{i} v_{ij} & j \in J' \end{cases};\quad A^{-} = \{v_{1}^{-}, \ldots, v_{n}^{-}\},\ v_{j}^{-} = \begin{cases} \min_{i} v_{ij} & j \in J \\ \max_{i} v_{ij} & j \in J' \end{cases} Anchor: Hwang & Yoon 1981, p.130 Eqs.(4.3)-(4.4)
5. 5.Adım 5 (F5): Step 4: Euclidean separation of each alternative from A⁺ and A⁻. Formül: d_{i}^{+} = \sqrt{\sum_{j=1}^{n}(v_{ij}-v_{j}^{+})^{2}},\quad d_{i}^{-} = \sqrt{\sum_{j=1}^{n}(v_{ij}-v_{j}^{-})^{2}} Anchor: Hwang & Yoon 1981, p.130 Eqs.(4.5)-(4.6)
6. 6.Adım 6 (F6): Step 5: Relative closeness to the ideal solution; rank by C* descending. Formül: C_{i}^{*} = \dfrac{d_{i}^{-}}{d_{i}^{+}+d_{i}^{-}},\quad 0 \le C_{i}^{*} \le 1 Anchor: Hwang & Yoon 1981, p.131 Eq.(4.7)

Commonly paired with

• •n_a + D-TOPSIS (common)

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