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

Overview

D-TOPSIS (Fei et al. 2016) handles incomplete/uncertain information via D-numbers (Deng 2012). Each matrix entry D_{ij} = {(B_k, v_k)} with Σ v_k ≤ 1. Step F0 converts each D-number to its expected crisp value E(D) = Σ_k v_k·x_k / Σ_k v_k; if multiple experts, first combine via Dempster orthogonal sum. Steps F1-F5 then run standard TOPSIS on the crisp expected-value matrix. Higher C* = closer to ideal.

Output

utility, higher is better

Data

D-Number, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

D-Numbers MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   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}.

   $$E(D_{ij})=\sum _{k=1}^{n_{ij}}{v}_k\cdot \frac{{\sum }_{x\in B_k}x}{|B_k|},D_1\oplus D_2(B)=\frac{{\sum }_{B_i\cap C_j=B}{D}_1(B_i)D_2(C_j)}{1-{\sum }_{B_i\cap C_j=\varnothing }{D}_1(B_i)D_2(C_j)}$$

   Fei 2016 §3 Steps 1-3; Deng 2012

2. 2

   Step 1 - Vector normalisation on the D-number-derived crisp matrix: scale each column to unit Euclidean norm.

   $$r_{ij}=\frac{x_{ij}}{\sqrt{{\sum }_{i=1}^m{x}_{ij}^2}},i=1,\dots ,m;j=1,\dots ,n$$

   Fei 2016 §3 Step 4; Hwang & Yoon 1981 Eq.(4.1)

3. 3

   Step 2 - Weighted normalised matrix: multiply each column by its weight.

   $$v_{ij}=w_j\cdot r_{ij}$$

   Hwang & Yoon 1981, p.130 Eq.(4.2)

4. 4

   Step 3 - Determine the positive-ideal A⁺ and negative-ideal A⁻ solutions per criterion direction.

   $$A^{+}=\{v_1^{+},\dots ,v_n^{+}\},v_j^{+}=\{\begin{array}{ll}{\max }_i{v}_{ij}& j\in J\\ {\min }_i{v}_{ij}& j\in J^{\prime }\end{array};A^{-}=\{v_1^{-},\dots ,v_n^{-}\},v_j^{-}=\{\begin{array}{ll}{\min }_i{v}_{ij}& j\in J\\ {\max }_i{v}_{ij}& j\in J^{\prime }\end{array}$$

   Hwang & Yoon 1981, p.130 Eqs.(4.3)-(4.4)

5. 5

   Step 4 - Euclidean separation of each alternative from A⁺ and A⁻.

   $$d_i^{+}=\sqrt{\sum _{j=1}^n(v_{ij}-v_j^{+}{)}^2},d_i^{-}=\sqrt{\sum _{j=1}^n(v_{ij}-v_j^{-}{)}^2}$$

   Hwang & Yoon 1981, p.130 Eqs.(4.5)-(4.6)

6. 6

   Step 5 - Relative closeness to the ideal solution; rank by C* descending.

   $$C_i^{\ast }=\frac{d_i^{-}}{d_i^{+}+d_i^{-}},0\le C_i^{\ast }\le 1$$

   Hwang & Yoon 1981, p.131 Eq.(4.7)

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• •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}.

• Incomplete D-number (Σ v_k < 1): normalize by Σ v_k before computing E(D), otherwise expectation is underestimated.

• Confusing D-numbers with Z-numbers: D:Ω→[0,1] (evidence theory, Deng 2012) ≠ Z=(A,B) (restriction+reliability pair, Zadeh 2011). They are distinct uncertainty frameworks.

• Dempster combination step (F0) is only needed when multiple experts provide separate D-number assessments for the same cell. Skip if single expert.

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