IF-ENTROPY - Intuitionistic Fuzzy Entropy Weight Method (Vlachos-Sergiadis 2007 entropy measure as applied by Hung-Chen 2010)

1. 1. Validate IFN axioms (μ+ν ≤ 1) on every matrix entry. If multi-DM input is supplied, also verify DM weights λ on the l-simplex (Σλ=1). Construct the IF decision matrix (or k matrices for group decision making).

   $$R^{(k)}=(r_{ij}^{(k)}{)}_{m\times n},r_{ij}^{(k)}=({\mu }_{ij}^{(k)},{\nu }_{ij}^{(k)}),\mu +\nu \le 1,\sum _k{\lambda }_k=1$$

2. 2. (Multi-DM only; identity for single-DM) Aggregate the k individual DM matrices R^(1), ..., R^(k) into a single collective IF decision matrix D̃ = (r̃_ij) using the Xu 2007 IFWA operator with DM weights λ. For single-DM input (dm_count=1) this step is the identity.

   $${\tilde{r}}_{ij}={\text{IFWA}}_{\lambda }(r_{ij}^{(1)},\dots ,r_{ij}^{(k)})=\left(1-\prod _k(1-{\mu }_{ij}^{(k)}{)}^{{\lambda }_k},\prod _k({\nu }_{ij}^{(k)}{)}^{{\lambda }_k}\right)$$

3. 3. Compute the IF entropy per criterion E^IFS_LT(C_j) using the Vlachos-Sergiadis 2007 measure (Hung-Chen 2010 Eq. 7). For each criterion j: sum over alternatives i of the four-term integrand [μ ln μ + ν ln ν − (1−π) ln(1−π) − π ln 2], scaled by −1/(m ln 2) so that E ∈ [0, 1]. Convention: x ln x = 0 when x = 0 (limit as x→0⁺).

   $$E_{LT}^{\text{IFS}}(C_j)=-\frac{1}{m\ln 2}\sum _{i=1}^m\left[{\mu }_{ij}\ln {\mu }_{ij}+{\nu }_{ij}\ln {\nu }_{ij}-(1-{\pi }_{ij})\ln (1-{\pi }_{ij})-{\pi }_{ij}\ln 2\right]$$

4. 4. Compute the divergence (information content) per criterion d_j = 1 − E^IFS(C_j). Higher d_j means lower entropy, hence higher information content and discrimination power.

   $$d_j=1-E_{LT}^{\text{IFS}}(C_j),j=1,\dots ,n$$

5. 5. Normalise the divergence vector to crisp weights on the n-simplex: w_j = d_j / Σ_k d_k. The output is a crisp weight vector w ∈ Δ^{n-1} satisfying Σw_j = 1, w_j ≥ 0.

   $$w_j=\frac{d_j}{{\sum }_{k=1}^n{d}_k},\sum _j{w}_j=1,w_j\ge 0$$