ENTROPY - Shannon Entropy Weighting Method

Overview

Entropy assigns higher weights to criteria with higher discrimination power (more variation across alternatives). A criterion where all alternatives perform equally (E_j = 1) gets w_j = 0. It is fully data-driven with no DM input. Sensitive to the cost-criterion inversion strategy - ensure consistent treatment.

Output

Weight, higher is better

Data

Crisp, complete numeric matrix

Weights

Derived internally, no weight source needed

Size

2+ alternatives, 3-12 criteria works best

Used for

Objective weighting (any domain), Performance benchmarking

How it works

1. 1

   Step 0 - Cost-criterion inversion: for j ∈ J⁻ apply chosen strategy (default min_j(x_kj)/x_ij). For j ∈ J⁺ leave unchanged.

   $$r_{ij}=\{\begin{array}{ll}{x}_{ij}& j\in J^{+}\\ {\min }_k{x}_{kj}/x_{ij}& j\in J^{-}\text{ (default)}\\ 1/x_{ij}& j\in J^{-}\text{ (reciprocal)}\\ {\max }_k{x}_{kj}-x_{ij}+\epsilon & j\in J^{-}\text{ (max-minus)}\end{array}$$

   Hwang & Yoon 1981 §3.3 (cost inversion convention)

2. 2

   Step 1 - Column-sum probability normalisation: p_ij = r_ij / Σ_i r_ij.

   $$p_{ij}=\frac{r_{ij}}{{\sum }_{i=1}^m{r}_{ij}}$$

   Hwang & Yoon 1981 §3.3 Eq.(3.3.1)

3. 3

   Step 2 - Shannon entropy per criterion: E_j = -k Σ_i p_ij ln(p_ij), k = 1/ln(m). Convention: p ln(p) = 0 when p = 0.

   $$E_j=-\frac{1}{\ln (m)}\sum _{i=1}^m{p}_{ij}\ln (p_{ij}),E_j\in [0,1]$$

   Shannon 1948 (entropy definition); MCDM adaptation per Hwang & Yoon 1981 §3.3

4. 4

   Step 3 - Divergence (information variability) d_j = 1 − E_j.

   $$d_j=1-E_j$$

   Hwang & Yoon 1981 §3.3

5. 5

   Step 4 - Normalised entropy weights w_j = d_j / Σ_k d_k; Σ w_j = 1.

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

   Hwang & Yoon 1981 §3.3

Look elsewhere when

Assumptions to verify

• Decision matrix exists with measurable criteria
• Sufficient inter-alternative variation per criterion

Edge cases and pitfalls

• •default min_j(x_kj)/x_ij). For j ∈ J⁺ leave unchanged.
• •when p = 0.

• Zero probability p_ij=0: ln(0) undefined - treated as 0 (lim p→0 p·ln(p) = 0).

• Cost inversion: adding 1/x_ij vs (max+min−x_ij) vs min/x_ij can give different weights - be consistent.

Works with

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