LMAW - Logarithm Methodology of Additive Weights

Overview

Q_i ∈ [0, n] where n = number of criteria. Higher Q means better alternative. Q_i is the sum over criteria of the sigmoid-weighted ξ_ij ∈ (0, 2). Direction is handled in F1 via Pamucar 2021 Eq.(2): benefit standardisation (x+max)/max, cost standardisation (x+min)/x - both produce ϑ_ij > 1 with best alternative receiving largest ϑ. F1 → F2 (log-norm) → F3 (sigmoid weighting) → F4 (sum) is the canonical pipeline.

Output

utility, higher is better

Data

Crisp, complete numeric matrix

Weights

Needs a weight source

Size

2+ alternatives, 3-12 criteria works best

Used for

Alternative selection, Supplier evaluation

How it works

1. 1

   Direction-aware standardisation ϑ_ij. Benefit: ϑ_ij = (x_ij + max_i x_ij)/max_i x_ij. Cost: ϑ_ij = (x_ij + min_i x_ij)/x_ij.

   $${\vartheta }_{ij}=\{\begin{array}{ll}\frac{x_{ij}+{\max }_i{x}_{ij}}{{\max }_i{x}_{ij}},& j\in J^{+}(\text{benefit})\\ \frac{x_{ij}+{\min }_i{x}_{ij}}{x_{ij}},& j\in J^{-}(\text{cost})\end{array}$$

   Pamucar et al. 2021 Facta Universitatis 19:361-380, Eq.(2)

2. 2

   Logarithmic transformation ϕ_ij = ln(ϑ_ij)/ln(Π_i ϑ_ij).

   $${\varphi }_{ij}=\frac{\ln ({\vartheta }_{ij})}{\ln \left({\prod }_{i=1}^m{\vartheta }_{ij}\right)}$$

   Pamucar et al. 2021 Facta Universitatis 19:361-380, Eq.(7)

3. 3

   Sigmoid-form weighted aggregation ξ_ij = 2·ϕ_ij^{w_j} / [(2−ϕ_ij)^{w_j} + ϕ_ij^{w_j}].

   $${\xi }_{ij}=\frac{2{\varphi }_{ij}^{w_j}}{(2-{\varphi }_{ij}{)}^{w_j}+{\varphi }_{ij}^{w_j}}$$

   Pamucar et al. 2021 Facta Universitatis 19:361-380, Eq.(6)

4. 4

   Aggregation Q_i = Σ_j ξ_ij; descending ranking (higher Q = better).

   $$Q_i=\sum _{j=1}^n{\xi }_{ij}$$

   Pamucar et al. 2021 Facta Universitatis 19:361-380, Eq.(8)

Look elsewhere when

Assumptions to verify

• Criteria preferences are independent (no synergistic interactions)
• Compensation is acceptable: high score on one criterion can offset low on another
• Decision matrix is complete (no missing values)

Edge cases and pitfalls

• Zero or negative values: F2 logarithm requires ϑ_ij > 0; E-2 enforces x_ij > 0.

• F1 direction handling is mandatory - without F1, the algorithm cannot distinguish benefit vs cost criteria.

Works with

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