LOPCOW - LOgarithmic Percentage Change-driven Objective Weighting

Overview

LOPCOW assigns higher weight to criteria with larger logarithmic spread across alternatives - similar to MEREC but uses absolute log of the ratio-based normalised value rather than the full log-utility difference. Criteria where all alternatives perform similarly get very low weight.

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

Any (objective weighting)

How it works

1. 1

   Construct the initial decision matrix X = [x_ij]_{m×n}.

   $$X=[x_{ij}{]}_{m\times n}$$

   Ecer-Pamucar 2022, p.4 Eq.(1); Çemrek 2025, Eq.(1)

2. 2

   Min-max normalisation per direction. Benefit: r_ij = (x_ij − x_min)/(x_max − x_min). Cost: r_ij = (x_max − x_ij)/(x_max − x_min).

   $$r_{ij}=\{\begin{array}{ll}\frac{x_{ij}-x_j^{\min }}{x_j^{\max }-x_j^{\min }}& j\in J^{+}\\ \frac{x_j^{\max }-x_{ij}}{x_j^{\max }-x_j^{\min }}& j\in J^{-}\end{array}$$

   Ecer-Pamucar 2022, p.5 Eqs.(2)-(3); Çemrek 2025, Eqs.(2)-(3)

3. 3

   Percentage value PV_j: logarithm of ratio between root-mean-square of normalised values across m alternatives and their standard deviation, scaled by 100.

   $$P{V}_j=\left|\ln \right(\frac{\sqrt{{\sum }_{i=1}^m{r}_{ij}^2/m}}{{\sigma }_j}\left)\right|\cdot 100$$

   Ecer-Pamucar 2022, Eq.(4); Saraç-Karamaşa 2025 Eq.(4), p.94

4. 4

   Normalised LOPCOW weights w_j = PV_j / Σ PV_k; Σ w_j = 1.

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

   Ecer-Pamucar 2022, p.5 Eq.(5); Çemrek 2025, Eq.(5)

Look elsewhere when

Assumptions to verify

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

Edge cases and pitfalls

• n_ij = 1 for the best alternative (by construction) → ln(1) = 0 → contributes 0 to PLC_j. Only deviations from the best matter.

Works with

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