GPU-TOPSIS: A Complete Vectorized and Parallel Reformulation of the TOPSIS Method for Large-Scale Multi-Criteria Decision Making

Abstract

The TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method is one of the most widely used multi-criteria decision-making (MCDM) approaches in industrial, financial, and scientific fields. However, its sequential computational cost of O(m × n), where m denotes the number of alternatives and n the number of criteria, becomes prohibitive when decision matrices have several million rows. Despite its geometric interpretability and simplicity, classical TOPSIS faces two key computational bottlenecks at scale: (i) Euclidean distance calculations O(m × n) dominating the total cost, and (ii) the O(m × log m) sorting step, both inherently sequential and memory-bound on CPUs. To overcome these limitations, we propose GPU-TOPSIS, a fully vectorized and parallel reformulation of TOPSIS based on tensor execution on graphics processing units (GPUs), whose main contributions are: (i) a formally correct reformulation of TOPSIS as a GPU tensor pipeline preserving mathematical fidelity to the original method; (ii) a two-pass fragment-processing algorithm guaranteeing exact mathematical equivalence with monolithic TOPSIS, while reducing the memory footprint from O(m × n) to O(mt × n), where mt < m is the size of each independently processed fragment; (iii) three independent implementations on CuPy, PyTorch, and TensorFlow, ensuring the framework’s portability and genericity. Experimental evaluations on real data from the Amazon Products 2023 dataset, using matrices of up to 200 million alternatives (via the 2-pass formulation), demonstrate speedups of up to 4.75× compared to the reference CPU implementation (NumPy), with inter-backend score differences below 5 × 10−8 and 100% ranking overlap across all tested Top-K thresholds. A perturbation sensitivity analysis of the criteria weights and cross-backend consistency tests confirms that GPU acceleration fully preserves robustness and decision reliability, making GPU-TOPSIS a practical, open, and reproducible solution for large-scale multi-criteria decision making in Big Data environments.