Enhanced Particle Swarm Optimization for Dual-Constraint Multi-Knapsack Problems: A Real-World Retail Packaging Application
Bibliographic record
Abstract
The Multi-Constraint Multi-Knapsack Parcel Optimization Problem (MCMKPOP) arises in retail logistics where products must be allocated to multiple packages under simultaneous weight and budget constraints.Conventional methods, including exact approaches and traditional metaheuristics, often address these constraints separately or rely on synthetic datasets, scalability and practical relevance.This study proposes an enhanced Particle Swarm Optimization (PSO) framework that introduces three key innovations: an assignment-based representation to encode discrete allocations, a repairbased constraint mechanism to ensure feasibility, and a hybrid value function that integrates price, weight, and efficiency trade-offs.Using a real-world dataset of 213 retail items and ten heterogeneous packaging configurations, the proposed approach was evaluated against Genetic Algorithm, Simulated Annealing, and Random Search.Experimental results from 30 independent runs show that PSO achieves 100% of the theoretical upper bound with a mean performance of 21.3083 ± 0.3145, balanced resource utilization (> 90%), and zero constraint violations.Statistical validation through parametric and non-parametric tests confirmed the robustness and significance of these findings.The novelty this work unified treatment of dual constraints, a multi-knapsackaware upper bound, and validation on real-world data, establishing PSO as both a methodological advance and a practical solution for intelligent retail packaging optimization.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".