📘 LPDM Framework

A Linear Partitioning Diversity Metric for Evaluation of Permutation-based Metaheuristic Algorithms

https://link.springer.com/article/10.1007/s12065-025-01035-9

✍️ Authors

• Majid Shahbazi
• Faramarz Safi-Esfahani
• Sara Shahbazi
• Seyedali Mirjalili

The LPDM Framework (Learning Phase Distribution Modeling) provides a hybrid experimentation environment designed to evaluate the performance of permutation-based metaheuristic algorithms using a novel diversity metric. This project supports both Python and C# implementations and includes empirical results, benchmark comparisons, and full methodology used in the related research paper:

📄 "A Linear Partitioning Diversity Metric for Evaluation of Permutation-based Metaheuristic Algorithms" (Shahbazi et al., 2025)

📁 Repository Structure

LPDM_Framework/ │ ├── Codes-Python/ # Python implementations of LPDM and benchmarks ├── Codes-C#/ # C# implementations for LPDM simulation and experiments ├── Experiment Results/ # Collected metrics and diversity analysis (CSV, XLSX) ├── README.md # Project overview and documentation ├── LICENSE # MIT License └── CITATION.cff # Citation information for this framework

🚀 Key Features

• Dual-language support: Implementations in Python and C#
• Diversity-aware evaluation: Incorporates the LPDM metric for understanding solution diversity in permutation-based algorithms
• Comprehensive experimental design: Includes 18 experiment configurations across multiple strategies, operators, and benchmark functions
• Benchmark Integration: Ready to integrate with standard metaheuristics (e.g., GA, PSO, SA, TS, ACO, etc.)
• Extendable and modular: Easy to add new algorithms or diversity measures
• Statistical Output: CSV/XLSX reports and visualizations for comparative analysis

📐 Methodology Overview

The LPDM Framework assesses diversity in metaheuristic search processes using a linear partitioning model over the solution space. The approach is broken into the following phases:

1. Search Space Encoding
   Permutation-based solutions are encoded and grouped via partitioned diversity spaces.

2. Diversity Metric Computation
   A linear diversity score is computed by mapping each solution to predefined partitions based on linear position indices.

3. Experimentation and Comparison
   The framework runs controlled experiments across standard metaheuristics with varied configurations and captures convergence behavior and diversity over time.

4. Evaluation
   Performance is analyzed using metrics such as solution quality, LPDM diversity, and statistical spread across runs.

For full methodological details, please refer to the accompanying paper.

📊 Experimental Results

The Experiment Results/ folder contains the outcomes of 18 structured experiments:

• Variations of selection strategies and neighborhood operators
• Impact of LPDM on convergence dynamics
• Benchmarks: Job Shop Scheduling Problems, Traveling Salesman Problems
• Results are provided in .csv and .xlsx formats with summaries of accuracy, diversity, and runtime.

🛠 Requirements (Python)

To run the Python-based modules:

bash pip install -r requirements.txt

📜 License

This project is licensed under the MIT License.

🔖 Citation

If you use this framework in your research, please cite the following:

```bibtex @article{shahbazi2025linear, author = {Shahbazi, M. and Safi-Esfahani, F. and Shahbazi, S., Mirjalili, S.}, title = {A linear partitioning diversity metric for evaluation of permutation-based metaheuristic algorithms}, journal = {Evolutionary Intelligence}, volume = {18}, pages = {69}, year = {2025}, doi = {10.1007/s12065-025-01035-9}, url = {https://doi.org/10.1007/s12065-025-01035-9} }

```

You can also refer to the citation file: CITATION.cff

🤝 Contribution

Contributions and extensions are welcome!
To report issues or propose enhancements, feel free to open an issue or fork the repository.

🧪 Explore the diversity, benchmark your algorithms, and improve your optimization strategies using LPDM!