Distribution-Shift: Regime-Switching Procurement Optimization

A regime-switching distributionally robust optimization (DRO) framework for procurement planning under demand uncertainty with Hidden Markov Model (HMM) state estimation.

🔍 Overview

This repository implements a procurement optimization model that:

• Identifies demand regimes using Hidden Markov Models (HMM)
• Learns transition dynamics between demand states
• Optimizes procurement decisions using distributionally robust optimization
• Compares regime-switching vs. pooled approaches for various sample sizes

📊 Key Features

1. Markov Chain Transition Matrix Learning

The system automatically learns transition probabilities between demand regimes:

# Example learned transition matrix (N=40 samples)
P = [[0.600, 0.400],     # From Low Demand  → [Low, High]
     [0.421, 0.579]]     # From High Demand → [Low, High]

Key Insights:

• Regime Persistence: 60% chance to stay in low demand, 58% chance to stay in high demand
• Switching Probability: ~40-42% chance of regime changes each period
• Long-run Equilibrium: 51.3% low demand, 48.7% high demand

2. Distributionally Robust Optimization

• McCormick relaxation for bilinear terms
• Wasserstein ambiguity sets with cross-validated epsilon tuning
• CVXPY implementation with multiple solver backends

3. Comprehensive Analysis Tools

• Transition matrix visualization and analysis
• Out-of-sample performance evaluation
• Regime forecasting and business insights

🚀 Quick Start

Prerequisites

# Activate your conda environment
conda activate Sai2501

# Required packages
pip install numpy pandas matplotlib seaborn cvxpy hmmlearn openpyxl

Basic Usage

# Run the main optimization with regime-switching analysis
python rs_base.py

# Analyze learned transition matrices
python transition_matrix_analysis.py

# Get business insights and forecasting applications
python transition_matrix_applications.py

# Create visualizations
python plot_base.py

📁 Repository Structure

├── rs_base.py                          # Main optimization script
├── oos_analys_RS.py                    # Out-of-sample analysis
├── transition_matrix_analysis.py       # Transition matrix deep dive
├── transition_matrix_applications.py   # Forecasting applications
├── transition_matrix_summary.py        # Business insights summary
├── plot_base.py                        # Visualization tools
├── visualization.py                    # Additional plotting
├── input_parameters.xlsx               # Model parameters
├── results/
│   ├── results_RS_vs_noRS.xlsx        # Comparison results
│   ├── results_comparison.xlsx        # Performance deltas
│   ├── transition_analysis_*.csv      # Learned matrices
│   └── transition_matrices_analysis.png # Visualizations
└── plots/                             # Generated plots

🔬 Methodology

1. Hidden Markov Model Training

def train_hmm_with_sorted_states(data, n_components=2, random_state=42):
    """
    Train Gaussian HMM and sort states by mean for stable indexing.
    Returns: model, remapped_states, sorted_transmat, sorted_means, state_counts
    """

Features:

• Gaussian emissions with configurable covariance
• State sorting by mean for interpretable results
• Cross-validation for robust parameter estimation

2. Demand Generation

Supports multiple demand distributions:

• Gaussian: Correlated demand with AR(1) structure
• Gamma: Heavy-tailed demand patterns
• Log-normal: Skewed demand distributions
• Weibull: Extreme value distributions
• Mixture: Combined distribution regimes

3. Optimization Framework

def optimize_McC_rs(self, K, epsilon, r, Nk, solver_preference=None):
    """
    McCormick DRO formulation with regime-switching.
    
    Args:
        K: Number of regimes
        epsilon: Wasserstein radius per regime
        r: Regime probability weights
        Nk: Sample counts per regime
    """

📈 Results Analysis

Transition Matrix Properties

Sample Size	Low→Low	Low→High	High→Low	High→High	Stationary [Low, High]
N=10	0.500	0.500	0.400	0.600	[0.444, 0.556]
N=20	0.500	0.500	0.308	0.692	[0.381, 0.619]
N=40	0.600	0.400	0.421	0.579	[0.513, 0.487]

Performance Comparison (RS vs No-RS)

N	Weight	RS Cost	No-RS Cost	Improvement
40	0.3	385,059	385,236	0.05%
40	0.5	381,088	381,283	0.05%
40	0.7	379,583	379,814	0.06%

🎯 Business Applications

1. Short-term Planning (1-2 periods)

• Regime persistence: ~58-60% probability
• Strategy: Plan for current regime, maintain flexibility
• Risk: Keep safety stock for potential switches

2. Medium-term Planning (3-5 periods)

• Convergence: Probabilities approach equilibrium
• Strategy: Balance high/low demand scenarios
• Implementation: Adaptive procurement strategies

3. Long-term Planning (10+ periods)

• Equilibrium: Use stationary distribution (51% Low, 49% High)
• Strategy: Design for balanced demand patterns
• Focus: Robustness over regime-specific optimization

4. Forecasting Example

Starting from high demand regime:

Period 1: 44.1% Low, 55.9% High
Period 2: 49.8% Low, 50.2% High
Period 3: 51.4% Low, 48.6% High
Period 5: 51.3% Low, 48.7% High (equilibrium)

🔧 Configuration

Model Parameters (input_parameters.xlsx)

• Costs: Holding cost (h=5), Backlog cost (b=20)
• Initial Inventory: I₀=1800, B₀=0
• Suppliers: Order costs, lead times, capacities, quality levels
• Prices: Time-varying supplier pricing

Experiment Settings

# Sample sizes for analysis
input_sample_no = [10, 20, 40]

# Cross-validation parameters
k_fold = 5
epsilon_grid = [0, 30, 60]

# Demand mixture weights for OoS evaluation
oos_weights = [0.3, 0.5, 0.7]

📊 Visualization Outputs

1. Transition Matrix Heatmaps: Visual representation of learned probabilities
2. Stationary Distribution Plots: Long-run regime equilibrium
3. Performance Comparison Charts: RS vs No-RS cost analysis
4. Regime Evolution Plots: Cross-validation stability analysis

🔍 Mathematical Properties

Transition Matrix Validation

• ✅ Row Stochastic: Each row sums to 1.0
• ✅ Irreducible: All states reachable from all other states
• ✅ Aperiodic: No cyclical patterns
• ✅ Fast Mixing: Quick convergence (mixing time < 1 period)

Stationary Distribution

Computed as the principal eigenvector of P^T:

eigenvals, eigenvecs = np.linalg.eig(P.T)
stationary = eigenvecs[:, 0].real / sum(eigenvecs[:, 0].real)

🛠️ Development

Adding New Demand Distributions

def get_custom_demand(time_horizon, params, M, seed=None):
    """Template for adding new demand distributions."""
    # Implementation here
    return pd.DataFrame(samples)

Extending Optimization Models

The framework supports adding new DRO formulations by extending the Models class in rs_base.py.

Custom Analysis Scripts

Use the transition matrix API for custom forecasting:

from rs_base import train_hmm_with_sorted_states
model, _, transmat, _, _ = train_hmm_with_sorted_states(data, n_components=2)

📚 References

1. Distributionally Robust Optimization: Wasserstein ambiguity sets
2. Hidden Markov Models: Regime identification and transition learning
3. McCormick Relaxation: Bilinear term linearization
4. Procurement Optimization: Multi-supplier, multi-period planning

🤝 Contributing

1. Fork the repository
2. Create feature branch: git checkout -b feature/amazing-feature
3. Commit changes: git commit -m 'Add amazing feature'
4. Push to branch: git push origin feature/amazing-feature
5. Open a Pull Request

📄 License

This project is licensed under the MIT License - see the LICENSE file for details.

🔗 Contact

• Repository: Distribution-Shift
• Author: Debdeep Paul
• Environment: Sai2501

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Generated on October 10, 2025 - Comprehensive analysis of regime-switching procurement optimization with learned Markov chain dynamics.