Mathematical Modeling
A comprehensive repository of mathematical formulations, algorithm implementations, and benchmarks for classical and modern Operations Research problems. Built with academic rigor — complete with references, complexity analysis, and scheduling notation.
View on GitHubProblem Families
Twelve families covering the breadth of Operations Research
Scheduling
Flow shop, job shop, parallel machine, single machine, flexible job shop, and resource-constrained project scheduling.
Explore FamilyRouting
Traveling salesman, capacitated vehicle routing, and vehicle routing with time windows. From exact DP to population-based metaheuristics.
Explore FamilyPacking & Cutting
0-1 knapsack, bin packing, and cutting stock. Classic combinatorial optimization with DP, B&B, and approximation algorithms.
Explore FamilyNetwork Flow & Design
Shortest path, maximum flow, minimum spanning tree, multi-commodity flow, and network design. Graph algorithms from Dijkstra to network simplex.
Explore FamilyStochastic & Robust
Newsvendor, two-stage SP, robust shortest path, stochastic knapsack, chance-constrained FL, robust portfolio, DRO, and more.
Explore FamilyCombinatorial
Graph coloring, graph partitioning, max independent set, vertex cover, max clique, maximum satisfiability, and job sequencing.
Explore FamilyInventory & Lot Sizing
EOQ variants, dynamic lot sizing (Wagner-Whitin, Silver-Meal), capacitated lot sizing (MIP + heuristics), multi-echelon inventory, safety stock, and vehicle loading.
Explore FamilyContinuous Optimization
Linear programming, quadratic programming, nonlinear programming, and semidefinite relaxation. Foundational OR methods with KKT conditions and duality.
Explore FamilyMulti-Objective
Bi-objective knapsack with epsilon-constraint, multi-objective TSP with weighted sum, and multi-objective shortest path with Pareto label-setting.
Explore FamilyAssignment & Matching
Linear assignment (Hungarian O(n³)), generalized assignment, quadratic assignment, and graph matching (Edmonds, Hopcroft-Karp).
Explore FamilyLocation & Covering
Facility location (1.488-approx), p-median, hub location, max coverage, set covering, and set packing.
Explore FamilyIntegrated Structural
Location-routing, inventory-routing, and assembly line balancing. Problems requiring joint optimization across families.
Explore FamilyReal-World Applications
See how Operations Research solves real problems across industries and scientific research — with interactive demos you can try
Problem Taxonomy
Complete classification of all implemented problems
Scheduling
Routing
Packing & Cutting
Assignment & Matching
Location & Covering
Network Flow & Design
Stochastic & Robust
Combinatorial
Inventory & Lot Sizing
Integrated Structural
Continuous Optimization
Multi-Objective
Featured Algorithms
A sample of key algorithms across exact, heuristic, and metaheuristic methods
| Algorithm | Type | Problem | Complexity | Reference |
|---|---|---|---|---|
| Johnson's Rule | Exact | F2||Cmax | O(n log n) | Johnson (1954) |
| NEH | Heuristic | Fm|prmu|Cmax | O(n² m) | Nawaz et al. (1983) |
| Iterated Greedy | Fm|prmu|Cmax | Iterative | Ruiz & Stützle (2007) | |
| Held-Karp DP | Exact | TSP | O(2ⁿ n²) | Held & Karp (1962) |
| Clarke-Wright Savings | Heuristic | CVRP | O(n² log n) | Clarke & Wright (1964) |
| Hungarian Method | Exact | LAP | O(n³) | Kuhn (1955) |
| Dijkstra's Algorithm | Exact | SPP | O((V+E) log V) | Dijkstra (1959) |
| Edmonds-Karp | Exact | Max-Flow | O(V E²) | Edmonds & Karp (1972) |
| Shifting Bottleneck | Heuristic | Jm||Cmax | O(m² n²) | Adams et al. (1988) |
| Bitmask DP | Exact | 0-1 Knapsack | O(n W) | Bellman (1957) |
| Kruskal's Algorithm | Exact | MST | O(E log E) | Kruskal (1956) |
| Critical Fractile | Exact | Newsvendor | O(S log S) | Arrow et al. (1951) |
Technology Stack
Built with Python and industry-standard scientific computing libraries