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Operations Research & Optimization

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.

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Problem Families

Twelve families covering the breadth of Operations Research

Scheduling

11 Problems · 600+ Tests

Flow shop, job shop, parallel machine, single machine, flexible job shop, and resource-constrained project scheduling.

Flow Shop Job Shop RCPSP FJSP
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Routing

8 Problems · 270+ Tests

Traveling salesman, capacitated vehicle routing, and vehicle routing with time windows. From exact DP to population-based metaheuristics.

TSP CVRP VRPTW
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Packing & Cutting

7 Problems · 230+ Tests

0-1 knapsack, bin packing, and cutting stock. Classic combinatorial optimization with DP, B&B, and approximation algorithms.

Knapsack Bin Packing Cutting Stock
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Network Flow & Design

5 Problems · 74 Tests

Shortest path, maximum flow, minimum spanning tree, multi-commodity flow, and network design. Graph algorithms from Dijkstra to network simplex.

Dijkstra Max Flow MST MCFP
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Stochastic & Robust

9 Problems · 120+ Tests

Newsvendor, two-stage SP, robust shortest path, stochastic knapsack, chance-constrained FL, robust portfolio, DRO, and more.

Newsvendor DRO SAA Robust
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Combinatorial

7 Problems

Graph coloring, graph partitioning, max independent set, vertex cover, max clique, maximum satisfiability, and job sequencing.

Graph Color MIS Clique MAX-SAT
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Inventory & Lot Sizing

7 Problems · 139 Tests

EOQ variants, dynamic lot sizing (Wagner-Whitin, Silver-Meal), capacitated lot sizing (MIP + heuristics), multi-echelon inventory, safety stock, and vehicle loading.

EOQ Lot Sizing CLSP Safety Stock
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Continuous Optimization

4 Problems

Linear programming, quadratic programming, nonlinear programming, and semidefinite relaxation. Foundational OR methods with KKT conditions and duality.

LP QP NLP SDP
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Multi-Objective

3 Problems

Bi-objective knapsack with epsilon-constraint, multi-objective TSP with weighted sum, and multi-objective shortest path with Pareto label-setting.

ε-Constraint Pareto Weighted Sum
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Assignment & Matching

4 Problems · 17 Tests

Linear assignment (Hungarian O(n³)), generalized assignment, quadratic assignment, and graph matching (Edmonds, Hopcroft-Karp).

LAP QAP Matching
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Location & Covering

6 Problems · 61 Tests

Facility location (1.488-approx), p-median, hub location, max coverage, set covering, and set packing.

CFLP p-Median Set Cover
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Integrated Structural

3 Problems · 47 Tests

Location-routing, inventory-routing, and assembly line balancing. Problems requiring joint optimization across families.

LRP IRP SALBP
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Problem Taxonomy

Complete classification of all implemented problems

Scheduling

11 problems · 879 tests
Permutation Flow ShopFm | prmu | Cmax265 tests
Parallel MachinePm, Qm, Rm | β | Cmax81 tests
Single Machine1 | β | γ99 tests
Job ShopJm | | Cmax74 tests
Flexible Job ShopFJm | | Cmax37 tests
RCPSPPS | prec | Cmax35 tests
Nurse SchedulingNSP12 tests
Assembly Line BalancingSALBP
Batch Scheduling1 | batch | ΣwjCj
Project SchedulingMPSP
Workforce SchedulingWS

Routing

8 problems · 498 tests
Traveling SalesmanTSP / ATSP92 tests
Capacitated VRPCVRP88 tests
VRP with Time WindowsVRPTW82 tests
Arc RoutingCARP
Chinese PostmanCPP
Dial-a-RideDARP
Multi-Depot VRPMDVRP
VRP Pickup & DeliveryVRPPD

Packing & Cutting

7 problems · 389 tests
0-1 KnapsackKP0187 tests
Bin PackingBPP1D56 tests
Cutting StockCSP1D47 tests
2D Bin Packing2D-BPP
Multidimensional KnapsackMdKP
Multiple KnapsackMKP
Strip Packing2D-SPP

Assignment & Matching

4 problems · 17 tests
Linear AssignmentLAP — Hungarian O(n³)17 tests
Quadratic AssignmentQAP
Generalized AssignmentGAP
Graph MatchingEdmonds / Hopcroft-Karp

Location & Covering

6 problems · 61 tests
Facility LocationUFLP16 tests
p-MedianPMP13 tests
Hub Locationp-Hub
Maximum CoverageMCLP
Set CoveringSCP
Set PackingSPP

Network Flow & Design

5 problems · 74 tests
Shortest PathSPP21 tests
Maximum FlowMax-Flow16 tests
Minimum Spanning TreeMST16 tests
Multi-Commodity FlowMCFP
Network DesignFCNDP

Stochastic & Robust

9 problems · 110 tests
NewsvendorNV | stochastic | E[cost]13 tests
Two-Stage SP2SSP10 tests
Robust Shortest PathRSP13 tests
Stochastic KnapsackSKP11 tests
Chance-Constrained FLCCFL11 tests
Robust PortfolioMarkowitz + Ω14 tests
Stochastic VRPSVRP13 tests
Robust Scheduling1 | ˜p | regret13 tests
DROWasserstein12 tests

Combinatorial

7 problems
Graph ColoringGCP
Graph PartitioningGPP
Max Independent SetMIS
Vertex CoverVC
Max CliqueMC
Maximum SatisfiabilityMAX-SAT
Job SequencingJS

Inventory & Lot Sizing

7 problems · 139 tests
Economic Order QuantityEOQ18 tests
Dynamic Lot SizingSilver-Meal / PPB18 tests
Wagner-WhitinWW — DP13 tests
Capacitated Lot SizingCLSP — MIP + Heuristic47 tests
Multi-Echelon InventoryBase Stock / Greedy17 tests
Safety StockAnalytical CSL14 tests
Vehicle Loading2D Bin Packing12 tests

Integrated Structural

3 problems · 47 tests
Inventory-RoutingIRP24 tests
Location-RoutingLRP23 tests
Assembly Line BalancingSALBP

Continuous Optimization

4 problems
Linear ProgrammingLP
Quadratic ProgrammingQP
Nonlinear ProgrammingNLP
Semidefinite RelaxationSDP / GW

Multi-Objective

3 problems
Bi-Objective Knapsackε-constraint
Multi-Objective TSPWeighted Sum
Multi-Objective Shortest PathPareto / Label-Setting

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 Metaheuristic 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

Python 3.10+
Modern type hints, dataclasses
NumPy
Numerical arrays & linear algebra
SciPy
Optimization solvers (HiGHS LP/MIP)
Pandas
Benchmark result analysis
Matplotlib
Visualization & plotting
Pytest
2,556 tests across all families
OR-Tools
CP-SAT constraint programming
Taillard Benchmarks
120 standard PFSP instances
 Portfolio
ESC