This volume studies decision-making under constraints using mathematical models.
This volume studies decision-making under constraints using mathematical models. It integrates optimization, stochastic models, and algorithmic methods.
Part I. Foundations
Chapter 1. Optimization Models
1.1 Decision variables 1.2 Objective functions 1.3 Constraints 1.4 Feasible regions 1.5 Examples
Chapter 2. Linear Programming
2.1 Standard form 2.2 Feasible solutions 2.3 Geometry of solutions 2.4 Applications 2.5 Examples
Chapter 3. Duality
3.1 Dual problems 3.2 Weak and strong duality 3.3 Complementary slackness 3.4 Economic interpretation 3.5 Examples
Part II. Algorithms for Linear Programming
Chapter 4. Simplex Method
4.1 Basic feasible solutions 4.2 Pivot operations 4.3 Degeneracy 4.4 Applications 4.5 Examples
Chapter 5. Interior-Point Methods
5.1 Barrier methods 5.2 Path-following algorithms 5.3 Convergence 5.4 Applications 5.5 Examples
Chapter 6. Large-Scale Optimization
6.1 Sparse systems 6.2 Decomposition methods 6.3 Column generation 6.4 Applications 6.5 Examples
Part III. Integer and Combinatorial Optimization
Chapter 7. Integer Programming
7.1 Formulations 7.2 Branch-and-bound 7.3 Cutting planes 7.4 Applications 7.5 Examples
Chapter 8. Combinatorial Optimization
8.1 Graph models 8.2 Shortest paths 8.3 Matching 8.4 Applications 8.5 Examples
Chapter 9. Network Flow Problems
9.1 Flow models 9.2 Max-flow min-cut theorem 9.3 Minimum cost flow 9.4 Applications 9.5 Examples
Part IV. Nonlinear Programming
Chapter 10. Nonlinear Optimization
10.1 Problem formulation 10.2 Optimality conditions 10.3 Local vs global optima 10.4 Applications 10.5 Examples
Chapter 11. Convex Optimization
11.1 Convex problems 11.2 Duality 11.3 Algorithms 11.4 Applications 11.5 Examples
Chapter 12. Heuristics and Metaheuristics
12.1 Greedy methods 12.2 Simulated annealing 12.3 Genetic algorithms 12.4 Applications 12.5 Examples
Part V. Stochastic Models
Chapter 13. Queueing Theory
13.1 Basic models 13.2 Birth-death processes 13.3 Performance measures 13.4 Applications 13.5 Examples
Chapter 14. Inventory Models
14.1 Deterministic models 14.2 Stochastic models 14.3 Optimization policies 14.4 Applications 14.5 Examples
Chapter 15. Markov Decision Processes
15.1 States and actions 15.2 Transition probabilities 15.3 Bellman equations 15.4 Applications 15.5 Examples
Part VI. Game Theory and Decision Analysis
Chapter 16. Game Theory
16.1 Strategic games 16.2 Nash equilibrium 16.3 Cooperative games 16.4 Applications 16.5 Examples
Chapter 17. Decision Theory
17.1 Utility functions 17.2 Risk analysis 17.3 Multi-criteria decision making 17.4 Applications 17.5 Examples
Chapter 18. Robust Optimization
18.1 Uncertainty models 18.2 Worst-case optimization 18.3 Applications 18.4 Examples 18.5 Connections
Part VII. Applications
Chapter 19. Logistics and Supply Chains
19.1 Transportation problems 19.2 Scheduling 19.3 Routing 19.4 Applications 19.5 Examples
Chapter 20. Finance and Economics
20.1 Portfolio optimization 20.2 Risk management 20.3 Pricing models 20.4 Applications 20.5 Examples
Chapter 21. Engineering Systems
21.1 Resource allocation 21.2 Control systems 21.3 Network design 21.4 Applications 21.5 Examples
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Large-scale optimization 22.2 Data-driven optimization 22.3 Online algorithms 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Complexity limits 23.2 Approximation algorithms 23.3 Uncertainty modeling 23.4 Computational challenges 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of operations research 24.2 Key contributors 24.3 Evolution of mathematical programming 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Optimization method summary B. Algorithm templates C. Proof techniques checklist D. Model formulation examples E. Cross-reference to other MSC branches