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Optimizing complex systems and decision-making.

Linear Programming

Linear Programming is a core technology topic with practical implementations, complexity analysis, and real-world applications.

Key points: Syntax and semantics, best practices, debugging techniques, and project-based application.

Simplex Algorithm

Simplex Algorithm is a core technology topic with practical implementations, complexity analysis, and real-world applications.

Key points: Time complexity (Big-O), space complexity, correctness proof, and practical implementation trade-offs.

Integer Programming

Integer Programming is a core technology topic with practical implementations, complexity analysis, and real-world applications.

Key points: Syntax and semantics, best practices, debugging techniques, and project-based application.

Network Flow Models

Network Flow Models is a core technology topic with practical implementations, complexity analysis, and real-world applications.

Key points: Core principles, practical applications, and advanced concepts that build deep understanding.

Queuing Theory

Queuing Theory is an important topic covering core concepts, practical applications, and advanced theory.

Key points: Core principles, practical applications, and advanced concepts that build deep understanding.

Game Theory

Game Theory is a core technology topic with practical implementations, complexity analysis, and real-world applications.

Key points: Core principles, practical applications, and advanced concepts that build deep understanding.

Core Concepts

Linear Programming

A mathematical method for maximizing or minimizing a linear objective function subject to linear constraints.

Simplex Algorithm

The most common method for solving linear programming problems, developed by George Dantzig.

Integer Programming

LP where some or all variables must be integers. Much harder to solve (NP-hard).

Network Flow Models

Problems involving flows through networks: shortest path, maximum flow, minimum cost flow.

Queuing Theory

The mathematical study of waiting lines, used to optimize service systems.

Game Theory

The study of strategic decision-making between competing actors.

Worked Examples & Applications

Transportation Problem: Minimize: ΣΣ Cᵢⱼ × Xᵢⱼ, Subject to: ΣXᵢⱼ = Sᵢ (supply), ΣXᵢⱼ = Dⱼ (demand), Xᵢⱼ ≥ 0. Solved using the Transportation Simplex Method.
Little's Law: L = λ × W, where L = average number of customers in system, λ = average arrival rate, W = average time in system. Example: If 10 customers arrive per hour and each stays 30 min, L = 10 × 0.5 = 5 customers on average.

Practice Questions

Q: What is the difference between LP and IP? A: In LP, variables can take any real value. In IP, some/all variables must be integers, making it NP-hard.
Q: State Little's Law. A: L = λW — The average number of items in a system equals the arrival rate multiplied by the average time spent in the system.

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