## Podcast Beta

## Questions and Answers

In the given equation (1), what does L.H.S. represent?

What is the relation between arithmetic mean and geometric mean?

What is the result when taking the geometric mean of two numbers?

What does the inequality 𝑥12 + 𝑦12 ≤ 𝑟 2 represent?

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What does 𝜆𝑥1 , 𝑦1 + 1 − λ 𝑥2 , 𝑦2 ∈ 𝐺 represent?

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In the primal simplex method, when is there an unbounded solution when maximizing the objective function $Z$?

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In the primal simplex method, when is there an unbounded solution when minimizing the objective function $Z$?

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In artificial starting solution for the primal simplex method, what are surplus variables introduced for?

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In a linear programming problem, what does it mean when constraints have no slack variables?

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In a linear programming problem, what does it mean when constraints have surplus variables?

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## Study Notes

### What is Operations Research?

- Operations Research is the scientific study of operations to make better decisions, maximizing benefits and minimizing effort and time.
- It is used to analyze complex real-life problems to improve or optimize performance.

### History of Operations Research

- Operations Research originated during World War II, when the UK used it to win the war by effectively using limited military resources.
- It was used to study strategic and tactical problems associated with air and land defense of the country.

### Convex Sets and Convex Functions

### Definition of Line

- A line is defined as a set of points 𝐿 = 𝑥 ȁ𝑥 = 𝜆𝑥1 + 1 − 𝜆 𝑥2 , 𝜆𝜖𝑅 that passes through two points 𝑥1 and 𝑥2 in 𝑆 ⊆ 𝑅𝑛.

### Definition of Line Segment

- A line segment is a set of points 𝐿 = 𝑥 ȁ𝑥 = 𝜆𝑥1 + 1 − 𝜆 𝑥2 , 0 ≤ 𝜆 ≤ 1 that lies on the line between two points 𝑥1 and 𝑥2 in 𝑆 ⊆ 𝑅𝑛.

### Definition of Convex Set

- A set 𝐾 ⊆ 𝑅𝑛 is convex if for each 𝑥1, 𝑥2 ∈ 𝐾, then 𝑥 ∈ 𝐾, where 𝑥 = 𝜆𝑥1 + 1 − 𝜆 𝑥2, 0 ≤ 𝜆 ≤ 1.

### Example of Convex Set

- The set 𝐺 = { 𝑥1, 𝑥2 : 𝑥12 + 𝑥22 ≤ 𝑟2 } is convex.

### Analytic Solution of Linear Programming Problems

### Types of Solutions in Primal Simplex Method

- Unique Optimal Solution: obtained when all coefficients of non-basic variables in the 𝑍-equation are positive or negative.
- Non-Unique Optimal Solution (infinite number of solutions): obtained when at least one coefficient of non-basic variables in the 𝑍-equation is zero, while all other coefficients are non-negative or non-positive.

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## Description

Explore the scientific study of operations and its application in solving complex real-life problems to optimize performance. Learn about the history of operations research and its significance, including its role in strategically using limited military resources effectively during World War II.