Podcast
Questions and Answers
In the given equation (1), what does L.H.S. represent?
In the given equation (1), what does L.H.S. represent?
- The left-hand side of the equation
- The sum of squares of arithmetic and geometric means
- A function of arithmetic and geometric means (correct)
- The right-hand side of the equation
What is the relation between arithmetic mean and geometric mean?
What is the relation between arithmetic mean and geometric mean?
- There is no specific relation between them
- They are equal for any two positive numbers
- Geometric mean is always greater than the arithmetic mean
- Arithmetic mean is always greater than the geometric mean (correct)
What is the result when taking the geometric mean of two numbers?
What is the result when taking the geometric mean of two numbers?
- The result is always greater than either number
- The result is always equal to either number
- The result may be greater, smaller, or equal to either number (correct)
- The result is always smaller than either number
What does the inequality 𝑥12 + 𝑦12 ≤ 𝑟 2 represent?
What does the inequality 𝑥12 + 𝑦12 ≤ 𝑟 2 represent?
What does 𝜆𝑥1 , 𝑦1 + 1 − λ 𝑥2 , 𝑦2 ∈ 𝐺 represent?
What does 𝜆𝑥1 , 𝑦1 + 1 − λ 𝑥2 , 𝑦2 ∈ 𝐺 represent?
In the primal simplex method, when is there an unbounded solution when maximizing the objective function $Z$?
In the primal simplex method, when is there an unbounded solution when maximizing the objective function $Z$?
In the primal simplex method, when is there an unbounded solution when minimizing the objective function $Z$?
In the primal simplex method, when is there an unbounded solution when minimizing the objective function $Z$?
In artificial starting solution for the primal simplex method, what are surplus variables introduced for?
In artificial starting solution for the primal simplex method, what are surplus variables introduced for?
In a linear programming problem, what does it mean when constraints have no slack variables?
In a linear programming problem, what does it mean when constraints have no slack variables?
In a linear programming problem, what does it mean when constraints have surplus variables?
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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