Operations Research: Making Better Decisions
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Questions and Answers

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?

  • 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?

  • 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?

    <p>Sum of squares of coordinates is less than or equal to 𝑟 2</p> Signup and view all the answers

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

    <p>A linear combination of points in set G with coefficients λ and (1-λ)</p> Signup and view all the answers

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

    <p>When one of the coefficients of the non-basic variable in $Z$-equation is negative, while all the corresponding entries in the constraint matrix are non-positive</p> Signup and view all the answers

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

    <p>When one of the coefficients of the non-basic variable in $Z$-equation is positive, while all the corresponding entries in the constraint matrix are non-positive</p> Signup and view all the answers

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

    <p>To transform a problem into standard form by converting inequality constraints to equality constraints</p> Signup and view all the answers

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

    <p>The problem is already formulated in standard form</p> Signup and view all the answers

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

    <p>There are excess resources or capacity available for each constraint</p> Signup and view all the answers

    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.

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