Slide 1

  • Linear Programming Problems - Corner Point Method

Slide 2

  • Linear programming is a mathematical technique used to determine the best possible outcome in a given mathematical model for a business or given constraint.
  • It involves optimizing an objective function subject to a set of constraints.
  • The corner point method is one approach to solving linear programming problems.

Slide 3

  • The corner points are the vertices (or corners) where the feasible region of the linear programming problem intersects.
  • These intersection points are checked to find the optimal solution.

Slide 4

  • Steps for solving linear programming problems using the corner point method:
    1. Identify the feasible region by graphing the system of inequalities.
    2. Determine the corner points by finding the intersection points of the boundary lines.
    3. Evaluate the objective function at each corner point.
    4. Compare the objective function values to find the optimal solution.

Slide 5

  • Example:
    • Maximize the objective function: Z = 2x + 3y
    • Subject to the constraints:
      • 2x + y ≤ 10
      • x + 3y ≤ 18
      • x, y ≥ 0

Slide 6

  • Step 1: Graphing the system of inequalities:

Slide 7

Graph

Slide 8

  • Step 2: Determine the corner points:
    • Corner point 1: (0, 0)
    • Corner point 2: (0, 10)
    • Corner point 3: (6, 0)
    • Corner point 4: (5, 2)
    • Corner point 5: (3, 3)

Slide 9

  • Step 3: Evaluate the objective function at each corner point:
    • Z(0, 0) = 2(0) + 3(0) = 0
    • Z(0, 10) = 2(0) + 3(10) = 30
    • Z(6, 0) = 2(6) + 3(0) = 12
    • Z(5, 2) = 2(5) + 3(2) = 19
    • Z(3, 3) = 2(3) + 3(3) = 15

Slide 10

  • Step 4: Compare the objective function values:
    • The maximum value of Z is 30 at the corner point (0, 10).
    • Therefore, the optimal solution is x = 0 and y = 10.

Slide 11

  • Linear Programming Problems - Corner Point Method

Slide 12

  • Linear programming is a mathematical technique used to determine the best possible outcome in a given mathematical model for a business or given constraint.
  • It involves optimizing an objective function subject to a set of constraints.
  • The corner point method is one approach to solving linear programming problems.

Slide 13

  • The corner points are the vertices (or corners) where the feasible region of the linear programming problem intersects.
  • These intersection points are checked to find the optimal solution.

Slide 14

  • Steps for solving linear programming problems using the corner point method:
    • Identify the feasible region by graphing the system of inequalities.
    • Determine the corner points by finding the intersection points of the boundary lines.
    • Evaluate the objective function at each corner point.
    • Compare the objective function values to find the optimal solution.

Slide 15

  • Example:
    • Maximize the objective function: Z = 2x + 3y
    • Subject to the constraints:
      • 2x + y ≤ 10
      • x + 3y ≤ 18
      • x, y ≥ 0

Slide 16

  • Step 1: Graphing the system of inequalities:

Slide 17

Graph

Slide 18

  • Step 2: Determine the corner points:
    • Corner point 1: (0, 0)
    • Corner point 2: (0, 10)
    • Corner point 3: (6, 0)
    • Corner point 4: (5, 2)
    • Corner point 5: (3, 3)

Slide 19

  • Step 3: Evaluate the objective function at each corner point:
    • Z(0, 0) = 2(0) + 3(0) = 0
    • Z(0, 10) = 2(0) + 3(10) = 30
    • Z(6, 0) = 2(6) + 3(0) = 12
    • Z(5, 2) = 2(5) + 3(2) = 19
    • Z(3, 3) = 2(3) + 3(3) = 15

Slide 20

  • Step 4: Compare the objective function values:
    • The maximum value of Z is 30 at the corner point (0, 10).
    • Therefore, the optimal solution is x = 0 and y = 10.

Slide 21

  • Quadratic Equations
    • A quadratic equation is a second-degree polynomial equation in a single variable with the form: ax^2 + bx + c = 0
    • The solutions to a quadratic equation can be found using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
  • Example: Solve the quadratic equation 2x^2 + 3x - 5 = 0

Slide 22

  • Complex Numbers
    • Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit (√(-1)).
    • The real part of a complex number is denoted by Re(z), and the imaginary part is denoted by Im(z).
  • Examples: Perform operations with complex numbers: (5 + 3i) + (2 - 4i), (3 + i)(2 - 3i)

Slide 23

  • Probability
    • Probability is the branch of mathematics that deals with the likelihood of an event occurring.
    • The probability of an event is a number between 0 and 1, where 0 represents impossibility and 1 represents certainty.
    • Probability can be calculated using various methods such as classical probability, relative frequency, and probability distributions.
  • Examples: Calculate the probability of rolling a 6 on a fair die, flipping a coin and getting heads.

Slide 24

  • Matrices and Determinants
    • A matrix is a rectangular array of elements arranged in rows and columns. It is often used to represent systems of linear equations or transformations.
    • The determinant of a square matrix can be used to find its inverse and solve systems of linear equations.
  • Examples: Find the determinant of a 2x2 matrix, solve a system of linear equations using matrices.

Slide 25

  • Vector Algebra
    • Vectors are quantities that have both magnitude and direction.
    • Vector addition, subtraction, and scalar multiplication can be performed to manipulate vectors.
    • Dot product and cross product are operations that can be performed on vectors to find their relationship and properties.
  • Examples: Find the dot product and cross product of two vectors.

Slide 26

  • Three-Dimensional Geometry
    • Three-dimensional geometry deals with the study of shapes and figures in three dimensions.
    • Topics include distance and midpoint formulas, equations of lines and planes in three dimensions, and finding intersections and angles between lines and planes.
  • Examples: Find the distance between two points in 3D space, find the equation of a line given a point and a direction vector.

Slide 27

  • Differential Calculus
    • Differential calculus involves the study of rates of change and slopes of curves.
    • Topics include limits, derivatives, and applications of derivatives such as finding maximum and minimum values and rates of change.
  • Examples: Find the derivative of a function, find the maximum and minimum values of a function.

Slide 28

  • Integral Calculus
    • Integral calculus involves the study of areas under curves and accumulation of quantities.
    • Topics include definite and indefinite integrals, techniques of integration, and applications of integrals such as finding areas and volumes.
  • Examples: Evaluate definite and indefinite integrals, find the area bounded by curves.

Slide 29

  • Differential Equations
    • Differential equations are equations that involve derivatives of an unknown function.
    • They are used to model and describe various natural phenomena in physics, engineering, and other fields.
    • Topics include first-order and second-order differential equations, separable and linear equations, and applications of differential equations.
  • Examples: Solve a first-order differential equation, solve a second-order homogeneous differential equation.

Slide 30

  • Probability Distributions
    • Probability distributions describe the likelihood of different outcomes in a random or uncertain process.
    • Common probability distributions include the binomial distribution, normal distribution, and exponential distribution.
    • Topics include probability mass functions, probability density functions, expected values, and standard deviations.
  • Examples: Calculate probabilities using the binomial distribution, find the mean and variance of a normal distribution.