Graph the System of Linear Inequalities

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Graph the System of Linear Inequalities

• x gt 5
• y lt 6
• y ? ½ x - 4

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3.4 Linear Programming

• Goal Solve Linear Programming Problems.

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Optimization

• Finding the MAXIMUM or MINIMUM of some Quantity.
• LINEAR PROGRAMMING is one type of Optimization
  Process.

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Linear Programming

• Process of Optimizing a LINEAR Objective
  Functions subject to a SYSTEM of linear
  inequalities called CONSTRAINTS.

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Constraints

• RESTRICTIONS on the values of one or more
  variables in a problem
• INEQUALITIES
• The graph of the System of Constraints is called
  the FEASIBLE REGION

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Objective Function

• Function you want to find the best combination of
  values for.
• May have a max., min., neither, or both.
• Example Highest profit
• or
• Lowest cost

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Vertex Principle

• The OPTIMAL SOLUTION will occur at a VERTEX of
  the feasible region.

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X ? 0 Y ? 0 X Y ? 8
C 3X 4Y
(0,8) (8,0) (0,0)
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Finding Maximum and Minimum

• Evaluate Objective Function at each Vertex
• (8,0)
• C 3(8) 4(0) 24
• (0,8)
• C 3(0) 4(8) 32
• (0,0)
• C 3(0) 4(0) 0

Maximum
Minimum
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A Region that is Unbounded

• Find the maximum and minimum value of
• C 5x 6y
• Subject to the following Constraints
• x ? 0
• y ? 0
• x y ? 5
• 3x 4y ? 16

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Use Linear Programming to find the Maximum Profit

• Two Manufacturing Plants make the same kind of
  bike. The table give the hours of general labor,
  machine time, and technician labor required to
  make one bike at each plant. For the two plants
  combined, the manufacturer can afford to use up
  to 4,000 hours of general labor, up to 1,500
  hours of machine time, and up to 2,300 hours of
  technical labor per week. Plant A earns a profit
  of 60 per bicycle and Plant B earns a profit of
  50 per bicycle. How many bicycles per week
  should the manufacturer make in each plant to
  maximize profit?

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Graph
Constraints Test Vertices Maximum
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Homework

• Pg. 166-167 9-17 all