Linear Programming

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Linear Programming
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Optimization -

• Finding the minimum or maximum value of some
  quantity.
• Linear programming is a form of optimization
  where you optimize an objective function with a
  system of linear inequalities called constraints.
• The overlapped shaded region is called the
  feasible region.

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Solving a linear programming problem

• Graph the constraints.
• Locate the ordered pairs of the vertices of the
  feasible region.
• If the feasible region is bounded (or closed), it
  will have a minimum a maximum.
• If the region is unbounded (or open), it will
  have only one (a minimum OR a maximum).
• 4. Plug the vertices into the linear equation
  (C) to find the min. and/or max.

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A note about Unbounded Feasible Regions

• If the region is unbounded, but has a top on it,
  there will be a maximum only.

• If the region is unbounded, but has a bottom,
  there will be a minimum only.

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Find the min. max. values of C-x3y subject to
the following constraints.

• Vertices of feasible region
• (2,8)
• C -23(8) 22
• (2,0)
• C -23(0) -2
• (5,0)
• C -53(0) -5
• (5,2)
• C -53(2) 1

• x ? 2
• x ? 5
• y ? 0
• y ? -2x12

Max. of 22 at (2,8) Min. of -5 at (5,0)
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Ex Cx5y Find the max. min. subject to the
following constraints

• x?0
• y?2x2
• 5?xy

• Vertices?
• (0,2)
• C05(2)10
• (1,4)
• C15(4)21
• Maximum only!
• Max of 21 at (1,4)

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Linear Programming

• Businesses use linear programming to find
  out how to maximize profit or minimize costs.
  Most have constraints on what they can use or
  buy.

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Find the minimum and maximumvalue of the
function f(x, y) 3x - 2y.

• We are given the constraints
• y 2
• 1 x 5
• y x 3

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1 x 5
8
7
6
5
4

• y 2

3

• y x 3

2
1
3
4
5
2
1
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Linear Programming

• The vertices of the quadrilateral formed are
• (1, 2) (1, 4) (5, 2) (5, 8)
• Plug these points into the function f(x, y) 3x
  - 2y

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Linear Programming

• f(x, y) 3x - 2y
• f(1, 2) 3(1) - 2(2) 3 - 4 -1
• f(1, 4) 3(1) - 2(4) 3 - 8 -5
• f(5, 2) 3(5) - 2(2) 15 - 4 11
• f(5, 8) 3(5) - 2(8) 15 - 16 -1

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Linear Programming

• f(1, 4) -5 minimum
• f(5, 2) 11 maximum

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Find the minimum and maximum value of the
function f(x, y) 4x 3y

• We are given the constraints
• y -x 2
• y x 2
• y 2x -5

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• y 2x -5

6
5
4
3

• y -x 2

2
1
3
4
1
2
5
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Vertices

• f(x, y) 4x 3y
• f(0, 2) 4(0) 3(2) 6
• f(4, 3) 4(4) 3(3) 25
• f( , - ) 4( ) 3(- ) -1

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Linear Programming

• f(0, 2) 6 minimum
• f(4, 3) 25 maximum

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Assignment