lecture 7.2 bt
TRANSCRIPT
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Today’s Agenda
Attendance / Announcements
Questions from Yesterday
Sections 7.2
Quiz Today
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Exam Schedule
Exam 4 (Ch 6,7)
Fri 11/15
Exam 5 (Ch 10)
Thur 12/5
Final Exam (All)
Thur 12/12
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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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Linear Programming
The Objective Function is
what we need to maximize or
minimize. For us, this will be a
function of 2 variables, f(x, y)
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Linear Programming
The Constraints are the
inequalities that provide us with
the Feasible Region.
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Linear Programming (pg 400)
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The general idea… (pg 398)Find max/min values of the objective
function, subject to the constraints.
yxyxf 52),(
0,0
1
842
623
yx
yx
yx
yx
Objective Function Constraints
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The general idea… (pg 398)
Graph the Feasible Region
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The general idea… (pg 398)
The Feasible Region makes up the possible inputs to the Objective Function
yxyxf 52),(
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Corner Point Thm (pg 400)
If a feasible region is bounded, then the objective function has both a maximum and minimum value, with each occurring at one or more corner points.
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Find the minimum and maximum
value 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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6
4
2
2 3 4
3
1
1
5
5
7
8
y ≤ x + 3
y ≥ 2
1 ≤ x ≤5
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6
4
2
2 3 4
3
1
1
5
5
7
8
y ≤ x + 3
y ≥ 2
1 ≤ x ≤5 Need to find corner
points (vertices)
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• The vertices (corners) of the
feasible region are:
(1, 2) (1, 4) (5, 2) (5, 8)
• Plug these points into the
function f(x, y) = 3x - 2y
Note: plug in BOTH x, and y values.
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Evaluate the function at each vertex
to find min/max values
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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So, the optimized solution is:
• 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
With the constraints:
52
24
1
2
xy
xy
xy
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6
4
2
53 4
5
1
1
2
3y ≥ -x + 2
y ≥ 2x -5
y ≤ 1/4x + 2
Need to find corner
points (vertices)
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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 = 7
3
1
3
1
3
7
3
28
3
25
3
Evaluate the function at each vertex
to find min/max values
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• f(0, 2) = 6 minimum
• f(4, 3) = 25 maximum
So, the optimized solution is:
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Classwork / Homework
• Page 403
•1, 3, 7, 9, 11