ma3264 mathematical modelling lecture 8
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MA3264 Mathematical Modelling Lecture 8. Chapter 7 Discrete Optimization Modelling. Example page 238. - PowerPoint PPT PresentationTRANSCRIPT
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MA3264 Mathematical ModellingLecture 8
Chapter 7
Discrete Optimization Modelling
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Example page 238
How many* tables and how many* bookcases should a carpenter make each week to maximize profit? He realizes a profit of $25 per table and $30 per bookcase. He has 600 feet of lumber per week and 40 hours of labor per week. Each table requires 20 feet of lumber and 5 hours of labor. Each bookcase requires 30 feet of lumber and 4 hours of labor. He has signed contracts to deliver 4 tables and 2 bookcases every week.
How many* : need not be integers since part of a table or bookcase can be made in a week
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Mathematical Formulation
Maximize
decisionvariables
21 3025 xx objectivefunction Subject to
6003020 21 xx4045 21 xx41 x22 x
constraints
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Mathematical Formulation
Maximize
decisionvariables
21 3025 xx objectivefunction Subject to
6003020 21 xx4045 21 xx41 x22 x
constraints
Question what real world entity does each decision variable represent ?
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Mathematical Formulation
Maximize
decisionvariables
21 3025 xx objectivefunction Subject to
6003020 21 xx4045 21 xx41 x22 x
constraints
Question what real world entity does the objective function represent ?
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Mathematical Formulation
Maximize
decisionvariables
21 3025 xx objectivefunction Subject to
6003020 21 xx4045 21 xx41 x22 x
constraints
Question what real world entity does each constraint represent ?
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Mathematical Formulation
Maximize
decisionvariables
21 3025 xx objectivefunction Subject to
6003020 21 xx4045 21 xx41 x22 x
constraints
Question what real world entities have been abstracted out of this formulation ?
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Solution
Maximize
decisionvariables
21 3025 xx objectivefunction Subject to
6003020 21 xx4045 21 xx41 x22 x
constraints
Question can we set the derivatives = 0 to solve this maximization problem ?
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Constraints
6003020 21 xx4045 21 xx
41 x22 x
constraints
Question what are the regions consisting of all 1x
2x
),( 21 xx that satisfypoints
the 1st , 2nd ,3rd, 4th constraint ?
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Constraint Line
1x
2x
)20,0()0,30(
Question What equation describes the doted line ?
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Constraint Line
6003020 21 xx
1x
2x
)20,0()0,30(
Answer The equation above describes the dotted line.
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Constraint Region
6003020 21 xx
1x
2x
)20,0()0,30(
Question What region is described by the inequality above ?
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Constraint Region
6003020 21 xx
1x
2x
)20,0()0,30(
THIS RED REGION
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Feasible Region
6003020 21 xx4045 21 xx
41 x22 x
constraints
Question what region consisting of all 1x
2x
),( 21 xx that satisfypoints
all four constraints ?
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Feasible Region
41 x
6003020 21 xx
1x
2x
)20,0(
)0,30(
)10,0(
)4,0(
)8,0(22 x
4045 21 xx is the red region
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Feasible Region
1x)2,4( )2,4.6(
)5,4(
2x
is both closed and bounded
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Objective Function
1x)2,4( )2,4.6(
)5,4(
2x
is continuous on the feasible region
21 3025 xx
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Objective Function
1x)2,4( )2,4.6(
)5,4(
2x
must have a maximum at some point p in the feasible region
21 3025 xx
not necessarily
unique
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Objective Function
1x)2,4( )2,4.6(
)5,4(
2x
A similar argument applied to edges (to be shown using visualizer) shows that f has a maximum at a vertex of the feasible region the simplex method
2121 3025),(f xxxx
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Suggested Reading
Introduction and Section 7.1 overview of discrete optimization modelling p. 236- 249
Linear Programming 1: Geometric Solutions p. 250-259
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Tutorial 8 Due Week 20–24 Oct
Problem 1. Page 245 Problem 1
Problem 2. Page 245 Problem 2
Problem 3. Page 258 Problem 3
Problem 4. Page 259 Problem 4, a
Problem 5. Page 259 Problem 4, b
Problem 6. Page 259 Problem 4, c