Math 407
Linear Optimization
Graphical solutions of two dimensional LPs
Solution Procedure
Step 1: Graph each of the linear constraints indication on which side of the constraint the
feasible region must lie. Dont forget the implicit constraints!
Step 2:
494
CHAPTER 9
LINEAR PROGRAMMING
9.3 THE SIMPLEX METHOD: MAXIMIZATION
For linear programming problems involving two variables, the graphical solution method introduced in Section 9.2 is convenient. However, for problems involving more than two variables o
Math 407A: Linear Optimization
Professor James Burke
Math Dept, University of Washington
Linear Algebra Review
Professor James Burke (Math Dept, University of Washington) Linear Optimization
Math 407A:
Linear Algebra Review
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Math 308 Review
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Line
Math 407: Linear Optimization
Lecture 14: Sensitivity Analysis
Concrete Products Corp
Math Dept, University of Washington
ecture 14: Sensitivity Analysis Concrete Products Corp (Math Linear University of Washington)
Math 407: Dept, Optimization
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FINAL EXAM OUTLINE FOR MATH 407
EXAM DATE: Thursday, August 18, 2011: 9:40am-11:40am
The nal exam for this course is set to be given on Thursday, August 18 at 9:40am-11:40am in
the same classroom that the course always meets in.
EXAM OUTLINE
The nal exam
Math 407A: Linear Optimization
Lecture 8: Initialization and the Two Phase Simplex Algorithm
Math Dept, University of Washington
Initialization
We have shown that if we are given a feasible dictionary (tableau)
for an LP, then the simplex algorithm will t
MATH 407 FINAL EXAM SAMPLE QUESTIONS
1. A farmer has three farms. He can grow three dierent crops on them. Data for the coming season
are given in the table below. To maintain a uniform work load among the three farms, the farmer
adopts the policy that th
SAMPLE QUESTIONS FOR FINAL
1. Model the following two problems as LPs.
(a) A company needs to lease warehouse storage space over the next 5 months. Just how
much space will be required in each of these months is known. However, since these space
requireme
FINAL EXAM OUTLINE FOR MATH 407
EXAM DATE: Monday, December 8, 2014: 8:30am-10:20am
The nal exam for this course is set to be given on Monday, December 8, 2014: 8:30am-10:20am
in the same classroom that the course always meets in.
EXAM OUTLINE
The nal exa
Math 407
Linear Optimization
Simplex Algorithm for Problems in Standard Form and having Feasible Origin
Solve the following LPs using the simplex algorithm in simplex tableau form. At each stage
of the simplex algorithm identify the BFS identied by the cu
Math 407
Linear Optimization
Transformation of LPs to Standard Form
Transform the following LPs to LPs in standard form.
1.
x1 12x2
minimize
subject to
5x1
2x1 +
x2 0
2.
maximize
minimize
2x3
x2
2x3 =
10
x2 20x3 30
, 1
x3 4
3x 12y +
4z
10z =
10
y 17z 1
MATH 407 Key Theorems
Theorem 0.1 (Weak Duality Theorem). If x Rn is feasible for P and y Rm is feasible for D, then
cT x y T Ax bT y.
Thus, if P is unbounded, then D is necessarily infeasible, and if D is unbounded, then P is necessarily
infeasible. More
Math 407A: Linear Optimization
Lecture 10: General Duality Theory
Math Dept, University of Washington
ecture 10: General Duality Theory (Math Dept, University of Washington)
Math 407A: Linear Optimization
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1
General Duality Theory
2
General Weak Dua
THE FAMILY FARM
Fred and Martha Schmertz work a 40 acre family farm outside of Athens
Georgia. They grow tubers. Specically, they grow potatoes, yams, beets,
and turnips. Planting season is coming around and they must decide how
many acres of each type of
486
CHAPTER 9
LINEAR PROGRAMMING
9.2 LINEAR PROGRAMMING INVOLVING TWO VARIABLES
Many applications in business and economics involve a process called optimization, in
which we are required to find the minimum cost, the maximum profit, or the minimum use
of
SECTION 9.4
32. The accounting firm in Exercise 31 raises its charge for an
audit to $2500. What number of audits and tax returns will
bring in a maximum revenue?
35. (Maximize)
Objective function:
z 2.5x1 x2
Constraints:
3x1 5x2 15
5x1 2x2 10
x1, x2 10
I
SECTION 9.5
9.5
THE SIMPLEX METHOD: MIXED CONSTRAINTS
521
THE SIMPLEX METHOD: MIXED CONSTRAINTS
In Sections 9.3 and 9.4, we looked at linear programming problems that occurred in standard form. The constraints for the maximization problems all involved in
Math 407A: Linear Optimization
Lecture 3: LP Modeling
Math Dept, University of Washington
LP Modeling
Model 10: Detergent Production
Model 9: Investing Over Time
Model 4: Blending
LP Modeling
The four basic steps of LP modeling.
LP Modeling
The four basic
Math 407A: Linear Optimization
Lecture 11: The Dual Simplex Algorithm
Math Dept, University of Washington
The Dual Simplex Algorithm
P
maximize
subject to
4x1 2x2 x3
x1 x2 + 2x3 3
4x1 2x2 + x3 4
x1 + x2 4x3 2
0 x1 , x2 , x3
P
maximize
subject to
4x1 2x2 x
Math 407A: Linear Optimization
Lecture 5: Simplex Algorithm I
Math Dept, University of Washington
1
Dictionaries, Augmented Matrices, the Simplex Tableau
Dictionaries
The Simplex Tableau
2
Basic Feasible Solutions (BFS)
3
The Grand Strategy: Pivoting
4
Th
Linear Programming
Lecture 7: Does the Simplex Algorithm Work?
Math Dept, University of Washington
Does the Simples Algorithm Work?
Choosing Entering and Leaving Variables
Unbounded LPs
Degeneracy
Overcoming Degeneracy
Cycling
The Basis-Dictionary Corresp
Linear Programming
Lecture 13: Sensitivity Analysis
Sensitivity Analysis
Silicon Chip Corporation
Break-even Prices and Reduced Costs
Range Analysis for Objective Coecients
Resource Variations, Marginal Values, and Range Analysis
Right Hand Side Perturbat
Math 407A: Linear Optimization
Lecture 12: The Geometry of Linear Programming
Math Dept, University of Washington
The Geometry of Linear Programming
Hyperplanes
Denition: A hyperplane in Rn is any set of the form
H (a, ) = cfw_x : aT x =
where a Rn \ cfw
Linear Programming
Lecture 6: The Simplex Algorithm
Language, Notation, and Linear Algebra
Math Dept, University of Washington
Dictionaries for LPs in Standard Form
The Simplex Algorithm via Matrix Multiplication
The Block Structure of the Simplex Algorit
Math 407A: Linear Optimization
Lecture 4: LP Standard Form
Math Dept, University of Washington
1
LPs in Standard Form
2
Minimization maximization
3
Linear equations to linear inequalities
4
Lower and upper bounded variables
5
Interval variable bounds
6
Fr
Math 407A: Linear Optimization
Lecture 9
The Fundamental Theorem of Linear Programming
The Strong Duality Theorem
Complementary Slackness
Math Dept, University of Washington
The Two Phase Simples Algorithm
The Fundamental Theorem of linear Programming
Dua
9.1 Systems of Linear
Inequalities
9.2 Linear Programming
Involving Two Variables
9.3 The Simplex Method:
Maximization
9.4 The Simplex Method:
Minimization
9.5 The Simplex Method:
Mixed Constraints
9
LINEAR
PROGRAMMING
J
John
von
Neumann
19031957
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