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8/3/2019 J02 LinAlg and LPs Rev
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MIT and James Orlin 2003
1
A brief review of Linear Algebra and
Linear Programming Models
Developed by James Orlin
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MIT and James Orlin 2003
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Review of Linear Algebra
Some elementary facts about vectors andmatrices.
The Gauss-Jordan method for solvingsystems of equations.
Bases and basic solutions and pivoting.
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Elementary Facts about Vectors
? A1 2 3 4v v v v v! is called a row vector.
1
2
3
4
! -
t
v
v
v
v
v
The transpose ofv is a column vector.
? A1 2 3 4w w w w w! is another row vector.
The inner productof vectors w and v is
given by:1 1 2 2 3 3 4 4
v w v w v w v w v w! $
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MIT and James Orlin 2003
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Matrix Multiplication
( )!ij
A a ( )!ij
B b ( )! ! vij
C c A B
1!! k kjn
ikji a bc
Suppose that A has n columns and B has n rows.
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Multiplying Matrices
Let C = (cij) = AB. Then cij is the innerproductof row iofA and column jofB.
21 22 23
11 12 13
31 32 33
a a a
A
a a a
a a a
! -
13
23
11 12
21 22
31 3332
b
b
b
b b
B b b
b b
! -
For example, what is c23?
23 21 13 22 23 23 33c a b a b a b!
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MIT and James Orlin 2003
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Multiplying Matrices
Let C = (cij) = A v B. Then each column ofC is obtained by adding multiples of
columns of A.
1 2 3 100 123
4 5 6 10 456
7 8 9 1 789
v !
- - -
1 2 3
100 4 10 5 6
7 8 9
!
- - -
Similarly, eachrow of C is
obtained byadding multiplesof rows of B.
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Elementary Facts about Solving
Equations
1 2 3
1 2 4 0
2 1 1 6 x x x
! - - - -
Find a linearcombination ofthe columns of
A that equals b.
Solve forAx= b, where
2v1
1 2 4
2 1 1A
! -
1
2
3
x
x
x
x
-
!0
6b
!
-
2v3 3v1
1 2 3
1 2 3
2 4 0
2 6
x x x
x x x
!
!
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MIT and James Orlin 2003
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Solving a
System of Equations
To solve a system of equations, use Gauss-Jordan
elimination.
x1 x2 x3 x4
1 2 4 1 = 0
2 1 -1 -1 = 6-1 1 2 2 = -3
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To solve the system of equations:
1
2
-1
2
1
1
4
-1
2
1
-1
2
=
=
=
0
6
-3
x1
x2 x3 x4
Go to the animation
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MIT and James Orlin 2003
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b1
b2
b3
=
=
=
a12
a22
a32
a14
a24
a34
a11
a21
a31
a13
a23
a33
The fundamental operation: pivoting
Pivot on a23
x1 x2 x3 x4
=
=
=
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MIT and James Orlin 2003
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b1
b2
b3
=
=
=
a12
a22
a32
a14
a24
a34
a11
a21
a31
a13
a23
a33
Pivot on a23
What will be the next coefficient of b1?
a32? of aij for i {2?
x1 x2 x3 x4
=
=
=
a22/a23 a24/a23a21/a23 1 b2/a23
Da11
Da11
=a11a13(a21/a23)
0
0
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MIT and James Orlin 2003
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-1
1
2
1
0
00
1
0
-1
1
0
=
=
=
1
0
0
Jordan Canonical Form
for an m x n matrix
The remaining variable(s) are called non-basic.
0
1
-1
x2x1 x3 x4
There are m columns that have been transformedinto unit vectors, one for each row. The variablesin these columns are called basic.
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MIT and James Orlin 2003
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-1
1
2
1
0
00
1
0
-1
1
0
=
=
=
1
0
0
Jordan Canonical Form
for an m x n matrix
In the basic solution, the non-basic variables areset to 0.
0
1
-1
x2x1 x3 x4
The basic solution is: x1 = 2, x2 = 1, x3 = -1, x4 = 0
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-1
1
0
4
-2
3
=
=
=
There is an easily determined solution
for every choice of non-basic variables.
If we set x4 = 2, what solution do we get?
0 -1
1 1
-1 2
x2x1 x3 x4
If we set x4 =
(, what solution do we get?
1
0
00
1
0
1
0
0
We get a feasible solution to the system of equations forevery choice of values for non-basic variables.
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MIT and James Orlin 20031
5
1
1
0
-1
1
0
4
-2
3
=
=
=
0
0
-2
3
-3
Another Jordan Canonical Form for
the same system of equations
What are the basicvariables?
1 0 -1
0 0 3
0 1 1
x2x1 x3 x4
1
What is the basic solution?
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Applications
A Financial Model
an investment model over multiple time
periods goal: optimize the total revenues at the end of
the time horizon
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A Financial Problem
Sarah has $1.1 million to invest in five differentprojects for her firm.
Goal: maximize the amount of money that isavailable at the beginning of 2006.
(Returns on investments are on the next slide).
At most $500,000 in any investment, except forCDs.
Can invest in CDs, at 5% per year.
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Return on investments
(undiscounted dollars)
A B C D E
Jan.2003
-1 - -1 -1 -
Jan.2004
.4 -1 1.2 - -
Jan.2005 .8 .4 - - -1
Jan.2006
- .8 - 1.5 1.2
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FormulateS
arahs problem as an LP
Payback for A: for every dollar invested inJanuary of 2003, Sarah receives $.40 in Januaryof 2004 and $.80 in January of 2005.
FORMULATION.
STEP 1. Choose the decision variables
Let xA denote the amount in millions of dollars investedin A.
Define xB, xC, xD, and xE similarly. Let x3 denote the amount put in a CD in 2003. (Define x4
and x5 similarly)
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Formulating the model
With your partner formulate the LP model.
Step 2. Formulate the objective function
put the objective function in words first. E.G.
we are minimizing cost or maximizingutility
Step 3. Formulate the constraints
Put the constraints in words first
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The model
Max z= .8 xB
+ 1.5 xD
+ 1.2 xE
+ 1.05 x5
Subject to xA + xC + xD + x3 = 1.1 (Year 2003)
.4 xA - xB + 1.2 xC + 1.05 x3 - x4 = 0 (Year 2004)
.8 xA + .4 xB - xE + 1.05 x4 - x5 = 0 (Year 2005)
0 e xj e .5 for j = A, B, C, D, E, 3, 4, 5
Excel Solution
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FAQ. Do the units matter? How does
one choose the units?
The units do not matter so long as one iscareful to use units correctly. It would be
possible to have xA be in $ millions and forxB to be in dollars.
But some choices of units are morenatural than others, and easier to use and
to communicate.
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MIT and James Orlin 200323
Addressing Problems in Practice
Understanding the model:
know the purpose of the model
vary numbers and assess the impact (sensitivityanalysis)
discuss ways of enhancing the model
Financial Problems in practice are much larger,involving lots of potential investments, and more
time periods. Data is often stored in a database.
Other (to be addressed soon)
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Understanding the model
How might management use this model inpractice?
Lets vary some of the data and see if we
can obtain insights: How will the return vary with the initial
investment?
How will the return vary with the CD rate ofreturn?
Use SolverTable or usual sensitivity analysis
Excel Solution
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Scaling up: dealing with large problems
n investments, over m time periods. Total budget B to invest in period 1.
The payback of $1 of investment j in period i is pij.If investment j starts in period i, then pij = -1,indicating that $1 is invested in that period.
Everything is reinvested.
Maximum of uj in investment j.
Maximize the total return in period m.
Work with your partner on formulating thegeneralization.
Let xj be the amount put into investment j
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The model
1
n
mj jjz p x
!! Maximize
11
n
j jjp x B
!! subject to
10
n
ij jjp x
!! for i = 2 to m-1
0j j
x ue e for j = 1 to n
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Enrichments of the model
Finance concentrators: have we madeassumptions that you would like to challenge?
Can we deal with a more realistic model?
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Summary
Gauss-Jordan solving of equations andother background in linear algebra
A financial problem
Note: Modeling in practice is an art form.It requires finding the right simplificationsof reality for a given situation.