J02 LinAlg and LPs Rev

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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2

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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3

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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4

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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5

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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6

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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8

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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9

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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10

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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11

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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12

-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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13

-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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14

-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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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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6

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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7

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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8

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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9

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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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.

J02 LinAlg and LPs Rev

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