FBA (1)

Author: Tõnis Aaviksaar

TALLINN 2006

CCFFT

2

Contents

• Intro• Systems of linear equations• Solution by row operations• Steady state mass balance• Linear Programming

3

Metabolic Networks• Metabolic networks consist of

reactions between metabolites

• Flux Balance Analysis (FBA) calculates flux patterns from a system of linear equations– Flux value = rate of reaction– Flux pattern is a collection of

flux valuesO2KG

OOXA

OAce-CoAOPYR

OPEP

CO2

CO2

PEPAce-CoA

Mal

Fum

Suc-CoAIcit

CO2

3PG O3PG

GAP DHAP

FBP

E4P

S7P

X5P

R5P

Ru5P

HOO4C 3CH22CH2

1COOH

1C

2CH3

O OH

1C

2CH3

O OH

2C

3CH3

O

1CO OH

HOO4C 3CH22C

O

1COOH

2CH23CH

HO 6COOH

HOO1C 4C

H H

5COOH

HOO5C 4CH23CH2

2C

O

1COOH

mdh

fumA

D

sucD

Eicd

acn

gltA

pckAacs

QACE I ppd

gpmeno

BPG

B

pgk

gapA

tpiA

fbaA1

F6P

fbp1pfk21

G6P

pgi

ppcoadA2

oadA1

GF

zwf

H1

rpe

rpiB

H2

SBP

fbp2

fbaA2

tal

rbcL

CO2

CO2

CO2

pfk22

CO2

OR5P

OE4P

ODHAP

OG6P

OF6P

ORu5P

OS7P

OGAP

4

v3

v4

v2

v1

A

B

C

....

........

v5

Flux Patterns

v3

v4

v2

v1

A

B

C

....

........

v5

v1

v2

v3

v4

v5

v =v3

v4

v2

v1

A

B

C

....

........

v5

5

Metabolic Networks• Metabolic networks consist of

reactions between metabolites– Thousands of metabolites– More reactions than metabolites

• Flux Balance Analysis (FBA) calculates flux patterns from a system of linear equations– Flux value = rate of reaction– Flux pattern is a collection of

flux valuesO2KG

OOXA

OAce-CoAOPYR

OPEP

CO2

CO2

PEPAce-CoA

Mal

Fum

Suc-CoAIcit

CO2

3PG O3PG

GAP DHAP

FBP

E4P

S7P

X5P

R5P

Ru5P

HOO4C 3CH22CH2

1COOH

1C

2CH3

O OH

1C

2CH3

O OH

2C

3CH3

O

1CO OH

HOO4C 3CH22C

O

1COOH

2CH23CH

HO 6COOH

HOO1C 4C

H H

5COOH

HOO5C 4CH23CH2

2C

O

1COOH

mdh

fumA

D

sucD

Eicd

acn

gltA

pckAacs

QACE I ppd

gpmeno

BPG

B

pgk

gapA

tpiA

fbaA1

F6P

fbp1pfk21

G6P

pgi

ppcoadA2

oadA1

GF

zwf

H1

rpe

rpiB

H2

SBP

fbp2

fbaA2

tal

rbcL

CO2

CO2

CO2

pfk22

CO2

OR5P

OE4P

ODHAP

OG6P

OF6P

ORu5P

OS7P

OGAP

6

FBA

• Steady-state mass balance equations• Weighted sums (linear combinations) of

– Reaction stoichiometries– Flux patterns

7

Contents

• Intro• Systems of linear equations• Solution by row operations• Steady state mass balance• Linear Programming

8

Linear equation

2x1 + 3x2 + 4x3 = 11

9

Linear equation

Linear equationa1x1 + a2x2 + a3x3 = b

in matrix form

x1

x2

x3

a1 a2 a3 b× =

2x1 + 3x2 + 4x3 = 11

10

Linear equation

Linear equationa1x1 + a2x2 + a3x3 = b

in matrix form

x1

x2

x3

a1 a2 a3 b× = a1x1 + a2x2 + a3x3 =

2x1 + 3x2 + 4x3 = 11

11

System of linear equations

System of linear equationsa11x1 + a12x2 + a13x3 = b1

a21x1 + a22x2 + a23x3 = b2

a31x1 + a32x2 + a33x3 = b3

in matrix form

x1

x2

x3

× =

a11 a12 a13

a21 a22 a23

a31 a32 a33

a11x1 + a12x2 + a13x3

a21x1 + a22x2 + a23x3

a31x1 + a32x2 + a33x3

=

b1

b2

b3

12

System of linear equations

System of linear equationsa11x1 + a12x2 + a13x3 = b1

a21x1 + a22x2 + a23x3 = b2

a31x1 + a32x2 + a33x3 = b3

in matrix form

x1

x2

x3

× =

a11 a12 a13

a21 a22 a23

a31 a32 a33

a11x1 + a12x2 + a13x3

a21x1 + a22x2 + a23x3

a31x1 + a32x2 + a33x3

=

b1

b2

b3

13

System of linear equations

System of linear equationsa11x1 + a12x2 + a13x3 = b1

a21x1 + a22x2 + a23x3 = b2

a31x1 + a32x2 + a33x3 = b3

in matrix form

x1

x2

x3

× =

a11 a12 a13

a21 a22 a23

a31 a32 a33

a11x1 + a12x2 + a13x3

a21x1 + a22x2 + a23x3

a31x1 + a32x2 + a33x3

=

b1

b2

b3

14

System of linear equations

System of linear equationsa11x1 + a12x2 + a13x3 = b1

a21x1 + a22x2 + a23x3 = b2

a31x1 + a32x2 + a33x3 = b3

in matrix form

x1

x2

x3

× =

a11 a12 a13

a21 a22 a23

a31 a32 a33

a11x1 + a12x2 + a13x3

a21x1 + a22x2 + a23x3

a31x1 + a32x2 + a33x3

=

b1

b2

b3

15

System of linear equations

System of linear equationsa11x1 + a12x2 + a13x3 = b1

a21x1 + a22x2 + a23x3 = b2

a31x1 + a32x2 + a33x3 = b3

in matrix form

x1

x2

x3

× =

b1

b2

b3

a11 a12 a13

a21 a22 a23

a31 a32 a33

16

Linear Combination of Columns

System of linear equationsa11x1 + a12x2 + a13x3 = b1

a21x1 + a22x2 + a23x3 = b2

a31x1 + a32x2 + a33x3 = b3

in matrix form

x1

x2

x3

× =

b1

b2

b3

a11 a12 a13

a21 a22 a23

a31 a32 a33

17

Linear Combination of Columns

System of linear equationsa11x1 + a12x2 + a13x3 = b1

a21x1 + a22x2 + a23x3 = b2

a31x1 + a32x2 + a33x3 = b3

in matrix form

=

a11

a21

a31

a12

a22

a32

a13

a23

a33

x1 + x2 + x3

x1

x2

x3

×

a11 a12 a13

a21 a22 a23

a31 a32 a33

18

Linear Combination of Columns

System of linear equationsa11x1 + a12x2 + a13x3 = b1

a21x1 + a22x2 + a23x3 = b2

a31x1 + a32x2 + a33x3 = b3

in matrix form

=

x1

x2

x3

×

a11 a12 a13

a21 a22 a23

a31 a32 a33

a11x1 + a12x2 + a13x3

a21x1 + a22x2 + a23x3

a31x1 + a32x2 + a33x3

19

Linear Combination of Columns

System of linear equationsa11x1 + a12x2 + a13x3 = b1

a21x1 + a22x2 + a23x3 = b2

a31x1 + a32x2 + a33x3 = b3

in matrix form

x1

x2

x3

× =

b1

b2

b3

a11 a12 a13

a21 a22 a23

a31 a32 a33

20

Contents

• Intro• Systems of linear equations

– Rows correspond to equations– Linear combination of columns

• Solution by row operations• Steady state mass balance• Linear Programming

21

Matrices

• Identity matrixI

• Inverse of a matrixAA-1 = Iif AB = I and BA = I then B = A-1 and A = B-1

• Solution to a system of linear equationsAx = bA-1Ax = A-1b

I =

1

1

1

22

Matrices

• Identity matrixI

• Inverse of a matrixAA-1 = Iif AB = I and BA = I then B = A-1 and A = B-1

• Solution to a system of linear equationsAx = bA-1Ax = A-1bIx = A-1bx = A-1b

I =

1

1

1

x1

x2

x3

×Ix =

1

1

1

=

x1

x2

x3

23

Matrices

• Identity matrixI

• Inverse of a matrixAA-1 = Iif AB = I and BA = I then B = A-1 and A = B-1

• Solution to a system of linear equationsAx = bA-1Ax = A-1bIx = A-1bx = A-1b

I =

1

1

1

x1

x2

x3

×Ix =

1

1

1

=

x1

x2

x3

24

Matrix Row Operations

–7 –6 –12 –33

5 5 7 24

1 4 5

1 6/7 12/7 33/7

5 5 7 24

1 4 5

1 6/7 12/7 33/7

5/7 –11/7 3/7

–6/7 16/7 2/7

· (–1 / 7)

– 5 · R1

– 1 · R1

25

Matrix Row Operations

–7 –6 –12 –33

5 5 7 24

1 4 5

1 6/7 12/7 33/7

5 5 7 24

1 4 5

· (–1 / 7)

– 5 · R1

– 1 · R1

1 −3

1 5

1 2

.….

1 6/7 12/7 33/7

5/7 –11/7 3/7

–6/7 16/7 2/7

· 7 / 5

26

Matrix Row Operations

• Equivalent systems of equations– Two systems of equations are equivalent if they have

same solution sets

• Row operations produce equivalent systems of equations– Changing the order of rows– Multiplication of a row by a constant

2x = 4 is equivalent to 4x = 8

– Addition of a row to another row2x1 + 3x2 = 5-x1 + 2x2 = 1

2x1 + 3x2 = 5x1 + 5x2 = 6

is equivalent to

27

Matrix Row Operations

• Equivalent systems of equations– Two systems of equations are equivalent if they have

same solution sets

• Row operations produce equivalent systems of equations– Changing the order of rows– Multiplication of a row by a constant

2x = 4 is equivalent to 4x = 8

– Addition of a row to another row2x1 + 3x2 = 5-x1 + 2x2 = 1

2x1 + 3x2 = 5x1 + 5x2 = 6

is equivalent to

28

Gaussian Elimination

• Given a system of linear equationsAx = b

• Matrix A is augmented by b[A | b]

• Which is then simplified by row operations to produce[I | c]

• Which corresponds to system of equationsIx = c

• Which is equivalent to the original systemAx = b

29

• System of equations−7x1 − 6x2 − 12x3 = −33

5x1 + 5x2 + 7x3 = 24

x1 + 4x3 = 5

• Row-Reduced [A | b] ~ [I | c] =

• Simplified equivalent system of equations, only one solutionx1 = −3

x2 = 5

x3 = 2

Row-Reduced [A | b] Examples

1 −3

1 5

1 2

−7 −6 −12 −33

5 5 7 24

1 4 5

[A | b] =

30

Row-Reduced [A | b] Examples

• System of equationsx1 − x2 + 2x3 = 1

2x1 + x2 + x3 = 8

x1 + x2 = 5

• Row-Reduced [A | b]

• Simplified equivalent system of equations, infinite number of solutions (solution space)x1 + x3 = 3

x2 − x3 = 2

1 1 3

1 −1 2

31

Row-Reduced [A | b] Examples

• System of equations2x1 + x2 + 7x3 − 7x4 = 2

−3x1 + 4x2 − 5x3 − 6x4 = 3

x1 + x2 + 4x3 − 5x4 = 2

• Row-Reduced [A | b]

• Inconsistent system, no solutions0x1 + 0x2 + 0x3 − 0x4 ≠ 1

1 3 −2

1 1 −3

1

32

Gaussian Elimination

• Carl Friedrich Gauss (1777–1855)

• Chiu-chang suan-shu or The Nine Chapters on the Mathematical Art, written around 250 BCE

33

Contents

• Intro• Systems of linear equations• Solution by row operations

– Equivalent linear systems– Reduced row echelon form

• Steady state mass balance• Linear programming

34

Steady State Approximation

• Steady state conditionFluxes ≠ 0

Concentrations = const

• Steady state mass balanceCompound production = consumption

Production – consumption = 0

v2

v1

A

....

....

35

Mass Balance Equations

v1 – v2 = 0

v2 – v3 – v5 = 0

v3 – v4 = 0

1 -1

1 -1 -1

1 -1

A

B

C

v1 v2 v3 v4 v5 v1

v2

v3

v4

v5

× =

Nv = 0

v3

v4

v2

v1

A

B

C

....

........

v5

36

Stoichiometry Matrix

…

…

…

…

…

H2O

O2

H2

N2

NH3

v1 v2 …

v1

v2

…

×v2

H2N2

1N2 + 3H2 = 2NH3

2H2O = 2H2 + 1O2

O2

NH3

H2Ov1

37

Stoichiometry Matrix

…

…

…

…

…

H2O

O2

H2

N2

NH3

v1 v2 …

v1

v2

…

×v2

H2N2

1N2 + 3H2 = 2NH3

2H2O = 2H2 + 1O2

O2

NH3

H2Ov1

38

Stoichiometry Matrix

-2 …

…

…

…

…

H2O

O2

H2

N2

NH3

v1 v2 …

v1

v2

…

×v2

H2N2

1N2 + 3H2 = 2NH3

2H2O = 2H2 + 1O2

O2

NH3

H2Ov1

39

Stoichiometry Matrix

-2 …

…

2 …

…

…

H2O

O2

H2

N2

NH3

v1 v2 …

v1

v2

…

×v2

H2N2

1N2 + 3H2 = 2NH3

2H2O = 2H2 + 1O2

O2

NH3

H2Ov1

40

Stoichiometry Matrix

-2 …

1 …

2 …

…

…

H2O

O2

H2

N2

NH3

v1 v2 …

v1

v2

…

×v2

H2N2

1N2 + 3H2 = 2NH3

2H2O = 2H2 + 1O2

O2

NH3

H2Ov1

41

Stoichiometry Matrix

-2 …

1 …

2 …

…

…

H2O

O2

H2

N2

NH3

v1 v2 …

v1

v2

…

×v2

H2N2

1N2 + 3H2 = 2NH3

2H2O = 2H2 + 1O2

O2

NH3

H2Ov1

42

Stoichiometry Matrix

-2 …

1 …

2 …

…

…

H2O

O2

H2

N2

NH3

v1 v2 …

v1

v2

…

×v2

H2N2

1N2 + 3H2 = 2NH3

2H2O = 2H2 + 1O2

O2

NH3

H2Ov1

43

Stoichiometry Matrix

-2 …

1 …

2 -3 …

-1 …

2 …

H2O

O2

H2

N2

NH3

v1 v2 …

v1

v2

…

×v2

H2N2

1N2 + 3H2 = 2NH3

2H2O = 2H2 + 1O2

O2

NH3

H2Ov1

44

Stoichiometry Matrix

-2 …

1 …

2 -3 …

-1 …

2 …

H2O

O2

H2

N2

NH3

v1 v2 …

v1

v2

…

×v2

H2N2

1N2 + 3H2 = 2NH3

2H2O = 2H2 + 1O2

O2

NH3

H2Ov1

45

Stoichiometry Matrix

-2

1

2

H2O

O2

H2

N2

NH3

v2

H2N2

1N2 + 3H2 = 2NH3

2H2O = 2H2 + 1O2

O2

NH3

H2Ov1

-3

-1

2

v1 + v2 + ….

v1 v2

46

Stoichiometry Matrix

-2

1

2

H2O

O2

H2

N2

NH3

v2

H2N2

1N2 + 3H2 = 2NH3

2H2O = 2H2 + 1O2

O2

NH3

H2Ov1

-3

-1

2

v1 + v2 + …. =

v1 = 3

v2 = 2

v1 v2

47

Stoichiometry Matrix

-6

3

6

H2O

O2

H2

N2

NH3

v2

H2N2

1N2 + 3H2 = 2NH3

2H2O = 2H2 + 1O2

O2

NH3

H2Ov1

-6

-3

6

v1 + v2 + …. =

v1 = 3

v2 = 2

v1 v2

48

Stoichiometry Matrix

-6

3

6

H2O

O2

H2

N2

NH3

v2

H2N2

1N2 + 3H2 = 2NH3

2H2O = 2H2 + 1O2

O2

NH3

H2Ov1

-6

-3

6

v1 + v2 + …. =

v1 = 3

v2 = 2

v1 v2

49

Calculable Fluxes

v1 – v2 = 0

v2 – v3 – v5 = 0

v3 – v4 = 0

v3

v4

v2

v1

A

B

C

....

........

v5

1 -1

1 -1 -1

1 -1

A

B

C

v1 v2 v3 v4 v5 v1

v2

v3

v4

v5

× =A

B

C

Nv = 0

50

Calculable Fluxes

v1 – v2 = 0

v2 – v3 – v5 = 0

v3 – v4 = 0

v3

v4

v2

v1

A

B

C

....

........

v5

1 -1

1 -1 -1

1 -1

A

B

C

v1 v2 v3 v4 v5 v1

v2

v3

v4

v5

× =A

B

C

Nv = 0

51

Calculable Fluxes

v3

v4

v2

v1

A

B

C

....

........

v5

1 -1

1 -1 -1

1 -1

A

B

C

v1 v2 v3 v4 v5 v1

v2

v3

v4

v5

× =

Nv = 0

A

B

C

-1

1 -1 -1

1

A

B

C

v2 v3 v5

v1

v4

+ =1

-1

v1 v4

v2

v3

v5

× ×

Nclcvclc + Nexpvexp = 0 v1 = 1.0

v4 = .4

A

B

C

52

Calculable Fluxes

v3

v4

v2

v1

A

B

C

....

........

v5

1 -1

1 -1 -1

1 -1

A

B

C

v1 v2 v3 v4 v5 v1

v2

v3

v4

v5

× =

Nv = 0

A

B

C

-1

1 -1 -1

1

A

B

C

v2 v3 v5

1.0

.4+ =

1

-1

v1 v4

v2

v3

v5

× ×

Nclcvclc + Nexpvexp = 0 v1 = 1.0

v4 = .4

A

B

C

53

Calculable Fluxes

v3

v4

v2

v1

A

B

C

....

........

v5

1 -1

1 -1 -1

1 -1

A

B

C

v1 v2 v3 v4 v5 v1

v2

v3

v4

v5

× =

Nv = 0

A

B

C

-1

1 -1 -1

1

A

B

C

v2 v3 v5

+ =

v2

v3

v5

×

Nclcvclc + Nexpvexp = 0

1.0

-.4

bexp

A

B

C

v1 = 1.0

v4 = .4

54

Calculable Fluxes

v3

v4

v2

v1

A

B

C

....

........

v5

1 -1

1 -1 -1

1 -1

A

B

C

v1 v2 v3 v4 v5 v1

v2

v3

v4

v5

× =

Nv = 0

A

B

C

Nclcvclc + Nexpvexp = 0

-1

1 -1 -1

1

A

B

C

v2 v3 v5

=-1.0

.4

A

B

C

bclc

v2

v3

v5

×

55

Row-Reduced [A | b] Examples

• System of equationsx1 − x2 + 2x3 = 1

2x1 + x2 + x3 = 8

x1 + x2 = 5

• Row-Reduced [A | b]

• Simplified equivalent system of equations, infinite number of solutions (solution space)x1 + x3 = 3

x2 − x3 = 2

1 1 3

1 −1 2

56

Dependent and Free Fluxes

-1 1

1 -1 -1

1 -1

A

B

C

v2 v3 v5 v1 v4 v2

v3

v5

v1

v4

× =A

B

C

1 -1

1 -1

1 -1 1

v2 v3 v5 v1 v4

v3

v4

v2

v1

A

B

C

....

........

v5

Nv = 0Nclcvclc + Nexpvexp = 0

57

Dependent and Free Fluxes

v1 v2 v5 v3 v4

1 -1 -1

1 -1 -1

1 -1

1 -1

1 -1 -1

1 -1

+ R3

+ R2

1 -1 -1

1 -1 -1

1 -1

+ R3v3

v4

v2

v1

A

B

C

....

........

v5

58

Dependent and Free Fluxes

v1 v2 v3 v4 v5

-1 1 1

1 -1 -1

1 -1

1 -1 -1

1 -1 -1

1 -1

+ R1

· (-1)

-1 1 1

-1 1

1 -1

v3

v4

v2

v1

A

B

C

....

........

v5

59

Dependent and Free Fluxes

v1 v2 v3 v4 v5

1 -1 -1

-1 1 1

1 -1

1 -1 -1

1 -1 -1

1 -1

+ R2

· (-1)

1 -1

-1 1 1

1 -1

v3

v4

v2

v1

A

B

C

....

........

v5

60

Contents

• Intro• Systems of linear equations• Solution by row operations• Steady state mass balance

– Steady state mass balance– Stoichiometry matrix– Dependent and free fluxes

• Linear Programming

61

Underdetermined Systems

• Be content with infinite solution space• Make more measurements• Assign “experimental” values• Assume that the microorganism “tries” to optimize

an objective– Maximize biomass production– Maximize ATP production– ….

62

The Simplex Method

• Objective function– A linear function

• Constraints– Linear inequalities

• Assumption that all variables are nonnegative– xi ≥ 0

• Solution space is a convex polytope– An optimal solution is a vertex– Move to neighboring vertex with highest objective

function value

63

Simplex Example (1)

v3

z

v2

v1

A

B

C

....

........

v4

+z −v3 =0C → max

64

Simplex Example (1)

v3

z

v2

v1

A

B

C

....

........

v4

+z −v3 =0

+v1 −v2 =0

+v2 −v3 −v4 =0

+v1 ≤5

C

A

B

→ max

65

Simplex Example (1)

v3

z

v2

v1

A

B

C

....

........

v4

+z −v3 =0

+v1 −v2 =0

+v2 −v3 −v4 =0

+v1 +x5 =5

C

A

B

→ max

66

Simplex Example (1)

v3

z

v2

v1

A

B

C

....

........

v4

+z −v3 =0

+v1 −v2 ≤0

+v1 −v2 ≥0

+v2 −v3 −v4 ≤0

+v2 −v3 −v4 ≥0

+v1 +x5 =5

C

A

A

B

B

→ max

67

Simplex Example (1)

v3

z

v2

v1

A

B

C

....

........

v4

+z −v3 =0

+v1 −v2 ≤0

−v1 +v2 ≤0

+v2 −v3 −v4 ≤0

−v2 +v3 +v4 ≤0

+v1 +x5 =5

C

A

A

B

B

→ max

68

Simplex Example (1)

v3

z

v2

v1

A

B

C

....

........

v4

+z −v3 =0

+v1 −v2 +x1 =0

−v1 +v2 +x2 =0

+v2 −v3 −v4 +x3 =0

−v2 +v3 +v4 +x4 =0

+v1 +x5 =5

C

A

A

B

B

→ max

69

Simplex Example (1)

v3

z

v2

v1

A

B

C

....

........

v4

+z −v3 =0

+v1 −v2 +x1 =0

−v1 +v2 +x2 =0

+v2 −v3 −v4 +x3 =0

−v2 +v3 +v4 +x4 =0

+v1 +x5 =5

C

A

A

B

B

→ max

70

Simplex Example (1)

v3

z

v2

v1

A

B

C

....

........

v4

+z −v2 +v4 +x4 =0

+v1 −v2 +x1 =0

−v1 +v2 +x2 =0

+x3 +x4 =0

−v2 +v3 +v4 +x4 =0

+v1 +x5 =5

C

A

A

B

B

→ max

71

Simplex Example (1)

v3

z

v2

v1

A

B

C

....

........

v4

+z −v2 +v4 +x4 =0

+v1 −v2 +x1 =0

−v1 +v2 +x2 =0

+x3 +x4 =0

−v2 +v3 +v4 +x4 =0

+v1 +x5 =5

C

A

A

B

B

→ max

72

Simplex Example (1)

v3

z

v2

v1

A

B

C

....

........

v4

+z −v1 +v4 +x2 +x4 =0

+x1 +x2 =0

−v1 +v2 +x2 =0

+x3 +x4 =0

−v1 +v3 +v4 +x2 +x4 =0

+v1 +x5 =5

C

A

A

B

B

→ max

73

Simplex Example (1)

v3

z

v2

v1

A

B

C

....

........

v4

+z −v1 +v4 +x2 +x4 =0

+x1 +x2 =0

−v1 +v2 +x2 =0

+x3 +x4 =0

−v1 +v3 +v4 +x2 +x4 =0

+v1 +x5 =5

C

A

A

B

B

→ max

74

Simplex Example (1)

v3

z

v2

v1

A

B

C

....

........

v4

+z +v4 +x2 +x4 +x5 =5

+x1 +x2 =0

+v2 +x2 +x5 =5

+x3 +x4 =0

+v3 +v4 +x2 +x4 +x5 =5

+v1 +x5 =5

C

A

A

B

B

→ max

75

Simplex Example (1)

v3

z

v2

v1

A

B

C

....

........

v4

+z +v4 +x2 +x4 +x5 =5

+x1 +x2 =0

+v2 +x2 +x5 =5

+x3 +x4 =0

+v3 +v4 +x2 +x4 +x5 =5

+v1 +x5 =5

C

A

A

B

B

→ max

76

Simplex Example (2)

z

v3

v1

A B

C

....

........

v4

v2

....

2A + 3B = 1C

+z −v3 =0

+v1 −2v3 −v4 +x1 =0

−v1 +2v3 +v4 +x2 =0

+v2 −3v3 +x3 =0

−v2 +3v3 +x4 =0

+v1 +x5 =1

+v2 +x6 =1

77

Simplex Example (2)

z

v3

v1

A B

C

....

........

v4

v2

....

2A + 3B = 1C

+z −v3 =0

+v1 −2v3 −v4 +x1 =0

−v1 +2v3 +v4 +x2 =0

+v2 −3v3 +x3 =0

−v2 +3v3 +x4 =0

+v1 +x5 =1

+v2 +x6 =1

78

Simplex Example (2)

+z −1/3v2 +1/3x4 =0

+v1 −2/3v2 −v4 +x1 +2/3x4 =0

−v1 +2/3v2 +v4 +x2 +2/3x4 =0

+x3 +x4 =0

−1/3v2 +v3 +1/3x4 =0

+v1 +x5 =1

+v2 +x6 =1

79

Simplex Example (2)

+z −1/3v2 +1/3x4 =0

+v1 −2/3v2 −v4 +x1 +2/3x4 =0

−v1 +2/3v2 +v4 +x2 +2/3x4 =0

+x3 +x4 =0

−1/3v2 +v3 +1/3x4 =0

+v1 +x5 =1

+v2 +x6 =1

80

Simplex Example (2)

+z −1/2v1 +1/2v4 +1/2x2 =0

+x1 +x2 =0

−3/2v1 +2/3v2 +3/2v4 +3/2x2 −x4 =0

+x3 +x4 =0

−1/2v1 +v3 +1/2v4 +1/2x2 =0

+v1 +x5 =1

+3/2v1 −3/2v4 −3/2x2 +x4 +x6 =1

81

Simplex Example (2)

+z −1/2v1 +1/2v4 +1/2x2 =0

+x1 +x2 =0

−3/2v1 +2/3v2 +3/2v4 +3/2x2 −x4 =0

+x3 +x4 =0

−1/2v1 +v3 +1/2v4 +1/2x2 =0

+v1 +x5 =1

+3/2v1 −3/2v4 −3/2x2 +x4 +x6 =1

82

Simplex Example (2)

+z +1/3x4 +1/3x6 =1/3

+x1 +x2 =0

+v2 +x6 =1

+x3 +x4 =0

+v3 +1/3x4 +1/3x6 =1/3

+v4 +x2 −2/3x4 +x5 −2/3x6 =1/3

+v1 −v4 −x2 +2/3x4 +2/3x6 =2/3

83

Simplex Example (2)

+z +1/3x4 +1/3x6 =1/3

+x1 +x2 =0

+v2+x6 =1

+x3 +x4 =0

+v3+1/3x4 +1/3x6 =1/3

+v4+x2 −2/3x4 +x5 −2/3x6 =1/3

+v1 −v4−x2 +2/3x4 +2/3x6 =2/3

z

v3

v1

A B

C

....

........

v4

v2

....

2A + 3B = 1C

84

Simplex Example (2)

+z +1/3x4 +1/3x6 =1/3

+x1 +x2 =0

+v2+x6 =1

+x3 +x4 =0

+v3+1/3x4 +1/3x6 =1/3

+v4+x2 −2/3x4 +x5 −2/3x6 =1/3

+v1 −v4−x2 +2/3x4 +2/3x6 =2/3

z

v3

v1

A B

C

....

........

v4

v2

....

2A + 3B = 1C

85

Simplex Example (2)

+z +1/3x4 +1/3x6 =1/3

+x1 +x2 =0

+v2+x6 =1

+x3 +x4 =0

+v3+1/3x4 +1/3x6 =1/3

+v4+x2 −2/3x4 +x5 −2/3x6 =1/3

+v1+x5 =1

z

v3

v1

A B

C

....

........

v4

v2

....

2A + 3B = 1C

86

Simplex Example (2)

+z +1/3x4 +1/3x6 =1/3

+x1 +x2 =0

+v2+x6 =1

+x3 +x4 =0

+v3+1/3x4 +1/3x6 =1/3

+v4+x2 −2/3x4 +x5 −2/3x6 =1/3

+v1+x5 =1

z

v3

v1

A B

C

....

........

v4

v2

....

2A + 3B = 1C

87

Contents

• Intro• Systems of linear equations• Solution by row operations• Steady state mass balance• Linear Programming

– Objective Function– Convex Solution Space– The Simplex Method– Multiple Optimal Solutions