Linear Programming - Standard Form

0,...,0,0

0,...,0,0

...

..

..

...

...

...

21

21

2211

22222121

11212111

332211

m

n

mnmnmm

nn

nn

nn

bbb

xxx

bxaxaxa

bxaxaxa

bxaxaxa

xcxcxcxcZMaximize (Minimize):

Subject to:

Linear Programming - Standard Form

0,...,0,0

0,...,0,0

...

..

..

...

...

...

21

21

2211

22222121

11212111

332211

m

n

mnmnmm

nn

nn

nn

bbb

xxx

bxaxaxa

bxaxaxa

bxaxaxa

xcxcxcxcZMaximize (Minimize):

Subject to:

ObjectiveFunction

Constraint Set

Non-negativeVariablesConstraint

Non-negativeRight-hand sideConstants

Linear Programming - Standard Form

0

0

b

x

bAx

cxZMaximize (Minimize):

Subject to:

Linear Programming - Standard Form

0

0

b

x

bAx

cxZMaximize (Minimize):

Subject to:

mnmm

n

n

aaa

aaa

aaa

...

..

...

...

21

22221

11211

A

mb

b

b

.2

1

b

nx

x

x

.2

1

x

]...[ 21 nccccwhere,

Linear Programming – Conversion to Standard Form

Inequality constraints: slack or surplus variables

10

10

8

8

21

1

21

1

xx

x

xx

xs.t.

=>

s.t.

=>

x2 – slack variable

x2 – surplus variable

Linear Programming – Conversion to Standard Form

Unrestricted variables: replace with 2 non-negative variables

0,,.s.t

2min

edunrestrictis,0.s.t

2min

432

243

431

12

21

xxx

xxxZ

xxx

xx

xxZ

set,

=>

Linear Programming – Conversion to Standard Form

Example:

edunrestrict

02,0

523

2

7..

32min

3

1

321

321

321

321

x

xx

xxx

xxx

xxxts

xxxZ

Linear Programming – Conversion to Standard Form

0,,,,,

5223

2

7..

332min

765421

5421

75421

65421

5421

xxxxxx

xxxx

xxxxx

xxxxxts

xxxxZ

Example:

Solving Systems of Linear Equations

22)2

73)1

5321

4321

xxxx

xxxx

Use the Gauss-Jordan elimination procedure to solve this series of linear equations.

Multiply row 1 by 2 and add to row 2.

Solving Systems of Linear Equations

16273)2

73)1

5431

4321

xxxx

xxxx

Divide row 2 by 3.

3

16

3

1

3

2

3

7)2

73)1

5431

4321

xxxx

xxxx

Multiply row 2 by –1 and add to row 1.

Solving Systems of Linear Equations

3

16

3

1

3

2

3

7)2

3

5

3

1

3

1

3

2)1

5431

5432

xxxx

xxxx

Solution: 0,0,0,3

5,

3

1654321 xxxxx

x1 and x2 are basic variables; x3 , x4 and x5 are non-basic variables

Solving Systems of Linear Equations

Solution:

is referred to as a basic solution since all non-basic variableshave been set to 0.

This solution is also referred to as a basic feasible solution since all basic variables are non-negative.

Every corner point of the feasible region corresponds to a basicFeasible solution – fundamental building block for the simplex method.

0,0,0,3

5,

3

1654321 xxxxx