1
15.053
February 21, 2002
z
Simplex Method Continued
Handouts:
Lecture Notes
Note:
This lecture is intended to be viewed as a slide
show
2
Today’s Lecture
z
Review of the simplex algorithm.
z
Formalizing the approach
z
Degeneracy and Alternative Optimal Solutions
z
Is the simplex algorithm finite?
(Answer, yes,
but only if we are careful)
3
3
3
4
2
0
0
3
2
1
LP Canonical Form
=
LP Standard Form + Jordan Canonical Form
=
=
2
6
x
1
x
2
z
=
0
The basic feasible solution is
x
1
= 0, x
2
= 0, x
3
= 6, x
4
= 2
The basic variables are x
3
and x
4
.
The nonbasic variables are x
1
and x
2
.
z is not a
decision
variable
1
0
0
1
0
0
x
4
x
3
4
Basic Variables
x
1
x
2
x
m+1
x
m
1
0
1
0
z
0
0
x
r
x
n
0
a
1,m+1
a
2,m+1
0
0
0
a
1,n
a
2,n
b
1
b
2
=
=
x
s
a
1,s
a
2,s
0
0
0
a
r,m+1
0
1
a
r,n
b
r
=
a
r,s
0
0
a
m,m+1
1
0
a
m,n
b
m
=
a
m,s
0
0
1
c
m+1
0
0
c
n

z
0
=
c
s
0
CV
m constraints, n variables
Nonbasic Variables
5
Notation
z
n
number of variables
z
m
number of constraints
z
s
index of entering variable
z
r
index of pivot row
–
note: the r
th
basic variable leaves the basis
z
The original data is c
j
, a
ij
, b
i
z
After performing pivots we represent the revised
coefficients
as
c
j
,
a
ij
,
b
i
6
The basic feasible solution
z
The current values are all nonnegative.
–
This is needed for canonical form
z
There is a basic variable associated with each
constraint.
–
in this case, the basic variable associated with
constraint i is x
i
.
z
There are nm nonbasic variables.
–
In this case, the nonbasic variables are
x
m+1
, … x
n
.
z
The bfs is as follows:
x
1
=
b
1
,
… , x
m
=
b
m
.
All other variables are 0.