102 The simplex algorithm.

1
10/2 The simplex algorithm. In an augmented
matrix, if a column has a 1 and all
other entries 0, it is said to be in solution.
The 1 is called a pivot and the associated
variable is a basic variable The pivot
operation is a combination of row operations
that brings a column into solution. You can
pivot on any nonzero entry. Example Perform a
pivot on the first row first column entry x
y s t rhs 2 5 1 0 10 1 2
0 1 20 The pivot can be used to go
from one solution to a system to another
solution.
2
Suppose we want to find the solution to the
system with augmented matrix which maximizes P
2x y and keeps x,y,s and t non-negative. x
y s t rhs 2 5 1 0 10 1
2 0 1 20 Right now the basic variables
are s and t. The nonbasic are x and y and the
solution is (0,0,10,20) with P0.
3
Using the simplex algorithm to solve linear
optimization problems. First example a problem
that can be worked graphically. (See lecture 9
for the graphical solution.) Maximize P 5x
8y 6 Subject to x,ygt 0 (1) x 4y lt4
and (2) y 4x lt 4. Introduce slack variables
s1 gt 0 and s2 gt 0 to turn inequalities (1) and
(2) in equations (1) and (2) (1) x 4y s1
4 and (2) y 4x s2 4 Now the problem is
Find x,y,s1,s2gt0 satisfying Equations (1) and
(2) for which P is as large As possible.
4

• Tableau for the problem
• P6 at (x,y,s1,s2)(0,0,4,4)
• x y s1 s2 P rhs
• 4 1 0 0 4
• 4 1 0 1 0 4
• -5 8 0 0 1 6

5
Here is a problem that is harder to work
graphically. Autoparts Inc produces 3 types of
parts (L, M, N). Each type requires (2,5,3)
units work on machine I And (2,2,5) units of work
on machine II. The machines have Respectively
(190, 150) units of work available. The
profit From each part is (8,5,11) dollars
respectively. Find a production schedule (x,
y, z) that maximizes profits. Set up Profit
function Machine I constraint Machine II
constraint Assume that the production of L
must be greater than or Equal to the total
production of M and N. The constraint is
6
Question How do you spot an unbounded problem
when you dont have a picture?
7
Setting up linear programming problems A
farmer has 150 acres of land suitable for crops A
and B. The cost of growing A is 40/acre. The
cost for B is 60/acre The farmer has 7600
captial available. Each acre of A takes 20 hrs
of labor and each acre of B takes 25 hrs of
labor. The farmer has 3300 hrs of labor
available. He expects to Make 150/acre for A
and 200/acre for B. How many acres of each crop
should he plant to maximize his profit?