Chapter 5 and 6

Use Desmos to graph and find points

Linear Programming Problem

Optimal Value
(Maximum or Minimum)

Methods of Solving

Geometic Method

P=80x1+60x2
2x1+x2=/<30
x1+2x2=/<20

2x1+x2 =30
x1+2x2 =20

x l y 80x1+60x2
0 l 0 0 Min:1200
0 l 20 1200 at (0,20) and (15,0)
10 l 10 1400 Max:1400
15 l 0 1200 at (10,10)

Standard Form

Check for optimal solutions.
If S is bounded, z will have both Max and Min
(Both will occur at corner points)

Evaluate z using corner points
by making a table

Using values from table, determine max and min

Standard Maximization Problem

Optimal Value
(Maximum or Minimum)

Methods of Solving

Simplex Method

Standard Form

Introduce slack variables
and find Initial System.

Create initial simplex tableau

Preform Pivot Operation until there are
no remaining negatives in bottom row of tableau

No negatives on bottom of
tableau means optimal solution
has been found

No positives above dotted line
in pivot column means there is
no optimal solution

P=80x1+60x2
2x1+x2=/<30
x1+2x2=/<20

2x1+x2+s1 =30
x1+2x2 +s2 =20
-80x1-60x2 +p=0

2 1 1 0 0 30 l 30
1 2 0 1 0 20 l 20
-80-60 0 0 1 0 l 0

1 0 2/3 -1/3 0 l 131/3
0 1 -1/3 2/3 0 l 31/3
0 0 331/3 131/3 1 1266.7

P =1266.7
x1= 131/3
x2= 31/3

-2 - 1 1 0 0 30 l 30
-1 -2 0 1 0 20 l 20
-80-60 0 0 1 0 l 0

No Optimal
Solution

P=80x1+60x2
2x1+x2=/<30
x1+2x2=/<20