Chapter 4
• Standard form
– All variables are nonnegative
– Pivot until all numbers in profit are positive
• Find pivot element and know how to pivot
• Standard form maximum
– Every constraint ≤ to a positive constraint
– Solution is read across BV column across to RHS column
• Standard form for minimum
– All other constraints > to a constant.
– Objective function has non negative coefficients.
– Solution is read down Slack variable rows down to profit
row
For #1-2 Determine if table is in final form, if it is
state solution if not state pivot element
1. maximum problem
BV P
x1
x2
s1
s2
RHS
1
0
1

5
1

20 

2
0 1 1 0 1 30 
2
4
S

P 1 1 2 1
1 120 

S1
2. minimum problem
BV P
S1
x2
2
1. Not final matrix – pivot 1 in s2/x2
P
x1
x2
s1
s2
RHS
1
0
1
0
1

2

2
0 1 1 0 1 10 
4
 2

1 2 0 4 5 20 
2. final matrix – x1 = 4 x2 = 5 C = 20
3. Is the following matrix in
standard form, if not what can be
modified to put in standard form
maximize : P  2 x1  4 x2  x3
subject to the constraints:
2 x1  3 x2  x3  8
3 x1  x2  2 x3  12
2 x1  x2  x3  10
x1  0
x2  0
x3  0
3. Not standard form, multiply equation 1
and 2 by negative 1
4. Write the duality
problem for the minimum
problem
Minimize C =6x1 + 3x2
Subject to:
x1 +x2 > 2
2x1 + 6x2 > 6
x1 > 0
x2 > 0
4. y1 + 2y2 < 6
y1 + 6y2 < 3
P = 2y1 + 6y2
Finance formulas
A  P  Pr t


i
P V 
n 
 1  (1  i ) 
R  L  Lrt
An  P (1  i )
n
(1  i )  1
AP
i
n
 1  (1  i )  n 
V  P

i


Chapter 5 w-up
1. If you put $50 in an account monthly earning 8%
interest compounded monthly. How much will she have in 4
years?
2. How long will it take an investment to quadruple if it is
invested at 3% compounded monthly?
3. Find the proceeds for a discounted loan of $2500 at 22.8%
interest rate for 8 months
4. Determine the monthly payment for a $7000 loan at
7.9%compounded monthly for 5 years.
1. $ 2817.50
2. 46.27 years
3. $2120
4. $141.60