Download Week 3 - Homework Assignment: The Simplex Method Complete the

Week 3 - Homework Assignment: The Simplex Method
Complete the following problems using your textbook:
4.1 Exercise Page 217----------2, 6 and 12
4.2 Exercise Page 234----------2
4.3 Exercise Page 242----------24
4.1 Exercise Page 217----------2, 6 and 12
In Exercises 1 –1 0, determine whether the given simplex tableau is in final form. If so, find
the solution to the associated regular linear programming problem. If not, find the pivot
element to be used in the next iteration of the simplex method.
2.
x
y
u
v
P
Constant
1
1
1
0
0
6
1
0
-1
1
0
2
3
0
5
0
1
30
Solution:
This is final form since there are no negative numbers present in the bottom row.
Solution will be x=0, y =6, u =0, v = 2, P = 30
6.
x
y
z
u
v
w
P
0
1
0
0
1
0
0
0
Constant
2
0
-1
0
3
0
0
6
1
0
30
0
1
63
Solution:
This is not the optimal solution because there are negative numbers present in the bottom row.
The most negative value is the -1 in the first column of the bottom row, and the smallest ratio is
30/2 = 15.
Pivot element for the next iteration is 2 which is in the first column of the third row.
In Exercises 11–25, solve each linear programming problem by the simplex method.
12.
Maximize
P = 5x + 3y
Subject to
x+y
80
3x
x
90
0, y
0
Solution:
Initial simplex tableau will be
Tableau #1
x
y
s1
s2
p
1
1
1
0
0
80
3
0
0
1
0
90
-5
-3
0
0
1
0
Pivot about the 3 in the first column of the second row:
R2-> (1/3)R2
Then perform the operations R1 -> R1-R2
And R3->R3+5R2
Tableau #2
x
y
s1
s2
p
0
1
1
-1/3 0
50
1
0
0
1/3 0
30
0
-3
0
5/3 1
150
Pivot about the 1 in the second column of the first row:
Perform the operation R3-> 3R1 + R3
Tableau #3
x
y
s1
s2
p
0
1
1
-1/3 0
50
1
0
0
1/3 0
30
0
0
3
2/3 1
300
There are no remaining negative numbers in the bottom row so this is in final form.
The optimal solution is P = 300, with x = 30, y = 50
Answer: P = 300; x = 30, y = 50
4.2 Exercise Page 234----------2
In Exercises 1 – 6, use the technique developed in this section to solve the minimization problem.
2.
Minimize
C = -2x – 3y
Subject to
3x + 4y
24
7x – 4y
16
x
0
0, y
Solution:
Let P = -C
Maximize P = 2x + 3y
Subject to
3x + 4y
24
7x – 4y
16
x
0
0, y
Initial simplex tableau will be
Tableau #1
x
y
s1
s2
p
3
4
1
0
0
24
7
-4
0
1
0
16
-2
-3
0
0
1
0
Pivot about the 4 in the second column of the first row:
Tableau #2
x
y
s2
P
3/4 1
1/4 0
0
10
1
0
1/4 0
s1
1
3/4 0
0
6
40
1
18
Optimal Solution: P = 18; x = 0, y = 6
Since C = -P = -18
The required solution will be C = -18, x = 0, y = 6
Answer: C = -18, x = 0, y = 6
4.3 Exercise Page 242----------24
24. FINANCE—INVESTMENTS Natsano has at most $50,000 to invest in the common stocks of two
companies. He estimates that an investment in company A will yield a return of 10%, whereas an
investment in company B, which he feels is a riskier investment, will yield a return of 20%. If he decides
that his investment in the stock of company A is to exceed his investment in the stock of company B by
at least $20,000, determine how much he should invest in the stock of each company in order to
maximize the return on his investment.
Solution:
Maximize R = 0.1x + 0.2y
Subject to
x+y
50000
x – y ≥ 20000
x
0, y
0
Initial simplex tableau will be
Tableau #1
x
y
s1
s2
R
1
1
1
0
0
50000
1
-1
0
-1
0
20000
-1/10 -1/5 0
0
1
0
Pivot about the 1 in the first column of the second row:
Tableau #2
x
y
s1
s2
R
0
2
1
1
0
30000
1
-1
0
-1
0
20000
0
-3/10 0
-1/10 1
2000
Pivot about the 2 in the second column of the first row:
Tableau #3
x
y
s1
s2
R
0
1
1/2 1/2 0
15000
1
0
1/2 -1/2 0
35000
0
0
3/20 1/20 1
6500
Optimal Solution: R = 6500; x = 35000, y = 15000
He should invest $35000 in company A and $15000 in company B to get the maximum return $6500.
Pivot Column
In the simplex method, how is a pivot column selected? A pivot row? A pivot element? Give examples
of each.
Solution:
Pivot column is the column with the most negative entry in the last row.
Example of pivot column:
z-column is the pivot column.
Pivot row is the row with the least non-negative ratio. Negative values or undefined values are ignored.
Now divide last column by the entries in pivot column, we will get
Thus first row is the pivot row.
Pivot element is the entry where the pivot column and pivot row intersect. Thus circled element is the
pivot element.