A.Consider the transportation problem where costs are to be minimized from locations (rows) 1, 2 and 3 to destinations (columns) A, B, C, and D according to the following cost and requirements table, where M is a very large number: Destination A B C D Supply Location 1 9 8 6 7 25 2 M 9 15 8 15 3 6 12 10 M 10 Demand: 5 20 15 10 Solve for the optimal solution using the transportation simplex method and the northwest corner rule to obtain an initial basic feasible solution. Be sure to identify the shipping assignments and compute total shipping costs for your solution. (15 points) B.Consider the following network problem: O T 5 8 3 4 8 5 8 7 6 4 3 6 7 5 4 3 6 5 4 8 3 5 6 5 9 7 a. Display the final solution and compute the shortest path. (5 points) O T 5 8 3 4 8 5 8 7 6 4 3 6 7 5 4 3 6 5 4 8 3 5 6 5 9 7 b. If the arc values represent distances, find the minimal spanning tree that would connect all the nodes. (Disregard the arc direction.) Display the final solution and compute the distance of the minimal spanning tree. (5 points) O T 5 8 3 4 8 5 8 7 6 4 3 6 7 5 4 3 6 5 4 8 3 5 6 5 9 7 c. If the arc values represent maximum capacities, find the maximum flow from the beginning arrow to the ending arrow. Display the final solution and compute the maximum flow through the network. (5 points) O T 5 8 3 4 8 5 8 7 6 4 3 6 7 5 4 3 6 5 4 8 3 5 6 5 9 7 C. Consider the following nonlinear programming problem: max Z = 54(x1) - 9(x1)2 + 78(x2) - 13(x2)2 s. t. x1 <= 4 2x2 <= 12 3x1 + 2x2 <= 18 x1, x2 >= 0 a. List the KKT conditions. 10 pts. b. Derive the optimal solution. 5 pts. D.Consider the assignment problem having the following cost and requirements table. Solve for the optimal solution, using the Northwest Corner Rule as the initial starting solution. (15 points) Assign A B C D 1 4 6 8 7 2 7 M 2 6 3 8 5 6 7 4 5 2 3 4 E.Four new products are being considered for production. The start-up costs and marginal revenues are shown below. Product 1 2 3 4 Start-up costs $30,000 $35,000 $40,000 $50,000 Marginal revenue $20 $25 $35 $30 Let x1, x2, x3, and x4 denote the continuous variables associated with the production levels of the four products. Formulate this problem as a mathematical programming problem, including the following additional constraints. 15 pts · No more than three of the prod.

1. A.Consider the transportation problem where costs are to be
minimized from locations (rows) 1, 2 and 3 to destinations
(columns) A, B, C, and D according to the following cost and
requirements table, where M is a very large number:
Destination
A
B C D
Supply
Location 1
9
8 6 7
25
2
M
9 15 8
15
3
6
12 10 M
10
Demand:
5
20 15 10
Solve for the optimal solution using the transportation simplex
method and the northwest corner rule to obtain an initial basic

2. feasible solution. Be sure to identify the shipping assignments
and compute total shipping costs for your solution. (15 points)

9. B.Consider the following network problem:
O
T
5
8
3
4

10. 8
5
8
7
6
4
3
6
7
5
4
3
6
5
4
8
3
5

11. 6
5
9
7
a. Display the final solution and compute the shortest path.
(5 points)
O
T
5
8
3
4
8
5
8
7

12. 6
4
3
6
7
5
4
3
6
5
4
8
3
5
6
5
9
7

13. b. If the arc values represent distances, find the minimal
spanning tree that would connect all the nodes. (Disregard the
arc direction.) Display the final solution and compute the
distance of the minimal spanning tree.
(5 points)
O
T
5
8
3
4
8
5
8
7
6
4
3
6

14. 7
5
4
3
6
5
4
8
3
5
6
5
9
7
c. If the arc values represent maximum capacities, find the
maximum flow from the beginning arrow to the ending arrow.
Display the final solution and compute the maximum flow
through the network.

15. (5 points)
O
T
5
8
3
4
8
5
8
7
6
4
3
6
7
5
4

16. 3
6
5
4
8
3
5
6
5
9
7
C. Consider the following nonlinear programming problem:
max Z = 54(x1) - 9(x1)2 + 78(x2) - 13(x2)2
s. t. x1 <= 4 2x2 <=
12 3x1 + 2x2 <= 18
x1, x2 >= 0
a. List the KKT conditions. 10 pts.

17. b. Derive the optimal solution. 5 pts.
D.Consider the assignment problem having the following cost
and requirements table. Solve for the optimal solution, using
the Northwest
Corner Rule as the initial starting solution. (15 points)
Assign
A B C D
1 4 6 8 7

18. 2 7 M 2 6
3 8 5 6 7
4 5 2 3 4
E.Four new products are being considered for production. The
start-up costs and marginal revenues are shown below.
Product 1 2 3 4 Start-up
costs $30,000 $35,000 $40,000 $50,000 Marginal
revenue $20 $25 $35 $30
Let x1, x2, x3, and x4 denote the continuous variables
associated with the production levels of the four products.
Formulate this problem as a mathematical programming
problem, including the following additional constraints.
15 pts
· No more than three of the products can be produced
· Product 1 is produced only if products 2 and 4 are produced •
Only two of the following constraints hold, not all three:
3x1 + 2x2 + x3 + 4x4 <= 3000
2x1 + 4x2 + x3 + 5x4 <= 3500 4x1 + 3x2 + x3
+ 3x4 <= 2500
2
10