## Transcribed Text

1. Consider the transportation problem where costs are to be minimized
from locations 1, 2 and 3 to destinations 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 6 3 M 7 8
2 5 2 5 8 9
3 3 6 7 M 5
Demand: 4 6 6 6
(a) 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.
6 3 M 7
5 2 5 8
3 6 7 M
6
3
M
7
5
2
5
8
3
6
7
M
6
3
M
7
5
2
5
8
3
6
7
M
6
3
M
7
5
2
5
8
3
6
7
M
6
3
M
7
5
2
5
8
3
6
7
M
6 3 M 7
5 2 5 8
3 6 7 M
6 3 M 7
5 2 5 8
3 6 7 M
(b) What would be the initial basic feasible solution for this problem
using Vogel’s Approximation? Show your work.
6 3 M 7
5 2 5 8
3 6 7 M
(c) What would be the initial basic feasible solution for this problem using
Russell’s Approximation? Show your work.
6 3 M 7
5 2 5 8
3 6 7 M
(d)What would be the initial basic feasible solution for this problem using
the minimum cost technique? Show your work.
6 3 M 7
5 2 5 8
3 6 7 M
2. Solve one iteration of the following problem using the Interior Point
Algorithm, with an initial starting solution of (12, 2) and α = 0.8.
Max Z = x1 + 4x2
s.t. x1 + x2 = 14
x1, x2 ≥ 0
Initial solution: x1 = 12, x2 = 2, Z = . .
Solution after one iteration: x1 = . ., x2 = . ., Z = . .
3. 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 $15,000 $20,000 $25,000 $30,000
Marginal revenue $25 $30 $45 $40
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.
• No more than three of the products can be produced
• Product 1 is produced only if products 3 and 4 are produced
• Only one of the following constraints holds, not both:
3x1 + 2x2 + x3 + 4x4 ≤ 2000
2x1 + 3x2 + x3 + 3x4 ≤ 2500
4. Consider the following network problem:
6 6
7
5
8 4 5
S 7 9 E
5 3 4 6
6
2 4
a. If the arc values represent distances, find the shortest path from S to E.
Display the final solution and compute the shortest
path.
6 6
7
5
8 4 5
S 7 9 E
5 3 4 6
6
2 4
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.
6 6
7
5
8 4 5
S 7 9 E
5 3 4 6
6
2 4
c. If the arc values represent maximum capacities, find the maximum flow
from S to E. Display the final solution and compute the maximum flow
through the network.
6 6
7
5
8 4 5
S 7 9 E
5 3 4 6
6
2 4

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