Reconsider Prob. 8.1-3. Suppose that the sales forecasts have been revised downward to 240, 400, and 320 units per day of products 1, 2, and 3, respectively, and that each plant now has the capacity to produce all that is required of any one product. Therefore, management has decided that each new product should be assigned to only one plant and that no plant should be assigned more than one product (so that three plants are each to be assigned one product, and two plants are to be assigned none). The objective is to make these assignments so as to minimize the total cost of producing these amounts of the three products.

(a) Formulate this problem as an assignment problem by constructing the appropriate cost table.

(b) Obtain an optimal solution.

(c) Reformulate this assignment problem as an equivalent transportation problem by constructing the appropriate parameter table.

(d) Starting with Vogel's approximation method, interactively apply the transportation simplex method to solve the problem as formulated in part (c).

Prob. 8.1-3

Four cargo ships will be used for shipping goods from one port to four other ports (labeled 1, 2, 3, 4). Any ship can be used for making any one of these four trips. However, because of differences in the ships and cargoes, the total cost of loading, transporting, and unloading the goods for the different ship-port combinations varies considerably, as shown in the following table:

The objective is to assign the four ships to four different ports in such a way as to minimize the total cost for all four shipments.

(a) Describe how this problem fits into the general format for the assignment problem.

(b) Obtain an optimal solution.

(c) Reformulate this problem as an equivalent transportation problem by constructing the appropriate parameter table.

(d) Use the northwest corner rule to obtain an initial BF solution for the problem as formulated in part (c).

(e) Starting with the initial BF solution from part (d), interactively apply the transportation simplex method to obtain an optimal set of assignments for the original problem.

(f) Are there other optimal solutions in addition to the one obtained in part (e)? If so, use the transportation simplex method to identify them.