Consider the following transportation problem:

The unit costs are given by the following matrix:

(a)    Solve the problem for the optimal flows (using any manual or computerized means that you have at your disposal. That is, you can solve it by hand, use MENU-OKF, or any other linear pro› gramming package available to you.) No matter how you solve the problem, present your solution in the form of a table as shown in Figure 2.9. Be sure to include not only the optimal flows, but also the dual variables and the dual slack values.

(b)   

(b) Suppose the supply at node ´ is increased to 191 and the supply at node D is decreased to 159. What is the change in the objective function? In general, if the supply at node i is increased by a small amount and the supply at node j is decreased by an equal amount, what is the change in the objective function in terms of the supplies, the costs, the dual variables, and the magnitude of the change in supplies? (Assume that the change is small enough that the basic variables in the primal solution thos e that are greater than zero remai n basic after the change.) Clearly define any notation that you use.