An automobile association is organizing a series of car races that will last for four days. The organizers know that rj ≥ 0 special tires in good condition will be required on each of the four successive days, j = 1, 2, 3, 4. They can meet these needs either by buying new tires at P dollars apiece or by reshaping used tires (reshaping is a technique by which the grooves on the tire are deepened, using a special profile-shaped tool). Two kinds of service are available for reshaping: normal service, which takes one full day at N dollars a tire, and quick service, which takes overnight at Q dollars a tire. How should the association, which starts out with no special tires, meet the daily requirements at minimal cost?
a) Formulate a mathematical model for the above problem. Does it exhibit the characteristics of a network problem? Why? (Hint. Take into account the fact that, at the end of day j, some used tires may not be sent to reshaping.)
b) If the answer to (a) is no, how can the formulation be manipulated to become a network problem? Draw the associated network. (Hint. Add a redundant constraint introducing a fictitious node.)
c) Assume that a tire may be reshaped only once. How does the above model change? Will it still be network problem?