What are the worst cases in constrained Last-In-First-Out pick-up and delivery problems?

Pick-up and delivery operations in transportation tasks are usually performed using stacks. The Last-In-First-Out nature of stacks imposes restrictions to the order in which pick-ups and deliveries are performed. In this paper, we study the problem of determining the minimum number of stacks needed to perform given sequences of pick-up and delivery operations. We consider both, limited and unlimited capacity stacks and allow some transformations on the delivery route. These problems amount to computing the smallest bounded chromatic number in a family of permutation graphs. We study the worst cases of these problems.