This dissertation investigates two research topics. The first one is the construction of explicit convex and concave envelopes of nonlinear functions through polyhedral subdivisions. The second topic is the design, analysis and management of Fee-For-Service contracts in pharmaceutical distribution supply chain.

For the first research topic, it is well known that, in order to solve a general nonconvex optimization problem to global optimality, finding strong convex relaxations is important. A significant amount of research has been devoted to developing concave overestimators and convex underestimators of non-linear functions f(x) over the hypercube. One motivation for this research is that, when solving an optimization problem with a convex feasible region that aims to maximize an objective function f(x), replacing f(x) by a concave overestimator yields a convex relaxation of the problem. Such a relaxation can then be used in branch-and-bound algorithms for global optimization. In this thesis, we derive explicit convex and concave envelopes of several nonlinear functions over subsets of hyper-rectangles. These envelopes are derived by identifying polyhedral subdivisions of the hyperrectangle over which the envelopes can be developed easily. In particular, we use a classical result of Lovász to construct the concave envelopes of concave-extendable supermodular functions. We then use orthogonal disjunctions theory to develop the convex envelopes of disjunctive convex functions. Finally, we utilize a result on convex extensions to convexify symmetric functions of binary variables. We show that many convex/concave envelope formulae in the literature are special cases of our constructions. We also describe the convex/concave envelopes of many new functions that have potential uses in factorable nonlinear programming.

For the second research topic, we study how to design, analyze and manage Fee-For-Service contracts in pharmaceutical distribution supply chain. Fee-For-Service (FFS) contracts, first introduced in 2004, dramatically changed the way the pharmaceutical distribution supply chains are designed, managed, and operated. Investment buying (IB), where buying occurs in anticipation of drug price increases, used to be the way distributors made most of their profits. FFS contracts limit the amount of inventory distributors can carry at any time (by imposing an inventory cap) and require inventory information sharing from the distributors to the manufacturers while compensating the distributors with a per-unit fee. In spite of its widespread popularity, FFS has never been rigorously analyzed and its effectiveness has not been carefully tabulated. In this thesis, we study the multi-period stochastic inventory problems faced by manufacturers and distributors under FFS and IB models. In particular, we derive optimal policies and develop procedures to compute the optimal policy parameters. We show that FFS contracts can improve the total supply chain profit. Moreover, the manufacturer and distributor are now able to share a larger profit. Thus, there exists a range of per-unit fees that leads to pareto-improvement. Simulation results show that (1) such improvement is of the order of 1.7% on average and as much as 5.5% and (2) the improvement increases as the inventory cap decreases. Determining the pareto-improving per-unit fees is a source of contention in FFS contract negotiation. We propose a simple, yet effective, heuristic for computing this fee. We believe that these results have the potential to improve the efficiency of pharmaceutical distribution supply chains, thus reducing the healthcare costs that are such a big burden on the U.S. economy.