We consider the inventory management problem of procurement and allocation for a non-stationary equipment overhaul problem with stochastic demands for multiple job types requiring different combinations of service parts over a finite horizon. Because the service parts involved tend to be expensive, inventory needs to be managed carefully in order to operate in a cost-effective manner. This problem is complex because the procurement lead times tend to be long. We incorporate into our model holding costs for unused service parts and backordering costs for outstanding jobs. Using linear programs to approximate the value function which appears in the dynamic programming formulation, we derive a non-standard allocation procedure which is neither first-come-firstserved nor myopic. To obtain procurement decisions, we decompose the original problem into multiple single-part-multi-job-type sub-problems using dual variables. These sub-problems may be tackled by another layer of decomposition using the classical result of Clark and Scarf (1960). Using independent order-up-to procurement and myopic allocation as the bench mark, our policy performs better for the canonical "N" system with two service parts and two job types and also for larger problems. In the second part of this dissertation, we focus on a single-part-single-jobtype problem where backorders accumulate increasing per-period backlogging charges. We show that the state space may still be collapsed into the single- dimensional inventory position using a non-traditional dynamic program formulation where the immediate cost function consists of all the expected holding and backordering charges associated with the procured units. We use stoppingtime random variables to capture the periods in which the procured units are matched up with customer requests. Using this alternate cost accounting mechanism, we independently show the optimality of base-stock policies, a result also obtained by Huh et al. (2011) using a more traditional approach. This dynamic program is suitable for computation and it allows us to compute the optimal base-stock levels. Finally, we extend these results to two-echelon inventory distribution systems with aging backorders. Because the first-come-first-served allocation of inventory is not necessarily optimal in a distribution network, we modify our cost-accounting mechanism at the level of the upper echelon and derive a dynamic program which is a lower bound for the exact problem. This lowerbound dynamic program has the advantage of using a state vector that does not need to distinguish among backorders of different ages. We decompose this lower-bound dynamic program using Clark and Scarf's (1960) approach to derive yet another lower bound consisting only of single-location problems with single-dimensional state variables. A numerical study is carried out to see how the performance of an operating policy driven by this decomposition compares with the lower bound obtained.