Title: Essays on Inventory Management and Object Allocation
Description: This dissertation consists of three essays. In the first, we establish a framework for proving equivalences between mechanisms that allocate indivisible objects to agents. In the second, we study a newsvendor model where the inventory manager has access to two experts that provide advice, and examine how and when an optimal algorithm can be efficiently computed. In the third, we study classical single-resource capacity allocation problem and investigate the relationship between data availability and performance guarantees. We first study mechanisms that solve the problem of allocating indivisible objects to agents. We consider the class of mechanisms that utilize the Top Trading Cycles (TTC) algorithm (these may differ based on how they prioritize agents), and show a general approach to proving equivalences between mechanisms from this class. This approach is used to show alternative and simpler proofs for two recent equivalence results for mechanisms with linear priority structures. We also use the same approach to show that these equivalence results can be generalized to mechanisms where the agent priority structure is described by a tree.

Second, we study the newsvendor model where the manager has recourse to advice, or decision recommendations, from two experts, and where the objective is to minimize worst-case regret from not following the advice of the better of the two agents. We show the model can be reduced to the class machine-learning problem of predicting binary sequences but with an asymmetric cost function, allowing us to obtain an optimal algorithm by modifying a well-known existing one. However, the algorithm we modify, and consequently the optimal algorithm we describe, is not known to be efficiently computable, because it requires evaluations of a function v which is the objective value of recursively defined optimization problems. We analyze v and show that when the two cost parameters of the newsvendor model are small multiples of a common factor, its evaluation is computationally efficient. We also provide a novel and direct asymptotic analysis of v that differs from previous approaches. Our asymptotic analysis gives us insight into the transient structure of v as its parameters scale, enabling us to formulate a heuristic for evaluating v generally.

This, in turn, defines a heuristic for the optimal algorithm whose decisions we find in a numerical study to be close to optimal. In our third essay, we study the classical single-resource capacity allocation problem. In particular, we analyze the relationship between data availability (in the form of demand samples) and performance guarantees for solutions derived from that data. This is done by describing a class of solutions called epsilon-backwards accurate policies and determining a suboptimality gap for this class of solutions. The suboptimality gap we find is in terms of epsilon and is also distribution-free. We then relate solutions generated by a Monte Carlo algorithm and epsilon-backwards accurate policies, showing a lower bound on the quantity of data necessary to ensure that the solution generated by the algorithm is epsilon-backwards accurate with a high probability. Combining the two results then allows us to give a lower bound on the data needed to generate an α-approximation with a given confidence probability 1-delta. We find that this lower bound is polynomial in the number of fares, M, and 1/α.