Essays on Stochastic Inventory Systems

University of Minnesota Ph.D. dissertation. July 2015. Major: Industrial and Systems Engineering. Advisor: Saif Benjaafar. 1 computer file (PDF); ix, 147 pages.This thesis consists of three essays in stochastic inventory systems. The first essay is on the impact of input price variability and correlation on stochastic inventory systems. For a general class of such systems, we show that the expected cost function is concave in the input price. From this, it follows that higher input price variability in the sense of the convex order always leads to lower expected cost. We show that this is true under a wide range of assumptions for price evolution, including cases with i.i.d. prices and cases where prices are correlated and evolve according to an AR(1) process, a geometric Brownian motion, or a Markovian martingale. In addition, the result holds in cases where there is just a single period. We also examine the impact of price correlation over time and across inputs, and we find that expected cost is increasing in price correlation over time and decreasing in price correlation across components. We present results of a numerical study that provide insights on how various parameters influence the effects of price variability and correlation. The second essay is on the optimal control of inventory systems with stochastic and independent leadtimes. We show that a fixed base-stock policy is sub-optimal and can perform poorly. For the case of exponentially distributed leadtimes, we show that the optimal policy is state-dependent and specified in terms of an inventory-dependent threshold function. Moreover, we show that this threshold function is non-increasing in the inventory level and characterized by at most m parameters. That is, once the threshold function starts to decrease it continues to decrease with a rate that is at least one. Taking advantage of this structure, we develop an efficient algorithm for computing these parameters. In characterizing the structure of the optimal policy, we rely on an application of the Banach fixed point theorem. We compare the performance of the optimal policy to that of simpler heuristics. We also extend our analysis to systems with lost sales and systems with order cancellations. The third essay is on the optimal policies for inventory systems with concave ordering costs. By extending the Scarf (1959} model to systems with piecewise linear concave ordering costs, we characterize the structure of optimal policies for periodic review inventory systems with concave ordering costs and general demand distributions. We show that, except for a bounded region, the generalized (s,S) policy is optimal. We do so by (a) introducing a conditional monotonicity property for the optimal order-up-to levels and (b) applying the notion of c-convexity. We also provide conditions under which the generalized (s, S) policy is optimal for all regions of the state space

The Big Data Newsvendor: Practical Insights from Machine Learning

We investigate the data-driven newsvendor problem when one has n observations of p features related to the demand as well as historical demand data. Rather than a two-step process of first estimating a demand distribution then optimizing for the optimal order quantity, we propose solving the “Big Data” newsvendor problem via single step machine learning algorithms. Specifically, we propose algorithms based on the Empirical Risk Minimization (ERM) principle, with and without regularization, and an algorithm based on Kernel-weights Optimization (KO). The ERM approaches, equivalent to high-dimensional quantile regression, can be solved by convex optimization problems and the KO approach by a sorting algorithm. We analytically justify the use of features by showing that their omission yields inconsistent decisions. We then derive finite-sample performance bounds on the out-of-sample costs of the feature-based algorithms, which quantify the effects of dimensionality and cost parameters. Our bounds, based on algorithmic stability theory, generalize known analyses for the newsvendor problem without feature information. Finally, we apply the feature-based algorithms for nurse staffing in a hospital emergency room using a data set from a large UK teaching hospital and find that (i) the best ERM and KO algorithms beat the best practice benchmark by 23% and 24% respectively in the out-of-sample cost, and (ii) the best KO algorithm is faster than the best ERM algorithm by three orders of magnitude and the best practice benchmark by two orders of magnitude

A capacitated commodity trading model with market power

This Master Thesis proposes an analytical model of commodity trading in the presence of market power, which is new to the literature. We establish the rational behavior of traders between two markets and derive price spread dynamics based on the trading volumes. We consider two commodity markets that quote different prices for the same product, and a trader capable of purchasing in one market to resell in the other, within a trading capacity. For simplicity, we assume that the inventory present between the two markets is constant, i.e., the amount stored is fixed (and is directly linked to the trading capacity). We first establish the structure of the optimal trading policy. This policy determines, given a realization of the current price spread between the two markets, how much (if any) volume must be bought/sold from/at the first market and sold/bought at/from the second market, so that the long-run average profit is maximized. When optimizing, the trader takes into account its impact on future spreads. We then describe the parameters of the optimal policy and we apply our model to kerosene prices in New York and Los Angeles, and illustrate the insights derived