Woonghee Tim Huh

(s, S) Optimality in Joint Inventory-Pricing Control: An Alternate Approach

We study a stationary, single-stage inventory system, under periodic review, with fixed ordering costs and multiple sales levers (such as pricing, advertising, etc.). We show the optimality of (s, S)-type policies in these settings under both the backordering and lost-sales assumptions. Our analysis is constructive and is based on a condition that we identify as being key to proving the (s, S) structure. This condition is entirely based on the single-period profit function and the demand model. Our optimality results complement the existing results in this area.

A Nonparametric Asymptotic Analysis of Inventory Planning with Censored Demand

We study stochastic inventory planning with lost sales and instantaneous replenishment where, contrary to the classical inventory theory, knowledge of the demand distribution is not available. Furthermore, we observe only the sales quantity in each period and lost sales are unobservable, that is, demand data are censored. The manager must make an ordering decision in each period based only on historical sales data. Excess inventory is either perishable or carried over to the next period. In this setting, we propose nonparametric adaptive policies that generate ordering decisions over time. We show that the T-period average expected cost of our policy differs from the benchmark newsvendor cost---the minimum expected cost that would have incurred if the manager had known the underlying demand distribution---by at most O(1/T0.5).

Pricing Multiple Products with the Multinomial Logit and Nested Logit Models: Concavity and Implications

We consider the problem of pricing multiple differentiated products with the nested logit model and, as a special case, the multinomial logit model. We prove that concavity of the total profit function with respect to market share holds even when price sensitivity may vary with products. We use this result to analytically compare the optimal monopoly solution to oligopolistic equilibrium solutions. To demonstrate further applications of the concavity result, we consider several multiperiod dynamic models that incorporate the pricing of multiple products in the context of inventory control and revenue management, and establish structural results of the optimal policies.

Adaptive Data-Driven Inventory Control with Censored Demand Based on Kaplan-Meier Estimator

Using the well-known product-limit form of the Kaplan-Meier estimator from statistics, we propose a new class of nonparametric adaptive data-driven policies for stochastic inventory control problems. We focus on the distribution-free newsvendor model with censored demands. The assumption is that the demand distribution is not known and there are only sales data available. We study the theoretical performance of the new policies and show that for discrete demand distributions they converge almost surely to the set of optimal solutions. Computational experiments suggest that the new policies converge for general demand distributions, not necessarily discrete, and demonstrate that they are significantly more robust than previously known policies. As a by-product of the theoretical analysis, we obtain new results on the asymptotic consistency of the Kaplan-Meier estimator for discrete random variables that extend existing work in statistics. To the best of our knowledge, this is the first application of the Kaplan-Meier estimator within an adaptive optimization algorithm, in particular, the first application to stochastic inventory control models. We believe that this work will lead to additional applications in other domains.

On the Optimal Policy Structure in Serial Inventory Systems with Lost Sales

We study a periodically reviewed, serial inventory system in which excess demand from external customers is lost. We derive elementary properties of the vector of optimal order quantities in this system. In particular, we derive bounds on the sensitivity (or, more mathematically, the derivative) of the optimal order quantity at each stage to the vector of the current inventory levels. Our analysis uses the concept of L-natural-convexity, which was studied in discrete convex analysis and recently used in the study of single-stage inventory systems with lost sales. We also remark on how our analysis extends to models with capacity constraints and/or backordering.

Asymptotic Optimality of Order-Up-To Policies in Lost Sales Inventory Systems

We study a single-product single-location inventory system under periodic review, where excess demand is lost and the replenishment lead time is positive. The performance measure of interest is the long-run average holding cost and lost sales penalty cost. For a large class of demand distributions, we show that when the lost sales penalty becomes large compared to the holding cost, the relative difference between the cost of the optimal policy and the best order-up-to policy converges to zero. For any given cost parameters, we establish a bound on this relative difference. Numerical experiments show that the best order-up-to policy performs well, yielding an average cost that is within 1.5% of the optimal cost when the ratio between the lost sales penalty and the holding cost is 100. We also propose a heuristic order-up-to level using two newsvendor expressions; in our experiments, the cost of this order-up-to policy is 2.52% higher, on an average, than the best order-up-to policy.

Multiresource Allocation Scheduling in Dynamic Environments

Motivated by service capacity-management problems in healthcare contexts, we consider a multiresource allocation problem with two classes of jobs elective and emergency in a dynamic and nonstationary environment. Emergency jobs need to be served immediately, whereas elective jobs can wait. Distributional information about demand and resource availability is continually updated, and we allow jobs to renege. We prove that our formulation is convex, and the optimal amount of capacity reserved for emergency jobs in each period decreases with the number of elective jobs waiting for service. However, the optimal policy is difficult to compute exactly. We develop the idea of a limit policy starting at a particular time, and use this policy to obtain upper and lower bounds on the decisions of an optimal policy in each period, and also to develop several computationally efficient policies. We show in computational experiments that our best policy performs within 1.8% of an optimal policy on average.

Technical note---Linear Inflation Rules for the Random Yield Problem: Analysis and Computations

In this paper, we propose a simple heuristic approach for the inventory control problem with stochastic demand and multiplicative random yield. Our heuristic tries to find the best candidate within a class of policies that are referred to in the literature as the linear inflation rule (LIR) policies. Our approach is computationally fast, easy to implement, and intuitive to understand. Moreover, we find that in a significant number of instances our heuristic performs better than several other well-known heuristics that are available in the literature.

Inventory Management with Auctions and Other Sales Channels: Optimality of (s, S) Policies

We study periodic-review inventory replenishment problems with fixed ordering costs, and show the optimality of (s, S) inventory replenishment policies. Inventory replenishment is instantaneous, i.e., the lead time is zero. We consider several sales mechanisms, e.g., auction mechanisms, name-your-own-price mechanisms, and multiple heterogeneous sales channels. We prove this result by showing that these models satisfy a recently-established sufficient condition for the optimality of (s, S) policies. Thus, this paper shows that the optimality of (s, S) policies extends well beyond the traditional sales environments studied so far in the inventory literature.

Average Cost Single-Stage Inventory Models: An Analysis Using a Vanishing Discount Approach

An important problem in the theory of dynamic programming is that of characterizing sufficient conditions under which the optimal policies for Markov decision processes (MDPs) under the infinite-horizon discounted cost criterion converge to an optimal policy under the average cost criterion as the discount factor approaches 1. In this paper, we provide, for stochastic inventory models, a set of such sufficient conditions. These conditions, unlike many others in the dynamic programming literature, hold when the action space is noncompact and the underlying transition law is weakly continuous. Moreover, we verify that these conditions hold for almost all conceivable single-stage inventory models with few assumptions on cost and demand parameters. As a consequence of our analysis, we partially characterize, for the first time, optimal policies for the following inventory systems under the infinite-horizon average-cost criterion, which have thus far been a challenge: (a) capacitated systems with setup costs, (b) uncapacitated systems with convex ordering costs plus a setup cost, and (c) systems with lost sales and lead times.