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We consider an inventory control problem with multiple products and stockout substitution. The firm knows neither the primary demand distribution for each product nor the customers’ substitution probabilities between products a priori, and it needs to learn such information from sales data on the fly. One challenge in this problem is that the firm cannot distinguish between primary demand and substitution (overflow) demand from the sales data of any product, and lost sales are not observable. To circumvent these difficulties, we construct learning stages with each stage consisting of a cyclic exploration scheme and a benchmark exploration interval. The benchmark interval allows us to isolate the primary demand information from the sales data, and then this information is used against the sales data from the cyclic exploration intervals to estimate substitution probabilities. Because raising the inventory level helps obtain primary demand information but hinders substitution demand information, inventory decisions have to be carefully balanced to learn them together. We show that our learning algorithm admits a worst-case regret rate that (almost) matches the theoretical lower bound, and numerical experiments demonstrate that the algorithm performs very well. This paper was accepted by J. George Shanthikumar, big data analytics.

In many production/distribution systems, materials flow in fixed lot sizes (e.g., in full truckloads or full containers) and under regular schedules (e.g., delivery every week). In this paper, we study a multiechelon serial system with batch ordering and fixed replenishment intervals. We derive the optimal inventory control policy, provide a distribution-function solution for its optimal control parameters, and present an efficient algorithm for computing those parameters. Further, we show that the optimal expected system cost is minimized when the ordering times for all stages are synchronized. In contrast to the known approach in the literature that develops a lower bound for the average cost of a given period for the classical serial system, we develop a lower bound for the average total cost over an appropriately defined cycle and then construct a policy that reaches the lower bound. We also discuss its extension to the nonlinear shortage cost case (i.e., the nonlinear cost case). This paper generalizes several recent results on the analysis of multiechelon systems.

Motivated by the challenges faced by the telecom industry during the past decade, in this paper we study a dynamic capacity expansion problem for service firms. There is a random demand for the firm's capacity in each period: the demand in excess of the capacity is lost, and revenue is generated for the fulfilled demand. At the beginning of each period, the firm might increase its capacity through purchasing equipment for immediate delivery, which is constrained by a random supply limit, or it might sign a future contract for equipment delivery in the following period. We assume that the firm's capacity might partially become obsolete due to natural deterioration or technology innovation. We aim at characterizing optimal capacity expansion strategies and comparing the profit functions as well as the optimal control policies of different options. Specifically, we show that the optimal capacity expansion policy for the current period is determined by a base-stock policy. Compared with the case where no future contracts are available, the optimal control parameters of capacity expansion are always smaller. We further show that when the obsolescence rate is deterministic, the optimal policy for capacity expansion through future contracts is also a base-stock type. The results are extended to the cases with stochastically dependent capacity supply limits and stochastically dependent demand processes, which establish the robustness of the optimal policy in various market conditions.

We consider a periodic-review, single-product inventory system with lost sales and positive lead times under censored demand. In contrast to the classical inventory literature, we assume the firm does not know the demand distribution a priori and makes an adaptive inventory-ordering decision in each period based only on the past sales (censored demand) data. The standard performance measure is regret, which is the cost difference between a learning algorithm and the clairvoyant (full-information) benchmark. When the benchmark is chosen to be the (full-information) optimal base-stock policy, Huh et al. [Huh WT, Janakiraman G, Muckstadt JA, Rusmevichientong P (2009a) An adaptive algorithm for finding the optimal base-stock policy in lost sales inventory systems with censored demand. Math. Oper. Res. 34(2):397–416.] developed a nonparametric learning algorithm with a cubic-root convergence rate on regret. An important open question is whether there exists a nonparametric learning algorithm whose regret rate matches the theoretical lower bound of any learning algorithms. In this work, we provide an affirmative answer to this question. More precisely, we propose a new nonparametric algorithm termed the simulated cycle-update policy and establish a square-root convergence rate on regret, which is proven to be the lower bound of any learning algorithm. Our algorithm uses a random cycle-updating rule based on an auxiliary simulated system running in parallel and also involves two new concepts, namely the withheld on-hand inventory and the double-phase cycle gradient estimation . The techniques developed are effective for learning a stochastic system with complex system dynamics and lasting impact of decisions. This paper was accepted by Yinyu Ye, optimization.

Transportation and production contracts often specify the frequency and volume reserved by the supplier for a particular customer's deliveries. This practice motivated Henig et al. (Henig, M., Y. Gerchak, R. Ernst, D. Pyke. 1997. An inventory model embedded in designing a supply contract. Management Sci. 43 184–189) to study a periodic-review inventory-control model where ordering cost is zero if the order quantity does not exceed a given contract volume and is linear in the excess quantity otherwise. This paper addresses the same problem but with a fixed cost if the order quantity is above the contract volume. The fixed cost may represent the cost of disruption for the supplier (finding more trucks, arranging extra processing capacity, persuading other customers to wait, etc.) as well as additional administrative costs. Also, suppliers may impose such costs simply to induce desired behavior by buyers. This order-cost function is neither convex nor concave. The classical inventory models with fixed costs are special cases with contract volume zero. We partially characterize the optimal policy for this system and develop a simple, effective heuristic policy. We also apply the model to a production-control problem in which an incentive is provided for not ordering over a certain quota.

Because of uncertainty in customer demand and lack of understanding in customer reactions to price changes, it is a challenge for many companies, such as manufacturers and retailers, to match supply and demand. Most of the models in the operations literature, however, have focused on the case in which the underlying customer demand information is known a priori, which is not true in many applications. In “Coordinating Pricing and Inventory Replenishment with Nonparametric Demand Learning,” B. Chen, X. Chao, and H. Ahn develop a data-driven algorithm for pricing and inventory decisions that learns the demand and customer information from sales data on the fly, and they show that the profit generated from the algorithm converges to the clairvoyant optimal profit at the quickest possible rate. We consider a firm (e.g., retailer) selling a single nonperishable product over a finite-period planning horizon. Demand in each period is stochastic and price sensitive, and unsatisfied demands are backlogged. At the beginning of each period, the firm determines its selling price and inventory replenishment quantity with the objective of maximizing total profit, but it knows neither the average demand (as a function of price) nor the distribution of demand uncertainty a priori; hence, it has to make pricing and ordering decisions based on observed demand data. We propose a nonparametric, data-driven algorithm that learns about the demand on the fly and, concurrently, applies learned information to make replenishment and pricing decisions. The algorithm integrates learning and action in a sense that the firm actively experiments on pricing and inventory levels to collect demand information with minimum profit loss. Besides convergence of optimal policies, we show that the regret of the algorithm, defined as the average profit loss compared with that of the optimal solution had the firm known the underlying demand information, vanishes at the fastest possible rate as the planning horizon increases.

We develop the first nonparametric learning algorithm for periodic-review perishable inventory systems. In contrast to the classical perishable inventory literature, we assume that the firm does not know the demand distribution a priori and makes replenishment decisions in each period based only on the past sales (censored demand) data. It is well known that even with complete information about the demand distribution a priori, the optimal policy for this problem does not possess a simple structure. Motivated by the studies in the literature showing that base-stock policies perform near optimal in these systems, we focus on finding the best base-stock policy. We first establish a convexity result, showing that the total holding, lost sales and outdating cost is convex in the base-stock level. Then, we develop a nonparametric learning algorithm that generates a sequence of order-up-to levels whose running average cost converges to the cost of the optimal base-stock policy. We establish a square-root convergence rate of the proposed algorithm, which is the best possible. Our algorithm and analyses require a novel method for computing a valid cycle subgradient and the construction of a bridging problem, which significantly departs from previous studies.

This paper studies the optimal control policy for capacitated periodic-review inventory systems with remanufacturing. The serviceable products can be either manufactured from raw materials or remanufactured from returned products; but the system has finite capacities in manufacturing, remanufacturing, and/or total manufacturing/remanufacturing operations in each period. Using L -natural convexity and lattice analysis, we show that, for systems with a remanufacturing capacity and a manufacturing/total capacity, the optimal remanufacturing policy is a modified remanufacture-down-to policy and the optimal manufacturing policy is a modified total-up-to policy. Our study reveals that the optimal policies always give production priority to remanufacturing for systems with a remanufacturing capacity and/or a total capacity; but this priority fails to hold for systems with a manufacturing capacity.

In this paper we study a stochastic production/inventory system with finite production capacity and random demand. The cumulative production and demand are modeled by a two-dimensional Brownian motion process. There is a setup cost for switching on the production and a convex holding and shortage cost, and our objective is to find the optimal production/inventory control that minimizes the average cost. Both lost-sales and backlog cases are studied. For the lost-sales model we show that, within a large class of policies, the optimal production strategy is either to produce according to an ( s , S ) policy, or never turn on the machine at all (thus it is optimal for the firm to not enter the business); whereas for the backlog model, we prove that the optimal production policy is always of the ( s , S ) type. Our approach first develops a lower bound for the average cost among a large class of nonanticipating policies and then shows that the value function of the desired policy reaches the lower bound. The results offer insights on the structure of the optimal control policies as well as the interplays between system parameters.

We develop the first approximation algorithm for periodic-review perishable inventory systems with setup costs. The ordering lead time is zero. The model allows for correlated demand processes that generalize the well-known approaches to model dynamic demand forecast updates. The structure of optimal policies for this fundamental class of problems is not known in the literature. Thus, finding provably near-optimal control policies has been an open challenge. We develop a randomized proportional-balancing policy (RPB) that can be efficiently implemented in an online manner, and we show that it admits a worst-case performance guarantee between 3 and 4. The main challenge in our analysis is to compare the setup costs between RPB and the optimal policy in the presence of inventory perishability, which departs significantly from the previous works in the literature. The numerical results show that the average performance of RPB is good (within 1% of optimality under i.i.d. demands and within 7% under correlated demands).