Dynamic learning,pricing, and ordering by a censored newsvendor

We address the problem of determining optimal ordering and pricing policies in a finite‐horizon newsvendor model with unobservable lost sales. The demand distribution is price‐dependent and involves unknown parameters. We consider both the cases of perishable and nonperishable inventory. A very general class of demand functions is studied in this paper. We derive the optimal ordering and pricing policies as unique functions of the stocking factor (which is a linear transformation of the safety factor). An important expression is obtained for the marginal expected value of information. As a consequence, we show when lost sales are unobservable, with perishable inventory the optimal stocking factor is always at least as large as the one given by the single‐period model; however, if inventory is nonperishable, this result holds only under a strong condition. This expression also helps to explain why the optimal stocking factor of a period may not increase with the length of the problem. We compare this behavior with that of a full information model. We further examine the implications of the results to the special cases when demand uncertainty is described by additive and multiplicative models. For the additive case, we show that if demand is censored, the optimal policy is to order more as well as charge higher retail prices when compared to the policies in the single‐period model and the full information model. We also compare the optimal and myopic policies for the additive and multiplicative models. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Dynamic coordination of production planning and sales admission control in the presence of a spot market

We consider the decision‐making problem of dynamically scheduling the production of a single make‐to stock (MTS) product in connection with the product's concurrent sales in a spot market and a long‐term supply channel. The spot market is run by a business to business (B2B) online exchange, whereas the long‐term channel is established by a structured contract. The product's price in the spot market is exogenous, evolves as a continuous time Markov chain, and affects demand, which arrives sequentially as a Markov‐modulated Poisson process (MMPP). The manufacturer is obliged to fulfill demand in the long‐term channel, but is able to rein in sales in the spot market. This is a significant strategic decision for a manufacturer in entering a favorable contract. The profitability of the contract must be evaluated by optimal performance. The current problem, therefore, arises as a prerequisite to exploring contracting strategies. We reveal that the optimal strategy of coordinating production and sales is structured by the spot price dependent on the base stock and sell‐down thresholds. Moreover, we can exploit the structural properties of the optimal strategy to conceive an efficient algorithm. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Inventory control under substitutable demand: A stochastic game application

Substitutable product inventory problem is analyzed using the concepts of stochastic game theory. It is assumed that there are two substitutable products that are sold by different retailers and the demand for each product is random. Game theoretic nature of this problem is the result of substitution between products. Since retailers compete for the substitutable demand, ordering decision of each retailer depends on the ordering decision of the other retailer. Under the discounted payoff criterion, this problem is formulated as a two‐person nonzero‐sum stochastic game. In the case of linear ordering cost, it is shown that there exists a Nash equilibrium characterized by a pair of stationary base stock strategies for the infinite horizon problem. This is the unique Nash equilibrium within the class of stationary base stock strategies. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 359–375, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10018     

On the (S − 1, S) lost sales inventory model with priority demand classes

In this paper an inventory model with several demand classes, prioritised according to importance, is analysed. We consider a lot‐for‐lot or (S ? 1, S) inventory model with lost sales. For each demand class there is a critical stock level at and below which demand from that class is not satisfied from stock on hand. In this way stock is retained to meet demand from higher priority demand classes. A set of such critical levels determines the stocking policy. For Poisson demand and a generally distributed lead time, we derive expressions for the service levels for each demand class and the average total cost per unit time. Efficient solution methods for obtaining optimal policies, with and without service level constraints, are presented. Numerical experiments in which the solution methods are tested demonstrate that significant cost reductions can be achieved by distinguishing between demand classes. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 593–610, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10032     

Dynamic inventory management with cash flow constraints

In this article, we consider a classic dynamic inventory control problem of a self‐financing retailer who periodically replenishes its stock from a supplier and sells it to the market. The replenishment decisions of the retailer are constrained by cash flow, which is updated periodically following purchasing and sales in each period. Excess demand in each period is lost when insufficient inventory is in stock. The retailer's objective is to maximize its expected terminal wealth at the end of the planning horizon. We characterize the optimal inventory control policy and present a simple algorithm for computing the optimal policies for each period. Conditions are identified under which the optimal control policies are identical across periods. We also present comparative statics results on the optimal control policy. © 2008 Wiley Periodicals, Inc. Naval Research Logistics 2008     

Optimal operating policies in a commodity trading market with the manufacturer's presence

With the help of the Internet and express delivery at relatively low costs, trading markets have become increasingly popular as a venue to sell excess inventory and a source to obtain products at lower prices. In this article, we study the operational decisions in the presence of a trading market in a periodic‐review, finite‐horizon setting. Prices in the trading market change periodically and are determined endogenously by the demand and supply in the market. We characterize the retailers'optimal ordering and trading policies when the original manufacturer and the trading market co‐exist and retailers face fees to participate in the trading market. Comparing with the case with no trading fees, we obtain insights into the impact of trading fees and the fee structure on the retailers and the manufacturer. Further, we find that by continually staying in the market, the manufacturer may use her pricing strategies to counter‐balance the negative impact of the trading market on her profit. Finally, we extend the model to the case when retailers dynamically update their demand distribution based on demand observations in previous periods. A numerical study provides additional insights into the impact of demand updating in a trading market with the manufacturer's competition. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Revenue management with aircraft reassignment flexibility

A well‐studied problem in airline revenue management is the optimal allocation of seat inventory among different fare‐classes, given a capacity for the flight and a demand distribution for each class. In practice, capacity on a flight does not have to be fixed; airlines can exercise some flexibility on the supply side by swapping aircraft of different capacities between flights as partial booking information is gathered. This provides the airline with the capability to more effectively match their supply and demand. In this paper, we study the seat inventory control problem considering the aircraft swapping option. For theoretical and practical purposes, we restrict our attention to the class of booking limit policies. Our analytical results demonstrate that booking limits considering the swapping option can be considerably different from those under fixed capacity. We also show that principles on the relationship between the optimal booking limits and demand characteristics (size and risk) developed for the fixed‐capacity problem no longer hold when swapping is an option. We develop new principles and insights on how demand characteristics affect the optimal booking limits under the swapping possibility. We also develop an easy to implement heuristic for determining the booking limits under the swapping option and show, through a numerical study, that the heuristic generates revenues close to those under the optimal booking limits. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Capacity investment decisions under risk aversion

This article studies the optimal capacity investment problem for a risk‐averse decision maker. The capacity can be either purchased or salvaged, whereas both involve a fixed cost and a proportional cost/revenue. We incorporate risk preference and use a consumption model to capture the decision maker's risk sensitivity in a multiperiod capacity investment model. We show that, in each period, capacity and consumption decisions can be separately determined. In addition, we characterize the structure of the optimal capacity strategy. When the parameters are stationary, we present certain conditions under which the optimal capacity strategy could be easily characterized by a static two‐sided (s, S) policy, whereby, the capacity is determined only at the beginning of period one, and held constant during the entire planning horizon. It is purchased up to B when the initial capacity is below b, salvaged down to Σ when it is above σ, and remains constant otherwise. Numerical tests are presented to investigate the impact of demand volatility on the optimal capacity strategy. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 218–235, 2016     

The initial provisioning decision for insurance type items

Initial provisioning decisions (inventory stocking requirements) for low demand items often have to be made without much knowledge of what future demand rates will be. When the nature of an item is such that little demand for it is expected, the problem of whether to stock initially or risk not stocking the item is most critical. This report discusses this problem and presents decision procedures which can be used to handle this aspect of initial provisioning. The procedures relate an item's provisioning desirability to its provisioning characteristics, such as expected cost, expected resupply time, current information on its likely demand rate, and to an overall operating policy or criterion. The criterion function measures the total system degredation as a function of the events of having items out of stock when demand occurs. Several different policy functions are discussed and the provisioning decision rules which apply to each are presented. Demand rate information is handled through a Bayesian type approach. The decision rules presented in this report can be utilized to either determine stocking requirements within a budgetary constraint, or determine the relative stocking desirability on an item-by-item basis.     

Optimal pricing and inventory control policy in periodic‐review systems with fixed ordering cost and lost sales

This paper studies a periodic‐review pricing and inventory control problem for a retailer, which faces stochastic price‐sensitive demand, under quite general modeling assumptions. Any unsatisfied demand is lost, and any leftover inventory at the end of the finite selling horizon has a salvage value. The cost component for the retailer includes holding, shortage, and both variable and fixed ordering costs. The retailer's objective is to maximize its discounted expected profit over the selling horizon by dynamically deciding on the optimal pricing and replenishment policy for each period. We show that, under a mild assumption on the additive demand function, at the beginning of each period an (s,S) policy is optimal for replenishment, and the value of the optimal price depends on the inventory level after the replenishment decision has been done. Our numerical study also suggests that for a sufficiently long selling horizon, the optimal policy is almost stationary. Furthermore, the fixed ordering cost (K) plays a significant role in our modeling framework. Specifically, any increase in K results in lower s and higher S. On the other hand, the profit impact of dynamically changing the retail price, contrasted with a single fixed price throughout the selling horizon, also increases with K. We demonstrate that using the optimal policy values from a model with backordering of unmet demands as approximations in our model might result in significant profit penalty. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

Newsvendor problems with sequentially revealed demand information

This article analyzes a capacity/inventory planning problem with a one‐time uncertain demand. There is a long procurement leadtime, but as some partial demand information is revealed, the firm is allowed to cancel some of the original capacity reservation at a certain fee or sell off some inventory at a lower price. The problem can be viewed as a generalization of the classic newsvendor problem and can be found in many applications. One key observation of the analysis is that the dynamic programming formulation of the problem is closely related to a recursion that arises in the study of a far more complex system, a series inventory system with stochastic demand over an infinite horizon. Using this equivalence, we characterize the optimal policy and assess the value of the additional demand information. We also extend the analysis to a richer model of information. Here, demand is driven by an underlying Markov process, representing economic conditions, weather, market competition, and other environmental factors. Interestingly, under this more general model, the connection to the series inventory system is different. © 2012 Wiley Periodicals, Inc. Naval Research Logistics 2012     

The one-period,N-location distribution problem

This paper studies the one-period, general network distribution problem with linear costs. The approach is to decompose the problem into a transportation problem that represents a stocking decision, and into decoupled newsboy problems that represent the realization of demand with the usual associated holding and shortage costs. This approach leads to a characterization of optimal policies in terms of the dual of the transportation problem. This method is not directly suitable for the solution for large problems, but the exact solution for small problems can be obtained. For the numerical solutions of large problems, the problem has been formulated as a linear program with column generation. This latter approach is quite robust in the sense that it is easily extended to incorporate capacity constraints and the multiproduct case.     

Lateral stock transshipments in a two‐location inventory system with fixed and joint replenishment costs

We address the problem of inventory management in a two‐location inventory system, in which the transshipments are carried out as means of emergency or alternative supply after demand has been realized. This model differs from previous ones as regards its replenishment costs structure, in which nonnegligible fixed replenishment costs and a joint replenishment cost are considered. The single period planning horizon is analyzed, with the form and several properties of the optimal replenishment and transshipment policies developed, discussed and illustrated. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 525–547, 1999     

Impact of reseller's and sales agent's forecasting accuracy in a multilayer supply chain

We consider a three‐layer supply chain with a manufacturer, a reseller, and a sales agent. The demand is stochastically determined by the random market condition and the sales agent's private effort level. Although the manufacturer is uninformed about the market condition, the reseller and the sales agent conduct demand forecasting and generate private demand signals. Under this framework with two levels of adverse selection intertwined with moral hazard, we study the impact of the reseller's and the sales agent's forecasting accuracy on the profitability of each member. We show that the manufacturer's profitability is convex on the reseller's forecasting accuracy. From the manufacturer's perspective, typically improving the reseller's accuracy is detrimental when the accuracy is low but is beneficial when it is high. We identify the concrete interrelation among the manufacturer‐optimal reseller's accuracy, the volatility of the market condition, and the sales agent's accuracy. Finally, the manufacturer's interest may be aligned with the reseller's when only the reseller can choose her accuracy; this alignment is never possible when both downstream players have the discretion to choose their accuracy. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 207–222, 2014     

Quantity discount pricing policies for heterogeneous retailers with price sensitive demand

Although quantity discount policies have been extensively analyzed, they are not well understood when there are many different buyers. This is especially the case when buyers face price‐sensitive demand. In this paper we study a supplier's optimal quantity discount policy for a group of independent and heterogeneous retailers, when each retailer faces a demand that is a decreasing function of its retail price. The problem is analyzed as a Stackelberg game whereby the supplier acts as the leader and buyers act as followers. We show that a common quantity discount policy that is designed according to buyers' individual cost and demand structures and their rational economic behavior is able to significantly stimulate demand, improve channel efficiency, and substantially increase profits for both the supplier and buyers. Furthermore, we show that the selection of all‐units or incremental quantity discount policies has no effect on the benefits that can be obtained from quantity discounts. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

A note on “resource flexibility with responsive pricing”

We revisit the capacity investment decision problem studied in the article “Resource Flexibility with Responsive Pricing” by Chod and Rudi [Operations Research 53, (2005) 532–548]. A monopolist firm producing two dependent (substitutable or complementary) products needs to determine the capacity of one flexible resource under demand risk so as to maximize its expected profit. Product demands are linear functions of the prices of both products, and the market potentials are random and correlated. We perform a comparative statics analysis on how demand variability and correlation impact the optimal capacity and the resulting expected profit. In particular, C&R study this problem under the following assumptions/approximations: (i) demand intercepts follow a bivariate Normal distribution; (ii) demand uncertainty is of an additive form; (iii) and under approximate expressions for the optimal capacity and optimal expected profit. We revisit Propositions 2, 3, 4, 5, and 10 of C&R without these assumptions and approximations, and show that these results continue to hold (i) for the exact expressions for the optimal expected profit and optimal capacity, and (ii) under any arbitrary continuous distribution of demand intercepts. However, we also show that the additive demand uncertainty is a critical assumption for the C&R results to hold. In particular, we provide a case of multiplicative uncertainty under which the C&R results (Propositions 2 and 3) fail. © 2010 Wiley Periodicals, Inc. Naval Research Logistics 2010     

Vertically restrictive pricing in supply chains with price‐dependent demand

We examine the behavior of a manufacturer and a retailer in a decentralized supply chain under price‐dependent, stochastic demand. We model a retail fixed markup (RFM) policy, which can arise as a form of vertically restrictive pricing in a supply chain, and we examine its effect on supply chain performance. We prove the existence of the optimal pricing and replenishment policies when demand has a linear additive form and the distribution of the uncertainty component has a nondecreasing failure rate. We numerically compare the relative performance of RFM to a price‐only contract and we find that RFM results in greater profit for the supply chain than the price‐only contract in a variety of scenarios. We find that RFM can lead to Pareto‐improving solutions where both the supplier and the retailer earn more profit than under a price‐only contract. Finally, we compare RFM to a buyback contract and explore the implications of allowing the fixed markup parameter to be endogenous to the model. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006.     

A Bayesian analysis on the effect of multiple supply options in a quick response environment

In the apparel industry, vendors often suffer from high mismatches in supply and demand. To cope with this problem, they procure the same style product from different suppliers with different manufacturing costs. Especially in the quick response environment, which allows vendors to monitor trends in customer demand and search for available suppliers through the electronic market, they have additional opportunities to improve their decision‐making. In this paper, we propose an analytical profit maximization model and develop efficient decision tools to help both the middle and lower level managers pursuing this strategy. Furthermore, we have shown how significantly the vendors' potential competitive edge can be improved by exploiting multiple supply options, even at the expense of high premium procurement costs for late orders. The effect is critical, especially in a highly competitive market, and it has important implications for the top managers. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2003     

On valuing appreciating human assets in services

This paper derives optimal policies for when to hire, train, and lay off employees in service organizations when worker productivity appreciates deterministically or stochastically under constant, monotonic, and cyclic service demand. The model, an extension of the machine replacement problem from the engineering economy literature, uses infinite horizon linear programming to produce optimal personnel planning strategies and dual prices that characterize the workforce's economic value. The effect of employee attrition is also considered. In its valuation of human assets, unlike pure accounting approaches that only try to measure, this paper proposes an approach to capture the interaction between firm decisions pertaining to human resources and human resource value. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Competition under time‐varying demands and dynamic lot sizing costs

We develop a competitive pricing model which combines the complexity of time‐varying demand and cost functions and that of scale economies arising from dynamic lot sizing costs. Each firm can replenish inventory in each of the T periods into which the planning horizon is partitioned. Fixed as well as variable procurement costs are incurred for each procurement order, along with inventory carrying costs. Each firm adopts, at the beginning of the planning horizon, a (single) price to be employed throughout the horizon. On the basis of each period's system of demand equations, these prices determine a time series of demands for each firm, which needs to service them with an optimal corresponding dynamic lot sizing plan. We establish the existence of a price equilibrium and associated optimal dynamic lotsizing plans, under mild conditions. We also design efficient procedures to compute the equilibrium prices and dynamic lotsizing plans.© 2008 Wiley Periodicals, Inc. Naval Research Logistics 2009