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This study is motivated by a fundamental question among researchers: do price data correspond to a pure strategy Nash equilibrium of an estimated econometric model of Bertrand competition? The affirmative answer to this question is essential for validating the estimated model. Despite the extensive empirical applications of Bertrand competition over the last three decades, answering the question remains a daunting task for the researchers. This study takes a step to fill this gap, identifying a set of easy‐to‐verify and easy‐to‐interpret conditions under which, by checking how closely the price data fit the system of first‐order conditions, one can easily obtain an answer to the question.

We study a monopolistic firm which introduces two product versions sequentially in two periods. We analyze and compare the firm's decisions on the innovation level of the new version, the production quantities and prices of both versions, and the associated profit in four settings: when the customers are myopic or strategic in period 1 and whether the leftover inventory of the old version is phased out from the market (single rollover strategy) or is sold in the market (dual rollover strategy). In period 2, newcomers who wish to buy the new version arrive in the market. We show that the firm can improve both its profit and its innovation level by adopting an appropriate rollover strategy when selling to strategic customers. This finding underscores the importance of choosing a rollover strategy. Interestingly and differently from the existing literature, we see that strategic waiting behavior can accelerate innovation. These analytical insights remain largely valid when some of the customers who cannot get the old version due to a stockout leave the market before the new version arrives, or when some of the newcomers are interested in the new version as well as the leftover old version.

Traditional take-or-pay contracts have been used to mitigate risks in energy supply chains for decades. More recently, the downstream segment of the natural gas industry has increased capital expenditure on storage facilities in order to enhance supply flexibility and minimize operational risks. In this paper, we study the enhanced value of a take-or-pay gas contract from a buyer’s perspective in the presence of spot market trading and local storage capability. We use a multistage stochastic program via a computationally efficient split-variable formulation and solve this procurement problem to delineate the impact of various key managerial levers on the design, valuation and usage of a take-or-pay contract. Among these, the net convenience yield, storage cost and take-or-pay contract terms are identified as being the most important. Further, we numerically show the subadditivity of values for take-or-pay and storage options. Practically relevant managerial insights are generated to assist decision makers in energy supply chains, where price and demand uncertainties are abundant.

Willingness To Pay (WTP) of customers plays an anchoring role in pricing. This study proposes a new choice model based on WTP, incorporating sequential decision making, where the products with positive utility of purchase are considered in the order of customer preference. We compare WTP-choice model with the commonly used (multinomial) Logit model with respect to the underlying choice process, information requirement, and independence of irrelevant alternatives. Using WTP-choice model, we find and compare equilibrium and centrally optimal prices and profits without considering inventory availability. In addition, we compare equilibrium prices and profits in two contexts: without considering inventory availability and under lost sales. One of the interesting results with WTP-choice model is the 'loose coupling' of retailers in competition; prices are not coupled but profits are. That is, each retailer should charge the monopoly price as the collection of these prices constitute an equilibrium but each retailer's profit depends on other retailers' prices. Loose coupling fails with dependence of WTPs or dependence of preference on prices. Also, we show that competition among retailers facing dependent WTPs can cause price cycles under some conditions. We consider real-life data on sales of yogurt, ketchup, candy melt, and tuna, and check if a version of WTP-choice model (with uniform, triangle, or shifted exponential WTP distribution), standard or mixed Logit model fits better and predicts the sales better. These empirical tests establish that WTP-choice model compares well and should be considered as a legitimate alternative to Logit models for studying pricing for products with low price and high frequency of purchase. [ABSTRACT FROM AUTHOR]

Inventory inaccuracy is common in many businesses. While retailers employ cash registers to enter incoming orders and outgoing sales, inaccuracy arises because they do not record invisible demand such as spoilage, damage, pilferage, or returns. This setting results in incomplete inventory and demand information. An important inventory control problem therefore is to maximize the total expected discounted profit under this setting. Allowing for dependence between demand and invisible demand, we obtain the associated dynamic programming equation with an infinite-dimensional state space, and reduce it to a simpler form by employing the concept of unnormalized probability. We develop an analytical upper bound on the optimal profit as well as an iterative algorithm for an approximate solution of the problem. We compare profits of the iterative solution and the myopic solution, and then to the upper bound. We see that the iterative solution performs better than the myopic solution, and significantly so in many cases. Furthermore, it gives a profit not far from the upper bound, and is therefore close to optimal. Using our results, we also discuss meeting inventory service levels.

Problem definition: We propose and study a dynamic upgrading framework as a way to mitigate supply-demand mismatches among products of different qualities. Dynamic upgrades from a regular product to a premium one are offered between booking and check-in times by a travel firm. A regular product purchaser receives an upgrade offer as a link to a website and may accept it after observing an upgrade fee dynamically determined by the firm on the website. These offers are time-limited; the firm can stop them by deactivating the links. We characterize the optimal timing, quantity and pricing of dynamic upgrades between the booking and check-in times. As a common practice, static upgrading has been implemented either at the booking or the check-in time. Travel firms have recently started to adopt dynamic upgrades, which attest to their importance in practice. Methodology/results: We formulate and solve the problem using dynamic programming with two profit maximization approaches: a capacitated (limited) or a costly upgrade model. We prove that two models are profit-equivalent and the optimal upgrade policies in both are of a pulsing type; either maintaining zero or the maximum number of upgrade links. Focusing on the easy-to-implement capacitated upgrade model, we propose four variations to incorporate relevant business rules: upgrade postponement, restricted upgrade fees, limited upgrade (de-)activation times, and stockout-induced upward substitution. Using multimodularity concepts, we compare optimal policies across model variations and establish monotonicity properties. Managerial implications: The equivalence of two dynamic upgrade formulations allows managers to adopt either one. The optimal policy structures and monotonicity properties help managers understand tradeoffs in dynamic upgrades. Through a comprehensive and systematic numerical study, we identify business contexts suitable for dynamic upgrades and quantify their profit improvements. Compared to the optimal profit with check-in upgrades, the profit improvement with dynamic upgrades, flexible upgrade fees, unlimited (de-)activations and upward substitutions is 5.52% on average and can be as high as 59.34%. We also provide an upgrade process framework that can be used in practice and discuss its customization and implementation.

In a centrally managed system, inventory at a retailer can be transshipped to a stocked-out retailer to meet demand. As the inventory at the former retailer may be demanded by future customers of that retailer and transshipment time/cost is non-negligible, it can be more profitable to not transship in some situations. When unsatisfied demand is backordered, reassignment of inventory to a previously backordered demand can perhaps become profitable as demand uncertainty resolves over time. Despite this intuition, we prove that no reassignments are necessary for cost optimality under periodic holding cost accounting in a two-retailer system. This remains valid for multi-retailer systems according to numerical analyses. When holding costs are accounted for only at the end of each replenishment cycle, reassignments are necessary for optimality but insignificant in reducing the total cost. In most instances tested, the decrease in total cost from reassignments is below 2% for end of cycle holding cost accounting. These results simplify transshipment policies and facilitate finding good policies in both implementation and future studies, as reassignments can be omitted from consideration in optimization models under periodic holding cost accounting and in approximation models under cyclical cost accounting. © 2013 Operational Research Society Ltd.

We study a discrete-time periodic-review inventory system where the unmet demand is lost, and the excess inventory is subject to shrinkage. We first derive the state evolution and then introduce unnormalized conditional probabilities to transform the nonlinear state evolution into a linear one. We then prove the existence and uniqueness of the solution for the Bellman equation in the case of unbounded costs and show that the solution yields the value function.

Information delays exist when the most recent inventory information available to the Inventory Manager (IM) is dated. In other words, the IM observes only the inventory level that belongs to an earlier period. Such situations are not uncommon, and they arise when it takes a while to process the demand data and pass the results to the IM. In this paper, we establish that the average cost optimal policy is of state-dependent basestock type with respect to the reference inventory position. We show that the optimal base stock depends on the age and the magnitude of the latest observed delay. We illustrate the results by solving an example with delays of 0 and 1, for which we are able to obtain formulas/bounds for the basestock levels.

We analyze a supply chain consisting of a supplier and a retailer. The supplier's unit production cost, which characterizes his type, is only privately known to him. When trading with the retailer, the supplier demands a reservation profit that depends on his unit production cost. We model this problem as a game of adverse selection. In this model, the retailer offers a menu of contracts, each of which consists of two parameters: the ordering quantity and the supplier's share of the channel profit. We show that the optimal contract depends critically on a surrogate measure—the ratio of the types’ reservation profit differential to their production cost differential. An important implication from our analysis is that information asymmetry alone does not necessarily induce loss in channel efficiency. The optimal contract can coordinate the supply chain as long as the low-cost supplier's cost efficiency is neither much overvalued nor much undervalued in the outside market. We further discuss the retailer's preference of the supplier's type under different market conditions, as well as evaluate the effects of the supplier's reservation profit, the retail price, and the demand uncertainty on the optimal contract.

This paper proposes a new methodology to solve partially observed inventory problems. Generally, these problems have infinite-dimensional states that are conditional distributions of the inventory level. Our methodology involves linearizing the state transitions via unnormalized probabilities. It then uses an appropriate functional basis to represent the state. Considering the speed and stability of computations, we choose truncated Chebyshev polynomials as the basis. We use Fast Fourier Transforms along with an appropriate discretization of inventory levels to speed up the computations. These main ideas are blended to obtain an iterative algorithm to solve a partially observed inventory model with rain checks. In this model, the inventory manager (IM) does not know the inventory level when it is positive. Otherwise, the IM fully observes it. This model provides a context to illustrate our methodology, which applies to other such models. Although this model has been studied mathematically in the literature, the use of our algorithm provides a numerical approximation of the optimal order quantities. These are compared to the orders released under a base mean-stock policy, where the IM replaces the unobserved inventory level with its mean and applies the well-known base stock policy. We show numerically that the optimal order quantity is very close to the base mean-stock order quantity, when the variance of the inventory distribution is small. When the mean of the inventory distribution is large, the optimal order quantity is more than the base mean-stock quantity, and it is the other way around when the mean is small or negative. These insights are explained via uncertainty and information effects and their interplay. We expect this interplay to show up in other partially observed inventory models.

We study a centralized inventory sharing system of two retailers that are replenished periodically. Between two replenishments, a unit can be transshipped to a stocked-out retailer from the other. It arrives a transshipment time later, during which the stocked-out retailer incurs backorder cost. Without transshipment, backorder cost is incurred until the next replenishment. Since the transshipment time is shorter than the time between two replenishments, transshipments can reduce the backorder cost at the stocked-out retailer and the holding costs at the other retailer. The system is directed by a centralized inventory manager, who minimizes the long-run average cost consisting of replenishment, holding, backorder, and transshipment costs. The transshipment policy is characterized by hold-back inventory levels, which are nonincreasing in the remaining time until the next replenishment. The transshipment policy differs from those in the literature because we allow for multiple transshipments between replenishments, positive transshipment times, and backorder costs. We also discuss the challenges associated with positive replenishment time and develop upper and lower bounds of average cost in this case. Bounds are numerically shown to have an average gap of 1.1%. A heuristic solution is based on the upper bound and differs from the optimal cost by at most this gap.

This paper studies the joint decisions of IDS configuration and alarm investigation capacity under active and passive responses. In active response, alarm events are blocked immediately, whereas alarm events are allowed to access the information assets in the passive response. Despite facilitating information flow, passive response exposes the assets to attacks while the security analysts investigate the alarms. On the other hand, active response may unnecessarily delay the benign traffic since alarm events are blocked. We find closed-form formulas for the optimal investigation capacity and show the optimal configuration under active response is smaller than under passive response. We also provide expressions that can be used to evaluate security costs and benefits under various configurations, capacities and responses. Numerical studies are done to illustrate the sensitivity of the optimal decisions.

This paper analyzes a stochastic inventory problem with an order-time constraint that restricts the times at which a manufacturer places new orders to a supplier. This constraint stems from the limited upstream capacity in a supply chain, such as production capacity at a supplier or transportation capacity between a supplier and a manufacturer. Consideration of limited upstream capacity extends the classical inventory literature that unrealistically assumes infinite supplier/transporter capacity. But this consideration increases the complexity of the problem. We study the constraint under a Poisson demand process and allow for a fixed ordering cost. In presence of the constraint, we establish the optimality of an (s,S) policy under both the discounted and average cost objectives. Under the average cost objective, we show the uniqueness of the order-up-to level S. We numerically compare our model with the classical unconstrained model. We report significant savings in costs that can be achieved by using our model when the order time is constrained.

In many real-life contexts, inventory levels are only incompletely observed due to non-observation of demand, discrepancies in transmitting sales data, transaction errors, spoilage, misplacement, or theft of inventory. We study a periodic review inventory system, where the demand is not observed and the unmet demand is backordered. As a result, the inventory manager cannot tell the exact quantities of inventories or backorders. However, by looking at the shelf, he knows whether the inventory is positive or nonpositive. Only with this information, the inventory manager must determine the order quantity in each period that would minimize the expected total discounted cost over an infinite-horizon. The dynamic programming formulation of this problem has an infinite-dimensional state space. We use the concept of unnormalized probability to establish the existence of an optimal feedback policy and the uniqueness of the solution of the dynamic programming equation when the periodic cost has linear growth.

This paper introduces two-dimensional (weight and volume) overbooking problems arising mainly in the cargo revenue management, and compares them with one-dimensional problems. It considers capacity spoilage and cargo offloading costs, and minimizes their sum. For one-dimensional problems, it shows that the optimal overbooking limit does not change with the magnitude of the booking requests. In two-dimensional problems, the overbooking limit is replaced by a curve. The curve, along with the volume and weight axes, encircles the acceptance region. The booking requests are accepted if they fall within this region. We present Curve (Cab) and Rectangle (Rab) models. The boundary of the acceptance region in the Cab (resp. Rab) model is a curve (resp. rectangle). The optimal curve for the Cab model is shown to be unique and continuous. Moreover, it can be obtained by solving a series of simple equations. Finding the optimal rectangle for the Rab model is more challenging, so we propose an approximate rectangle. The approximate rectangle is a limiting solution in the sense that it converges to the optimal rectangle as the booking requests increase. The approximate rectangle is numerically shown to yield costs that are very close to the optimal costs.

This paper revisits the finite-horizon model of a censored newsvendor by Ding et al. [Ding, X., M. L. Puterman, A. Bisi. 2002. The censored newsvendor and the optimal acquisition of information. Oper. Res. 50 517–527]. An important result claimed there without a proper proof is that the myopic order quantity is always less than or equal to the optimal order quantity. Lu et al. [Lu, X., J. S. Song, K. Zhu. 2008. Analysis of perishable inventory systems with censored demand data. Oper. Res. 56(4) 1034–1038.] supplied a correct proof of the result. We analyze the same model using the interesting concept of the unnormalized probability, which simplifies the dynamic programming equation considerably and facilitates the proof of the claim. Moreover, it produces the proof of the existence of an optimal solution for an infinite-horizon setting of the problem.

This paper analyzes intrusion detection decisions in the presence of multiple alarm types, which differ in occurrence probabilities, damage and investigation costs. Specifically, multi-period optimization models are used to study three critical decisions associated with intrusion detection: (i) Allocation of the investigation budget to different periods and to different alarm types; (ii) Configuration of an intrusion detection system (IDS), i.e. choosing a false alarm rate for a given IDS; and (iii) Allocation of an appropriate amount of the investigation budget in the presence of alternative investment opportunities. Three models that cascade onto each other are presented. We minimize the sum of security costs including damages, due to ignored alarms, the investigation cost and the undetected intrusion cost. We show that it can be optimal to ignore non-critical alarms in order to allocate more of the investigation budget to critical alarms that may occur in the future. We establish that the security costs decrease as the investigation budget increases. Our last model deals with security investments—in the form of an investigation budget. The investigation budget must be increased until the rate of increase in savings in security costs due to the additional budget are equal to the internal rate of return of an organization. These analyses are done with explicit (derived) cost functions, as opposed to implicit (assumed) cost functions. We conclude by providing additional managerial insights and numerical examples.

Information delays exist when the most recent inventory information available to the Inventory Manager (IM) is dated. Such situations arise when it takes a while to process the demand data, count the inventory, and pass the results to the IM. We show that the optimal total inventory-related cost decreases when the length of the information delay decreases. The amount of the decrease is an important datum for an IM interested in considering whether or not to invest in reducing the delay. The investment is required to finance design and acquisition of an information (collection and dissemination) system that can reduce the information delay. Such systems include phone calls, business meetings, and the use of information collection mechanisms such as radiofrequency identification tags.

The newsvendor problem is relatively easy to solve when the distribution of demand for newspapers is known. When the demand is unknown, the newsvendor faces a dual problem in the sense of Feldbaum (1960): to choose a decision variable that maximizes profit in the present period, and choose a large enough "observation window" to be able to view the process correctly so that consistent parameter estimation can occur. This is a difficult problem in general. In this paper, we treat a special case when the newsvendor faces exponential demand, independently and identically distributed with unknown mean.

This paper revisits the model of the censored newsvendor presented by Ding, Puterman and Bisi [8], We analyze that model in an infinite-horizon context by using the interesting concept of unnormalized probabilities. The unnormalized probabilities considerably simplify the dynamic programming equation and facilitate the proof of the existence of an optimal policy. They can also be used to give a simple, alternative proof to Ding et al.'s claim that the myopic order quantity is always less than or equal to the optimal order quantity. Importantly, the concept of unnormalized probabilities can be used to treat other important operations research problems with partial observations.

Price variability is one of the major causes of the bullwhip effect. This paper analyzes the impact of procurement price variability in the upstream of a supply chain on the downstream retail prices. Procurement prices may fluctuate over time, for example, when the supply chain players deploy auction type procurement mechanisms, or if the prices are dictated in market exchanges. A game theory framework is used here to model a serial supply chain. Sequential price game scenarios are investigated to show that there is an increase in retail price variability and an amplified reverse bullwhip effect on prices (RBP) under certain demand conditions.

In many inventory control contexts, inventory levels are only partially (i.e., not fully) observed. This may be due to nonobservation of demand, spoilage, misplacement, or theft of inventory. We study a discrete-time periodic-review inventory system where the unmet demand is backordered. When the inventory level is nonnegative, the inventory manager does not know the exact inventory level. Otherwise, inventory shortages occur, and the inventory manager issues rain checks to customers. The shortages are fully observed via the rain checks. The inventory manager determines the order quantity based on the partial information on the inventory level. The objective is to minimize the expected total discounted cost over an infinite horizon. The dynamic programming formulation of this problem has an infinite dimensional state space. We use the methodology of the unnormalized probability to establish the existence of an optimal feedback policy when the periodic cost has linear growth. Moreover, uniqueness and continuity of the solution to dynamic programming equations are proved when the discount factor is sufficiently small.

This paper considers two important issues related to security risk management. First, the presence of network externalities in security risks. Second, the distinction of general (network) and system-specific protection measures. We found the optimal allocation of security resources (investments) in protecting every system in an organization. The results show that the consideration of network externalities and layered protection changes the risk mitigation decisions significantly. In addition, accurate estimation of system risk plays a critical role in the success of risk management. Otherwise, the use of a uniform baseline protection approach may be more desirable when the misjudgment of relative system risks is likely to occur.

We consider a newsvendor problem with partially observed Markovian demand. Demand is observed if it is less than the inventory. Otherwise, only the event that it is larger than or equal to the inventory is observed. These observations are used to update the demand distribution from one period to the next. The state of the resulting dynamic programming equation is the current demand distribution, which is generally infinite dimensional. We use unnormalized probabilities to convert the nonlinear state transition equation to a linear one. This helps in proving the existence of an optimal feedback ordering policy. So as to learn more about the demand, the optimal order is set to exceed the myopic optimal order. The optimal cost decreases as the demand distribution decreases in the hazard rate order. In a special case with finitely many demand values, we characterize a near-optimal solution by establishing that the value function is piecewise linear.

Information delays exist when the most recent inventory information available to the inventory manager (IM) is dated; namely, the IM observes only the inventory level of an earlier period. We introduce information delays into the standard multiperiod stochastic inventory problem with backorders. We consider two types of information delays: (i) a constant delay and (ii) a random delay. We define an appropriate reference inventory position, which is a sufficient statistics for finding the optimal order quantity. We show that the optimal ordering policy is of base-stock type with respect to the reference inventory position and is of (s, S) type if there is also a fixed cost of ordering.

Capacity planning decisions affect a significant portion of future revenue. In equipment intensive industries, these decisions usually need to be made in the presence of both highly volatile demand and long capacity installation lead times. For a multiple product case, we present a continuous-time capacity planning model that addresses problems of realistic size and complexity found in current practice. Each product requires specific operations that can be performed by one or more tool groups. We consider a number of capacity allocation policies. We allow tool retirements in addition to purchases because the stochastic demand forecast for each product can be decreasing. We present a cluster-based heuristic algorithm that can incorporate both variance reduction techniques from the simulation literature and the principles of a generalized maximum flow algorithm from the network optimization literature. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2006

An automotive parts manufacturer produces a wide variety of parts in a job shop environment. Many of the manufacturing operations have substantial setups. When a client phones in an order, the manufacturer must decide quickly whether or not it has the capacity required to accept the order. We develop a simplified formulation of the order acceptance problem. We formulate the discrete-time version as an integer program. The problem is NP-hard, but in 51 out of 51 test problems the LP relaxation is tight. For larger problems we test several heuristics. Three of the heuristics look promising: simulated annealing, a genetic algorithm, and a linear-programming-based heuristic.

This Note introduces recent developments in the analysis of inventory systems with partial observations. The states of these systems are typically conditional distributions, which evolve in infinite dimensional spaces over time. Our analysis involves introducing unnormalized probabilities to transform nonlinear state transition equations to linear ones. With the linear equations, the existence of the optimal feedback policies are proved for two models where demand and inventory are partially observed. In a third model where the current inventory is not observed but a past inventory level is fully observed, a sufficient statistic is provided to serve as a state. The last model serves as an example where a partially observed model has a finite dimensional state. In that model, we also establish the optimality of the basestock policies, hence generalizing the corresponding classical models with full information. To cite this article: A. Bensoussan et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).

This paper studies capacity expansions for a production facility that faces uncertain customer demand for a single product family. The capacity of the facility is modeled in three tiers, as follows. The first tier consists of a set of upper bounds on production that correspond to different resource types (e.g., machine types, categories of manpower, etc.). These upper bounds are augmented in increments of fixed size (e.g., by purchasing machines of standard types). There is a second-tier resource that constrains the first-tier bounds (e.g., clean room floor space). The third-tier resource bounds the availability of the second-tier resource (e.g., the total floor space enclosed by the building, land, etc.). The second and third-tier resources are expanded at various times in various amounts. The cost of capacity expansion at each tier has both fixed and proportional elements. The lost sales cost is used as a measure for the level of customer service. The paper presents a polynomial time algorithm (FIFEX) to minimize the total cost by computing optimal expansion times and amounts for all three types of capacity jointly. It accommodates positive lead times for each type. Demand is assumed to be nondecreasing in a “weak” sense. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2004.

We consider a discrete-time capacity expansion problem involving multiple product families, multiple machine types, and non-stationary stochastic demand. Capacity expansion decisions are made to strike an optimal balance between investment costs and lost sales costs. Motivated by current practices in the semiconductor and other high-tech industries, we assume that only minimal amounts of finished-goods inventories are held, due to the risk of obsolescence. We assume that when capacity is in short supply, management desires to ensure that a minimal service level for all product families is obtained. Our approach uses a novel assumption that demand can be approximated by a distribution whose support is a collection of rays emanating from a point and contained in real multi-dimensional space. These assumptions allow us to solve the problem as a max-flow, min-cut problem. Computational experiments show that large problems can be solved efficiently.

We perform four experiments to evaluate several features of capacity planning approaches used in practice and make concrete suggestions for practitioners. We use an optimal capacity planning technique (FIFEX) with industrial demand and capacity data for realistic conclusions. The first experiment studies the cost effects of allowing for capacity expansion only at special times such as at the beginning of each month or each quarter. We find that these effects are not great if there is at least one capacity expansion opportunity about every six months. The second experiment determines the cost implications of capacity planning heuristics. We demonstrate that FIFEX delivers solutions which cost 5-10% less than heuristic solutions. The third experiment examines the sensitivity of costs against the frequency of forecasting and planning. Our experiments indicate that costs can be decreased by 2%-7% by doubling forecasting and planning frequency. We suggest that practitioners forecast and plan at least once every quarter. The final experiment compares joint optimization of tool and floor-shell expansions with sequential optimization. 3%-9% of costs can be saved by using the joint optimization technique FIFEX instead of sequential optimization. We note that although cost savings are relatively small in percentages, they correspond to tens of million dollars.

In the semiconductor industry, many critical decisions are based on demand forecasts. However, these forecasts are subject to random error. In this paper, we lay out a scheme estimating the variance and correlation of forecast errors (without altering given forecasts) and modeling the evolution of forecasts over time. Our scheme allows correlations across time, products and technologies. It also addresses the case of nonstationary errors due to ramps (technology migrations). It can be used to simulate chip demands for production planning/capacity expansion studies.

The processing time of large orders is, in many industries, longer than that of small orders. This renders supply lead times in such settings to be increasing in the order size. Yet that pattern is not re#ected in existing inventory control models, especially those allowing for random lead times. This work aims at rectifying the situation. Our setting is an order-quantity/reorder-point model with backordering, where the shortage penalty is incurred per unit per unit time. The processing time of each unit is random; the processing time of a lot is correlated with its size. For the case where lead time is proportional to the lot size, we obtain a closed-form solution. That is, unlike the classical (Q, r) model (where lead time is independent of lot size), no iterations are required here. We also analyze a case where the processing time exhibits economies of scale in the lot size. Finally, we consider a situation where a customer can secure shorter processing times by reserving capacity at the supplier’s manufacturing facility. ( 2000 Elsevier Science B.V. All rights reserved.

This paper considers inventory models of the order-quantity/order-point type, or ( Q , r ) models. In general, the control parameters ( Q and r ) depend on both the demand process and the replenishment lead time. Although many studies have treated lead time as constant, focusing solely on demand, a ( Q , r ) model with stochastic lead time could be a building block in Supply Chain Management. Variability in lead time between successive stages is often what disturbs supply chain coordination. In a two-stage system with a constant demand rate, we will concentrate on lead time as a random variable, and develop two probabilistic models. In the first, lead time T is exogenous. Lead time is made endogenous in the second stochastic model through an “expediting factor” τ , the constant of proportionality between random variables T (the expedited lead time) and T (ordinary lead time): T =τT . For expedited orders ( τ τ >1. The second model thus has three decision variables ( Q , r , τ ). For each model, we show that the expected cost per unit time is jointly convex in the decision variables and obtain the global minimizer. Numerical examples are given. Sensitivity analyses are conducted with respect to the cost parameters, and suggestions are made for future research

Willingness To Pay (WTP) of customers plays an anchoring role in pricing. This paper proposes a new choice model based on WTP, incorporating sequential decision making, where the products with positive utility of purchase are considered in the order of customer preference. We compare WTP-choice model with the commonly used (multinomial) Logit model with respect to the underlying choice process, information requirement, independence of irrelevant alternatives. Using WTP-choice model, we compare equilibrium prices, demands and profits of several contexts: without considering inventory and with stockouts - lost sales and backorders for "retailer favoring" customers and for "availability favoring" customers. One of the interesting results with WTP-choice model is the "loose coupling" of retailers; prices are not coupled but profits are. That is, each retailer should charge the monopoly price as the collection of these prices constitute an equilibrium but each retailer's profit depends on other retailers' prices. Loose coupling fails with dependence of WTPs, dependence of preference on prices or batch pricing for heterogeneous customers. We show that competition among retailers with dependent WTPs can cause price cycles under some conditions. We consider real-life data on sales of yogurt, ketchup, candy melt, and tuna, and check if a version of WTP-choice model (with uniform, triangle or shifted exponential WTP distribution), standard or mixed Logit model fits better and predicts the sales better. These empirical tests establish that WTP-choice model compares well and should be considered as a legitimate alternative to Logit models for studying competitive pricing for products with low price and high frequency of purchase.

Sunil Chopra and Peter Meindl have recently updated their Supply Chain Management book (2. edition published by Prentice Hall). The new edition is more comprehensive than the previous edition; it includes basically two new chapters on Sourcing and Pricing and Revenue Management. Discussion of design options for a distribution network of Chapter 4 is also new material. The rest of the new edition is very similar to the first edition. This similarity between editions hopefully justifies me as a legitimate reviewer because I have used only the first edition in three sections of an elective 3-credit course at a business school. Practically, the class populations were split equally between MBA students and MS students. The students generally had above-average quantitative background with regard to ordinary business school students. My perspective on the book is partly shaped by my students’ reactions. I believe that this perspective is a healthy one as a “good book” should teach well to a novice reader. Below, I go over the book in detail and discuss what readers can and cannot find in the book. At times, I take the liberty of suggesting additions/deletions of some topics or reordering some topics. The new edition, like the old one, examines supply chains in six parts. The first part deals with Building a Strategic Framework to Analyze Supply Chains. This indeed is a good start to give a top-down view to readers. It starts a smooth transition from the qualitative discussions of an average business school course to more quantitative discussions of a supply chain course. Moreover, the students warm up with the subject matter while the three chapters of Part I are discussed. In Part I, the concept of a strategic fit between customer wishes and supply chain capabilities is very important. On the other hand, one can quickly skim over the sections of Process view of supply chain and Expanding strategic scope. The second part is mainly on logistics: location and transportation analysis. This part used to come after inventory chapters in the first edition. Since outcomes of infrequent decisions typically become inputs for frequent decisions, infrequent decisions should be discussed first, e.g. location decisions should come before inventory decisions. Ordering decisions (effectively chapters) according to frequency eliminates potential confusion as to how decisions are related and what is their relative chronology. Hence, I applaud authors’ decision to discuss logistics earlier. Location chapter includes interesting examples of inventory and transportation cost trade offs. These examples deserve a careful reading because they emphasize the concept of integration (system-view) which is a keystone for supply chain management. On the other hand, location (Chapter 5) and transportation (Chapter 14) could have remained next to each other. The relegation of transportation may be due to its inclusion of routing type combinatorial topics that may not be appropriate/interesting for some readers. Those readers may skip parts or all of the transportation chapter but the others can read the transportation immediately after the location. Speaking of the placement of chapters, I agree with the authors that network design decisions under uncertainty belong to Part II. This discussion was provided under the Financial evaluation chapter in the first edition. Authors use the methodology of decision trees to solve two examples here. With these examples, instructors may also illustrate how simulation can be used instead of decision trees. This is now made possible by an appendix in the new edition: Simulation using spreadsheets. If this appendix is pulled into Part II which seems possible with little effort, readers can be presented with a comparative discussion of decision trees and simulation. Such a comparison adds to the value of the book. It

This paper studies spare part transshipments between two service part facilities whose demands are correlated. Transshipments are used to reduce severity of part stock outs. Facilities are run by an inventory manager (IM) who minimizes replenishment, transshipment, and inventory costs. We show that the optimal transshipment policy is an inventory hold-back type; if the part inventory at a facility is less than or equal to its hold-back level, a transshipment request made for that part by a stocked out retailer is rejected. The hold-back levels increase toward the next replenishment of partsThis implies that transshipment requests are initially accepted until a critical time and afterwards they are rejected. A heuristic is designed using this critical time as the single decision variable. It performs within 0.7-1.8% of the optimal cost. Heuristic policies of no inventory sharing and complete sharing, respectively, perform within 3% and 2% of the optimal cost. Since the computation of hold-back levels and implementation of the optimal transshipments, respectively, require limited resources and little IM oversight, we advocate for the use of the optimal transshipment policy.

We develop a general methodology for a partially observed stochastic control problem. The dynamics is governed by a discrete-time Markov process. We describe an application to an inventory system with possibility of shrinkage, and introduce unnormalized conditional probabilities to transform the nonlinear state evolution into a linear one. We then prove the existence and uniqueness of the solution for the Bellman equation between the floor and the ceiling functions in the case of unbounded costs.

Inventory inaccuracy is common at retailers. At many retailers, a cash register records incoming orders and outgoing sales, but not the demand or the shrinkage. The shrinkage refers to spoilage or pilferage of inventory. The demand differs from the sales in the periodic-review lost-sales inventory model presented in this paper. When the inventory exceeds the demand, the remaining inventory is subject to a shrinkage unobserved by the inventory manager. Otherwise, the unmet demand is lost and unobserved. Our objective is to minimize the expected discounted cost over an infinite horizon. We use dynamic programming along with the concept of unnormalized probability, and prove the existence of an optimal feedback policy and that the value function is the solution of the dynamic programming equation. We obtain a theoretical lower bound for the cost via the formulation of a fictitious inventory problem. For discrete-valued inventories, we provide the analogous formulation, develop an iterative algorithm, and compare its solution to a myopic solution and a lower bound. We identify many cases of parameter values for which the solution of the iterative algorithm performs significantly better than the myopic solution. Moreover, the achieved cost is close to the lower bound.

This paper proposes a new methodology to solve partially observed inventory problems. Generally, these problems have infinitedimensional states that are conditional distribution of the inventory level. Our methodology involves linearizing the state transitions via unnormalized probabilities. It then uses an appropriate functional basis to represent the state. Considering the speed and stability of computations, we choose truncated Chebyshev polynomials as the basis. We use Fast Fourier Transforms along with an appropriate discretization of inventory levels to speed up the computations. These main ideas are to obtain an iterative algorithm to solve a partially observed inventory model with rain checks. In this model, the inventory manager (IM) does not know the inventory level when it is positive. Otherwise, the IM fully observes it. This model provides a context to illustrate our methodology, which applies to other such models. Although this model has been studied mathematically in the literature, the use of our algorithm provides a numerical approximation of the optimal order quantities.These are compared to the orders released under a base mean-stock policy, where the IM replaces the unobserved inventory level with its mean and applies the well-known base stock policy. We show numerically that the optimal order quantity is very close to the base mean-stock order quantity, when the variance of the inventory distribution is small. When the mean of the inventory distribution is large, the optimal order quantity is more than the base mean-stock quantity, and it is the other way around when the mean is small or negative. These insights are explained via uncertainty and information effects and their interplay. We expect this interplay to show up in other partially observed inventory models.

Frequent product introductions emphasize the importance of product rollover strategies. With single rollover, when a new product is introduced, the old product is phased out from the market. With dual rollover, the old product remains in the market along with the new product. Anticipating the introduction of the new product and the potential markdown of the old product, strategic customers may delay their purchases. We study the interaction between product rollover strategies and strategic customer purchasing behavior and find that single rollover is more valuable when the new product's innovation is low and the number of strategic customers is high. Interestingly and counter to intuition, the firm may have to charge a lower price for the old product as well as receive a lower profit with a higher value disposal (outside) option for the old product under single rollover. Facing a market composed of both strategic and myopic customers, the firm does not necessarily reduce the stocking level as more myopic customers become strategic. This paper was accepted by Yossi Aviv, operations management.

This paper studies spare part transshipments between two service part facilities whose demands are correlated. Transshipments are used to reduce severity of part stock outs. Facilities are run by an inventory manager (IM) who minimizes replenishment, transshipment, and inventory costs. We show that the optimal transshipment policy is an inventory hold-back type; if the part inventory at a facility is less than or equal to its hold-back level, a transshipment request made for that part by a stocked out retailer is rejected. The hold-back levels increase toward the next replenishment of partsThis implies that transshipment requests are initially accepted until a critical time and afterwards they are rejected. A heuristic is designed using this critical time as the single decision variable. It performs within 0.7-1.8% of the optimal cost. Heuristic policies of no inventory sharing and complete sharing, respectively, perform within 3% and 2% of the optimal cost. Since the computation of hold-back levels and implementation of the optimal transshipments, respectively, require limited resources and little IM oversight, we advocate for the use of the optimal transshipment policy.

Under many circumstances, demand observations are often censored due to the lack of tracking lost sales caused by stockouts. To understand the impact of the lost sales information on the ordering decisions, a periodic-review inventory model is formulated in which only the sales information is obtained immediately upon the realization of the demand. This is equivalent to observing the demand when the sales are less than the available stock and to inferring that the demand is higher than the stock when there is a stockout. Subsequently, the lost sales information is obtained after a delay. In the resulting model, an optimal policy, if exists, reveals a very complex structure. By decomposing the derivative of the value function, we demonstrate two different roles of inventory in our model: satisfying the demand and extracting the demand information. We show that the optimal inventory levels under the delayed observation of the lost sales are always higher than those for which the demands are fully observed. Moreover, as illustrated in numerical examples, the optimal policy possesses a counterintuitive behavior with respect to the problem parameters. To understand the key drivers of the optimal decisions, we further compare the costs under different demand observations. Two important observations are made. First, a lower cost is obtained when the realized demand is observed than when the demand is only observed to be higher than the inventory level, and, furthermore, the cost difference represents the value of demand information. Second, while a higher inventory level induces a more accurate demand forecast, the value of exact demand observation is not monotone in the procurement cost. Consequently, the optimal ordering quantity is not always decreasing in the procurement cost.

In many inventory control contexts, inventory levels are only partially (i.e., not fully) observed. This may be due to non-observation of demand, spoilage, misplacement, or theft of inventory. We study a periodic review inventory system where the unmet demand is backordered. When inventory level is nonnegative, the inventory manager does not know the exact inventory level. Otherwise, inventory shortages occur and the inventory manager issues rain checks to customers. The shortages are fully observable via the rain checks. The inventory manager determines the order quantity based on the partial information on the inventory level. The objective is to minimize the expected total discounted cost over an infinite horizon. The dynamic programming formulation of this problem has an infinite dimensional state space. We use the methodology of the unnormalized probability to establish the existence of an optimal feedback policy when the periodic cost has linear growth. Moreover, uniqueness and continuity of the solution to dynamic programming equations are proved when the discount factor is sufficiently small.

We develop a general filtering framework for the problem of estimating the state of a system whose dynamics is governed by a discrete-time Markov process. We describe applications to inventory control systems with partial observations. We introduce conditional distributions and unnormalized conditional probabilities to transform nonlinear transition equations into linear ones. Moreover, this transformation greatly facilitates our study of the stochastic optimal control problems governed by nonlinear transition equations.

This paper introduces recent developments in the analysis of inventory systems with partial observations. The states of these systems are typically conditional distributions, which evolve in infinite dimensional spaces over time. Our analysis involves introducing unnormalized probabilities to transform nonlinear state transition equations to linear ones. With the linear equations, the existence of the optimal feedback policies are proved for two models where demand and inventory are partially observed. In a third model where the current inventory is not observed but a past inventory level is fully observed, a sufficient statistic is provided to serve as a state. The last model serves as an example where a partially observed model has a finite dimensional state. In that model, we also establish the optimality of the basestock policies, hence generalizing the corresponding classical models with full information.