Comments on “Convergence Properties of the Likelihood of Computed Dynamic Models”

We show by counterexample that Proposition 2 in Fernández‐Villaverde, Rubio‐Ramírez, and Santos (Econometrica (2006), 74, 93–119) is false. We also show that even if their Proposition 2 were corrected, it would be irrelevant for parameter estimates. As a more constructive contribution, we consider the effects of approximation error on parameter estimation, and conclude that second order approximation errors in the policy function have at most second order effects on parameter estimates.     

Optimal Inventory Control with Retail Pre‐Packs

A pre‐pack is a collection of items used in retail distribution. By grouping multiple units of one or more stock keeping units (SKU), distribution and handling costs can be reduced; however, ordering flexibility at the retail outlet is limited. This paper studies an inventory system at a retail level where both pre‐packs and individual items (at additional handling cost) can be ordered. For a single‐SKU, single‐period problem, we show that the optimal policy is to order into a “band” with as few individual units as possible. For the multi‐period problem with modular demand, the band policy is still optimal, and the steady‐state distribution of the target inventory position possesses a semi‐uniform structure, which greatly facilitates the computation of optimal policies and approximations under general demand. For the multi‐SKU case, the optimal policy has a generalized band structure. Our numerical results show that pre‐pack use is beneficial when facing stable and complementary demands, and substantial handling savings at the distribution center. The cost premium of using simple policies, such as strict base‐stock and batch‐ordering (pre‐packs only), can be substantial for medium parameter ranges.     

Mitigating Inventory Overstocking: Optimal Order‐up‐to Level to Achieve a Target Fill Rate over a Finite Horizon

Service level agreements (SLAs) are widely adopted performance‐based contracts in operations management practice, and fill rate is the most common performance metric among all the measurements in SLAs. Traditional procedures characterizing the order‐up‐to level satisfying a specified fill rate implicitly assume an infinite performance review horizon. However, in practice, inventory managers are liable to maintain and report fill rates over a finite performance review horizon. This horizon discrepancy leads to deviation between the target fill rate and actual achieved fill rate. In this study, we first examine the behavior of the fill rate distribution over a finite horizon with positive lead time. We analytically prove that the expected fill rate assuming an infinite performance review horizon exceeds the expected fill rate assuming a finite performance review horizon, implying that there exists some inventory “waste” (i.e., overstocking) when the traditional procedure is used. Based on this observation and the complexity of the problem, we propose a simulation‐based algorithm to reduce excess inventory while maintaining the contractual target fill rate. When the lead time is significant relative to the length of the contract horizon, we show that the improvement in the inventory system can be over 5%. Further, we extend our basic setting to incorporate the penalty for failing to meet a target, and show how one can solve large‐scale problems via stochastic approximation. The primary managerial implication of our study is that ignoring the performance review horizon in an SLA will cause overstocking, especially when the lead time is large.     

Lowest Cost May Not Lower Total Cost: Using “Spackling” to Smooth Mass‐Customized Production

Consider a manufacturer who mass customizes variants of a product in make‐to‐order fashion, and also produces standard variants as make‐to‐stock. A traditional manufacturing strategy would be to employ two separate manufacturing facilities: a flexible plant for mass‐customized items and an efficient plant for standard items. We contrast this traditional focus strategy with an alternative that better utilizes capacity by combining production of mass‐customized and standard items in one of two alternate spackling strategies: (1) a pure‐spackling strategy, where the manufacturer produces everything in a (single) flexible plant, first manufacturing custom products as demanded each period, and then filling in the production schedule with make‐to‐stock output of standard products; or (2) a layered‐spackling strategy, which uses an efficient plant to make a portion of its standard items and a separate flexible plant where it spackles. We identify the optimal production strategy considering the tradeoff between the cost premium for flexible (versus efficient) production capacity and the opportunity costs of idle capacity. Spackling amortizes fixed costs of capacity more effectively and thus can increase profits from mass customization vis‐à‐vis a focus strategy, even with higher cost production for the standard goods. We illustrate our framework with data from a messenger bag manufacturer.     

Some Limitations of “Risk = Threat × Vulnerability × Consequence” for Risk Analysis of Terrorist Attacks

Several important risk analysis methods now used in setting priorities for protecting U.S. infrastructures against terrorist attacks are based on the formula: Risk=Threat×Vulnerability×Consequence. This article identifies potential limitations in such methods that can undermine their ability to guide resource allocations to effectively optimize risk reductions. After considering specific examples for the Risk Analysis and Management for Critical Asset Protection (RAMCAP?) framework used by the Department of Homeland Security, we address more fundamental limitations of the product formula. These include its failure to adjust for correlations among its components, nonadditivity of risks estimated using the formula, inability to use risk‐scoring results to optimally allocate defensive resources, and intrinsic subjectivity and ambiguity of Threat, Vulnerability, and Consequence numbers. Trying to directly assess probabilities for the actions of intelligent antagonists instead of modeling how they adaptively pursue their goals in light of available information and experience can produce ambiguous or mistaken risk estimates. Recent work demonstrates that two‐level (or few‐level) hierarchical optimization models can provide a useful alternative to Risk=Threat×Vulnerability×Consequence scoring rules, and also to probabilistic risk assessment (PRA) techniques that ignore rational planning and adaptation. In such two‐level optimization models, defender predicts attacker's best response to defender's own actions, and then chooses his or her own actions taking into account these best responses. Such models appear valuable as practical approaches to antiterrorism risk analysis.     

Cross‐Training with Imperfect Training Schemes

Cross‐training workers is one of the most efficient ways of achieving flexibility in manufacturing and service systems for increasing responsiveness to demand variability. However, it is generally the case that cross‐trained employees are not as productive on a specific task as employees who were originally trained for that task. Also, the productivity of the cross‐trained workers depends on when they are cross‐trained. In this work, we consider a two‐stage model to analyze the effects of variations in productivity levels on cross‐training policies. We define a new metric called achievable capacity and show that it plays a key role in determining the structure of the problem. If cross‐training can be done in a consistent manner, the achievable capacity is not affected when the training is done, which implies that the cross‐training decisions are independent of the opportunity cost of lost demand and are based on a trade‐off between cross‐training costs at different times. When the productivities of workers trained at different times differ, there is a three‐way trade‐off between cross‐training costs at different times and the opportunity cost of lost demand due to lost achievable capacity. We analyze the effects of variability and show that if the productivity levels of workers trained at different times are consistent, the decision maker is inclined to defer the cross‐training decisions as the variability of demand or productivity levels increases. However, when the productivities of workers trained at different times differ, an increase in the variability may make investing more in cross‐training earlier more preferable.     

A Note on “Discrete Sequential Search with Group Activities”

This note comments on a paper published by Wagner and Davis [Decision Sciences (2001), 32(4), 557–573]. These authors present an integer‐programming model for the single‐item discrete sequential search problem with group activities. Based on their experiments, they conjecture that the problem can be solved as a linear program. In this note, we provide a counterexample for which the optimal value of the linear program they propose is different from the optimal value of the integer‐programming model, hence contradicting their conjecture for the specific linear program that they specify. To the best of our knowledge, the conjecture settled in this note was still an open question.     

Joint Initial Stocking and Transshipment—Asymptotics and Bounds

A common problem faced by many firms in their supply chains can be abstracted as follows. Periodically, or at the beginning of some selling season, the firm needs to distribute finished goods to a set of stocking locations, which, in turn, supply customer demands. Over the selling season, if and when there is a supply‐demand mismatch somewhere, a re‐distribution or transshipment will be needed. Hence, there are two decisions involved: the one‐time stocking decision at the beginning of the season and the supply/transshipment decision throughout the season. Applying a stochastic dynamic programming formulation to a two‐location model with compound Poisson demand processes, we identify the optimal supply/transshipment policy and show that the optimal initial stocking quantities can be obtained via maximizing a concave function whereas the contribution of transshipment is of order square‐root‐of T. Hence, in the context of high‐volume, fast‐moving products, the initial stocking quantity decision is a much more important contributor to the overall profit. The bounds also lead to a heuristic policy, which exhibits excellent performance in our numerical study; and we further prove both the bounds and the heuristic policy are asymptotically optimal when T approaches infinity. Extension to multiple locations is also discussed.     

“These Power Lines Make Me Ill”: A Typology of Residents’ Health Responses to a New High‐Voltage Power Line

Little attention has been devoted to the potential diversity in residents’ health responses when exposed to an uncertain environmental health risk. The present study explores whether subgroups of residents respond differently to a new high‐voltage power line (HVPL) being put into operation. We used a quasi‐experimental prospective field study design with two pretests during the construction of a new HVPL, and two posttests after it was put into operation. Residents living nearby (0–300 m, n = 229) filled out questionnaires about their health and their perception of the environment. We applied latent class growth models to investigate heterogeneity in the belief that health complaints were caused by a power line. Classes were compared on a wide range of variables relating to negative‐oriented personality traits, perceived physical and mental health, and perceptions of the environment. We identified five distinct classes of residents, of which the largest (49%) could be described as emotionally stable and healthy with weak responses to the introduction of a new power line. A considerable minority (9%) responded more strongly to the new line being activated. Residents in this class had heard more about the health effects of power lines beforehand, were more aware of the activation of the new line, and reported a decrease in perceived health afterwards. Based on our findings we can conclude that there is a considerable heterogeneity in health responses to a new HVPL. Health risk perceptions appear to play an important role in this typology, which has implications for risk management.     

Staffing Call Centers with Uncertain Arrival Rates and Co‐sourcing

In a call center, staffing decisions must be made before the call arrival rate is known with certainty. Once the arrival rate becomes known, the call center may be over‐staffed, in which case staff are being paid to be idle, or under‐staffed, in which case many callers hang‐up in the face of long wait times. Firms that have chosen to keep their call center operations in‐house can mitigate this problem by co‐sourcing; that is, by sometimes outsourcing calls. Then, the required staffing N depends on how the firm chooses which calls to outsource in real time, after the arrival rate realizes and the call center operates as a M/M/N + M queue with an outsourcing option. Our objective is to find a joint policy for staffing and call outsourcing that minimizes the long‐run average cost of this two‐stage stochastic program when there is a linear staffing cost per unit time and linear costs associated with abandonments and outsourcing. We propose a policy that uses a square‐root safety staffing rule, and outsources calls in accordance with a threshold rule that characterizes when the system is “too crowded.” Analytically, we establish that our proposed policy is asymptotically optimal, as the mean arrival rate becomes large, when the level of uncertainty in the arrival rate is of the same order as the inherent system fluctuations in the number of waiting customers for a known arrival rate. Through an extensive numerical study, we establish that our policy is extremely robust. In particular, our policy performs remarkably well over a wide range of parameters, and far beyond where it is proved to be asymptotically optimal.     

Optimal Inventory Control with Dual‐Sourcing,Heterogeneous Ordering Costs and Order Size Constraints

We consider a dual‐sourcing inventory system, where procuring from one supplier involves a high variable cost but negligible fixed cost whereas procuring from the other supplier involves a low variable cost but high fixed cost, as well as an order size constraint. We show that the problem can be reduced to an equivalent single‐sourcing problem. However, the corresponding ordering cost is neither concave nor convex. Using the notion of quasi‐convexity, we partially characterize the structure of the optimal policy and show that it can be specified by multiple thresholds which determine when to order from each supplier and how much. In contrast to previous research, which does not consider order size constraints, we show that it is optimal to simultaneously source from both suppliers when the beginning inventory level is sufficiently low. We also show that the decision to source from the low‐cost supplier is not monotonic in the inventory level. Our results require that the variable costs satisfy a certain condition which guarantees quasi‐convexity. However, extensive numerical results suggest that our policy is almost always optimal when the condition is not satisfied. We also show how the results can be extended to systems with multiple capacitated suppliers.     

Vote‐Buying and Reciprocity

While vote‐buying is common, little is known about how politicians determine who to target. We argue that vote‐buying can be sustained by an internalized norm of reciprocity. Receiving money engenders feelings of obligation. Combining survey data on vote‐buying with an experiment‐based measure of reciprocity, we show that politicians target reciprocal individuals. Overall, our findings highlight the importance of social preferences in determining political behavior.     

Assemble‐to‐Order Inventory Management via Stochastic Programming: Chained BOMs and the M‐System

We study an inventory management mechanism that uses two stochastic programs (SPs), the customary one‐period assemble‐to‐order (ATO) model and its relaxation, to conceive control policies for dynamic ATO systems. We introduce a class of ATO systems, those that possess what we call a “chained BOM.” We prove that having a chained BOM is a sufficient condition for both SPs to be convex in the first‐stage decision variables. We show by examples the necessity of the condition. For ATO systems with a chained BOM, our result implies that the optimal integer solutions of the SPs can be found efficiently, and thus expedites the calculation of control parameters. The M system is a representative chained BOM system with two components and three products. We show that in this special case, the SPs can be solved as a one‐stage optimization problem. The allocation policy can also be reduced to simple, intuitive instructions, of which there are four distinct sets, one for each of four different parameter regions. We highlight the need for component reservation in one of these four regions. Our numerical studies demonstrate that achieving asymptotic optimality represents a significant advantage of the SP‐based approach over alternative approaches. Our numerical comparisons also show that outside of the asymptotic regime, the SP‐based approach has a commanding lead over the alternative policies. Our findings indicate that the SP‐based approach is a promising inventory management strategy that warrants further development for more general systems and practical implementations.     

Mass Customization and Guardrails: “You Can Not Be All Things to All People”

A mass customization strategy enables a firm to match its product designs to unique consumer tastes. In a classic horizontal product‐differentiation framework, a consumer's utility is a decreasing function of the distance between their ideal taste and the taste defined by the most closely aligned product the firm offers. A consumer thus considers the taste mismatch associated with their purchased product, but otherwise the positioning of the firm's product portfolio (or, “brand image”) is immaterial. In contrast, self‐congruency theory suggests that consumers assess how well both the purchased product and its overall brand image match with their ideal taste. Therefore, we incorporate within the consumer utility function both product‐specific and brand‐level components. Mass customization has the potential to improve taste alignment with regard to a specific purchased product, but at the risk of increasing brand dilution. Absent brand dilution concerns, a firm will optimally serve all consumers’ ideal tastes at a single price. In contrast, by endogenizing dilution costs within the consumer utility model, we prove that a mass‐customizing firm optimally uses differential pricing. Moreover, we show that the firm offers reduced prices to consumers with extreme tastes (to stimulate consumer “travel”), with a higher and fixed price being offered to those consumers having more central (mainstream) tastes. Given that a continuous spectrum of prices will likely not be practical in application, we also consider the more pragmatic approach of augmenting the uniformly priced mass customization range with preset (non‐customized) outlying designs, which serve customers at the taste extremes. We prove this practical approach performs close to optimal.     

Optimal Policies for a Two‐Product Inventory System under a Flexible Substitution Scheme

We study a two‐product inventory model that allows substitution. Both products can be used to supply demand over a selling season of N periods, with a one‐time replenishment opportunity at the beginning of the season. A substitution could be offered even when the demanded product is available. The substitution rule is flexible in the sense that the seller can choose whether or not to offer substitution and at what price or discount level, and the customer may or may not accept the offer, with the acceptance probability being a decreasing function of the substitution price. The decisions are the replenishment quantities at the beginning of the season, and the dynamic substitution‐pricing policy in each period of the season. Using a stochastic dynamic programming approach, we present a complete solution to the problem. Furthermore, we show that the objective function is concave and submodular in the inventory levels—structural properties that facilitate the solution procedure and help identify threshold policies for the optimal substitution/pricing decisions. Finally, with a state transformation, we also show that the objective function is ‐concave, which allows us to derive similar structural properties of the optimal policy for multiple‐season problems.     

A Reply to Williamson's “Outsourcing …”

A recent article by Oliver Williamson essentially comprises a critique of supply chain management (SCM) from the perspective of his own field, transaction‐cost economics. Here is one reader's response. SCM can indeed be faulted for inflated rhetoric, among other sins. I believe, however, that the two fields have much to learn from each other.     

Competition and Coordination in Two‐Tier Public Service Systems under Government Fiscal Policy

As a result of government budgetary limits and rapid market growth, many public service systems—such as health care—are characterized by extensive customer wait times that have become a serious problem. This problem might be solved by allowing private firms to enter these markets, which would provide customers with a choice between a free (governmental) public service provider (SP) and a fee‐charging (or “toll”) private SP. In such a two‐tier service system, the two SPs are differentiated by service quality and cost efficiency. This study focuses on the competition and coordination issues for two‐tier service systems with customers who are sensitive to both service quality and delay. The free system attempts to maximize its expected total customer utility with limited capacity, whereas the toll system attempts to maximize its profit. Neither goal is aligned with the social welfare goal of the public service. To achieve the social welfare goal, the government plays a crucial role in coordinating the two‐tier service system via the budget, the tradeoff of social members' goals, and tax‐subsidy policies. Using a mixed duopoly game, we establish Nash equilibrium strategies and identify the conditions for the existence of the two‐tier service system. We employ several interesting and counter‐intuitive managerial insights generated by the model to show that the public service can be delivered more efficiently via customer choice and SP competition. In addition, we show that a relatively low tax‐subsidy rate can almost perfectly coordinate the two SPs to achieve most of the maximum possible benefit of the two‐tier service system.     

The Effect of Earnings‐Based Metrics on Vertical Efficiency

The tendency to rely on accounting earnings as the primary metric of corporate performance has been subject to criticism in recent times. A key concern is that earnings misrepresent changes in value in that cash outlays occur upfront but expenses are recognized only over time. While recognized expenses indeed add up to the initial cash outflow, the equality holds only in undiscounted (nominal), not discounted (real), terms. Accordingly, corporate earnings figures suffer from a form of money illusion. In this paper, we demonstrate that such money illusion can have an upside when it is present in vertical relationships subject to self‐interest. In particular, a buyer who focuses on earnings has incentives to increase purchases since it does not immediately encounter the full cost of cash outflows. These added incentives can promote more efficient trade. We also show that the increased incentives to buy can also lead to Pareto improvements by spurring the supplier to invest more in developing technology. Finally, we demonstrate that judiciously chosen inventory valuation rules can lead to efficient supply chain outcomes. Thus, efficiency can be achieved when supply chain parties freely trade and regulators specify only the accounting rules under which they operate.     

Optimal Temporal Customer Purchasing Decisions Under Trade‐In Programs with Up‐Front Fees

To entice customers to purchase both current and new generation products over time, many firms offer different trade‐in programs including programs that require customers to pay an up‐front fee. To examine the effectiveness of the trade‐in programs, we develop a two‐period model in which a firm sells the first generation product in the first period and the second generation product in the second period; however, the firm offers a trade‐in program that customers can participate in when purchasing the first generation product in the first period. To participate, each customer has to pay a nonrefundable fee in the first period so that she has the option to trade‐in her first generation product and receive a prespecified trade‐in value to be used for the purchase of the second generation product in the second period. To capture market heterogeneity and market uncertainty, we examine the case when the valuation of the first generation product varies among customers and the valuation of the second generation product is uncertain a priori. By analyzing a two‐period game, we determine the optimal purchasing behavior of each rational customer, and we show that the firm is always better off by offering its own trade‐in programs. Also, our numerical analysis reveals that trade‐in programs can benefit the firm significantly especially when (i) the residual value of the first generation product is high; (ii) the expected incremental value of the second generation product is high; or (iii) the valuation of the second generation product is highly uncertain.     

Experimental Results Indicating Lattice‐Dependent Policies May Be Optimal for General Assemble‐To‐Order Systems

We consider an assemble‐to‐order (ATO) system with multiple products, multiple components which may be demanded in different quantities by different products, possible batch ordering of components, random lead times, and lost sales. We model the system as an infinite‐horizon Markov decision process under the average cost criterion. A control policy specifies when a batch of components should be produced, and whether an arriving demand for each product should be satisfied. Previous work has shown that a lattice‐dependent base‐stock and lattice‐dependent rationing (LBLR) policy is an optimal stationary policy for a special case of the ATO model presented here (the generalized M‐system). In this study, we conduct numerical experiments to evaluate the use of an LBLR policy for our general ATO model as a heuristic, comparing it to two other heuristics from the literature: a state‐dependent base‐stock and state‐dependent rationing (SBSR) policy, and a fixed base‐stock and fixed rationing (FBFR) policy. Remarkably, LBLR yields the globally optimal cost in each of more than 22,500 instances of the general problem, outperforming SBSR and FBFR with respect to both objective value (by up to 2.6% and 4.8%, respectively) and computation time (by up to three orders and one order of magnitude, respectively) in 350 of these instances (those on which we compare the heuristics). LBLR and SBSR perform significantly better than FBFR when replenishment batch sizes imperfectly match the component requirements of the most valuable or most highly demanded product. In addition, LBLR substantially outperforms SBSR if it is crucial to hold a significant amount of inventory that must be rationed.