The Economic Efficiency of Sampling Size: The Case of Beef Trim Revisited

A recent paper by Ferrier and Buzby provides a framework for selecting the sample size when testing a lot of beef trim for Escherichia coli O157:H7 that equates the averted costs of recalls and health damages from contaminated meats sold to consumers with the increased costs of testing while allowing for uncertainty about the underlying prevalence of contamination. Ferrier and Buzby conclude that the optimal sample size is larger than the current sample size. However, Ferrier and Buzby's optimization model has a number of errors, and their simulations failed to consider available evidence about the likelihood of the scenarios explored under the model. After correctly modeling microbial prevalence as dependent on portion size and selecting model inputs based on available evidence, the model suggests that the optimal sample size is zero under most plausible scenarios. It does not follow, however, that sampling beef trim for E. coli O157:H7, or food safety sampling more generally, should be abandoned. Sampling is not generally cost effective as a direct consumer safety control measure due to the extremely large sample sizes required to provide a high degree of confidence of detecting very low acceptable defect levels. Food safety verification sampling creates economic incentives for food producing firms to develop, implement, and maintain effective control measures that limit the probability and degree of noncompliance with regulatory limits or private contract specifications.     

Optimal Food Safety Sampling Under a Budget Constraint

Much of the literature regarding food safety sampling plans implicitly assumes that all lots entering commerce are tested. In practice, however, only a fraction of lots may be tested due to a budget constraint. In such a case, there is a tradeoff between the number of lots tested and the number of samples per lot. To illustrate this tradeoff, a simple model is presented in which the optimal number of samples per lot depends on the prevalence of sample units that do not conform to microbiological specifications and the relative costs of sampling a lot and of drawing and testing a sample unit from a lot. The assumed objective is to maximize the number of nonconforming lots that are rejected subject to a food safety sampling budget constraint. If the ratio of the cost per lot to the cost per sample unit is substantial, the optimal number of samples per lot increases as prevalence decreases. However, if the ratio of the cost per lot to the cost per sample unit is sufficiently small, the optimal number of samples per lot reduces to one (i.e., simple random sampling), regardless of prevalence. In practice, the cost per sample unit may be large relative to the cost per lot due to the expense of laboratory testing and other factors. Designing effective compliance assurance measures depends on economic, legal, and other factors in addition to microbiology and statistics.     

Optimal Inspection of Imports to Prevent Invasive Pest Introduction

The United States imports more than 1 billion live plants annually—an important and growing pathway for introduction of damaging nonnative invertebrates and pathogens. Inspection of imports is one safeguard for reducing pest introductions, but capacity constraints limit inspection effort. We develop an optimal sampling strategy to minimize the costs of pest introductions from trade by posing inspection as an acceptance sampling problem that incorporates key features of the decision context, including (i) simultaneous inspection of many heterogeneous lots, (ii) a lot‐specific sampling effort, (iii) a budget constraint that limits total inspection effort, (iv) inspection error, and (v) an objective of minimizing cost from accepted defective units. We derive a formula for expected number of accepted infested units (expected slippage) given lot size, sample size, infestation rate, and detection rate, and we formulate and analyze the inspector's optimization problem of allocating a sampling budget among incoming lots to minimize the cost of slippage. We conduct an empirical analysis of live plant inspection, including estimation of plant infestation rates from historical data, and find that inspections optimally target the largest lots with the highest plant infestation rates, leaving some lots unsampled. We also consider that USDA‐APHIS, which administers inspections, may want to continue inspecting all lots at a baseline level; we find that allocating any additional capacity, beyond a comprehensive baseline inspection, to the largest lots with the highest infestation rates allows inspectors to meet the dual goals of minimizing the costs of slippage and maintaining baseline sampling without substantial compromise.     

Optimizing multi-stage production with constant lot size and varying number of unequal sized batches

This paper describes a model for a multi-stage production/inventory system in which a uniform lot size is produced through all stages with a single setup and without interruption at each stage. Partial lots, called batches, may be transported to the next stage upon completion. The number of the unequal sized batches may differ across stages. Considering setup costs, inventory holding costs, and transportation costs, an optimization method is developed to determine the economic lot size and the optimal batch sizes for each stage. The method is illustrated by a computational example and further numerical simulations.     

Optimal Cost Sampling for Decision Making with Multiple Regression Models

This paper develops an explicit relationship between sample size, sampling error, and related costs for the application of multiple regression models in observational studies. Graphs and formulas for determining optimal sample sizes and related factors are provided to facilitate the application of the derived models. These graphs reveal that, in most cases, the imprecision of estimates and minimum total cost are relatively insensitive to increases in sample size beyond n=20. Because of the intrinsic variation of the regression model, even if larger samples are optimal, the relative change in the total cost function is small when the cost of imprecision is a quadratic function. A model-utility approach, however, may impose a lower bound on sample size that requires the sample size be larger than indicated by the estimation or cost-minimization approaches. Graphs are provided to illustrate lower-bound conditions on sample size. Optimal sample size in view of all considerations is obtained by the maximin criterion, the maximum of the minimum sample size for all approaches.     

Optimal Purchase and Transportation Cost Lot Sizing for a Single Item

A review of the literature indicates that the traditional approach for evaluating quantity discount offerings for purchased items has not adequately considered the effect that transportation costs may have on the optimal order quantity; despite the general fact that purchased materials must bear transportation charges. The transportation cost structure for less-than-truckload (LTL) shipments reflects sizable reductions in freight rates when the shipment size exceeds one of the nominal rate breakpoints. However, the shipper must also be aware of the opportunity to reduce total freight costs by artificially inflating the actual shipping weight to the next rate breakpoint, in order that a lower marginal tariff is achieved for the entire shipment. Such over-declared shipments result in an effective freight rate schedule that is characterized by constant fixed charge segments in addition to the nominal marginal rates. Over-declared shipments are economical when the shipment volume is less than the rate breakpoint, but greater than a cost indifference point between the two adjacent marginal rates. This paper presents a simple analytical procedure for finding the order quantity that minimizes total purchase costs which reflect both transportation economies and quantity discounts. After first solving for the series of indifference points that apply to a particular freight rate schedule, a total purchase cost expression is presented that properly accounts for the actual transportation cost structure. The optimal purchase order quantity will be one of the four following possibilities: (1) the valid economic order quantity (EOQ), QC; (2) a purchase price breakpoint in excess of QC; (3) a transportation rate breakpoint in excess of QC; and (4) a modified EOQ which provides an over-declared shipment in excess of QC. Finally, an algorithm which systematically explores these four possibilities is presented and illustrated with a numerical example.     

Optimal two-stage production-inventory policy with whole lot and split lots

This paper deals with the problem of determining the optimal lot sizes for a two-stage production-inventory system. The first stage has a large lot to be processed in a single batch and is called the ‘whole lot’; this is split in the second stage into smaller lots of equal size called ‘split lots’. The lot in stage 1 is inspected for its conformity to standards before it is passed on to stage 2. We use a single sampling attribute plan for inspection and examine the consequences of rejecting or accepting the lot. The rejected lots are 100 % inspected and defectives are eliminated..The accepted lots contain defectives in the uninspected part which are identified during stage 2 processing. We have combined these aspects of inspection with the usual inventory costs and developed a model to determine the optimal sizes of the whole lot and split lots     

Optimizing multi-stage production with constant lot size and varying numbers of batches

The model presented in this paper assumes that a uniform lot size is produced through a series of manufacturing stages, with a single set-up and without interruption at each stage. Transportation of partial lots, called batches, is allowed between stages after the whole batch is completed. The batch sizes must be equal at any particular stage, but the optimal number of equal-sized batches may differ across stages. Of course, the set-up costs, the inventory-holding costs and the transportation costs influence both the optimal batch-sizes at the various stages and the uniform lot size. An optimization method for this deterministic model is developed and is illustrated by an example.     

The Bootstrap and the Edgeworth Correction for Semiparametric Averaged Derivatives*

In a number of semiparametric models, smoothing seems necessary in order to obtain estimates of the parametric component which are asymptotically normal and converge at parametric rate. However, smoothing can inflate the error in the normal approximation, so that refined approximations are of interest, especially in sample sizes that are not enormous. We show that a bootstrap distribution achieves a valid Edgeworth correction in the case of density‐weighted averaged derivative estimates of semiparametric index models. Approaches to bias reduction are discussed. We also develop a higher‐order expansion to show that the bootstrap achieves a further reduction in size distortion in the case of two‐sided testing. The finite‐sample performance of the methods is investigated by means of Monte Carlo simulations from a Tobit model.     

Salmonella Prevalence Alone Is Not a Good Indicator of Poultry Food Safety

Salmonella is a leading cause of foodborne illness (i.e., salmonellosis) outbreaks, which on occasion are attributed to ground turkey. The poultry industry uses Salmonella prevalence as an indicator of food safety. However, Salmonella prevalence is only one of several factors that determine risk of salmonellosis. Consequently, a model for predicting risk of salmonellosis from individual lots of ground turkey as a function of Salmonella prevalence and other risk factors was developed. Data for Salmonella contamination (prevalence, number, and serotype) of ground turkey were collected at meal preparation. Scenario analysis was used to evaluate effects of model variables on risk of salmonellosis. Epidemiological data were used to simulate Salmonella serotype virulence in a dose‐response model that was based on human outbreak and feeding trial data. Salmonella prevalence was 26% (n = 100) per 25 g of ground turkey, whereas Salmonella number ranged from 0 to 1.603 with a median of 0.185 log per 25 g. Risk of salmonellosis (total arbitrary units (AU) per lot) was affected (p ≤ 0.05) by Salmonella prevalence, number, and virulence, by incidence and extent of undercooking, and by food consumption behavior and host resistance but was not (p > 0.05) affected by serving size, serving size distribution, or total bacterial load of ground turkey when all other risk factors were held constant. When other risk factors were not held constant, Salmonella prevalence was not correlated (r = ?0.39; p = 0.21) with risk of salmonellosis. Thus, Salmonella prevalence alone was not a good indicator of poultry food safety because other factors were found to alter risk of salmonellosis. In conclusion, a more holistic approach to poultry food safety, such as the process risk model developed in the present study, is needed to better protect public health from foodborne pathogens like Salmonella.     

The Backroom Effect in Retail Operations

Traditional inventory models fail to take into account the dynamics between the retail sales floor and the backroom, commonly used by retailers for extra storage. When a replenishment order for a given item arrives at a retail store, it may not fit on the allocated shelf space, making backroom storage necessary. In this article, we introduce the backroom effect (BRE) as a consequence of misalignment of case pack size, shelf space, and reorder point. This misalignment results from the fragmented nature of inventory policy decision making in the retail industry and affects basic trade‐offs in inventory models. We specify conditions under which the BRE exists, quantify the expected amount of backroom inventory, derive an optimal short‐term inventory policy, and assess the impact of the BRE on the optimal inventory policy and total costs. Our results indicate that ignoring the BRE leads to artificially high reorder points and higher total costs. The paper concludes with a discussion of theoretical and managerial implications.     

RELIABILITY-CONSTRAINED PRICING,CAPACITY, AND QUALITY: AN ILLUSTRATION OF ECONOMIC CONCEPTS TO POSTAL SERVICES

A model is developed from which welfare-optimal prices, capacities, and reliabilities for a service provider are simultaneously determined. Solutions are determined under conditions of stochastic demand subject to a reliability constraint on service quality. Both quality of service provided, as well as price, impact on demand for services rendered. Results indicate that (i) optimal prices are equated to the reliability-constrained marginal costs, (ii) optimal reliabilities require that the marginal benefits of increasing reliability are equated to the marginal costs of doing so, and (iii) optimal capacity allocation involves minimizing the system's expected costs subject to meeting the prespecified reliability constraint for service quality. The model is applied to postal delivery services in light of the growing competition that has emerged in this industry.     

Optimal Bandwidth Selection in Heteroskedasticity–Autocorrelation Robust Testing

This paper considers studentized tests in time series regressions with nonparametrically autocorrelated errors. The studentization is based on robust standard errors with truncation lag M=bT for some constant b∈(0, 1] and sample size T. It is shown that the nonstandard fixed‐b limit distributions of such nonparametrically studentized tests provide more accurate approximations to the finite sample distributions than the standard small‐b limit distribution. We further show that, for typical economic time series, the optimal bandwidth that minimizes a weighted average of type I and type II errors is larger by an order of magnitude than the bandwidth that minimizes the asymptotic mean squared error of the corresponding long‐run variance estimator. A plug‐in procedure for implementing this optimal bandwidth is suggested and simulations (not reported here) confirm that the new plug‐in procedure works well in finite samples.     

The Firm as a Means of Concealing Information

The aim of this paper is to demonstrate that a firm may owe its continued existence to its attempts to conceal information from its competitors about the unknown characteristics of a certain factor, not just to its savings on market transaction costs, its team‐working, risk‐sharing, or the encouragement of ex ante specific investment. This is because the existence of a firm contract severs the relationship between the factor market and the product market, thereby making it difficult for outsiders to observe the marginal contribution of the intermediate factor and make statistical inferences about the factor's unknown characteristics. Furthermore, an optimal contract is determined by a trade‐off not only between traditional risk‐sharing and incentive, but also between the incentive and information concealing. Finally, we show that this latter kind of trade‐off also affects the position of the optimal boundary of the firm.     

The Impact of Capacity Costs on Product Differentiation in Delivery Time,Delivery Reliability,and Price

We develop an analytical framework for studying the role capacity costs play in shaping the optimal differentiation strategy in terms of prices, delivery times, and delivery reliabilities of a profit‐maximizing firm selling two variants (express and regular) of a product in a capacitated environment. We first investigate three special cases. The first is an existing model of price and delivery time differentiation with exogenous reliabilities, which we only review. The second focuses on time‐based (i.e., length and reliability) differentiation with exogenous prices. The third deals with deciding on all features for an express variant when a regular product already exists in the marketplace. We subsequently address the integrative framework of time‐and‐price‐based differentiation for both products in a numerical study. Our results shed light on the role that customer preferences towards delivery times, reliabilities and prices, and the capacity costs (absolute and relative) have on the firm's optimal product positioning policy.     

Variable Royalty Rates for Improving Franchise Channel Coordination*

A business format franchisor obtains a major part of its revenues from franchise royalties, which are typically a fixed percentage of franchisee gross sales. When a fixed royalty rate is used and the marginal costs of operating the franchise are increasing, the franchisee does not have an incentive to increase sales beyond a certain “optimal” volume. We present a model that recommends the use of a variable franchise royalty rate for extending this optimal sales volume. For a general convex cost function, we show that a new lower rate can be applied to incremental sales beyond the original optimal level. We show that this new rate should be less than half of the original rate when a quadratic cost function is applicable. Adopting a variable royalty rate increases franchisor royalty revenues and franchisee profits.     

基于厂商理论的区域产业边界和最优要素配置结构

区域经济发展核心是产业结构优化升级和资源的优化配置。在全球经济结构亟待优化升级和要素资源呈现不同层次紧缺的大背景下,如何合理配置和引导区域资源配置,优化产业发展方向,确定包括新兴产业在内的产业发展战略,是每一个地区新形势下面临的重要问题。一方面,在经济全球化和零边际成本趋势驱动下,区域产业发展不再局限于区域内部资源禀赋,资源流动性加强,区域产业发展的选择更加广泛和灵活;另一方面,资源日益短缺和经济快速发展推动了劳动力成本和资本成本上升,产业的生产要素配置也随之变化。基于厂商理论,将区域作为市场经济中的生产主体,在考虑区域交易成本和生产要素成本的基础上,构建区域产业边界模型和产业的最佳要素配置结构模型,开创了资源和商品全球化趋势下的产业升级战略的研究范式,探索了成本约束下的产业生产要素优化配置模式,为经济新常态下的区域产业升级战略和资源配置研究提供理论指引,为区域制定个性化的产业发展战略提供实践依据,同时对零边际成本社会趋势下的产业格局进行了初步的探索。研究得出：产业边界能够在一定程度下反映区域产业长期演化趋势,在完全市场经济下产业边界是产业发展的最优规模;区域产业的要素配置格局取决于产业的资源占用和资源贡献情况,最优配置结构受各要素的相对贡献率和相对成本影响;产业边界与成本的关系取决于规模经济与否,在规模经济状态下与生产成本成正相关关系,与交易成本成负相关关系;大部分地区的农业有较大提升空间,最佳资源配置方案是加大机械动能投入。     

THE CAPITAL MARKET VALUE OF A MULTIPERIOD INVESTMENT WITH THE OPTION OF PREMATURE ABANDONMENT

Conventional approaches to determining optimal abandonment of a project under uncertainty either assume risk-neutrality or impose a mean-variance criterion. Risk-neutrality is unrealistic while the mean-variance criterion precludes determination of the optimal strategy without consideration of covariances of returns among projects. Further, the use of variance of present value as a risk measure may result in the “optimality” of a time 0 strategy that involves maintaining a position at time t that will be “suboptimal” and would not be maintained. The use of the multiperiod capital asset pricing model (CAPM) as a decision criterion is consistent with contemporary theory of market behavior and remedies the deficiencies of the mean-variance approach noted above. Computationally, the optimal strategy for abandonment, when the commitment must be made at time 0 (a lease, say), can be determined with little difficulty beyond that of mean-variance models. When time of abandonment can remain unspecified, the value of the prospect that abandonment will occur at the optimal time can be determined, though the technique necessary is considerably more complicated. In both cases, the marginal costs of commitments that limit discretion over abandonment can be determined and attributed to those commitments.     

Leveraging risk assessment for foodborne outbreak investigations: The Quantitative Risk Assessment-Epidemic Curve Prediction Model

Root cause analysis can be used in foodborne illness outbreak investigations to determine the underlying causes of an outbreak and to help identify actions that could be taken to prevent future outbreaks. We developed a new tool, the Quantitative Risk Assessment-Epidemic Curve Prediction Model (QRA-EC), to assist with these goals and applied it to a case study to investigate and illustrate the utility of leveraging quantitative risk assessment to provide unique insights for foodborne illness outbreak root cause analysis. We used a 2019 Salmonella outbreak linked to melons as a case study to demonstrate the utility of this model (Centers for Disease Control and Prevention [CDC], 2019). The model was used to evaluate the impact of various root cause hypotheses (representing different contamination sources and food safety system failures in the melon supply chain) on the predicted number and timeline of illnesses. The predicted number of illnesses varied by contamination source and was strongly impacted by the prevalence and level of Salmonella contamination on the surface/inside of whole melons and inside contamination niches on equipment surfaces. The timeline of illnesses was most strongly impacted by equipment sanitation efficacy for contamination niches. Evaluations of a wide range of scenarios representing various potential root causes enabled us to identify which hypotheses, were likely to result in an outbreak of similar size and illness timeline to the 2019 Salmonella melon outbreak. The QRA-EC framework can be adapted to accommodate any food–pathogen pairs to provide insights for foodborne outbreak investigations.     

INVENTORY MANAGEMENT AND CAPITAL BUDGETING: A PEDAGOGICAL NOTE

This paper shows, through an alternative development of the lot size model, how the methods of capital budgeting can be logically applied to the determination of optimal inventory levels. There is no reason why inventory management needs be treated any differently than the management of fixed assets. The rule that the stock of fixed assets should be expanded until the marginal rate of return equals the marginal cost of capital results from maximizing the present value of profits as a function of the investment level. In a like manner, the “square root” formula from inventory theory results from maximizing the present value of profits as a function of the investment in inventory. In addition, formulating the inventory problem as a capital budgeting problem has advantages for incorporating deterioration, obsolescence, and other costs into the model in a more logical and less intuitive way.