In this article, we investigate the ( R, S) periodic review, order‐up‐to level inventory control system with stochastic demand and variable leadtimes. Variable leadtimes can lead to order crossover, in which some orders arrive out of sequence. Most theoretical studies of order‐up‐to inventory systems under variable leadtimes assume that crossovers do not occur and, in so doing, overestimate the standard deviation of the realized leadtime distribution and prescribe policies that can inflate inventory costs. We develop a new analytic model of the expected costs associated with this system, making use of a novel approximation of the realized (reduced) leadtime standard deviation resulting from order crossovers. Extensive experimentation through simulation shows that our model closely approximates the true expected cost and can be used to find values of R and S that provide an expected cost close to the minimum cost. Taking account of, as opposed to ignoring, crossovers leads, on average, to substantial improvements in accuracy and significant cost reductions. Our results are particularly useful for managers seeking to reduce inventory costs in supply chains with variable leadtimes.

In this paper we consider an inventory management problem motivated by a specific practical context that concerns the commonly used periodic review, reorder point, order-up-to-level control system. In particular, the reorder point and order-up-to level have to be selected so as to satisfy two management-specified constraints. First, from a marketing (or, more generally, a business) perspective, the actual fill rate achieved must be no lower than a prescribed value. Second, for production or supply purposes, the average time between consecutive replenishments must be as close as possible to a target value. The demand distribution considered is the negative binomial. An efficient procedure is developed with illustrations.

In this paper, we address an important practical situation, namely where the usual replenishment lead time (when the supplier's production facility is operating) is a random variable and the supplier shuts down for an interval of known duration (for maintenance, vacation, etc.) each year. The demand rate is constant and any demand when out of stock is assumed to be lost. Under such circumstances we develop a heuristic procedure to decide when to initiate replenishment as well as the associated order-up-to-levels. Through the use of simulation (which accurately estimates the average costs per unit time), the heuristic is shown to perform excellently in a selection of small size problems when one can find the optimal solution. For a large number of problems of more realistic size, the use of simulation reveals that the heuristic achieves substantial cost savings when compared with a simpler, baseline approach. The heuristic itself does not require the use of simulation. The sensitivity of total expected costs to various parameters (such as the length of the shutdown interval and characteristics of the lead time distribution) is discussed.

In this paper, we consider a periodic review order-up-to-level (or base stock) inventory control system under normally distributed demand. For such circumstances, an expression for the exact fill rate (fraction of demand satisfied without backordering) has been available in the literature, but has not been widely known, let alone used by practitioners. In this paper, we redevelop the expression and contrast our derivation with the earlier published one. The paper has two purposes. First, we hope that the reappearance of the exact result in this journal will lead to its wider adoption. Second, showing two contrasting approaches to obtaining the same result may be useful for both research and pedagogical purposes.

Encouraging interest in inventory management necessitates that instructors overcome concerns that the subject is too abstract or conceptual. To aid in this process, we describe a competition engaging students in concepts, including demand estimation, demand uncertainty, and costs of inventory and shortages. The competition simulates a multi-item newsvendor problem employing participant-generated data. We present results from use of the exercise in multiple class settings over the past decade. A number of possible extensions of the basic competition are discussed. Data collection and analysis materials are available to interested readers.

Despite several decades of research in psychology and mathematics education pointing to the importance of learning mathematics with understanding, other research on teachers’ instructional practice in mathematics classrooms has found a remarkably consistent characterization of mathematics teaching in the United States as generally doing little to help students develop a deep understanding of mathematical ideas. Because the practice of teaching mathematics for understanding is so rarely encountered, it has not been extensively studied empirically. This paper summarizes the findings of an analysis of selected mathematical and pedagogical features of the lesson materials found in the portfolio entries submitted by candidates seeking certification by the National Board for Professional Teaching Standards in the area of Early Adolescence/Mathematics. These lessons were selected by teachers and were intended to display “best practice” examples of their teaching mathematics for understanding. Some implications for further research and for teacher education are also discussed.

Teacher educators, professional developers, and researchers have recently shown great interest in the design and facilitation of an approach to mathematics teacher education that is commonly called practice-based professional development (PBPD). At the conceptual and operational heart of PBPD one finds professional learning tasks (PLTs)—activities that are situated in and organized around components and artifacts of instructional practice that replicate or resemble the work of teaching. PLTs are often built around artifacts of practice such as curriculum materials, video or narrative records of classroom teaching episodes, and samples of student work. In particular, video and narrative cases (e.g., Smith, Silver, & Stein, 2005) have been extensively used in PBPD.

The authors present an analysis of portfolio entries submitted by candidates seeking certification by the National Board for Professional Teaching Standards in the area of Early Adolescence/Mathematics. Analyses of mathematical features revealed that the tasks used in instruction included a range of mathematics topics but were not consistently intellectually challenging. Analyses of key pedagogical features of the lesson materials showed that tasks involved hands-on activities or real-world contexts and technology but rarely required students to provide explanations or demonstrate mathematical reasoning. The findings suggest that, even in lessons that teachers selected for display as best practice examples of teaching for understanding, innovative pedagogical approaches were not systematically used in ways that supported students’ engagement with cognitively demanding mathematical tasks.

In this paper we consider a periodic review, reorder point, order-up-to-level system, a type commonly used in practice. Motivated by a specific practical context, we present a novel approach to determining the reorder point and order-up-to-level (for a given review interval) so as to target desired values of (i) customer fill rate and (ii) average time between consecutive replenishments. Specifically, by using a diffusion model (producing normally distributed demand) we convert a periodic review, constant lead time setting into one having continuous review and a random lead time. The method is simple to implement and produces quite reasonable results.

This paper is concerned with movement from a current operating point so as to reach a 2D efficient frontier. After a discussion of different criteria for deciding on which point on the frontier to target, we focus, as an illustration, on a particular inventory management context and the use of the criterion of minimum distance from the current point to the frontier. Specifically, the efficient frontier turns out to be a hyperbola in a 2D representation of total (across a population of items) average stock (in monetary units) versus total fixed costs of replenishments per year. Any current (or proposed) operating strategy, differing from the class along the frontier, is located above the frontier. Finding the minimum distance from the current point to the frontier requires determining the smallest root of a quartic equation within a restricted range.