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Order crossover research: a 60-year retrospective to highlight future research opportunities.

Abstract

We review the literature spanning 60 years of efforts to model the phenomenon of order crossover. The importance of this research has increased due to today's longer ocean supply chains, with their greater inherent uncertainty. The incidence of order crossover is higher in these supply chains, and ignoring it will result in overestimation of safety stock. The literature is grouped into four areas, which are described separately, and is mapped to reveal gaps. Gaps exist for combinations of lead time distributions, demand distributions, and inventory policies where extant methods do not address order crossover. The effective lead time approach has proven flexible enough to be used in a wide range of periodic review situations. However, our results show that its accuracy is undermined by bimodal skewed lead-time distributions, especially when lower bounds are nonzero. Future research needs to concentrate on exploring nonparametric methods or similar methods robust to different lead time distribution assumptions to provide a method of setting safety stock in "real world" situations where orders cross.

Keywords

Systematic literature review, inventory management, order crossover, transit time variability, global supply chains

Introduction

Uncertain transit times are a primary cause of stochastic order lead times. The popular Hadley and Whitin (1963) (HW) model provides an easy-to-use closed-form method of calculating the inventory needed to achieve desired service levels and fill rates for a single item when lead times are stochastic. HW makes a critical assumption that is often violated in practice: no orders cross; that is, orders arrive in the sequence in which they were placed. In supply chains with multiple outstanding orders and stochastic lead times, the probability of order crossover is high. Hence, the HW-based inventory models that are widely taught in our undergraduate, graduate, and executive programs and in common use by practitioners today will yield inaccurate results. Developing an exact closed-form solution to determine reorder point(s) and order up to point (OUP) presents a serious challenge.

Order crossover occurs when identically independently distributed (iid) lead times result in a more recent order in a time series of issued orders crossing and arriving before older orders placed at earlier times. Note that orders may be from one or more suppliers. Hayya et al. (2008) describe two predominant "schools of thought" for circumventing the issue of order crossovers: (1) the HW school, which assumes that the probability of order crossover is very small and thus ignores order crossing, (1) and (2) the Song and Zipkin (1996) school, where the occurrence of order crossovers is obviated due to the highly stylized assumption of an infinite number of identical independent processors simultaneously processing orders, which do not interact with each other. We focus on the HW school, which is in predominant use in academia and by practitioners, and where there is plenty of evidence that the no-order-crossover assumption can cause significant overestimation of inventory and higher inventory holding costs (Hayya et al. 1995; Robinson, Bradley, and Thomas 2001; Srinivasan, Novack, and Thomas 2011).

The literature highlights numerous logistics scenarios that can lead to order crossover. These include short frequent ordering (JIT systems), variable supply lead times, use of multiple suppliers, and use of multiple modes of transportation (Robinson et al. 2001). The growth of global supply chains using ocean shipping has more recently attracted attention in the order crossover literature (cf. Disney et al. 2016). Ocean supply chains are susceptible to a wide variety of delays and disruptions, including those from congestion in and disruption of seaport operations, which can be exacerbated by missed intermodal connections (Lewis et al. 2013). Consequently, there is a high probability of order crossover in these ocean supply chains with more variable lead time (Bischak et al. 2014; Disney et al. 2016; Srinivasan et al. 2011). Furthermore, the prevalence of continuous, positively skewed bimodal lead time distributions with nonzero lower bounds in ocean shipping makes the problem of order crossover in global supply chains omnipresent and harder to solve (Das, Kalkanci, and Caplice 2014; Saldanha et al. 2009). In our own research efforts, which follow up work on seaport disruptions by Lewis et al. (2013), we encountered the problem of being unable to satisfactorily account for order crossovers. Order crossover arises in cases of seaport disruption because newer orders that need to be expedited or diverted at a closed port, cross older orders that remained on vessels at anchor, or in the ports during or soon after the disruption while terminals battled congestion trying to clear the backlog.

Several approaches have been developed that compute close-to-optimal inventory policy parameters for the single-item and single- or multiple-supplier scenarios in the presence of order crossovers. However, these approaches are restricted to specific inventory policies and stylized assumptions for policy parameters, lead time, and/or demand distributions. In this article, we conduct an extensive systematic literature review going back 60 years to Takacs (1956), the first known reference from which the concept of order crossover was developed. There have been a few excellent reviews of the literature on order crossover that provide a survey of methods and a typology of order crossover (Hayya et al. 2008; Riezebos 2006), including two doctoral dissertations that provided updates of the state of the art (He 1992; Srinivasan 2007). This literature review provides a much-needed update and an in-depth and comprehensive categorization of the extant methods, identifying those methods most relevant to setting inventory policies in the presence of order crossover. This includes a primer for their implementation that is a necessary step as we establish the boundary conditions limiting their use. Therefore, our contributions delineate for (1) the logistics researcher the extant knowledge as it maps to logistics scenarios where order crossover can be encountered, with the objective of identifying promising areas for research, and for (2) the logistics practitioner some of the problems with the existing HW-based models for setting inventory policies when order crossover is expected and some areas where extant approaches may or may not be used to overcome this problem.

Literature Review

The classical HW sequential approach uses the convolution of two random variables (rv), lead time (L) and demand (D), to obtain the rv lead time demand (X), which we denote as [LTD.sub.HW]. The rv X of [LTD.sub.HW] is used to set inventory policies for different customer service criteria. This approach is at its core predicated on a single order cycle analysis, where each order is received in the sequence in which it is ordered; therefore, each order cycle is independent of the others. This cyclical structure allows [LTD.sub.HW] to be used as a theoretical device to arrive at a closed form solution for setting inventory policies. However, when order cycles overlap and orders cross and interact with each other, the sequential cyclical structure breaks down. While some researchers have attempted to work within the theoretical structure of sequential order cycles, others have sought to find theoretical devices that are alternatives to the traditional [LTD.sub.HW] approach. One approach is to avoid the sequential cyclical structure entirely, relying on a queueing-theoretic approach.

The queueing-theoretic approach originated in the seminal work of Takacs (1956), which modeled a telephone system with stochastic service times, which is similar to an inventory system with stochastic lead times for unit-sized orders. An approach within the sequential tradition that seeks to explain the apparent paradox of iid lead times where orders do not cross is Washburn's (1973) so-called noninterchangeability hypothesis. Another theoretical device within the sequential ordering tradition is the shortfall distribution (SF), rooted in Zalkind's (1978) seminal work. A closely related third theoretical device is effective lead time (ELT), which originated in the work of Sculli and Wu (1981) and was popularized by Hayya and colleagues, most notably in Hayya et al. (2008).

Consequently, we see that, as illustrated in figure 1, the order crossover literature can be broadly bifurcated into queueing theory and the classical sequential approach. The sequential approach can be further trifurcated into Washburn's (1973) noninterchangeability hypothesis, Zalkind's (1978) SF approach, and Hayya's ELT approach. Figure 1 shows each stream through its historic development, A chronological literature review is provided in the Appendix outlining contributions and limitations, and classifying each paper according to its branch of order crossover literature.

Queueing-Theoretic Approach

The earliest order crossover research follows the queueing-theoretic study by Takacs (1956) of a telephone exchange with stochastic service times, which assumes a Markovian (Poisson) arrivals process and a general service process with n servers (M/G / n), assuming order quantity Q = 1. Later studies assumed an exponential service process (M/M/n) or an infinite number of servers (M/G/[infinity]).

The approach of Takacs (1956) allowed the estimation of the optimum number of lines to meet a specified service criterion (fraction of calls not receiving an open line) for an exchange, depending upon the volume of incoming calls. The key similarity between the telephone and inventory queueing systems is the relationship between outstanding orders and the number of busy lines, which subsequent studies have used to calculate the steady state probability of the number of orders outstanding. For example, Scarf (1958) extended the Takacs (1956) telephone system model to a continuous review--reorder point, fixed order quantity--(s, Q) inventory policy with Q [greater than or equal to] 1, where the steady state probabilities are used to calculate the service level and total cost of the inventory system. Galliher, Morse, and Simond (1959) further built on this early work for order sizes Q [greater than or equal to] 1 using an E/M/n inventory queueing system where arrivals are distributed according to an Erlang distribution with Poisson demands. Later Gross and Harris (1973) extended the earlier work to an M / M / n inventory queueing system for which Wijngaard (1978) employed the properties of the negative exponential distribution to provide simplified expressions for the steady state probability of the number of outstanding orders.

Thereafter, the queueing approach fell out of favor due to its intractability, except when demand follows a strict Poisson process with general or exponential lead times accompanied by the = 1 or Q [greater than or equal to] 1 assumption, respectively. Asserting this point, Hayya et al. (1995, 281) state that "the probabilistic analysis associated with the queueing approach becomes quite cumbersome and impractical in many situations, thereby restricting its application to only certain inventory systems such as the M/G/[infinity] type queues." Consequently, for the rest of this review and paper, we focus our attention on the three sequential approaches.

Sequential Approaches

The earliest work in the sequential tradition is by Hayya et al. (1995), who studied a single- and dual-source order-splitting scenario under an (s, Q policy with iid exponential lead times. They provide closed form expressions for the probabilities of order crossover for both fixed and exponential demand in single- and dual-vendor models. In an attempt to work within the order cycle framework, He et al. (1998) extend the single-cycle structure to a multicycle structure in the context of an (s, Q inventory system. However, to maintain analytical tractability, they assume constant demand and iid exponential and uniform lead times. Further, they restrict s [less than or equal to] Q so that only pairwise crossings can be studied.

Analytical intractability remains a problem with trying to resolve the problem of order crossover under the sequential ordering tradition. Bashyam and Fu (1998) have to restrict their periodic review policy to R = 1 to maintain analytical tractability for their simulation-based algorithm adapted from nonlinear programming. Moreover, while they are able to accommodate iid lead times and stochastic demands, they have to provide an upper bound to the lead time to ensure tractability. Under these restrictive conditions, their approach offers an improvement of close to 5% of the optimal 95% of the time, which compares favorably with the HW approach, which overestimates costs by more than 15% over 60% of the time.

Following Riezebos' (2006) call for research on order crossovers resulting from different lead time processes, Riezebos and Gaalman (2009) focus on the dynamic-deterministic process with multiple suppliers, each with a different constant lead time, where orders cross due to orders placed with different suppliers at different epochs. As a result, the magnitude and timing of order crossover are known a priori. Nevertheless, their stochastic dynamic programming approach is still restricted with respect to the decomposability of the problem. Riezebos and Zhu (2015) apply this analysis to the MRP situation to determine the optimal ordering policy as forecast updates of gross requirements become available over time.

In general, studying order crossover using [LTD.sub.HW] within the classical sequential tradition is very complicated, necessitating impractical assumptions to maintain analytical tractability. Consequently, researchers have turned to other theoretical devices to study this phenomenon within the sequential tradition.

Noninterchangeability

Washburn's noninterchangeability hypothesis is the first of the sequential approaches to deal with order crossover and is predicated on the idiosyncratic assumption that "each unit ordered can satisfy only one particular unit of demand" (Washburn 1973, 3). In other words, each unit demanded is customized to a particular customer's needs, effectively tightly coupling orders to customers, rendering order crossover immaterial. The practical implication of this assumption is the same as that of the HW low probability of order crossover; that is, it circumvents the problem of order crossover. Liberatore (1979) developed an (s, Q inventory model for the upper bounds of total costs using noninterchangeability. Sphicas (1982) and Sphicas and Nasri (1984) extended Liberatore's (1979) work. They demonstrated two distinct and complementary cases for optimal values of the decision variables for calculating the optimal policy and the calculation of the probability of order crossover between two consecutive orders when lead times are bounded. Ramasesh et al. (1991) assume noninterchangeability to examine the statistical economies of scale realized from using dual sourcing with order splitting. To maintain analytical tractability, constant demand is an assumption made by all of the noninterchangeability research (Appendix). This undermines the practical relevance of the noninterchangeability hypothesis in all but the most esoteric supply chain contexts. Hence, noninterchangeability has not been used in over two decades, and so we drop it from further consideration as a viable theoretical device for future research.

Shortfall Distribution

Zalkind (1978) pioneered the conceptualization of the SF distribution, which is the distribution of orders outstanding (placed but not yet received), including demand that has occurred since the last order. Zalkind's (1978) seminal work using the base stock (R, S) provided a solution for calculating the probability of order crossover and the distribution of the SF (H), from which the OUP (S) can be calculated. These results can be derived for review periods R [greater than or equal to] i with iid discrete bounded lead times and stochastic discrete bounded demand. Relaxing these exact assumptions results in cumbersome and unwieldy computations necessitating complex algorithms that were heretofore unavailable for practitioners to implement. Moreover, when the distributions of L and/or D are assumed to be continuous and unbounded, the analytical integrations required to derive the expression for the SF distribution become intractable.

Robinson et al. (2001) extend Zalkind's calculation of the distribution of H to an infinite upper bound for R = 1. From the unpublished work of Zalkind (1976), they derive the mean ([[mu].sub.H]) and standard deviation ([[sigma].sub.H]) of the SF distribution, proving that when order crossover is possible, [[mu].sub.H] =/[[mu].sub.X] and [[sigma].sub.H] < [[mu].sub.X] (where [[sigma].sub.X]and ax is the mean and standard deviation of [LTD.sub.HW] respectively). Similarly to X, H is a random sum of the rv D and the number of orders outstanding (K). Their heuristic solution uses the variance of K ([[sigma].sup.2.sub.K]) instead of [[sigma].sup.2.sub.L] in the popular HW formula to calculate the variance for H ([[sigma].sup.2.sub.H]). Robinson et al. (2001) demonstrate that [[sigma].sub.H] can be used analogously to [[sigma].sub.X] in calculating S for the (R = 1, 5) policy. An interesting result from Robinson et al. (2001) is that even if X is bimodal, H is unimodal.

Because, [[sigma].sup.2.sub.K] requires the computation of an infinite sum, Robinson et al. (2001) provide a heuristic solution that estimates the variance of K as a function of the first two moments of L. Bradley and Robinson (2005) further extend this work by providing tight upper bounds for the heuristic approach. Finally, Robinson and Bradley (2008) calculate even tighter upper bounds for the variance of the number of orders outstanding. This time, however, the tighter bounds are found by assuming that H is distributed according to the four-parameter family of beta distributions.

While Robinson and Bradley (2008) demonstrate that their approach has the best performance, their expressions can only be derived when R = 1. Although SF is unimodal when R = 1, the SF distribution is not necessarily unimodal when R > 1. Therefore, we cannot employ conventional approaches approximating the SF with a known distribution form to estimate the OUP as the inverse cumulative distribution function (CDF) for a given probability of no stockout (PNS) customer service criterion (S = [F.sup.-1.sub.H][PNS]) (Zalkind 1978).

Other studies follow in the SF tradition, assuming R = 1, and therefore suffer from the same shortcomings. Srinivasan et al. (2011) use a dynamic programing model to compute the optimal OUP level for the (R = 1, S) inventory policy with order crossing. Following Zalkind's (1978) approach, their optimal policy uses the inventory position as well as the arrival probabilities of future orders to calculate the OUP. In addition to the foregoing limitations, their approach also suffers from the extensive computational requirements to implement their algorithm. Hence, problems of dimensionality restrict their approach to consider demands no greater than 40 units and lead times no longer than 5 periods. Disney et al. (2016) present a new method for determining the distribution of the number of open orders for the (R = 1, 5) inventory policy. They demonstrate that their proportional OUP approximation is marginally better (<0.8%) than the OUP with R=1. Humair et al. (2013) consider (R = 1, S) using the SF approach to account for order crossover with stochastic lead times in a multiechelon inventory system.

Effective Lead Time

Sculli and Wu (1981) is the first published reference to ELT. They defined this construct as the minimum of the set of random variables representing lead times for multiple suppliers. Their objective is not to model order crossover but to demonstrate the statistical scale economies when orders are split between multiple suppliers. Pan et al. (1991) were the first to use order statistics of a random sample of lead times to develop the distribution of ELT to study order crossover, albeit with constant demand. Similarly, continuing the order splitting research, Ramasesh et al. (1991) incorporate multiple considerations, including ELT for order crossover, with the objective that even with order crossover there is still an advantage to order splitting.

Much later, Hayya et al. (2008) were the first to explicitly use the ELT concept to set inventory policies in the presence of order crossover and show the close correspondence of the ELT to Zalkind's SF distribution. Building on the order statistic concept used by Pan et al. (1991), they define ELT as the time interval between the ith order placement and the ith order arrival, with the index i not tagged to a particular order. They use a simulation approach to show that in an (R = 1, S) inventory system with iid lead times and stochastic demand, the variance of ELT is very close to that of the number of orders outstanding as defined by Zalkind (1978) and Robinson et al. (2001).

Hayya, Harrison, and Chatfield (2009) examine the (s, Q.) policy with iid normal demand and iid exponential lead time. Because this problem is analytically intractable due to order crossovers, they use simulated data generated from factorial experiments to develop a convolution of demand during ELT and then use regression expressions to estimate the optimal parameters that minimize cost. Muharremoglu and Yang (2010) study (R = X, S) base stock policies using an indexing device similar to that of ELT, where each unit is indexed and related to a unique customer index. They remove the sequential assumption by allowing unit-customer pairings to update every period, thus taking account of order crossover. The ordered list of order lead times then provides the distribution of ELT. They describe how all that is needed is the random vector of order release and arrival times, which is available from most firms' information systems. Interestingly, their results show that the accuracy of the ELT method is affected by the shape of the distribution. They corroborate the finding of Hayya et al. (2008) that for R = 1 the ELT has the same distribution as the SF. Under restrictive conditions, they show the calculation of (R = 1, S) policies for three- and five-stage supply chains and how relaxing those conditions produces close-to-optimal results with a single-stage approximation. Hayya, Harrison, and He (2011) use deterministic demand and exponentially distributed lead times to show that reducing the mean of lead time reduces the variance caused by order crossover in the (R = 1, S) policy. Hayya et al. (2013) use simulations to extract the distribution of ELT directly from the parent lead time and show that because ELT is a time series; that is, a joint distribution of mixtures of random variables, deriving a statistical distribution of ELT would be impossible.

Taking a slightly different approach, Bischak et al. (2014) use simulations to generate the distributions of ELT from an experimental range of parent gamma lead times. The standard deviation of ELT is estimated for each of these distributions. These are used to develop a nonlinear regression model using the ratio of the review period to the standard deviation of the parent lead time to predict the standard deviation of ELT. The ELT can then be used in place of the standard deviation of the parent lead time to estimate the gamma-distributed effective lead-time demand ([LTD.sub.E] across a wide range of review periods. The benefit of Bischak's approach, besides its simplicity, is it applicability to base stock policies with R > 1. In addition, Bischak et al. (2014) show that their approach works with first-order auto-correlated or AR(i) lead times, for which they derive a similar expression to estimate the standard deviation of ELT.

Wensing and Kuhn (2015) argue that the ELT is superior to the SF distribution for developing time-based customer service measures such as waiting time per order. Using convolutions to determine the probability distributions of ELT for non-negative and finite bounded discrete lead times, they illustrate their point by enumerating different service levels in base stock policies, with R [greater than or equal to] 1 representing customer Service measures. However, they do not provide any way to compute [LTD.sub.E] and set the OUP.

The ELT approach, like the closely related SF one, has fewer restrictive assumptions than the other approaches. Yet, unlike the SF, it has shown promise for applicability across both the continuous review and base stock (R [greater than or equal to] 1) inventory policies. Nevertheless, many of the approaches used thus far in the literature have to resort to simplifying assumptions, such as restricting inventory parameters like the length of the review period. These assumptions restrict their use to fundamental discovery, such as understanding the underlying properties of the [LTD.sub.E] distribution.

The applicability of these approaches is often bounded by conditions that are not found in many of today's supply chains. For example, lead time distributions may not follow the neat shape of known statistical distributions and can often be bimodal, particularly in global ocean shipping supply chains (cf. Das et al. 2014; Saldanha et al. 2009), unless order quantities exceed lead time demand, in which case order crossing is minimal and the HW approach may be used (Ramasesh et al. 1991). However, as Saldanha et al. (2009) explain, in global supply chains, liner shipping schedules necessitate using base stock policies with review periods of seven days or more (R > 1). Because those shipments can arrive at any time during that review period, an approximation like that of Disney et al. (2016) assuming a unit review period base stock policy, which effectively puts all the intervening periods' order arrivals into a single bucket, will result in errors (Zalkind 1978). Moreover, in global multistage supply chains, concepts such as logistics postponement exploiting statistical scale economies and transloading cannot be handled by currently available methods (Jula and Leachman 2011). Consequently, we now map the boundary conditions of the extant literature to identify where currently available approaches can be applied and the gaps that need to be filled by future research.

Boundary Conditions

Table 1 distills the limiting conditions for each study in the Appendix to delineate the boundary conditions of the extant order-crossing methods and map the areas that are ripe for future research. The salient boundary conditions in the order-crossing literature are the distributional assumptions underlying demand and lead time. Hence, table 1 displays the literature mapped against the distributional assumptions of demand and lead time. Each article is represented in the cells formed by the intersection of its specific distributional assumptions using the first two letters of the first author's name and the publication year (e.g., Hayya et al. 2008 is Hao8). The font format represents the theoretical device in the sequential tradition used to address order crossing, unformatted representing ELT, double underlined representing [LTD.sub.HW] , and bold representing SF. Besides distributional assumptions, researchers often confine their work to a specific inventory policy, which is shown by the shaded backgrounds, with clear for continuous review, grey for base stock with R = 1, and black for base stock with R [greater than or equal to] 1.

We have greyed out the areas with constant demand and lead time, as these have little practical relevance. The light cells delineate the area of greatest practical relevance, with iid demand and lead times. Only Bischak et al. (2014) consider AR(1) lead times, which may occur during transportation delays due to congested networks during peak season, such as during the summer peak import season from Asia, and during and immediately following major supply chain disruptions. While several studies focus on specific distributional forms, the ELT heuristic has been prescribed for any, or what we term as general iid lead times. Hayya et al. (2008) demonstrate its use only for the base stock policy with R = 1. Bischak et al. (2014) extend the ELT approach to the more general R [greater than or equal to] 1 base stock policy for a broad family of bell-shaped and right-skewed distributional forms, including the gamma, lognormal, and Weibull.

Zalkind's SF is applicable to the base stock policy with R [greater than or equal to] 1 with a general iid discrete bounded lead time distribution. While Robinson, Bradley, and Thomas (2001) extend this to an infinite upper bound for discrete iid lead times, they need to restrict the base stock policy to R = 1 to maintain analytical tractability.

Table 1 indicates that the majority of the research is for single sourcing, with only Hayya et al. (1995) providing a stylized example of dual sourcing. The other approach, using dual and multiple sourcing, is that of Riezebos and Gaalman (2009) and Riezebos and Zhu (2015), who employ deterministic lead times due to their primary objective of dynamic lead times arising from orders arriving from multiple suppliers.

Thus, from table 1, the gaps in the literature are clear, namely in the areas for general iid continuous and discrete lead times and AR(i) lead times. This is especially true for the continuous review and base stock policies with R > 1. The SF and ELT approaches appear to be the most popular and have the most external validity worth pursuing for research purposes, while being accessible to practitioners. Staying in the single-source, single-item context, we now attempt to demonstrate the implementation of the SF and ELT approaches. This will provide greater granularity of their boundary conditions and the extent of their applicability, which further highlights the gaps in the literature.

Shortfall

Zalkind's (1978) method for calculating the SF from which the OUP can be directly obtained for base stock models with R [greater than or equal to] 1 can be used only for discrete, non-negative, bounded lead time and demand distributions. The method involves computing the SF distribution, from which the OUP can be directly obtained using a simple quantile function. To observe space constraints, an example of this method for (R = 1, S) policy with L ~ U[0,4] and D~U[0,3] is provided in an Online Appendix. (2) This is available upon request from the first author, but the results are provided here. Figure 2 provides the results of the computations for the SF distribution (table B5 of the Online Appendix), which is compared to the [LTD.sub.HW] distribution for our example. The horizontal axis provides the values of the [LTD.sub.HW] (X) and SF (H) distributions. The probability mass function (pmf) and cumulative distribution function (CDF) are represented on the primary and secondary vertical axes, respectively. The pmf and CDF for the [LTD.sub.HW] were obtained by simulating a million draws to calculate the random sum of demand over a random draw of lead time in Wolfram Mathematica 10.4.

A comparison of the CDFs of the SF and [LTD.sub.HW] in figure 2 clearly shows that for any PNS = [P.sub.1] > 75%, [F.sup.-1.sub.H] ([P.sub.1]) > [F.sup.-1.sub.H] ([P.sub.1]); that is, the OUP calculated with [LTD.sub.HW] is greater than that calculated with the SF, leading to excess safety stock. From figure 2, it is apparent that even for this very simple example, the difference in the OUP can be over 18% for a PNS = 99%. The differences in safety stock (SS) are greater; for example, for an average lead-time demand of [[mu].sub.X] = 3 for our illustration in figure 2 and a PNS of, say, 95%, SS is reduced by 20%, nearly double the difference for OUP at that PNS when using the SF versus [LTD.sub.HW].

From table 1, it should be noted that this is the only method available to calculate the OUP from the SF with the base stock policy when R > 1. Yet the method has its limitations, namely extensive, cumbersome computations, especially as the maximum lead time (M) becomes large, which makes the method inaccessible, particularly for practitioners. While we have attempted to address some of these shortcomings by providing algorithms for computing the shortfall distributions for any discrete bounded lead time and demand distributions in our Online Appendix, it is quickly apparent that the computational cost versus inventory accuracy trade-off can make this method unattractive.

To obtain the pmf of the SF, we must permute all possible orders outstanding and then convolve these with the demand distribution to arrive at the SF distribution. Consider the simple case of R = 1 with a nominal M = 10 periods. We must permute 1,024 unique patterns of the number of orders outstanding (K) to calculate the pmf of K (details in Online Appendix footnote 1). The number of permutations is considerably less for R > 1. For example, the likely base stock scenario in a global ocean shipping supply chain with R = 7 and maximum lead times of, say, 50 days requires only 256 permutations. In the second step, the convolutions can be calculated on a computer with the help of a simple algorithm (Online Appendix), based on Zalkind's (1978, 1388-89) equation 8 (9) for R = 1 (R > 1). Consider again the R = 1 example with M = 10 and a very conservative maximum demand (n) of 10 units. The maximum number of convolutions in a single iteration required to compute the probability of units on order will be [n.sup.M+1], requiring a matrix with [n.sup.M+1] rows, that is, 100 billion rows. For our global ocean shipping supply chain, where M = 50 and R = 7, the rows of the matrix can be a large number that grows exponentially in M, that is, [n.sup.48]. Even for smaller-dimensional matrices that can be accommodated in most office software today, say 1 x [10.sup.6] (for M = 5, n = 10 with R = 1), an appropriate sorting algorithm needs to be included in our algorithm to effectively compute the probability distribution of the number of units on order, which is required to compute the pmf of the SF. The quality and appropriateness of the sorting algorithm will determine the feasibility of this method. There are a number of sorting algorithms, ranging from the general and simplistic (bubble sort) to the specific and sophisticated (block sort). However, sorting algorithms are beyond the scope of this article. Interested readers can see Evans and Leemis (2004), who discuss a combination convolution-sorting algorithm. Hence, we can conclude that except for small upper bounds on lead time and demand, the Zalkind approach is generally impractical.

Given the limitations of Zalkind's approach, the extension to the infinite upper bound by Robinson et al. (2001) offers a significant enhancement, albeit restricted to R = 1. As explained in the literature review, they show that using [[sigma].sup.2.sub.K] instead of [[sigma].sup.2.sub.L] in the HW expression for [[sigma].sup.2.sub.X], we can obtain [[sigma].sup.2.sub.H] = [[mu].sub.L][[sigma].sup.2.sub.D] + [[sigma].sup.2.sub.D] [[sigma].sup.2.sub.K], which can be used just like [[sigma].sup.2.sub.K] for estimating the OUR Robinson et al. (2001) and Bradley and Robinson (2005) show that the SF distribution is best approximated by a negative binomial distribution H ~ NB([r.sub.H], [p.sub.H]), where [r.sub.H] = [[mu].sup.2.sub.H]([[sigma].sup.2.sub.H] - [[sigma].sub.H]), the number of successes, and [p.sub.H] = [[mu].sub.H]/[[sigma].sup.2.sub.H], the probability of success on each trial. Therefore, with PNS =P1, the OUP can be calculated as S = [F.sup.-1.sub.B:H]([P.sub.1]). Due to the infinite sum required to calculate [[sigma].sup.2.sub.K], Robinson et al. (2001) offer a heuristic solution that is improved upon by Bradley and Robinson (2005), where [[??].sup.2.sub.K] = min {[[sigma].sup.2.sub.L], [[mu].sub.L], [[sigma].sub.L]/[square root of 3]}. Substituting [[??].sup.2.sub.K] for [[sigma].sup.2.sub.K] gives us an upper bound for the OUP, assuming H~NB. The disadvantage of using the negative binomial approximation for H is that popular spreadsheet programs in common use, such as Microsoft's Excel, do not have an inverse function for the negative binomial distribution. Instead, the analyst can construct a table of the CDF for the integer value number of failures (representing the OUP) before the specified number of successes (rH), distributed according to the negative binomial distribution, using the Excel function = NEGBINOM.INV(OUP, [r.sub.H], [p.sub.H], TRUE). Then the OUP can be imputed by a lookup function for where the PNS just exceeds the CDF to return the number of failures as the correct OUP.

Robinson and Bradley (2008) provide an even tighter upper bound heuristic solution using [[??].sup.2.sub.K] = min {[[sigma].sup.2.sub.L], ([CV.sup.2] + 1/9/[CV.sup.2] + 1) [[mu].sub.L], [[sigma].sub.L]/[square root of 3]}, where CV = [[sigma].sub.L]/[[mu].sub.L], the coefficient of variation of lead time, and H is distributed according to the four-parameter family of beta distributions. In their online supplement, Robinson and Bradley (2008) show how the third (skewness) and fourth (kurtosis) moments of the SF distribution can be used to estimate the parameters of the four-parameter beta function. To do so the analyst needs to compile a sample of lead times, which can be used as an input to the Excel functions SKEW() and KURT(), to compute skewness and kurtosis, respectively. Note that the KURT() function must be centered by adding 3[(n-1).sup.2]/(n-2)(n~3) to the result (n is the sample size). Using these parameters, it is a simple matter to then use the Excel function = BETA.INV([P.sub.1];[alpha],[beta],l,u) to compute the OUP for the required PNS, where I and u are the lower and upper bounds of the beta function, respectively. Robinson and Bradley (2008) show that the beta approximation is the best method for calculating OUP using the SF approach, with the negative binomial coming in a close second. Nevertheless, they note that more research needs to be done to generalize their findings, which may be specific to their test bed of parameters. But the most effective extension would be the ability to extend this to a base stock policy for R > 1. That is a clear advantage of the ELT approach.

Effective Lead Time

From table 1, it is clear that the ELT method has the broadest applicability of all the approaches for the base stock policy. This is extended to R > 1 by Bischak et al. (2014). The heuristic solution is based on gamma-distributed lead times, effective lead times, and effective lead-time demands ([LTD.sub.E]), for which, as Swan and Tyworth (2001, 19) point out, "The gamma distribution is theoretically appealing because it has non-negative values and can approximate both bell-shaped and skewed distributions." Although Bischak et al. (2014, 3) developed their heuristic using gamma-distributed lead times, the same approach can be applied to lognormal and Weibull-distributed lead times. Nevertheless, the robustness of this heuristic has not been tested with bimodal distributions.

We conduct a set of experiments to test the robustness of the Bischak et al. (2014) heuristic over a range of bimodal distributions as well as bell-shaped and positively skewed distributions. For the gamma experiments, we selected 12 levels of [[mu].sub.L] ranging from 1 to 34 in increments of three periods. Two levels of coefficient of variation of lead time (CVL), 0.1 (bell-shaped) and 0.5 (positive skew because of the zero lower bound), were used to arrive at 24 combinations each of [[mu].sub.L] and [[sigma].sub.L] for the gamma experiments in total. Following a similar procedure from Das et al. (2014), we combined a lognormal and normal distribution to generate a set of bimodal distributions that closely approximated the mean levels of the unimodal experiments ([[mu].sub.L] approximately ranging from 4 to 34 in three-period increments). By varying the distances of each component distribution's mean, we maintained a unit normalized difference between the two means of the component lognormal and normal distributions (distance between the two means divided by the resultant mean of the bimodal distribution) and a CVL in the range 0.3-0.5. The lognormal is the first distribution sampled, to ensure non-negative samples. Mix rates of 0.7 and 0.9 were used to represent positive skew and low and high levels of bimodality, respectively. This resulted in 22 unique bimodal distributions.

We conducted the experiments in Wolfram Mathematica 10.4, where for each of the 46 lead time distributions or experimental treatments, we generated a lead time vector of length 100,000. We used the Hayya et al. (2008, 2009, 2011, 2013) algorithm for generating ELT vectors from the random draws of lead time for three review periods, R = 1, 7, 30, representing typical supply chain practices of daily, weekly, and monthly inventory reviews, respectively. Assuming gamma-distributed demand with mean ([[mu].sub.D]) of 1,000 units and 10 levels of standard deviation ([[mu].sub.D]) determined by coefficient of variation of demand (CVD) ranging from 0.1 to 1, we generate 10 vectors of LTDC for each treatment of ELT. The [LTD.sub.E] is calculated as a random sum of demand over the random draw of the rv L of ELT and the review period ([??] + R). The vectors were then used to generate histograms, from which we extracted the OUP corresponding to a PNS, [P.sub.1] = 0.70, 0.75, 0.80, 0.85, 0.90, 0.95, and 0.99, as [F.sup.-1]([P.sub.1]). This is compared with the OUP estimated from the Bischak et al. (2014) heuristic used to calculate the Standard deviation of ELT [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This result is then used to estimate the first two moments of the gamma distributed rv [??] of [LTD.sub.E], as demonstrated in Swan and Tyworth (2001) for the analogous gamma-distributed [LTD.sub.HW] (X), where [??] ~ ([alpha], [beta]) with parameters [alpha] = [[mu].sup.2.sub.[??]]/ [[sigma].sup.2.sub.[??]] and [beta] = [[sigma].sup.2.sub.[??]]/[[mu].sub.[??]]. Using the popular HW method, we calculate the mean of [LTD.sub.E] as [[mu].sub.[??]] = ([[mu].sub.L] + R) [[mu].sub.D] and the standard deviation as [[sigma].sub.x] = [square root of ([[mu].sub.L] + R) [[sigma].sup.2.sub.D] + [[mu].sup.2.sub.D] [[sigma].sup.2.sub.[??]]]. Note that [[mu].sub.L] = [[mu].sub.[??]] for all R [greater than or equal to] 1. These parameters for [??] are then used to calculate the OUP as the [P.sub.1] th quantile for the gamma distribution, where S = [F.sup.-1.sub.[gamma]:[??]]([P.sub.1]). Note that from Tyworth, Guo, and Ganeshan (1996), the calculation of the OUP for the gamma-distributed [LTD.sub.E] can be carried out in Excel as S = GAMMA.INV([P.sub.1];[alpha], [beta]).

In general, we found that the Bischak et al. (2014) heuristic performed well for bell-shaped and right-skewed distributions. However, there were some significant departures of the Bischak et al. (2014) estimated OUP from the true OUP obtained from the bimodal and even the gamma lead times. As Muharremoglu and Yang (2010) point out, the accuracy of the ELT method is dependent upon the shape of the ELT distribution. In figure 3, we provide an example of a lead time distribution with high bimodality at low CVD (panel A) and high CVD (panel B) for R = 7. This results in a highly bimodal distribution of actual LTDE at low CVD that is very far off the Bischak et al. (2014) estimated distribution. Comparing the CDFs at low CVD, it is very clear that the estimated OUP can be far off the actual. This means that the difference in safety stock will be even more substantial.

On the other hand, as CVD increases the high bimodality of the lead time notwithstanding, the distribution of [LTD.sub.E] becomes unimodal, as is seen in panel B of figure 3. Consequently, the Bischak et al. (2014) heuristic works better at high CVD. Hence, we will focus on the low CVD for examining the other results of interest.

In table 2, we list the results for the difference in OUP and safety stock between the estimated and actual distributions for selected levels of the high bimodality and gamma lead time distributions for all experimental levels of PNS and review period (R). We have eliminated lead time levels where these differences are not great, especially with no or low probability of order crossing, where the distribution of [LTD.sub.E] is very close to that of [LTD.sub.HW]. Yet the problem of setting inventory parameters that rely on the normal or gamma approximation with bimodal or positively skewed lead time and lead-time demand distributions remains, which, while beyond the scope of this article, still needs to be addressed.

The first takeaway we want to highlight in table 2 is that no pattern of differences in the OUP and safety stock between the estimated and actual LTDr distributions can be inferred across the levels of PNS. This is because those differences depend upon the shape of each distribution relative to the other, which can be quite different, depending upon the parameters of lead time and demand distribution. For example, for [[mu].sub.L] = 3.999 with R = 7, the difference in OUP goes from 14.11% at [P.sub.1] = 85% to 0.25% at [P.sub.1] = 90% and back up to 15.49% at [P.sub.1] = 95%. This corresponds to the figure of CDFs in figure 3, panel A, where the CDFs are seen crossing each other near 90%. Note that this is the reason the direction of the differences is not shown (differences are shown in absolute percentages). Second, while the difference in OUP may not be large, the difference in safety stock can be substantial; see in our previous example that for a 14.11% difference in the OUP the difference in safety stocks is 88.49%. The findings from the low-bimodality distributions are the same. Even for gamma-distributed lead times, the differences can be substantial, although not as wide-spread as the bimodal. The largest difference in safety stock is 18.2% for a review period of 7, mean lead time of 4, and PNS of 70%.

Several studies have shown that global supply chain lead times have a nonzero lower bound (Das et al. 2014; Disney et al. 2016; Saldanha et al. 2009). Even in domestic and regional supply chains, there are very rarely situations where lead times are zero. It is a straightforward exercise to show that the Bischak et al. (2014) heuristic model in its current form would be inaccurate and new parameters would need to be empirically derived for each new lower bound, which would make this approach cumbersome. In some experiments we conducted, where we shifted each of our experimental distributions by a third of their mean (e.g., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] --the gamma parameters for X were estimated using the shifted mean), we saw significant differences in the estimated and actual OUP and safety stocks.

What becomes immediately apparent is that we are only just beginning to understand the extent to which the ELT, particularly the Bischak et al. (2014) heuristic approach, can be used to address order crossing. There are clearly areas where the Bischak et al. (2014) approach can be applied even for nonstandard positive-skewed bimodal distributions, for example, at higher CVD. However, there are situations even for gamma-distributed lead times where errors in setting OUP and safety stock are possible. Hence, more work needs to be done to formulate a general theory of LTDp beyond our results and the limited test bed of experimental distributions based on the Bischak et al. (2014) heuristic.

Discussion and Future Research Opportunities

Most of the work conducted on order crossing addresses the base stock policy due to the deterministic nature of the order-generating period, that is, the review period. From table 1, it can be seen that more work is needed to address order crossing when the continuous review (s, Q) policy is in use, particularly in understanding the conditions under which the probability of order crossover is high, low, or absent. Even for base stock policies, there are numerous areas that are ripe for extensions such as the Bischak et al. (2014) heuristic, which is not accurate at low CVD, particularly for nonstandard, positively skewed, bimodal lead time distributions. Fast-moving steady demand items are characterized by low CVD, while global ocean shipping lead times are often characterized by nonstandard, bimodal right and left skew distributions (Das et al. 2014; Saldanha et al. 2009). This critical gap needs to be addressed by future research. Note that there is an upper bound for R/[[sigma].sub.L] (Bischak et al. 2014 report this as R/[[sigma].sub.L] [congruent to] 10 for their test bed of gamma distributions), which indicates when the distribution of [LTD.sub.E] is equivalent to that of [LTD.sub.HW], that is, the probability of order crossing is very low or zero and so the [LTD.sub.HW] is appropriate. In our research, this upper bound varied and could be as low as R/[[sigma].sub.L] [congruent to] 4 depending upon the skew and bimodality of distribution; more work needs to be done to understand this relationship. As pointed out by Muharremoglu and Yang (2010), the shape of the ELT distribution affects its accuracy in estimating inventory parameters with known distributions for approximating [LTD.sub.E]. Our findings show that there is still no way of determining the parameters or shape of the resulting gamma-distributed [LTD.sub.E] vis-a-vis that of the true [LTD.sub.E]; hence, the results of the Bischak et al. (2014) heuristic may vary even for bell-shaped and right or left skewed distributions.

Our experimental test bed is limited as a function of the objective of systematically reviewing the literature and investigating the methods available to explore order crossover. This test bed can be systematically explored in future research to establish sound boundary conditions for the applicability of the Bischak et al. (2014) heuristic and other methods. In addition, while Muharremoglu and Yang (2010) find that for R = 1 base stock policies the distribution of ELT is the same as that for the SF, this fact still needs to be established for R > 1.

Nevertheless, at least for the base stock policy, Muharremoglu and Yang (2010) point out that the random vector of order release and order arrival times can be extracted from most firms' information systems. In this case, even the Bischak et al. (2014) heuristic is unnecessary and the ELT can be used directly to derive the OUP, as pointed out earlier in this article. However, more work needs to be done to determine whether a gamma approximation or another nonparametric approach for setting OUP is necessary, especially in global ocean shipping supply chains, where lead times take on nonstandard forms, including high bimodality. For example, consider the bootstrap approach implemented by Bookbinder and Lordahl (1989) and the quantile approach implemented by Lordahl and Bookbinder (1994), these methods can be applied to the order crossing setting.

Although we test only positively (right) skewed bimodal lead time distributions, Das et al. (2014) provide evidence of negatively (left) skewed bimodal and multimodal lead time distributions in global supply chains employing ocean shipping. The results of our experiments with positively skewed bimodal lead time distributions indicate that negative skew distributions could lead to inaccuracies with the Bischak et al. (2014) approach, but multimodal lead time distributions may not, which needs to be investigated further. Other extensions include testing the efficacy of the Bischak et al. (2014) heuristic for accommodating order crossing for base stock policies with R > 1 in a multistage global supply chain. This includes logistics postponement and transloading strategies and nonzero (shifted) lower-bound lead times.

Considering the dynamic stochastic lead times that are typical in today's domestic and global supply chains, the general order crossing problem becomes more complex (Riezebos 2006). With shortages in land transportation capacity, shippers are forced to contract out, or worse still, go to the market for some of their transportation needs. Hence, transportation lead times are not stationary but dynamic. In global ocean supply chains, factors such as liner shipping capacity and market conditions can induce liner firms to change their port-to-port transit times, for example, to reduce operating costs with reduced fuel burn. Also, shippers renegotiate contracts with the same liner firm with potentially different service parameters or with different liner firms that have different service speeds and service parameters. General order crossover is an area that has been paid no attention in the literature and is ripe for future research, particularly using nonparametric bootstrap and order statistic approaches to address this very practical but complex problem.

The literature has focused on the probability of order crossing, which determines the number of orders that we can expect to cross in a time period. However, in our experiments, we also observed another phenomenon, the magnitude of order crossing, which also impacts the overestimation of safety stock. The magnitude of order crossing is the number of periods on average by which a newer order crosses older orders. Consider two different-shaped lead time distributions with the same probability of order crossover and the HW approach for setting inventory parameters. We conjecture that higher safety stock (compared to actual) and a higher probability of order crossover will be realized for the more positively skewed distribution, which results in a higher magnitude of order crossover, vis-a-vis the less positively skewed (or bell-shaped) distribution, which will have a lower magnitude of order crossover. Intuitively, we can employ the ELT concept to see that a positive skew distribution will result in more samples of longer lead times, which will bring about a greater reduction in the standard deviation of ELT. Therefore, the difference of safety stock set by HW as compared to the ELT approach will be much greater for the positively skewed versus the bell-shaped distribution.

Conclusion

The last 60 years have seen much progress in modeling order crossover. Quick calculations of the effect of order crossing and the resulting optimal safety stock are possible under narrowly defined conditions involving restricted distributions of lead time and/or demand. Using existing methods to set inventory policies that optimize total costs for "real-world" supply chains in the presence of order crossover remains a challenge. Measuring actual lead-time demand, or more directly, effective lead-time demand remains the only way to accurately plot its distribution in many situations when orders cross. Even then, the original problem of the best way to model the shape of lead-time demand to accurately set safety stocks still exists.

The Bischak et al. (2014) heuristic using effective lead-time demand to set inventory parameters under base stock policies comes closest to modeling order crossover under real world conditions. Yet even this method falls short, especially for products with low CVD and nonstandard (bimodal) lead time distributions with nonzero lower bounds, which can dynamically change over time. In addition, the effect of order crossover has been largely ignored under numerous inventory policy and real world conditions; for example, for (s, Q) continuous review systems and multistage supply chains.

An obvious next step is to find more accurate ways to set OUP and safety stock using flexible methods that are robust under numerous distributional assumptions and real world considerations. One such set of methods is the bootstrap and order-statistic approaches demonstrated by Bookbinder and Lordahl (1989) and Lordahl and Bookbinder (1994), respectively. These methods are flexible to deal with nonstandard distributional forms and can be implemented using current office software in simple routines as part of a firm's information systems to dynamically update parameters in response to changing data. It is with that aspiration that we anticipate this article to be a first step in opening up this fascinating area for new research to build on the rich 60-year order crossover research tradition. Toward this end, here are some directions for future research.

Little is known about order crossover under the continuous review policy. We need to understand the conditions where order crossing is likely to be high, low, or absent. We need to find methods to set reorder points and safety stocks in the presence of order crossing. For example, can the Bischak et al. (2014) approach be applied to the continuous review context using the order quantity consumption rate and reorder quantity consumption rate as defined by Hayya et al. (1995)? Or are bootstrap and order-statistic nonparametric approaches better suited to study order crossover for continuous review policies?

As shown in our limited experiments, the Bischak et al. (2014) approach has limitations even for gamma lead times. Further research needs to be undertaken to clearly define the boundary conditions and extend this work for ELT under base stock policies to nonstandard left and right skew distributions. This may include expanding the Bischak et al. (2014, 764) analytic relationship between [[sigma].sub.L]/[[sigma].sub.L] and R/[[sigma].sub.L] to a broad family of distributions that include bimodal and right- and left-skewed distributions with nonzero lower bounds.

As we discuss, stochastic dynamic order crossover as defined by Riezebos (2006), especially in multistage global supply chains with logistics postponement and transloading requirements, is a very real problem that is not addressed by any of the current approaches. Nonparametric bootstrap and order statistic approaches incorporating dynamic updating from firms' information systems appear to offer a reasonable approach to addressing this problem.
Appendix

Literature Review Table

Study             Order Processing       Order      Inventory
                                        Crossing      policy

Takacs             Queueing: M/C/n        Yes          Not
(1956) (a)                                          applicable

Scarf (1958)       Queueing: M/M/n        Yes         (s, Q)

Galliher,        Queueing: E/M/n (b)      Yes         (s, Q)
Morse, and
Simond (1959)

Washburn              Queueing:           Noc         (s, Q)
(1973)           Noninterchangeable

Gross and          Queueing: M/M/n        Yes         (s, S)
Harris (1973)

Wijngaard          Queueing: M/M/n        Yes         (s, Q)
(1978)

Zalkind              Sequential           Yes         (R,Q)
(1978)

Llberatore       Noninterchangeable        No         (s, Q)
(1979)

Sphicas          Noninterchangeable        No         (s, Q)
(1982)

Sphicas and      Noninterchangeable        No         (s, Q)
Nasri (1984)

Ramasesh         Noninterchangeable       Yes         (s, Q)
et al.              / Sequential
(1991) (5)

Pan et al.           Sequential           Yes         (s, Q)
(1991) (5)

Hayya et al.         Sequential           Yes         (s, Q)
(1995) (e)

He et al.            Sequential           Yes         (s, Q)
(1998)                                 (pairwise)

Bashyam and          Sequential           Yes       (R, s, S)
Fu (1998)

Robinson,            Sequential           Yes         (R, S)
Bradley, and
Thomas (2001)

Bradley and          Sequential           Yes         (R, S)
Robinson
(2005)

Robinson and         Sequential           Yes         (R, S)
Bradley
(2008)

Hayya et al.         Sequential           Yes         (R, S)
(2008)

Hayya,               Sequential           Yes         (s, Q)
Harrison, and
Chatfield
(2009)

Riezebos and         Sequential           Yes         (R, S)
Caalman
(2009) (e)

Muharremoglu         Sequential           Yes         (R, S)
and Yang
(2010) (e)

Hayya,               Sequential           Yes         (R, S)
Harrison, and
He (2011)

Srinivasan,          Sequential           Yes         (R, S)
Novack, and
Thomas (2011)

Hayya et             Sequential           Yes         (R, S)
al. (2013)

Humair et            Sequential           Yes         (R, S)
al. (2013)

Bischak et           Sequential           Yes         (R, S)
al. (2014)

Wensing and          Sequential           Yes         (R, S)
Kuhn (2015)

Riezebos and         Sequential           Yes         (R, S)
Zhu (2015)
(5)

Disney et            Sequential           Yes         (R, S)
al. (2016)

Study             Review Period       Demand       Lead time

Takacs             Continuous      Unit Poisson    Stochastic
(1956) (a)                           process

Scarf (1958)       Continuous      Convolutions   Exponential
                                     of unit
                                     Poisson
                                     process

Galliher,          Continuous        Poisson      Exponential
Morse, and
Simond (1959)

Washburn           Continuous          Unit           iid
(1973)                               Constant

Gross and          Continuous        Poisson      Exponential
Harris (1973)

Wijngaard          Continuous        Poisson        Negative
(1978)                                            Exponential

Zalkind            R [greater      Discrete iid   Discrete iid
(1978)            than or equal      bounded        bounded
                      to] 1

Llberatore         Continuous        Constant      Stochastic
(1979)

Sphicas            Continuous        Constant      Stochastic
(1982)

Sphicas and        Continuous        Constant      Stochastic
Nasri (1984)                                        bounded

Ramasesh           Continuous        Constant         iid
et al.
(1991) (5)

Pan et al.         Continuous        Constant       Uniform
(1991) (5)                                        Exponential
                                                     Normal

Hayya et al.       Continuous      Normal Gamma   Exponential
(1995) (e)

He et al.          Continuous        Constant       Uniform
(1998)                                            Exponential

Bashyam and           R = l            iid            iid
Fu (1998)

Robinson,             R = 1        Discrete iid   Discrete iid
Bradley, and
Thomas (2001)

Bradley and           R = l        Discrete lid   Discrete iid
Robinson
(2005)

Robinson and          R = l        Discrete iid   Discrete iid
Bradley
(2008)

Hayya et al.          R = l            iid            iid
(2008)

Hayya,             Continuous         Normal        Poisson,
Harrison, and                                     Exponential,
Chatfield                                            Gamma
(2009)

Riezebos and          R = l         Stochastic      Constant
Caalman
(2009) (e)

Muharremoglu          R = 1        Discrete iid   Discrete iid
and Yang                             bounded        bounded
(2010) (e)

Hayya,                R = 1          Constant     Exponential
Harrison, and
He (2011)

Srinivasan,           R = 1          Discrete     Discrete iid
Novack, and                          Uniform        bounded
Thomas (2011)

Hayya et              R = 1          Constant         iid
al. (2013)

Humair et             R = l            iid        Discrete iid
al. (2013)

Bischak et       R [greater than       iid         Gamma-iid
al. (2014)       or equal to] 1                   Gamma-AR(1)

Wensing and      R [greater than    Stochastic      Discrete
Kuhn (2015)      or equal to] 1        iid        bounded iid

Riezebos and          R = l         Stochastic      Constant
Zhu (2015)
(5)

Disney et             R = l        Discrete iid   Discrete iid
al. (2016)

Study            Approach    Multiple     Dynamic/
                            suppliers    Static Lead
                                            Times

Takacs           Queueing      Not           Not
(1956) (a)                  applicable   applicable

Scarf (1958)     Queueing       No         Static

Galliher,        Queueing       No         Static
Morse, and
Simond (1959)

Washburn           LTD                     Static
(1973)

Gross and        Queueing       No         Static
Harris (1973)

Wijngaard        Queueing       No         Static
(1978)

Zalkind             SF          No         Static
(1978)

Llberatore       LTD (d)        No         Static
(1979)

Sphicas          LTD (d)        No         Static
(1982)

Sphicas and      LTD (d)        No         Static
Nasri (1984)

Ramasesh         LTD/ELT       Two         Static
et al.
(1991) (5)

Pan et al.         ELT       Multiple      Static
(1991) (5)

Hayya et al.       LTD         Two         Static
(1995) (e)

He et al.        LTD (4)        No         Static
(1998)

Bashyam and      LTD (d)        No         Static
Fu (1998)

Robinson,           SF          No         Static
Bradley, and
Thomas (2001)

Bradley and         SF          No         Static
Robinson
(2005)

Robinson and        SF          No         Static
Bradley
(2008)

Hayya et al.       ELT          No         Static
(2008)

Hayya,             ELT          No         Static
Harrison, and
Chatfield
(2009)

Riezebos and     LTD (d)       Yes         Dynamic
Caalman
(2009) (e)

Muharremoglu     LTD (d)        No         Static
and Yang
(2010) (e)

Hayya,             ELT          No         Static
Harrison, and
He (2011)

Srinivasan,         SF          No         Static
Novack, and
Thomas (2011)

Hayya et           ELT          No         Static
al. (2013)

Humair et           SF          No         Static
al. (2013)                                 (Multi-
                                          echelon)

Bischak et         ELT          Ni         Static
al. (2014)

Wensing and      SF / ELT       No         Static
Kuhn (2015)

Riezebos and     LTD (d)       Yes         Dynamic
Zhu (2015)
(5)

Disney et           SF          No         Static
al. (2016)

(a) Study of a telephone system with stochastic service times
that has similarities to an inventory system with stochastic
lead times. There is only a unit size outstanding equivalent
order in the telephone system.

(b) Distinct from the typical Markov process queueing model,
E represents an Erlang arrival distribution where magnitudes
(demands) are generated by a Poisson process.

(c) Crossovers, while allowed, do not affect the analyses,
as the noninterchangeability property renders all crossovers
ineffectual.

(d) Lead time demand is implied by the product of the
demand and lead time processes.

(e) Order splitting is allowed.


<ADD> John Patrick Saldanha Corresponding Author West Virginia University john.saldanha@mail.wvu.edu Peter Swan Penn State Harrisburg Transportation Journal, Vol. 56, No. 3, 2017 Copyright @ 2017 The Pennsylvania State University, University Park, PA </ADD>

Notes

(1.) Equivalently, we can assume that the maximum lead time is less than or equal to the period between orders (e.g., lead times are less than or equal to the review period [R] for the base stock policy). Alternatively, Bischak et al. (2014) show that for the base stock policy and gamma-distributed lead time when R/[[sigma].sub.L] = 4, order crossover is rare, and for R/[[sigma].sub.L] = 6, it is eliminated. For the continuous review (s, Q) policy, the incidence of order crossover is less well defined and understood. Under restrictive conditions of deterministic demand and stochastic lead times, He et al. (1998) restrict their analysis to s [less than or equal to] Q, allowing at most pairwise order crossover, while Ramasesh et al. (1991) avoid order crossover by constraining s [less than or equal to] [[mu].sub.x].

(2.) The Online Appendix is available at the first author's researchgate.net profile as a resource under the full text of this article in the contribution section. Alternatively, it can be requested directly from the first author.

(3.) There is a typo in Bischak et al. (2014, 3): the LHS of equation 1 is erroneously inverted. The correct ratio is a [[sigma].sub.L]/[ [sigma].sub.L] [right arrow] 1 as R [right arrow] [infinity]; i.e., the probability of order crossing decreases as [[sigma].sub.L] [right arrow] [[sigma].sub.L]. This correct ratio is also seen in Figure 1 of Bischak et al. (2014,3).

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Caption: Figure 1 Classification of Order Crossover Literature

Caption: Figure 2 Comparison of the Shortfall and the Lead Time Demand Distributions for L ~ U[0,4] and D ~ U[0,3]

Caption: Figure 3 Actual Effective Lead Time Demand ([LTD.sub.E]) versus Estimated Effective [LTD.sub.E]: When Lead Time is Bimodal, [[mu].sub.L] = 4.0, Mix Rate = 0.9, Skew = 2.62 (High Bimodality) for a Base Stock Policy with R = 7
Table 1/Map of the Order-Crossing Literature Classified According
to the Approach, Demand, and Lead Time Type and Inventory Policy

Demand [down arrow]

             Constant   Ha08       Ha08       Ha08   Ha11
                                   He98       He98
                                   Ra91 (1)   Ra91 (1)
                                              Pa91 (1)

             Normal     Hao8       Ha08       Ha08
                                              Ha95 (1)

             Gamma      Ha08       Ha08       Ha08
iid                                           Ha95 (1)
Continuous                                    Hal3

             General    Ha08       Ha08       Ha08
                        Ri09 (1)              Ha09
                        Ri15 (1)

iid          General    Ha08       Ha08       Ha08
Discrete

iid          General    Ha08       Ha08       Ha08
Discrete
Bounded

Lead Time               Constant   Uniform    Exponential
[right arrow]
                                   iid Continuous

             Constant   Ha08    Ha08      Bi14
                        Bi14

             Normal     Ha08    Ha08      Bi14
                        Bi14

             Gamma      Ha08    Ha08      Bi14
iid                     Bi14
Continuous

             General    Ha08    Ha08      Bi14
                        Ha09
                        Bi14

iid          General    Ha08    Hao8      Bi14
Discrete                Bi14

iid          General    Ha08    Ha08      Bi14
Discrete                Bi14
Bounded

Lead Time               Gamma   General   Gamma   General
[right arrow]
                        iid Continuous    AR(1) Continuous
                                          (2)

             Constant   Ha08      Ha08

             Normal     Ha08      Ha08

             Gamma      Ha08      Ha08
iid
Continuous

             General    Ha08      Ha08
                        Ha09      Hul3
                                  Ba98 (4)

iid          General    Ha08      Ha08   Hul3 (5)
Discrete                          Ro01   Dil6
                                  Br05
                                  Ro08

iid          General    Ha08      Ha08
Discrete
Bounded

Lead Time               Poisson   General
[right arrow]
                        iid Discrete

             Constant   Ha08

             Normal     Ha08

             Gamma      Ha08
iid
Continuous

             General    Hao8

iid          General    Ha08
Discrete

iid          General    Ha08   Mu10 (5)
Discrete                Za78
Bounded                 Sr11
                        Wel5 (3)
Lead Time               General           General
[right arrow]

                        iid Discrete      AR(1)
                        Bounded           Discrete (2)

(1) Sourcing from more than one supplier with order splitting

(2) AR(1) is 1st order autocorrelation

(3) Wensing and Kuhn (2015) compared time based customer service
metrics between the SF and ELT approaches

(4) Bashyam and Fu (1998) more specifically use an (R, s, S)
inventory policy

(5) Humair et al. (2013) study order crossing in multi-echelon
supply chains

Table 2/The Absolute Percentage Difference Between the Bischak
et al. (2014) Heuristic and the True Distribution of Effective
Lead Time Demand (LTDE) for Selected Experiments

I

     PNS            70%                75%

R   ([mu]L)    OUP       SS       OUP       SS

1
     3.999    1.86%    21.32%    2.17%    19.05%
    (2.03)

7
     3.999    10.10%   125.67%   11.53%   110.27%
    (2.03)
     6.994    5.74%    77.05%    6.24%    64.48%
    (2.40)
     9.994    3.12%    37.56%    2.85%    26.41%
    (3.59)
    13.000    1.34%    15.59%    0.98%    8.74%
     (482)

30
    19.000    4.21%    61.61%    4.05%    45.69%
    (7.44)
    21.999    4.19%    56.68%    3.86%    40.21%
    (8.76)
   25.0127    4.31%    54.51%    3.99%    38.84%
   (10.21)
    28.001    3.82%    46.20%    3.48%    32.31%
   (11.60)
    31.064    3.26%    37 92%    2.79%    24.88%
   (13.08)
    34.009    2.48%    28.16%    1.99%    17.34%
   (14.41)

      PNS           80%               85%

R   ([mu]L)    OUP       SS      OUP       SS

1
     3.999    2.36%    16.57%   2.51%    14.44%
    (2.03)

7
     3.999    12.92%   99.05%   14.11%   88.49%
    (2.03)
     6.994    6.45%    53.44%   5.91%    40.05%
    (2.40)
     9.994    2.23%    16.55%   1.06%    6.41%
    (3.59)
    13.000    0.74%    5-33%    0.60%    3.53%
     (482)

30
    19.000    3.45%    31.23%   1.56%    11.54%
    (7.44)
    21.999    3.08%    25.73%   0.97%    6.64%
    (8.76)
   25.0127    3.00%    23.39%   0.40%    2.55%
   (10.21)
    28.001    2.43%    18.09%   0.22%    1.35%
   (11.60)
    31.064    1.68%    12.03%   0.77%    4.51%
   (13.08)
    34.009    1.05%    7.30%    0.48%    2.74%
   (14.41)

      PNS          90%          95%

R   ([mu]L)    OUP      SS      OUP

1
     3.999    2.50%   11.86%   2.86%
    (2.03)

7
     3.999    0.25%   1.28%    1549%
    (2.03)
     6.994    2.44%   13.57%   8.89%
    (2.40)
     9.994    0.43%   2.13%    1.69%
    (3.59)
    13.000    0.53%   2.54%    0.58%
     (482)

30
    19.000    4.66%   28.23%   7.42%
    (7.44)
    21.999    5.20%   29.14%   7.36%
    (8.76)
   25.0127    5.43%   28.44%   6.26%
   (10.21)
    28.001    4.58%   22.93%   4.96%
   (11.60)
    31.064    3.21%   15.47%   3.79%
   (13.08)
    34.009    2.06%   9.69%    2.69%
   (14.41)

      PNS      95%          99%

R   ([mu]L)     SS      OUP      SS

1
     3.999    10.90%   3.33%   9.60%
    (2.03)

7
     3.999    64.00%   9.16%   28.44%
    (2.03)
     6.994    39.53%   5.52%   18.36%
    (2.40)
     9.994    6.76%    1.64%   4.95%
    (3.59)
    13.000    2.26%    0.11%   0.31%
     (482)

30
    19.000    35.91%   4.54%   16.38%
    (7.44)
    21.999    32.99%   3.27%   10.97%
    (8.76)
   25.0127    26.32%   1.96%   6.19%
   (10.21)
    28.001    19.92%   0.88%   2.67%
   (11.60)
    31.064    14.68%   0.10%   0.30%
   (13.08)
    34.009    10.16%   0.46%   1.31%
   (14.41)

II

      PNS          70%              75%

R   ([mu]L)    OUP      SS      OUP      SS

1
        10    1.12%   15.55%   1.20%   12.82%

7
         4    0.64%   18.20%   0.55%   12.23%
         7    1.32%   16.55%   1.07%   10.39%

30
        22    1.09%   13.24%   0.81%   7.55%
        25    0.90%   10.30%   0.48%   4.26%

      PNS          80%              85%

R   ([mu]L)    OUP      SS      OUP     SS

1
        10    1.24%   10.64%   1.26%   8.81%

7
         4    0.43%   7.60%    0.27%   3.84%
         7    0.71%   5-47%    0.15%   0.95%

30
        22    0.41%   3.07%    0.21%   1.27%
        25    0.06%   0.43%    0.61%   3.50%

      PNS          90%         95%

R   ([mu]L)    OUP     SS      OUP

1
        10    1.23%   7.08%   1.03%

7
         4    0.08%   0.94%   0.71%
         7    0.84%   4.39%   2.17%

30
        22    1.29%   6.50%   2.34%
        25    1.16%   5.53%   1.34%

      PNS      95%         99%

R   ([mu]L)    SS      OUP      SS

1
        10    4.74%   0.35%   1.20%

7
         4    6.55%   2.53%   17.00%
         7    9.06%   3.24%   10.15%

30
        22    9.42%   2.62%   7.95%
        25    5.10%   0.87%   2.51%

Notes: Top half (I) is for bimodal lead times with 0.9 mix
rate and with CVD = 0.1. Bottom half (II) is for gamma lead
times with CVL = 0.5 and with CVD = 0.1
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