Large Lead Times Research Articles

Prospects and status of forecasting monthly mean subregional rainfall during the Indian summer monsoon using the coupled Unified Model

AbstractWhile there is huge demand for regional forecasts, information needed for selection of the most appropriate temporal and spatial scales is not available. The objective of this study is to demonstrate the basis of forecasting monthly mean rainfall over homogeneous regions by analyzing the forecasting skill and source of predictability. Reforecasts generated at the National Center for Medium Range Weather Forecasting (NCMRWF) for the period 1993–2015 using the coupled Unified Model are used in this study. Analysis of the forecasting skill over increasingly large lead times, averaging periods and spatial scales, is carried out to compare the skill at different time‐scales and to highlight the effect of spatial averaging over regions of coherent rainfall characteristics. Analysis of probabilistic forecasts is carried out to further demonstrate the usefulness of monthly mean forecasts. The influence of forcings on rainfall is studied both in model and in observations to understand the model's skill in representing interannual variability of monthly mean rainfall. Multiple regression analyses carried out for rainfall using climate indices as independent variables shows that the extent of forcings can largely explain the high variability of rainfall during the onset and withdrawal phase compared to the peak phase of monsoons. ENSO‐related subsidence is found to influence mainly the southern peninsular region, while tropical sea surface temperatures (SSTs) in the Indian Ocean are found to influence rainfall over northwest and central India by forcing circulation patterns typically associated with circumglobal teleconnections (CGTs) which are strongest during the month of June. Interestingly, the influence of CGTs on rainfall in the northeast is opposite to its influence on other homogeneous regions, which explains the contrast in influence of the North Indian Ocean SSTs on rainfall over the northeast and over All India. The model representation of influence of forcings and strength of teleconnections is better for specific region–month pairs, which is seen to influence the monthly variations in skill of forecasting rainfall over homogeneous regions.

Asymptotic Optimality of Semi-Open-Loop Policies in Markov Decision Processes with Large Lead Times

A generic way to verify asymptotic optimality of semi-open-loop policies for a wide class of MDPs with large lead times. In many real-life inventory models, order lead times can result in uncertain effects of inventory decisions. However, as the lead time grows large, one would naturally postulate that the effect of the delayed order depends weakly on the current inventory level and, thus, intuit that decoupling the delayed order with the current inventory level may provide good heuristics. Motivated by these examples, in “Asymptotic Optimality of Semi-open-Loop Policies in Markov Decision Processes with Large Lead Times,” Bai et al. consider a generic Markov decision process (MDP) with one delayed control and one immediate control. For MDPs defined on general spaces with uniformly bounded cost functions and a fast mixing property, they construct a periodic semi-open-loop policy for each lead time value and show that these policies are asymptotically optimal as the lead time goes to infinity. For MDPs defined on Euclidean spaces with linear dynamics and convex structures, they impose another set of conditions under which constant delayed-control policies are asymptotically optimal.

Asymptotic Optimality of Semi-Open-Loop Policies in Markov Decision Processes with Large Lead Times

A generic way to verify asymptotic optimality of semi-open-loop policies for a wide class of MDPs with large lead times. In many real-life inventory models, order lead times can result in uncertain effects of inventory decisions. However, as the lead time grows large, one would naturally postulate that the effect of the delayed order depends weakly on the current inventory level and, thus, intuit that decoupling the delayed order with the current inventory level may provide good heuristics. Motivated by these examples, in “Asymptotic Optimality of Semi-open-Loop Policies in Markov Decision Processes with Large Lead Times,” Bai et al. consider a generic Markov decision process (MDP) with one delayed control and one immediate control. For MDPs defined on general spaces with uniformly bounded cost functions and a fast mixing property, they construct a periodic semi-open-loop policy for each lead time value and show that these policies are asymptotically optimal as the lead time goes to infinity. For MDPs defined on Euclidean spaces with linear dynamics and convex structures, they impose another set of conditions under which constant delayed-control policies are asymptotically optimal.

1.79-Approximation Algorithms for Continuous Review Single-Sourcing Lost-Sales and Dual-Sourcing Inventory Models

Stochastic inventory systems with lead times are often challenging to optimize, including single-sourcing lost-sales and dual-sourcing systems. Recent numerical results suggest that capped policies demonstrate superior performance over existing heuristics. However, the superior performance lacks a theoretical foundation. In “1.79-Approximation Algorithms for Continuous Review Single-Sourcing Lost-Sales and Dual-Sourcing Inventory Models,” the author provides a theoretical foundation for this phenomenon in two classical inventory models. First, in a continuous review lost-sales model with lead times and Poisson demand, he proves that a capped base-stock policy has a worst-case performance guarantee of 1.79 by conducting an asymptotic analysis under a large penalty cost and lead time. Second, in a more complex continuous review dual-sourcing model with general lead times and Poisson demand, he proves that a similar capped dual-index policy has a worst-case performance guarantee of 1.79 under large lead time and ordering cost differences. The results provide a deeper understanding of the superior numerical performance of capped policies and present a new approach to proving worst-case performance guarantees of simple policies in hard inventory problems.

Beyond probabilities: A possibilistic framework to interpret ensemble predictions and fuse imperfect sources of information

AbstractEnsemble forecasting is widely used in medium‐range weather predictions to account for the uncertainty that is inherent in the numerical prediction of high‐dimensional, nonlinear systems with high sensitivity to initial conditions. Ensemble forecasting allows one to sample possible future scenarios in a Monte‐Carlo‐like approximation through small strategical perturbations of the initial conditions, and in some cases stochastic parametrization schemes of the atmosphere–ocean dynamical equations. Results are generally interpreted in a probabilistic manner by turning the ensemble into a predictive probability distribution. Yet, due to model bias and dispersion errors, this interpretation is often not reliable and statistical postprocessing is needed to reach probabilistic calibration. This is all the more true for extreme events which, for dynamical reasons, cannot generally be associated with a significant density of ensemble members. In this work we propose a novel approach: a possibilistic interpretation of ensemble predictions, taking inspiration from possibility theory. This framework allows us to integrate in a consistent manner other imperfect sources of information, such as the insight about the system dynamics provided by the analogue method. We thereby show that probability distributions may not be the best way to extract the valuable information contained in ensemble prediction systems, especially for large lead times. Indeed, shifting to possibility theory provides more meaningful results without the need to resort to additional calibration, while maintaining or improving skills. Our approach is tested on an imperfect version of the Lorenz '96 model, and results for extreme event prediction are compared against those given by a standard probabilistic ensemble dressing.

A Survey of Recent Progress in the Asymptotic Analysis of Inventory Systems

It has long been recognized that many inventory models most relevant to practice are inherently high‐dimensional, and hence generally believed to become computationally intractable as certain problem parameters grow large (suffering from the “curse of dimensionality”). In the last decade, asymptotic analysis has shown that in many interesting settings such problems can actually be well‐approximated by much simpler optimization problems, leading to new algorithms and insights. In this survey, we review the state‐of‐the‐art as regards applying asymptotic analysis to such challenging inventory problems. In addition to surveying the literature, we present a detailed introduction to the relevant tools and methodologies through three in‐depth case studies in which asymptotic analysis has recently led to major progress: lost‐sales models, dual‐sourcing models, and Assemble‐to‐Order systems in the presence of large lead times.

Review of Multi-Supplier Inventory Models in Supply Chain Management: An Update

This paper presents a typology and classifies the literature on inventory models with multiple sourcing options. The categories for classification include the number of considered echelons, the source of uncertainty, the model structure, as well as the general research approach. By means of the classification, the existing research is reviewed and insights and gaps are identified. The literature shows that stochasticity, constraints, cost structures and lead times have a significant impact on the sourcing decision. High demand volatility as well as high uncertainty in the lead time, yield or disruption of a primary source indicates a significant value of a second, reliable, fast but more expensive source. The same is true if a buyer has limited storage space or if the primary supplier is constrained in capacity or order size. If the emergency supplier has a minimum order requirement or the buyer is limited financially, the opposite is true and single sourcing is viable. Furthermore, besides a small emergency cost premium, large lead time differences between suppliers and short emergency lead times favor multiple sourcing. The review identifies several research gaps. Firstly, one important gap is on multi-echelon supply chain structures. Secondly, simple policy structures for more realistic and complex problem settings with multiple suppliers, multiple decision criteria and multiple, possibly correlated, sources of uncertainty should be investigated. Lastly, data driven approaches in industry applications could be a focal point of future research efforts.

Asymptotic Optimality of Semi-Open-Loop Policies in Markov Decision Processes with Large Lead Times

We consider a generic Markov decision process (MDP) with two controls: one control taking effect immediately and the other control whose effect is delayed by a positive lead time. Computing the optimal policy of this MDP is difficult when the lead time is large. Interestingly, as the lead time grows, one would naturally expect that the effect of the delayed action only weakly depends on the current state, and intuitively decoupling the delayed action from the current state could provide good controls. The purpose of this paper is to substantiate this decoupling intuition for some MDPs by establishing asymptotic optimality of the semi-open-loop policies, which specify open-loop controls for the delayed action and closed-loop controls for the immediate action. Specifically, we show that for an MDP with a fast mixing property and uniformly bounded cost functions, certain periodic semi-open-loop policies are asymptotically optimal. For a classical lost-sales inventory model with divisible products, we provide an elementary proof of asymptotic optimality of constant-order policies. For the same model with indivisible products and integral order quantities, we prove that a special integral open-loop policy, referred to as bracket policy, is asymptotically optimal. Our approach relies on a natural lower bound, provided by the optimal semi-open-loop policies for a finite-horizon problem with a horizon length equal to the original model lead time. We show that as the horizon length becomes large, the long-run average cost incurred by some specific semi-open-loop policies becomes close to the lower bound.

Lead Time Independent Joint Inventory and Pricing Controls for Lost Sales System with Probabilistic Service Level Constraints

We consider the problem of joint inventory and pricing control in a finite-horizon setting with Poisson demand, positive lead time, and lost sales. We study the performance of a lead time independent heuristic control, which we call Dynamic Batch Pricing (DBP), and show that it is asymptotically optimal when the annual expected demand is large. Moreover, with a proper tuning of the control parameters, we also show that DBP has a robust performance with respect to the magnitude of lost sales cost, which is important in applications such as retails where the targeted service level is very high (equivalently, the lost sales cost is very large). Our result for the lost sales system complements the recent result of Chen et al. (2019), who show that a lead time independent heuristic can be near optimal in the backorder system with a large lead time. Together, both our result and Chen et al. (2019) reinforce the message that simple lead time independent heuristics can perform sufficiently well, at least in some practically relevant cases if not in all cases. This insight is important given that the task of solving the joint inventory and pricing problem (under either backorder or lost sales system) with positive lead time for the most general model remains very challenging.

A 1.79-approximation Algorithm for a Continuous Review Lost-Sales Inventory Model

Stochastic inventory systems with lead times are often challenging to optimize, including single-sourcing lost-sales and dual-sourcing inventory systems. Recent numerical results have suggested that capped policies demonstrate superior performance over existing heuristics. However, the superior performance lacks a theoretical foundation, and why such policies generally perform so well remains a major open question. In this paper, we provide a theoretical foundation for this phenomenon in two classical inventory models with lead times. First, in a continuous review lost-sales inventory model with lead times and Poisson demand, we prove that a so-called capped base-stock policy has a worst-case performance guarantee of 1.79, by conducting an asymptotic analysis under a large penalty cost and lead time. Second, we extend the analysis to a more complex continuous review dual-sourcing inventory model with general lead times and Poisson demand, and also prove that a so-called capped dual-index policy has a worst-case performance guarantee of 1.79 under large lead time and ordering cost differences. Our results provide a deeper understanding of the superior numerical performance of capped policies, and present a new approach to proving worst-case performance guarantees of simple policies in hard inventory problems.

Manufacturing Application-Driven Foveated Near-Eye Displays.

Traditional optical manufacturing poses a great challenge to near-eye display designers due to large lead times in the order of multiple weeks, limiting the abilities of optical designers to iterate fast and explore beyond conventional designs. We present a complete near-eye display manufacturing pipeline with a day lead time using commodity hardware. Our novel manufacturing pipeline consists of several innovations including a rapid production technique to improve surface of a 3D printed component to optical quality suitable for near-eye display application, a computational design methodology using machine learning and ray tracing to create freeform static projection screen surfaces for near-eye displays that can represent arbitrary focal surfaces, and a custom projection lens design that distributes pixels non-uniformly for a foveated near-eye display hardware design candidate. We have demonstrated untethered augmented reality near-eye display prototypes to assess success of our technique, and show that a ski-goggles form factor, a large monocular field of view (30o×55o), and a resolution of 12 cycles per degree can be achieved.

Ionospheric anomalies detected by ionosonde and possibly related to crustal earthquakes in Greece

Abstract. Ionosonde data and crustal earthquakes with magnitude M≥6.0 observed in Greece during the 2003–2015 period were examined to check if the relationships obtained earlier between precursory ionospheric anomalies and earthquakes in Japan and central Italy are also valid for Greek earthquakes. The ionospheric anomalies are identified on the observed variations of the sporadic E-layer parameters (h′Es, foEs) and foF2 at the ionospheric station of Athens. The corresponding empirical relationships between the seismo-ionospheric disturbances and the earthquake magnitude and the epicentral distance are obtained and found to be similar to those previously published for other case studies. The large lead times found for the ionospheric anomalies occurrence may confirm a rather long earthquake preparation period. The possibility of using the relationships obtained for earthquake prediction is finally discussed. Keywords. Ionosphere (Ionospheric disturbances)

On the Performance of Tailored Base-Surge Policies: Theory and Application at Walmart.Com

Problem Definition: Many import items sold at Walmart.com have two suppliers: one is faster and has a random production capacity limit while the other is slower and has infinite capacity. The lead time difference of the two suppliers could be as large as 12 weeks. Walmart.com faces an important and challenging supply chain problem and must make routine inventory replenishment decisions. Academic/Practical Relevance: Dual-sourcing inventory models have been extensively studied in academia and dual-sourcing strategies have been widely implemented in practice. Recently, the so-called Tailored Base-Surge (TBS) policy has begun to receive more attention and demonstrated good performance in existing dual-sourcing models. TBS is intuitive and easy to manage: a constant-order is placed from the slow (regular) supplier every period and a base-stock policy is adapted from the fast (express) supplier. Methodology: This paper develops a new dual-sourcing inventory model incorporating large lead time difference as well as random capacity of the express supplier, and investigates the performance of TBS both theoretically and empirically by using Walmart’s data. Results: We prove that a TBS policy is asymptotically optimal as the regular lead time grows, extending the previous asymptotic optimality of TBS policies in the dual-sourcing system without random capacities. We complement the theory and compare TBS with three well-known heuristics in a “simulated” real-world setting through model calibration by using historical sales data from Walmart.com. Our results demonstrate that TBS outperforms others as for minimizing the long-run average inventory cost. In particular, the inventory cost would be reduced by at least 33% if adapting a TBS policy. Our results also suggest that at an aggregate level, TBS could reduce Weeks of Supply by 2.7 weeks while increasing Met Demand by 3.3% compared with the single-sourcing policy using only the import supplier. Managerial Implications: Our results show that if the proposed dual-sourcing model and algorithm are adapted, Walmart’s inventory system would be significantly improved especially for import items.

Constant-Order Policies for Lost-Sales Inventory Models with Random Supply Functions: Asymptotics and Heuristic

We consider an infinite-horizon lost-sales inventory model where the supply takes positive lead times and is a random function of the order quantity (e.g., random yield/capacity). The optimal policy for this model is computationally intractable; and no heuristic has been proposed in the literature. In this paper, we focus on a simple class of constant-order policies (COP) that place the same order in every period, regardless of the system state. Under some assumptions on the random supply function, we prove that the best COP is asymptotically optimal with large lead times and the optimality gap converges to zero exponentially fast in the lead time. We also prove that, if the mean supply capacity is less than the mean demand, then the best COP is also asymptotically optimal with large penalty costs; otherwise, the long-run average cost of the best COP asymptotically increases at the rate of square-root of the penalty cost. Further, we construct a simple heuristic COP and show that it performs very close to the best COP. Finally, we provide a numerical study to derive further insights into the performance of the best COP.

Asymptotic Optimality of Tailored Base-Surge Policies in Dual-Sourcing Inventory Systems

Dual-sourcing inventory systems, in which one supplier is faster (i.e., express) and more costly, while the other is slower (i.e., regular) and cheaper, arise naturally in many real-world supply chains. These systems are notoriously difficult to optimize because of the complex structure of the optimal solution and the curse of dimensionality, having resisted solution for over 40 years. Recently, so-called tailored base-surge (TBS) policies have been proposed as a heuristic for the dual-sourcing problem. Under such a policy, a constant order is placed at the regular source in each period, while the order placed at the express source follows a simple order-up-to rule. Numerical experiments by several authors have suggested that such policies perform well as the lead time difference between the two sources grows large, which is exactly the setting in which the curse of dimensionality leads to the problem becoming intractable. However, providing a theoretical foundation for this phenomenon has remained a major open problem. In this paper, we provide such a theoretical foundation by proving that a simple TBS policy is indeed asymptotically optimal as the lead time of the regular source grows large, with the lead time of the express source held fixed. Our main proof technique combines novel convexity and lower-bounding arguments, an explicit implementation of the vanishing discount factor approach to analyzing infinite-horizon Markov decision processes, and ideas from the theory of random walks and queues, significantly extending the methodology and applicability of a novel framework for analyzing inventory models with large lead times recently introduced by Goldberg and coauthors in the context of lost-sales models with positive lead times. This paper was accepted by Gad Allon, operations management.

Optimality Gap of Constant-Order Policies Decays Exponentially in the Lead Time for Lost Sales Models

Inventory models with lost sales and large lead times have traditionally been considered intractable due to the curse of dimensionality. Recently, Goldberg and coauthors laid the foundations for a new approach to solving these models, by proving that as the lead time grows large, a simple constant-order policy is asymptotically optimal. However, the bounds proven there require the lead time to be very large before the constant-order policy becomes effective, in contrast to the good numerical performance demonstrated by Zipkin even for small lead time values. In this work, we prove that for the infinite-horizon variant of the same lost sales problem, the optimality gap of the same constant-order policy actually converges exponentially fast to zero, with the optimality gap decaying to zero at least as fast as the exponential rate of convergence of the expected waiting time in a related single-server queue to its steady-state value. We also derive simple and explicit bounds for the optimality gap, and demonstrate good numerical performance across a wide range of parameter values for the special case of exponentially distributed demand. Our main proof technique combines convexity arguments with ideas from queueing theory.

Asymptotic Optimality of Constant-Order Policies for Lost Sales Inventory Models with Large Lead Times

Lost sales inventory models with large lead times, which arise in many practical settings, are notoriously difficult to optimize due to the curse of dimensionality. In this paper, we show that when lead times are large, a very simple constant-order policy, first studied by Reiman, performs nearly optimally. The main insight of our work is that when the lead time is very large, such a significant amount of randomness is injected into the system between when an order for more inventory is placed and when the order is received, that “being smart” algorithmically provides almost no benefit. Our main proof technique combines a novel coupling for suprema of random walks with arguments from queueing theory.

Repair of a mirror coating on a large optic for high laser damage applications using ion milling and over-coating methods

When an optical coating is damaged, deposited incorrectly, or is otherwise unsuitable, the conventional method to restore the optic often entails repolishing the optic surface, which can incur a large cost and long lead time. We propose three alternative options to repolishing, including (i) burying the unsuitable coating under another optical coating, (ii) using ion milling to etch the unsuitable coating completely from the optic surface and then recoating the optic, and (iii) using ion milling to etch through a number of unsuitable layers, leaving the rest of the coating intact, and then recoating the layers that were etched. Repairs were made on test optics with dielectric mirror coatings according to the above three options. The mirror coatings to be repaired were quarter wave stacks of HfO2 and SiO2 layers for high reflection at 1054 nm at 45 deg incidence in P-polarization. One of the coating layers was purposely deposited incorrectly as Hf metal instead of HfO2 to evaluate the ability of each repair method to restore the coating’s high laser-induced damage threshold (LIDT) of 64.0 J/cm2. The repaired coating with the highest resistance to laser-induced damage was achieved using repair method (ii) with an LIDT of 49.0 to 61.0 J/cm2.

Price and lead time differentiation, capacity strategy and market competition

We study a duopoly market in which customers are heterogeneous, and can be segmented as price or time sensitive. Each firm tailors (differentiates) its products/services for the two customer classes solely based on guaranteed lead time and the corresponding price. Our objective is to understand how competition affects price and lead time differentiation of the firms in the presence of different operations strategy (shared versus dedicated capacity), product substitution and asymmetry between the competing firms. Our results suggest that when firms use dedicated resources to serve the two market segments, pure price competition always tends to decrease individual prices as well as price differentiation, irrespective of the market behaviour. Further, the effect of competition is more pronounced when customers are allowed to self-select, thereby introducing substitutability between the two product options. On the other hand, when firms compete in time, in addition to price, the effect of competition on product differentiation depends crucially on the behaviour of the market. Our results further suggest that the firm with a larger market base should always maintain a larger price and lead time differentiation between the two market segments. Similarly, the firm with a capacity cost advantage should also maintain a larger lead time differentiation.

Combining STEREO SECCHI COR2 and HI1 images for automatic CME front edge tracking

COR2 coronagraph images are the most commonly used data for coronal mass ejection (CME) analysis among the various types of data provided by the STEREO (Solar Terrestrial Relations Observatory) SECCHI (Sun-Earth Connection Coronal and Heliospheric Investigation) suite of instruments. The field of view (FOV) in COR2 images covers 2–15 solar radii (Rs) that allow for tracking the front edge of a CME in its initial stage to forecast the lead-time of a CME and its chances of reaching the Earth. However, estimating the lead-time of a CME using COR2 images gives a larger lead-time, which may be associated with greater uncertainty. To reduce this uncertainty, CME front edge tracking should be continued beyond the FOV of COR2 images. Therefore, heliospheric imager (HI1) data that covers 15–90 Rs FOV must be included. In this paper, we propose a novel automatic method that takes both COR2 and HI1 images into account and combine the results to track the front edges of a CME continuously. The method consists of two modules: pre-processing and tracking. The pre-processing module produces a set of segmented images, which contain the signature of a CME, for both COR2 and HI1 separately. In addition, the HI1 images are resized and padded, so that the center of the Sun is the central coordinate of the resized HI1 images. The resulting COR2 and HI1 image set is then fed into the tracking module to estimate the position angle (PA) and track the front edge of a CME. The detected front edge is then used to produce a height-time profile that is used to estimate the speed of a CME. The method was validated using 15 CME events observed in the period from January 1, 2008 to August 31, 2009. The results demonstrate that the proposed method is effective for CME front edge tracking in both COR2 and HI1 images. Using this method, the CME front edge can now be tracked automatically and continuously in a much larger range, i.e., from 2 to 90 Rs, for the first time. These improvements can greatly help in making the quantitative CME analysis more accurate and have the potential to assist in space weather forecasting.