Lead Time Demand Distribution Research Articles

Supply chain model having stochastic lead time demand with variable production rate and demand dependent on price and advertisement

In this present study, a single-manufacturer and single-retailer supply chain management model are formulated for a single product. This study specifically looks at a supply chain with variable production rate, stochastic lead time demand, and price- and advertisement-dependent demand. By incorporating these complex aspects into a model, which enables to examine their combined effects on supply chain performance, this study adds to the body of knowledge. The study reveals unique insights into the complex interplay between pricing tactics, advertising efforts, production dynamics, and the variability brought on by stochastic lead times through meticulous study and modelling. Finally, the total system profit is calculated and optimized with all the decision variables. A classical approach is performed to obtain the optimized solution of the joint profit function along with the decision variables. Two models are discussed in the study: (1) the model with normally distributed lead time demand and (2) the model with distribution-free lead time demand. The joint profit of the supply chain is found to be lesser by 1% for the normally distributed lead time demand than the distribution free pattern. The comparison of the shipment policies and the safety factors for the different distribution patterns of the lead time demand are shown. Though the huge increment in the safety factor for unknown leadtime demand distribution may help in reducing the uncertainity factor and disruptions in the supply chain, but also it may unnecessarily tie up more capital which can be invested in other sectors of the supply chain.

Estimating the reorder point for a fill-rate target under a continuous review policy in the presence of non-standard lead-time demand distributions

Skewed and multimodal lead-time demand (LTD) distributions can be present in supply chains. Under these conditions, logistics managers find that conventional approaches, which assume standard LTD distribution shapes, yield unreliable estimates of a single item’s reorder point (ROP) in relation to meeting fill-rate targets under a continuous review inventory policy. Furthermore, logistics managers have limited data available from which to determine the true shape of the underlying LTD distribution. In this study, I present a non-parametric bootstrap approach to set a least-biased estimate of the ROP. In addition, to deriving bootstrap expressions for non-standard LTD shapes, I derive expressions for standard distribution shapes to estimate the ROP across a range of fill rates. Thus, eliminating the double counting of stockouts with conventional fill-rate measures. Using Monte Carlo simulation experiments, I show that in comparison to the conventional state-of-the-art approaches the non-parametric bootstrap approach yields the least-biased ROP estimates robust to both the shape of the LTD distribution and the sample size. This research offers more accurate ROP estimators, which can serve as a basis for expanding the scope of future inventory management research and enabling logistics managers to reduce inventory while maintaining and even improving fill rates.

The Research on Distribution of Lead-Time Demand on the Basis of Multivariate Higher-Order Markov Chain

Inventory management is an important part of supply chain management: inventory shortages could result in reduced delivery speeds and response speeds while excess inventory could lead to increased inventory and operating costs. Therefore, finding ways to efficiently control inventory has become an issue companies are most concerned about. Choosing a proper inventory management method based on the lead-time demand distribution fitted from historical data has become the key criteria to solve this issue. However, it is difficult to determine the lead-time distribution based on the limited amount of historical data directly. Thus, the method this report introduces uses a multivariate higher-order Markov chain to reconstruct historical data in order to expand the amount of data used to fit the lead-time distribution of demand, which is significant for inventory management.

RNN / LSTM with modified Adam optimizer in deep learning approach for automobile spare parts demand forecasting

The spare parts demand forecasting is very much essential for the organizations to minimize the cost and prevent the stock outs. The demand of spare parts/ car sales distribution is an important factor in inventory control. The valuation of the demand is challenging as the automobile spare parts/car sales demand are often recurrent. The renowned empirical method adopts historical demand data to create the distribution of lead time demand. Although it works reasonably well when service requirements are relatively low, it has difficulty reaching high target service levels. In this paper, we proposed Recurrent Neural Networks/ Long-Short Term Memory (RNN / LSTM) with modified Adam optimizer to predict the demand for spare parts. In this LSTM, weight vectors are generated respectively. These weights are optimized using the Modified-Adam algorithm. The accuracy of the forecast and the performance of the inventory are considered in the experimental result. Experimental results confirm that RNN / LSTM with a Modified-Adam works well with minimal error compared to other existing methods. We conclude that the proposed RNN/LSTM with Modified-Adam algorithm is well suited for the prediction of automobile spare parts.

Coordination Supply Chain Management Under Flexible Manufacturing, Stochastic Leadtime Demand, and Mixture of Inventory

The necessity of coordination among entities is essential for the success of any supply chain management (SCM). This paper focuses on coordination between two players and cost-sharing in an SCM that considers a vendor and a buyer. For random demand and complex product production, a flexible production system is recommended. The study aims to minimize the total SCM cost under stochastic conditions. In the flexible production systems, the production rate is introduced as the decision variable and the unit production cost is minimum at the obtained optimal value. The setup cost of flexible systems is higher and to control this, a discrete investment function is utilized. The exact information about the probability distribution of lead time demand is not available with known mean and variance. The issue of unknown distribution of lead time demand is solved by considering a distribution-free approach to find the amount of shortages. The game-theoretic approach is employed to obtain closed-form solutions. First, the model is solved under decentralized SCM based on the Stackelberg model, and then solved under centralized SCM. Bargaining is the central theme of any business nowadays among the players of an SCM to make their profit within a centralized and decentralized setup. For this, a cost allocation model for lead time crashing cost based on the Nash bargaining model with the satisfaction level of SCM members is proposed. The cost allocation model under Nash bargaining achieves exciting results in SCM coordination.

On the distribution-free continuous review (Q, r, L) inventory model with lead-time-dependent partial backlogging

ABSTRACTIn this article, we study a distribution-free continuous review inventory system that assembles lost sales and backorders. The order quantity, the reorder point and lead-time are decision variables of the problem. The study under consideration assumes that purchasing cost is order dependent, partial backlogging is lead-time dependent and the distribution of lead-time demand is known partially. The objective of the paper is twofold. First, in the random framework, we provide the optimal order quantity, reorder level and lead-time simultaneously to acquire significant savings in the total costs of the model than Sana and Goyal ((q, r, l) model for stochastic demand with lead-time dependent partial backlogging. Annals of Operations Research. 233(1), 401–410, 2015). Second, we extend the crisp model to the fuzzy random environment employing the uncertain demand rate as a fuzzy random variable. We develop the mathematical methodologies to obtain the optimal solutions such that the total expected cost of the inventory system is minimized in both the cases. An algorithm is proposed with the help of the developed methodologies. We employ the proposed algorithm to solve two numerical examples to demonstrate the effectiveness of the methods.

An investigation on bootstrapping forecasting methods for intermittent demands

Many bootstrapping approaches have been proposed in the academic literature for non-parametric demand forecasting. Two approaches have been developed to deal particularly with intermittent demands. A first approach that samples demand data by using a Markov chain to switch between no demand and demand periods and a second approach that separately samples demand intervals and demand sizes. The relevant studies have claimed improvements over parametric approaches when estimating the lead-time demand distribution. However, it should be noted that the outperformance of the two bootstrapping approaches has been shown under a limited set of control parameters and assumptions. The purpose of this paper is to broaden the empirical and numerical settings when analyzing the performance of the two bootstrapping approaches and the parametric one. Hence, more exhaustive assumptions (i.e. generated demand distributions) and a wider range of control parameters (i.e. length of demand histories, length of lead-times, demand distribution parameters and cost parameters) are considered in the numerical analysis. More empirical data with a wider range of demand patterns are also used for the purpose of the empirical investigation. The results show that for highly intermittent demands and small values of lead-times, contrary to what has been claimed in the literature, the parametric approach outperforms the bootstrapping approach that separately samples demand intervals and sizes.

A general method for addressing forecasting uncertainty in inventory models

In practice, inventory decisions depend heavily on demand forecasts, but the literature typically assumes that demand distributions are known. This means that estimates are substituted directly for the unknown parameters, leading to insufficient safety stocks, stock-outs, low service, and high costs. We propose a framework for addressing this estimation uncertainty that is applicable to any inventory model, demand distribution, and parameter estimator. The estimation errors are modeled and a predictive lead time demand distribution obtained, which is then substituted into the inventory model. We illustrate this framework for several different demand models. When the estimates are based on ten observations, the relative savings are typically between 10% and 30% for mean-stationary demand. However, the savings are larger when the estimates are based on fewer observations, when backorders are costlier, or when the lead time is longer. In the presence of a trend, the savings are between 50% and 80% for several scenarios.

How inventory management systems mistreat retail project quantity items and other bimodally distributed products

To insure appropriate inventory levels in retail outlets, replenishment managers concern themselves with both timing of shipments and quantity of product available within the store. While organizations regularly use inventory management or enterprise resource planning systems to procure inventories, we find that these systems can incorrectly manage a sizable group of products found within retail organizations. To illustrate, we explore a collection of goods called retail project quantity items, subsequently explaining how customers purchase and use these items in a fashion that leads to a bimodal distribution of demand. We then employ simulations to demonstrate how the use of a bimodal lead-time demand distribution, rather than a normal or Poisson distribution, improves respective product service levels. Based on these results, we offer a modified reorder point heuristic and discuss how inventory managers should alter existing replenishment processes to insure sufficient stocks of retail project quantity items. Lastly, we discuss broader implications for inventory management systems and bimodally distributed products.

Spare parts inventory management: New evidence from distribution fitting

Spare parts are necessary for ensuring the functioning of the critical equipment of many companies, and as such, they play a central role in these companies’ operations. Inventory control of spare parts is particularly challenging due to the nature of their demand, which is usually slow-moving, erratic and lumpy. As inventory policies rely on the forecasted lead-time demand distribution and this choice impacts the performance of the system, an ill-suited hypothesized distribution may result in high preventable costs. In this study, we contribute to the empirical literature by analyzing what distributions best fit spare parts demand. We use the Kolmogorov Smirnov (K–S) goodness-of-fit test to find the best-fitting distributions to our data and compare our results to those in the literature. Furthermore, we implement a slightly modified K–S test that places greater emphasis on differences in the right tail of the distribution, mirroring real-world inventory applications, and less emphasis on the left tail. Finally, we link the goodness-of-fit of the distributions to their inventory performance. Our first dataset comes from the German renewable energy industry and is composed of the weekly demand for more than 4000 items over the period 2011–2013. The second dataset comes from the Royal Air Force. It is composed of monthly demand for 5000 items over the period 1996–2002.

Order Crossover Research: A 60‐Year Retrospective to Highlight Future Research Opportunities

AbstractWe 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.

A periodic review integrated inventory model with controllable safety stock and setup cost under service level constraint and distribution-free demand

In this study, we consider a stochastic integrated manufacturer-retailer inventory model with service level constraint. The model analyzed in this article considers the situation in which the vendor and the buyer establish a long-term contract and strategic partnership to jointly determine the best strategy. The lead time and setup cost are assumed can be controlled by an additional crashing cost and an investment, respectively. It is assumed that shortages are allowed and partially backlogged on the buyer’s side, and that the protection interval (i.e., review period plus lead time) demand distribution is unknown but has given finite first and second moments. The objective is to apply the minmax distribution free approach to simultaneously optimize the review period, the lead time, the setup cost, the safety factor, and the number of deliveries in order to minimize the joint total expected annual cost. The service level constraint guarantees that the service level requirement can be satisfied at the worst case. By constructing Lagrange function, the analysis regarding the solution procedure is conducted, and a solution algorithm is then developed. Moreover, a numerical example and sensitivity analysis are given to illustrate the proposed model and to provide some observations and managerial implications.

An improved method for forecasting spare parts demand using extreme value theory

Inventory control for spare parts is essential for many organizations due to the trade-off between preventing high holding cost and stockouts. The lead time demand distribution plays a central role in inventory control. The estimation of this distribution is problematic as the spare part demand is often intermittent, and as a consequence often only a limited number of non-zero data points are available in practice. The well-known empirical method uses historical demand data to construct the lead time demand distribution. Although it performs reasonably well when service requirements are relatively low, it has difficulties in achieving high target service levels. In this paper, we improve the empirical method by applying extreme value theory to model the tail of the lead time demand distribution. To make the most out of a limited number of demand observations, we establish that extreme value theory can be applied to lead time demand periods computed over overlapping intervals. We consider two service levels: the expected waiting time and cycle service level. Our experiments show that our method improves the inventory performance compared to the empirical method and is competitive with the WSS method, Croston’s method and SBA for a range of demand distributions.

Optimization Models for the Continuous Review Inventory System

Businesses that carry inventory as a means of responding to customer demand must decide when to place inventory orders and how much to order. A model for selecting an optimal order quantity and reorder point in the continuous review system is implemented in spreadsheet software. Historical data on period demand and lead time is processed by performing a statistical test to select a mixture distribution for lead time demand. This distribution is incorporated into an expected shortage calculation and cost model where shortages incur lost sales and/or goodwill costs. The lead time demand distribution selection is evaluated through experiments. In examples, employing an incorrect distribution leads to expenditures between one and eighteen percent higher than those experienced with the correct model. A mismatch of the backorder or lost sales model with the correct underlying assumptions leads to expenses between four and thirteen percent larger than necessary in an example problem.

Manufacturing quality improvement and setup cost reduction in a vendor-buyer supply chain model

Quality improvement and setup cost reduction of any production system are endless procedure. Customer's demand is always intended to have the best quality product and the industries always try to improve the quality of products. This paper develops a two-echelon supply chain model with quality improvement of products and setup cost reduction under controllable lead time. The lead time demand follows a normal distribution and in the second case, it does not consider any specific distribution except a mean and standard deviation. Both models are solved analytically to obtain global solution. Two improved iterative algorithms are developed in order to obtain the optimal results of decision variables numerically to minimise the total system cost. The expected value of additional information is calculated to show the financial effect for collecting the information about lead time demand distribution. Some numerical examples and sensitivity analysis are given to illustrate the model. [Received 8 December 2016; Revised 27 March 2017; Accepted 21 April 2017]

Controllable lead time, service level constraint, and transportation discounts in a continuous review inventory model

This paper studies two models based on the distribution of lead time demand. The first model assumes a normally distributed lead time demand and the second assumes that there is no specific distribution for lead time demand, but it is with known mean and standard deviation. The continuous-review inventory model is used for both cases. Transportation cost is dependent on the ordered quantity i . e ., how much quantity buyer orders for delivery, based on that, a transportation discount is used to reduce the total cost. Service level constraint is included in this model to avoid backorder cost. Two efficient lemmas are established to obtain the optimum solution of the model. The expected value of additional information (EVAI) is calculated to show the excess amount needed for the distribution free case. Some numerical examples and sensitivity analysis are given to illustrate the model.

Variabilidad de la demanda del tiempo de entrega, existencias de seguridad y costo del inventario

The objective of this research is to investigate the impacts of the lead-time demand variability on the safety stock and the inventory cost.The analysis is applied to two products for which data is available, one with a normal distribution of lead time demand and the other with uniform distribution. The costs analyzed are the safety stock maintenance and the shortages when the consumer demand exceeds availability.The safety stock is set at the value that minimizes the sum of the two previously mentioned costs and is then analyzed how the demand variability affects this stock.The results were similar for the two products tested. With more demand variability, safety stock increases linearly; this behavior is maintained, even if the cost structure of maintenance or shortages are modified. Likewise, with more demand variability the minimum cost increases to a greater extent for the item with a normal distribution. The service level of the minimum cost is located at a value below 100% for both items, 94.5% for the normal distribution item and 96% for the uniform distribution item.In summary, the two articles exhibited an inverse relationship between the service level of the minimum cost and the ratio of the maintenance cost and the shortages.

Lead time uncertainty and supply chain coordination in lost sales inventory models

The effects of lead time uncertainty on optimal inventory policies in a two-echelon supply chain operating under continuous review inventory assumptions are examined. In this system, the buyer selects an inventory order quantity and reorder point to maximise its profits, while the seller selects an order quantity from its external supplier that is an integer multiple of the buyer's order quantity; unsatisfied demand is accounted for as lost sales and may also incur a penalty cost representing lost customer goodwill. Normally distributed demand per unit time is assumed and lead time can follow any discrete probability distribution, resulting in a mixture of Gaussians distribution for lead time demand. Examples where lead time is discrete and not well-approximated by a continuous distribution are examined, and use of the mixture of Gaussians approach versus a single normal distribution for lead time demand in these cases reduces supply chain costs by between 12% and 13%.

Stochastic Reorder Point-Lot Size (r,Q) Inventory Model under Maximum Entropy Principle

This paper takes into account the continuous-review reorder point-lot size (r,Q) inventory model under stochastic demand, with the backorders-lost sales mixture. Moreover, to reflect the practical circumstance in which full information about the demand distribution lacks, we assume that only an estimate of the mean and of the variance is available. Contrarily to the typical approach in which the lead-time demand is supposed Gaussian or is obtained according to the so-called minimax procedure, we take a different perspective. That is, we adopt the maximum entropy principle to model the lead-time demand distribution. In particular, we consider the density that maximizes the entropy over all distributions with given mean and variance. With the aim of minimizing the expected total cost per time unit, we then propose an exact algorithm and a heuristic procedure. The heuristic method exploits an approximated expression of the total cost function achieved by means of an ad hoc first-order Taylor polynomial. We finally carry out numerical experiments with a twofold objective. On the one hand we examine the efficiency of the approximated solution procedure. On the other hand we investigate the performance of the maximum entropy principle in approximating the true lead-time demand distribution.

Inventory management for new products with triangularly distributed demand and lead-time

This paper proposes a computational methodology to deal with the inventory management of new products by using the triangular distribution for both demand per unit time and lead-time. The distribution for demand during lead-time (or lead-time demand) corresponds to the sum of demands per unit time, which is difficult to obtain. We consider the triangular distribution because it is useful when a distribution is unknown due to data unavailability or problems to collect them. We provide an approach to estimate the probability density function of the unknown lead-time demand distribution and use it to establish the suitable inventory model for new products by optimizing the associated costs. We evaluate the performance of the proposed methodology with simulated and real-world demand data. This methodology may be a decision support tool for managers dealing with the measurement of demand uncertainty in new products.