Replenishment Lead Time Research Articles

The Repair Kit Problem with positive replenishment lead times and fixed ordering costs

The Repair Kit Problem (RKP) concerns the determination of a set of items taken by a service engineer to perform on-site product support. Such a set is called a kit. Models developed in the literature have always ignored the lead times associated with delivering items to replenish the kit, thereby limiting the practical relevance of the proposed solutions. Motivated by a real life case, we develop a model with positive lead times to control the replenishment quantities of the items in the kit, and study the performance of (s, S) policies under a service objective. The choice for (s, S) policies is made in order to accommodate fixed ordering costs. We present a method to calculate job fill rates with exact expressions, and discuss a heuristic approach to optimize the reorder level and order-up-to level for each item in the kit. The empirical utility of the model is assessed on real world data from an equipment manufacturer and useful insights are offered to after-sales managers.

A study on the effectiveness of inventory management and control system in a milk producer organisation

Inventory management is mainly concerned about identifying the amount and the position of the goods that a firm has in their inventory. Inventory management is imperative as it helps to defend the intended course of production against the chance of running out of important materials or goods. Inventory management also includes making essential connections between the replenishment lead time of goods, asset management, and the carrying costs of inventory, future inventory price forecasting, physical inventory, and available space for inventory, demand forecasting and much more. The major objective of the study is to examine to the techniques used by the company to control the inventory. It also includes studying the effectiveness of the technique adopted. In this report we can come to know the background information and the objective of the industry as a whole and also develop the project procedures and various method performed in order to get a detailed report on the industry.

A Simple Heuristic for Joint Inventory and Pricing Models with Lead Time and Backorders

We study a joint inventory and pricing problem in a single-stage system with a positive lead time. We consider both additive and multiplicative demand forms. This problem is, in general, intractable due to its computational complexity. We develop a simple heuristic that resolves this issue. The heuristic involves a myopic pricing policy that generates each period’s price as a function of the initial inventory level and a base-stock policy for inventory replenishment. In each period, the firm monitors its so-called price-deflated inventory position and places an order to reach a target base-stock level. The price-deflated inventory position weights the on-hand and pipeline inventory according to a factor that reflects the sensitivity of price to the net inventory level. To assess the effectiveness of our heuristic, we construct an upper bound to the exact system. The upper bound is based on an information-relaxation approach and involves a penalty function derived from the proposed heuristic. A numerical study suggests that the heuristic is near-optimal. The heuristic approach can be applied to a wide variety of inventory systems, such as systems with fixed ordering costs or fixed batch sizes. The heuristic enables us to explore the use of price as a lever to balance supply and demand. Our findings indicate that a responsive strategy (that effectively reduces the replenishment lead time) leads to a more stable pricing policy and that the value of dynamic pricing increases with lead time. This paper was accepted by Martin Lariviere, operations management.

Impacts of retailer’s risk averse behaviors on quick response fashion supply chain systems

Supply chain systems for fashion apparel products face a high level of risk as the market demand is very volatile and unpredictable. In order to cope with demand volatility, the quick response system which aims to shorten replenishment lead time has been well-established. With a shortened lead time, retailers can postpone the ordering decision and improve their demand forecast by gathering updated market information. However, there is a limit for quick response in which the market demand forecast is never fully accurate and uncertainty can never be fully eliminated. Thus, to keep the level of risk under control, retailers tend to possess a risk-averse behaviour in making their respective inventory decisions. In this paper, we explore the make-to-order quick response fashion supply chain with a risk averse retailer. We employ the mean-risk framework to incorporate the retailer’s risk averse behavior into the optimization model. With the focal point on uncovering the impacts brought by the retailer’s risk averse behavior to the quick response fashion supply chain system, we analytically derive important theoretical insights regarding the retailer’s optimal decisions, the implied inventory service levels, the values of quick response, and the contractual arrangements to attain Pareto improvement when the retailer is risk averse.

Reservation policies of advance orders in the presence of multiple demand classes

We consider an inventory system where customers provide advance order information. Specifically each customer order contains two attributes: a request date when the order is received and a due date specifying when the customer wants the order delivered. We assume M different customer classes whose demands are generated by independent Poisson processes. All customers in the same class have the same constant demand lead-time (the difference between the due date and the request date), and all demand lead-times are less than the replenishment lead-time of the inventory system. The inventory system is operated as a base-stock system where a replenishment order is issued instantaneously upon the receipt of a customer order. A key characteristic is that customers with a long demand lead-time can be overtaken by customers with a shorter demand lead-time. Therefore, a potentially important issue is: should items in stock be reserved for specific order requests, and if so, when? Here it is understood that after the reservation, the item cannot be used to fill other customer requests. We describe a general reservation policy, meaning that to each customer class a given reservation delay arises between receipt of an order and the reservation. Assuming that the revenue of an order depends on the demand lead-time, we propose a profit optimization model where the expected profit is the difference between the expected revenue and the expected inventory holding costs. We derive the profit function for the general reservation policy and find the optimal base stock level and the reservation policy by exhaustive search. In our numerical results we find cases where, by using the general reservation policy, there are improvements by more than 5% compared to either making no reservations at all, or making a complete reservation (that is, reservation immediately upon receipt) of items to incoming orders. Our numerical results also show that instead of using a general reservation policy, one can, without much loss of optimality, resort to a simpler reservation policy, denoted Backward Delay, BD.

Weibull vs. normal distribution of demand to determine the safety stock level when using the continuous-review (S, s) model without backlogs

Determining safety stock is one of the most critical issues in inventory and logistics management. This paper presents a cost-based analytical model to determine the best safety stock and customer service level for uncertain demand of products during generic replenishment lead times and a continuous-review inventory system. The proposed model uses the Weibull distribution to model the generic stochastic demand. α and β are the scale parameter and the shape parameter, respectively. An experimental analysis based on the assumption of different distributions of demand of product demonstrates the effectiveness of the proposed model compared to the so-called normal-based traditional model. Significantly, the proposed Weibull-based model is a comprehensible and user-friendly tool that can be useful for production system managers and practitioners.

A robust inventory routing policy under inventory inaccuracy and replenishment lead-time

In this paper, we study an inventory routing problem under replenishment lead-time and inventory inaccuracy, which exist extensively in distribution systems for fresh products, but are often ignored in existing research. To solve the problem, a robust inventory routing policy is developed in three steps. At first, we propose the methods of updating the probability of the current net inventory and predicting those in future periods. For each candidate route, we develop a Robust TQL Algorithm to optimize the replenishment time, replenishment quantity and replenishment stage length. Finally, a genetic algorithm-based method is developed to optimize the delivery route.

The value of information sharing in a serial supply chain with AR(1) demand and non-zero replenishment lead times

This paper analyzes the value of information sharing, in terms of reduction in the demand variance and average (on-hand) inventory level, in a single product multi-stage (i.e., serial) supply chain with non-zero replenishment lead times. An order-one auto-regressive, AR(1), process characterizes the customer demand. We quantify the reduction in the demand variance and average (on-hand) inventory level in a multi-stage supply chain considering two information sharing scenarios: (1) supply-chain-wide information sharing; and (2) Vendor-Managed Inventory (VMI). In contrast to the related existing literature on a three-stage supply chain with non-zero replenishment lead times, we prove for a multi-stage supply chain that there exists no difference, in terms of the expectation and the variance of total demand over lead time, between supply-chain-wide information sharing and VMI. When there is an instantaneous replenishment across all firms, our analytical results are in agreement with the existing literature on a multi-stage supply chain. Further, we show that the value of information sharing is always greater than or equal to those obtained by an existing study. We also observe that the variance of total demand over lead time is reflected in the variance of the inventory level, derived using inventory balance equation. Furthermore, we carry out a comparative study with respect to the benefits of information sharing under three different supply chain settings, and find that the value of information sharing is more for upstream firms, when demand correlation over a period is high or when lead times are high or both.

Admission and inventory control of a single-component make-to-order production system with replenishment setup cost and lead time

We consider the admission control and inventory management problems of a single-component make-to-order production system. Components are purchased from suppliers in batches of fixed size subject to stochastic lead times and setup costs. A control policy specifies when a batch of components is purchased, and whether the demand for each MTO production is accepted upon arrival. We formulate the problem as a Markov decision process (MDP) model, and characterize the structure of optimal admission control and inventory replenishment policies. We show that a state dependent base-stock policy is optimal for the inventory replenishment, although the MDP value function is not necessarily convex. We also show that the optimal admission control can be identified as a lattice dependent policy. A sensitivity analysis is conducted to show how the optimal policy changes as a function of the system parameters. To effectively coordinate admission and inventory control decisions, we propose simple, implementable, and yet effective heuristic policies. Our extensive numerical results suggest that the proposed heuristics can greatly help firms to effectively coordinate their admission and inventory control activities.

Inventory management with variable lead-time dependent procurement cost

A reduction in the inventory replenishment lead-time allows reducing safety stock requirements and improving customer service. However, it might be accompanied by increased procurement costs because of premium charges imposed by suppliers, or higher transportation costs. This paper studies a single-stage variable lead-time inventory system with lead-time dependent procurement cost. Selection of the lead-time value represents finding the trade-off between benefits of lead-time reduction and increase in the procurement cost. A model for joint optimization of inventory and procurement costs is developed. Numerical studies are conducted to identify conditions under which lead-time reduction is favorable compared to procuring at the lowest cost.

Optimal and asymptotically optimal policies for assemble-to-order n- and W-systems

We consider two specially structured assemble-to-order (ATO) systems—the N- and W-systems—under continuous review, stochastic demand, and nonidentical component replenishment leadtimes. Using a hybrid approach that combines sample-path analysis, linear programming, and the tower property of conditional expectation, we characterize the optimal component replenishment policy and common-component allocation rule, present comparative statics of the optimal policy parameters, and show that some commonly used heuristic policies can lead to significant optimality loss. The optimality results require certain symmetry in the cost parameters. In the absence of this symmetry, we show that, for systems with high demand volume, the asymptotically optimal policy has essentially the same structure; otherwise, the optimal policies have no clear structure. For these latter systems, we develop heuristic policies and show their effectiveness. © 2016 Wiley Periodicals, Inc. Naval Research Logistics, 2016

Enhancement of Inventory Management for the Wafer Manufacturing Industry by Combining Market Demand Forecast and Demand-Pull Replenishment

Abstract Theory of constraints (TOC) uses a demand-pull replenishment strategy with buffer management (DPBM) to effectively manage inventory. This strategy has been demonstrated to perform better than traditional inventory strategies, such as (s, S), (s, Q), (R, S), (R, s, S) policies. However, the replenishment lead time is long, and demand changes rapidly in the semiconductor industry. These characteristics cause problems in managing semiconductor inventory and complicate the inventory replenishment strategy. In order to effectively solve the issue of inventory management, this paper proposes a novel approach to improve the classical DPBM by using market demand forecast information and demand-pull replenishment to effectively manage inventory in the semiconductor industry. In numerical verification, this paper used real data provided by a wafer foundry in Taiwan to demonstrate the feasibility and effectiveness of the proposed approach. Simulated scenarios that account for different patterns of demand and different degrees of accuracy of the demand forecasts were generated to test the performance of the proposed approach. After comparing the result that was obtained from the proposed approach with the one from the classical DPBM, it was found that the proposed approach can reduce the average inventory and does not adverse the level of service as long as the demand forecasts are not too far from the real demand.

An $(s,S)$ inventory model with level-dependent $G/M/1$-Type structure

Inventory models are widely used in a variety of real-world applications. In particular, inventory systems with perishable items have received a significant amount of attention. We consider an $(s,S)$ continuous inventory model with perishable items, impatient customers, and random lead times. Two characteristic behaviors of impatient customers are balking and reneging. Balking is when a customer departs the system if the item they desire is unavailable. Reneging occurs when a waiting customer leaves the system if their demand is not met within a set period of time. The proposed system is modeled as a two-dimensional Markov process with level-dependent $G/M/1$-type structure. We also consider independent and identically distributed replenishment lead times that follow a phase-type distribution. We find an efficient approximation method for the joint stationary distribution of the number of items in the system, and provide formulas for several performance measures. Moreover, to minimize system costs, we find the optimal values of $s$ and $S$ numerically and perform a sensitivity analysis on key parameters.

A note on the rationing policies of multiple demand classes with lost sales

We study inventory rationing in a system with multiple demand classes and lost sales. It is assumed to have at most one outstanding order, resulting in two periods in an order cycle separated by the time of order release. We review the most related work by Melchiors (2001, 2003) (Ph.D. thesis, University of Aarhus, Int. J. Prod. Econ. 81–82, (11), 461–468), and find that the existing approximated and optimal policies are not easy to obtain due to computational complexity. Also as the rationing issue before order release is not well addressed in literature, in this paper we prove the static rationing being optimal. Furthermore in such a system with two distinct periods, the optimal rationing policy is a combination of a dynamic policy during the replenishment lead time and a static policy before order release. In order to make the rationing policies to be readily used in practice, we introduce two approximated methods for calculating the rationing levels in two periods, respectively. The results, in particular the combination of static and dynamic rationing, outperform the existing approximations in literature. In addition, the computation is obviously simplified due to the efficient algorithm of dynamic rationing and the explicit expressions of static rationing.

A Model to Analyze Airbase Oil Inventory System with Endogenous Lead Time

In this paper, we analyze airplane oil inventory problem of a base in the Republic of Korea Air Force (ROKAF) which should keep maintaining an appropriate inventory level of oil for the efficient operation of flights. In ROKAF, airplane oil supply is ordered (or delivered) at a constant time period, which means reorder point is not stochastic but deterministic. Since the airplane oil inspection time occupies large proportion of time between a reorder point and the time the oil is available for usage, it is imperative to analyze the inspection time which causes delay in fulfilling an order. In addition, if demands increase dramatically because of a war or an unexpected training, the reorder point could be shortened than in normal situations. In such abnormal circumstances, the shortened reorder point may require more delay for order fulfillment. In this setting, the inspection time endogenously determines the replenishment lead time (or supply lead time) and the pattern of the inventory. Meanwhile, since the procedure of airplane oil inspection is composed of several steps, it may be better that the distributions of inspection times are treated as phase-type distributions. In this context, we use D/PH/1/m queue and counting process of D-BMAP (Discrete-time Batch Markovian Arrival Process) with 2-class to analyze the delay (lead time) for order fulfillment. Using this model, we find a way to derive decision variables such as the number of inventory replenishments and average fill rate.

Optimal and Asymptotically Optimal Policies for Assemble-to-Order N- and W-Systems

We consider two specially structured assemble-to-order (ATO) systems -- the N- and W-systems -- under continuous review, stochastic demand, and non-identical component replenishment leadtimes. Using a hybrid approach that combines sample-path analysis, linear programming, and the tower property of conditional expectation, we characterize the optimal component replenishment policy and common-component allocation rule, present comparative statics of the optimal policy parameters, and show that some commonly used heuristic policies can lead to significant optimality loss. The optimality results require a certain symmetry in the cost parameters. In the absence of this symmetry, we show that, for systems with high demand volume, the asymptotically optimal policy has essentially the same structure; otherwise, the optimal policies have no clear structure. For these latter systems, we develop heuristic policies and show their effectiveness.

Modelling Bullwhip Effect in a Closed Loop Supply Chain with ARMA Demand

The growing importance of environmental concerns and focus on recycling has encouraged our research efforts to study the bullwhip effect on Closed Loop Supply Chain (CLSC). This article attempts to measure bullwhip effect in a six echelon CLSC with recycling as a reprocessing option for materials like paper and plastics (perhaps the first time in CLSC literature). The proposed model considers the effect of two critical parameters of CLSC: quality of recyclable raw material (in terms of its yield) and degree of segregation at source. It is assumed that the echelons in CLSC employ an Order-Up-To (OUT) inventory policy with Minimum Mean Square Error (MMSE) forecasting scheme, and that the demand forecast is performed via a first order Auto Regressive Moving Average model (ARMA [1, 1]). The model developed is employed to investigate the impact of Auto Regressive (AR) and Moving Average (MA) parameter, yield of recycled material, degree of segregation at source and the lead-time on the bullwhip effect. The model enables the CLSC managers to anticipate the downstream demand. In order to avoid order-process instability, a careful selection of Auto Regressive (AR) and the MA parameters is advised. Sensitivity analysis on replenishment lead-times provides managerial insights for effective design of recycling—distribution system, with constant accumulated lead-time. Further, the analysis reveals that increased degree of segregation at the source reduces the bullwhip effect.

COORDINATED PRICING AND INVENTORY CONTROL WITH BATCH PRODUCTION AND ERLANG LEADTIMES

This paper addresses a joint pricing and inventory control problem for a batch production system with random leadtimes. Assume that demand arrives according to a Poisson process with a price-dependent arrival rate. Each replenishment order contains a single batch of a fixed lot size. The replenishment leadtime follows an Erlang distribution, with the number of completed phases recording the delivery state of outstanding orders. The objective is to determine an optimal inventory-pricing policy that maximizes total expected discounted profit or long-run average profit. We first show that when there is at most one order outstanding at any point in time and that excess demand is lost, the optimal reorder policy can be characterized by a critical stock level and the optimal pricing decision is decreasing in the inventory level and delivery state. We then extend the analysis to mixed-Erlang leadtime distribution which can be used to approximate any random leadtime to any degree of accuracy. We further extend the analysis to allowing three outstanding orders where the optimal reorder point becomes state-dependent: the closer an outstanding order is to its arrival or the more orders are outstanding, the lower selling price is charged and the lower reorder point is chosen. Finally, we address the backlog case and show that the monotone pricing structure may not be true when the optimal reorder point is negative.

Impact of Order Splitting on Bullwhip Effect in Supply Chain: Case of Identical Lead Time at Distributors-Retailer Links

This research work attempts to establish the bullwhip effect measure under the dual sourcing environment in which the lead time periods of two distributors to fulfill the retailer's orders are identical. Our model was based on the simple three-echelon supply chain with one supplier, two distributors and one retailer for a stationary first-order autoregressive, i.e., AR(1), incoming demand process. It was assumed that the minimum mean-square error forecasting technique and the order-up-to inventory policy were employed in all stages. The impacts of the autoregressive coefficient, the replenishment lead time and the proportion of order quantities placed by the retailer with the two distributors were investigated. A detailed comparison of the bullwhip effect of dual sourcing and that of single sourcing was also provided.

Serial Inventory Systems with Markov-Modulated Demand: Solution Bounds, Asymptotic Analysis, and Insights

We study inventory control of serial supply chains with continuous, Markov-modulated demand (MMD). Our goal is to simplify the computational complexity by resorting to certain approximation techniques, and, in doing so, to gain a deeper understanding of the problem. First, we perform a derivative analysis of the problem's optimality equations, and develop general, analytical solution bounds for the optimal policy. This leads to simple-to-compute near-optimal heuristic solutions, which also reveal an intuitive relationship with the primitive model parameters. Second, we establish an MMD central limit theorem under long replenishment lead time through asymptotic analysis. We show that the relative errors between our heuristic and the optimal solutions converge to zero as the lead time becomes sufficiently long, with the rate of convergence being the square root of the lead time. Third, we show that, by leveraging the Laplace transform, the computational complexity of our heuristic is superior to the existing methods. Finally, we provide the first set of numerical study for serial systems under MMD. The numerical results demonstrate that our heuristic achieves near-optimal performance even under short lead times and outperforms alternative heuristics in the literature. In addition, we observe that, in an optimally run supply chain under MMD, the internal fill rate can be high and the demand variability propagating upstream can be dampened, both different from the system behaviors under stationary demand.