Constant Lead Time Research Articles

Closed-Form Approximations for Optimal (r, q) and (S, T) Policies in a Parallel Processing Environment

We consider a single-item continuous-review (r, q) inventory system with a renewal demand process and independent, identically distributed stochastic lead times. Using a stationary marked-point process technique and a heavy-traffic limit, we prove a previous conjecture that inventory position and inventory on-order are asymptotically independent. We also establish closed-form expressions for the optimal policy parameters and system cost in heavy-traffic limit, the first of their kind, to our knowledge. These expressions sharpen our understanding of the key determinants of the optimal policy and their quantitative and qualitative impacts. For example, the results demonstrate that the well-known square-root relationship between the optimal order quantity and demand rate under a sequential processing environment is replaced by the cube root under a stochastic parallel processing environment. We further extend the study to periodic-review (S, T) systems with constant lead times. The electronic companion is available at https://doi.org/10.1287/opre.2017.1623 .

Performance Bounds and Asymptotic Optimality of Modified (r, Q) Policies for Stochastic Distribution Inventory Systems

We study a classic one-warehouse multi-retailer distribution system, in which any inventory replenishment at each location incurs a fixed-plus-variable cost and takes a constant lead time. The optimal policy is unknown and even if it exists, must be extremely complicated. The goal of this paper is to identify an easy-to-compute heuristic policy within the class of modified echelon (r, Q) policies that does not require an integer-ratio property or a synchronized, nested ordering property, yet has certain performance bounds. We first develop a cost upper bound for any given modified echelon (r, Q) policy. Computation of the bound does not require an exact evaluation of the system-wide cost, which is notoriously difficult. We next adopt parameters of the heuristic by minimizing the cost upper bound, which is equivalent to solving a set of independent single-stage (r, Q) systems. With a cost lower bound that has been established in the literature, we then develop easy-to-compute performance bounds for the heuristic policy. Finally, using those bounds, we show that the proposed modified echelon (r, Q) heuristic policy is asymptotically optimal as a pair of system parameters is scaled up, e.g., when the ratios of the fixed cost of the warehouse over those of the retailers become large. Numerical study demonstrates that our proposed heuristic performs well and tends to outperform the echelon-stock (r, nQ) heuristic policy studied in the literature.

Inventory Control System for a Healthcare Apparel Service Centre with Stockout Risk: A Case Analysis.

Based on the real-world inventory control problem of a capacitated healthcare apparel service centre in Hong Kong which provides tailor-made apparel-making services for the elderly and disabled people, this paper studies a partial backordered continuous review inventory control problem in which the product demand follows a Poisson process with a constant lead time. The system is controlled by an (Q,r) inventory policy which incorporate the stockout risk, storage capacity, and partial backlog. The healthcare apparel service centre, under the capacity constraint, aims to minimize the inventory cost and achieving a low stockout risk. To address this challenge, an optimization problem is constructed. A real case-based data analysis is conducted, and the result shows that the expected total cost on an order cycle is reduced substantially at around 20% with our proposed optimal inventory control policy. An extensive sensitivity analysis is conducted to generate additional insights.

Closed-Form Approximations for Optimal (r,q) and (S,T) Policies in a Parallel Processing Environment

We consider a single-item continuous-review (r,q) inventory system with a renewal demand process and i.i.d. stochastic leadtimes. Using a stationary marked point process technique and a heavy traffic limit, we prove a previous conjecture that inventory position and inventory on-order are asymptotically independent. We also establish closed-form expressions for the optimal policy parameters and system cost in heavy traffic limit, the first of their kind to our knowledge. These expressions sharpen our understanding of the key determinants of the optimal policy and their quantitative and qualitative impacts. For example, the results demonstrate that the well-known square-root relationship between the optimal order quantity and demand rate under a sequential processing environment is replaced by the cube root under a stochastic parallel processing environment. We further extend the study to periodic-review (S,T) systems with constant leadtimes.

Computationally efficient optimization of stock pooling and allocation levels for two-demand-classes under general lead time distributions

ABSTRACTIn this article we develop a procedure for estimating service levels (fill rates) and for optimizing stock and threshold levels in a two-demand-class model managed based on a lot-for-lot replenishment policy and a static threshold allocation policy. We assume that the priority demand classes exhibit mutually independent, stationary, Poisson demand processes and non-zero order lead times that are independent and identically distributed. A key feature of the optimization routine is that it requires computation of the stationary distribution only once. There are two approaches extant in the literature for estimating the stationary distribution of the stock level process: a so-called single-cycle approach and an embedded Markov chain approach. Both approaches rely on constant lead times. We propose a third approach based on a Continuous-Time Markov Chain (CTMC) approach, solving it exactly for the case of exponentially distributed lead times. We prove that if the independence assumption of the embedded Markov chain approach is true, then the CTMC approach is exact for general lead time distributions as well. We evaluate all three approaches for a spectrum of lead time distributions and conclude that, although the independence assumption does not hold, both the CTMC and embedded Markov chain approaches perform well, dominating the single-cycle approach. The advantages of the CTMC approach are that it is several orders of magnitude less computationally complex than the embedded Markov chain approach and it can be extended in a straightforward fashion to three demand classes.

On the calculation of safety stocks when demand is forecasted

The inventory control literature generally assumes that the demand distribution and all its parameters are known. In practical applications it is often suggested to estimate the demand variance either directly or based on the one-period ahead forecast errors. The variance of the lead time demand, essential for safety stock calculations, is then obtained by multiplying the estimated per-period demand variance by the length of the lead time. However, this is flawed, since forecast errors for different periods of the lead time are positively correlated, even if the demand process itself does not show (process) auto-correlation. As a result these traditional procedures lead to safety stocks that are too low. This paper presents corrected lead time demand variance expressions and reorder levels for inventory systems with a constant lead time where demand fluctuates around a constant level. Firstly, we derive the exact lead time forecast error of mean demand conditional on the true demand variance. Secondly, we derive for normally distributed demand the correct reorder level under uncertainty of both the demand mean and variance. We show how the results can be implemented in inventory models, and particularly discuss batch ordering policies combined with moving average and exponential smoothing forecasts. We find that traditional approaches can lead to safety stocks that are up to 30 percent too low and service levels that are up to 10 percent below the target.

Approximated closed-form minimum-cost solution to the (r, q) policy with complete backordering and further developments

This study considers the continuous-review (r, q) inventory policy with complete backordering, a deterministic and constant lead time, and a Gaussian lead-time demand. A closed-form solution to the first-order conditions of optimality is not available, so this system can only be optimized using an iterative method. However, an iterative optimization approach may still limit the application of the model itself in a practical context with numerous items and the need to frequently recalculate the control variables for the respective inventory policies. In fact, if a closed-form optimal solution is not available, the computational effort required by an iterative method may not be negligible and its implementation can actually be technically difficult. Thus, to overcome the characteristic limitations of an iterative optimization approach, we develop a closed-form near-optimal solution in this study, which is derived via an ad hoc logistic approximation of the standard normal loss function. We then provide an alternative formulation of the expected total cost function. In particular, by introducing a parameter for the number of admissible stockouts per time unit, we can ensure the functional dependence of the service level on the order quantity. This allows us to optimize the order quantity, where the safety stock and the expected shortage are considered by the constraint mentioned above. We present numerical tests to investigate the efficiency of the approximated solution method and to study the sensitivity of the new cost model with respect to some fundamental parameters.

Approximated closed-form minimum-cost solutions to the (S − 1, S) policy with complete backordering

In this paper, we consider the continuous review (S − 1, S) policy with complete backordering, deterministic and constant lead-time, and with a Poissonian lead-time demand. Moreover, we take into account two different expressions of the total expected cost depending on the customer-service criterion adopted. First, we approximate the Poissonian lead-time demand with a well-suited Gaussian random variable. Although this system can be easily solved with a simple algorithmic approach, it is interesting to be aware of the direct analytical functional relationship between the optimal value of the decision variable and the parameters of the model. Hence, the second step consists in providing some approximated closed-form minimum-cost solutions for both cost models considered under a Gaussian lead-time demand. An extensive numerical study is finally shown to characterise the precision achieved by the approximations developed.

Inventory models with nonlinear shortage costs and stochastic lead times; applications of shape properties of randomly stopped counting processes

AbstractIn this article, we study generalizations of some of the inventory models with nonlinear costs considered by Rosling in (Oper. Res. 50 (2002) 797–809). In particular, we extend the study of both the periodic review and the compound renewal demand processes from a constant lead time to a random lead time. We find that the quasiconvexity properties of the cost function (and therefore the existence of optimal (s, S) policies), holds true when the lead time has suitable log‐concavity properties. The results are derived by structural properties of renewal delayed processes stopped at an independent random time and by the study of log‐concavity properties of compound distributions. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 345–356, 2015

A lost sales (r, Q) inventory control model for perishables with fixed lifetime and lead time

We consider a perishable inventory system that operates under stochastic demand, constant lifetime and a constant lead time. The system employs a continuous review (r,Q) inventory control policy where unfilled demands are lost. We investigate the properties of the cost function and present an approximation procedure to find the parameters r and Q that minimize the total cost. We then conduct a numerical analysis to examine the performance of the proposed model and study the sensitivity to changes in the system parameters. We demonstrate the suitability of the proposed approximations compared to optimal (r,Q) parameters obtained by simulation and show that our proposal outperforms another approximation procedure from the literature, in particular for increasing ordering cost and demand variability. The proposed model contributes to the literature by providing a simple and efficient algorithm to compute the best (r, Q) parameters that minimize the total cost. Besides, it can be used in automated store ordering systems.

Exact and heuristic linear-inflation policies for an inventory model with random yield and arbitrary lead times

We investigate a periodic-review inventory system for a single item with stochastic demand and random yield. Since the optimal policy for such a system is complicated, we study the class of stationary linear-inflation policies where orders are only placed if the inventory position is below a critical stock level, and where the order quantity is controlled by a yield inflation factor. We consider two different models for the uncertain supply, namely binomial and stochastically proportional yield, and we allow positive and constant lead times as well as asymmetric demand and yield distributions. In this paper we propose two novel approaches to derive optimal and near-optimal numerical values for the critical stock level, minimizing the average holding and backorder cost for a given inflation factor. First, we present a Markov chain approach which is exact in case of negligible lead time. Second, we provide a steady-state analysis to derive approximate closed-form expressions for the optimal critical stock level. We conduct an extensive numerical study to test the performance of our approaches. The numerical experiments reveal an excellent performance of both approaches. Since our derived formulas are easily implementable and highly accurate they are very valuable for practical application.

A computational study on fill rate expressions for single-stage periodic review under normal demand and constant lead time

This paper reviews fill rate expressions for a single stage periodic review inventory system under normal demand and constant lead time, discusses the relationship among all expressions in the literature, and evaluates their robustness and accuracy. Monte Carlo simulation is used to numerically compare all expressions. We present conditions under which some expressions produce higher values than others.

One-for-one-period ordering policy for inventory systems with unidirectional lateral transshipments

Consider an inventory system with one supplier, two retailers, Poisson demand, and constant lead times. One retailer has a higher demand rate and can ship items to the other retailer when facing a stock out, for example. Arriving demands to any retailer are lost if it is out of stock, and it is not possible to initiate a transshipment. Retailers apply one-for-one-period ordering, (1, T), policy. We derive the long-run average cost, prove its convexity, and show that it is independent of the lead time. We present an algorithm to find the optimal solution and minimize the system cost. We show that for long lead times, applying one-for-one-period ordering policy results in a lower cost than the base stock, (S-1, S), policy, so it is preferred in long lead time condition.

Due-date assignment for multi-server multi-stage assembly systems

In this paper, we attempt to present a constant due-date assignment policy in a multi-server multi-stage assembly system. This system is modelled as a queuing network, where new product orders are entered into the system according to a Poisson process. It is assumed that only one type of product is produced by the production system and multi-servers can be settled in each service station. Each operation of every work is operated at a devoted service station with only one of the servers located at a node of the network based on first come, first served (FCFS) discipline, while the processing times are independent random variables with exponential distributions. It is also assumed that the transport times between each pair of service stations are independent random variables with generalised Erlang distributions. Each product's end result has a penalty cost that is some linear function of its due date and its actual lead time. The due date is calculated by adding a constant to the time that the order enters into the system. Indeed, this constant value is decided at the beginning of the time horizon and is the constant lead time that a product might expect between the time of placing the order and the time of delivery. For computing the due date, we first convert the queuing network into a stochastic network with exponentially distributed arc lengths. Then, by constructing an appropriate finite-state continuous-time Markov model, a system of differential equations is created to find the manufacturing lead-time distribution for any particular product, analytically. Finally, the constant due date for delivery time is obtained by using a linear function of its due date and minimising the expected aggregate cost per product.

Critical spare parts ordering decisions using conditional reliability and stochastic lead time

Asset-intensive companies face great pressure to reduce operation costs and increase utilization. This scenario often leads to over-stress on critical equipment and its spare parts associated, affecting availability, reliability, and system performance. As these resources impact considerably on financial and operational structures, the opportunity is given by demand for decision-making methods for the management of spare parts processes. We proposed an ordering decision-aid technique which uses a measurement of spare performance, based on the stress–strength interference theory; which we have called Condition-Based Service Level (CBSL). We focus on Condition Managed Critical Spares (CMS), namely, spares which are expensive, highly reliable, with higher lead times, and are not available in store. As a mitigation measure, CMS are under condition monitoring. The aim of the paper is orienting the decision time for CMS ordering or just continuing the operation. The paper presents a graphic technique which considers a rule for decision based on both condition-based reliability function and a stochastic/fixed lead time. For the stochastic lead time case, results show that technique is effective to determine the time when the system operation is reliable and can withstand the lead time variability, satisfying a desired service level. Additionally, for the constant lead time case, the technique helps to define insurance spares. In conclusion, presented ordering decision rule is useful to asset managers for enhancing the operational continuity affected by spare parts.

Optimal policy for brownian inventory models with general convex inventory cost

We study an inventory system in which products are ordered from outside to meet demands, and the cumulative demand is governed by a Brownian motion. Excessive demand is backlogged. We suppose that the shortage and holding costs associated with the inventory are given by a general convex function. The product ordering from outside incurs a linear ordering cost and a setup fee. There is a constant leadtime when placing an order. The optimal policy is established so as to minimize the discounted cost including the inventory cost and ordering cost.

Performance Evaluation of Modified Echelon <i>(r, Q)</I> Policies for Stochastic Serial Inventory Systems

We consider the classic continuous-review N-stage serial inventory system with a homogeneous Poisson demand arrival process at the most downstream stage (Stage 1). Any shipment to each stage, regardless of its size, incurs a positive fixed setup cost and takes a positive constant leadtime. The optimal policy for this system under the long-run average cost criterion is unknown. Finding a good worst-case performance guarantee for this problem remains an open problem. We tackle this problem by introducing a class of modified echelon (r, Q) policies that do not require Q_{i 1}/Q_i to be a positive integer: Stage i 1 ships to Stage i based on its observation of the echelon inventory position at Stage i; if it is at or below r_i and Stage i 1 has positive on-hand inventory, then a shipment is sent to Stage i to raise its echelon inventory position to r_i Q_i as close as possible. We construct a heuristic policy within this class of policies, which has the following features: First, it has provably primitive-dependent performance bounds. In a two-stage system, the performance of the heuristic policy is guaranteed to be within (1 K_1/K_2) times of the optimal cost, where K_1 is the downstream fixed cost and K_2 is the upstream fixed cost. We also provide an alternative performance bound, which depends on efficiently-computable optimal (r, Q) solutions to N single-stage systems, but tends to be tighter. Second, the heuristic is simple, efficiently computable and numerically performs well, likely to even outperform the optimal integer-ratio echelon (r,Q) policies when K_1 is dominated by K_2. Third, the heuristic is asymptotically optimal, as we take some dominant relationships, between the setup or holding cost primitives at an upstream stage and its immediate downstream stage, to the extreme, e.g., when h_2/h_1\rightarrow0, where h_1 is the downstream holding cost parameter and h_2 is the upstream holding cost parameter.

The order and volume fill rates in inventory control systems

This paper differentiates between an order (line) fill rate and a volume fill rate and specifies their performance for different inventory control systems. When the focus is on filling complete customer orders rather than total demanded quantity the order fill rate would be the preferred service level measure. The main result shows how the order and volume fill rates are related in magnitude. Earlier results derived for a single-item, single-stage, continuous review inventory system with backordering and constant lead times controlled by a base-stock policy are extended in different directions. Demand is initially assumed to be generated by a compound renewal process. An important generalization then concerns the class of customer order-size distributions, i.e. compounding distributions, with increasing failure rate for which the volume fill rate always exceeds the order fill rate. Other extensions consider more general inventory control review policies with backordering, as well as some relations between service measures. A particularly important result in the paper concerns an alternative service measure, the customer order fill rate, and shows how this measure always exceeds the other two more well-known service measures, viz. the order fill rate and the volume fill rate.

Modified Echelon (r, Q) Policies with Guaranteed Performance Bounds for Stochastic Serial Inventory Systems

We consider the classic continuous-review Nstage serial inventory system with a homogeneous Poisson demand arrival process at the most downstream stage (Stage 1). Any shipment to each stage, regardless of its size, incurs a positive fixed setup cost and takes a positive constant lead time. The optimal policy for this system under the long-run average cost criterion is unknown. Finding a good worst-case performance guarantee remains an open problem. We tackle this problem by introducing a class of modified echelon (r, Q) policies that do not require Qi + 1/Qi to be a positive integer: Stage i + 1 ships to Stage i based on its observation of the echelon inventory position at Stage i; if it is at or below ri and Stage i + 1 has positive on-hand inventory, then a shipment is sent to Stage i to raise its echelon inventory position to ri + Qi as close as possible. We construct a heuristic policy within this class of policies, which has the following features: First, it has provably primitive-dependent performance bounds. In a two-stage system, the performance of the heuristic policy is guaranteed to be within (1 + K1/K2) times the optimal cost, where K1 is the downstream fixed cost and K2 is the upstream fixed cost. We also provide an alternative performance bound, which depends on efficiently computable optimal (r, Q) solutions to N single-stage systems but tends to be tighter. Second, the heuristic is simple, it is efficiently computable and it performs well numerically; it is even likely to outperform the optimal integer-ratio echelon (r,Q) policies when K1 is dominated by K2. Third, the heuristic is asymptotically optimal when we take some dominant relationships between the setup or holding cost primitives at an upstream stage and its immediate downstream stage to the extreme, for example, when h2/h1 → 0, where h1 is the downstream holding cost parameter and h2 is the upstream holding cost parameter.

Lead time estimation in MRP/ERP for make-to-order manufacturing systems

Abstract In this paper we consider the problem of dynamically updating the manufacturing lead times estimates that are used in MRP systems. Once a new order with specific and known processing requirements enters a make-to-order manufacturing system, an exact completion time estimate is assigned to it that is based on the system's current status. The approach that is suggested addresses multi-machine, multi-product manufacturing environments with no special configuration of their resources: To this objective, simple, iterative algorithms are used to substitute the fixed lead time estimates of typical MRP systems. The results that are presented underline the significant improvement of the proposed approach over static, constant MRP lead time estimates that are as yet exclusively resorted to. The case study that is presented also supports the method's ease of applicability in real-life Make-To-Order environments.