Deterministic Lead Time Research Articles

A measure of the bullwhip effect in supply chains with stochastic lead time

In this paper, we exactly quantify the bullwhip effect, the variance amplification in replenishment orders, for cases of stochastic demand and stochastic lead time in a simple two-stage supply chain with one supplier and one retailer. In most of the previous research, the impact of order lead time on the bullwhip effect in supply chains with pre-specified demand processes is investigated mostly for cases of deterministic lead time. In this paper, we deal with a first-order autoregressive, AR(1), demand process and investigate the behavior of a measure for the bullwhip effect with respect to autoregressive coefficient and stochastic order lead time. Extension to a mixed first-order autoregressive-moving average, ARMA(1,1), demand process is also considered.

Managing quality improvement, setup cost and lead time reductions in a just-in-time integrated inventory model

This article investigates a just-in-time integrated inventory model with quality improvement, setup cost and lead time reduction. The objective is to simultaneously optimise the order quantity, setup cost, process quality level, number of deliveries and lead time. A solution procedure is suggested for solving the proposed model and a numerical example is considered to illustrate the impact of setup cost reduction with capital investment. The current model predicts a lower total cost, order quantity and number of deliveries compared with Yang, J-S. and Pan, J.C-H. (2004) 'Just-in-time purchasing: an integrated inventory model involving deterministic variable lead time and quality improvement investment', Int. J. Production Research, Vol. 42, pp.853-863. An integrated inventory model is helpful in breaking down the traditional barriers as it results in lower total relevant cost compared to the independent inventory policies of both purchaser and vendor. These results can be used to promote cooperation and to minimise the total system cost of integrated enterprises.

Stochastic leadtimes in a one-warehouse, N-retailer inventory system with the warehouse not carrying stock

This study extends upon a multi-echelon inventory model developed by Graves, introducing in the one-warehouse, N-retailer case—as Graves suggested—stochastic leadtimes between the warehouse and the retail sites in place of the original deterministic leadtimes. Effects of stochastic leadtimes on required base stock levels at the retail sites in the case where the warehouse carries no stock (e.g., serves as a cross-dock point) were investigated analytically. Two alternative treatments of stochastic leadtime distributions were considered. Using as a baseline Graves’ computational study under deterministic leadtimes, results of the current study suggest that it may be better to use the deterministic model with an accurately estimated mean leadtime than a stochastic model with a poorly estimated mean leadtime.

A mutually beneficial coordination mechanism for a one-supplier multi-retailers supply chain

In this study, we investigate devising a mutually beneficial coordination mechanism for a supply chain system consisting of a single supplier and a group of heterogeneous retailers. We employ the assumptions in the basic EOQ model, namely, deterministic demand, no shortages, and deterministic lead-time in our study. We also assume that the supplier has complete knowledge of the relevant cost parameters of the retailers. The supplier's (finite) production capacity is sufficient to meet the retailer's demand. All of the retailers follow an integer-ratio policy, i.e., each retailer purchases at an integer or reciprocal of an integer multiple of the supplier's setup interval. We assume that the maximum cost a retailer is prepared to incur cannot exceed its budget constraint when adopting the supplier's production setup interval. In this mutually beneficial coordination mechanism, the supplier offers a price discount to compensate the cost increase a retailer encounters from adopting the supplier's production setup interval. Therefore, our price–discount strategy encourages the retailers to coordinate their replenishment orders to accommodate the supplier's cost saving policies. Also, we propose a savings–sharing policy that asks the supplier to share part of its saving as retailers are willing to implement the designated replenishment schedule. Numerical experiments show that our mutually beneficial coordination mechanism could efficiently generate further cost savings for both the supplier and the retailers in the supply chain. The results also indicate that the cost savings for the supplier may not be sufficient to compensate the cost increase for the retailers when the major setup cost is below a threshold value.

A discrete‐time order‐replacement model with time discounting and spare part provisioning

PurposeTo determine the optimal spare part order‐replacement policy for any high cost single unit complex system in a discrete‐time circumstance.Design/methodology/approachThe expected total discounted cost over an infinite planning horizon is taken as a criterion of optimality as it allows us to put emphasis on the present behavior of the system.FindingsThe problem under consideration is a two‐dimensional discrete optimization problem (regular ordering time and inventory time limit for the spare are decision variables) which is difficult to handle, in general. However, it is explored that the problem can be reduced to a simple one‐dimensional one and the optimal ordering time is to be determined under the two extreme situations: no replacement of the spare until the original unit fails and replacement of the spare as soon as it is delivered.Research limitations/implicationsFor modeling simplicity, deterministic lead time is considered for both regular and expedited orders. A more appropriate assumption would be to consider randomized lead time for both the orders.Practical implicationsThe research provides a useful order‐replacement strategy for a single‐unit system where the failure of the unit is better measured by the number of cycles completed before failure rather than the instant of failure.Originality/valueThe work done in this paper carries certain values as any continuous time model for the problem under consideration can be regarded as only an approximate model.

An inventory system with periodic regular review and flexible emergency review

We analyze a discrete time inventory model with two supply modes. Regular orders are placed periodically and arrive after a deterministic lead time. Emergency orders, characterized by a shorter deterministic lead time but higher purchase costs, can be placed in case of imminent stock-outs. We propose and analyze flexible order-up-to policies with emergency orders that arrive one, two, ⃛, up to some fixed number of time units before a regular order arrives. Based on an approximate cost model, we develop two sets of approximate explicit optimality conditions. The first set of conditions are rather complicated, but simulation shows that they do lead to near-optimal policies in most cases. The second set of conditions are very simple and seem more practical, but they only lead to a near-optimal policy if there are enough emergency supply opportunities in a review period to prevent most backorders.

A Periodic Review Inventory System with Emergency Replenishments

This paper proposes and analyzes a periodic review inventory system with two replenishment modes. Regular orders are placed periodically following a base stock policy on inventory position, and arrive at the stocking location after a deterministic lead time. The location also has the option of placing emergency orders, characterized by a shorter lead time but higher acquisition cost, in case of imminent stockouts. Thus, at some appropriate time in the replenishment cycle, the necessity and size of an emergency order is determined according to a base stock policy on net stock. The timing of the emergency order is such that this order arrives and can be used to satisfy the demand in the time period just before the arrival of a regular order, when the likelihood of a stockout is highest. An approximate cost model is developed which can easily be optimized with respect to the order-up-to parameters. This model is used as the basis for a heuristic algorithm, which leads to solutions that are very close to the exact optimal solutions determined through simulation. It is shown that the proposed system offers substantial cost savings relative to a system without the emergency replenishment option.

An inventory system with periodic regular review and flexible emergency review

We analyze a discrete time inventory model with two supply modes. Regular orders are placed periodically and arrive after a deterministic lead time. Emergency orders, characterized by a shorter deterministic lead time but higher purchase costs, can be placed in case of imminent stock-outs. We propose and analyze flexible order-up-to policies with emergency orders that arrive one, two, ..., up to some fixed number of time units before a regular order arrives. Based on an approximate cost model, we develop two sets of approximate explicit optimality conditions. The first set of conditions are rather complicated, but simulation shows that they do lead to near-optimal policies in most cases. The second set of conditions are very simple and seem more practical, but they only lead to a near-optimal policy if there are enough emergency supply opportunities in a review period to prevent most backorders.

Inventory rationing in an (s, Q) inventory model with lost sales and two demand classes

Whenever demand for a single item can be categorised into classes of different priority, an inventory rationing policy should be considered. In this paper we analyse a continuous review (s, Q) model with lost sales and two demand classes. A so-called critical level policy is applied to ration the inventory among the two demand classes. With this policy, low-priority demand is rejected in anticipation of future high-priority demand whenever the inventory level is at or below a prespecified critical level. For Poisson demand and deterministic lead times, we present an exact formulation of the average inventory cost. A simple optimisation procedure is presented, and in a numerical study we compare the optimal rationing policy with a policy where no distinction between the demand classes is made. The benefit of the rationing policy is investigated for various cases and the results show that significant cost reductions can be obtained.

Addendum to “A Single-Item Inventory Model for a Nonstationary Demand Process”

In preparing a review I recently discovered an important reference for a key result in Graves (1999). Wecker (1979) had previously derived the variance for demand over a deterministic lead-time for an IMA (0, 1, 1) demand process. I develop effectively the same result, given as Equation (8) in Graves (1999). I state this as the variance of the inventory random variable for an inventory system that is subject to an IMA (0, 1, 1) demand process, a deterministic replenishment lead-time and an adaptive base-stock control policy given by (7). But given these assumptions, the variance of the inventory is the same as the variance of the demand over the lead-time. Although the Wecker manuscript has not been published, Eppen and Martin (1988) reference it and use the key result as part of their research.

Optimal replacement for a one‐unit system subject to delivery and test

Notes that the so‐called age replacement model was first proposed to determine the optimum age limit at which the system is replaced after installation. It was then extended by further assuming that the one‐unit system is supported by a buffer which stores only one spare for replacement, and that whenever each replacement starts, one new unit is ordered and then arrives at the buffer in a random delivery time. Others have studied the same model but with the restriction that each new unit will be ordered in a deterministic lead time later, after each replacement starts. This article extends the age replacement model by taking into consideration the possibility that the unit, on arrival, may not be acceptable after test. The expression of the long‐run average cost per unit time is then derived. How to find the optimum age limit which minimizes such an expression is studied and two numerical examples are presented.

Simple optimal replenishment and disposal policies for a product recovery system with leadtimes

The paper addresses a problem of product recovery management where a single product is stocked in order to fulfill a stochastic demand of customers who may return products after usage, thus generating also stochastic product returns. The material flow can be controlled by procuring new products on the one hand, and by remanufacturing or disposal of returned items on the other. A situation is considered where all costs are proportional and where remanufacturing as well as procurement needs a fixed deterministic leadtime which can be different for both activities. For periodic review control it is shown how the optimal decision rules for procurement, remanufacturing and disposal can be evaluated by exploiting the functional equations of a dynamic programming formulation. The serious impact of leadtimes on the complexity of the control rule is elaborated, and it is demonstrated for which leadtime situations simple optimal policies can be derived.

Safety Stock Determination with Serially Correlated Demand in a Periodic-Review Inventory System

We consider a periodic-review inventory replenishment model with an order-up-to-R operating doctrine for the case of deterministic lead times and a covariance-stationary stochastic demand process. A method is derived for setting the inventory safety stock to achieve an exact desired stockout probability when the autocovariance function for Gaussian demand is known. Because the method does not require that parametric time-series models be fit to the data, it is easily implemented in practice. Moreover, the method is shown to be asymptotically valid when the autocovariance function of demand is estimated from historical data. The effects on the stockout rate of various levels of autocorrelated demand are demonstrated for situations in which autocorrelation in demand goes undetected or is ignored by the inventory manager. Similarly, the changes to the required level of safety stock are demonstrated for varying levels of autocorrelation.

Service Constrained (s, S) Inventory Systems with Priority Demand Classes and Lost Sales

This paper presents a model of an (s, S) inventory system in which there are two priority classes of customers. The model treats excess demands as lost sales and can accommodate an arbitrary deterministic lead time. After considering the associated Markov-chain model, an approximate, renewal-based model is derived. This approximation is used to develop a greedy heuristic which minimizes expected costs subject to a fill-rate service constraint. The paper concludes with the results of an extensive numerical test of both the accuracy of the approximation and the performance of the heuristic with respect to the true optimal solution. Results indicate good performance which deteriorates as the fill rate requirement and lead time increase.

A probabilistic scheduling period inventory model for deteriorating items with lead time

A probabilistic scheduling period inventory model is developed for continuously decaying items. The model assumes no shortages, deterministic lead time and a general deterioration function. The developed model is shown to be related to the similar model without lead time and also to the similar model for non-deteriorating items. Two special cases are considered and an example is also furnished.

Planning Horizons for a Stochastic Lead-Time Inventory Model

The planning-horizon literature has thus far focused on the analysis of deterministic lead-time inventory systems. We show that under certain conditions several planning-horizon theorems hold in the stochastic lead-time case as well. Specifically, demands are assumed non-interchangeable and deterministic; thus production runs are assigned to sets of specific demands and can only be used in the satisfaction of those demands. When holding, backlogging, setup (ordering), and variable production costs are stationary over time, and when the lead-time distribution is invariant over time, it is always optimal to produce for sets of consecutive integral demands. Production dates for specific shipments follow from convexity properties. Building upon these results, we prove several planning-horizon theorems (of the Wagner-Whitin and Zabel varieties). A forward dynamic programming recursion is given. These results are shown to generalize those of the basic dynamic lot-size model. We present a numerical example that illustrates the sensitivity of the optimal policy to changes in lead-time variance.

A Dynamic Inventory Model with Stochastic Lead Times

A study is made of a dynamic inventory model with stochastic lead times. A probability model is developed for the arrival of outstanding orders in which it is assumed that orders do not cross in time and that the arrival probabilities are independent of the number and size of outstanding orders. With these assumptions, it is shown that the sequential multidimensional minimization problem normally associated with the random lead time model can be reduced to a sequence of one-dimensional minimizations. The minimizations are a function of a variable representing the sum of stock on hand plus all outstanding orders. Optimal ordering policies are characterized under the assumptions of convex expected holding and shortage costs, a linear ordering cost and a fixed setup cost (greater than or equal to zero) paid when the order is placed. These policies are shown to be quite similar to those obtained with deterministic lead times but some differences in the behavior of the single-period critical numbers (when the setup cost is zero) are noted.