Integrated vendor--buyer inventory system with sublot sampling inspection policy and controllable lead time

This article investigates the impact of inspection policy and lead time reduction on an integrated vendor--buyer inventory system. We assume that an arriving order contains some defective items. The buyer adopts a sublot sampled inspection policy to inspect selected items. The number of defective items in the sublot sampling is a random variable. The buyer's lead time is assumed reducible by adding crash cost. Two integrated inventory models with backorders and lost sales are derived. We first assume that the lead time demand follows a normal distribution, and then relax the assumption about the lead time demand distribution function and apply the minimax distribution-free procedure to solve the problem. Consequently, the order quantity, reorder point, lead time and the number of shipments per lot from the vendor to the buyer are decision variables. Iterative procedures are developed to obtain the optimal strategy.     

A note on defective units in an inventory model with sub-lot sampling inspection for variable lead-time demand with the mixture of free distributions

In a recent paper Wu and Ouyang (2000) assumed that an arriving order lot may contain some defective items and considered that the number of defective items in the sub‐lot sampled to be a random variable. They derived a modified mixture inventory model with backorders and lost sales, in which the order quantity, re‐order point, and the lead‐time were decision variables. In their studies they assumed that the lead‐time demand followed a normal distribution for the first model and relaxed the assumption about the form of the distribution function of the lead‐time demand for the second model. When the demand of the different customers is not identical with regard to the lead‐time, then one cannot use only a single distribution (such as Wu and Ouyang (2000) ) to describe the demand of the lead‐time. Hence, we extend and correct the model of Wu and Ouyang (2000) by considering the lead‐time demand with the mixed normal distributions (see Everitt and Hand (1981) , and Wu and Tsai (2001) ) for the first model and the lead‐time demand with the mixed distributions for the second model. And we also apply the minimax mixed distributions free approach to the second model. Moreover, we also develop an algorithm procedure to obtain the optimal ordering strategy for each case.     

Computational procedure of optimal inventory model involving controllable backorder rate and variable lead time with defective units

This article considers that the number of defective units in an arrival order is a binominal random variable. We derive a modified mixture inventory model with backorders and lost sales, in which the order quantity and lead time are decision variables. In our studies, we also assume that the backorder rate is dependent on the length of lead time through the amount of shortages and let the backorder rate be a control variable. In addition, we assume that the lead time demand follows a mixture of normal distributions, and then relax the assumption about the form of the mixture of distribution functions of the lead time demand and apply the minimax distribution free procedure to solve the problem. Furthermore, we develop an algorithm procedure to obtain the optimal ordering strategy for each case. Finally, three numerical examples are also given to illustrate the results.     

The distribution free continuous review inventory system with a service level constraint

The stochastic inventory models require the information on the lead time demand. However, the distributional information of the lead time demand is often limited in practice. We relax the assumption that the cumulative distribution function, say F, of the lead time demand is completely known and merely assume that the first two moments of F are known and finite. The distribution free approach for the inventory model consists of finding the most unfavorable distribution for each decision variable and then minimizing over the decision variable. We apply the distribution free approach to the continuous review inventory system with a service level constraint. We develop an iterative procedure to find the optimal order quantity and reorder level.     

Mixture inventory model involving variable lead time and controllable backorder rate

This paper allows the backorder rate as a control variable to widen applications of Ouyang et al.'s model [J. Oper. Res. Soc. 47 (1996) 829]. In this study, we assume that the backorder rate is dependent on the length of lead time through the amount of shortages. We discuss two models that are perfect and partial information about the lead time demand distribution, that is, we first assume that the lead time demand follows a normal distribution, and then remove this assumption by only assuming that the first and second moments of the probability distribution of lead time demand are known. For each case, we develop an algorithm to find the optimal ordering strategy. Three numerical examples are given to illustrate solution procedure.     

On the distribution free continuous-review inventory model with a service level constraint

We consider a continuous-review (Q, r) inventory model with a fill rate service constraint and relax the assumption that the distribution of lead time demand is known. We adopt a distribution free approach: We assume that only the first two moments of the lead time demand distribution are known, and then, optimize the policy parameters against the worst possible distribution. We are able to derive closed-form expressions for the optimal order quantity and reorder point.     

Inventory management with log-normal demand per unit time

This paper examines optimal policies in a continuous review inventory management system when demand in each time period follows a log-normal distribution. In this scenario, the distribution for demand during the entire lead time period has no known form. The proposed procedure uses the Fenton-Wilkinson method to estimate the parameters for a single log-normal distribution that approximates the probability density function (PDF) for lead time demand, conditional on a specific lead time. Once these parameters are determined, a mixture of truncated exponentials (MTE) function that approximates the lead time demand distribution is constructed. The objective is to include the log-normal distribution in a robust decision support system where the PDF that best fits the historical period demand data is used to construct the lead time demand distribution. Experimental results indicate that when the log-normal distribution is the best fit, the model presented in this paper reduces expected inventory costs by improving optimal policies, as compared to other potential approximations.     

Order overlapping: A practical approach for preventing shortages during screening

We develop an EOQ-type model with unreliable supply, where each order contains a random fraction of imperfect quality items with a known distribution. Upon receiving an order, a screening process is conducted to identify imperfect items. These imperfect items are salvaged as a single batch at the end of the screening process. To prevent shortages during the screening process, an order is placed when the inventory level is just enough to cover the demand during the screening period. Then, the demand during the screening period of an order is met from the inventory of the “previous” order. This represents an improvement in customer service level over earlier literature, where demand is assumed to be met during the screening period from the order being screened leading to potential shortages. We show that this improvement in service level comes at a modest cost. We further analytically and numerically study our model and draw useful insights.     

Economic ordering policy for an item with imperfect quality subject to the in-house inspection

This article considers a production/inventory system where each lot of items received or produced contains a random proportion of defective units, items of imperfect quality. The purchaser contacts a 100% inspection in order to identify the perfect (acceptable) quality items. The model examines the following two options for the imperfect quality items: sell them to a secondary market, as a single batch and at a price lower to that of new ones, or rework them at some cost and then use them as new ones to satisfy demand. After inspection, the good quality items are sent to the working inventory warehouse in batches of equal size. For both of these cases, the optimal ordering lot size and the optimal number of batches are obtained. A numerical example illustrates the solution procedure and sensitivity analysis results are reported.     

Computing optimal base-stock levels for an inventory system with imperfect supply

We study a single-item, single-site, periodic-review inventory system with negligible fixed ordering costs. The supplier to this system is not entirely reliable, such that each order is a Bernoulli trial, meaning that, with a given probability, the supplier delivers the current order and any accumulated backorders at the end of the current period, resulting in a Geometric distribution for the actual resupply lead time. We develop a recursive expression for the steady-state probability vector of a discrete-time Markov process (DTMP) model of this imperfect-supply inventory system. We use this recursive expression to prove the convexity of the inventory system objective function, and also to prove the optimality of our computational procedure for finding the optimal base-stock level. We present a service-constrained version of the problem and show how the computation of the optimal base-stock level using our DTMP method, incorporating the explicit distribution of demand over the lead time plus review (LTR) period, compares to approaches in the literature that approximate this distribution. We also show that the version of the problem employing an explicit penalty cost can be solved in closed-form for the optimal base-stock level for two specific period demand distributions, and we explore the behavior of the optimal base-stock level and the corresponding optimal service level under various values of the problem parameters.     

Some remarks on an optimal order quantity and reorder point when supply and demand are uncertain

In the reports in the literature on inventory control, the effects of the random capacity on an order quantity and reorder point inventory control model have been integrated with lead time demand following general distribution. An iterative solution procedure has been proposed for obtaining the optimal solution. However, the resulting solution may not exist or it may not guarantee to give a minimum to the objective cost function, the expected cost per unit time. The aim of this study was to introduce a complete solution of the order quantity/reorder point problem, optimality, properties and bounds on the optimal order quantity and reorder point. The two most appealing distributions of lead time demand, normal and uniform distributions, in conjunction with an exponentially distributed capacity, are used to illustrate our findings in determining the optimal order quantity and reorder point.     

Quality improvement and backorder price discount under controllable lead time in an inventory model

The proposed study investigates a continuous review inventory model with order quantity, reorder point, backorder price discount, process quality, and lead time as decision variables. An investment function is used to improve the process quality. Two models are developed based on the probability distribution of lead time demand. The lead time demand follows a normal distribution in the first model and in the second model it does not follow any specific distribution but mean and standard deviation are known. We prove two lemmas to obtain optimal solutions for the normal distribution model and distribution free model. Finally, some numerical examples are given to illustrate the model.     

Mixture inventory model with back orders and lost sales for variable lead time demand with the mixtures of normal distribution

In recent papers by Ben-Daya and Raouf and by Ouyang et al. a continuous review inventory model is presented in which they considered both the lead time and the order quantity as decision variables. When the demands of the different customers do not have identical lead times, then we cannot use only a distribution (such as Ouyang et al. who used a normal distribution) to describe the demand of the lead time. Hence, we have extended the model of Ouyang et al. by considering the mixtures of normal distribution (see the book by Everitt and Hand). In addition, we also still assume that shortages are allowed. Moreover, the total amount of stock-out is considered as a mixture of back orders and lost sales during the stock-out period. Moreover, we also develop an algorithmic procedure to find the optimal order quantity and optimal lead time; the effects of parameters are also studied.     

Classification on defective items using unidentified samples

A set of unlabelled items is used to establish a decision rule to classify defective items. The lifetime of an item has an exponential distribution. It is known that the Bayes decision rule, which classifies good and defective items, gives a minimum probability of misclassification. The Bayes decision rule needs to know the prior probability (defective percentage) and two mean lifetimes. In the set of unidentified samples, the defective percentage and two mean lifetimes are unknown. Hence, before we can use the Bayes decision rule, we have to estimate the three unknown parameters. In this study, a set of unlabelled samples is used to estimate the three unknown parameters. The Bayes decision rule with these estimated parameters is an empirical Bayes (EB) decision rule. A stochastic approximation procedure using the set of unidentified samples is established to estimate the three unknown parameters. When the size of unlabelled items increases, the estimates computed by the procedure converge to the real parameters and hence gradually adapt our EB decision rule to be a better classifier until it becomes the Bayes decision rule. The results of a Monte Carlo simulation study are presented to demonstrate the convergence of the correct classification rates made by the EB decision rule to the highest correct classification rates given by the Bayes decision rule.     

A replenishment policy for items with price-dependent demand,time-proportional deterioration and no shortages

In this article, an order-level inventory system for deteriorating items has been developed with demand rate as a function of selling price. The demand and the deterioration rate are price dependent and time proportional, respectively. We have considered a perishable item that follows a three-parameter Weibull distribution deterioration. Shortages are not permitted in our model. The optimal solution is illustrated with a numerical example and the sensitivity analysis of parameters is carried out.     

A Bayesian approach to a dynamic inventory model under an unknown demand distribution

In this paper, the Bayesian approach to demand estimation is outlined for the cases of stationary as well as non-stationary demand. The optimal policy is derived for an inventory model that allows stock disposal, and is shown to be the solution of a dynamic programming backward recursion. Then, a method is given to search for the optimal order level around the myopic order level. Finally, a numerical study is performed to make a profit comparison between the Bayesian and non-Bayesian approaches, when the demand follows a stationary lognormal distribution. A profit comparison is also made between the stationary and non-stationary Bayesian approaches to observe whether the Bayesian approach incorporates non-stationarity in the demand. And, it is observed whether stock disposal reduces the losses due to ignoring non-stationarity in the demand.Scope and purposeIn the context of inventory models, one of the crucial factors to determine an optimal inventory policy, is the accurate forecasting or estimation of the demand for items in the inventory. The assumption of a constant demand is seriously questioned in recent times, since in reality the demand is generally uncertain and may even vary with time. For instance, the demand for new products, spare parts, or style goods, is likely to fluctuate widely, the average demand is quite likely to be low, and may exhibit a trend. In such situations, the Bayesian approach is a very useful tool for demand estimation, which is applicable even when past observations are scarce. In this paper, we use this approach to estimate the demand for an item, and obtain the expressions for finding the optimal inventory policies. We give a simpler method to find the optimal inventory policy, since the procedure to obtain the optimal inventory policy in the Bayesian framework, is quite tedious especially for long planning horizons, and in cases where the future demand becomes unpredictable. To widen the application of the method, we have given a general procedure which is not restricted to any particular probability distribution for the demand. We compare the Bayesian approach with the corresponding non-Bayesian approach, in terms of the optimum expected profits, when the demand follows a lognormal distribution. We also investigate how well the Bayesian approach incorporates non-stationarity in the demand.     

Economic production quantity (EPQ) models under an imperfect production process with shortages backordered

In this paper, we develop economic production quantity (EPQ) models to determine the optimal production lot size and backorder quantity for a manufacturer under an imperfect production process. The imperfect production process is characterised by the fraction of defective items at the time of production γ. The paper considers different cases of the EPQ model depending on (1) whether γ is known with certainty or is a random variable, and (2) whether imperfect items are drawn from inventory (a) as they are detected, (b) at the end of each production period or (c) at the end of each production cycle. Straightforward convexity results are shown and closed-form solutions are provided for the optimal order and backorder quantities for each of the cases we considered. We provide two numerical examples: one in which the defective probability follows a uniform distribution and the second which we assume follows a beta distribution, to illustrate the effects of yield variability and timing of the withdrawal of defectives on the optimal solutions. We obtain similar results for both numerical examples, which show that both the yield variability and the withdrawal timing are not critical factors.     

A continuous review inventory model with the controllable production rate of the manufacturer

Optimal operating policy in most deterministic and stochastic inventory models is based on the unrealistic assumption that lead‐time is a given parameter. In this article, we develop an inventory model where the replenishment lead‐time is assumed to be dependent on the lot size and the production rate of the manufacturer. At the time of contract with a manufacturer, the retailer can negotiate the lead‐time by considering the regular production rate of the manufacturer, who usually has the option of increasing his regular production rate up to the maximum (designed) production capacity. If the retailer intends to reduce the lead‐time, he has to pay an additional cost to accomplish the increased production rate. Under the assumption that the stochastic demand during lead‐time follows a Normal distribution, we study the lead‐time reduction by changing the regular production rate of the manufacturer at the risk of paying additional cost. We provide a solution procedure to obtain the efficient ordering strategy of the developed model. Numerical examples are presented to illustrate the solution procedure.     

Two EPQ models with imperfect production processes,inspection errors,planned backorders,and sales returns

In practice the items received in a lot may contain defective items, and during the screening process to eliminate the defective items, the inspector may incorrectly classify a non-defective item as defective (a Type I error) or incorrectly classify a defective item as non-defective (a Type II error). In this paper, we develop two economic production quantity models with imperfect production processes, inspection errors, planned backorders, and sales returns. A closed form solution is obtained for the optimal production lot size and the maximum shortage level for both models. We provide two numerical examples, one in which the defective probability and Type I and Type II inspection errors follow uniform distributions, and the second in which we assume they follow beta distributions. Sensitivity analyses are performed to see the impact of the defective probability, the probability of the Type I inspection error, the probability of the Type II inspection error, the holding cost, and the backordering cost on the optimal solutions. We obtain similar results on the sensitivity analyses for both numerical examples. The results show that the time factor of when to sell the defective items has a significant impact on the optimal production lot size and the backorder quantity. The results also show that if customers are willing to wait for the next production when a shortage occurs, it is profitable for the company to have planned backorders although it incurs a penalty cost for the delay.     

Batching work and rework processes with limited deterioration of reworkables

We study a deterministic problem of planning the production of new and recovering defective items of the same product manufactured on the same facility. Items of the product are produced in batches. The processing of a batch includes two stages. In the first work stage, all items of a batch are manufactured and good quality items go to the inventory to satisfy given demands. In the second rework stage, some of the defective items of the same batch are reworked. Each reworked item has the required good quality. While waiting for rework, defective items deteriorate. There is a given deterioration time limit. A defective item, that is decided not to be reworked or cannot be reworked because its waiting time will exceed the deterioration time limit, is disposed of immediately after its work operation completes. Deterioration results in an increase in time and cost for performing rework processes. It is assumed that the percentage of defective items is the same in each batch, and that they are evenly distributed in each batch. A setup time as well as a setup cost is required to start batch processing and to switch from production to rework. The objective is to find batch sizes and positions of items to be reworked such that a given number of good-quality items is produced and total setup, rework, inventory holding, shortage and disposal cost is minimized. A polynomial dynamic programming algorithm is presented to solve this problem.