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.     

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.     

Periodic review fuzzy inventory model with variable lead time and fuzzy demand

This paper investigates a periodic review fuzzy inventory model with lead time, reorder point, and cycle length as decision variables. The main goal of this study is to minimize the expected total annual cost by simultaneously optimizing cycle length, reorder point, and lead time for the whole system based on fuzzy demand. Two models are considered in this paper: one with normal demand distribution and another with a distribution‐free approach. The model assumes a logarithmic investment function for lost‐sale rate reduction. Furthermore, two separate efficient computational algorithms are explained to obtain the optimal solution. Some numerical examples are given to illustrate the model.     

(Q,r,L) Inventory model with defective items

This paper assumes that an arrival order lot may contain some defective items, and the number of defective items is a random variable. We derive a modified mixture inventory model with backorders and lost sales, in which the order quantity, the reorder point and the lead time are decision variables. In our studies, we first assume that the lead time demand follows a normal distribution, and then relax the assumption about the form of the distribution function of the lead time demand and apply the minimax distribution-free procedure to solve the problem. We develop an algorithm procedure to obtain the optimal ordering strategy. Furthermore, the effects of parameters are also included.     

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.     

A periodic review inventory model involving variable lead time with a service level constraint

In this study, we consider a mixture periodic review inventory model in which both the lead time and the review period are considered as decision variables. Instead of having a stock-out term in the objective function, a service level constraint is added to the model. In our paper, we first assume that the protection interval (i.e. the review period plus the lead time) demand follows a normal distribution, and then we relax this assumption and only assume that the first two moments of the protection interval demand are given. For each case, we develop an algorithm to find the optimal review period and optimal lead time. Furthermore, a sensitivity analysis is also performed.     

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.     

可控前置时间的连续盘点联合库存策略研究

鉴于控制前置时间对精益生产系统的重要性,在考虑买方与卖方合作的同时,扩展Goyal生产批量交货的假设,假设需求服从正态分布,以订购数量、运送次数与前置时间为决策变量,建立前置时间可控制的联合库存模型以确定适当的库存水平,使得库存总成本最小化,且可以通过协商在买卖双方之间进行节省成本的分配。进行了数值范例,并将联合库存模型与Banerjee模型、Goyal模型进行了比较。     

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.     

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.     

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.     

Lot size reorder point inventory model with controllable lead time and set-up cost

This paper deals with the lead time and set-up cost reductions problem on the modified lot size reorder point inventory model in which the production process is imperfect. We consider that the lead time can be shortened at an extra crashing cost, which depends on the length of lead time to be reduced and the ordering lot size. The option of investing in reducing set-up cost is also included. Two commonly used investment cost functional forms, logarithmic and power, are employed for set-up cost reduction. We assume that the stochastic demand during lead time follows a Normal distribution. The objective is simultaneously to optimize the lot size, reorder point, set-up cost and lead time. An algorithm of finding the optimal solution is developed, and two numerical examples are given to illustrate the results.     

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.     

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.     

可修备件需求量的通用拟生灭过程模型

This paper aims at two problems which exist in most of repairable spare part demand models at present: the exponential distribution as the basic assumption and one typical distribution corresponding to a model. A general repairable spare part demand model built on quasi birth-and- death process is developed. This model assumes that both the operational time of the unit and the maintenance time of the unit follow the continuous time phase type distributions. The first passage time distribution to be out of spares, the first mean time to be out of spares, and an algorithm to get the minimal amount of spares under certain restrictions are obtained. At the end of this paper, a numerical example is given.     

An alternative simple solution algorithm of an inventory model with fixed and variable lead time crash costs under unknown demand distribution

Inventory models with controllable lead time both for known and unknown demand distributions have been proposed in the literature. A model is useless unless it is formulated correctly and feasible. A simple solution procedure of a model also plays an important role in its application. This article highlights an erroneous formulation of an inventory model developed with fixed and variable lead time crash costs under unknown demand distribution, and also demonstrates its infeasibility. To attain feasibility we extend the model to include a constraint. Then, we present an alternative simple solution technique of the modified model and carry out a comparative study on a numerical example to show its potential significance.     

An integrated approach for the optimization of tolerance design and quality cost

Different normality-based optimization strategy (NBOS) methods have been developed and used to perform quality improvement in the past few decays. Improving the quality of a production process using a NBOS method possibly incurs misleading results if the quality measurements follow a skewed distribution. An integrated model, with components of a tolerance cost model for the determinations of optimal tolerance limits and a quality investment model for the identification of optimal investment level, is applied to establish a new optimization strategy method for the skew normal distribution (SND), named SNDOS method. The SND generalizes the normal distribution to include skewed distributions as members, and hence the SNDOS method is applicable for quality improvement either the distribution of quality measurements follow a symmetric or skewed distribution. Two examples about car seat production process are used to illustrate the application of the SNDOS method. The sensitivity of the SNDOS method to the loss coefficient of the integrated model is evaluated for different inputs of the skewness parameter of the SND through a numerical study.     

A hybrid approach to designing inbound-resupply strategies

We model the behavior of the demand distribution and lead-time distribution separately and determine the value of demand during lead time by discrete-event simulation. A new methodology determines the optimal inbound-resupply strategy when suppliers ship goods to receivers. An evolutionary metaheuristic uses discrete-event simulation to calculate safety stock levels and minimize total logistics costs.     

求解动态维修资源优化调度的多目标进化算法

为解决维修资源调度过程中出现的维修资源预测不准、资源冲突的问题,本文建立了不同作战阶段的多供应中心?多需求点的的动态维修资源优化调度模型,使得多个供应中心可以及时、高效地对需求点进行维修资源调度,减少了资源调度时间和每个需求点的维修资源不满足量。为了更好地求解提出的模型,本文提出了一种改进的多目标进化算法,在经典的多目标进化算法的基础上,使用正态分布交叉算子、全局探索增强型差分进化算子和自适应变异算子的协同进化策略,提高了算法的局部搜索能力和种群的多样性。仿真实验表明,本文提出的算法具有良好的收敛性和分布均匀性,并且具有较高的求解效率。     

Inventory control with general demand and lead time distributions

This paper extends an earlier paper in this journal by Murphy (1975) by presenting practical approaches for determining the expected lost sales for a reorder level system of inventory control. First, a set of formulas are derived for computing the first four moments (f.f.m.) of the lead time demand, these formulas are applicable to any distribution forms of daily demand and lead time. By fitting the lead time demand's f.f.m. to either a beta or a Schmeiser-Deutsch distribution, this paper then demonstrates how the expected lost sales can be easily computed. The approaches presented provide accurate modelling for diversified situations and the computations can be easily performed manually.