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.     

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.     

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.     

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.     

Lot sizing under (Q, R) policy in a capcity constrained manufacturing facility

Traditional approaches to lot sizing and inventory control consider uncertainties in either the demand or manufacturing process but not both. In these models it is usually assumed that the lead times are independent and identically distributed, but this is not realistic in many practical instances. In this paper we consider the lot sizing problem for items with stochastic demands and manufacturing lead times. It is assumed that the inventory of the finished product is controlled by continuous review policy of Q,R type—order quantity, order point system—and the problem is to determine optimal Q and R. We examine the decision parameters under a variety of conditions using exact and approximate methods.     

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.     

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.     

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.     

Design of Stochastic Distribution Networks Using Lagrangian Relaxation

This paper addresses the design of single commodity stochastic distribution networks. The distribution network under consideration consists of a single supplier serving a set of retailers through a set of distribution centers (DCs). The number and location of DCs are decision variables and they are chosen from the set of retailer locations. To manage inventory at DCs, the economic order quantity (EOQ) policy is used by each DC, and a safety stock level is kept to ensure a given retailer service level. Each retailer faces a random demand of a single commodity and the supply lead time from the supplier to each DC is random. The goal is to minimize the total location, shipment, and inventory costs, while ensuring a given retailer service level. The introduction of inventory costs and safety stock costs leads to a nonlinear NP-hard optimization problem. A Lagrangian relaxation approach is proposed. Computational results are presented and analyzed showing the effectiveness of the proposed approach.     

A minimax distribution free procedure for mixed inventory model involving variable lead time with fuzzy demand

In a recent paper, Ouyang and Wu applied the minimax decision approach to solve a continuous review mixed inventory model in which the lead time demand distribution information is unknown but the annual demand is fixed and given. However, in the practical situation, the annual demand probably incurs disturbance due to various uncertainties. In this article, we attempt to modify Ouyang and Wu's model by considering two fuzziness of annual demand (i.e., fuzzy number of annual demand and statistic-fuzzy number of annual demand) and to investigate a computing schema for the continuous review inventory model in the fuzzy sense. We give an algorithm procedure to obtain the optimal ordering strategy for each case.Scope and purposeIn most of the early literature dealing with inventory problems, either using deterministic or probabilistic models, lead time is viewed as a prescribed constant or a stochastic variable. Recently, some researchers (e.g., Liao and Shyu, Ben-Daya and Raouf, and Ouyang and Wu) incorporated the crashing lead time idea to continuous review inventory models, in which the annual demand is given and fixed. However, in the real situation, the annual demand will probably have a little disturbance due to various uncertainties. The purpose of this article is to modify the Ouyang and Wu's model to accommodate this reality, specifically, we apply the fuzzy set concepts to deal with the uncertain annual demand. We first consider a case where the annual demand is treated as the triangular fuzzy number. Then, we employ the statistical method to construct a confidence interval for the annual demand, and through it to establish the corresponding fuzzy number (namely, the statistic-fuzzy number). For each fuzzy case, we investigate a computing schema for the new model and develop an algorithm to find the optimal ordering strategy.     

(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.     

随机中断环境下的库存控制研究

考虑一需求为复合Poisson分布、提前期为指数分布和短缺损失的连续检查库存系统. 在假设供应商和零售商工作和中断的持续时间服从独立指数分布条件下, 利用水平穿越法, 确定零售商库存水平的平稳分布函数, 在此基础上, 构建长程平均费用率模型, 并利用交叉熵法得到最优库存控制策略. 最后, 通过仿真实验, a分析了中断强度和系统参数对最优库存策略和平均费用率的影响.     

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.     

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。     

Efficient inventory management in multi-echelon distribution systems

In this paper, an improved DRP method to schedule multi-echelon distribution network is proposed such that order-quantities and order points are dynamically obtained to meet the demand in just in time concept and minimize the out-of-stock probability. The order scheduling method reflects the dynamic characteristics of inventory level changes in the regional distribution centers and the central distribution center. The experiment has been done with various demand distributions, forecast error distributions and lead times. The proposed method was compared with the traditional DRP-based scheduling methods which use different lot-sizing and order point decision techniques. From the result, it is found that the proposed heuristic method yields preferable lot-sizing schedules.     

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.     

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.     

Application of fuzzy sets to manufacturing/distribution planning decisions in supply chains

In the real-world manufacturing/distribution planning decision (MDPD) integration problems in supply chains, the environmental coefficients and parameters are normally imprecise due to incomplete and/or unavailable information. This work presents a fuzzy linear programming approach based on the possibility theory. It applies this approach to solve multi-product and multi-time period MDPD problems with imprecise goals and forecast demand by considering the time value of money of related operating cost categories. The proposed approach attempts to minimize the total manufacturing and distribution costs by considering the levels of inventory, subcontracting and backordering, the available machine capacity and labor levels at each source, forecast demand and available warehouse space at each destination. This study utilizes an industrial case study to demonstrate the feasibility of applying the proposed approach to practical MDPD problems. The primary contribution of this paper is a fuzzy mathematical programming methodology for solving the MDPD integration problems in uncertain environments.     

Manufacturing setup cost reduction and quality improvement for the distribution free continuous-review inventory model with a service level constraint

Over several decades, the continuous-review inventory model has been widely studied based on various assumptions and restrictions such as those related with quality improvement, service level constraint, and setup cost reduction. We extend Moon and Choi's [1] model by assuming setup cost reduction and quality improvement. A distribution free approach is employed such that only mean and standard deviation need to be known. The total system cost is minimized with respect to decision variables against the worst possible distribution scenario. The benefit of using quality improvement and setup cost reduction in this model is shown. Numerical examples show that this model offers significant improvements over existing models. Finally, sensitivity analysis of the key parameters is also presented.     

A linear programming formulation for an inventory management decision problem with a service constraint

Inventory systems with uncertainty go hand in hand with the determination of a safety stock level. The decision on the safety stock level is based on a performance measure, for example the expected shortage per replenishment period or the probability of a stock-out per replenishment period. The performance measure assumes complete knowledge of the probability distribution during lead time, which might not be available. In case of incomplete information regarding the lead-time distribution of demand, no single figure for the safety stock can de determined in order to satisfy a performance measure. However, an optimisation model may be formulated in order to determine a safety stock level which guarantees the performance measure under the worst case of lead-time demand, of which the distribution is known in an incomplete way. It is shown that this optimisation problem can be formulated as a linear programming problem.