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.     

A periodic review inventory model involving fuzzy expected demand short and fuzzy backorder rate

The purpose of this paper is to extend [Ouyang, L. Y., Chuang, B. R. (2001). A periodic review inventory-control system with variable lead time. International Journal of Information and Management Sciences, 12, 1–13] periodic review inventory model with variable lead time by considering the fuzziness of expected demand shortage and backorder rate. We fuzzify the expected shortage quantity at the end of cycle and the backorder (or lost sales) rate, and then obtain the fuzzy total expected annual cost. Using the signed distance method to defuzzify, we derive the estimate of total expected annual cost in the fuzzy sense. For the proposed model, we provide a solution procedure to find the optimal review period and optimal lead time in the fuzzy sense so that the total expected annual cost in the fuzzy sense has a minimum value. Furthermore, a numerical example is provided and the results of fuzzy and crisp models are compared.     

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.     

IPA Derivatives for Make-to-Stock Production-Inventory Systems With Lost Sales

This note applies the stochastic fluid model (SFM) paradigm to a class of single-stage, single-product make-to-stock (MTS) production-inventory systems with stochastic demand and random production capacity, where the finished-goods inventory is controlled by a continuous-time base-stock policy and unsatisfied demand is lost. This note derives formulas for infinitesimal perturbation analysis (IPA) derivatives of the sample-path time averages of the inventory level and lost sales with respect to the base-stock level and a parameter of the production rate process. These formulas are comprehensive in that they are exhibited for any initial inventory state, and include right and left derivatives (when they differ). The formulas are obtained via sample path analysis under very mild regularity assumptions, and are inherently nonparametric in the sense that no specific probability law need be postulated. It is further shown that all IPA derivatives under study are unbiased and fast to compute, thereby providing the theoretical basis for online adaptive control of MTS production-inventory systems.     

Der Einfluss von Such- und Empfehlungssystemen auf den Absatz im Electronic Commerce

The Internet and related technologies have vastly expanded the variety of products that can be profitably promoted and sold by online retailers. Furthermore, search and recommendation tools reduce consumers’ search costs in the Internet and enable them to extend their search from a few easily found best-selling products (blockbusters) to a large number of less frequently selling items (niches). As a result, Long Tail sales distribution patterns emerge that illustrate an increasing demand in niches. We show in this article how different classes of search and recommendation tools affect the distribution of sales across products, total sales, and consumer surplus. We hereby use an agent-based simulation which is calibrated based on real purchase data of a video-on-demand retailer. We find that a decrease in search costs through improved search technology can either shift demand from blockbusters to niches (search filters and recommendation systems) or from niches to blockbusters (charts and top lists). We break down demand changes into substitution and additional consumption and show that search and recommendation technologies can lead to substantial profit increases for retailers. We also illustrate that decreasing search costs through search and recommendation technologies always lead to an increase in consumer surplus, suggesting that retailers can use these technologies as competitive advantage.     

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.     

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.     

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.     

Continuous review inventory model with controllable lead time,lost sales rate and order processing cost when the received quantity is uncertain

In practice the quantity received may not match the quantity ordered due to worker's strike, rejection during inspection, damage during transportation, human errors in counting, etc. Accordingly, the managers often must make decisions under uncertain quantity received circumstances. In this study, we investigate the continuous review inventory model with shortages include the case where the quantity received is uncertain, in which the lead time, lost sales rate and order processing cost are decision variables. Here, we consider the lead time crashing cost is an exponential function of lead time, and the order processing cost and lost sales rate are logarithmic functions of capital investment. The objective of this study is to minimize the total relevant cost by simultaneously optimizing the order quantity, lost sales rate and order processing cost. In addition, an efficient algorithm is developed to determine the optimal policy, and our approach is illustrated through a numerical example. From the results of numerical example, it can be shown that, the significant savings can be achieved through the reductions of order processing cost and lost sales rate.     

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.     

Distribution-free approach for stochastic Joint-Replenishment Problem with backorders-lost sales mixtures,and controllable major ordering cost and lead times

In this paper, we study the periodic-review Joint-Replenishment Problem (JRP) with stochastic demands and backorders-lost sales mixtures. We assume that lead times aare made of two major components: a common part to all items and an item-specific portion. We further suppose that the item-specific component of lead times and the major ordering cost are controllable. To reflect the practical circumstance characterized by the lack of complete information about the demand distribution, we adopt the minimax distribution-free approach. That is, we assume that only the mean and the variance of the demand can be evaluated. The objective is to determine the strict cyclic replenishment policy, the length of (the item-specific component of) lead times, and the major ordering cost that minimize the long-run expected total cost. To approach this minimization problem, we present a first optimization algorithm. However, numerical tests highlighted how computationally expensive this algorithm would be for a practical application. Therefore, we then propose two alternative heuristics. Extensive numerical experiments have been carried out to investigate the performance of the developed algorithms. Results have shown that the proposed alternative heuristics are actually efficient and seem therefore promising for a practical application.     

Supply chain coordination using time-based temporary price discounts

In this paper, the issue of supply chain coordination (SCC) in a buyer–seller supply chain (SC) with an order size constraint is investigated. The buyer keeps safety stock to cope with lead time demand uncertainties from customers’ side. Unsatisfied demand will be lost. Therefore the whole SC sales volume depends on the service level provided by the buyer. By proposing a time-based temporary price discount in each replenishment cycle, the seller intends to convince the buyer to optimize its safety stock globally. Maximum and minimum discounts, which are acceptable for both parties, are determined and an appropriate discount schedule is derived. A set of numerical experiments are conducted to show performance of the proposed model. The results show that the safety stock coordination is profitable; the proposed model is capable of coordinating supply chain. In addition, the model can share extra benefits between SC members fairly.     

Sole versus dual sourcing in a continuous-review inventory system with lost sales

While existing cost analysis studies on the problem of sole versus dual sourcing only deal with the backorder case, we present an exact treatment of this problem in the context of a lost sales inventory system with compound Poisson demand and exponentially distributed lead times. The stationary distribution of the inventory level (stock on hand) and the corresponding cost functions with/without a service level constraint are derived using a level-crossing methodology, and numerical results are presented.     

The (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>, <Emphasis Type="Italic">r</Emphasis>, <Emphasis Type="Italic">l</Emphasis>) model for stochastic demand under Intuitionistic fuzzy aggregation with Bonferroni mean

This paper investigates a hill type economic production-inventory quantity (EPIQ) model with variable lead-time, order size and reorder point for uncertain demand. The average expected cost function is formulated by trading off costs of lead-time, inventory, lost sale and partial backordering. Due to the nature of the demand function, the frequent peak (maximum) and valley (minimum) of the expected cost function occur within a specific range of lead time. The aim of this paper is to search the lowest valley of all the valley points (minimum objective values) under fuzzy stochastic demand rate. We consider Intuitionistic fuzzy sets for the parameters and used Intuitionistic Fuzzy Aggregation Bonferroni mean for the defuzzification of the hill type EPIQ model. Finally, numerical examples and graphical illustrations are made to justify the model.     

按单装配系统中组件生产和库存分配控制策略研究

针对由两种组件、三类顾客需求组成的按单装配系统, 本文研究了其中的组件生产控制与库存分配问题. 在各类顾客需 求是泊松到达过程, 各种组件加工时间服从指数分布的假设下, 我们运用马尔科夫决策理论建立了无限期折扣总成本模型, 根据Lippman转换得到了相应归一化后的离散最优方程, 在此基础之上分析了生产和库存分配联合最优控制策略的结构性质. 本文证明了最优策略是依赖于系统状态的动态策略. 组件的最优生产策略是动态基库存策略, 其中基库存水平是关于系统中其他组件库存水平的非减函数. 而最优的分配策略是动态的阈值策略, 对于只需一种组件构成的顾客需求, 组件的分配阈值是系统中另一组件库存水平的增函数; 而对于同时需要两种组件组成的顾客需求, 其各组件的分配阈值是另一组件库存水平的减函数. 最后通过数值试验给出了各个参数对联合最优控制策略的影响, 并得到了相应的管理启示.     

RNN / LSTM with modified Adam optimizer in deep learning approach for automobile spare parts demand forecasting

The spare parts demand forecasting is very much essential for the organizations to minimize the cost and prevent the stock outs. The demand of spare parts/ car sales distribution is an important factor in inventory control. The valuation of the demand is challenging as the automobile spare parts/car sales demand are often recurrent. The renowned empirical method adopts historical demand data to create the distribution of lead time demand. Although it works reasonably well when service requirements are relatively low, it has difficulty reaching high target service levels. In this paper, we proposed Recurrent Neural Networks/ Long-Short Term Memory (RNN / LSTM) with modified Adam optimizer to predict the demand for spare parts. In this LSTM, weight vectors are generated respectively. These weights are optimized using the Modified-Adam algorithm. The accuracy of the forecast and the performance of the inventory are considered in the experimental result. Experimental results confirm that RNN / LSTM with a Modified-Adam works well with minimal error compared to other existing methods. We conclude that the proposed RNN/LSTM with Modified-Adam algorithm is well suited for the prediction of automobile spare parts.

     

A perishable inventory system with modified (S−1,S) policy and arbitrary processing times

In this article we analyze a lost sales (S−1,S) perishable system, under Poisson demands and exponential lifetimes, in which the reorders are placed at every demand epoch so as to take the inventory position back to its maximum level S. The items are replenished one at a time and the resupply time has arbitrary distribution. The various operating characteristics are obtained using Markov renewal techniques. A matrix recursive scheme is developed to determine the stationary distribution of the underlying Markov chain. The efficiency of this procedure in the determination of optimal S that minimizes the long run expected cost rate is discussed. Sensitivity analysis of various system parameters is also carried out.Scope and purposeThe analysis of perishable inventory systems, which has potential applications in various sectors of industry, is far more difficult than their infinite lifetime counterparts. In the study of perishable systems under traditional (S−1,S) policy, reorders for items are placed at demand as well as failure epochs. However, it will be more meaningful not to place orders at failure epochs, which is also practical and economical since it avoids continuous monitoring of items. This paper deals with a continuous review perishable system under a more realistic base stock policy, with variable ordering quantity and arbitrary unit resupply times. The operating characteristics of this complex model are derived using the techniques of semi-regenerative processes. A matrix recursive scheme developed to determine the stationary distribution has enabled us to code an efficient numerical program for cost optimization.     

An economic order quantity model of imperfect quality items with partial backlogging

This article deals with an economic order quantity (EOQ) model in which a certain percentage of a lotsize is of imperfect quality products. This percentage follows a uniform distribution function. During the inspection of the total lot-size, a stock-out situation may occur. In a stock-out situation, a partial fraction of the demand is adjusted by partial back ordering and the rest of the demand is considered as a case of lost sales. The associated integrated expected profit function is generalised for the general distribution function of imperfect quality products. Then the objective function is maximised. Also, three special cases of the general model are studied. A suitable numerical example is provided to illustrate the model and the solution procedure. Comparison between the general and special cases are also shown with the help of numerical examples. Sensitivity analysis of the optimal solutions with respect to all individual parameters of the general model is carried out.     

Stochastic (Q, r) inventory model with customer reneging

This paper presents a stochastic inventory model for situations in which, during a stockout period, unsatisfied demands are initially backordered but may renege (leave) probabilistically at the rate ρ to be filled elsewhere. The model is suggested by the customers' different reactions to a stockout condition: some patient customers wait until their demand is satisfied, while other impatient or urgent customers cannot wait long and have to fill their demand from another source. The cost of a backorder is assumed to be proportional to the length of time for which the backorder exists, and a fixed penalty cost is incurred per unit of lost demand. Based on a heuristic treatment of a lot-size reorder-point policy, a mathematical model representing the average annual cost of operating the inventory system is developed. The optimal operating policy variables minimizing the average annual cost can be calculated iteratively. At the extremes when ρ = 0 and ρ = ξ, the model presented reduces to the usual backorders and lost sales case, respectively.