A queuing approach for inventory planning with batch ordering in multi-echelon supply chains

This paper presents stylized models for conducting performance analysis of the manufacturing supply chain network (SCN) in a stochastic setting for batch ordering. We use queueing models to capture the behavior of SCN. The analysis is clubbed with an inventory optimization model, which can be used for designing inventory policies . In the first case, we model one manufacturer with one warehouse, which supplies to various retailers. We determine the optimal inventory level at the warehouse that minimizes total expected cost of carrying inventory, back order cost associated with serving orders in the backlog queue, and ordering cost. In the second model we impose service level constraint in terms of fill rate (probability an order is filled from stock at warehouse), assuming that customers do not balk from the system. We present several numerical examples to illustrate the model and to illustrate its various features. In the third case, we extend the model to a three-echelon inventory model which explicitly considers the logistics process.     

Controlling setup cost in (Q, r, L) inventory model with defective items

This study discusses a mixture inventory model with back orders and lost sales in which the order quantity, reorder point, lead time and setup cost are decision variables. It is assumed that an arrival order lot may contain some defective items and the number of defective items is a random variable. There are two inventory models proposed in this paper, one with normally distributed demand and another with distribution free demand. Finally we develop two computational algorithms to obtain the optimal ordering policy. A computer code using the software Matlab is developed to derive the optimal solution and present numerical examples to illustrate the models. Additionally, sensitivity analysis is conducted with respect to the various system parameters.     

Cost Comparisons for Out-of-Phase Inventory Models

Inventory costs for a fixed time period have traditionally been determined by allocating total costs per cycle uniformly throughout that cycle as well as any partial cycles. This procedure for cost allocation has led to the solution of numerous inventory problems, most notable of which is the anticipated price-increase model. When comparing two out-of-phase inventory models, if costs are accounted for when they occur over a fixed planning horizon, inventory policies should be changed to reflect the impact of this different cost-allocation procedure. For the anticipated price-increase model, the ‘optimal’ order quantity as well as the implied savings in inventory costs will be different when cost models are developed based on these different cost-allocation methods. If the objective is to maximize over a fixed planning horizon the actual savings in inventory costs as they occur, the cost models presented here should be used.     

Theory of exchange

In the context of production activity, several parameters play an important role in the total cost estimation and its optimization. These parameters include facility setup cost, inventory carrying cost, and manufacturing cost for the basic model. Shortages can be incorporated in certain environment and costs associated with shortages need to be included in the analysis. It is expected that the industries will run their manufacturing facility at an optimum level. In the multi-product manufacture, optimum common cycle time approach is usually adopted and all the items are produced in each cycle. A situation may occur in the real world, in which a particular parameter concerning an item is exchanged with that of another item. It is of interest to examine the aftereffects. Otherwise also, for the purpose of internal benchmarking, a deliberate exchange of parameters can take place. This can be implemented in case of cost improvement. A generalized approach is presented and discussion is made with reference to various parameters.     

考虑通货膨胀与最长顾客等待时间的变质性物品的库存模型

本文建立了一种考虑通货膨胀与时间价值的变质性物品的库存模型,在模型中允许短缺发生且拖后的需求速率与在缺货期间已经发生的缺货量有关.和已有相关模型的主要区别在于本模型把一个可重复的订货周期内的最大平均利润的净现值作为目标函数,且增加了在缺货期间最长顾客等待时间的限制,以确保库存系统拥有较高的服务水平.然后讨论了模型最优解的存在性与唯一性,并提供了寻求模型整体最优解的算法.最后用实例说明了此模型在实际中的应用.     

Single Supplier�CBuyer Integrated Inventory Model Under Multiple JIT Delivery and Stock-Dependent Demand

In this paper, an integrated inventory model for a supply chain comprising of single buyer and single supplier is studied when demand is stock-dependent and units in inventory deteriorate at a constant rate. The total cost of the integrated system consists of the transportation cost, inspection cost and the cost of less flexibility under the assumption of JIT deliveries. The total integrated cost of single-supplier and single-buyer is minimized with respect to number of inspections and deliveries, the cycle time of deliveries and the delivery size for the replenishment time. A numerical example is given to validate the model. The sensitivity analysis carried out suggests that the unit inspection cost, deterioration rate of units in inventory and stock-dependent parameter are the critical factors.     

Optimum inventory policies with discrete time proportional demand,discrete replenishment opportunities and different optimizing criteria

This paper extends the deterministic, single product, dynamic E0Q model to the case where demand increases linearly with time but at discrete time points and where the number of replenishments is also discrete. The problem is to find the number of orders and the replenishment schedule that will either maximize the return on the investment on inventory or minimize inventory costs. The proposed solution to either problem requires to first find the replenishment schedule that will minimize the total inventory throughout the planning horizon, for a given number of orders and then find the optimal number of replenishment points. The solution algorithms exploit the discrete nature of the demand and do not require the decomposability property of dynamic programming. This is particularly important in the return on investment case, where decomposability cannot be achieved.     

A dynamic inventory model with supplier selection in a serial supply chain structure

Considering the inherent connection between supplier selection and inventory management in supply chain networks, this article presents a multi-period inventory lot-sizing model for a single product in a serial supply chain, where raw materials are purchased from multiple suppliers at the first stage and external demand occurs at the last stage. The demand is known and may change from period to period. The stages of this production–distribution serial structure correspond to inventory locations. The first two stages stand for storage areas for raw materials and finished products in a manufacturing facility, and the remaining stages symbolize distribution centers or warehouses that take the product closer to customers. The problem is modeled as a time-expanded transshipment network, which is defined by the nodes and arcs that can be reached by feasible material flows. A mixed integer nonlinear programming model is developed to determine an optimal inventory policy that coordinates the transfer of materials between consecutive stages of the supply chain from period to period while properly placing purchasing orders to selected suppliers and satisfying customer demand on time. The proposed model minimizes the total variable cost, including purchasing, production, inventory, and transportation costs. The model can be linearized for certain types of cost structures. In addition, two continuous and concave approximations of the transportation cost function are provided to simplify the model and reduce its computational time.     

A deteriorating inventory model with time-varying demand and shortage-dependent partial backlogging

Recently, Chu et al. [P. Chu, K.L. Yang, S.K. Liang, T. Niu, Note on inventory model with a mixture of back orders and lost sales, European Journal of Operational Research 159 (2004) 470–475] presented the necessary condition of the existence and uniqueness of the optimal solution of Padmanabhan and Vrat [G. Padmanabhan, P. Vrat, Inventory model with a mixture of back orders and lost sales, International Journal of Systems Science 21 (1990) 1721–1726]. However, they included neither the purchase cost nor the cost of lost sales into the total cost. In this paper, we complement the shortcoming of their model by adding not only the cost of lost sales but also the non-constant purchase cost, and then extend their model from a constant demand function to any log-concave demand function. We also provide a simple solution procedure to find the optimal replenishment schedule. Further, we use a couple of numerical examples to illustrate the results and conclude with suggestions for future research.     

Inventory control in the presence of an electronic marketplace

This paper studies a periodic review inventory model in the presence of an electronic marketplace (EM). Emergency orders can be placed in the EM for additional cost, and excess inventory can be sold to the EM. When the order leadtime from the supplier is one period, the optimal inventory control policy is developed from a dynamic programming model of the problem. The policy is characterized by three critical inventory levels. When the order leadtime from the supplier is longer than one period, an EM policy is developed to determine the quantities of inventory to purchase from and sell to the EM in each period. Based on this EM policy, three ordering policies are proposed to determine the order quantity from the supplier. Numerical results show that significant cost reductions can be obtained by using the EM to adjust the inventory level in each period. The amount of cost reduction is greatly affected by system parameters, especially the order leadtime from the supplier and the costs for transactions in the EM.     

Global optimal policy for vendor–buyer integrated inventory system within just in time environment

Traditionally, inventory problems for the vendor and the buyer are treated separately. In modern enterprises, however, the integration of vendor–buyer inventory system is an important issue. This co-operative approach to inventory management contributes to the success of supply chain management by minimizing the joint inventory cost. The joint inventory cost and the response time can further be reduced when the buyer orders and the vendor replenishes the required items just in time (JIT) for their consumption. The inclusion of the JIT concept in this model contributes significantly to a joint inventory cost reduction. A numerical example and sensitivity analysis are carried out. The derived results show an impressive cost reduction when compared with Goyal’s model.     

Balancing capacity and lot sizes

In many companies there is an on-going discussion about capacity, capacity utilization and capital tied up in inventories. However, traditional models such as the EOQ model only include capacity considerations in the set-up cost, or in the cost of a replenishment order. This implies e.g. that they do not consider the set-up time as a capacity constraint. Furthermore, in these models the set-up cost is usually treated as a constant, even though the opportunity cost for capacity in general is dependent upon the capacity utilization.The purpose of this paper is to derive an analytical model for the balancing of capacity and lot sizes. The model includes costs for capacity, work-in-process (queueing, set-up, and processing time), and finished goods inventory. The total costs are minimized with respect to capacity. Then, the corresponding, recommended lot sizes are determined. The model was tested with data from a Swedish manufacturing company. The results turned out to coincide with experiences of the company in many important respects. The model offers production management an opportunity to discuss the relationship between capacity, work-in-process, and lot sizes.     

A real-time decision rule for an inventory system with committed service time and emergency orders

In this paper, we study the inventory system of an online retailer with compound Poisson demand. The retailer normally replenishes its inventory according to a continuous review (nQ, R) policy with a constant lead time. Usually demands that cannot be satisfied immediately are backordered. We also assume that the customers will accept a reasonable waiting time after they have placed their orders because of the purchasing convenience of the online system. This means that a sufficiently short waiting time incurs no shortage costs. We call this allowed waiting time “committed service time”. After this committed service time, if the retailer is still in shortage, the customer demand must either be satisfied with an emergency supply that takes no time (which is financially equivalent to a lost sale) or continue to be backordered with a time-dependent backorder cost. The committed service time gives an online retailer a buffer period to handle excess demands. Based on real-time information concerning the outstanding orders of an online retailer and the waiting times of its customers, we provide a decision rule for emergency orders that minimizes the expected costs under the assumption that no further emergency orders will occur. This decision rule is then used repeatedly as a heuristic. Numerical examples are presented to illustrate the model, together with a discussion of the conditions under which the real-time decision rule provides considerable cost savings compared to traditional systems.     

需求率依赖于库存水平的离散性库存费的库存模型

传统的库存控制模型都视需求率为固定不变的,放松了这个假定,通过考虑库存费为存储时间的阶梯函数的情形:(1)全单位库存费用,(2)增量库存费用,并且在需求率依赖于库存水平,当库存水平下降到一定程度时,需求率变为常数的形式下,把变化的订购费引入,发展了两个离散性库存费的变质物品的库存控制模型。在模型中允许周期末库存水平不为零,并且提出了最优解的算法。     

Joint pricing and ordering policy for an advance booking system with partial order cancellations

In this paper, an inventory model for deteriorating items with price-dependent demand is developed. The cycle is divided into two periods, where an advance sales period is followed by a spot sales period. In practice, customers with reservations may cancel their orders before receiving them. During the advance sales period, the rate of reservations which will not be cancelled is dependent on the length of the waiting time for the receiving order. During the spot sales period, all customers receive their orders at the time of the purchase. We prove the existence of the realistic relationship that the advance sales price is smaller than the spot sales price. We also develop some useful properties and provide an iterative procedure for solving the maximization problem. Numerical examples are given to demonstrate the effectiveness of the proposed approach and we conclude the paper with suggestions for possible future research.     

在中心仓库有损失销售的二级分布库存管理模型的研究

通过对一个中心仓库和N个零售商的二级分布库存系统进行分析,采用基本(S-1,S)库存策略,综合运用了排队法和M ETR IC近似法,提出了一种在中心仓库有损失销售的二级库存管理模型,该模型描述在中心仓库缺货情况下,多数零售商不等待延期付货,而直接与供应商订货,导致中心仓库就会因损失销售而产生机会成本.该模型可达到二级分布库存系统的总成本最小.     

On an EPQ model for deteriorating items under permissible delay in payments

This paper derives a production model for the lot-size inventory system with finite production rate, taking into consideration the effect of decay and the condition of permissible delay in payments, in which the restrictive assumption of a permissible delay is relaxed to that at the end of the credit period, the retailer will make a partial payment on total purchasing cost to the supplier and pay off the remaining balance by loan from the bank. At first, this paper shows that there exists a unique optimal cycle time to minimize the total variable cost per unit time. Then, a theorem is developed to determine the optimal ordering policies and bounds for the optimal cycle time are provided to develop an algorithm. Numerical examples reveal that our optimization procedure is very accurate and rapid. Finally, it is shown that the model developed by Huang [1] can be treated as a special case of this paper.     

Economic order quantity under advance payment

Though advance payment is widely used in practice, its influences on buyer’s inventory policy are rarely discussed. This paper investigates the buyer’s inventory policy under advance payment, including all payment in advance and partial-advanced–partial-delayed payment. The buyer’s ordering policy is derived by minimizing his total inventory costs including inventory holding cost, ordering cost, and interest cost caused by advance payment or delayed payment. The conclusions show that when all the payment is paid in advance, the buyer’s optimal replenishment cycle is influenced only by the price discount associated with advance payment, and the length of advance payment has no effect. For the partial-advanced–partial-delayed payment case, the buyer’s replenishment cycle is also not influenced by the length of advance period. However, in this situation, the delayed period and the price discount may have impacts on the inventory policy. We also use discounted cash flow (DCF) model to derive the buyer’s replenishment cycle and show that the replenishment cycle is negatively related to the length of advance period. Numerical examples are presented to illustrate the results.     

Inventory control with two switch-over levels for a class of M/G/1 queueing systems with variable arrival and service rate

This paper deals with inventory control in a class of M/G/1 queueing systems. At each point of time the system can be switched from one of two possible stages to another. The rate of arrival process and the service rate depend on the stage of the system. The cost structure imposed on the model includes both fixed switch-over costs and a holding cost at a general rate depending on the stage of the system. The rule for controlling the inventory is specified by two switch-over levels.Using an embedding approach, we will derive a formula for the long-run average expected costs per unit time of this policy. By an appropriate choice of the cost parameters, we may obtain various operating characteristics for the system amongst which the stationary distribution of the inventory and the average number of switch-overs per unit time.     

Evaluation of cycle-count policies for supply chains with inventory inaccuracy and implications on RFID investments

Inventory record inaccuracy leads to ineffective replenishment decisions and deteriorates supply chain performance. Conducting cycle counts (i.e., periodic inventory auditing) is a common approach to correcting inventory records. It is not clear, however, how inaccuracy at different locations affects supply chain performance and how an effective cycle-count program for a multi-stage supply chain should be designed. This paper aims to answer these questions by considering a serial supply chain that has inventory record inaccuracy and operates under local base-stock policies. A random error, representing a stock loss, such as shrinkage or spoilage, reduces the physical inventory at each location in each period. The errors are cumulative and are not observed until a location performs a cycle count. We provide a simple recursion to evaluate the system cost and propose a heuristic to obtain effective base-stock levels. For a two-stage system with identical error distributions and counting costs, we prove that it is more effective to conduct more frequent cycle counts at the downstream stage. In a numerical study for more general systems, we find that location (proximity to the customer), error rates, and counting costs are primary factors that determine which stages should get a higher priority when allocating cycle counts. However, it is in general not effective to allocate all cycle counts to the priority stages only. One should balance cycle counts between priority stages and non-priority stages by considering secondary factors such as lead times, holding costs, and the supply chain length. In particular, more cycle counts should be allocated to a stage when the ratio of its lead time to the total system lead time is small and the ratio of its holding cost to the total system holding cost is large. In addition, more cycle counts should be allocated to downstream stages when the number of stages in the supply chain is large. The analysis and insights generated from our study can be used to design guidelines or scorecard systems that help managers design better cycle-count policies. Finally, we discuss implications of our study on RFID investments in a supply chain.