Evaluating shortfall distributions in periodic inventory systems with stochastic endogenous demands and lead-times

This paper addresses a multi-period production/inventory problem with two suppliers, where demand sizes and supplier lead time are stochastic and correlated. A discrete time, single item inventory system is considered, where inventory levels are reviewed periodically and managed using a base-stock policy. At the end of each period, a replenishment order is placed, which enters a queue at the buffer stage and is consequently forwarded to the first available supplier. We present a mathematical model of this inventory system and determine optimal safety stock levels for it, in closed form, using matrix analytic techniques and the properties of phase type distributions. To account for the effect of order crossovers, which occur whenever replenishment orders do not arrive in the sequence in which they were placed, the inventory shortfall distribution is analyzed. Finally, a set of numerical experiments with a system with two suppliers is presented, where the proposed model is compared to other existing models.     

Multi-item inventory control with full truckloads: A comparison of aggregate and individual order triggering

In this paper, we consider the stochastic joint replenishment problem in an environment where transportation costs are dominant and full truckloads or full container loads are required. One replenishment policy, taking into account capacity restrictions of the total order volume, is the so-called QS policy, where replenishment orders are placed to raise the individual inventory positions of all items to their order-up-to levels, whenever the aggregate inventory position drops below the reorder level. We first provide a method to compute the policy parameters of a QS policy such that item target service levels can be met, under the assumption that demand can be modeled as a compound renewal process. The approximation formulas are based on renewal theory and are tested in a simulation study which reveals good performance. Second, we compare the QS policy with a simple allocation policy where replenishment orders are triggered by the individual inventory positions of the items. At the moment when an individual inventory position drops below its item reorder level, a replenishment order is triggered and the total vehicle capacity is allocated to all items such that the expected elapsed time before the next replenishment order is maximized. In an extensive simulation study it is illustrated that the QS policy outperforms this allocation policy since it results in lower inventory levels for the same service level. Although both policies lead to similar performance if items are identical, it can differ substantially if the item characteristics vary.     

An Example of Disbenefits of Centralized Stocking

This paper deals with a single-period, single-product inventory model. It is a multilocation problem with an opportunity for centralization. In the centralized system, the location's demands are satisfied from one central warehouse whose orders are processed on a first come, first served basis. The inventory size will be constrained to meet a specific maximum probability of being out of stock at each location. Under these assumptions, Stulman argues that a centralized system requires a lower total starting inventory than the equivalent decentralized one. We shall present a counter example showing that, under conditions where an 86% probability of stockouts is the maximum acceptable, a higher total starting inventory may be required by centralizing two locations' stock-holdings.     

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.     

Control policies for single-stage production systems with perishable inventory and customer impatience

We consider problems of inventory and admission control for make-to-stock production systems with perishable inventory and impatient customers. Customers may balk upon arrival (refuse to place orders) and renege while waiting (withdraw delayed orders) during stockouts. Item lifetimes and customer patience times are random variables with general distributions. Processing, setup, and customer inter-arrival times are however assumed to be exponential random variables. In particular, the paper studies two models. In the first model, the system suspends its production when its stock reaches a safety level and can resume later without incurring any setup delay or cost. In the second model, the system incurs setup delays and setup costs; during stockouts, all arriving customers are informed about anticipated delays and either balk or place their orders but cannot withdraw them later. Using results from the queueing literature, we derive expressions for the system steady-state probabilities and performance measures, such as profit from sales and costs of inventory, setups, and delays in filling customer orders. We use these expressions to find optimal inventory and admission policies, and investigate the impact of product lifetimes and customer patience times on system performance.     

A multi-product continuous review inventory system in stochastic environment with budget constraint

Common characteristics of inventory systems include uncertain demand and restrictions such as budgetary and storage space constraints. Several authors have examined budget constrained multi-item stochastic inventory systems controlled by continuous review policies without considering marginal review shortage costs. Existing models assume that purchasing costs are paid at the time an order is placed, which is not always the case since in some systems purchasing costs are paid when order arrive. In the latter case the maximum investment in inventory is random since the inventory level when an order arrives is a random variable. Hence payment of purchasing costs on delivery yields a stochastic budget constraint for inventory. In this paper with mixture of back orders and lost sales, we assume that mean and variance of lead time demand are known but their probability distributions are unknown. After that, we apply the minimax distribution free procedure to find the minimum expected value of the random objective function with budget constraint. The random budget constraint is transformed to crisp budget constraint by chance-constraint technique. Finally, the model is illustrated by a numerical example.     

Feasibility principles for Downstream Demand Inference in supply chains

Many companies are adopting strategies that enable Demand Information Sharing (DIS) between the supply chain links. Recently, a steady stream of research has identified mathematical relationships between demands and orders at any link in the supply chain. Based on these relationships and strict model assumptions, it has been suggested that the upstream member can infer the demand at the downstream member from their orders. If this is so, DIS will be of no value. In this paper, we argue that real-world modelling requires less restrictive assumptions. We present Feasibility Principles to show that it is not possible for an upstream member to accurately infer consumer demand under more realistic model assumptions. Thus, we conclude that DIS has value in supply chains. We then move our focus to the supply chain model assumptions in the papers arguing that there is value in sharing demand information. Using a simulation experiment, we show that the value of sharing demand information in terms of inventory reductions will increase under more realistic supply chain model assumptions.     

A multi-product continuous review inventory system with stochastic demand,backorders, and a budget constraint

Common characteristics of inventory systems include uncertain demand and restrictions such as budgetary or storage space constraints. Several authors have examined budget constrained multi-item stochastic inventory systems controlled by continuous review policies without considering marginal shortage costs. Existing models assume that purchasing costs are paid at the time an order is placed, which is not always the case since in some systems purchasing costs are paid when orders arrive. In the latter case the maximum investment in inventory is random since the inventory level when an order arrives is a random variable. Hence payment of purchasing costs on delivery yields a stochastic budget constraint for inventory. This paper models a multi-item stochastic inventory system with backordered shortages when estimation of marginal backorder cost is available, and payment is due upon order arrival. The budget constraint can easily be converted into a storage constraint.     

Inventory rationing in an (<Emphasis Type="Italic">s,Q</Emphasis>) inventory model with lost sales and two demand classes

Whenever demand for a single item can be categorised into classes of different priority, an inventory rationing policy should be considered. In this paper we analyse a continuous review (s, Q) model with lost sales and two demand classes. A so-called critical level policy is applied to ration the inventory among the two demand classes. With this policy, low-priority demand is rejected in anticipation of future high-priority demand whenever the inventory level is at or below a prespecified critical level. For Poisson demand and deterministic lead times, we present an exact formulation of the average inventory cost. A simple optimisation procedure is presented, and in a numerical study we compare the optimal rationing policy with a policy where no distinction between the demand classes is made. The benefit of the rationing policy is investigated for various cases and the results show that significant cost reductions can be obtained.     

基于选址效益的联合覆盖模型研究

传统的覆盖模型含有“全有全无”和“单一覆盖”两个假设,即假设需求点在设施的服务半径内才被覆盖,否则不被覆盖;需求点只能被最近的设施覆盖。这两条假设在实际应用中均存在不合理之处。松弛了这两条假设,研究逐渐覆盖和联合覆盖。在保证每个需求点都享受到最低服务水平的情况下,提出了选址效益最大化的联合覆盖模型。由于目标函数中含有分式,通过引入辅助变量的方法,将含有分式目标函数的非线性规划转化成等价的线性规划。最后,通过数值算例分析了最低服务水平限制对最佳选址方案的影响,并得到选址成本、总服务水平和单位成本服务水平随最低服务水平限制的变化,同时对影响模型的重要参数做了敏感性分析。     

Order-Up-To policies in Information Exchange supply chains

Collaboration in Supply Chains (SC) is concerned with the alignment of the decision making process amongst SC partners. This is crucial in the planning and inventory management area where this alignment is enabled by the exchange of information. Several benefits deriving from such effective collaboration exist, such as: excess inventory elimination, lead times reduction, improved customer service, efficient product development, etc. Operations Management literature proclaims the virtues of collaboration and information sharing but academicians and practitioners have recently identified various gaps that still need further work. More specifically it has been shown that several deleterious phenomena as the bullwhip effect; inventory instability and intermittent orders are not completely eliminated in Information Exchange supply chains. The reason is mainly because companies adopt order policies that are prone to create instability along the SC. In this paper we show how the performance of an Information Exchange SC can be improved by shifting from a myopic periodic review Order-Up-To policy to a periodic review Order-Up-To with feedback gain. To do so, we model the SC structure through difference equations and study the system response in term of internal process efficiency and customer service level.     

Base stock policy with retrial demands

In this work, we consider a continuous review base stock policy inventory system with retrial demands. The maximum storage capacity is S. It is assumed that primary demand is of unit size and primary demand time points form a Poisson process. A one-to-one ordering policy is adopted. According to this policy, orders are placed for one unit, as and when the inventory level drops due to a demand. We assume that the demands occur during the stock-out periods enter into the orbit of infinite size. The lead time is assumed to be exponential. The joint probability distribution of the inventory level and the number of demands in the orbit are obtained in the steady state case. Various system performance measures in the steady state are derived. The results are illustrated with suitable numerical examples.     

Sourcing from suppliers with random yield for price-dependent demand

We study a multi-period inventory planning problem. In each period, the firm under consideration can source from two possibly unreliable suppliers for a price-dependent demand. Our analysis suggests that the optimal procurement policy is neither a simple reorder-point policy nor a complex one without any structure, as previous studies suggest. Instead, we prove the existence of a reorder point for each supplier. No order is placed to that supplier for any inventory level above the reorder point and a positive order is issued to that supplier for almost every inventory level below the reorder point. We characterize conditions under which the optimal policy reveals monotone response to changes in the inventory level. Furthermore, two special cases of our model are examined in detail to demonstrate how our analysis generalizes a number of well-known results in the literature.     

Continuous-review inventory problem with random supply interruptions

This paper considers a continuous-review stochastic inventory problem with random demand and random lead-time where supply may be disrupted due to machine breakdowns, strikes or other randomly occurring events. The supplier availability is modelled as a semi-Markov process (more specifically, as an alternating renewal process). The standard (q, r) policy is used when the supplier is available (ON), i.e., when the inventory position reaches the reorder point r, q units are ordered to raise the inventory position to the target level of R = q + r. The form of the policy changes when the supplier becomes unavailable (OFF) in which case orders cannot be placed when the reorder point r is reached. However, as soon as the supplier becomes available again one orders enough to bring the inventory position up to the target level of R. The regenerative cycles are identified by observing the inventory position process. We construct the average cost per time objective function using the renewal reward theorem. It is assumed that the duration of the ON period is E_k (i.e., k-stage Erlangian) and the OFF period is general. In analogy with queuing notation we call this an E_k/G system. By employing the ‘method of stages’, we obtain a problem with a larger state space for the ON/OFF stochastic process; but the resulting ON process can now be analyzed using Markovian techniques. For asymptotic values of q, the objective function assumes a particularly simple form which is shown to be convex under mild restrictions on the density functions of demand. Numerical examples illustrate the results.     

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.     

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.     

Modified base-stock policies for semiconductor production system with dependent yield rates

We consider a two-stage production system faced by semiconductor manufacturing which produces a hierarchy of multiple grades of outputs. In the first stage, a single type of input (wafer) is used to produce multiple types of semi-finished parts with dependent yield rates, and in the second stage, each type of semi-finished parts can be transformed into a corresponding type of final products, or downgraded to a type of lower grade final products. Random customer demands are faced on the final products, and demands of different types of final products are not allowed to be substituted. The advantage of this production system is that it can prevent unhealthy ordering from customers who intentionally send out false demand signals for high grade products and revise the orders to lower grade products when the delivery time is close, which was observed in semiconductor manufacturing. The objective of the study is to plan the quantity of the input at the first stage and the respective downgrade quantities at the second stage so as to meet the required service level at the minimum cost. With some common assumptions, we propose a modified base-stock policy for this two-stage production system and show that the occurrence of nil excess inventory above the base-stock level follows a renewal process. We further extend the modified base-stock policy to a better policy that invokes risk pooling over multiple grade products. The performance of these two polices are evaluated via simulation to provide managerial insights.     

Effect of deterioration on the instantaneous replenishment model with imperfect quality items

The assumptions required to justify the use of the economic order quantity model (EOQ) are rarely met. To provide mathematical models that more closely represent real-life situations, these assumptions must be relaxed. Among these assumptions are, first, items stocked are of perfect quality, and second, they preserve their characteristics during their stay in inventory. This paper considers a modified EOQ-type inventory model for a deteriorating item with unreliable supply. That is, a percentage of the on-hand inventory is wasted due to deterioration. Moreover, orders may contain a random proportion of defective items, which follow a known distribution. As soon as an order is received, a retailer conducts a screening process to identify imperfect quality items, which are salvaged as a single batch at the end of the screening process. First, a mathematical model is developed, assuming that no shortages are allowed. For that, it is assumed that the inventory level when placing an order is just enough to cover the demand during the screening period. The concavity of the profit function is established and sensitivity analysis is provided to analyze the impact of changing various model parameters on the optimal order quantity and profit. Then, the assumption of no shortages is relaxed, and a model is developed to incorporate backorders. We analyze the model with backorders numerically and provide managerial insights.     

Modeling and analysis for determining optimal suppliers under stochastic lead times

The policy of simultaneously splitting replenishment orders among several suppliers has received considerable attention in the last few years and continues to attract the attention of researchers. In this paper, we develop a mathematical model which considers multiple-supplier single-item inventory systems. The item acquisition lead times of suppliers are random variables. Backorder is allowed and shortage cost is charged based on not only per unit in shortage but also per time unit. Continuous review (s,Q)$(s\text{,}Q)$ policy has been assumed. When the inventory level depletes to a reorder level, the total order is split among n suppliers. Since the suppliers have different characteristics, the quantity ordered to different suppliers may be different. The problem is to determine the reorder level and quantity ordered to each supplier so that the expected total cost per time unit, including ordering cost, procurement cost, inventory holding cost, and shortage cost, is minimized. We also conduct extensive numerical experiments to show the advantages of our model compared with the models in the literature. According to our extensive experiments, the model developed in this paper is the best model in the literature which considers order splitting for n-supplier inventory systems since it is the nearest model to the real inventory system.     

Optimising a general repair kit problem with a service constraint

Field services are a particular type of after-sales service performed at the customer’s location where technicians repair malfunctioning machines. The inventory decisions about which spare part types to take to the repair site and in what quantities is called the repair kit problem. This problem is characterized by an order-based performance measure since a customer is only satisfied when all required spare parts are available to fix the machine. As a result, the service level in the decision making process is defined as a job fill rate. In this paper we derive a closed-form expression for the expected service level and total costs for the repair kit problem in a general setting, where multiple units of each part type can be used in a multi-period problem. Such an all-or-nothing strategy is a new characteristic to investigate, but commonly used in practice. Namely, items are only taken from the inventory when all items to perform the repair are available in the right quantity. We develop a new algorithm to determine the contents of the repair kit both for a service and cost model while incorporating this new expression for the job fill rate. We show that the algorithm finds solutions which differ on average 0.2% from optimal costs. We perform a case study to test the performance of the algorithm in practice. Our approach results in service level improvements of more than 30% against similar holding costs.