An optimal ordering policy for a spare unit with lead time

We consider a generalised ordering policy in which a spare unit for replacement can be delivered only by order after a constant lead time. Introducing the costs for ordering, shortage and holding, we derive the expected cost per unit time in the steady-state. We discuss the optimal ordering policy which minimises the expected cost, and show in a main theorem that the optimal policy is reduced to either one of two typical ordering policies depending on some conditions. We further discuss a similar ordering policy with varying lead times.     

Optimal Ordering Policies for Inventory Systems with Emergency Ordering

This paper considers dynamic single- and multi-product inventory problems in which the demands in each period are independent and identically distributed random variables. The problems considered have the following common characteristics. At the beginning of each period two order quantities are determined for each product. A “normal order” quantity with a constant positive lead time of λ _n periods and an “emergency order” quantity with a lead time of λ _e periods, where λ _e = λ _n - 1. The ordering decisions are based on linear procurement costs for both methods of ordering and convex holding and penalty costs. The emergency ordering costs are assumed to be higher than the normal ordering costs. In addition, future costs are discounted.For the single-product problem the optimal ordering policy is shown to be the same for all periods with the exception of the last period in the N-period problem. For the multi-product problem the one- and N-period optimal ordering policy is characterized where it is assumed that there are resource constraints on the total amount that can be ordered or produced in each period.     

Generalized spare ordering policies with random lead time

Generalized ordering policies for a spare unit with general life time and lead time distributions are considered. Introducing the costs of ordering, shortage, and holding, we derive the expected cost rate and the cost effectiveness. We show that the generalized ordering policy minimizing the cost rate or maximizing the cost effectiveness is either one of the two typical ordering policies: (1) the original unit is replaced as soon as the ordered spare is delivered, or (2) the delivered spare is put into inventory until the original unit fails. Some properties regarding the optimum ordering policy and the optimum ordering time are also presented.     

Optimal spare ordering policy under a rebate warranty

This paper develops, from the customer’s perspective, the optimal spare ordering policy for a non-repairable product with a limited-duration lifetime and under a rebate warranty. The spare unit for replacement is available only by order and the lead time for delivery follows a specified probability distribution. Through evaluation of gains due to the rebate and the costs due to ordering, shortage, and holding, we derive the expected cost per unit time and cost effectiveness in the long run and examine the optimal ordering time by minimizing or maximizing these cost expressions. We show that there exists a unique optimum solution under mild assumptions. We provide a numerical example and illustrate sensitivity analysis.     

具有Weibull生存死亡特征的冷链品库存补货定价策略研究

建立了无限期内冷链品具有Weibull生存死亡特征、随机需求且受售价影响的库存补货定价模型,其中售价是连续变化的,需求率是售价的指数函数,变质率服从的三参数Weibull分布,提前期固定。系统以利润最大化为目标函数,在(r,Q)库存策略下,建立库存模型,采用直接法,对模型近似求解,得到最优补货定价策略。利用Matlab进行算例模拟和灵敏度分析发现:补货提前期和单位仓储成本对补货定价策略影响较大,二者增大会导致系统利润降低;单位处理成本的增加,在一定程度上使得系统降低最优补货量,使系统利润增加;保鲜期固定的前提下,受冷链品的流动环境因子和存储环境因子影响的变质率对补货定价策略影响较大,它的增大会使系统利润降低。这些发现能够帮助优化系统模型,对现实问题具有一定的指导意义。     

Effect of declining selling price: profit analysis for a single period inventory model with stochastic demand and lead time

In this paper, considering the empirical trend for sales and price of fashion apparels as prototype, optimal ordering policy for a single period stochastic inventory model is investigated. The impact of the presence of random lead time and declining selling price on the profitability of the retailer is explored. Existence of unique optimal solutions for net profit functions is proved. Numerical examples are presented to illustrate the method of identifying profitable levels of inventory holding and penalty costs. Percentage profit per unit investment in inventory is obtained in order to assist managers in taking business decisions, specifically to the extent of whether or not to take up a particular business under known constraints. It is demonstrated that the optimal inventory policy in the absence of price decline and lead time differs considerably from that when lead time and price decline are simultaneously considered.     

Inventory control with indivisible units of stock transfer

We consider an infinite horizon, single item inventory model with backorders and a fixed lead time. Demand is stationary stochastic and review is periodic. Inventory may only be replenished in multiples of a fixed package size q but demands may be of any size. Ordering costs are linear and combined holding and shortage costs can be expressed as a convex function of the inventory position. The control policy is defined as (s, S, q), where an order is placed if the inventory position falls to or below s and the order size is the largest multiple of q which results in the inventory position not exceeding S. The parameters s and S are restricted to be multiples of q. The objective is to find the control policy that minimizes the long run average cost per unit time. The optimal solution procedure requires renewal theory and a structured search. Fortunately, a heuristic based on the ‘quantized ordering’ approach of Zheng and Chen provides solutions that are near optimal over a broad range of parameter values.     

Optimality of D-Policies for an M/G/1 Queue with a Removable Server

We consider an M/G/1 queue with a removable server. When a customer arrives, the workload becomes known. The cost structure consists of switching costs, running costs, and holding costs per unit time which is a nonnegative nondecreasing right-continuous function of a current workload in the system. We prove an old conjecture that D-policies are optimal for the average cost per unit time criterion. It means that for this criterion there is an optimal policy that either runs the server all the time or switches the server off when the system becomes empty and switches it on when the workload reaches or exceeds some threshold D.     

Optimal spare ordering policy for preventive replacement under cost effectiveness criterion

This paper presents a spare ordering policy for preventive replacement with age-dependent minimal repair and salvage value consideration. The spare unit for replacement is available only by order and the lead-time for delivering the spare due to regular or expedited ordering follows general distributions. To analyze the ordering policy, the failure process is modelled by a non-homogeneous Poisson process. By introducing the costs due to ordering, repairs, replacements and downtime, as well as the salvage value of an un-failed system, the expected cost effectiveness in the long run are derived as a criterion of optimality. It is shown, under certain conditions, there exists a finite and unique optimum ordering time which maximizes the expected cost effectiveness. Finally, numerical examples are given for illustration.     

A heuristic for triggering emergency orders in an inventory system

This paper considers a single-echelon inventory system with a warehouse facing compound Poisson customer demand. Normally the warehouse replenishes from an outside supplier according to a continuous review reorder point policy. However, it is also possible to use emergency orders. Such orders incur additional costs but have a much shorter lead time. We consider standard holding and backorder costs as well as ordering costs. A heuristic decision rule for triggering emergency orders is suggested. The decision rule minimizes the expected costs under the assumption that there is only a single possibility for an emergency replenishment, but the rule is used repeatedly as a heuristic. Given a certain reorder point policy for normal replenishments, our decision rule will always reduce the expected costs. A simulation study illustrates that the suggested technique performs well under different conditions.     

A continuous review replenishment–disposal policy for an inventory system with autonomous supply and fixed disposal costs

In this study, we analyze an inventory system facing stochastic external demands and an autonomous supply (independent return flow) in the presence of fixed disposal costs and positive lead times under a continuous review replenishment–disposal policy. We derive the analytical expressions of the operating characteristics of the system; and, construct the objective function to minimize the total expected costs of ordering, holding, purchasing and disposal per unit time subject to a fill rate constraint. An extensive numerical analysis is conducted to study the sensitivity of the policy parameters and the benefit of employing a policy which allows for disposal of excess stock in this setting. We model the net demand process as the superposition of normally distributed external demand and inflows, which is expressed as a Brownian motion process. Our findings indicate that the disposal option results in considerable savings even (i) in the presence of non-zero fixed disposal costs, (ii) large actual demand rates with high return ratios (resulting in small net demands) and (iii) for moderate return ratios with high demand variability.     

Analysis of a two-echelon inventory system with two supply modes

In this paper, we consider a serial two-echelon periodic review inventory system with two supply modes at the most upstream stock point. As control policy for this system, we propose a natural extension of the dual-index policy, which has three base-stock levels. We consider the minimization of long run average inventory holding, backlogging, and both per unit and fixed emergency ordering costs. We provide nested newsboy characterizations for two of the three base-stock levels involved and show a separability result for the difference with the remaining base-stock level. We extend results for the single-echelon system to efficiently approximate the distributions of random variables involved in the newsboy equations and find an asymptotically correct approximation for both the per unit and fixed emergency ordering costs. Based on these results, we provide an algorithm for setting base-stock levels in a computationally efficient manner. In a numerical study, we investigate the value of dual-sourcing in supply chains and illustrate that dual-sourcing can lead to significant cost savings in cases with high demand uncertainty, high backlogging cost or long lead times.     

Scheduling production of multiple part-types in a system with pre-known demands and deterministic inactive time intervals

We introduce and formalise a scheduling problem that consists in determining an optimum policy (i.e. one that minimises total inventory holding costs) to produce n part-types using a system that is able to share its capacity at all times among these part-types and that switches between an active and an inactive state for pre-known periods of time. Consequently, when active the system must produce enough reserves to meet the demand during the inactive interval. We show that there is always a simple optimum policy in which the production of the part-types is prioritised and, provided the units are properly defined, the optimum priority ordering corresponds to a non-decreasing sequence of the unit holding costs of the part-types.     

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.     

Continuous-review,lost-sales inventory models with Poisson demand,a fixed lead time and no fixed order cost

This paper considers continuous-review lost-sales inventory models with no fixed order cost and a Poisson demand process. There is a holding cost per unit per unit time and a lost sales cost per unit. The objective is to minimise the long run total cost. Base stock policies are, in general, sub-optimal under lost sales. The optimal policy would have to take full account of the remaining lead times on all the orders currently outstanding and such a policy would be too complex to analyse, let alone implement. This paper considers policies which make use of the observation that, for lost sales models, base stock policies can be improved by imposing a delay between the placement of successive orders. The performance of these policies is compared with that of the corresponding base stock policy and also with the policy of ordering at fixed and regular intervals of time.     

Demand seasonality in retail inventory management

We investigate the value of accounting for demand seasonality in inventory control. Our problem is motivated by discussions with retailers who admitted to not taking perceived seasonality patterns into account in their replenishment systems. We consider a single-location, single-item periodic review lost sales inventory problem with seasonal demand in a retail environment. Customer demand has seasonality with a known season length, the lead time is shorter than the review period and orders are placed as multiples of a fixed batch size. The cost structure comprises of a fixed cost per order, a cost per batch, and a unit variable cost to model retail handling costs. We consider four different settings which differ in the degree of demand seasonality that is incorporated in the model: with or without within-review period variations and with or without across-review periods variations. In each case, we calculate the policy which minimizes the long-run average cost and compute the optimality gaps of the policies which ignore part or all demand seasonality. We find that not accounting for demand seasonality can lead to substantial optimality gaps, yet incorporating only some form of demand seasonality does not always lead to cost savings. We apply the problem to a real life setting, using Point-of-Sales data from a European retailer. We show that a simple distinction between weekday and weekend sales can lead to major cost reductions without greatly increasing the complexity of the retailer’s automatic store ordering system. Our analysis provides valuable insights on the tradeoff between the complexity of the automatic store ordering system and the benefits of incorporating demand seasonality.     

Profit maximization in an inventory system with time-varying demand,partial backordering and discrete inventory cycle

In this paper, an inventory problem where the inventory cycle must be an integer multiple of a known basic period is considered. Furthermore, the demand rate in each basic period is a power time-dependent function. Shortages are allowed but, taking necessities or interests of the customers into account, only a fixed proportion of the demand during the stock-out period is satisfied with the arrival of the next replenishment. The costs related to the management of the inventory system are the ordering cost, the purchasing cost, the holding cost, the backordering cost and the lost sale cost. The problem is to determine the best inventory policy that maximizes the profit per unit time, which is the difference between the income obtained from the sales of the product and the sum of the previous costs. The modeling of the inventory problem leads to an integer nonlinear mathematical programming problem. To solve this problem, a new and efficient algorithm to calculate the optimal inventory cycle and the economic order quantity is proposed. Numerical examples are presented to illustrate how the algorithm works to determine the best inventory policies. A sensitivity analysis of the optimal policy with respect to some parameters of the inventory system is developed. Finally, conclusions and suggestions for future research lines are given.

     

Sensitivity of continuous review stochastic (s,S) inventory systems to ordering delays

This paper examines continuous review stochastic (s, S) inventory systems with ordering delays. That is, systems where the difference between the time the order should be placed and the time the order is actually placed is non-trivial. The traditional inventory ordering, holding and penalty costs are included and the average cost for an (s, S) policy is developed and examined. Computational results are presented for two cases. In the first case, the manager is aware of the delay and uses the policy that minimizes all costs. We present the increase in cost due to having a delay. In the second case the manager is unaware of the delay and uses the (integer) square root formula. We present the increase in cost due to using the square root formula when it is inappropriate and in fact our computational results indicate that there may very well be a large increase in cost due to being unaware of the ordering delay.     

Effect of coordinated replenishment policies on quality

Coordinated replenishment is a supply chain policy that affects many operational performance measures, including cost, lead time, and quality. In this paper, we develop a mathematical model of a simplified supply chain in which conformance quality is one of the supplier's decision variables and both the supplier and its customer are trying to minimize expected annual cost. Our expected cost model includes the important quality costs (appraisal, prevention, internal failure, and external failure) as well as holding, set-up, and ordering costs. Our results indicate that coordination leads to a decline in total cost but that coordination does not necessarily lead to an improvement in quality. In other words, buyers who are using coordinated replenishment may be trading higher quality for lower cost.