直接配送下随机需求库存-路径问题最优平稳策略及其算法

直接配送策略下随机需求库存-路径问题（Stochastic Demand Inventory Routing Problem with Direct Deliveries, SDIRPDD）由于其需求的不确定性、决策的长期性以及其最优策略形式对求解其他库存-路径问题（IRP）的参考价值,使得对SDIPRDD问题的研究成为物流、供应链优化领域研究的一个热点。文章首先证明了无约束SDIRPDD的最优平稳策略为（s,S）形式,并通过分析车辆数约束对客户单阶段期望成本函数的影响,给出了存在车辆数和客户库存容量约束时SDIRPDD问题的最优平稳策略形式,进而提出了一种求解有约束SDIRPDD问题最优平稳策略的近似算法。最后,通过数值算例验证了算法的有效性并分析了结果的现实意义。     

Multi-item inventory model with quantity-dependent inventory costs and demand-dependent unit cost under imprecise objective and restrictions: A geometric programming approach

A multi-item inventory model with constant demand and infinite replenishment is developed under the restrictions on storage area, total average shortage cost and total average inventory investment cost. These restrictions may be precise or imprecise. Here, it is assumed that inventory costs are directly proportional to the respective quantities, and unit purchase/production cost is inversely related to the demand. Restricted shortages are allowed but fully backlogged. First, the problem is formulated in crisp environment taking the deterministic and precise inventory parameters. It is solved by both geometric programming (GP) and gradient-based non-linear programming (NLP) methods. Later, the problem is formulated with fuzzy goals on constraints and objectives where impreciseness is introduced through linear membership functions. It is solved using the fuzzy geometric programming (FGP) method. The inventory models are illustrated with numerical values and compared with the crisp results. A sensitivity analysis on the optimum order quantity and average cost is also presented due to the variation in the tolerance of total average inventory investment cost and total average shortage cost following Dutta et al., 1993, Fuzzy Sets and Systems, 55, 133-142.     

A competitive multiple-product newsboy problem with partial product substitution

This paper studies a multi-product competitive newsboy problem with shortage penalty cost and partial product substitution. We characterize the unique Nash equilibrium of the competitive model and analyze some properties of the equilibrium. An iterative algorithm is developed on the basis of approximating the effective demand as well as the expected profit function for each product. Numerical experiments are conducted to illustrate the impacts of product substitution, demand correlation and demand variation on the optimal order quantities and the corresponding expected profits, and to compare the total optimal inventory level of the competitive case with that of the centralized case. The conclusion that competition always results in a higher total inventory level, even under the effect of product substitution is drawn in the symmetric case.     

A newsboy problem with an emergency order under a general backorder rate function

In this paper, a generalization is presented of the newsboy problem where an emergency lot can be ordered to provide for a certain fraction of shortage. This fraction is described by a general backorder rate function which is non-increasing with respect to the unsatisfied demand. An exponential distribution for the demand during the selling season is assumed. An expression is obtained in a closed form for the optimal lot size and the maximum expected profit. A general sensitivity analysis of the optimal policy with respect to the backorder rate function and the parameters of the inventory system is developed. When the backorder rate function is described by some particular functions, its behavior is analyzed with respect to changes in the parameters. To illustrate the theoretical results, some numerical examples are also given.     

A new MRP/GT lot sizing heuristic: A simulation study

This paper studies the lot-sizing problem in Material Requirements Planning/Group Technology (MRP/GT) systems. A GT production cell is designed to produce many families of components. A major setup is required when switching from manufacturing one family of components to another family, and a minor setup is needed when switching from manufacturing a component type to another component type within the same family. Inventory holding cost is incurred if inventory level is positive, and inventory shortage cost is incurred if inventory level is negative, that is, backordering. The objective of the proposed lot-sizing problem is to minimize the sum of major and minor setup costs, holding and shortage costs, and regular production cost, and to meet simultaneously the demand requirements. The proposed problem is modelled into a linear integer program, a heuristic method to solve the problem is proposed, and a simulation experiment conducted to evaluate the performance of the proposed heuristic and some existing heuristics. The computational results show that the proposed heuristic is useful to reduce the total cost significantly over a wide variety of simulated environments.     

持有成本和变质率时变的非立即变质品库存策略

本文主要研究非立即变质品的库存策略设计问题。以平均利润最大化为决策目标,构建了一个一般化的库存模型,假设需求受即时库存水平影响且在保鲜期和变质期内库存水平对需求的影响系数不同,持有成本和变质期内的变质率均随时间发生变化,系统允许缺货且短缺量部分延迟订购。进一步证明了当参数满足一定条件时,模型存在唯一的最优解。最后,采用牛顿法给出一些具体的数值算例,并对模型中主要参数的灵敏度进行了分析。结果显示:延长物品的保鲜期将有助于增加系统的平均利润;增加变质期内的需求比增加保鲜期内的需求更有利于增加系统的平均利润;对于变质率高的产品来说,其变质处理成本对最优策略和平均利润的影响不容忽视。此外,本文所构建的模型具有一般性,因此,其应用范围更加广泛。     

A Note on the Effect of Sudden Obsolescence on the Optimal Lot Size

In most management situations, the decision to hold inventory involves some element of obsolescence risk The traditional approach to this problem, in both theory and practice, is to incorporate obsolescence costs in the inventory holding cost. This note explores this issue and demonstrates that the traditional approach is appropriate for items subject to sudden obsolescence; that is, for items with demand lifetimes that are exponentially distributed.     

基于可控提前期和延迟供货的短生命周期产品库存模型

本文用斜坡型函数来描述短生命周期产品需求变化特征,用持有成本不断增加的形式来表示存货的无形变质所带来的损失,通过向供应商支付额外的赶工成本来缩短提前期,同时在允许缺货的情况下设置合理的缺货期价格折扣来控制订单的流失,即以价格折扣的损失来挽留住更多的顾客,使其总成本到达最优。本文在此基础上建立了短生命周期产品的库存模型,并提出了最优解的求解方法。     

短生命周期产品的三种库存模型的比较

讨论短生命周期产品库存模型,考虑无形变质因素,并假定无形变质率与需求率成反比关系.在线性需求和常数生产率的条件下,假设,若需求率为1,则视作产品的市场生命周期结束.并且,在产品生命周期的前期不存在缺货的情况下,给出了短生命周期产品分别在理想状态、允许缺货以及价格折扣导致需求率变化等3种状态下的库存模型,最后在相同参数条件下对这3种模型进行比较,找出了单位时间内平均总成本最小的最优库存模型,厂商可以根据自身的情况选择合适的库存策略.     

A hybrid inventory management system responding to regular demand and surge demand

This paper proposes a hybrid policy for a stochastic inventory system facing regular demand and surge demand. The combination of two different demand patterns can be observed in many areas, such as healthcare inventory and humanitarian supply chain management. The surge demand has a lower arrival rate but higher demand volume per arrival. The solution approach proposed in this paper incorporates the level crossing method and mixed integer programming technique to optimize the hybrid inventory policy with both regular orders and emergency orders. The level crossing method is applied to obtain the equilibrium distributions of inventory levels under a given policy. The model is further transformed into a mixed integer program to identify an optimal hybrid policy. A sensitivity analysis is conducted to investigate the impact of parameters on the optimal inventory policy and minimum cost. Numerical results clearly show the benefit of using the proposed hybrid inventory model. The model and solution approach could help healthcare providers or humanitarian logistics providers in managing their emergency supplies in responding to surge demands.     

Multiechelon (R,S) Inventory Model*

Most studies in multiechelon inventory systems have concentrated on understanding the specific aspects of a system's behavior. The problem of optimal policy computation has largely been ignored. In this paper, we investigate a two-echelon inventory system experiencing stochastic demand and a pull system of inventory allocation. Both echelons use an order-up-to-level type control policy. A mathematical model is developed to determine the optimal order level at all echelons and validated through simulation. Two simple algorithms to locate the optimum solution are presented. The use of graphical tools in optimal policy calculation is also discussed.     

A New Two‐Bin Policy for Inventory Systems with Differentiated Demand Classes

We consider an inventory system under continuous review with two demand classes that are different in terms of service level required (or penalty cost incurred for backordering of demand). Prior literature has proposed the critical level rationing (CLR) policy under which the demand from the lower priority class is backordered once inventory falls below the critical level. While this reduces the penalty cost for the higher demand class, the fill rate achieved for the lower priority demand class gets compromised. In this study, we propose a new class of two‐bin (2B) policy for the problem. The proposed 2B policy assigns separate bins of inventory for the two demand classes. The demand for each class is fulfilled from its assigned bin. However, when the bin intended for the higher demand class is empty, the demand from the higher class can still be fulfilled with the inventory from the other bin. The advantage of the 2B policy is that better fill rates are achieved, especially for the lower demand class. Computational results show that the proposed policy is able to provide a much higher service level for the lower priority class demand without increasing the total cost too much and without affecting the service level for the higher priority class. When a service level constrained optimization problem is considered, the 2B policy dominates the CLR policy when the service level difference for the two classes is not too high or the service levels required for both the classes are relatively lower.     

Joint Lot Sizing and Scheduling of Multiple Items with Sequence-Dependent Setup Costs

The objective of this research is to investigate the effects of setup-cost estimating methods on the lot sizing and scheduling of multiple products in multiple periods. These initial setup cost estimators (ISCEs) are used to estimate sequence-independent initial setup costs from sequence-dependent setup costs. A search of the literature reveals that, although sequence-dependent setup costs are frequently found in practice and ISCEs are frequently used, there is a dearth of research concerning the effect of using ISCEs. After a review of the literature, a mixed integer formulation of the joint problem of lot sizing and scheduling is presented, followed by a discussion of the difficulty in solving the formulation. Next, the six ISCEs evaluated are presented. These ISCEs range from simple (select the minimum setup cost) to complex (use the branch-and-bound solution to a traveling salesman-type problem). Each ISCE is evaluated using a full factorial design with five independent variables: demand distribution (three levels), demand trend (three levels), setup to inventory level (six levels), setup distribution (three levels), and setup variability (two levels). Two hypotheses are researched. Do the more computationally complex ISCEs produce lower overall costs than do the simpler ISCEs? Does the reduction in total cost justify the additional computation cost? The results of this study demonstrate that it may be incorrect to use “conventional wisdom'’when selecting an ISCE.     

STOCK LEVELS AND DELIVERY RATES IN VENDORMANAGED INVENTORY PROGRAMS

Using the latest information technology, powerful retailers like Wal‐Mart have taken the lead in forging shorter replenishment‐cycles, automated supply systems with suppliers. With the objective to reduce cost, these retailers are directing suppliers to take full responsibility for managing stocks and deliveries. Suppliers' performance is measured according to how often inventory is shipped to the retailer, and how often customers are unable to purchase the product because it is out of stock. This emerging trend also implies that suppliers are absorbing a large part of the inventory and delivery costs and, therefore, must plan delivery programs including delivery frequency to ensure that the inherent costs are minimized. With the idea to incorporate this shift in focus, this paper looks at the problem facing the supplier who wants quicker replenishment at lower cost. In particular, we present a model that seeks the best trade‐off among inventory investment, delivery rates, and permitting shortages to occur, given some random demand pattern for the product. The process generating demand consists of two components: one is deterministic and the other is random. The random part is assumed to follow a compound Poisson process. Furthermore, we assume that the supplier may fail to meet uniform shipping schedules, and, therefore, uncertainty is present in delivery times. The solution to this transportationinventory problem requires determining jointly delivery rates and stock levels that will minimize transportation, inventory, and shortage costs. Several numerical results are presented to give a feel of the optimal policy's general behavior.     

Testing for Nonlinear Dependence in Inventory Data

This paper describes the properties of nonlinear dynamical systems in demand data and assesses the implications for inventory management. A test from Brock, Dechert, and Scheinkman (BDS) [6] for detecting nonlinearities is presented. This test assists inventory managers with two issues. First, it can be used to detect departures from the independence and stationarity assumptions of particular inventory models, thereby identifying the need for more suitable models. Second, the BDS test can determine whether forecasting model improvements are possible, and can measure improvements in the forecasting model process. The corresponding reductions in forecast errors improve inventory management and lead to reduced inventory costs. Using actual weekly demand for oil filters, this paper demonstrates the use of the BDS test and the effects of nonlinearities in demand data on inventory model performance.     

Cost and Shortage Trade-Offs in Aggregate Production Planning*

It is exceedingly difficult, if not impossible, to measure shortage costs. To bypass this difficulty in aggregate production planning, this paper develops an optimal policy function (piecewise linear or a curve) for trade-offs between shortages and the sum of production and inventory costs. The optimal management decision is based on this function. It should be of major interest in production planning since similar functions for inventory management have been successfully applied in practice.     

Shelf Space Management When Demand Depends on the Inventory Level

Two factors that their influence on the demand has been investigated in many papers are (i) the shelf space allocated to a product and to its complement or supplement products and (ii) the instantaneous inventory level seen by customers. Here we analyze the joint shelf space allocation and inventory decisions for multiple items with demand that depends on both factors. The traditional approach to solve inventory models with a state‐dependent demand rate uses a time domain approach. However, this approach often does not lead to closed‐form expressions for the profit rate with both dependencies. We analyze the problem in the inventory domain via level crossing theory. This approach leads to closed‐form expressions for a large set of demand rate functions exhibiting both dependencies. These closed‐form expressions substantially simplify the search for optimal solutions; thus we use them to solve the joint inventory control and shelf space allocation problem. We consider examples with two products to investigate the significance of capturing both demand dependencies. We show that in some settings it is important to capture both dependencies. We consider two heuristics, each one of them ignores one of the two dependencies. Using these heuristics it seems that ignoring the dependency on the shelf space might be less harmful than ignoring the dependency on the inventory level, which, based on computational results, can lead to profit losses of more than 6%. We demonstrate that retailers should use their operational control, e.g., reorder point, to promote higher demand products.     

提前期、构建成本和短缺量拖后率均可控的EOQ模型

传统库存模型通常将提前期和构建成本视为不可控制。事实上可以通过追加投资缩短提前期和降低构建成本。缺货期间,为减少订单丢失量和补偿顾客的损失,供应商会给予一定的价格折扣。现实库存系统中,容易得到需求的期望值和标准差,但较难得到其分布规律。基于此,考虑短缺量拖后率与价格折扣和缺货期间库存水平相关,提出了一种需求为任意分布且提前期和构建成本均可控的EOQ模型,证明了模型存在唯一最优解,给出了一种寻优算法。数值仿真分析表明,一般情况下,压缩提前期和降低构建成本能降低订购批量和安全库存,降低库存总成本;短缺量拖后系数和缺货概率对库存总成本影响较大,企业应尽量降低缺货概率,尤其在短缺量拖后系数较小时。     

The effects of component commonality in an infinite horizon inventory model

This paper studies the effects of component commonality in the context of an infinite horizon inventory model. Three models are proposed that are characterized by different degrees of component commonality. Assuming the three models all follow the same inventory policy, exact service level measures are derived and incorporated into cost optimization problems. With the infinite horizon assumption, potential setup cost reductions can be evaluated due to the inclusion of common components. The results indicate that, as expected, commonality incurs significant cost savings; what is new and unique is that setup cost may increase or decrease when commonality is present. In addition, when the behaviour of the optimal solutions is examined, it is found that some of the well-known properties suggested by the existing one-period models do not hold for this infinite horizon model.     

EOQ FORMULA: IS IT VALID UNDER INFLATIONARY CONDITIONS?

The classical analysis of the economic order quantity (EOQ) problem ignores the effect of inflation. When a firm's cost factors are expected to rise at an annual rate of 10 percent or more, what adjustments in order quantities should the firm make to control its lot-size inventory (or cycle stock)? Using a model that includes both inflationary trends and time discounting, it is concluded that inflation brings no incentive either to increase or to decrease order quantities. In addition, order quantities can be computed using the classical EOQ formula under inflationary conditions, provided that the cost of capital invested in inventory is interpreted as an inflation-free cost. This interpretation implies that changes in the inflation rate should not affect the cost of capital that is utilized in the EOQ formula for determining order quantities.