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.     

Optimal policy for a dynamic,non-stationary,stochastic inventory problem with capacity commitment

This paper studies a single-product, dynamic, non-stationary, stochastic inventory problem with capacity commitment, in which a buyer purchases a fixed capacity from a supplier at the beginning of a planning horizon and the buyer’s total cumulative order quantity over the planning horizon is constrained with the capacity. The objective of the buyer is to choose the capacity at the beginning of the planning horizon and the order quantity in each period to minimize the expected total cost over the planning horizon. We characterize the structure of the minimum sum of the expected ordering, storage and shortage costs in a period and thereafter and the optimal ordering policy for a given capacity. Based on the structure, we identify conditions under which a myopic ordering policy is optimal and derive an equation for the optimal capacity commitment. We then use the optimal capacity and the myopic ordering policy to evaluate the effect of the various parameters on the minimum expected total cost over the planning horizon.     

需求分布未知且提前期可控的联合库存决策

在提前期内需求为自由分布且提前期依赖于订购批量和生产率的条件下,建立了同时考虑供需双方成本的联合库存决策模型,运用最小最大准则分析了模型最优解的存在性,设计了高效的最优解搜索算法.通过数值算例说明算法的有效性,分析了获取需求分布信息的代价与收益;并与供需独立的决策模型比较,表明了联合库存决策可以显著地降低供应链成本.     

Optimality of Base-Stock and (s, S) Policies for Inventory Problems with Information Delays

Information delays exist when the most recent inventory information available to the inventory manager (IM) is dated; namely, the IM observes only the inventory level of an earlier period. We introduce information delays into the standard multiperiod stochastic inventory problem with backorders. We consider two types of information delays: (i) a constant delay and (ii) a random delay. We define an appropriate reference inventory position, which is a sufficient statistics for finding the optimal order quantity. We show that the optimal ordering policy is of base-stock type with respect to the reference inventory position and is of (s, S) type if there is also a fixed cost of ordering.All authors were supported in part by NSF Grant DMS-0509278.     

A stochastic program based lower bound for assemble-to-order inventory systems

In this paper we introduce a multi-stage stochastic program that provides a lower bound on the long-run average inventory cost of a general class of assemble-to-order (ATO) inventory systems. The stochastic program also motivates a replenishment policy for these systems. Our lower bound generalizes a previous result of Do?ru et al. (2010) [3] for systems with identical component replenishment lead times to those with general deterministic lead times. We provide a set of sufficient conditions under which our replenishment policy, coupled with an allocation policy, attains the lower bound (and is hence optimal). We show that these sufficient conditions hold for two examples, a single product system and a special case of the generalized W model.     

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.     

A new algorithm for a multi-item periodic review inventory system

In this paper, we deal with a multi-item, stochastic, periodic review inventory system with general cost structure which permits partial or complete backlogging of unfilled demand. Since both the (, S) policy and the mixed reorder policy are not optimal, we derive several properties of an optimal ordering policy and propose a new algorithm for computing it. This algorithm is based on the policy iteration method (PIM), but reduces substantially computation times in the policy evaluation and improvement routines of the PIM.     

Some stochastic inventory models with deterministic variable lead time

In this paper we determine optimal reduction in the procurement lead time duration for some stochastic inventory models, jointly with the optimal ordering decisions. The models are developed with complete and partial information about the lead time demand distribution. The stochastic models analyzed in this paper are the classical continuous and periodic review models with a mixture of backorders and lost sales and the base stock model. For each of these models, we provide sufficient conditions for the uniqueness of the optimal operating policy. We also develop algorithms for solving these models and provide illustrative numerical examples.     

不确定性对混合CVaR约束库存系统的影响

运用应用概率中的随机占优研究需求不确定性对混合CVaR约束库存系统最优订购量和最优利润的影响。引入刻画决策者风险态度的“风险偏好系数”,得到系统最优订购量和最优利润关于风险偏好系数的单调性。研究表明随机大需求总会导致系统较高的最优订购量和最优利润;在割准则序意义下,最优订购量可能随需求可变性的增加而增加也可能随需求可变性的增加而减少;在二阶随机占优且风险偏好系数大于等于1的情况下系统最优利润具有随机单调性,然而当风险偏好系数小于1时最优利润在二阶随机占优意义下的结论不一定成立,我们通过一个数值例子来说明。     

Optimal policy for brownian inventory models with general convex inventory cost

We study an inventory system in which products are ordered from outside to meet demands, and the cumulative demand is governed by a Brownian motion. Excessive demand is backlogged. We suppose that the shortage and holding costs associated with the inventory are given by a general convex function. The product ordering from outside incurs a linear ordering cost and a setup fee. There is a constant leadtime when placing an order. The optimal policy is established so as to minimize the discounted cost including the inventory cost and ordering cost.     

EOQ holds under stochastic demand,a technical note

We consider a variant of the economic order quantity (EOQ) model. Mainly, we assume that demand occurs at random, one unit at a time, and is characterized by independent and identically distributed times between two demand epochs. We also assume that the ordering policy is characterized by ordering the same amount whenever the inventory level drops to zero, and a demand occurs. Surprisingly, we show that the optimal order quantity that minimizes the expected inventory cost follows the familiar EOQ formula.     

A scheduling policy for adjusting economic lot quantities to a feasible solution

A heuristic scheduling policy is introduced for a multi-item, single-machine production facility. The scheduling policy uses the presumed optimal order quantities derived from solving an Economic Lot Size Problem and checks that the quantities obtain a feasible production schedule according to current inventory levels and expected demand rates. If not, the scheduling policy modifies the order quantities to achieve a possible solution without shortages. The scheduling policy is inspired by modification of the similar heuristic Dynamic Cycle Lengths Policy by Leachman and Gascon from 1988, 1991. The main characteristics of this scheduling policy are successive batches of the same item are treated explicitly, due to that it is quite possible that one item be manufactured several times before one other item is manufactured once more; the batches are ordered in increasing run-out time; if the existing situation creates stock-outs with ordinary order quantities, then the order quantities are decreased with a common scaling factor to try to prevent inventory shortages; in case the decrease of the order quantities changes expected run-out times, the batches are reordered after new run-out times; no filling up to an explicit inventory level is done, the filling up is done by the desirable order quantity; to prevent possible excess inventory the policy suggests time periods where no production should be performed. The scheduling policy contains no economical evaluation; this is supposed to be done when the order quantities are calculated, the policy prevents shortages and excess inventory. A numerical example illustrates the suggested scheduling policy. Finally, it is discussed as to how the policy can also take into account stochastic behaviour of the demand rates and compensate the schedule by applying appropriate safety times.     

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.     

需求依赖于服务水平的易变质品库存策略研究

本文研究需求依赖于上一周期服务水平、缺货时订单部分损失的两周期易变质品库存问题。分别考虑一次订货和多次订货两种情况,以平均利润最大化为目标构建库存模型,证明了模型解的存在性和唯一性,得到了最优库存服务水平和最优补货策略。最后,通过算例给出两个模型的应用,对重要参数进行了灵敏度分析,并且将两种模型的结果进行了对比分析。结果表明:订单损失率的增加会提高服务水平,但会使得利润降低;顾客期望服务水平的提高会降低第一阶段的服务水平,同时使利润减少;单位库存持有成本或变质率的增加会降低服务水平和平均利润。通常情况,企业通过多次订货能获得更大的利润,而只有当库存持有成本极小时,一次订购才能够获得更大的利润。同时,结果也表明:服务水平对库存策略有较大的影响,因此在进行库存决策时考虑服务水平具有重要的作用。     

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.     

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.     

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.     

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。     

Analysis of a single-stage,single-product,stochastic, MRP-controlled assembly system

This paper models a single-stage, single-product, stochastic assembly system, operating according to an Materials Requirements Planning controlled (MRP) ordering philosophy. It deals explicitly with the underlying stochastic process that describes the end-product inventory position, enabling production lead times to be treated as independent and generally distributed random variables. The inventory position process is identified as a Markov renewal process, and this structure is exploited to determine system performance measures such as average inventory level, average backorder level, and the probability distribution of the end-product inventory position. An example, which demonstrates the type of analysis possible, focuses on quantifying the effect of kitting on the availability of end-products.