Analysis of an order-up-to-level policy for perishables with random issuing

Much of the research in perishable items inventory management has focused on the first-in-first-out issue process. However, motivated by the technical characteristics of the blood unit issue process, we model an order-up-to-level policy under periodic review setting with random issue of items from inventory. We provide empirical evidence in support of the random issuing assumption using real data on serial numbers of blood units issued from a blood bank. For general demand distribution we derive exact expressions for per period expected shortage, expected wastage and expected cost as functions of the policy parameters R (order-up-to-level) and T (review period). Since the exact model becomes computationally burdensome with increase in the number of periods of life of the perishable item an approximate model for the random issuing process is developed. The accuracy of the approximation is affirmed using simulation analysis. A gradient search-based heuristic is provided to identify the optimum policy parameters for the approximate model. A real life application of the model is demonstrated in determining the optimum frequency and order-up-to-level for blood collection at a blood bank.     

Optimal inventory policy for specialty chemical products in multiple packages

Many products are inventoried and sold in multiple outer packages, which causes the manufacturer or distributor to maintain a segmented inventory. Oftentimes, this scenario leads to shortages in some packages and over-stock in other package types. The shortages are usually linked to specific shortage costs, and the over-stock is always associated with some sort of unnecessary holding cost that either erodes profit margins or results in increased costs for the consumer. In this paper, the inventory policy for a specialty chemical manufacturer's cleaner, which is inventoried in four unique packages, is studied. A mathematical model is developed to account for the costs associated with initial procurement, holding and repackaging (shortage), which are the primary costs associated with the inventory system. Based upon this model, an optimal inventory policy is generated that reduces the need for repackaging, without requiring a prohibitive amount of safety stock. A specific historical case is summarized, and the results of this example are compared with the results that the new inventory policy would have produced, given the same circumstances.     

Optimal control policy for a standing order inventory system

In this paper, we consider a standing order inventory system in which an order of fixed size arrives in each period. Since demand is stochastic, such a system must allow for procurement of extra units in the case of an emergency and sell-offs of excess inventory. Assuming the average-cost criterion, Rosenshine and Obee (Operations Research 24 (1976) 1143–1155) first studied such a system and devised a 4-parameter inventory control policy that is not generally optimal. The current paper uses dynamic programming to determine the optimal control policy for a standing order system, which consists of only two operational parameters: the dispose-down-to level and order-up-to level. Either the average-cost or discounted-cost criterion can be assumed in the proposed model. Also, both the backlogged and lost-sales problems are investigated in this paper. By using a convergence theorem, we stop the dynamic programming computation and obtain the two optimal parameters.     

带有两货栈及时变需求的变质性物品的最优EOQ模型(英)

本文针对一般的时变需求与两个货栈（自己货栈和租用货栈），建立了变质性物品的最优确定性EOQ模型，提供了用来寻求最优补充策略的方法，并就线性需求出示了两个数值例子．     

Optimal policy for deteriorating items with trapezoidal type demand and partial backlogging

Inventory model for time-dependent deteriorating items with trapezoidal type demand rate and partial backlogging is considered in this paper. The demand rate is defined as a continuous trapezoidal function of time, and the backlogging rate is a non-increasing exponential function of the waiting time up to the next replenishment. We proposed an optimal replenishment policy for such inventory model, numerical examples to illustrate the solution procedure.     

基于MDP的血小板库存最优订货策略和使用策略研究

血液是一种特殊的稀缺资源,临床上通过输血可以治疗一些血液疾病甚至挽救生命。目前血液尚不能人工制造,只能通过献血者捐献的方式获得,因此需要合理的库存管理来有效利用有限的血液资源。近期研究表明,血小板生命周期较短,且其输注效果会随着储存时间的延长而降低。因此,本文针对血小板的库存控制问题,考虑短缺、浪费和输血新鲜度之间的平衡,以效用最大化为目标建立有限时域的马尔科夫决策过程模型,探讨最优的血小板订货策略和使用策略。根据最优策略数值解的主要特征提出了启发式策略,数值实验结果表明所提启发式控制策略性能接近于最优控制策略,且优于传统策略。     

A note on “Two-warehouse inventory model with deterioration under FIFO dispatch policy”

A recently published paper by Lee [C.C. Lee, Two-warehouse inventory model with deterioration under FIFO dispatching policy, European Journal of Operational Research 174 (2006) 861–873] considers different dispatching models for the two-warehouse inventory system with deteriorating items, in which Pakkala and Achary’s LIFO (last-in–first-out) model [T.P.M. Pakkala, K.K. Achary, A deterministic inventory model for deteriorating items with two warehouses and finite replenishment rate, European Journal of Operational Research 57 (1992) 71–76] is first modified, and then the author concludes that the modified LIFO model always has a lower cost than Pakkala and Achary’s LIFO model under a particular condition specified by him. The present note points out that this conclusion is incorrect and misleading. Alternatively, we provide a new sufficient condition such that the modified LIFO model always has a lower cost than Pakkala and Achary’s model. Besides, we also compare Pakkala and Achary’s original LIFO model with Lee’s FIFO (first-in–first-out) model for the special case where the two warehouses have the same deteriorating rates. Finally, numerical examples are provided to investigate and examine the impact of corresponding parameters on policy choice. The results in this note give a much clearer picture than those at Lee’s paper about the relationships between the different dispatching policies for the two-warehouse inventory system with deterioration items.     

Inventory policy for products with price and time-dependent demands

This paper investigates the problem of jointly determining the order size and optimal prices for a perishable inventory system under the condition that demand is time and price dependent. It is assumed that a decision-maker has the opportunity to adjust prices before the end of the sales season to influence demand and to improve revenues. A mathematical model is developed to find the optimal number of prices, the optimal prices and the order quantity. Analytical results show that a stationary solution to the Kuhn–Tucker necessary conditions can be found and it is shown to be the optimal solution. The analytical results lead us to derive a solution procedure for determining the optimal order size and prices.     

An optimal replenishment policy for deteriorating items with stock-dependent demand and relaxed terminal conditions under limited storage space

This paper deals with the problem of determining the optimal replenishment policy for deteriorating items with stock-dependent demand in which the terminal condition of zero-ending inventory is relaxed. In the model, shortages are allowed and partial backlogging/lost sales. That is, the zero/non-zero ending inventory models are considered simultaneously. The items in stock are displayed to the customers in shelves with limited storage capacity. In theoretical analysis, the necessary and sufficient conditions of the existence and uniqueness of the optimal solutions under various cases are shown. We then also provide a simple algorithm to find the optimal solutions for various situations. Further, a couple of numerical examples are presented to demonstrate the developed model and solution procedure, and several management insights are obtained from the numerical examples. Finally, sensitivity analysis of the optimal solution with respect to major parameters is also carried out.     

Optimal inventory policy under power demand pattern and partial backlogging

In this paper, we study an inventory model with a power demand pattern that allows shortages. It is assumed that only a fraction of demand is backlogged during the shortage period and the remainder is considered lost sales. The aim of the paper is to determine the lot size and the length of the inventory cycle that maximize the total inventory profit per unit time. A general approach to obtain the optimal solution of the inventory problem and the maximum associated profit is developed. Some inventory models proposed in the literature are particular cases of the model analyzed here. Numerical examples are included to complement the theoretical results.     

需求受库存影响下具有固定保质期物品的最优订货策略

带有固定保质期物品的订货是供应链终端销售系统的一个重要决策问题,假设需求依赖库存展示水平并考虑"后进先出"的销售策略而建立了相应的库存决策模型,其中物品在固定保质期内仍具有常数的变质速率.然后以系统平均利润最大化为目标讨论了模型最优解的存在性及唯一性,并提供了寻求模型整体最优解的简单方法.最后给出应用实例,并分析了模型参数变化对最优订货策略的影响.     

An optimal ordering and issuing policy for a two-stage inventory system for perishable products

This paper considers the problem of ordering and issuing policies arising in controlling finite-life-time fresh-meat-carcass inventories in supermarkets. A supermarket orders a product, which constitutes a set of sub-products of fixed proportion, from a vendor at the beginning of each time cycle. After it is received from the vendor, the product is stored in the cool-room, before being issued to the display shelves. The sub-products then satisfy random customer demand. After passing the life-time, sub-products are salvaged. In this system, the sub-products are issued to the display shelves according to theorder-up-to level policies at the beginning of every period. The decisions to be taken to solve this problem are the product-ordering quantity from the outside vendor and the order-up-to issuing quantities for each sub-product. The objective function to be maximized is the expected profit per unit time, consisting of revenue from sales and salvage, and the cost of ordering, processing (or issuing), inventory holding, emergency issuing, and shortage. In this paper we first develop a mathematical model describing actual operations and then simplify the sub-product runout period so that optimal ordering and issuing policies are easily established. We then carry out extensive numerical experiments for a case of two sub-products in order to ascertain the properties and the behavior of the optimal solutions.     

Profitability ratio maximization in an inventory model with stock-dependent demand rate and non-linear holding cost

This paper studies a deterministic inventory model with a stock-dependent demand pattern where the cumulative holding cost is a non-linear function of both time and stock level. When the monetary resources are limited and the inventory manager can invest his/her money in buying different products, it seems reasonable to select the ones that provide a higher profitability. Thus, a new approach with the aim of maximizing the profitability ratio (defined as the profit/cost quotient) is considered in this paper. We prove that the profitability ratio maximization is equivalent to minimizing the inventory cost per unit of an item. The optimal policy is obtained in a closed form, whose general expression is a generalization of the classical EOQ formula for inventory models with a stock-dependent demand rate and a non-linear holding cost. This optimal solution is different from the other policies proposed for the problems of minimum cost or maximum profit per unit time. A complete sensitivity analysis of the optimal solution with respect to all the parameters of the model is developed. Finally, numerical examples are solved to illustrate the theoretical results and the solution methodology.     

Optimal price and lot size determination for a perishable product under conditions of finite production, partial backordering and lost sale

The paper describes an EOQ model of a perishable product for the case of price dependent demand, partial backordering which depends on the length of the waiting time for the next replenishment, and lost sale. The model is solved analytically to obtain the optimal price and size of the replenishment. In the model, the customers are viewed to be impatient and a fraction of the demand is backlogged. This fraction is a function of the waiting time of the customers. In most of the inventory models developed so far, researchers considered that inventory accumulates at the early stage of the inventory and then shortage occurs. This type of inventory is called IFS (inventory followed by shortage) policy. In the present model we consider that shortage occurs before the starting of inventory. We have proved numerically that instead of taking IFS, if we consider SFI (shortage followed by inventory) policy, we would get better result, i.e., a higher profit. The model is extended to the case of non-perishable product also. The optimal solution of the model is illustrated with the help of a numerical example.     

基于二层信用支付的变质物品库存模型

基于时变需求的库存问题一直是库存管理者关注的重点之一,大多数基于二层信用支付的库存模型都是假设需求率为常数.假设需求率是时间的指数函数,建立了二层信用支付条件下的变质物品库存模型,并证明了最优解是存在且唯一的,给出了确定最优补货策略的算法步骤,最后通过数值例子对主要参数进行了灵敏度分析.     

An optimal batch size for integrated production–inventory policy in a supply chain

This study presents a new integrated production–inventory policy under a finite planning horizon and a linear trend in demand. We assume that the vendor makes a single product and supplies it to a buyer with a non-periodic and just-in-time (JIT) replenishment policy in a supply chain environment. The objective is to minimize the joint total costs incurred by the vendor and the buyer. In this study, first, we develop a mathematical model and prove that it has the optimal solution. Then, we describe an explicit solution procedure for obtaining the optimal solution. Finally, we provide two numerical examples to illustrate both increasing and decreasing demands in our proposed model, and we show that the performance of the integrated consideration is better than the performance of any independent decision from either the buyer or the vendor.     

An Inventory Model for Deteriorating Items with?Two?Levels of Trade Credit Taking Account of Time Discounting

This paper presents an extended inventory model of Huang (J. Oper. Res. Soc. 54, 1011–1015, 2003), which investigated the retailer’s optimal inventory policy under two levels of trade credit. Herein, we consider the impact of a replenishment policy on the timing of the cash flows associated with payments to suppliers and revenue streams from customers. That is, the same cash amount will possess different money value at different future time. To see this, we adopt the more appropriate net present value (NPV) object instead of the average cost objective. In addition, the deteriorating effects will be incorporated in this inventory model, and the time to deterioration of each item follows an exponential distribution. The discounted cash flow (DCF) approach is used to derive the optimal solution in this study. Furthermore, we first show that the optimal solution not only exists bus also is unique. Then, we provide a theorem to locate the optimal ordering policy. Finally, a numerical example for illustration is provided.     

Optimal policy and stability regions for the single product periodic review inventory problem,with stationary demands

In this paper we present rules concerning the optimal policy and stability regions for the single product periodic review inventory problem with stationary demands, over a finite horizon. The key parameter to the whole study is the Lot-Sizing Index (LSI) introduced by Blackburn and Millen. Two algorithms are presented. The first one constructs stability regions which are expressed as intervals of the LSI parameter, covering the whole range of its values. The proposed algorithm is very simple to understand and implement, and most importantly, it provides a solution table which can be used by the decision maker to easily determine the optimal policy for any problem with a given horizon and any possible combination of its cost parameters, namely any LSI value. The second proposed algorithm determines the optimal policy for any given LSI value; it constitutes a completely different approach to that of the Wagner–Whitin algorithm and requires very little computational effort.     

Economic design of an inventory policy for non-instantaneous deteriorating items under permissible delay in payments

In this paper, Economic Order Quantity (EOQ) based model for non-instantaneous deteriorating items with permissible delay in payments is proposed. This model aids in minimizing the total inventory cost by finding an optimal replenishment policy. In this model shortages are allowed and partially backlogged. The backlogging rate is variable and dependent on the waiting time for the next replenishment. Some useful theorems have been framed to characterize the optimal solutions. The necessary and sufficient conditions of the existence and uniqueness of the optimal solutions are also provided. An algorithm is designed to find the optimal replenishment cycle time and order quantity under various circumstances. Numerical examples are given to demonstrate the theoretical results. Sensitivity analysis of the optimal solution with respect to major parameters of the system has been carried out and the implications are discussed in detail. In the discussions, suggestions are given to minimize the total cost of the inventory system.