Economic order quantity models for ameliorating/deteriorating items under inflation and time discounting

The items that incur a gradual loss in quality or quantity over time while in inventory are usually called deteriorating items. In reality, there are some items whose value or utility or quantity increase with time and those items can be termed as ameliorating items. In this paper, an effort has been made to incorporate these two opposite physical characteristics of stored items into inventory model. We develop models for ameliorating/deteriorating items with time-varying demand pattern over a finite planning horizon, taking into account the effects of inflation and time value of money. Optimal solutions of the proposed models are derived and the effects of amelioration/deterioration on the inventory replenishment policies are studied with the help of numerical examples.     

基于生长价值跟踪的生鲜农产品采收销售模型

生鲜农产品的生长增值性对于提升渠道价值具有重要贡献。为此,将影响生鲜农产品需求的成熟度进行数学刻画并引入增值性变质库存模型,针对生鲜农产品育成育肥后定时采收销售和达到临界成熟后适时采收销售两种情形建立了农户利润模型,求解出了最优采收销售时间。研究得出,农户适时采收销售获得的利润更大;产品达到临界成熟后,农户无需再付出生产努力。     

An inventory model for ameliorating/deteriorating items with trapezoidal demand and complete backlogging under inflation and time discounting

Recently, numerous inventory models were developed for ameliorating items (say, fish, ducklings, chicken, etc.) considering the constant demand rate. However, such types of problems are not useful in the real market. The demand rate of ameliorating items is fluctuates in their life‐period. The consumption and demand of ameliorating items are not generally steady. In a few seasons, the demand rate increases; ordinarily, it is static, and sometimes, it declines. With the outcome that their demand rate can be properly portrayed by a trapezoidal‐type. In the proposed model, an inventory model for ameliorating/deteriorating items are considered with inflationary condition and time discounting rate. Additionally, having shortages that is completely backlogged. The demand rate is taken as the continuous trapezoidal‐type function of time. The amelioration and deterioration rate are considered as Weibull distribution. To obtain the minimum cost, mathematical formulation of the proposed model with solution procedure is talked about. Numerical cases are given to be checked the optimal solution. Additionally, we have talked about the convexity of the proposed model through graphically. Conclusion with future worked are clarified appropriately. Copyright © 2016 John Wiley & Sons, Ltd.     

Retailer’s optimal ordering policy for deteriorating items with maximum lifetime under supplier’s trade credit financing

Many products such as fruits, vegetables, pharmaceuticals, volatile liquids, and others not only deteriorate continuously due to evaporation, obsolescence, spoilage, etc. but also have their expiration dates (i.e., a deteriorating item has its maximum lifetime). Although numerous researchers have studied economic order quantity (EOQ) models for deteriorating items, few of them have taken the maximum lifetime of a deteriorating item into consideration. In addition, a supplier frequently offers her/his retailers a permissible delay in payments in order to stimulate sales and reduce inventory. There is no interest charge to a retailer if the purchasing amount is paid to a supplier within the credit period, and vice versa. In this paper, we propose an EOQ model for a retailer when: (1) her/his product deteriorates continuously, and has a maximum lifetime, and (2) her/his supplier offers a permissible delay in payments. We then characterize the retailer’s optimal replenishment cycle time. Furthermore, we discuss a special case for non-deteriorating items. Finally, we run several numerical examples to illustrate the problem and provide some managerial insights.     

Ordering Policies of Deteriorating Items under Permissible Delay in Payments

In developing mathematical models in inventory control it is assumed that payment will be made to the supplier for the goods immediately after receiving the consignment. However, in practice, it is found that the supplier allows a certain fixed period to settle the account. During this fixed period no interest is charged by the supplier, but beyond this period interest is charged under the terms and conditions agreed upon and, moreover, interest can be earned on the revenue received during the credit period. In this paper an attempt has been made to obtain the optimum order quantity of deteriorating items under a permissible delay in payments. A numerical example is also given.     

信用期下时滞变质品供应链库存和定价的Stackelberg均衡策略

在允许缺货和考虑资金机会成本情况下,根据时滞变质品的基本库存模型,分别构建了信用期下供应商为领导者(SL)和零售商为领导者(RL)的Stackelberg博弈模型。通过分析SL和RL下的Stackelberg博弈模型唯一均衡解,得到两个模型中均衡解的解析表达式。最后,根据数值算例分析得出:(1)在SL供应链中信用期并不总使整个供应链协调,然而在RL供应链中信用期的协调效果较好;(2)延长信用期或增加零售价格均能刺激零售商多订货;(3)在两个模型中,零售价格均随变质时刻递增,且整条供应链达到Pareto改进;(4)SL中供应商变动信用期与RL中零售商变动价格相比,SL供应链收益更高;而当信用期和价格固定,其他参数变动时,RL供应链收益更大。     

易变质产品的可替代库存模型的最优订购策略研究

研究了易变质产品的可替代库存模型.在有限计划期内,供应商面临两种不同产品的需求,当一种产品发生短缺时,另一种产品可以以一定的替代率代替短缺产品.通过分析系统的总成本函数的性质,提出了最优订购策略.最后通过算例验证了算法的最优性.     

A note on an EOQ model for deteriorating items under supplier credit linked to ordering quantity

This paper explores the inventory replenishment policy for deteriorating items in which the supplier provides a permissible delay to the purchaser if the order quantity is greater than or equal to a predetermined quantity. As a matter of fact, the inventory system discussed by this paper is the same as that of Chang et al. [C.T. Chang, L.Y. Ouyang, J.T. Teng, An EOQ model for deteriorating items under supplier credit credits linked to ordering quantity, Appl. Math. Model. 27 (2003) 983–996]. However, their approach in solving the problems needs further analysis. This article deals with an alternative approach to present a simple procedure in order to determine the optimal ordering policy when the supplier provides a permissible delay in payments linked to order quantity. Numerical examples reveal that the solution algorithm described in this paper is accurate and rapid.     

A cooperative inventory policy with deteriorating items for a two-echelon model

A cooperative inventory policy between supplier and buyer is proposed. Unlike other studies, we consider the case of deteriorating items and permit the completed back-order in the problem. We solve the problem without the condition of equal replenishments periods during a specified planning horizon and present a procedure to find the optimal solution. A case is presented to demonstrate the application of the proposed approach. The sensitivity analysis for a cooperation policy between supplier and buyer also are explored.     

A note on the inventory models for deteriorating items with ramp type demand rate

In this research we study the inventory models for deteriorating items with ramp type demand rate. We first clearly point out some questionable results that appeared in (Mandal, B., Pal, A.K., 1998. Order level inventory system with ramp type demand rate for deteriorating items. Journal of Interdisciplinary Mathematics 1, 49–66 and Wu, K.S., Ouyang, L.Y., 2000. A replenishment policy for deteriorating items with ramp type demand rate (Short Communication). Proceedings of National Science Council ROC (A) 24, 279–286). And then resolve the similar problem by offering a rigorous and efficient method to derive the optimal solution. In addition, we also propose an extended inventory model with ramp type demand rate and its optimal feasible solution to amend the incompleteness in the previous work. Moreover, we also proposed a very good inventory replenishment policy for this kind of inventory model. We believe that our work will provide a solid foundation for the further study of this sort of important inventory models with ramp type demand rate.     

An inventory control model with stochastic review interval and special sale offer

Periodic review inventory models are widely used in practice, especially for inventory systems in which many different items are purchased from the same supplier. However, most of periodic review models have assumed a fixed length of the review periods. In practice, it is possible that the review periods are of a random (stochastic) length. This paper presents an inventory control model in the case of random review intervals and special sale offer from the supplier. The replenishment interval is assumed to obey from two different distributions, namely, exponential and uniform distributions. Also, shortages are allowed in the term of partial backordering. For this model, its convexity condition is discussed and closed form solutions are proposed.     

An EOQ model for deteriorating items under supplier credits linked to ordering quantity

In the classical inventory economic order quantity (or EOQ) model, it was assumed that the purchaser must pay for the items received immediately. However, in practices, the supplier usually is willing to provide the purchaser a permissible delay of payments if the purchaser orders a large quantity. As a result, in this paper, we establish an EOQ model for deteriorating items, in which the supplier provides a permissible delay to the purchaser if the order quantity is greater than or equal to a predetermined quantity. We then characterize the optimal solution and provide an easy-to-use algorithm to find the optimal order quantity and replenishment time. Finally, several numerical examples are given to illustrate the theoretical results.     

The optimal order and payment policies for deteriorating items in discount cash flows analysis under the alternatives of conditionally permissible delay in payments and cash discount

This study investigates a deteriorating inventory problem in which the supplier simultaneously offers the retailer either a conditionally permissible delay in payments or a cash discount. In the case of a conditionally permissible delay, if the retailer orders more than a predetermined quantity, then he/she has a grace period to make the full payment. Otherwise, he/she must pay the payment for goods of certain proportion first while receiving the goods and has a grace period to pay off the rest. As to a cash discount, if the retailer pays for the entire amount of the order within a certain short period, then he/she will receive a cash discount from the supplier. In additions, from a financial standpoint, all cash outflows related to the inventory control that occur at different points of time have different values. Hence, it is necessary to take account of the factor of time value of monetary when drafting the replenishment policy. In a word, this paper uses an alternate approach-discount cash flow (DCF) analysis to establish an inventory problem for deteriorating items in which the supplier provides the retailer either a conditionally permissible delay or a cash discount. We then study the necessary and sufficient conditions for finding the optimal solution. Furthermore, we establish several theoretical results to obtain the solution that provides the smallest present value of all future cash flows. Finally, several numerical examples are given to illustrate the results and obtain some managerial insights.     

基于信用支付和现金折扣的变质物品库存模型

本文在供应商提供给零售商定期信用支付和现金折扣情况下,研究了零售商的变质物品最优库存问题。基于信用支付和现金折扣的两种支付条件下,分四种情况建立库存模型,并给出了寻求变质物品最优订购周期和最优付款时间的有效算法。最后,给出算例以及最优解,以说明本模型及求解过程。     

基于二层信用策略的两货栈库存模型

假设供应商向零售商提供信用支付期的同时,零售商也向顾客提供信用支付期,研究了两货栈的变质物品库存模型,并讨论了模型最优解的唯一性,最后给出了最优订购策略的算法步骤与数值例子.     

需求率基于库存的变质物品的两货栈模型

针对需求受库存水平影响的变质品的两货栈系统,本文首先考虑了允许缺货但缺货要补情形下,分别给出了L₁系统(库存系统只用自己的货栈存放物品)和L₂系统(库存系统使用自己货栈和租用货栈存放物品)时,销售商的库存和运输策略模型。其次给出了允许缺货但缺货不补L₁系统和L₂系统时,销售商的库存和运输策略,并对模型最优解作了相关分析,最后给出算例,给销售商实际的库存管理提供了理论依据。     

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.     

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.     

Credit financing in a two-warehouse environment for deteriorating items with price-sensitive demand and fully backlogged shortages

In the current global market, organizations use many promotional tools in order to increase their sales. One such tool is permissible delay in payments, i.e., the buyer does not have to pay for the goods purchased immediately rather can defer the payment for a prescribed period given by the supplier. This phenomenon motivates the retailer/buyer to order a large inventory lot so as to take full benefit of credit period. But the well decorated showroom (OW) with modern facilities has a limited storage capacity. Thus the retailer has to hire a rented warehouse to store the excess units. In this scenario, retailer usually adopts two types of dispatch policy: FIFO & LIFO, depending upon the situation, e.g., nature of items/deteriorating items, location of warehouse. Further in order to survive in the market, the retailer dynamically adjusts the prices of the goods to boost the demand and enhance the revenues.In the light of these facts, this paper develops an inventory model for deteriorating items with price-sensitive demand under permissible delay in payment in a two warehouse environment. Shortages are allowed and fully backlogged. The objective of this study is to find the optimal inventory and pricing policies so as to maximize the total average profit. Further, the different trade credit scenario has been exhibited with the help of a numerical example. A comprehensive sensitivity analysis has also been carried out to advocate the implication of FIFO and LIFO dispatch policy.     

Analytical models for supplier selection and order quantity allocation

In the literature, decision models and techniques for supplier selection do not often consider inventory management of the items being purchased as part of the analysis. In this article, two mixed integer nonlinear programming models are proposed to select the best set of suppliers and determine the proper allocation of order quantities while minimizing the annual ordering, inventory holding, and purchasing costs under suppliers’ capacity and quality constraints. The first model allows independent order quantities for each supplier while the second model restricts all order quantities to be of equal size, as it would be required in a multi-stage (supply chain) inventory model. Illustrative examples are used to highlight the advantages of the proposed models over a previous model introduced in the literature.