Optimal selling price and lotsize with time varying deterioration and partial backlogging

The article deals with an EOQ (economic order quantity) model over an infinite time horizon for perishable items where demand is price dependent and partial backorder is permitted. The rate of deterioration is taken to be time proportional and it is assumed that shortage occurs at starting of the inventory cycle. Based on the partial backlogging and lost sale cases, the author develops the criterion for the optimal solution for the replenishment schedule, and proves the optimal ordering policy is unique. Moreover, the article suggests to new functions regarding price-dependent demand and time varying deterioration rate. Finally, numerical examples are illustrated to test the model in various issues.     

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.     

部分延迟订购的易变质品联合定价与生产策略

构建了一个需求同时依赖于销售价格和库存水平,生产率和变质率均为常数,允许缺货且缺货量部分延迟订购的易变质品联合定价与生产控制模型。首先证明了在销售价格给定的情况下,系统的总利润函数是关于生产计划的严格凹函数,平均利润函数是严格的伪凹函数,即存在唯一的最优解,并给出其充分条件。接着给出问题的一个数值求解算法。最后通过算例,展示了模型及相关算法的应用,并对相关参数进行了灵敏度分析,结果显示:当产品的生产成本、缺货成本和机会成本增加时,系统的平均利润将下降;生产成本和延迟订购阻力系数对最优定价和生产策略以及平均利润的影响较大。     

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.     

需求随价格变化的具有折扣的易变质物品的库存模型

本文研究了在需求随价格变化及物品易变质的条件下，当供应商给予数量折扣时的库存问题。证明了当供应商给予数量折扣时，零售商的需求量是增大的，并给出了供应商给予数量折扣时零售商的订货量和订货周期的计算方法。对物品变质率和需求价格敏感系数对零售商的订货量、订货周期、出售价格和单位时间利润的影响进行了数值分析，并给出了数值算例。     

Inventory returns and special sales in a lot-size system with constant rate of deterioration

The problem of returning or of selling the inventory excess to optimal stock level is considered for deteriorating items. Two inventory models, viz. the infinite and the finite horizon models, are developed, in which the deterioration is assumed to be a constant fraction of the on hand inventory. Both the models are developed under the assumptions of instantaneous delivery and no shortages. When there is no deterioration, the developed models are related to the corresponding inventory models for non-deteriorating items. Examples are given to illustrate the derived results.     

An Economic order quantity model for Items with Three-parameter Weibull distribution Deterioration,Ramp-type Demand and Shortages

In this paper, we develop an economic order quantity inventory model for items with three-parameter Weibull distribution deterioration and ramp-type demand. Shortages are allowed in the inventory system and are completely backlogged. The demand rate is deterministic and varies with time up to a certain point and eventually stabilized and becomes constant. The instantaneous rate of deterioration is an increasing function of time. We provide simple analytical tractable procedures for deriving the model and give numerical examples to illustrate the solution procedure. Our adoption of ramp-type demand reflects a real market demand for newly launched product.     

Optimal EOQ Models for Deteriorating Items with Time-Varying Demand

In this paper, optimal inventory lot-sizing models are developed for deteriorating items with general continuous time-varying demand over a finite planning horizon and under three replenishment policies. The deterioration rate is assumed to be a constant fraction of the on-hand inventory. Shortages are permitted and are completely backordered. The proposed solution procedures are shown to generate global minimum replenishment schedules for both general increasing and decreasing demand patterns. An extensive empirical comparison using randomly generated linear and exponential demands revealed that the replenishment policy which starts with shortages in every cycle is the least cost policy and the replenishment policy which prohibits shortages in the last cycle exhibited the best service level effectiveness. An optimal procedure for the same problem with trended inventory subject to a single constraint on the minimum service level (maximum fraction of time the inventory system is out of stock during the planning horizon) is also proposed in this paper.     

A two-warehouse inventory model for deteriorating items under conditionally permissible delay in payment

For the capacity of any warehouse is limited, it has to rent warehouse (RW) for storing the excess units over the fixed capacity W of the own warehouse (OW) in practice. The RW is assumed to offer better preserving facilities than the OW resulting in a lower rate of deterioration and is assumed to charge higher holding cost than the OW. In this paper, a two-warehouse inventory model for deteriorating items is considered with constant demand under conditionally permissible delay in payment. The purpose of this study is to find the optimal replenishment policies for minimizing the total relevant inventory costs. Useful theorems to characterize the optimal solutions have been derived. Furthermore, numerical examples are provided to illustrate the proposed model, sensitivity analysis of the optimal solutions with respect to major parameters is carried out and some managerial inferences are obtained.     

Predictive control of periodic-review production inventory systems with deteriorating items

In this paper a predictive control strategy is applied to a periodic-review dynamic inventory system with deteriorating items. Given the current inventory level, we determine the optimal production rates to be implemented at the beginning of each of the following periods over the control horizon. The effectiveness of this approach is the use of future information of the inventory target level and the desired production rate, which are available, along the fixed horizon. The deterioration coefficient may be known or unknown and both cases are considered. In the case where it is unknown, the self-tuning predictive control is applied. The proposed control algorithms are illustrated by simulations.     

需求离散的变质性物品库存补充策略优化研究

研究了在不允许缺货情况下需求为离散的变质性物品的库存补充策略问题.在假定变质率为常数的情况下,建立了有限时域内变质性物品的补充策略模型,并给出了求最优补充策略的方法.     

An analysis of production-inventory models with deteriorating items in a two-echelon supply chain

This paper revisits two previous studies that addressed the integrated production–inventory problem for deteriorating items in a two-echelon supply chain, where the item’s deterioration rate is a constant or follows a continuous probability distribution function. The aim of this study is to present an improved solution procedure to determine the delivery lot size and the number of deliveries per production batch cycle that minimizes the total cost of the entire supply chain. The performance of the proposed methodology is illustrated analytically and numerically.     

Partial backlogging inventory lot-size models for deteriorating items with fluctuating demand under inflation

In this paper, the traditional inventory lot-size model is extended to allow not only for general partial backlogging rate but also for inflation. The assumptions of equal cycle length and constant shortage length imposed in the model developed by Moon et al. [Moon, I., Giri, B.C., Ko, B., 2005. Economic order quantity models for ameliorating/deteriorating items under inflation and time discounting, European Journal of Operational Research 162(3), 773–785] are also relaxed. For any given number of replenishment cycles the existence of a unique optimal replenishment schedule is proved and further the convexity of the total cost function of the inventory system in the number of replenishments is established. The theoretical results here amend those in Yang et al. [Yang, H.L., Teng, J.T., Chern, M.S., 2001. Deterministic inventory lot-size models under inflation with shortages and deterioration for fluctuating demand, Naval Research Logistics 48(2), 144–158] and provide the solution to those two counterexamples by Skouri and Papachristos [Skouri, K., Papachristos, S., 2002. Note on “deterministic inventory lot-size models under inflation with shortages and deterioration for fluctuating demand” by Yang et al. Naval Research Logistics 49(5), 527–529.]. Finally we propose an algorithm to find the solution, and obtain some managerial results by using sensitivity analyses.     

(<Emphasis Type="Italic">T</Emphasis>, <Emphasis Type="Italic">S</Emphasis><Subscript><Emphasis Type="Italic">i</Emphasis></Subscript>) Policy Inventory Model for Deteriorating Items with Time Proportional Demand

An EOQ model is reconsidered here in which the demand rate is changing linearly with time and the deterioration is assumed to be a constant fraction of the onhand inventory. The planning horizon is finite and known and the replenishment periods are assumed to be constant. The problem is to find the optimal number of replenishments, which are instantaneous. When there is no deterioration, the model developed is related to the corresponding model for nondeteriorating items. An example followed by sensitivity analysis is given to illustrate the derived results.     

An inventory model of deteriorating items with lot-size dependent replenishment cost and a linear trend in demand

This paper presents an inventory model for deteriorating items over a finite time horizon where the demand increases linearly with time. The method is developed by assuming that the successive replenishment cycle lengths are the same. Many O.R. scientists/researchers obtained an optimal replenishment schedule where the replenishment cost is constant in each cycle length over the finite time horizon. In this paper, we relax the assumption of fixed replenishment cost. The replenishment cost per replenishment is taken to be linearly dependent on the lot-size of that replenishment. Shortages are allowed and are fully backlogged. As a special case, the results for the model without shortages are derived. Finally, two numerical examples are presented to illustrate the model.     

Weibull变质率和短缺量拖后的易变质物品EOQ模型

基于需求和采购价格均为时变的EOQ模型,考虑物品的变质率呈更符合现实情况的三参数Weibull分布,同时考虑短缺量拖后和资金时值对易变质物品库存管理的影响,构建了相应的EOQ模型.应用数学软件Matlab对该库存模型进行仿真计算和主要影响参数的灵敏度分析.结果表明,该模型存在最优解,且各主要影响参数对最优库存控制各有不同程度的影响,资金时值对库存总成本净现值的影响程度要甚于短缺量拖后的影响,故在制定科学的库存策略时资金时值需要更加关注.     

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.     

Retailer’s optimal replenishment and payment policies in the EPQ model under cash discount and two-level trade credit policy

The main purpose of this paper is to investigate the retailer’s optimal cycle time and optimal payment time under the supplier’s cash discount and trade credit policy within the economic production quantity (EPQ) framework. In this paper, we assume that the retailer will provide a full trade credit to his/her good credit customers and request his/her bad credit customers pay for the items as soon as receiving them. Under this assumption, we model the retailer’s inventory system as a cost minimization problem to determine the retailer’s optimal inventory cycle time and optimal payment time under the replenishment rate is finite. Then, an algorithm is established to obtain the optimal strategy. Finally, numerical examples are given to illustrate the theoretical results and obtain some managerial phenomena.     

需求受库存水平影响的短生命周期物品延迟订货的库存管理

像计算机、电视机、空调等这类具有物理变质的可能性很小,但生命周期较短、不断更新换代、价值不断贬值的电子产品会发生无形变质的现象.在假设无形变质率与需求率负相关,同时在产品的存储过程中,考虑库存水平对销售量的影响情况下,研究需求受库存水平影响、且缺货时存在延迟订货的短生命周期物品的库存管理问题.创新之处在于考虑了产品的缺货问题;在允许缺货的条件下,建立了短生命周期物品的库存模型;运用数值算例进行了求解和验证;并对各参数进行了敏感性分析.     

Determining optimal selling price and lot size with a varying rate of deterioration and exponential partial backlogging

In this paper, a deterministic inventory model for deteriorating items with price-dependent demand is developed. The demand and deterioration rates are continuous and differentiable function of price and time, respectively. In addition, we allow for shortages and the unsatisfied demand is partially backlogged at a negative exponential rate with the waiting time. Under these assumptions, for any given selling price, we first develop the criterion for the optimal solution for the replenishment schedule, and prove that the optimal replenishment policy not only exists but also is unique. If the criterion is not satisfied, the inventory system should not be operated. Next, we show that the total profit per unit time is a concave function of price when the replenishment schedule is given. We then provide a simple algorithm to find the optimal selling price and replenishment schedule for the proposed model. Finally, we use numerical examples to illustrate the algorithm.