The simplified solution procedure for deteriorating items under stock-dependent demand and two-level trade credit in the supply chain management

Min et al. [1] (J. Min, Y.W. Zhou, J. Zhao, An inventory model for deteriorating items under stock-dependent demand and two-level trade credit, Appl. Math. Model. 34 (2010) 3273–3285.) develop an inventory model for deteriorating items under stock-dependent demand and two-level trade credit. They provide the necessary and sufficient conditions of the existence and uniqueness of the optimal solutions that could maximize the retailer’s average profit per unit time. Basically, their paper is correct and interesting. Recently, several researchers have been showing a huge interest in developing simple and easy to implement solution procedures in management science. Therefore this paper indicates that Min et al.’s solution procedure can be further improved and simplified. So, the main purpose of this paper is to present simple and easy to understand solution procedures to locate the optimal solutions of an inventory model that considers deteriorating items under stock-dependent demand and two-level trade credit.     

考虑保质期的非即时易腐品最优订购与定价决策

针对有保质期约束的非即时易腐品,零售商可以在产品非腐败阶段和腐败阶段进行差异化定价来调整市场需求。即零售商可以调整产品非腐败阶段的定价策略,使得非即时易腐品在进入腐败阶段之前全部售出;或者在产品腐败阶段降低零售价格以刺激市场需求。因此,在需求依赖价格的假设下,本文研究了零售商关于有保质期约束的非即时易腐品的最优订购与定价决策。通过分析零售商单位时间利润函数的理论性质,得到了零售商的最优订购与定价策略。基于线性和指数型两种需求函数形式进行数值算例,可以发现针对保质期较长或者非腐败阶段较长的易腐品,零售商会延长订货周期来增加销售收入。特别是在指数型需求函数情境下,当市场需求与零售价格高度相关时,零售商更有意愿降价来刺激销量,从而使得易腐品在非腐败阶段内销售完毕。     

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。     

Optimal Ordering Policy for Deteriorating Items with Partial Backlogging under Permissible Delay in Payments

In 1985, Goyal developed an Economic order quantity (EOQ) model under conditions of permissible delay in payments. Jamal et al. then generalized Goyal’s model for deteriorating items with completely backlogging. However, they only ran several simulations to indicate that the total relevant cost may be convex. Recently, Teng amended Goyal’s model by considering the difference between unit price and unit cost, and provided an alternative conclusion that it makes economic sense for some retailers to order less quantity and take the benefits of the permissible delay more frequently. However, he did not consider deteriorating items and partial backlogging. In this paper, we establish a general EOQ model for deteriorating items when the supplier offers a permissible delay in payments. For generality, our model allows not only the partial backlogging rate to be related to the waiting time but also the unit selling price to be larger than the unit purchase cost. Consequently, the proposed model includes numerous previous models as special cases. In addition, we mathematically prove that the total relevant cost is strictly pseudo-convex so that the optimal solution exists and is unique. Finally, our computational results reveal six managerial phenomena.     

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。     

允许缺货且带数量折扣的腐烂物质库存模型

在短缺量拖后率是等待时间的负指数函数、订购成本是批量的线性函数的条件下,建立了带数量折扣的腐烂物质库存模型,目标是优化总平均利润.在给定销售价格的情况下,证明了库存系统的最优补货策略存在且唯一;且若采用最优补货策略,平均利润函数是销售价格的凹函数;最后给出了模型的算法,并用数值例子说明了模型和算法的有效性.     

一类时变需求且存货影响销售率的EOQ模型

从实际背景出发,在R AM P型需求和存货影响销售率条件下,讨论了一类以缺货开始的变质性物品的EOQ模型.给出了模型的求解方案及单位时间内平均利润最大化的两个充分条件,并用数值算例进行了验证.     

多货栈及变质情形下两种可替代物品的经济订货批量模型

研究了多货栈及变质情形下两种可替代物品的经济订货批量问题.在计划期内,若某一种易变质物品发生缺货,则可以被另一种易变质物品以一定的替代率代替补充,不同物品有不同的变质率,且要决定租用货栈的数量.以库存系统的总费用最小为目标函数,分别对货栈容量无限与有限的情形建立模型,证明了最优策略存在的唯一性,并分别给出了求解最优订购策略的算法,最后通过一个算例验证了算法的最优性.     

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.     

Optimal pricing,production, and inventory for deteriorating items under demand uncertainty: The finite horizon case

This paper aims to investigate the joint dynamic pricing and production decisions of deteriorating items with uncertain demand over a finite selling season, where the demand is price sensitive and the potential demand is characterized by a stochastic process. The stocks deteriorate physically at a constant fraction of the on-hand inventory. A joint dynamic pricing and production problem to maximize the total expected profit is modeled as a stochastic optimal control problem. We derive the closed-form solutions, which are in time-dependent linear feedback form of the inventory level when it is either positive or negative. It is shown that the manufacturer always benefits from a reduction in the volatility of potential market demand. In addition, to highlight the effectiveness of the joint dynamic strategy, we also consider the case of optimal production with a static price. A numerical example is presented to illustrate the validity of the optimal control policy, and sensitivity analysis on major parameters is performed to provide more managerial insights into deteriorating items.     

价格和库存均影响需求、部分拖后供给的易变质品生产库存

考虑产品的需求率受库存和销售价格的影响,拖后供给因子与需求得到满足的实际等待时间有关,建立了易变质品生产库存模型,给出了寻求最优生产策略和销售价格的方法,并分析参数变化对于平均利润、销售价格和服务率的影响.     

基于收益共享的多周期易变质产品供应链协调模型

研究了单个供应商与单个零售商组成的易变质产品供应链系统在有限计划期内的协调问题。考虑到市场需求会同时受到多种因素的影响,构建了市场需求率依赖于销售价格与当前库存水平且随时间呈现一般连续变化的需求函数。将零售商的订购次数、销售价格以及供应商的批发价格作为决策变量,分别求解了集中式与分散式供应链系统下的最优策略。通过与分散式决策的利润值进行比较,得出集中式决策能够使得供应链系统的利润值至少增加1/3。利用收益共享契约机制协调此系统,求解了供应链实现完美协调时收益共享因子的取值区间。最后通过算例验证了理论结果并分析了相关参数变化对系统协调的影响。     

Optimal pricing and ordering policy for non-instantaneous deteriorating items under inflation and customer returns

This paper deals with an economic production quantity inventory model for non-instantaneous deteriorating items under inflationary conditions considering customer returns. We adopt a price- and time-dependent demand function. Also, the customer returns are considered as a function of both price and demand. The effects of time value of money are studied using the Discounted Cash Flow approach. The main objective is to determine the optimal selling price, the optimal replenishment cycles, and the optimal production quantity simultaneously such that the present value of total profit is maximized. An efficient algorithm is presented to find the optimal solution. Finally, numerical examples are provided to solve the presented inventory model using our proposed algorithm, which is further clarified through a sensitivity analysis. The results of analysing customer returns provide important suggestions to financial managers who use price as a control to match the quantity sold to inventory while maximizing revenues. The paper ends with a conclusion and an outlook to future studies.     

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.     

Two-warehouse production model for deteriorating inventory items with stock-dependent demand under inflation over a random planning horizon

This paper develops a production-inventory model for a deteriorating item with stock-dependent demand under two storage facilities over a random planning horizon, which is assumed to follow exponential distribution with known parameter. The effects of learning in set-up, production, selling and reduced selling is incorporated. Different inflation rates for various inventory costs and time value of money are also considered. A hybrid genetic algorithm is designed to solve the optimization problem which is hard to solve with existing algorithms due to the complexity of the decision variable. To illustrate the model and to show the effectiveness of the proposed approach a numerical example is provided. A sensitivity analysis of the optimal solution with respect to the parameters of the system is carried out.     

Optimal Production and Marketing Planning

This paper presents an integrated production, marketing and inventory model which determines the production lot size, marketing expenditure and products selling price. Our model is highly nonlinear and non-convex and cannot be solved directly. Therefore, Geometric Programming (GP) is used to locate the optimal solution of the proposed model. In our GP implementation, we use a transformed dual problem in order to reduce the model to an optimization of an unconstrained problem in a single variable and the resulting problem is solved using a simple line search. We analyze the solution in different cases in order to study the behaviour of the model and for each case, a numerical example is used to demonstrate the implementation of our analysis.     

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.     

Technical note on inventory model with trapezoidal type demand

This note is a response to optimal policy for deteriorating items with trapezoidal type demand and partial backlogging by Cheng et al. [4]. In the above mentioned paper, a new inventory model was created, but both their model and their solution procedure contained some questionable results. In this note a detailed examination of their paper will be provided, offering an enhancement to their important inventory model and solution procedure. Numerical examples and detailed analysis of the four scenarios are used to illustrate our findings.     

A multi-item newsvendor problem with preseason production and capacitated reactive production

Given items with short life cycles or seasonal demands, one can potentially improve profits by producing during the selling season, especially when its production capacity is substantial. We develop a two-stage, multi-item model incorporating reactive production that employs a firm’s internal capacity. Production occurs in an uncapacitated preseason stage and a capacitated reactive stage. Demands occur in the reactive stage. Reactive capacities are pre-allocated to each item in the preseason stage and cannot be changed during the reactive stage. Reactive production occurs during the selling season with full knowledge of demands. The objective is expected profit maximization. Unsatisfied demand is lost. The revenue, salvage value, and production and lost sales costs are proportional. Assuming no fixed costs, we present a simple algorithm for computing optimal policies. For a model with fixed costs for allocating preseason stage production and reactive stage capacity to product families, we characterize optimal policies and develop optimal and heuristic algorithms.     

Optimal production time and number of maintenance actions for an imperfect production system under equal-interval maintenance policy

This paper deals with the optimal production/maintenance (PM) policy for a deteriorating production system which may shift from the in-control state to the out-of-control state while producing items. The process is assumed to have a general shift distribution. Under the commonly used maintenance policy, equal-interval maintenance, the joint optimizations of the PM policy are derived such that the expected total cost per unit time is minimized. Different conditions for optimality, lower and upper bounds and uniqueness properties on the optimal PM policy are provided. The implications of another commonly used policy, to perform a maintenance action only at the end of the production run, are also discussed. Structural properties for the optimal policy are established so that an efficient solution procedure is obtained. In the exponential case, some extensions of the results obtained previously in the literature are presented. A numerical example is provided to illustrate the solution procedure for the optimal production and maintenance policy.