Optimal dynamic pricing for deteriorating items with reference-price effects

In this paper, a dynamic pricing problem for deteriorating items with the consumers’ reference-price effect is studied. An optimal control model is established to maximise the total profit, where the demand not only depends on the current price, but also is sensitive to the historical price. The continuous-time dynamic optimal pricing strategy with reference-price effect is obtained through solving the optimal control model on the basis of Pontryagin's maximum principle. In addition, numerical simulations and sensitivity analysis are carried out. Finally, some managerial suggestions that firm may adopt to formulate its pricing policy are proposed.     

Dynamic joint pricing and production policy for perishable products

A joint dynamic pricing and production problem for perishable products without shortages is considered. The demand rate is price‐dependent and time‐varying. This paper constructs an optimal control model to maximize the total profit under a general nonlinear production cost function. The feature of the optimal joint dynamic pricing and production policy is analyzed by solving the corresponding optimal control problem on the basis of improved Pontryagin's maximum principle. Then, an effective algorithm is designed to obtain the optimal joint policy. The case of the joint static optimal policy is also investigated and compared with the dynamic one. Finally, numerical examples are presented to illustrate the effectiveness of the proposed methods, and some managerial implications are provided for the management of perishable items.     

变质品生产、库存控制与预防维修联合建模与优化

针对产品变质带来库存损耗的问题,在经济生产批量模型的基础上加入零库存时间,提出新的库存控制策略.该策略以预防维修周期和零库存时间为联合决策变量,利用更新定理建立了单位时间内总成本的期望模型.为了提高对此期望的计算速度,设计基于矩阵运算的蒙特卡罗仿真算法,并与遗传算法结合,大大提高了模型的优化速度.通过比较研究,验证了仿真算法的显著优势,并通过敏感性分析明确了相关参数的变化对最优策略的影响.     

Optimal replenishment policy for perishable items with stock-dependent selling rate and capacity constraint

In this paper, a deterministic inventory model is developed for deteriorating items with stock-dependent demand and finite shelf/display space. Furthermore, we allow for shortages and the unsatisfied demand is partially backlogged at the exponential rate with respect to the waiting time. We provide solution procedures for finding the maximum total profit per unit time. In a specific circumstance, the model will reduce to the case with no shortage. Further, we use numerical examples to illustrate the model.     

两阶段需求相关的动态易逝品最优订货策略

以动态市场环境下的易逝品为研究对象,针对两阶段需求相关、每次订货时带有固定的订购费用以及未满足需求部分延期供给等实际现象,建立了动态易逝品销售商两阶段订货决策模型.对模型的最优解进行了理论分析,并提出了寻求两阶段订货策略的简单方法.以朗科优盘经销商的采购实例来说明该模型的应用价值.最后,通过探讨订货决策与需求波动、两阶段需求相关系数以及延期供给率等因素的关系,得出了一些重要的管理启示.     

持续补货模式下的改良品库存模型研究

改良品的订货问题一直没有引起足够的重视。分别建立持续补货模式下不允许缺货与允许缺货的改良品库存模型,并分析改良率与补货周期、各种费用参数的相关关系,研究表明：改良率与补货周期、单位时间存储费用呈负相关关系,与单位时间订货费用、单位时间改良费用、单位时间缺货费用、单位时间总费用呈正相关关系。同时,对不允许缺货与允许缺货的改良品库存模型单位时间总费用进行比较,并分析库存模型各参数对两种库存模型单位时间总费用差值的影响方式,研究表明：单位时间总费用差值与改良率、补充速率、单次订货费用、单位存储费用、单位改良费用呈正相关关系,与单位缺货费用成呈负相关关系,与需求速率的相关关系取决于补充速率与需求速率的大小关系。     

库存水平影响需求下变质品订购、定价和保鲜技术投资的联合决策

研究变质品的订购、定价和保鲜技术投资联合决策问题.假设产品需求同时受价格和库存水平的影响,系统不允许缺货并放松期末库存水平为零的约束,零售商拥有有限的货架空间或存储空间,同时考虑零售商可以通过投资保鲜技术减低产品的变质率,以平均利润最大化为目标构建库存水平影响需求下变质品的订购、定价和保鲜技术投资联合决策模型.首先证明最优策略的存在性和唯一性,并给出零售商建立期末库存的条件;然后利用最优解的相关性质设计一个求解模型的多阶段迭代算法;最后通过具体算例验证展示模型和算法的可行性和实用性,并完成相关参数的敏感性分析,获得一定的管理启示.     

Optimizing the pricing and replenishment policy for non-instantaneous deteriorating items with stochastic demand and promotional efforts

Retailer promotional activity has become prevalent in the business world. Promotional efforts impact the replenishment policy and the sale price of goods. In this paper, the problem of replenishment policy and pricing for non-instantaneous deteriorating items subject to promotional effort is considered. We adopt a price dependent stochastic demand function in which shortages are allowed and partially backlogged. The major objective is to simultaneously determine the optimal selling price, the optimal replenishment schedule, and the optimal order quantity to maximize the total profit. First, we prove that a unique optimal replenishment schedule exists for any given selling price. Second, we prove that the total profit is a concave function of price. Third, we present an algorithm to obtain the optimal solution and solve a numerical example. Last, we extend the numerical example by performing a sensitivity analysis of the model parameters and discuss specific managerial insights.     

Coordinating replenishment and pricing policies for non-instantaneous deteriorating items with price-sensitive demand

This article will formulate and solve an inventory system with non-instantaneous deteriorating items and price-sensitive demand. The purpose of this study is to determine the optimal selling price and the length of replenishment cycle such that the total profit per unit time has a maximum value for the retailer. We first establish a proper model for a mathematical formulation. Then we develop several theoretical results and provide the decision-maker with an algorithm to find the optimal solution. Finally, two numerical examples are provided to illustrate the solution procedure, and a sensitivity analysis of the optimal solution with respect to major parameters is carried out.     

An EPQ model for deteriorating items with inventory-level-dependent demand and permissible delay in payments

This article develops an inventory model for exponentially deteriorating items under conditions of permissible delay in payments. Unlike the existing related models, we assume that the items are replenished at a finite rate and the demand rate of the items is dependent on the current inventory level. The objective is to determine the optimal replenishment policies in order to maximise the system's average profit per unit of time. A simple method is shown for finding the optimal solution of the model based on the derived properties of the objective function. In addition, we deduce some previously published results as the special cases of the model. Finally, numerical examples are used to illustrate the proposed model. Some managerial insights are also inferred from the sensitive analysis of model parameters.     

Joint replenishment with imperfect items and price discount

A joint replenishment problem is presented to determine the ordering policy for multiple items having a certain percentage of defective units. The purpose of this paper is to study the impact of the percentage of defective units on the ordering policy. Two different scenarios are presented for joint replenishment problem: (1) without price discount and (2) with price discount. For each scenario, the total expected cost per unit time is derived and algorithms are presented to determine the family cycle length and the integer number of intervals that the replenishment quantity of each item will last. Numerical examples are presented and the results are discussed.     

Joint replenishment policy with backordering and special sale

In this paper, an inventory control model with a joint replenishment policy and a temporary discount is developed. We assume that shortage is allowed and buyer uses an economic order quantity inventory control model. Different cases based on ordering policies for the first joint replenishment if the special order is not taken, and coincidence of a special period length with a positive or negative inventory level of the last regular period length, are investigated. Furthermore, several theorems are proved through which closed-form solutions are obtained. At the end, two numerical examples illustrate the different situations that the buyer may face and sensitivity analyses for both examples are reported.     

Optimal pricing and lot-sizing for perishable inventory with price and time dependent ramp-type demand

Product perishability is an important aspect of inventory control. To minimise the effect of deterioration, retailers in supermarkets, departmental store managers, etc. always want higher inventory depletion rate. In this article, we propose a dynamic pre- and post-deterioration cumulative discount policy to enhance inventory depletion rate resulting low volume of deterioration cost, holding cost and hence higher profit. It is assumed that demand is a price and time dependent ramp-type function and the product starts to deteriorate after certain amount of time. Unlike the conventional inventory models with pricing strategies, which are restricted to a fixed number of price changes and to a fixed cycle length, we allow the number of price changes before as well as after the start of deterioration and the replenishment cycle length to be the decision variables. Before start of deterioration, discounts on unit selling price are provided cumulatively in successive pricing cycles. After the start of deterioration, discounts on reduced unit selling price are also provided in a cumulative way. A mathematical model is developed and the existence of the optimal solution is verified. A numerical example is presented, which indicates that under the cumulative effect of price discounting, dynamic pricing policy outperforms static pricing strategy. Sensitivity analysis of the model is carried out.     

An inventory replenishment policy for deteriorating items with shortages and partial backlogging

It has long been assumed that the shortages in inventory systems are either completely backlogged or totally lost. However, it is more reasonable to characterize that the longer the waiting time for the next replenishment, the smaller the backlogging rate would be. Moreover, the opportunity cost due to lost sales should be considered since some customers would not like to wait for backlogging during the shortage periods. Without considering these two realistic conditions, study on the inventory modeling for deteriorating items with shortages and partial backlogging cannot be complete and general. In the present article we define an appropriate time-dependent partial backlogging rate and introduce the opportunity cost due to lost sales. Numerical examples are also presented to illustrate the effects of changes in backlogging parameter and unit opportunity cost on total cost and the optimal number of replenishments.Scope and purposeIn a recent article published in this Journal, Giri et al. (Comput. Oper. Res. 27 (2000) 495–505) implemented an existing procedure to the inventory problem of Hariga and Al-Alyan (Comput. Oper. Res. 24 (1997) 1075–83) which concerns with lot-sizing heuristic for deteriorating items with shortages allowed in all cycles except the last one. Giri et al. deviated from the traditional practice and suggested a new policy allowing shortages in all cycles over a finite planning horizon. Their numerical results indicated the proposed policy is cheaper to operate with a cost reduction up to 15%. However, they did not consider the opportunity cost due to lost sales that happen because customers would not like to wait for backlogging. Moreover, for many products with growing sales, the length of the waiting time for the next replenishment is the main factor for determining whether the backlogging will be accepted or not, and the backlogging rate is expected to be time-dependent. Thus the assumption made in Giri et al. that the backlogging rate is a fixed fraction of the total amount of shortages is not reasonable.The purpose of this paper is to present a more realistic discussion of the inventory problem for deteriorating items with time-varying demands and shortages over a finite planning horizon. In contrast to the model by Giri et al., we define an appropriate partial backlogging rate and introduce the opportunity cost due to lost sales. We attempt to complement their model as a practical and general solution for inventory replenishment problems. With these extensions, the scope of applications of the present results is expanded.     

Optimal joint replenishment,delivery and inventory management policies for perishable products

In this paper we analyze the optimal joint decisions of when, how and how much to replenish customers with products of varying ages. We discuss the main features of the problem arising in the joint replenishment and delivery of perishable products, and we model them under general assumptions. We then solve the problem by means of an exact branch-and-cut algorithm, and we test its performance on a set of randomly generated instances. Our algorithm is capable of computing optimal solutions for instances with up to 30 customers, three periods, and a maximum age of two periods for the perishable product. For the unsolved instances the optimality gap is always small, less than 1.5% on average for instances with up to 50 customers. We also implement and compare two suboptimal selling priority policies with an optimized policy: always sell the oldest available items first to avoid spoilage, and always sell the fresher items first to increase revenue.     

Modelling of the inventory replenishment problem with a two-segment piecewise linear demand

The classical inventory replenishment problem with a linear function in demand uses a ‘single-segment’ linear function as its demand and can be modelled by a simple algorithm. Moreover, this article extends the algorithm to provide a heuristic solution for the inventory replenishment model with a two-segment linear function in demand called the ‘two-segment piecewise linear demand model’. In addition, this article proposes a general procedure for solving both models. Meanwhile, several examples taken from the literature illustrate our algorithm for these two models with convincing results. Furthermore, this study shows that when the demand is a two-segment piecewise linear function over time, it is better to use the proposed algorithm rather than devising a decoupled solution approach by treating segments separately. Finally, a sensitivity analysis of two factors, demand and cost, is performed. The model is highly extensible and applicable, so it can serve as an inventory planning tool to solve the replenishment problem.     

需求依赖库存且有时变短缺拖后率的非立即变质物品库存模型

研究了时变短缺部分拖后条件下非立即变质性物品的库存补给模型,其中物品的变质率随时间变化而变化。当库存水平为正值时,市场需求受销售价格影响;当库存为负值时,不能满足的需求被部分拖后,拖后率与在缺货期间已经发生的缺货量有关。通过考虑短缺拖后率和变质率同时随时间变化对库存补给策略的影响,建立具有短缺量部分拖后的非立即变质性物品的库存模型,并且给出模型最优解存在的必要条件,得到一类更加符合实际情形的库存模型。最后,用数值算例说明模型的实际应用。     

A supplier-selection model with classification and joint replenishment of inventory items

Since inventory costs are closely related to suppliers, many models in the literature have selected the suppliers and also allocated orders, simultaneously. Such models usually consider either a single inventory item or multiple inventory items which have independent holding and ordering costs. However, in practice, ordering multiple items from the same supplier leads to a reduction in ordering costs. This paper presents a model in capacity-constrained supplier-selection and order-allocation problem, which considers the joint replenishment of inventory items with a direct grouping approach. In such supplier-selection problems, the following items are considered: a fixed major ordering cost to each supplier, which is independent from the items in the order; a minor ordering cost for each item ordered to each supplier; and the inventory holding and purchasing costs. To solve the developed NP-hard problem, a simulated annealing algorithm was proposed and then compared to a modified genetic algorithm of the literature. The numerical example represented that the number of groups and selected suppliers were reduced when the major ordering cost increased in comparison to other costs. There were also more savings when the number of groups was determined by the model in comparison to predetermined number of groups or no grouping scenarios.     

Managing dual warehouses with an incentive policy for deteriorating items

Distributors in a supply chain usually limit their own warehouse in finite capacity for cost reduction and excess stock is held in a rent warehouse. In this study, we examine inventory control for deteriorating items in a two-warehouse setting. Assuming that there is an incentive offered by a rent warehouse that allows the rental fee to decrease over time, the objective of this study is to maximise the joint profit of the manufacturer and the distributor. An optimisation procedure is developed to derive the optimal joint economic lot size policy. Several criteria are identified to select the most appropriate warehouse configuration and inventory policy on the basis of storage duration of materials in a rent warehouse. Sensitivity analysis is done to examine the results of model robustness. The proposed model enables a manufacturer with a channel distributor to coordinate the use of alternative warehouses, and to maximise the joint profit of the manufacturer and the distributor.     

Optimal order policy in response to announced price increase for deteriorating items with limited special order quantity

When a supplier announces an impending price increase due to take effect at a certain time in the future, it is important for each retailer to decide whether to purchase additional stock to take advantage of the present lower price. This study explores the possible effects of price increases on a retailer's replenishment policy when the special order quantity is limited and the rate of deterioration of the goods is assumed to be constant. The two situations discussed in this study are as follows: (1) when the special order time coincides with the retailer's replenishment time and (2) when the special order time occurs during the retailer's sales period. By analysing the total cost savings between special and regular orders during the depletion time of the special order quantity, the optimal order policy for each situation can be determined. We provide several numerical examples to illustrate the theories in practice. Additionally, we conduct a sensitivity analysis on the optimal solution with respect to the main parameters.