Optimal markdown pricing for holiday basket with customer valuation

We consider a problem in which a retailer who plans to sell holiday basket during a holiday and needs to determine an optimal markdown price for remaining inventory after the holiday. The retailer also needs to determine an optimal inventory level to maximise the total expected profit during and after the holiday. We assume customers’ total intended spending on the holiday basket is a random variable that is realised during the holiday and customers purchase the holiday basket based on their valuations. We formulate the problem as a two-stage stochastic programming model with the first-stage decision determining an optimal inventory level and the second stage determining an optimal markdown price based on remaining inventory and the realisation of customers’ total intended spending. We show that we can derive a unique optimal markdown price corresponding to the realisation of customers’ total intended spending. At last, we provide numerical results to illustrate the impact of holiday price of the basket, the highest customer valuation as well as the salvage value of the basket, all of which affect the retailer’s decision on markdown price.     

Optimal pricing policy for a deteriorating product by dynamic tracking control

We study the optimal selling price of a deteriorating product under a deterministic situation in a finite time horizon where the time horizon is either known or unknown. Inventory holding cost is expressed as a quadratic function of the current inventory level. Given a known time horizon, we develop a model by considering the deterioration dynamics of the product, and show its equivalence to a generalised optimal control problem of a linear quadratic form, i.e. an optimal dynamic tracking problem with constraints on the control variable. An optimal pricing policy is derived based on the maximum value principle. The control policy takes a state feedback form; it exhibits a closed-loop relationship between the optimal selling price (control variable) and the optimal inventory level (state variable). Given an unknown time horizon, an optimal pricing policy is derived through a similar approach when the initial inventory level meets certain conditions. Numerical situations are conducted to illustrate the effectiveness of the derived price control policies. Some interesting managerial insights are discussed.     

Pricing and lot-sizing decisions in a two-echelon system with transportation costs

We consider a single-buyer single-supplier system. The market demand is sensitive to the selling price set by the buyer. Both the buyer and the supplier operate with unit product costs, inventory holding costs, and order placement costs. In addition, the buyer is responsible for the freight cost. We formulate a model for determining the optimal lot-sizing and pricing decisions. Existing models for the problem do not consider the transportation costs with price sensitive market demand, and determine the optimal decisions through an exhaustive search. We propose an approximate solution procedure, and report the computational results on the effectiveness of the proposed procedure.     

Manufacturing lot-sizing,procurement and delivery schedules over a finite planning horizon

In this study, an integrated manufacturing system for technology-related companies whose products are experiencing continuous price decrease during the life cycle is studied for optimal procurement, production and delivery schedules over a finite planning horizon. The model considers the inventory cost both at manufacturing and at delivery from supplier. Since the price is continuously decreasing, a manufacturing firm delivers the finished goods in small quantities frequently. Frequent deliveries in small lots are effective to reduce the total cost of the supply chain. The key for high-tech industries is to reduce the inventory holding time since the component prices are continuously decreasing, and this can only be achieved by implementing an efficient supply chain. Therefore, the main purpose of this paper is to develop an integrated inventory model for high-tech industries in JIT environment under continuous price decrease over finite planning horizon while effectively and successfully accomplishing supply chain integration so that the total cost of the system is minimal. An efficient algorithm is developed to determine the optimal or near-optimal lot sizes for raw material procurement, and manufacturing batch under a finite planning horizon. Finally, the solution technique developed for the model is illustrated with numerical examples.     

价格增长下两阶段存货影响销售率EOQ模型

考虑产品采购价随时间指数增长和两阶段存货影响销售,建立了一个价格增长和两阶段存货影响销售率EOQ模型。采用实际调查数据对模型进行了验算,分析了农产品和食品价格大幅度上涨对经销商订货策略的影响。研究结果表明：初始采购价和采购价增长因子对订购策略和各项成本影响较大,存货影响销售率因子和存货影响销售率临界点对订购策略和各项成本有一定影响。     

Joint pricing and inventory strategies in a supply chain subject to inventory inaccuracy

This paper focuses on pricing strategies, inventory policies for a supply chain when Radio Frequency Identification (RFID) technology is adopted to cope with inventory inaccuracy. The supply chain consists of one supplier and one retailer, in which the RFID tag price is shared between the supplier and the retailer. We present and compare the performance differences between a wholesale price contract and a consignment contract when the retailer is the Stackelberg leader and the supplier is the follower. Based on the optimal pricing and inventory decisions, an interesting observation of contract selection is that there are two critical values of inventory available rate such that when the inventory availability is less than the lower value, both the supplier and the retailer prefer a consignment contract; when the inventory availability is greater than the upper value, a wholesale price contract is their best choice; when the inventory availability is between the two values, the supplier prefers a wholesale price contract and the retailer prefers a consignment contract. Additionally, there exist threshold values of RFID tag price and sharing rate to determine the contract preference for the retailer. Furthermore, the profits of both the supplier and the retailer are independent of the RFID tag price sharing rate in a wholesale price contract, and the supplier has the incentive to invest in RFID tag cost in a consignment contract.     

周期可变下考虑多车型的库存路径优化研究

目的 研究基于供应商管理库存模式下，由一个供应商和多个零售商组成的二级供应链配送系统的库存路径优化问题，确定计划期内各配送周期的长度、数量，对供应链中库存策略与配送方案问题进行协调优化，使系统总成本最低。方法 考虑的成本包括库存持有成本、缺货成本和配送成本。为降低供应链系统的总成本，提出全新的周期可变策略。结果 采用遗传算法求解得出最佳方案。将不同策略的算例结果进行对比分析，结果表明周期可变策略与周期固定策略相比，系统总成本最低节约比例为1.7%，最高节约比例为42.3%。结论 通过对计划期内各配送周期的长度及数量进行划分调整可以有效地节约系统总成本，同时，采用多车型的配送方案明显优于采用同车型的配送方案。     

供应商管理库存的价格效应研究

通过建立数学模型,论证了VMI在供应链中收益的非负性;在此基础上还给出了供求均衡价格与订货成本比、库存成本比的函数关系,该公式说明当订货成本比与库存成本比相差比较大时,VMI可以有效降低均衡价格,否则均衡价格将会上升。     

Agent-based simulation of pricing strategy for agri-products considering customer preference

Agri-products typically have short life cycles and are perishable. The price of perishable goods is influenced by many factors. Customers in a market have different preferences with regard to price, the quality of the product and the brand. Though studies on pricing strategies that consider customer preference are rare, this paper addresses the problem of optimal pricing strategy for retailers considering customer preferences. Traditional mathematical methods cannot adequately describe the complexities of customer preference. Due to these complexities, this paper proposes an agent-based simulation model composed of six retailers and hundreds of customers, each with personal preferences. The different retailers set prices according to freshness, inventory, cost and other factors. Due to the perishable nature of the products considered, this paper proposes a new categorising price strategy that sets prices according to different degrees of treatments. By comparing the final profit of all retailers at the end of a simulation, the categorising price strategy is demonstrated to be the optimal strategy if customers with different preferences are randomly distributed. Furthermore, based on the model, the paper studies how optimal strategies are influenced by the proportion of customers with different preferences.     

服务员休假的生产服务库存模型的稳态分析及最优生产策略

将服务员休假与生产服务库存相结合，研究带有服务员休假的 M/M/1 生产服务库存系统模型。需求 (顾客) 的到达过程服从 Poisson 过程，顾客的服务时间和每个产品的生产时间都服从指数分布。当系统库存为零时，服务员开始随机长度的休假。假设休假时间服从指数分布。首先，利用拟生灭过程理论给出了系统的稳态平衡条件。其次，对忽略服务时间的生产服务库存系统模型进行了稳态分析，得到了此系统的稳态概率的明显的解析表达式，进而证明了系统的稳态概率分布具有乘积解形式的结构。在此基础上，进一步得到了系统的一些稳态性能指标和费用函数的计算公式，数值求解了模型的最优 $(s,S)$ 库存策略，并研究系统的一些参数对性能指标、最优策略和最优费用的影响。     

短生命周期产品的延期交货和价格折扣模型

用斜坡型函数来描述短生命周期产品需求变化特征,用持有成本不断增加表示短生命周期产品无形变质所带来的损失,研究在允许缺货情况下,如何设置合理价格折扣来控制订单的流失,确定各周期的最佳延迟供货量、最优价格折扣、最优订货时间和订货量,从而实现企业总成本的最优。给出了一种迭代逼近的方法来寻求在有限时域内的最优订购策略,通过数值分析验证了该模型的有效性和可操作性,对相应的参数做出了灵敏度分析。该模型能够确定出企业最佳的价格折扣以挽留住更多的顾客,使其总利润达到最优。     

回购与缺货成本下的连续时间报童模型

对带回购策略与缺货成本的连续时间报童模型,研究了其最优订购策略与批发价策略,使得生产商和零售商期望收益最大化。在零售价格依赖需求过程与零售价格外生的两种情形下,讨论了带回购策略和缺货成本对连续时间报童均衡策略的共同影响。运用随机最大值原理研究了以上问题均衡策略的存在唯一性,讨论了均衡策略满足的条件,在价格外生情形下得出结论：回购策略与缺货成本的共同作用,使得零售商的最优订购量和供应商的最优批发价格均提高很多。最后,通过数值算例做了最优批发价格与最优订购策略的敏感度分析,具体例子验证了已得结论。     

Optimization and optimality of a joint pricing and inventory control policy in periodic-review systems with lost sales

This paper studies a periodic review dynamic pricing and inventory control problem with fixed ordering cost included. Demand is uncertain and price-sensitive in a general form. At the end of each period, all unmet demand is lost. We focus on the optimization and optimality of an (s, S, p)-type policy, which operates as follows: whenever the inventory level is less than or equal to s, an order is placed to order-up-to level S; when the inventory level is larger than s, no order is issued; the price p is specified by the inventory level. We further numerically investigate the optimal solutions and their sensitivity to cost parameters and demand uncertainty.     

基于双渠道联合库存策略的生鲜品产销

生鲜厂商采用“线上”“线下”两种销售渠道时,采用联合库存策略可以有效降低库存节约成本。考虑影响各渠道需求的各种因素和消费者的选择行为,建立了基于生鲜产品双渠道联合库存策略下的多周期产销最大收益模型。已知线下渠道销售价格,假设库存优先满足线下渠道销售,从生鲜厂商的角度给出了最优线上销售价格和最优生产量。算例表明,该策略下保质期短的生鲜农产品线上定价较高,选择线下自提取货的消费者比例越高则收益越大,需求不确定性增大则会使得利益受损,为生鲜产商通过双渠道销售生鲜农产品提供理论支持。     

短缺量拖后率与价格折扣相关变质品VMI模型

考虑短缺量拖后率与订货商给予顾客的价格折扣正线性相关,提出一种需求指数时变的变质物品供应商管理库存模型,研究了供应链最优库存策略。数值仿真和主要参数灵敏度分析表明,订货商在缺货期间向顾客提供价格折扣有利于降低丢单损失和系统库存总成本;当需求增长因子变化时,VMI系统应保持库存控制策略不变;当变质系数和拖后率上限变大时,VMI系统则应保持订货商补货次数和价格折扣不变,同时适当降低订货商服务水平。     

Optimal production and pricing policies in a combined make-to-order/make-to-stock system

In this paper, we consider a production system which is capable to produce two types of products. The first type of products is make-to-order, while the second type is make-to-stock whose demand is satisfied by the on-hand inventory. The demand arrival rates of both types of products are price-sensitive. The excess demand that cannot be satisfied immediately is either backlogged or lost. The system costs include the holding costs of product inventories and shortage costs of unsatisfied demand. The objective is to maximise the total discounted profit over an infinite planning horizon by coordinating the production process and pricing decisions. By analysing the properties of objective functions, we characterise the optimal control policy by two switch curves and the optimal price is also given for different ordering and inventory levels. We also explore the monotonicity of both switch curves which will reduce the computation effort. Numerical experiments are conducted to demonstrate the use of the switch curves in managing the production system and illustrate that compared with the static pricing policy, the optimal integrated price and inventory control policy can result in a significant profit improvement in the make-to-order/make-to-stock system that is much higher than in a single-product system.     

时变变质率产品的订购-定价策略

研究时变变质率产品的订购与定价联合决策问题。以系统平均利润最大化为目标函数且将已变质产品的变质处理成本考虑在内,构建了相应的订购与定价联合决策模型,其中需求同时依赖于价格和库存水平、系统允许缺货且缺货量部分延迟订购。接着,对于已知的定价策略,证明并给出了最优补货策略唯一存在的充分条件;另一方面,对于已知补货策略的情况也证明了最优定价策略的存在性。此外,证明了在联合订购与定价决策下得到的最优销售价格总是大于单独进行定价决策时得到的最优价格。进而,在模型证明和分析的基础上给出一个寻求最优解的算法。最后,通过数值模拟的方法对模型中主要参数的灵敏度进行了分析,并给出相应的管理建议。结果表明：如果零售商忽略库存水平对需求的影响或变质处理成本,制定的销售价格将偏低;对于价格弹性较高的产品,零售商应采取低价销售策略;当顾客的等待耐心较差或产品的替代性较强时,最优策略应该是提高产品的销售定价并缩短缺货期。     

Optimal two-level piecewise-constant price discrimination for a storable perishable product

Price differentiation over time is an additional policy that firms might consider when determining prices for perishable products. The common policy of a fixed price regardless of freshness might result in leaving some expired inventory unsold. Price differentiation can impact the demand for perishable products, which declines as the expiration date approaches. We develop an optimisation model with the goal of evaluating the monetary effectiveness of the strategy of simultaneously combining price discrimination across heterogeneous consumers with price differentiation over time for perishable inventory under separable multiplicative demand factors of price and time. Necessary optimality equations are derived, and their solutions are proved to constitute a unique global optimal solution. It is proved that an optimal pricing policy is to implement price discrimination with respect to consumers’ sensitivity to freshness, while dynamically changing the price over time, starting with a lower price at the early stages of the product’s shelf life and increasing it at a later stage. The monetary benefit that the retailer and consumer can derive from the suggested pricing policy is evaluated by comparing the model to other models in which price discrimination or dynamic pricing are not implemented. A numerical example that illustrates the significance is introduced. From the analysis of a numerical illustration of the model, it is concluded that a dynamic price discrimination policy can be approximated by an identical-to-all dynamic pricing policy in order to maximise the retailer’s profit and thus, mitigate the retailer’s risk from failing in the process of implementing price discrimination.     

Optimal inventory policies for uniform replenishment systems with time-dependent demand

This study develops an analysis of lot size inventory systems where the replenishment rate is uniform and demand follows a power demand pattern. Shortages are not allowed. Holding cost, replenishing cost and purchasing cost are considered in inventory system control. The objective of the study is to find the economic production quantity that minimises total inventory cost per unit of time. We conclude that optimal inventory policies depend on the demand pattern index chosen to represent customer demand. Theoretical results are illustrated with a business case study. A sensitivity analysis is proposed to describe the optimal policy behaviour.     

Optimal stocking strategies for inventory mechanism with a stochastic short-term price discount and partial backordering

Price discount is an important research topic in the field of inventory management. The existing research on this topic mainly considers fixed price discount, but ignores the situation in which stochastic short-term price discount may be involved. In this paper, we study an inventory problem considering stochastic short-term price discount and partial backordering. To address this problem, we propose an optimal replenishment and stocking model to maximise the retailers' profit. After that, a cost–benefit analysis-based heuristic method for solving the developed model is presented by considering two scenarios depending on whether a replenishment point belongs to a discount period or not. Furthermore, an algorithm is provided to elicit an optimal ordering policy from multiple solutions derived from the given heuristic solution method. Finally, a real case is offered to demonstrate the application of the proposed model, followed by a sensitivity analysis. The results indicate that a retailer can identify the optimal replenishment policy with the aim of achieving maximal profit in situations where stochastic short-term price discount and partial backordering are considered for certain inventory problems at hand. In addition, sensitivity analysis illustrates a fact that different values of the introduced parameters may influence the optimal replenishment policy.