Joint pricing and procurement of substitutable products with random demands – A technical note

Single-period joint pricing and procurement of substitutable products entails one time procurement and pricing decisions for substitutable products that face price dependent stochastic demands. Recently, Karakul and Chan [Karakul, M., Chan, L., 2008. Analytical and managerial implications of integrating product substitutability in the joint pricing and procurement problem. European Journal of Operational Research 190, 179–204] considered this problem for two one-way substitutable products. Authors model the demands for each product in the well known additive form, where the mean demands are linear functions of the price of the high grade new product plus an additive stochastic noise term. By assuming that the noise term for the low grade existing product follows a general discrete distribution and the noise term for the high grade product follows a general continuous distribution, authors are able to show the unimodality of the expected profit function with respect to the procurement quantities and the price of the new product. In this paper, we extend this result to the case where the noise term in the demand of the low grade product follows a general continuous distribution as well.     

The effect of supply uncertainty in price-setting newsvendor models

We consider a price-setting newsvendor model in which a firm needs to make joint inventory and pricing decisions before the selling season. The supply process is uncertain such that the received quantity is the product of the order quantity and a random yield rate. Two cost structures are investigated, the in-house production case in which the firm pays for the input quantity and the procurement case in which the firm pays for the quantity received only. Our objective is to investigate the effect of yield randomness on optimal decisions and expected profit. By using the theory of stochastic comparisons, we find that under both cost structures, a less variable yield rate leads to a lower optimal price and a higher expected profit. Moreover, we show that in the in-house production case, a stochastically larger yield rate also results in a lower optimal price and a higher profit, but this is not true in the procurement case. Examples show that the effect of supply uncertainty on optimal order quantity is not universal.     

A study on the effect of yield uncertainty in price-setting newsvendor models with additive-multiplicative demand

Random yield and uncertain demand usually both exist in many industries, such as the semiconductor industry. In this paper, the price-setting newsvendor model is studied which involves a single manufacturer and a single retailer with random yield and uncertain demand respectively. Under the condition of additive-multiplicative demand, we investigate the varying effects of random yield on the optimal price, order quantity, and expected profit in two situations with different cost structures. The first case is an in-house production case where the firm pays for the raw material quantity it has ordered, and the second one is a procurement case where the firm pays for the real product quantity it receives only. By using the theory of stochastic comparisons, we find that a less variable and a stochastically larger yield rate both lead to a lower optimal price and a higher expected profit for the in-house production case. Moreover, a less variable yield rate also results in a lower optimal price and a higher profit for the procurement case, but this is not true for a stochastically larger yield rate. Numerical examples illustrate that the effect of yield randomness on the optimal order quantity is not general.     

A production–recycling–inventory system with imprecise holding costs

This paper presents an optimal control recycling production inventory system in fuzzy environment. The used items are bought back and then either put on recycling or disposal. Recycled products can be used for the new products which are sold again. Here, the rate of production, recycling and disposal are assumed to be function of time and considered as control variables. The demand inversely depends on the selling price. Again selling price is serviceable stock dependent. The holding costs (for serviceable and non-serviceable items) are fuzzy variables. At first we define the expected values of fuzzy variable, then the system is transferred to the fuzzy expected value model. In this paper, an optimal control approach is proposed to optimize the production, recycling and disposal strategy with respect so that expected value of total profit is maximum. The optimum results are presented both in tabular form and graphically.     

不确定需求与产出供应链回购契约决策

针对市场需求的不确定性和供应商产出产品合格率的不确定性,提出供应商产出量小于销售商订购量时需再生产满足销售商的订购,以及供应商生产过量和回购的产品通过折扣市场出售,建立以供应链期望利润为目标函数,以回购单价为决策变量的供应链回购契约决策模型.通过模型的分析,证明了模型存在最优的期望利润,给出了最优回购单价满足的方程;同时,也证明了最优订购量,最优生产量和销售商最优期望利润是回购单价的增函数.最后,通过算例验证了模型的有效性.     

Time-dependent and independent control rules for coordinated production and pricing under demand uncertainty and finite planning horizons

We address the effect of uncertainty on a manufacturer’s dynamic production and pricing decisions over a finite planning horizon. The demand for products, which depends on their price, is characterized by two stochastic processes: potential demand and customer price sensitivity. An optimal policy for coordinating production and pricing is a time-dependent feedback rule with respect to the state of the manufacturer’s inventories. We show that when the volatility of customer sensitivity to the product price is negligible, the optimal policy can be obtained analytically. Moreover, our simulations demonstrate that the volatility of stochastic customer price sensitivity does not have a strong effect on the manufacturer’s expected profit. Therefore, the solution derived for the case of customer price sensitivity with zero volatility can serve as a good approximation heuristic for the optimal policy if the true volatility of customer price sensitivity is within 40 % of its mean and the volatility of potential demand is within 25 % of its mean. Moreover, under these conditions, a simplified, time-independent control rule deteriorates expected profits by only 1.5 %.     

Optimal capacity for substitutable products under operational postponement

We consider a monopolist producing two substitutable products with one flexible (shared) capacity. The demand of each product is a linear function of the prices of both products, and is subject to an additive shock. We study the impact of two key drivers, namely the degree of substitution between the products and the level of operational postponement, on the optimal capacity and the resulting expected profit. We show that the relationship between the optimal capacity and the degree of product substitution is not impacted by the different postponement strategies the firm can utilize or by the different settings (forced clearance versus holdback) considered in the previous literature. On the other hand, how capacity is affected by postponement critically depends on how closely substitutable the products are. In particular, we show that the well-known result that operational postponement and capacity are strategic complements in a single-product setting (Van Mieghem and Dada, 1999) no longer holds in our setting, because the two substitutable products are now linked through consumer-driven substitution, which the firm can influence through pricing. In particular, capacity and operational postponement (in the form of quantity postponement) can be either strategic substitutes or strategic complements, and this depends on both the firm’s cost structure and the degree of substitution between the products. We also study the impact of forced clearance on the firm’s expected profit and find that clearance deteriorates the firm’s earnings more when the products it offers are highly differentiated.     

An EOQ model for salesmen’s initiatives, stock and price sensitive demand of similar products - A dynamical system

The paper deals with an inventory model to determine the retailer’s optimal order quantity for similar products. It is assumed that the amount of display space is limited and the demand of the products depends on the display stock level where more stock of one product makes a negative impression of the another product. Besides it, the demand rate is also dependent on selling price and salesmen’s initiatives. Also, the replenishment rate depends on the level of stock of the items. The objective of the model is to maximize the profit function, including the effect of inflation and time value of money by Pontryagin’s Maximal Principles. The stability analysis of the concerned dynamical system has been done analytically.     

Optimal warranties,reliabilities and prices for durable goods in an oligopoly

This paper addresses the problem of durable goods manufacturers in an oligopoly seeking optimal values for three decision variables: product warranty, reliability and price. Each firm seeks a warranty-reliability-price combination that maximizes expected profit subject to quite general constraints on the firm's decision variables. Warranty serves as a signal of product reliability, which is not observable by consumers. We present a game-theoretic model of warranty-reliability-price competition in such a market and examine Nash equilibria for this game. We show that under fairly general assumptions each firm can optimally set its warranty and reliability independently of price and competitors' actions. In addition, we show that optimal warranties and reliabilities are complementary, and we explore the impact of different market factors on the optimal warranty and reliability. Finally, we show that optimal warranties are longer and products more reliable when consumers are risk averse.     

单向可替代产品价格和库存的联合决策

考虑单周期问题中零售商同时销售两种可单向替代的产品,以期望利润为目标函数建立数学模型.将库存和替代价格共同作为零售商决策变量,证明其目标函数是凹函数,并得到求模型最优解的充要条件及解存在的范围.最后假设产品需求为正态分布,通过数值实验对模型的最优解进行分析,结果表明:实行最优替代价格策略可以有效提高零售商期望利润;允许替代销售不一定提高市场服务水平;被替代产品的销售价格和残值对零售商的最优替代价格决策没有显著影响.     

Newsvendor with multiple options of expediting

We consider a manufacturer facing single period inventory planning problem with uncertain demand and multiple options of expediting. The demand comes at a certain time in the future. The manufacturer may order the product in advance with a relatively low cost. She can order additional amount by expediting after the demand is realized. There are a number of expediting options, each of which corresponds to a certain delivery lead time and a unit procurement price. The unit procurement price is decreasing over delivery lead time. The selling price is also decreasing over time. In this paper, we assume that the manufacturer must deliver all products to the customer in a single shipment. The problem can be formulated as a profit maximization problem. We develop structural properties and show how the optimal solution can be identified efficiently. In addition, we compare our model with the classical newsvendor model and obtain a number of managerial insights.     

质量约束下闭环供应链差异决策研究

针对由一原始设备制造商和一独立再制商所组成的闭环供应链系统,考虑到新品质量水平对独立再制商可回收量的质量约束效应,以及新品与再制品的价格及质量竞争,分别构建了原始设备制造商领导、独立再制商领导、无领导三种不同市场结构下闭环供应链决策模型,探讨了质量约束效应对闭环供应链运营的影响。研究表明:(1)三种市场结构下,新品和再制品两者在“质量水平-价格”两维度均有所差异,且市场地位的不对等有利于产品质量水平的共同提升;(2)质量约束效应的存在会对产品的价格和质量同时产生影响;(3)再制品具有质量劣势,且质量约束效应的增强,再制品质量劣势越大;(4)市场结构的变化及质量约束效应的强弱并不会影响市场总需求量,仅仅会影响成员企业的市场份额;在大多数情况下,消费者对新品需求更为活跃。但当原始设备制造商为市场领导者时,质量约束效应低于一阈值水平时,再制品销量反而高于新品销量;(5)无论从成员个体,还是从供应链整体,原始设备制造商为领导者的市场结构最有效,且新品质量约束效应并不会影响供应链总利润,仅影响成员企业在总利润中的分成;新品质量约束效应的增强更有利于原始设备制造商利润的改善,但同时会损害独立再制商利润。     

Optimal procurement strategies for online spot markets

Spot markets have emerged for a broad range of commodities, and companies have started to use them in addition to their traditional, long-term procurement contracts (forward contracts). In comparison to forward contracts, spot markets offer products at essentially negligible lead time, but typically command a higher expected price for this added flexibility while also exhibiting substantial price uncertainty. In our research, we analyze the resulting procurement challenge and quantify the benefits of using spot markets from a supply chain perspective. We develop and solve mathematical models that determine the optimal order quantity to purchase via forward contracts and the optimal quantity to purchase via spot markets. We analyze the most general situation where commodities can be both bought and sold via a spot market and derive closed-form results for this case. We compare the obtained results to the reference scenario of pure contract sourcing and we include results for situations where the use of spot markets is restricted to either buying or selling only. Our approaches can be used by decision makers to determine optimal procurement strategies based on key parameters such as, demand and spot price volatilities, correlation between demand and spot prices, and risk aversion. The results of our analysis demonstrate that significant profit improvements can be achieved if a moderate fraction of the commodity demand is procured via spot markets. The results also show that companies who use spot markets can offer a higher expected service level, but that they might experience a higher variability in profits than companies who do not use spot markets. We illustrate our analytical results with numerical examples throughout the paper.     

Incorporating one-way substitution policy into the newsboy problem with imprecise customer demand

This paper presents an approach for solving an inventory model for single-period products with maximizing its expected profit in a fuzzy environment, in which the retailer has the opportunity for substitution. Though various structures of substitution arise in real life, in this study we consider the fuzzy model for two-item with one-way substitution policy. This one-way substitutability is reasonable when the products can be stored according to certain attribute levels such as quality, brand or package size. Again, to describe uncertainty usually probability density functions are being used. However, there are many situations in real world that utilize knowledge-based information to describe the uncertainty. The objective of this study is to provide an analysis of single-period inventory model in a fuzzy environment that enables us to compute the expected resultant profit under substitution. An efficient numerical search procedure is provided to identify the optimal order quantities, in which the utilization of imprecise demand and the use of one-way substitution policy increase the average expected profit. The benefit of product substitution is illustrated through numerical example.     

基于视情维修的质保政策设计研究

随着产品同质化程度不断提升,完善的质保服务已成为厂商提升品牌形象、刺激用户购买需求和全面反馈市场信息的主要途径。本文以具有退化特性的耐用品为研究对象,建立了视情维修条件下性能退化的维纳过程模型,考虑产品价格、质保期长度和维修费用承担比例对产品需求的影响,以最大化厂商利润为目标,确定了最优的质保期长度和维修费用承担比例。结合算例,比较了无视情维修和提供视情维修两种情形下对应的厂商利润,并分析了产品退化速率、成本参数和产品价格对利润的影响。研究结果表明,在质保服务范围内提供合理的视情维修服务既可将产品可靠性维持在一个较高水平,又可显著提升厂商利润。     

Analysis and design of returns policies from a supplier's perspective

This paper considers the problem of designing a returns policy in a supply chain from a supplier's perspective. The supply chain considered here is assumed to have one supplier and one retailer who serves a random demand of a product with a short life cycle. The retailer can return all the unsold products to the supplier with a partial refund. We found that if the retailer behaviour is rational, that is, ordering the optimal quantity to maximize its expected profit, then both retailer and supplier could benefit from the returns policy. Furthermore, we established that the optimal buyback price is independent of the mean of the random demand, but the variance of the demand has a significant impact on setting the optimal buyback price. The higher the variance the higher the optimal buyback price and the larger the profit gain of both parties. Numerical studies are employed to help understand the benefits of returns policies for the supplier, the retailer, and the whole supply chain.     

Optimal bidding in auctions of mixed populations of bidders

A mixed population of bidders consists of two asymmetric groups. Members of the first group are game-theoretic players, who maximize their expected profit and incorrectly believe that their opponents act similarly. The second group of bidders adopts an irrational strategy: they either choose their bids randomly following a given probability distribution, in a “naïve” form of bidding, or follow a decision-theoretic approach, maximizing their expected profit under the assumption that all other bids are random. In a sealed bid private-value procurement auction we examine the optimal strategy of a new player, who has perfect knowledge of the structure of the mixed bidder population and enters the auction. The optimal bid of the new bidder is derived when the cost and mark-up follow a uniform distribution in [0, 1]. The effect of the relative size of the group of game-theoretic bidders and the population size on the optimal bid price is established.     

Modeling and analysis of a warranty policy using new and reconditioned parts

Remanufacturing processes such as refurbishing and reconditioning can extend the life of a product returned from the field. This provides financial opportunities and allows manufacturers to engage in sustainable practices. However, the inability to access a sufficient quantity of reconditioned components from end‐of‐life products can force the concurrent utilization of new components. This paper deals with the determination of an optimal warranty policy where a mixture of new and reconditioned components are used to carry out replacements upon failure for products under warranty. A mathematical optimization model is developed to maximize the manufacturer's expected total profit based on four decision variables: the warranty length, the sale price, the age of reconditioned components, and the proportion of reconditioned components to be used. A numerical procedure is used to compute the optimal solution. Numerical results are provided and discussed to demonstrate the validity and the added value of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

考虑消费者具有质量和价格偏好的双渠道供应链最优决策

产品价格和质量是消费者永恒的关注点。虽然多渠道给消费者带来了便捷的购物途径, 但是不同渠道的产品存在质量和价格分差异, 消费者在购买时须面对一个渠道、价格和质量的协调选择问题。本文提出了消费者对产品质量和价格偏好的双渠道定价决策模型, 并讨论了以制造商为领导者的分散决策和集中决策情形下的最优策略。比较了消费者偏好及产品质量水平对不同决策情形下的供应链决策的影响, 并通过设计收益共享契约机制实现了渠道的协调并证明了其有效性。研究表明:当产品质量在一定范围内时, 制造商和零售商可以获得最大利润;当消费者对质量的偏好逐渐增加时, 制造商的利润和零售商的利润随着消费者对质量的偏好提高而下降;制造商和零售商可以通过协调销售价格消除供应链的双重边际效应,从而实现供应链的协调。最后利用算例分析了消费者的质量、价格偏好对总利润的影响并给出了相应的管理启示。     

Optimal control of price and production in an assemble-to-order system

We study the optimal control of an assembly system that produces one assembled-to-order final product with multiple made-to-stock components and sells it at variable price. It is shown that a threshold control on component production, product price, and product orders maximizes total discounted profit over an infinite horizon.