Optimal Planning Quantities for Product Transition

The replacement of an existing product with a new one presents many challenges. In particular, uncertainties in a new product introduction often lead to extreme cases of demand and supply mismatches. This paper addresses inventory planning decisions for product upgrades when there is no replenishment opportunity during the transition period. We allow product substitution: when a company runs out of the old product, a customer may be offered the new product as a substitute. We show that the optimal substitution decision is a time‐varying threshold policy and establish the optimal planning policy. Further, we determine the optimal delay in a new product introduction, given the initial inventory of the old product.     

Joint Stocking and Product Offer Decisions Under the Multinomial Logit Model

This article studies a joint stocking and product offer problem. We have access to a number of products to satisfy the demand over a finite selling horizon. Given that customers choose among the set of offered products according to the multinomial logit model, we need to decide which sets of products to offer over the selling horizon and how many units of each product to stock so as to maximize the expected profit. We formulate the problem as a nonlinear program, where the decision variables correspond to the stocking quantity for each product and the duration of time that each set of products is offered. This nonlinear program is intractable due to its large number of decision variables and its nonseparable and nonconcave objective function. We use the structure of the multinomial logit model to formulate an equivalent nonlinear program, where the number of decision variables is manageable and the objective function is separable. Exploiting separability, we solve the equivalent nonlinear program through a dynamic program with a two dimensional and continuous state variable. As the solution of the dynamic program requires discretizing the state variable, we study other approximate solution methods. Our equivalent nonlinear program and approximate solution methods yield insights for good offer sets.     

无外部市场条件下中间产品转移价格的博弈分析与决策

本文首先用反例证明了文^[1]中在各分厂成本函数和最终产品需求函数已知的条件下,给出的中间产品最优转移价格的两个计算公式并不总是成立的,给出了这两个计算公式成立的充要条件,并分析了公司关于中间产品转移价格决策的动态一致性问题。接着,利用完全信息动态博弈对中间产品转移价格的确定问题做了进一步的分析研究。最后,针对文^[1]中给出的中间产品最优转移价格的两个计算公式不成立的情况,讨论了中间产品转移价格的确定问题。     

Competitive Pricing in a Multi‐Product Multi‐Attribute Environment

We address the problem of simultaneous pricing of a line of several products, both complementary products and substitutes, with a number of distinct price differentiation classes for each product (e.g., volume discounts, different distribution channels, and customer segments) in both monopolistic and oligopolistic settings. We provide a generic framework to tackle this problem, consider several families of demand models, and focus on a real‐world case‐study example. We propose an iterative relaxation algorithm, and state sufficient conditions for convergence of the algorithm. Using historical sales and price data from a retailer, we apply our solution algorithm to suggest optimal pricing, and report on numerical results.     

考虑参照依赖与产品展示的体验式商品定价与订购策略

消费者在购买体验式商品时面临着产品价值的不确定性,因而会产生参照依赖行为。本文研究了考虑消费者参照依赖行为的定价与订购问题,并分析了产品展示策略的影响。研究发现,给定产品满足率时,只有当消费者获得高价值的概率大于某个临界值时,参照依赖下的最优价格才会高于没有参照依赖下的最优价格,并且产品价值维度与产品价格维度的参照依赖对最优价格起着相反的作用。进一步给出了最优订购量满足的条件,并发现在一定条件下最优价格随着订购量的增大而增大。当企业采取产品展示策略之后,会产生两方面的效应,一是消费者数量减少,二是剩余消费者的保留价格增大,此时最优价格随着展示系数的增大而增大。最后,通过数值分析得到了更多的管理启示。     

基于MNL选择模型的替代产品定价和库存决策研究

受现实中生产商和大型零售商不同契约关系的启发,本文以两个生产商和一个零售商组成的系统为对象,从顾客在两种替代性产品之间的选择行为出发,分别研究了零售商定价和生产商定价模式下的最优产品定价和库存决策。研究表明,在MNL选择模型下,零售商联合定价时应当选择使得产品单位利润相等的定价策略;生产商定价博弈中存在唯一的Nash均衡。数值实验分别考察了"生产商强势"和"零售商强势"两种情形下的最优决策和利润分配情况;实验结果能给相关企业的销售模式选择带来有益的管理启示。     

Measuring the Impact of Increased Product Substitution on Pricing and Capacity Decisions Under Linear Demand Models

We consider two substitutable products and compare two alternative measures of product substitutability for linear demand functions that are commonly used in the literature. While one leads to unrealistically high prices and profits as products become more substitutable, the results obtained using the other measure are in line with intuition. Using the more appropriate measure of product substitutability, we study the optimal investment mix in flexible and dedicated capacities in both monopoly and oligopoly settings. We find that the optimal investment in manufacturing flexibility tends to decrease as the products become closer substitutes; this is because (1) pricing can be used more effectively to balance supply and demand, and (2) the gains obtained by shifting production to the more profitable product are reduced due to increased correlation between the price potentials of the substitutable products. The value of flexibility always increases with demand variability. We also show that, as long as the optimal investments in dedicated capacity for both products are positive, the optimal expected prices and production quantities do not depend on the cost of the flexible capacity. Manufacturing flexibility simply allows the firm to achieve those expected values with lower capacity, while leading to higher expected profits.     

可加效用下基于消费者惰性的易逝品动态定价

本文在考虑销售商的风险规避态度和消费者惰性的情况下,研究了易逝品的动态定价策略。首先给出风险规避的销售商实施动态定价的马尔可夫决策过程,并运用MNL随机效用模型来刻画惰性消费者的购买决策,在此基础上,以风险规避销售商的期望效用最大化为目标,利用动态规划方法建立了可加效用下的易逝品动态定价模型,并探讨了消费者的惰性行为和销售商的风险规避态度对最优价格的影响。结论表明：在销售商的风险态度为风险规避的情况下,考虑消费者惰性行为的最优价格随库存水平的增加而降低,随剩余销售时间的增加而提高,消费者的惰性行为和销售商的风险规避态度对最优价格都存在负向影响,即最优价格随惰性深度的增加而降低,随惰性宽度的增加而降低,同时,销售商的风险规避态度越强烈,其制定的产品价格越低。     

On the Benefit of Inventory‐Based Dynamic Pricing Strategies

We study the optimal pricing and replenishment decisions in an inventory system with a price‐sensitive demand, focusing on the benefit of the inventory‐based dynamic pricing strategy. We find that demand variability impacts the benefit of dynamic pricing not only through the magnitude of the variability but also through its functional form (e.g., whether it is additive, multiplicative, or others). We provide an approach to quantify the profit improvement of dynamic pricing over static pricing without having to solve the dynamic pricing problem. We also demonstrate that dynamic pricing is most effective when it is jointly optimized with inventory replenishment decisions, and that its advantage can be mostly realized by using one or two price changes over a replenishment cycle.     

To Preannounce or Not: New Product Development in a Competitive Duopoly Market

In this study, we consider the issue of preannouncing or not preannouncing the development of a new product. Our research is motivated by contrasting views in the literature and varying actions observed in practice. We develop and analyze a game theoretic model that examines the effect of a firm's preannouncement of its product development. Our model is based on a durable goods duopoly market with profit‐maximizing firms. The first firm is an innovator who initially begins developing the product; the second firm is an imitator that begins developing a competing product as soon as it becomes aware of the innovator's product. We assume that consumers are rationally expectant and purchase at most one unit of the product when they have maximum positive utility surplus that is determined by the characteristics of the product, the consumer's marginal utility, and the consumer's discounted utility for future expected products. The innovator firm can release information about its product when it begins developing the product or can guard information about its product until it introduces the product into the market. Our analysis and numerical tests show that, under some conditions, the innovator firm can benefit by preannouncing its product and giving the imitator firm additional time to differentiate its product. We discuss these conditions and their implications for new product development efforts.     

Competing on Time: An Integrated Framework to Optimize Dynamic Time‐to‐Market and Production Decisions

This study develops a comprehensive framework to optimize new product introduction timing and subsequent production decisions faced by a component supplier. Prior to market entry, the supplier performs process design activities, which improve manufacturing yield and the chances of getting qualified for the customer's product. However, a long delay in market entry allows competitors to enter the market and pass the customer's qualification process before the supplier, reducing the supplier's share of the customer's business. After entering the market and if qualified, the supplier also needs to decide how much to produce for a finite planning horizon by considering several factors such as manufacturing yield and stochastic demand, both of which depend on the earlier time‐to‐market decision. To capture this dependency, we develop a sequential, nested, two‐stage decision framework to optimize the time‐to‐market and production decisions in relation to each other. We show that the supplier's optimal market entry and qualification timing decision need to be revised in real time based on the number of qualified competitors at the time of market‐entry decision. We establish the optimality of a threshold policy. Following this policy, at the beginning of each decision epoch, the supplier should optimally stop preparing for qualification and decide whether to enter the market if her order among qualified competitors exceeds a predetermined threshold. We also prove that the supplier's optimal production policy is a state‐dependent, base‐stock policy, which depends on the time‐to‐market and qualification decisions. The proposed framework also enables a firm to quantify how market conditions (such as price and competitor entry behavior) and operating conditions (such as the rate of learning and inventory/production‐related costs) affect time‐to‐market strategy and post‐entry production decisions.     

A Single‐Period Assortment Optimization Model

This paper studies a single‐period assortment optimization problem with unequal cost parameters. The consumer choice process is characterized by a Multinomial Logit (MNL) model. When the store traffic is a continuous random variable, we explicitly derive the structure of the optimal assortment. Our approach is to use a comprehensive measure–profit rate to evaluate the profitability of each variant and then determine which product should be offered. We demonstrate that the optimal assortment contains the few items that have the highest profit rate. When the store traffic is discrete, the optimal solution is difficult to obtain. We propose a “profit rate” heuristic, which is inspired by the result for the case of continuous store traffic. In a special case with equal cost parameters and normal demand distribution, the profit rate heuristic is indeed optimal. Using randomly generated data, we test the effectiveness of the heuristic and find that the average percentage error is less than 0.1% and that the hit rate is above 90%. Our research provides managerial insights on assortment planning and accentuates the importance of measuring the profitability of each product when the demand is random and cannibalization among different products exists.     

Inventory‐Based Dynamic Pricing with Costly Price Adjustment

We study an average‐cost stochastic inventory control problem in which the firm can replenish inventory and adjust the price at anytime. We establish the optimality to change the price from low to high in each replenishment cycle as inventory is depleted. With costly price adjustment, scale economies of inventory replenishment are reflected in the cycle time instead of lot size—An increased fixed ordering cost leads to an extended replenishment cycle but does not necessarily increase the order quantity. A reduced marginal cost of ordering calls for an increased order quantity, as well as speeding up product selling within a cycle. We derive useful properties of the profit function that allows for reducing computational complexity of the problem. For systems requiring short replenishment cycles, the optimal solution can be easily computed by applying these properties. For systems requiring long replenishment cycles, we further consider a relaxed problem that is computational tractable. Under this relaxation, the sum of fixed ordering cost and price adjustment cost is equal to (greater than, less than) the total inventory holding cost within a replenishment cycle when the inventory holding cost is linear (convex, concave) in the stock level. Moreover, under the optimal solution, the time‐average profit is the same across all price segments when the inventory holding cost is accounted properly. Through a numerical study, we demonstrate that inventory‐based dynamic pricing can lead to significant profit improvement compared with static pricing and limited price adjustment can yield a benefit that is close to unlimited price adjustment. To be able to enjoy the benefit of dynamic pricing, however, it is important to appropriately choose inventory levels at which the price is revised.     

Pricing and Logistics Decisions for a Private‐Sector Provider in the Cash Supply Chain

We study a problem faced by a secure‐logistics provider (SLP) of maximizing profit by jointly pricing the services of fit‐sorting and transporting cash along with the design of the supporting logistics network, in a market consisting of a population of Depository Institutions (DIs). The need to jointly price the services assumes significance because they are partial substitutes of one another. Our study finds that the influence of the logistics network on prices is especially strong when there are non‐linearities in the cost of provisioning the logistics services. Furthermore, the impact of logistics decisions on different types of pricing schemes (e.g., volume discount, bundled pricing) is different, both in its structure and extent. In present times, when the market for the fit‐sorting service is relatively immature, our findings have major implications to the way an SLP's business is managed.     

A Nonatomic‐Game Approach to Dynamic Pricing under Competition

We study a revenue management problem involving competing firms. We assume the presence of a continuum of infinitesimal firms where no individual firm has any discernable influence over the evolution of the overall market condition. Under this nonatomic‐game approach, the unanimous adoption of an equilibrium pricing policy by all firms will yield a market‐condition process that in turn will elicit the said policy as one of the best individual responses. For both deterministic‐ and stochastic‐demand cases, we show the existence of equilibrium pricing policies that exhibit well‐behaving monotone trends. Our computational study reveals many useful insights, including the fact that only a reasonable number of firms are needed for our approach to produce near‐rational pricing policies.     

策略型消费者创新感知与产品定价换代策略研究

科学技术迅猛发展促使厂商频繁推出创新型产品以抢占市场份额,产品定价和换代策略成为厂商经营成败的重要因素。本文在产品技术创新视角下构建双寡头厂商三阶段产品最优定价与换代决策模型,结合效用函数和博弈分析方法从同质产品市场拓展为异质产品市场,求解出单品换代策略及共生换代策略下新品上市时点和旧产品退出市场时点的显示解,并通过仿真模拟探讨产品技术创新水平、消费者创新感知度对最优产品定价和换代策略的影响。研究表明产品技术创新水平和消费者创新感知度对厂商产品定价和换代策略有重要影响。同质产品市场下,市场潜在需求越大,产品初始定价越高。厂商面临内外部产品竞争时,市场潜在需求较小且两代产品处于共生状态下,为减弱内部竞争新旧产品最优定价差距不大;当市场潜在需求较大,厂商新品定价由技术创新水平决定,消费者技术创新感知值越高,新品定价越高,反之越低。当厂商技术创新水平处于劣势时,两厂商技术创新水平差距越大,劣势厂商越应提早退出市场。     

Static and Dynamic Pricing of Excess Capacity in a Make‐to‐Order Environment

We consider a make‐to‐order manufacturer that serves two customer classes: core customers who pay a fixed negotiated price, and “fill‐in” customers who make submittal decisions based on the current price set by the firm. Using a Markovian queueing model, we determine how much the firm can gain by explicitly accounting for the status of its production facility in making pricing decisions. Specifically, we examine three pricing policies: (1) static, state‐independent pricing, (2) constant pricing up to a cutoff state, and (3) general state‐dependent pricing. We determine properties of each policy, and illustrate numerically the financial gains that the firm can achieve by following each policy as compared with simpler policies. Our main result is that constant pricing up to a cutoff state can dramatically outperform a state‐independent policy, while at the same time achieving most of the increase in revenue achievable from general state‐dependent pricing. Thus, we find that constant pricing up to a cutoff state presents an attractive tradeoff between ease of implementation and revenue gain. When the costs of policy design and implementation are taken into account, this simple heuristic may actually out‐perform general state‐dependent pricing in some settings.     

The Retail Space‐Exchange Problem with Pricing and Space Allocation Decisions

We consider retail space‐exchange problems where two retailers exchange shelf space to increase accessibility to more of their consumers in more locations without opening new stores. Using the Hotelling model, we find two retailers’ optimal prices, given their host and guest space in two stores under the space‐exchange strategy. Next, using the optimal space‐dependent prices, we analyze a non‐cooperative game, where each retailer makes a space allocation decision for the retailer's own store. We show that the two retailers will implement such a strategy in the game, if and only if their stores are large enough to serve more than one‐half of their consumers. Nash equilibrium for the game exists, and its value depends on consumers’ utilities and trip costs as well as the total available space in each retailer's store. Moreover, as a result of the space‐exchange strategy, each retailer's prices in two stores are both higher than the retailer's price before the space exchange, but they may or may not be identical.     

Opt‐Out Options in New Product Co‐development Partnerships

We study three contractual arrangements—co‐development, licensing, and co‐development with opt‐out options—for the joint development of new products between a small and financially constrained innovator firm and a large technology company, as in the case of a biotech innovator and a major pharma company. We formulate our arguments in the context of a two‐stage model, characterized by technical risk and stochastically changing cost and revenue projections. The model captures the main disadvantages of traditional co‐development and licensing arrangements: in co‐development the small firm runs a risk of running out of capital as future costs rise, while licensing for milestone and royalty (M&R) payments, which eliminates the latter risk, introduces inefficiency, as profitable projects might be abandoned. Counter to intuition we show that the biotech's payoff in a licensing contract is not monotonically increasing in the M&R terms. We also show that an option clause in a co‐development contract that gives the small firm the right but not the obligation to opt out of co‐development and into a pre‐agreed licensing arrangement avoids the problems associated with fully committed co‐development or licensing: the probability that the small firm will run out of capital is greatly reduced or completely eliminated and profitable projects are never abandoned.