Dynamic pricing of limited inventories for multi-generation products

In this research, we consider a retailer selling products from two different generations, both with limited inventory over a predetermined selling horizon. Due to the spatial constraints or the popularity of a given product, the retailer may only display goods from one specific generation. If the transaction of the displayed item cannot be completed, the retailer may provide an alternative from another generation. We analyze two models - posted-pricing-first model and negotiation-first model. The former considers negotiation as being allowed on the price of the second product only and in the latter, only the price of the first product is negotiable. Our results show that the retailer can adopt both models effectively depending on the relative inventory levels of the products. In addition, the retailer is better off compared to the take-it-or-leave-it pricing when the inventory level of the negotiable product is high.     

Pricing and lot-sizing for a deteriorating item in a periodic review inventory system with shortages

We consider a single product that is, subject to continuous decay, a multivariate demand function of price and time, shortages allowed and completely backlogged in a periodic review inventory system in which the selling price is allowed to adjust upward or downward periodically. The objective of this paper is to determine the periodic selling price and lot-size over multiperiod planning horizon so that the total discount profit is maximized. The proposed model can be used as an add-in optimizer like an advanced planning system in an enterprise resource planning system that coordinates distinct functions within a firm. Particular attention is placed on investigating the effect of periodic pricing jointly with shortages on the total discount profit. The problem is formulated as a bivariate optimization model solved by dynamic programming techniques coupled with an iterative search process. An intensive numerical study shows that the periodic pricing is superior to the fixed pricing in profit maximization. It also clarifies that shortages strategy can be an effective cost control mechanism for managing deterioration inventory.     

A pricing model for clearing end-of-season retail inventory

The problem of setting prices for clearing retail inventories of fashion goods is a difficult task that is further exacerbated by the fact that markdowns enacted near the end of the selling season have a smaller impact on demand. In this article, we present discrete-time models for setting clearance prices in such an environment. When demand is deterministic, we compute optimal prices and show that decreasing reservation prices lead to declining optimal prices. When demand is stochastic and arbitrarily correlated across planning periods, we obtain bounds on the optimal expected revenue and on optimal prices. We also develop a heuristic procedure for finding near-optimal prices and test its accuracy through numerical experiments. These experiments reveal new insights for practitioners. For example, the penalty for choosing clearance price once and keeping it unchanged for the remainder of the selling season is found to be small when either the mean reservation prices do not change appreciably over time or when they drop sharply after the first period.     

Ordering and pricing of service products in an advance sales system with price-dependent demand

This investigation addresses a service inventory control problem in which a firm orders and sells a service which will be used or consumed by customers on a specific future date. The firm sells the product through an advance booking system, aiming to optimize product price to maximize the total expected profit. Considering situations in which product demand is price-dependent and customers with reservations may cancel advance orders, this work develops a continuous-time model to simultaneously determine the order quantity and selling prices. The analytical results reveal that the optimal ordering quantity and prices are derived via closed-form solutions. In addition, sensitivity analysis of the optimal prices with respect to the system parameters is also conducted to illustrate optimal decision characteristics.     

Competitive Multi-period Pricing for Perishable Products: A Robust Optimization Approach

We study a multi-period oligopolistic market for a single perishable product with fixed inventory. Our goal is to address the competitive aspect of the problem together with demand uncertainty using ideas from robust optimization and variational inequalities. The demand function for each seller has some associated uncertainty and we assume that the sellers would like to adopt a policy that is robust to adverse uncertain circumstances. We believe this is the first paper that uses robust optimization for dynamic pricing under competition. In particular, starting with a given fixed inventory, each seller competes over a multi-period time horizon in the market by setting prices and protection levels for each period at the beginning of the time horizon. Any unsold inventory at the end of the horizon is worthless. The sellers do not have the option of periodically reviewing and replenishing their inventory. We study non-cooperative Nash equilibrium policies for sellers under such a model. This kind of a setup can be used to model pricing of air fares, hotel reservations, bandwidth in communication networks, etc. In this paper we demonstrate our results through some numerical examples.     

Bundling decisions in supply chains

Firms often sell products in bundles to extract consumer surplus. While most bundling decisions studied in the literature are geared to integrated firms, we examine a decentralized supply chain where the suppliers retain decision rights. Using a generic distribution of customers’ reservation price we establish equilibrium solutions for three different bundling scenarios in a supply chain, and generate interesting insights for distributions with specific forms. We find that (i) in supply chain bundling the retailer’s margin equals the margin of each independent supplier, and it equals the combined margin when the suppliers are in a coalition, (ii) when the suppliers form a coalition to bundle their products the bundling gain in the supply chain is higher and retail price is lower than when the retailer bundles the products, (iii) the supply chain has more to gain from bundling relative to an integrated firm, (iv) the first-best supply chain bundling remains viable over a larger set of parameter values than those in the case of the integrated firm, (v) supplier led bundling is preferable to separate sales over a wider range of parameter values than if the retailer led the bundling, and (vi) if the reservation prices are uniformly distributed bundling can be profitable when the variable costs are low and valuations of the products are not significantly different from one another. For normally distributed reservation prices, we show that the bundling set is larger and the bundling gain is higher than that for a uniform distribution.     

Periodic pricing and replenishment policy for continuously decaying inventory with multivariate demand

This paper deals with the joint decisions on pricing and replenishment schedule for a periodic review inventory system in which a replenishment order may be placed at the beginning of some or all of the periods. We consider a single product which is subject to continuous decay and a demand which is a function of price and time, without backlogging over a finite planning horizon. The proposed scheme may adjust periodically the selling price upward or downward that makes the pricing policy more responsive to structure changes in supply or demand. The problem is formulated as a dynamic programming model and solved by numerical search techniques. An extensive numerical study is conducted to attend qualitative insights into the structures of the proposed policy and its sensitivity with respect to major parameters. The numerical result shows that the solution generated by the periodic policy outperforms that by the fixed pricing policy in maximizing discount profit.     

Price markdown scheme in a multi-echelon supply chain in a high-tech industry

This paper studies the price markdown scheme in a supply chain that consists of a supplier, a contract manufacturer (CM), and a buyer (retailer). The buyer subcontracts the production of the final product to the CM. The CM buys the components from the supplier and charges the buyer a service fee for the final product produced. The price markdown is made possible by the supplier with the development of new manufacturing technologies that reduce the production cost for the sourced component. Consequently, the buyer adjusts the retail price in order to possibly stimulate stronger demand that may benefit both the supplier and the buyer. Under this scenario, we identify the optimal discount pricing strategies, capacity reservation, and the stocking policies for the supplier and the buyer. We also investigate the optimal inventory decision for the CM to cope with the price discount by considering both demand and delivery uncertainties. Our results suggest that higher production cost accelerates the effects of higher price sensitivity on lowering the optimal capacity and stocking policies in the supply chain. The effect of mean demand error on the optimal prices is relatively marginal compared with that from price sensitivity. We also found that increasing the standard deviation of the random demand does not necessarily increase the stocking level as one would predict. The results show that delivery uncertainty plays an important role in the inventory carried beyond the price break. We discuss potential extensions for future research.     

Two-period dynamic versus fixed-ratio pricing in a capacity constrained duopoly

This paper analyzes the impact of dynamic and fixed-ratio pricing policies on firm profits and equilibrium prices under competition. Firms that have equal inventories of perfectly substitutable and perishable products compete for customer segments that demand the product at different times. In each period, customers first purchase from the low price firm and then from the high price firm up to their inventories, provided the prices are lower than the maximum they are willing to pay. The main conclusions of this paper are as follows: although dynamic pricing is a more sophisticated policy than fixed-ratio pricing, it may lead to decreased equilibrium profits; under both pricing policies, one firm assumes the role of a low-cost high-output firm while the other assumes the role of a high-cost low-output firm; and, the supply demand ratio has more impact on the outcome of the competition than the heterogeneity in consumer reservation prices.     

Dynamic pricing of inventory/capacity with infrequent price changes

We consider a problem of dynamically pricing a single product sold by a monopolist over a short time period. If demand characteristics change throughout the period, it becomes attractive for the company to adjust price continuously to respond to such changes (i.e., price-discriminate intertemporally). However, in practice there is typically a limit on the number of times the price can be adjusted due to the high costs associated with frequent price changes. If that is the case, instead of a continuous pricing rule the company might want to establish a piece-wise constant pricing policy in order to limit the number of price adjustments. Such a pricing policy, which involves optimal choice of prices and timing of price changes, is the focus of this paper.We analyze the pricing problem with a limited number of price changes in a dynamic, deterministic environment in which demand depends on the current price and time, and there is a capacity/inventory constraint that may be set optimally ahead of the selling season. The arrival rate can evolve in time arbitrarily, allowing us to model situations in which prices decrease, increase, or neither. We consider several plausible scenarios where pricing and/or timing of price changes are endogenized. Various notions of complementarity (single-crossing property, supermodularity and total positivity) are explored to derive structural results: conditions sufficient for the uniqueness of the solution and the monotonicity of prices throughout the sales period. Furthermore, we characterize the impact of the capacity constraint on the optimal prices and the timing of price changes and provide several other comparative statics results. Additional insights are obtained directly from the solutions of various special cases.     

基于在线捆绑的易逝品动态定价模型研究

运用动态规划和组合优化理论,建立了基于在线捆绑的易逝品动态标价模型.针对是否补充缺货分别建立了紧急补货模型和失销模型.将有限的销售期限分割成N个时间决策单元,提出了每个决策单元的捆绑结构和捆绑包价格的确定方法.证明了紧急补货模型价值函数的可分解性、最优捆绑价格在时间上的非递减性和在存量上的非递增性;证明了失销模型最优期望收益在存量上的非递减性.该模型有助于实践中运用在线捆绑策略的易逝品生产和服务企业对捆绑包结构和捆绑包价格做出正确的决策.     

An EOQ model with stock and price sensitive demand

Many business practices show that the presence of a larger quantity of goods displayed may attract more customers than that with a smaller quantity of goods. This phenomenon implies that the demand may have a positive correlative with stock level. Under such a circumstance, a firm should seriously consider its pricing and ordering strategy since the demand for their goods may be affected by their selling prices and inventory level. This paper aims to develop a continuous inventory model for finding the strategy for a firm that sells a seasonal item over a finite planning time. The purpose of this firm is to maximize its expected profit by determining the optimal ordering quantity and price setting/changing strategy. Some sufficient conditions are found for finding the optimal decision rules.     

Integration of pricing and capacity allocation for perishable products

For years pricing and capacity allocation decisions in most revenue management models have been carried out independently. This article presents a comprehensive model to integrate these two decisions for perishable products. We assume that the supplier sells the same products to different micro-markets at distinct prices. Throughout the sales season, the supplier faces decisions as to which micro-markets or customer classes should be served and at what prices. We show that (i) at any time, a customer class is active (being served) if and only if the price offered is over a threshold level, but the optimal price may not be the highest one of the supplier’s choice; (ii) when the price decision is made in conjunction with inventory, it is similar to the procedure shown in pure pricing models, i.e., the optimal price comes from a subset of prices that forms a maximum increasing concave envelope; (iii) because of dynamic changes in the optimal prices, the nested-price structure does not necessarily hold in general and needs to be redefined; and (iv) the optimal pricing and capacity control policy is based on a sequence of threshold points that incorporate inventory, price and demand intensity. Numerical examples are provided.     

Optimal policies for inventory usage,production and pricing of fashion goods over a selling season

A short selling season and highly uncertain demands prior to the season characterize production and selling of fashion goods. Once the season starts and demands turn up with a peak interest in the beginning, monopoly becomes under tremendous pressure to produce the required amount so as not to disappoint its customers. It motivates the monopoly to prepare significant inventories by the opening day. Unfortunately, even the most advanced techniques for demand forecasting are likely to induce either an overestimate or underestimate of the initial inventories. Both affect the monopoly's profit. Overestimation results in surplus, which may never be sold, and excessive inventory holding costs. Underestimation implies sales as well as customer loyalty losses. Given inventory level at the beginning of the selling season, we derive policies of handling this inventory, production capacity and product prices in order to maximize the profit and thus diminish the effect of inherent inaccuracy of initial inventory estimation of fashion goods. A case of bookstore management illustrates the effectiveness of the suggested strategies.     

Production planning and pricing policy in a make-to-stock system with uncertain demand subject to machine breakdowns

We consider a make-to-stock system served by an unreliable machine that produces one type of product, which is sold to customers at one of two possible prices depending on the inventory level at the time when a customer arrives (i.e., the decision point). The system manager must determine the production level and selling price at each decision point. We first show that the optimal production and pricing policy is a threshold control, which is characterized by three threshold parameters under both the long-run discounted profit and long-run average profit criteria. We then establish the structural relationships among the three threshold parameters that production is off when inventory is above the threshold, and that the optimal selling price should be low when inventory is above the threshold under the scenario where the machine is down or up. Finally we provide some numerical examples to illustrate the analytical results and gain additional insights.     

Dynamic pricing of inventory with cancellation demand

When an inventory item has such a limited selling period that only a single supply order can be placed to satisfy future demand, a decision-maker must determine the quantity of the order to meet future demand and how to price this stock. Although this problem has received considerable attention, related investigations typically view the demand and selling price as exogenous parameters and assume that customers cannot cancel an order or return the product after purchasing the item. Pricing is, however, an important pervasive marketing vehicle that affects demand, and customers indeed cancel or return their orders after placing them. The newsboy problem is extended here so that demand is price-dependent and customers may cancel their orders. This paper seeks to develop decision rules to maximize the total expected profit over a given planning period. Analysis results demonstrate the feasibility of applying the order-up-to structure to yield the order quantity.     

Speculative production and anticipative reservation of reactive capacity by a multi-product newsvendor

In this paper the optimal sourcing decisions of a multi-product newsvendor prior to the selling season of the products are studied. To satisfy the uncertain demands, the newsvendor can either utilize speculative production, or anticipatively reserve capacity. During the selling season when demand has become known, the newsvendor can utilize its reserved capacity and reactively satisfy demand uncovered by its speculative production. For the case where capacity for speculative production may be limited, but potential reservation of reactive capacity is unlimited two capacity reservation settings are analyzed and compared. In the first one capacity for each product has to be reserved separately, while in the second setting one joint capacity reservation for all products is permitted which can then be allocated to the different products optimally during the selling season. For the case of separate individual reservations the optimal strategies are analytically derived and structural insights concerning their existence are presented. As the model allowing for joint reservation can not be tackled analytically in general an approximation based on an LP formulation is used. Through a numerical example insights on the value of the increased flexibility induced by joint reservation, the cost-premium acceptable for joint reservation and the relative levels of capacity reservation in the two settings are given.     

Capacitated procurement planning with price-sensitive demand and general concave-revenue functions

This paper develops effective solution methods for discrete-time, finite-horizon procurement planning problems with economies of scale in procurement, price-sensitive demand, and time-invariant procurement capacities. Our models consider general concave-revenue functions in each time period, and seek to maximize total revenue less procurement and inventory holding costs. We consider the case in which prices may vary dynamically, as well the important practical case in which a constant price is required during the planning horizon. Under mild conditions on the revenue function properties, we provide polynomial-time solution methods for this problem class. The structural properties of optimal solutions that lead to efficient solution methods also serve to sharpen intuition regarding optimal demand management strategies in complex planning situations.     

具有定价和开机成本的生产-库存系统的最优(s,d,S)策略

考虑一个具有有限容量和开机成本的连续盘点生产-库存系统, 其控制策略为(s,d,S)策略. 未被满足的需求都会丢失. 当机器处于关闭状态时,库存产品可以两个不同的价格进行销售. 当机器处于开机状态时,库存只能以较高的价格进行销售. 研究了如何发现该系统下的最优(s,d,S)策略,并开发了用于计算最优控制参数的有效算法.     

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。