Quantity Decisions Research Articles

Allocating Cost of Service to Customers in Inventory Routing

Vendor-managed inventory (VMI) replenishment is a collaboration between a supplier and its customers, where the supplier is responsible for managing the customers' inventory levels. In the VMI setting we consider, the supplier exploits synergies between customers, e.g., their locations, usage rates, and storage capacities, to reduce distribution costs. Due to the intricate interactions between customers, calculating a fair cost-to-serve for each customer is a daunting task. However, cost-to-serve information is useful when marketing to new customers or when revisiting routing and delivery quantity decisions. We design mechanisms for this cost allocation problem and determine their characteristics both analytically and computationally.

Price competition and coordination of dual-channel supply chains

The dual-channel is a more prevalent channel structure in current marketing systems, it including the traditional retail channel and the direct channel through web-based sales. And price competition and channel conflict are inevitable between retail channel and the manufacturer's direct sale channel. This paper develops pricing decisions and coordination models in a one-manufacturer-one retailer dual-channel supply chain. We fully characterise partners' optimal decisions by adopting a customer's utility function. We discuss the optimal decisions in decentralised dual-channel supply chain including market-like setting without revenue sharing contract and contract-like setting, and obtain the optimal prices and quantity decisions in centralised and decentralised dual-channel supply chain. Moreover, we propose the feasible revenue-sharing contracts to coordinate the dual-channel supply chain, and obtain the optimal parameters of the revenue sharing contract. It is always beneficial for both the partners and system to adopt a revenue sharing contract. Finally, the proposed models are further analysed through numerical examples.

How can we improve the performance of supply chain contracts? An experimental study

Although optimal forms of supply chain contracts have been widely studied in the literature, it has also been observed that decision makers fail to make optimal decisions in these contract setups. In this research, we propose different approaches to improve the performance of supply chain contracts in practice. We consider revenue sharing and buyback contracts between a rational supplier and a retailer who, unlike the supplier, is susceptible to decision errors. We propose five approaches to improve the retailer’s decisions which are in response to contract terms offered by the supplier. Through laboratory experiments, we examine the effectiveness of each approach. Among the proposed approaches, we observe that offering free items can bring the retailer’s effective order quantity close to the optimal level. We also observe that the retailer’s learning trend can be improved by providing him with collective feedbacks on the profits associated with his decisions. ► We propose five approaches to improve order quantity decisions in supply chain contracts. ► We show the coordinating form of revenue sharing contracts designed for a risk-averse retailer. ► We show coordinating forms of supply chain contracts when free items offered by the supplier. ► We examine the effectiveness of each proposed approach through laboratory experiments. ► Offering free items and providing collective feedback can improve the performance of contracts.

Sequential price and quantity decisions under supply and demand risks

Firms need to deal with not only risks from stochastic demand but also risks from supply side. The supply side risk may be due to parts/service outsourcing, third party logistics, or random yield in production processes. In this paper, we study how firms sequentially make price and quantity decisions under these two risks. The first question we try to answer is how these two risks affect the decisions and profits of the firm. We find that increased supply risk usually causes increased quantity/stocking decision, however, there exists a threshold level of supply risk above which the firm reduces quantity/stocking amount as supply risk increases. This observation may be used in a supply chain setting, where reduction of the supply risk can cause higher delivered quantity and improve supply chain performance. This observation also provides support and insights on prioritizing the risk reduction efforts from marketing and operations to achieve better coordination. At the same time, reduction of the risks help not only firms but also consumers as the optimal price decreases. To further improve decision making process under both uncertainties, we study the impact from information revelation and postponement of decisions. We compare results from different sequential decision making cases. As illustrated in the paper, firms gain competing advantage when decision postponement is available and this advantage becomes further significant as the risks increase. Our numerical examples also indicate that price postponement strategy is usually preferred but the relative profit difference between price postponement and quantity postponement become smaller as consumers become more sensitive to the price.

Impact of loss aversion on the newsvendor game with product substitution

Abstract This paper studies a newsvendor game in which two substitutable products are sold by two different retailers (newsvendors) with loss-averse preferences. Each loss-averse retailer facing stochastic customer demand and deterministic substitution rate will make an order quantity decision to maximize his expected utility. Since product substitution causes two retailers to make decisions in a competitive environment, game theory is used to find the retailers' optimal order quantities. It is shown that under certain conditions, there exists a unique Nash equilibrium in the newsvendor game. Under a symmetry assumption, each retailer's equilibrium order quantity is decreasing in the loss aversion coefficient and increasing in the substitution rate. Further, if the effect of loss aversion on the order quantity is strong enough to dominate the effect of competition, the total inventory level of a decentralized supply chain will be lower than that of a centralized supply chain. Numerical experiments are conducted to illustrate our results.

Pricing and Production Quantity Decisions in a Supply Chain with Demand-Stimulating Service under Demand Disruptions

Pricing and Production Quantity Decisions in a Supply Chain with Demand-Stimulating Service under Demand Disruptions

A parsimonious multivariate poisson model for market basket analysis

Intensified research on multivariate Poisson models offers new opportunities for the analysis of purchase quantities in market basket data. The investigation of positive or negative correlations in quantity decisions among product categories facilitates a deeper understanding of consumer purchase behavior. The applied multivariate log-normal Poisson model introduces interdependencies between categories with multivariate normal-distributed latent effects by means of a covariance matrix. As the size of this covariance matrix depends on the number of categories in the model, its ation may become tedious. Furthermore, we assume that quantity decisions do not interact for all pairs of categories. That is why we propose to use covariance selection to derive a parsimonious representation of the correlation structure. For two market basket data sets, we show that the vast majority of off-diagonal elements in the covariance matrix are irrelevant. For a data set with product categories, the model with a partly restricted covariance matrix achieves a better fit to the holdout data than the model with full covariance matrix. For a data set with subcategories of the broader category beverage, the proposed model with restricted covariance outperforms the model with full covariance matrix even on the calibration data. We conclude that interactions of quantity decisions are overall the exception, even for complements-in-use.

A Multicategory Model of Consumers' Purchase Incidence, Quantity, and Brand Choice Decisions: Methodological Issues and Implications on Promotional Decisions

The authors propose a multicategory model of consumers' purchase incidence, quantity, and brand choice decisions. The model specification allows for cross-category promotion effects in both components of the primary demand (incidence and quantity decisions) and uses a flexible functional form of consumer's utility to accurately measure those cross-category effects. To demonstrate the importance of the methodology, the authors investigate two issues of relevance to retailers, namely, how retailers should (1) allocate promotional expenditures across brands in a category and (2) coordinate timing of promotions of brands across categories. The authors estimate the proposed model using consumers' purchases in pasta sauce and pasta categories. The results reveal that using restrictive functional forms of utilities or ignoring cross-category effects in incidence and quantity decisions leads to incorrect assessments on relative allocation of promotional expenditures across brands. Furthermore, retailers are better off contemporaneously promoting brands across the two categories than promoting them in different periods, and ignoring cross-category effects in quantity decisions leads to the opposite inference, namely, that retailers are better off promoting brands across the two categories in different periods.

Price, rebate and order quantity decisions in a newsvendor framework with rebate-dependent recapture of lost sales

Abstract This paper analyses a single-period decision of a retailer facing uncertain and price dependent demand. The typical modeling of the problem in a newsvendor framework assumes the unfulfilled demand to be lost once and for all. However, in reality, there may be an opportunity to backlog the lost sales, by offering some incentive for waiting. Nevertheless, the retailer's procurement price may be higher, due to the likely cost increase of the emergency purchase. Further, not all the customers that could not buy in the first instance may avail the rebate offer and buy. The backlog fill rate is modeled as a function of the proportion of the rebate to the price. Then the retailer has to decide ahead of the realization of the demand the quantity to be ordered, the price and the rebate to be offered for backlogged sales that will maximize its expected profit. Numerical examples are presented to highlight model sensitivities to parametric changes.

Quantity decisions in a supply chain with early returns remanufacturing

Consider a supply chain where the retailer faces a stochastic demand and orders from the supplier, while the supplier manufactures new products and also remanufactures early returns to meet the order. The order and manufacturing quantity decisions are studied under three decentralised cases: (1) the Stackelberg case, where the manufacturing quantity is determined by the supplier after realising the order quantity from the retailer; (2) the Nash case, where the manufacturing and order quantities are determined simultaneously; (3) the inaccessible return information case, where the retailer does not have the distribution information of upstream returns. We find that in the Stakelberg case the order and manufacturing quantities are larger than in the inaccessible return information case, and the profits for the supplier and retailer are also higher. In contrast, the quantities and profits in the Nash case are the lowest of the three cases when the returns are negatively correlated to the demand. The centralised system is also explored by solving a two-step dynamic program. Computational results are reported to show the effects of system parameters.

Investments in Ski Areas: Effects of Compensation Payments in a Complementary Value Network

This study focuses on a complementary value network consisting of a snow sports company operating in a ski resort and accommodation businesses located in that resort. If the snow sports company increases the resort's attractiveness by investing in new infrastructure, such as ski lifts or snowmaking facilities, its investment has positive external effects on the revenue of local accommodation businesses. If the cost of these external effects is not compensated by the accommodation businesses, this situation typically creates an underinvestment problem at the expense of the snow sports company. In this context, the authors analyse the extent to which revenue-based and quantity-based compensation payments can mitigate the problem. First, they identify conditions under which such payments increase the profits of both the snow sports company and the accommodation businesses. They further show that a revenue-based compensation payment dominates a quantity-based compensation payment with respect to company profit maximization, because it distorts neither the price nor the quantity decisions of the accommodation businesses.

Negotiating Transfer Prices

We consider a team-investment setting in which transfer prices between two divisions are negotiable. Investments are made independently and simultaneously after the bargaining stage, i.e. with a given transfer price ‘on the table’. Both divisions’ investments jointly affect the sales price of the final product and total revenue. We analyze two transfer-pricing schemes and their corresponding bargaining problems. Both bargaining settings exhibit non-transferable utility because the transfer price not only allocates corporate profit but also affects corporate profit through the incentives it creates for the divisions’ investment and quantity decisions. In particular, we discuss how concepts from bargaining theory can be use used to determine a ‘fair’ agreement concerning the transfer price.

Signaling to Partially Informed Investors in the Newsvendor Model

We investigate a phenomenon in which firms may attempt to influence their market valuation by choosing an inventory stocking quantity which does not optimize expected profits. We employ the newsvendor model within a signaling game to examine a relatively common scenario in which the firm's equity holder has incomplete information concerning the demand for the firm's product. We apply a perfect Bayesian equilibrium solution and identify ranges of model parameters where the firm's stocking quantity decision does not maximize expected profits. This includes instances in which a firm facing high demand chooses a lower stocking quantity than that which would optimize expected profits and a firm facing low demand chooses a higher stocking quantity than that which would optimize expected profits. This result contrasts with prior research, which has shown that when equity holders have incomplete information about the quality of the firm's opportunities, high quality firms will consistently overinvest and low quality firms will invest to optimize expected profits. We show an extreme example of this behavior in which a high demand firm chooses that stocking quantity which would have been optimal under complete information for a low demand firm.

Information in Cournot: Signaling with incomplete control

We embed signaling in the classical Cournot model in which several firms sell a homogeneous good. The quality is known to all the firms, but only to some buyers. The quantity-setting firms can manipulate the price to signal quality. Because there is only one price in a market for a homogeneous good, each firm incompletely controls the price-signal through the quantity decision. We characterize the unique signaling Cournot equilibrium in which the price signals quality to the uninformed buyers. We then compare the signaling Cournot equilibrium with the full-information Cournot equilibrium. Signaling is shown to increase the equilibrium price. Moreover, under certain conditions regarding the composition of buyers, the number of firms, and the distribution of costs across firms, the effects of signaling and market externality cancel each other. In other words, the profits under signaling Cournot equal the profits of a cartel in a full-information environment.

TECHNICAL NOTE—Decentralized Inventory Sharing with Asymmetric Information

We study the information asymmetry issues in a decentralized inventory-sharing system consisting of a manufacturer and two independent retailers, who privately hold demand information, noncooperatively place their orders, but cooperatively share inventories with each other. We find that although the manufacturer needs retailers' mean demand and standard deviation for her wholesale price decision, each retailer only needs to know the other retailer's demand standard deviation for his order quantity decision. However, an incentive compatibility analysis shows that retailers have incentives to share their demand information untruthfully. Although a truth-inducing scheme can be developed for a system with symmetric retailers who share information between themselves, no such scheme can be developed to ensure truth-telling to the manufacturer. Further, we develop a coordination mechanism (CIS) for the decentralized inventory-sharing system, considering information asymmetry. We show that CIS coordinates the manufacturer-retailers system and leads to an all-win situation under complete information. More importantly, CIS minimizes the value of information such that each party can obtain expected profits very close to their first-best profits even under asymmetric information and hence indirectly solves the information asymmetry problem. To our knowledge, this work is the first to study decentralized inventory sharing and its coordination considering asymmetric information.

Coping with Gray Markets: The Impact of Market Conditions and Product Characteristics

Gray markets, also known as parallel imports, are marketplaces for trading genuine products that are diverted from authorized distribution channels. They have created fierce competition for manufacturers in many industries and each year billions of dollars worth of products are traded in these markets. Using a game-theoretic model, we analyze the impact of parallel importation on a price-setting manufacturer that serves two markets with uncertain demand. We characterize the optimal joint price and quantity decisions of the manufacturer which determine whether the manufacturer should ignore, block, or allow parallel importation. We also show that parallel importation forces the manufacturer to reduce her price gap while demand uncertainty forces her to lower prices in both markets. Moreover, we observe that parallel importation may force the manufacturer to exit the low-profit market. Through extensive numerical experiments, we explore the impact of market conditions (size and price elasticity) and product characteristics (a fashion item or a commodity) on the manufacturer's reaction to parallel importation. In addition, we provide interesting insights about the value of strategic pricing for coping with gray markets versus the uniform pricing policy that has been adopted by some companies to eliminate gray markets.

Customer rebates and retailer incentives in the presence of competition and price discrimination

Promotions are important tools for matching supply and demand in many industries. In the United States automotive industry, promotions are frequently offered, which may be given directly to customers (rebates) or given to dealers (incentives) to stimulate demand. We analyze the performance of customer rebate and retailer incentive promotions under competition. We study a setting with two manufacturers making simultaneous pricing and promotion decisions, and with two price-discriminating retailers as Stackelberg followers making simultaneous order quantity decisions. In the benchmark case with no promotions, we characterize the equilibria in closed form. We find that retailer incentives can be used by manufacturers to simultaneously improve each of their profits but can potentially lead to lower retailer profits. When manufacturers use customer rebates, we show that a manufacturer is able to decrease the profit of her competitor while increasing her own profit, although she is also at risk for her competitor to use rebates in a similar fashion. Unlike the monopoly case where the manufacturers are always better off with retailer incentives, customer rebates can be more profitable under some cases in the presence of competition. Using numerical examples we generate insights on the manufacturers’ preference of promotions in different market settings.

“No More” leads to “Want More,” but “No Less” leads to “Want Less”: Consumers' counterfactual thinking when faced with quantity restriction discounts

AbstractThe current research examines the impact of point‐of‐purchase (POP) discounts on consumers' counterfactual thinking (CFT). Study 1 reveals that consumers tend to engage in upward CFT (what might have been better) rather than downward CFT (what might have been worse) in response to POP discounts. Study 2 shows that upward CFT depends on how the discount information is framed. A discount with a lower‐quantity restriction (e.g., “X % off if you buy at least Y items”) leads consumers to counterfactually wish to buy more, but a discount with an upper‐quantity restriction (“X % off – limit Y items per customer”) leads consumers to wish to buy less. Study participants in both conditions report they would buy the same POP‐suggested amount, but for completely opposite reasons. In Study 3, this convergence effect in purchase quantity disappears when the maximum and minimum restrictions are lifted, suggesting that quantity restrictions in POP discounts guide quantity decisions. Copyright © 2011 John Wiley & Sons, Ltd.

Ordering quantity decisions considering uncertainty in supply-chain logistics operations

This research seeks to determine the optimal order amount for a retailer given uncertainty in a supply-chain's logistics network due to unforeseeable disruption or various types of defects (e.g., shipping damage, missing parts and misplacing products). Mixture distribution models characterize problems from solitary failures and contingent events causing network to function ineffectively. The uncertainty in the number of good products successfully reaching distribution centers (DCs) and retail stores poses a challenge in deciding product-order amounts. Because the commonly used ordering plan developed for maximizing expected profits does not allow retailers to address concerns about contingencies; this research proposes two improved procedures with risk-averse characteristics towards low probability and high impact events. Several examples illustrate the impact of a DC's operation policies and model assumptions on a retailer's product-ordering plan and resulting sales profit.

The newsvendor problem: Review and directions for future research

In this paper, we review the contributions to date for analyzing the newsvendor problem. Our focus is on examining the specific extensions for analyzing this problem in the context of modeling customer demand, supplier costs, and the buyer risk profile. More specifically, we analyze the impact of market price, marketing effort, and stocking quantity on customer demand; how supplier prices can serve as a coordination mechanism in a supply chain setting; integrating alternative supplier pricing policies within the newsvendor framework; and how the buyer’s risk profile moderates the newsvendor order quantity decision. For each of these areas, we summarize the current literature and develop extensions. Finally, we also propose directions for future research.