Production, Manufacturing and Logistics
The multi-item newsvendor model with cross-selling and the solution when demand is jointly normally distributed

https://doi.org/10.1016/j.ejor.2014.01.006Get rights and content

Highlights

  • We model the multi-item newsvendor problem considering cross-selling effect.

  • We derive the first-order optimality conditions for the centralized model.

  • The pure-strategy Nash equilibrium conditions of the competitive model are developed.

  • Develop efficient algorithms to solve the centralized and the competitive models.

  • Provide insights into the influences of cross-selling on newsvendor decision.

Abstract

Products are often demanded in tandem because of the cross-selling effect. The demand for an item can increase if sales of its cross-selling-associated items are achieved or decrease when the associated items are out of stock, resulting in lost sales. Therefore, a joint inventory policy should be pursued in a cross-selling system. This paper introduces customer-driven cross-selling into centralized and competitive newsvendor (NV) models by representing an item’s effective demand as a function of other items’ order quantities. We derive first-order optimality conditions for the centralized model in addition to pure-strategy Nash equilibrium conditions and uniqueness conditions of the equilibria for the competitive model. We further develop gradient-based (GB) and iteration-based (IB) algorithms to solve the centralized and competitive models, respectively. A computational study verifies the effectiveness of the proposed algorithms. The computational results show that a larger cross-selling effect leads to a larger order quantity in a centralized NV model but a smaller order quantity in a competitive NV model, and a larger positive correlation between items’ demands leads to higher profits with smaller order quantities in both models. Moreover, NVs will order more items if the demand variance is greater, however resulting in lower profits. In a competitive situation, one will prefer smaller order quantities than in a centralized decision situation.

Introduction

The classical newsvendor (NV) model is a single-period problem that maximizes the expected profit under stochastic demand by determining a product’s optimal order quantity. The NV model typically assumes that inventory remaining at the end of a planning period can be disposed of at a price lower than the unit purchasing cost. In recent decades, the academic literature on single-period inventory management has been replete with newsvendor problems, although new models continue to emerge. A comprehensive review of newsvendor research prior to 1999 is available in Khouja (1999), and a recent review that extends the work of Khouja (1999) by considering several specific extensions can be found in Qin, Wang, Vakharia, Chen, and Seref (2011).

There are two types of models that extend the classical NV model to analyze multi-item inventory systems. One is a NV model with constraints, e.g., storage, budget, or weight constraints, integrating multiple items together for a joint decision (Abdel-Malek and Montanari, 2005a, Abdel-Malek and Montanari, 2005b, Chen and Chen, 2010, Khouja and Mehrez, 1996). The other type of model considers the demand relationships between items. One of the most popular topics is the newsvendor problem with substitutions. However, cross-selling was rarely introduced into newsvendor formulations, even though it creates an important mutual relationship between items. This paper investigates the newsvendor problem by introducing cross-selling into multi-item NV models. As an initial comparison, we briefly review the NV model with substitutions.

An unsatisfied customer may select another product that is similar to the original choice in function, color, style, size, or price as a substitute. Assuming that each item is a substitute for another item, Parlar and Goyal (1984) proposed a two-item newsboy model. Khouja, Mehrez, and Rabinowitz (1996) developed the lower and upper bounds of order quantities for a two-item NV model and presented a numerical algorithm employing a Monte Carlo simulation. Rajaram and Tang (2001) proposed a service rate heuristic for solving the centralized NV model with normally distributed demand for substitutable items. Parlar (1988) extended the problem to consider the competitive situation and proposed a two-item competitive model with substitutions. Lippman and McCardle (1997) further developed the multi-item competitive NV model with substitutions and proved the existence of the pure-strategy Nash equilibrium. Netessine and Rudi (2003) derived first-order conditions for both the multi-item centralized and competitive NV models with demand substitutions and proved the uniqueness of the Nash equilibrium. Huang, Zhou, and Zhao (2011) reinvestigated the competitive NV model with demand substitutions by considering the shortage penalty cost and proposed an iterative algorithm. For more extensive literature related to this topic, please see Bitran and Dasu, 1992, Bassok et al., 1999, Smith and Agrawal, 2000, Mahajan and van Ryzin, 2001, and Dutta and Chakraborty (2010).

As for the mutual effect among the demands for different items, the ability to substitute is obviously a promoting factor for the substitutable item when original choices are out of stock. However, the existence of cross-selling may produce opposing effects in contrast to the substitution. Cross-selling implies that “a customer who has purchased a particular product may also be willing to purchase a related product” (Shen & Su, 2007) or that some items are always purchased together due to their unknown interior associations (Anand, Hughes, Bell, & Patrick, 1997). If one of a set of jointly demanded items is out of stock, the sales pattern for the associated items will be altered and demand for the other item in this set may decrease as a result (Wong et al., 2005, Zhang, 2012, Zhang et al., 2011). Cross-selling can be seen as a type of complementary pattern that is the distinct opposite of substitution.

Although the cross-selling effect may result in sales losses when some items are out of stock, one can also take advantage of the cross-selling effect to create new sales opportunities (Kamakura et al., 1991, Knott et al., 2002, Kotler and Connor, 1977). In this case, the cross-selling is driven by vendors. Many studies on cross-selling have emerged in subject areas such as sales marketing, financial services, customer behavior, and data mining. One can refer to Yasar and Andy, 2002, Kamakura et al., 2003, Ansell et al., 2007, Salazar et al., 2007, Chuang and Lin, 2008, Günes et al., 2010, and Li, Sun, and Montgomery (2011) for the related topics. However, academic research on cross-selling in operations management does not account for a substantial proportion of the literature.

Akşin and Harker (1999) explored the trade-offs between service and sales in the operation of call centers in retail banking. They indicate that “cross-selling can have detrimental effects on customer service due to the additional load it creates on the system”. Güneş and Akşin (2004) further studied the worth of undertaking a value-creation initiative such as cross-selling. Byers and So (2007) proposed a queuing model for optimizing control policies for cross-selling in telephone service centers, where the probability of a successful cross-sale for the existing customer is uniformly distributed. Gurvich, Armony, and Maglaras (2009) investigated a call center with cross-selling capability and studied the operational decisions for staffing, call routing, and cross-selling according to various forms of customer segmentation. They assumed that the probability of a customer accepting a cross-selling offer was an arbitrary non-increasing continuous function of the customer wait time. Aksin, Armony, and Mehrotra (2007) administered a state-of-the-art survey and reported that cross-selling activities were a prevalent practice in call center operations management.

Some studies proposed selling bundled items as multi-product packets (packaged goods) because of negative correlations in demand and complementary consumption with the same outcome as cross-selling. Ernst and Kouvelis (1999) investigated a business practice where two items could not directly substitute for each other, while packaged goods could substitute for individual items and vice versa. Netessine, Savin, and Xiao (2006) researched the revenue management problem of cross-selling products or services in an e-commerce setting, assuming that the probability of a customer buying a two-item packet was a function of the synthetic price of the two packaged items. However, these studies focused on bundled goods being substitutable and did not correlate the cross-selling effect among the individual items.

Research on cross-selling is sparse in the existing literature on inventory management. Zhang et al. (2011) presented a deterministic two-item EOQ model with correlated demand caused by cross-selling, which was later extended to a case of multiple items (Zhang, 2012). These papers showed that lost sales of the major item can affect the demand for the minor item, wherein they introduced cross-selling into the EOQ model of inventory management.

Another relevant paper is the Netessine and Zhang (2005) investigation of the newsvendor problem with complementary items. In their paper, the demand for an item can be increased by the sale of other items, which is defined as complementarity. Thus, the effective demand for an item is equal to the sum of its original demand and the demand derived by the sales of its complementary items. Their model assumes that the original “bare” demand must be distinguishable from the derived demand. This is a rigorous condition that is not easily satisfied in the cross-selling environment in practice (this point will be examined in Section 2.1). This paper investigates NV models in the cross-selling environment without identifying original demand.

The paper contributes in the following aspects: (1) we investigate the centralized newsvendor problem with cross-selling and propose optimality conditions, bounds on the optimal solution, and a gradient-based (GB) algorithm for solving the model; (2) we propose bounds on the equilibrium order quantities for a competitive NV model and an iteration-based (IB) algorithm; and (3) we conduct numerical experiments to establish how the parameters affect newsvendor decisions in the cross-selling environment.

The structure of this paper is as follows. Section 2 formulates the centralized NV model with cross-selling, develops the first-order conditions for optimality, and proposes the lower and upper bounds of the optimal order quantities. Section 3 presents the competitive version of the model, develops the first-order and uniqueness conditions for the Nash equilibrium, and establishes the lower and upper bounds of the equilibrium order quantities; theoretical comparisons between the two models are subsequently presented. Section 4 presents the gradient-based (GB) and iteration-based (IB) algorithms for the centralized and competitive models, respectively. Section 5 conducts a computational study through extensive numerical examples and provides some observations. Section 6 concludes the paper.

Section snippets

Notations and assumptions

Suppose that two items, i and j (i, j = 1,  , n), are associated with each other because of cross-selling. Let Di denote the total demand for item i with the probability function Pr(Di) and the probability density function f(Di) (we assume that Pr(Di) and f(Di) are continuous). Total demand implies that no items are out of stock over the entire planning horizon, where the demand for an item will not be decreased because of losing the cross-sales associated with its cross-selling items. In a

Model and equilibrium analysis

Consider a competitive situation where each of n items is sold by a given firm, forming a non-cooperative inventory game. Suppose that each firm has complete information on cross-selling among all the items. Therefore, for a newsvendor that makes the decision individually, the expected profit can be easily written as πi=EpiminQi,Die-ciQi+siQi-Die+, i.e.,πi=EuiDie-uiDie-Qi+-oiQi-Die+.

According to Leibniz’s rule, we have πi/Qi=ui-(ui+oi)PrDie<Qi and 2πi/Qi2=-(ui+oi)fDie(Qi)<0. Thus, πi is

Algorithms

Deriving a closed-form solution for a multi-item newsvendor problem with arbitrary distributions of demand remains a challenge because analytically solving the first-order conditions is complex. This paper employs an approximation of effective demand to define the probability of Die and applies this method to the model with jointly normally distributed demand to develop efficient algorithms for the solution.

Outline of numerical examples

Let jirji<1 and ijrji<1 hold for all i and j, ensuring that the equilibrium is unique. We then generate two-item, three-item and four-item problems with asymmetric parameters, as well as two-item and three-item problems with symmetric parameters. In the two-item symmetric examples, the demand is correlated to each other and we let the demand correlation coefficient change from −1 to 1. For the other examples, the demand is independent. All the numerical examples are detailed below.

Conclusions

There have been numerous NV models based on multi-item inventory problems with substitutions. While cross-selling is a frequent practice, it has rarely been introduced into NV models as a complementary relationship among items. This paper developed multi-item centralized and competitive NV models for a newsvendor problem considering the cross-selling effect. The first-order optimality conditions of the centralized model and the equilibrium conditions of the competitive model show that the

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grants Nos. 71271010, 71031001, 71271089, 71071059, and NCET Program of China (NCET-12-0191). We wish to thank the two anonymous reviewers for their constructive comments.

References (47)

  • A. Knott et al.

    Next-product-to-buy models for cross-selling applications

    Journal of Interactive Marketing

    (2002)
  • Y. Qin et al.

    The newsvendor problem: Review and directions for future research

    European Journal of Operational Research

    (2011)
  • K. Rajaram et al.

    The impact of product substitution on retail merchandising

    European Journal of Operational Research

    (2001)
  • R.-Q. Zhang

    An extension of partial backordering EOQ with correlated demand caused by cross-selling considering multiple minor items

    European Journal of Operational Research

    (2012)
  • R.-Q. Zhang et al.

    Deterministic EOQ with partial backordering and correlated demand caused by cross-selling

    European Journal of Operational Research

    (2011)
  • N. Agrawal et al.

    Estimating negative binomial demand for retail inventory management with unobservable lost sales

    Naval Research Logistics

    (1996)
  • O.Z. Aksin et al.

    The modern call center: A multi-disciplinary perspective on operations management research

    Production and Operations Management

    (2007)
  • O.Z. Akşin et al.

    To sell or not to sell: Determining the trade-offs between service and sales in retail banking phone centers

    Journal of Service Research

    (1999)
  • Anand, S. S., Hughes, J. G., Bell, D. A., & Patrick, A. R. (1997). Tackling the cross-sales problem using data mining....
  • J. Ansell et al.

    Identifying cross-selling opportunities, using lifestyle segmentation and survival analysis

    Marketing Intelligence and Planning

    (2007)
  • Y. Bassok et al.

    Single-period multiproduct inventory models with substitution

    Operations Research

    (1999)
  • G.R. Bitran et al.

    Ordering policies in an environment of stochastic yields and substitutable demands

    Operations Research

    (1992)
  • R.E. Byers et al.

    A mathematical model for evaluating cross sales policies in telephone service centers

    Manufacturing and Service Operations Management

    (2007)
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