Innovative Applications of O.R.
Modeling a multi-attribute utility newsvendor with partial backlogging

https://doi.org/10.1016/j.ejor.2012.02.027Get rights and content

Abstract

Traditional newsvendor models usually focus on single profit maximization or cost minimization approaches. However, making a monetary estimate of the consequences of lost sales as a result of shortages is often a difficult task for many practitioners. Besides, there is still a lack of an explicit account of decision-making judgments on the multiple consequences of making decisions with regard to order quantities. In order to deal with this problem, this paper presents a multi-attribute utility model for the newsvendor problem with regard to profit, the impacts of service level on corporate image and on customers’ goodwill, and the impact on the environment arising from the disposal of unsold products. Demand is partially backlogged according to a decreasing exponential function of the waiting time. The fundamental principles and limitations related to the application of the model built are also discussed.

Highlights

► This paper presents a multiple criteria model for the newsvendor problem. ► The model considers profit, impacts of service level and environmental issues. ► This newsvendor model contributes to overcoming limitations of previous models. ► The model explores a wider decision-making point-of-view. ► Results indicate a better balance on profit, service level and environmental issues.

Introduction

Inventory management in complex environments is a process that generally presents direct impacts on the total costs and service level of a company (and of the supply chain). This problem, and the entire supply chain management under a more comprehensive point-of-view, has been studied by means of Operations Research techniques for many years. In order to explore factors such as synergy, conflicts of interest, and the effects of trade-off on inventory holding costs, shortage costs and service level, simpler Economic Order Quantity models (Hillier and Lieberman, 2005) have evolved into more complex models which address uncertainty, multiple objectives, coordination, incomplete and asymmetric information and so on (Agrell, 1995, Benton and Park, 1996, Weng and McClurg, 2003, Zhou et al., 2004, Sucky, 2006, Jaber and Goyal, 2008).

A reference problem within the context of inventory/ordering management is the classical analysis on how much to order/produce for a single selling period, in order to minimize costs or to maximize profit, by taking into consideration that demand is uncertain and that the stocks remaining after that period must be discarded or sold for a given bargain price (Kumar, 1988). This problem is widely known as the newsvendor problem or single-period problem (Hillier and Lieberman, 2005). As Keren and Pliskin (2006) stress, since its introduction by Whitin (1955), this problem has been the subject of numerous research studies, which have developed several extensions to its classical formulation.

Even though it has been widely studied, newsvendor models remain an important research topic and a useful theoretical tool for analyzing ordering policies as well as supply chain coordination initiatives (Zhou and Wang, 2009, Hau, 2010). The management of newsvendor-type products is a field of study that attracts researchers and practitioners, especially those with regard to highly perishable items or environments in which technology develops quickly, in both of which life cycles get shorter and shorter. For these sorts of products, the shorter the life cycle, the more similar to typical single-period items inventory behavior becomes (Khouja, 1999, Zhou and Wang, 2009).

Several models and extensions for the newsvendor problem are found in the literature. In a reference work, Khouja (1999) presents a taxonomy of a wide range of contributions, which include: extensions for objectives other than profit (such as maximizing the probability of achieving a target profit); extensions for different pricing and discounting policies; extensions for different utility functions; and extensions for more complex logistic frameworks, such as multi-product, multi-stage and multi-location models.

One of the trends in the studies of newsvendor/inventory management is to relax the premise of complete lost sales after a stock-out, and to consider the possibility that part of, or even all the demand, should be backlogged. As San José et al. (2006) point out, many researchers have paid attention to demand backlogging on inventory management models. Zhou et al. (2004) present an inventory model, where demand is a deterministic function of time. The model assumes multiple and successive periods, and the replenishment cost per order is considered as dependent on lot size. Demand is partially backlogged, and it is modeled as a deterministic function of the waiting time until the next replenishment. Weng (2004) develops a newsvendor model to analyze the coordination effects of decisions on order size taken between a manufacturer and a reseller in order to maximize their joint expected profit. In his model, Weng addresses two scenarios of demand: in the first, all excess demand is backlogged and must be satisfied; in the second, it is assumed that all customers with unsatisfied orders are willing to wait, but the reseller may choose not to meet any additional demand if the resulting profit is negative. For that situation, a threshold value q for the additional demand is set in order to start a second replenishment order. Zhou and Wang (2009) extend the coordination analysis of Weng’s work to a partial backlogging situation. The fraction of the unfilled demand that is willing to wait for the next replenishment is assumed to be known and fixed.

San José et al. (2006) present an inventory model, where excess demand is partially backlogged according to a negative exponential function of the time that customers will wait until their orders are fulfilled. While Zhou et al. (2004) address the minimization of a total cost function, the problem in San José et al. (2006) was set in order to maximize the total profit per cycle. Total costs in their model comprise ordering costs, holding costs, backorder costs and costs of lost sales, which are assumed to be known parameters.

Lodree (2007) presented a two-level supply chain within a newsvendor framework, incorporating within shortages a combination of backorders and lost sales, and also allowing the buyer to start emergency replenishments. The problem was addressed with the objective of minimizing the total expected costs of the supply chain, within which a cost of “lost contract” between buyer and supplier, depending on the magnitude of shortage, was also incorporated. Later, Lodree et al. (2008) presented a newsvendor model with a similar objective function. However, they considered a complete backlogging situation for which backorder costs were proportional to the lead-time for the emergency fulfillment, and the emergency replenishment process was characterized by multiple shipments.

Lee and Lodree (2010) studied the impatience behaviors of different customers in newsvendor contexts under a condition of time-dependent partial backlogging. The problem was formulated in order to minimize total expected costs, and different backorder rate functions were used to classify customers as impatient, neutral or patient regarding their willingness to wait for an emergency replenishment after a shortage.

Even if the backlogging phenomenon is not taken into consideration, and hence backorder costs are not calculated, several authors recognize that estimating cost parameters in newsvendor models may be a difficult task for managers, especially for those costs related to shortages (Moon and Choi, 1994, Agrell, 1995, Petrovic et al., 1996, Chen and Chuang, 2000, Abad, 2001). As Moon and Choi (1994) stated, one basic reason is that stock-out costs are related to intangible components including impacts on customers’ goodwill, and delays in other parts of the system. Agrell (1995) criticized the approach of minimizing total costs, and pointed out that a consistent monetary transformation for a mono-objective treatment of the inventory problem requires an adequate knowledge of the costs involved in stock-outs, which is hard to find in practice.

In order to deal with the absence of precise information on the cost parameters of shortages, Moon and Choi (1994) presented a distribution-free stochastic inventory model with a constraint on the service level, while Chen and Chuang (2000) developed a newsvendor model with a shortage level constraint in order to determine the optimal order quantity and optimal timing of an order. Agrell (1995) proposed an inventory model formulated with three objectives: minimization of total expected (regular) costs, minimization of expected number of stock-outs, and minimization of expected size of shortages. Petrovic et al. (1996) presented two fuzzy newsvendor models, where inventory costs, as well as demand, were represented by fuzzy sets. The optimal order size for a fixed period was set to minimize the total “possible” cost. Abad (2001) approached the newsvendor problem with backlogging in order to determine an optimal price and an optimal order quantity, but backlogging costs and lost sales costs were ignored on that model because of the difficulty of estimating these parameters in practice. The problem was set to maximize profit per period, and the backlogged demand was a strictly decreasing function of the waiting time.

Among most newsvendor models proposed hitherto, only single profit maximization or cost minimization approaches have been considered. Besides the difficulties in estimating the costs of lost sales as well as other monetary parameters, we still note a lack of an explicit account of decision-making assessments and preference judgments on multiple consequences that making decisions on one-period order quantities may bring. Current models do not directly explore decision-maker’s preferences and value judgments on non-monetary factors such as customer goodwill, corporate image under stock-out, pollution, and possible environmental impacts due to the disposal of unsold products, for example. Also, they do not take into consideration value tradeoffs between these factors and under-stock/over-stock costs. Instead, non-monetary outcomes are often assumed to be implicitly considered within monetary parameters. As previously stated, these parameters are usually very hard to estimate, so such outcomes are converted into constraints or into a mono-attribute utility scale.

This paper differs from the existing literature on newsvendor topics by proposing a multi-attribute utility model for the newsvendor problem based on the methodological background of Multi-Attribute Utility Theory – MAUT (Keeney and Raiffa, 1976). A model was developed to explicitly represent the decision-maker’s preferences and behavior regarding risk caused by the uncertainty of demand. An exponential backlogging rate is considered. Within a wider decision-making framework, the newsvendor problem is addressed as a multiple criteria decision problem concerning other factors besides profit maximization or cost minimization. In this model, impacts of lost sales on corporate image and on customers’ goodwill, as well as environmental impacts that may appear from scrapping unsold products, are also assessed in the decision-making process of choosing an order quantity Q. This multi-attribute model also helps to overcome difficulties involving monetary transformations and assessments of costs due to the consequences of shortages in complex contexts, where the use of economic analysis alone may induce a myopic representation of the aspects involved.

This paper is organized as follows: In Section 2, some basic features of the newsvendor problem are presented; the backlogging condition is discussed; and a multi-attribute decision model is proposed. An illustrative numerical experiment is presented in Section 3, and a sensitivity analysis is discussed. Then, a discussion on results, in Section 4, highlights some aspects related to the background of the model and its application. Lastly, conclusions are drawn and remarks made on this research theme in Section 5.

Section snippets

The multi-attribute newsvendor model

This section introduces a multi-attribute expected-utility model for the newsvendor problem. We consider a newsvendor decision-maker (DM) who sells a short life-cycle product during a selling season. At the beginning of that season, the newsvendor faces the choice of ordering a number of Q products for sale. The entire order quantity is received at the same time. Demand is stochastic, and this choice must be made before demand realization x, which is known at the end of the period.

The DM must

Numerical experiment and sensitivity analysis

In this section, we present a numerical study in order to illustrate this multi-attribute approach for the newsvendor problem. It is generally not feasible to obtain a closed-form expression for the compromise solution of this model; therefore, a computer tool has been used to find the results.

Consider a retail company which is going to trade a computer product with a short-life-cycle, during a selling season, in a newsvendor context. After this period, it is expected that this product will be

Discussion of model and results

This section presents a discussion of fundamental principles related to the model built, connected to its application. Also, some limitations are discussed pointing out alternatives of implementation.

A successful use of this multi-attribute newsvendor model relies on the assumption that the distribution of demand is known, and that parameter α from the backlogging function may be obtained. Regarding the value of α, for newsvendor-type products with previous sale seasons, historical records of

Conclusions

A multi-attribute utility approach for the newsvendor problem or single-period problem has been presented in this paper. We discussed that traditional newsvendor models are mostly focused on profit maximization or cost minimization objectives, and therefore they face two main limitations: first, shortage costs due to lost sales and due to loss of customers’ goodwill are difficult for managers to estimate in practical situations. Secondly, single-objective models do not adequately address

Acknowledgments

The authors’ research is partially supported by the Brazilian Research Council (CNPq).

The authors would like to acknowledge the Editor and the anonymous reviewers for considering the potential of a previous version of this paper and for encouraging its improvement, with the insightful critique and valuable suggestions.

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