Discrete Dynamics in Nature and Society

Volume 2016, Article ID 1907680, 14 pages

http://dx.doi.org/10.1155/2016/1907680

## Joint Inventory, Pricing, and Advertising Decisions with Surplus and Stockout Loss Aversions

^{1}Department of Information Management and Decision Sciences, School of Business Administration, Northeastern University, Shenyang 110167, China^{2}State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110819, China^{3}Department of Economics and Business Economics, School of Business and Social Sciences, Aarhus University, 8210 Aarhus, Denmark

Received 29 December 2015; Accepted 26 April 2016

Academic Editor: Francisco R. Villatoro

Copyright © 2016 Bing-Bing Cao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The newsvendor models considering decision-makers’ behavioral factors remain a fruitful research area in operation management field in past decade. In this paper, we further extend the current literatures to look into joint inventory, pricing, and advertising decisions considering loss aversion effects under the newsvendor setting. The purpose is to explore how the loss aversions affect the optimal policy of order quantity, price, and advertising effort level. We present an integrated utility model to measure both economic payoff and loss aversion utility of the newsvendor, where surplus loss aversion and stockout loss aversion are first separately defined and quantified. Then, we analyze the optimal solution conditions of the integrated model under exogenous and endogenous price cases, respectively. Under exogenous price case, we find that the uniquely optimal policy exists and is presented in the closed form. Under endogenous price case, the optimal policy is determined under mild conditions; we also provide the solutions when order quantity factor or advertising effort level is fixed in this case. In addition, the sensitivity analysis shows that the loss aversions affect the optimal decisions of order quantity, price, and advertising effort level in a systematic way.

#### 1. Introduction

Both business practices and academic research show that the decisions on operations management (OM) in real world often deviate from the optimal solutions of the traditional analytical models in operations research (see [1–3]). This is because the traditional analytical models are usually based on a strong assumption of the newsvendor’s perfect rationality. In reality, the newsvendor usually cannot exhibit perfect rationality but can exhibit some psychological behaviors which cause the derivation of the decisions from the traditional optimal solutions. Hence, it is difficult to describe the newsvendor’s real decision-making process using the traditional analytical models. In order to bridge the gap between traditional models and real world situations, some scholars conduct studies on the integration of behavioral factors into the traditional analytical models (see [4–6]).

The behavioral factors which have been identified in the OM area include bounded rationality (see [6, 7]), reference dependence (see [8]), decision bias (see [9]), fairness concern (see [10, 11]), overconfidence (see [12]), mental accounting (see [13]), and loss aversion (see [5, 14, 15]). We will concentrate on the behavioral factor of loss aversion. The loss aversion implies that the newsvendor is more sensitive to the gains than to the absolutely commensurable losses. It is reasonable in the real business environment because the decision-makers may perform different preferences from risk-neutrality. It can be seen from the existing literatures that the loss aversion effect has been identified and addressed in the OM area (see [16–18]). Although the existing studies have made great contributions to the OM problem considering behavioral factors, the study on the joint inventory, pricing, and advertising decisions considering the loss aversion is still lacking. In this paper, we look into joint inventory, pricing, and advertising decisions considering the loss aversion under the newsvendor setting. Given that the retailer may have different sensitivity to the loss caused by the overstock and the loss caused by the out-of-stock, the loss aversion is distinguished as the surplus loss aversion and the stockout loss aversion. We focus on discovering the impact of the two loss aversions on the joint inventory, pricing, and advertising effort level decisions. Since price and advertising are the most important and direct marketing tools to balance demand and supply when facing a stochastic demand, we consider a price and advertising effort level dependent demand function with a stochastic demand factor. The loss-averse newsvendor needs to determine the optimal order quantity, retail price, and the advertising effort level before the beginning of the selling season.

Given the great complexity of the joint decision-making problem, we first analyze the economic payoff and loss aversion utility of the newsvendor separately and then establish an integrated utility function based on Bell’s integrated model (see [19]). In integrated function, the economic payoff is measured by the profit function; the loss aversion utility consists of two parts: a surplus loss aversion utility and a stockout loss aversion utility. The loss aversion utilities are measured by a linear function. Furthermore, the optimal solution conditions on the inventory, price, and advertising effort level are presented by analyzing the characteristics of the integrated utility function under the exogenous price case and the endogenous price case.

The main contribution of the study is extending the existing loss-averse newsvendor models and joint inventory and pricing models to be more realistic settings. In detail, we first identify and quantify two types of loss aversions, that is, stockout loss aversion and surplus loss aversion, in the newsvendor environment. Next, based on the utility maximization theory, we integrate the economic payoff and the loss aversion utility and determine a total utility model. Furthermore, we provide the structural properties of the optimal solutions to the integrated model under the exogenous price case and the endogenous price case. Under the exogenous price case, the optimal order quantity and advertising effort level exist and are given in the closed form. Under the endogenous case, the optimal order quantity, price, and advertising effort level can be determined simultaneously under mild conditions. In addition, we also provide the optimal solutions to the joint decision model when the order quantity factor or the advertising effort level is fixed. Moreover, the sensitivity analysis shows the robustness of research results. Finally, we give a numerical example and show that both the stockout loss aversion and the surplus loss aversion affect the optimal order quantity, price, and advertising effort level in a systematic way.

The rest of the paper is organized as follows. Section 2 outlines related literatures. Section 3 describes the newsvendor’s utility framework and constructs the utility model. Section 4 solves the loss-averse newsvendor problem with advertising effect under the exogenous price case. Section 5 solves the problem under the endogenous price case and provides the optimal solutions when the order quantity factor or the advertising effort level is fixed. Section 6 concludes with a brief discussion of future research directions. All proofs are provided in the technical appendix.

#### 2. Literature Review

Our study is closely related to three streams of literatures: the loss-averse newsvendor models, the newsvendor and pricing models, and the advertising optimization models. There is a great amount of studies in these areas, and it is difficult to exhaust the literatures. For the sake of brevity, we only focus on the latest and most representative studies here.

Extensive behavioral experiments show that the psychological behaviors play an important role in newsvendor’s decisions under uncertainty (see [6, 8, 19–24]). The loss-averse newsvendor problem has been a fruitful research topic in past few years. Lee et al. [5] analyze the impact of the loss aversion of the newsvendor on his/her optimal options decisions. They find that a loss-averse newsvendor will order less without supplying options. Herweg [14] extends the classical newsvendor problem with the expectation-based loss aversion. They state the order quantity for the loss-averse newsvendor is less than that for the risk-neutral newsvendor. Wang and Webster [17] focus on the loss aversion in classic newsvendor settings and find that loss aversion affects the optimal inventory policy. They also find that optimal order quantity may increase in wholesale price but decrease in retail price in this situation. Wang [18] extends the standard newsvendor problem into the game setting where the multiple loss-averse newsvendors and one risk-neutral supplier are considered and shows that the newsvendors’ total order quantity increases with the increase of loss aversion. Nagarajan and Shechter [23] address the newsvendor problem based on the prospect theory through an experimental study. They maintain that the real order quantity deviates from the theoretical optimal order quantity, and the prospect theory cannot explain the reason of the deviation. Ma et al. [25] study the loss-averse newsvendor problem with two ordering opportunities and market information updating and build a penalty model for the loss-averse newsvendor to obtain the target profit. Xu et al. [26] focus on the optimal decision for the loss-averse newsvendor problem under conditional value at risk. They introduce the legacy loss into the analysis of the loss-averse newsvendor problem and analyze the effect of the legacy loss on the optimal order quantity.

In addition, the newsvendor problem is also analyzed with other psychological behaviors such as reference dependence, decision bias, bounded rationality, and inequality aversion. Interested readers may please refer to the recently published papers for a thorough review [2, 3, 6, 22, 27, 28]. Although the above studies have made great contributions to the newsvendor model with loss aversion, they seldom consider the price and the advertising effect simultaneously, and they do not describe clearly the impacts of the aversions to the surplus loss and stockout loss on the optimal policy.

The newsvendor and pricing problem is the most typical topic in the interface between the OM and marketing. It is one of the extensions of the classical newsvendor model by considering the endogenous price and refers to the determination of the order quantity and price in order to maximize the newsvendor’s expected profit in an uncertain demand framework (see [29, 30]). The newsvendor and pricing model is a fundamental and significant model in OM (see [5]) and has attracted continuously extensive attention from both the academia and the practice. We refer the interested readers to [29, 31–33] for detailed literature review.

Advertising effort is another decision variable in our study and works as one of the indispensable marketing tools to increase demands. Recently, there are increasing interests from operation researchers about the joint OM and advertising decisions. For detailed survey of the advertising effect and its extensions, we refer the interested readers to [34–40]. However, how to integrate the newsvendor problem, pricing, and advertising effort level with the loss aversion behavior remains unresolved. But it can be seen that several studies attempt the integration of OM decisions, advertising effort, and behavioral factors. For example, Zhang et al. [41] study the cooperative advertising with reference price effect in a vertical supply chain and find that the firm will invest more in national advertising if impact of the reference price on the optimal policy is larger. Yang et al. [42] introduce the inequality aversion into the research on the cooperative advertising in a distribution channel. By equivalent analysis, they state that the channel coordination can be achieved under the mild conditions.

Our study is a realistic extension of the aforementioned papers, but it differs from them significantly in that the psychological behaviors, that is, loss aversions, of the newsvendor and advertising effect are simultaneously taken into account and that solutions for both the exogenous price case and the endogenous price case are presented.

#### 3. The Formulations

We consider joint inventory, pricing, and advertising decisions for a loss-averse newsvendor with newsvendor settings. In this problem, apart from the traditional business objective of economic payoff, the newsvendor is driven by the economic payoff and loss aversion. Here, we apply the classic weighted sum utility model proposed by Bell [19] to integrate the economic payoff and loss aversion utility; it is shown below: where the economic payoff can be measured by a newsvendor’s profit during the selling season; the psychological satisfaction means also loss aversion utility and can be measured by the psychological differences between the realized profit and the expected profit of the newsvendor. We further present the profit and loss aversion utility of the newsvendor in detail in the following sections.

##### 3.1. Profit: Economic Payoff of the Newsvendor

In the joint inventory, pricing, and advertising decisions, the newsvendor places an order of quantity at a unit purchasing cost and sells at price . The inventory cannot be replenished during the selling season. In addition, the newsvendor also does advertising to promote the products, and the advertising effort level depends on the newsvendor’s advertising investment. The price, the advertising effort level, and the market uncertainty can affect the demand. Without loss of generality, consider that the demand is composed of the two parts (see [43, 44]). One is the deterministic part which is related to the price and advertising effort level. Usually, this part is nonincreasing in the price (see [29, 45, 46]) and nondecreasing in the advertising effort level (see [47–49]). The other is the stochastic part which is denoted by a random factor , . Let and denote the probability density function and the cumulative distribution function of the random factor , respectively, and and denote the mean and the standard deviation, respectively. The demand function can be additive or multiplicative (see [29, 45]). Since the optimal policies for the additive demand function can be easily adapted to the ones for the multiplicative demand function (see [46]) and the model is tractable for the additive demand function, we use the linear additive demand function; it is given bywhere , and denote the market size and the price sensitivity, respectively, , , denotes the advertising effort level, , and denotes the advertising sensitivity, . We assume that the advertising cost is convex in the advertising effort level and the cost function of the advertising effort level is . It is commonly used in literatures (see [42, 50–52]). The parameters should be properly chosen to assure a positive demand for some range of and . In addition, if there is unsatisfied customer demand at the end of the selling season, a shortage cost incurs, and if there is excess stock by end of the selling season, an salvage value incurs, where .

Therefore, the profit function of the newsvendor can be written as

##### 3.2. Utility of Loss Aversion: Psychological Satisfaction

Loss aversion is first recognized by Kahneman and Tversky [53] in the framework of prospect theory, and it is an important psychological concept which receives increasing attention in the OM, especially in behavioral OM in recent years (see [5, 9, 16–18, 53–56]). Loss aversion implies that the newsvendor has different sensitivity to the perceived losses and the perceived gains (see [16, 17, 57]). Specifically, if the realized profit of the newsvendor is less than his expectation, then the newsvendor may feel extra loss beyond the actual economic lost sales. In fact, the newsvendor is often averse to the loss at the decision making phase.

Moreover, loss aversion is directly related to a reference point, denoted by . Generally, reference point can be the expected profit of the newsvendor (see [5, 16, 17, 55]). In fact, the selection of reference point is also a subjective choice of the newsvendor and it is often relevant to the market environment, newsvendor’s business strategies, and the competitive position. Theoretically, the reference point may be any arbitrary value in the profit range of the newsvendor, where denotes the theoretical minimum of the profit of the newsvendor, it may be negative, and denotes the theoretical maximum of the profit of the newsvendor, . Although, theoretically, the expected profit may be negative, the newsvendor usually does not choose a negative reference point since the businesses are always profit driven. Hence, the reference point is usually determined in the range of .

Based on the above analysis, the newsvendor’s perception on the gain and the loss can be described in Figure 1 (see [9, 16, 17]). It is easy to see that the newsvendor perceives loss when the profit is less than the reference point , and the utility caused by the loss decreases faster than the utility caused by the gain increases when the profit is greater than the reference point.