On the Preorder Strategy for Loss-Averse Newsvendor Model under CVaR Measure

Huang, Jingfu; Feng, Bo; Wang, Chengfei

doi:https://doi.org/10.1155/2022/7353083

Mathematical Problems in Engineering

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Abstract Introduction Notation Model Formulation Numerical Example Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Models and Methods for Risk Management under Emergencies

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Research Article | Open Access

Volume 2022 | Article ID 7353083 | https://doi.org/10.1155/2022/7353083

On the Preorder Strategy for Loss-Averse Newsvendor Model under CVaR Measure

Jingfu Huang,¹Bo Feng,²and Chengfei Wang¹

Academic Editor: Xinsheng Xu

Received20 Oct 2021

Accepted03 Jan 2022

Published02 Feb 2022

Abstract

Due to rapid development of manufacturing and online retail, the life cycle of many products is becoming shorter and shorter. Hence, manufacturers may launch new products in selling season to maintain market share and attract new customers. Under these circumstances, manufacturers may release a presale information before the selling season, and the retailer correspondingly makes a preorder according to the estimation to the demand. For the case that the wholesale price rises gradually with time during the preselling period and the market demand is stochastic, based on minimizing the legacy loss via CVaR measure in risk management, we establish a loss-averse newsvendor’s preordering decision model. By model analysis, we establish the closed form solution to the model and provide the optimal preordering time and preordering quantity to the retailer. Some numerical experiments are made to show the validity of the model, and some managerial insights are explored through the numerical experiments.

1. Introduction

As a promotion strategy in modern market, preselling mechanism is widely used by many enterprises in which the manufacturer releases a preselling information when the products are still in the production process [1–4]. For example, Apple provides customers a preordering service before a new generation of products comes to market [5] and the millet mobile phone industry adopted presale mechanism when it published the first generation cell phone [6]. Now, the preselling mechanism has been widely used in service industry, electronic manufacturing, and online retailing [7–12].

For the preselling mechanism, from the manufacturer’s perspective, it can collect new product’s demand information and reduce inventory risk in today’s competitive market, and hence it can help the manufacturer predict the market potential demand and reduce inventory risk. For this, Cachon [8] compared inventory risks under push, pull, and preorder discount contracts between suppliers and retailers, Gilbert and Cvsa [13] examined manufacturers’ wholesale price commitment decisions as well as issues related to presale timing, Boyaci and Özer [7] established a profit-maximization model in which a manufacturer collected advance sales information periodically prior to the regular sales season for a capacity decision, and Nasiry and Popescu [14] characterized the effect of anticipated regret on consumer decisions and on firm profits and policies in an advance selling context where buyers have uncertain valuations.

For the preselling mechanism, from the retailer’s perspective, the retailer can receive a price discount for accepting the manufacturer’s preselling, which will reduce the retailer’s purchasing cost. For example, Amazon offered a 20% preorder discount for all new-to-be-released video games from 2016 to 2018 [12]. Furthermore, preselling guarantees that the retailer receives the items earlier, which is particularly valuable when the product may be hard to find in the spot selling period due to its popularity [5, 15, 16]. A second benefit of preselling is that the selling season demand can be more accurately forecasted because orders received in the preselling period may be correlated with the demand in the selling season. The retailer can update its demand forecast and adjust the order quantity accordingly to reduce excess inventory and stock out risks. In addition to reducing inventory risks, preselling takes advantage of changes in consumers product valuation uncertainty over time as noted by Xie and Shugan [17], Shugan and Xie [18], and Gundepudi et al. [19].

It should be noted that most research studies on this issue assume that supply chain participants are risk-neutral, whereas some scholars showed through empirical research that not all the decision-makers are of this type [20]. Thus, it is essential to take the retailers’ behavior preference into considerations [21–23]. Due to this, the loss aversion theory is posed and is applied in many fields, such as portfolio optimization and supply chain management. In particular, the CVaR is a popular measure in describing decision-maker’s behavior preference for risk [24, 25] and has made a good success in quantifying and mitigating the potential risk in newsvendor problem [26–35].

In this paper, we consider a loss-averse newsvendor model in which the manufacturer will provide a preselling strategy to the retailer before the selling season. In the preselling period, the wholesale price increases with time. Since the demand during the selling season is stochastic, to minimize the retailer’s legacy loss, the retailer should determine the preordering time and preordering quantity according to the estimation to the demand in the selling season. For this, we establish an optimal preordering decision model. By modeling analysis, we obtain the retailer’s optimal preordering time and preordering quantity.

The remainder of this paper is organized as follows. Section 2 presents the notations and provides some necessary assumptions needed on the concerned problem. In Section 3, we establish an optimization model for the concerned problem. Section 4 provides the model solution and gives the retailer’s optimal preorder policy. Section 5 presents numerical examples to illustrate the validity of the model. Some conclusions are drawn in the last section.

2. Notation and Assumptions

In this paper, we consider the newsvendor model with a preselling strategy and the retailer is risk-averse. More precisely, the planning horizon consists of two periods—preselling period and spot selling period. During the preselling period, the wholesale price of the item gradually increases with time (see Figure 1). Thus, to enjoy the lower wholesale price provided by the supplier, the retailer may make a preorder. However, the prepaid purchase cost would result in certain interest cost during the preordering period. Thus, the retailer should make a tradeoff between enjoying the benefit provided by the preselling strategy and bearing the interest cost of prepaid for the preorder. With this, the retailer should determine the preordering time and the preorder quantity so that his profit is maximized. To this end, we need the notations listed in Table 1.

For the concerned model, we assume that(1)The retailer is risk-averse(2)The whole price during the preordering period is for , where is a constant(3)The demand during the spot selling period is random which obeys probability distribution with probability density function and probability distribution function

3. Model Formulation

For the concerned inventory mechanism, since the wholesale price in the preselling period gradually increases with time and it is lower than the ordinary wholesale price, the retailer should determine the preordering time and the preorder quantity. Further, since the demand during the spot selling period is random, it is impossible to make a preorder such that the order quantity is equal to the demand; this means that there is always a risk of loss for any preorder. Then, the retailer’s preorder decision will be affected by his attitude to the risk of loss. For this, we use the popular CVaR measure to describe his behavior preference for risk [24, 25]. To this end, we first compute the retailer’s loss function.

Suppose the preorder is made at with quantity . Then, from the assumption, the retailer’s profit iswhere , the first term is sale profit, the second term is the wholesale cost, the second term is the loan interest, and the last term is recycling profit.

Certainly, if the retailer’s preorder quantity exactly meets the demand, i.e., , then the retailer can obtain the maximal profit . Then, for preorder made at with quantity , the retailer’s loss is

Using the fact that , the function can be written as

For the confidence level , the value-at-risk of the retailer’s loss iswhich represents the minimum loss under the confidence level . As a risk-averse decision maker, the retailer may take VaR as the targeted loss and pay more attention to the loss above a given target level [24, 25]. This yields the following CVaR measure to describe the retailer’s behavior preference for risk:

To facilitate computing of the CVaR, Rockafellar and Uryasev [25] introduced the following auxiliary function:which transforms the optimization problemas the following minimization problem:

In the following, we will give a closed form solution to the optimization problem and hence present the retailer’s optimal preordering time and optimal preorder quantity.

4. Model Solution

Certainly, problem (1) can be written as

In the following, we first solve the inner problem w.r.t. and then solve the outer problem w.r.t. . For the inner problem, i.e.,we have the following conclusion.

Theorem 1. For fixed preordering time and confidence level , the retailer optimal preorder quantity iswithwhere .

Proof. For problem (3), we first consider the minimization problem w.r.t. for fixed and then solve the problem w.r.t. .
By , the objective function of problem (3) can be written asWe break the discussion into to two cases.

Case 1. . In this case,Computing the partial derivative respect to yieldsFrom the monotony increasing property of and , we conclude that for sufficiently large . If , then the stationary point is the minimum point of the objective function. Otherwise, reaches the minimum in the interval , and this boils down to Case 2 discussed as follows.

Case 2. . In this case,Then,Certainly, . If , thenThe case boils down to Case 1. If , then the minimum will be attained at the stationary point of the objective function, i.e., the root of the following equation denoted by :So, for any fixed , the optimal solution of the optimization problem isNext, we consider minimization problem:For this, we break the discussion into two cases.

Case 3. . In this case, it follows from thatUsing the fact thatwe conclude that the function is monotonically decreasing in . Hence, function reaches minimum in interval , and this boils down to Case 4 discussed as follows.

Case 4. . In this case, . Then,andSolving equation , i.e.,yields the optimal preorder quantityandThis completes the proof of the assertion.
Now, consider the out problem of optimization problem (2), i.e., minimization problemFor this, we have the following conclusion.

Theorem 2. For the risk-averse retailer with confidence level , the retailer’s optimal preordering time iswhereand is the root of the equation . Correspondingly, the optimal preorder quantity is

Proof. From Theorem 1, we know that the objection function of minimization problem (4) can be written asTo compute its minimum, we consider its derivativeIf equation is inconsistent, then function reaches its minimum at 0 or ; otherwise, we denote . Then, if , function reaches its minimum at 0 or ; if , then reaches its minimum at 0 or or .
From the discussion above, we can obtain the retailer’s optimal preordering timeand the optimal preorder quantityFrom Theorems 1 and 2, we can present the following algorithm for computing the retailer’s optimal preordering time and optimal preordering quantity.
In the next section, we will make some numerical experiments to test the efficiency of the proposed algorithm.

	Step 1. Input parameters .
	Step 2. Define function
and solve equation to obtain . If or , set .
	Step 3. If , then set , else set , goto next step.
	Step 4. Compute
	Step 5. Output , .

5. Numerical Example

In this section, we test the efficiency of the proposed model by three numerical examples and make some numerical sensitivity analysis to test the influence of the involved parameters on the retailer’s optimal preorder strategy. Some data of the considered examples are taken from [31].

Example 1. Considering the preorder inventory mechanism with , , , , , , and , the demand obeys the exponential distribution with probability density function , .
For this problem, by Algorithm 1, retailer’s optimal preordering time is , and optimal preordering quantity is .
To better understand the influence of the parameters involved in the model on the retailer’s optimal preorder strategy, we conduct some parameters sensitivity analysis on the model.
For inventory mechanism of Example 1, we let retailer’s confidence level increase from 0.90 to 0.99, while keeping other parameters unchanged. The numerical results are shown in Figures 2 and 3 from which we see that with the increase in the retailer’s confidence level of , the optimal preordering time is decreasing and the retailer’s optimal preordering quantity is also increasing.
For the inventory mechanism, we let the selling price increase from 65 to 75, while keeping other parameters unchanged. The numerical results are shown in Figures 4 and 5, from which we can see that, when increases from 65 to 75, the retailer’s optimal preordering time is also increasing, while the optimal preordering quantity is decreasing.
For the inventory mechanism considered in Example 1, we let increase from 20 to 30, while keeping other parameters unchanged. The numerical results are shown in Figures 6 and 7, from which we can see that, when increases from 20 to 30, the retailer’s optimal preordering time is decreasing, while the optimal preordering quantity is increasing.
For the inventory mechanism considered in Example 1, we let increase from 0.07 to 0.09, while keeping other parameters unchanged, the numerical results are shown in Figures 8 and 9, from which we can see that, if goes from 0.05 to 0.09, the retailer’s preordering time is increasing in and the retailer’s optimal preordering quantity is also increasing in .

Example 2. Considering the preorder inventory system with , , , , , and , we let increase from 45 to 55, while the market demand obeys the exponential distribution with probability density function , .
For this inventory mechanism, the numerical results are shown in Figures 10 and 11, from which we can see that, when goes from 45 to 55, the retailer’s preordering time is increasing in and the retailer’s optimal preordering quantity is also increasing in .

Example 3. Considering the preorder inventory system with , , , , , and , we let increase from 0.2 to 0.4, while the market demand obeys the exponential distribution with probability density function , .
For this inventory mechanism, the numerical results are shown in Figures 12 and 13, from which we can see that, when goes from 0.2 to 0.4, the retailer’s preordering time is decreasing in , and the retailer’s optimal preordering quantity is also increasing in , and the increase rate is slower and slower.
From the numerical experiments on the involved parameter sensitivity analysis, we can see that the retailer’s confidence level , item selling price , item initial wholesale price , and loan interest rate have a larger influence on retailer’s preordering strategy, while the end time of preselling period and wholesale price increasing coefficient have a less influence on the retailer’s preordering strategy. The retailer’s optimal preordering time is decreasing in , , and , while increasing in . The retailer’s optimal preordering quantity is increasing in , , and , while decreasing in .

6. Conclusion

For the concerned inventory mechanism, the planning horizon consists of two periods—preselling period and spot selling period. During the preselling period, the wholesale price of the item gradually increases with time; based on minimizing the legacy loss via CVaR measure, we established a risk-averse’ optimal preorder strategy for the newsvendor model and gave the retailer’s optimal preordering strategy. Some numerical examples were provided to show the sensitivity of the involved parameters in the model, in which can see that the retailer’s confidence level, item selling price, item initial wholesale price, and the loan interest rate are crucial ingredients in the retailer’s preordering problem.

From the result of numerical experiments in this study, the high confidence level will make the retailer preorder earlier and preorder more items. If the initial wholesale price is high or the wholesale price increases rapidly, the retailer should make preorder earlier and order more items to reduce the legacy loss. The high selling price will make the retailer preorder later and preorder more items to reduce the legacy loss. The high loan interest rate will make retailer preorder earlier and order less items. The longer the duration of the preselling period is, more preorder items should be made.

Certainly, the preordering model considered in this paper assumes that the probability distribution of the demand during the preselling period is fixed. However, the demand distribution may be more and more clear with the time during the period. Thus, the research of the issue that the fluctuation of demand is getting weaker and weaker with the time is more significant.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors equally contributed to this paper and read and approved the final manuscript.

Acknowledgments

This project was supported by the National Natural Science Foundation of China (12071250).

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Copyright © 2022 Jingfu Huang 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.

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