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

Authors: Jingfu Huang; Bo Feng; Chengfei Wang
Journal: Mathematical Problems in Engineering
 (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.


---

## Body

## 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. Section2 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 Figure1). 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.Figure 1
The variation of the wholesale price in the planning horizon.Table 1
Notations.
SymbolDescriptionT0The ending time of presellingc0Wholesale price at the beginning of presellingbItem selling price in spot marketrItem recycling priceτThe wholesale price increasing coefficientDMarket demand in spot marketQdRetailer’s preordering quantity (variable)tdRetailer’s preordering time (variable)γThe loan interest rate∗The optimal valueFor the concerned model, we assume that(1)
The retailer is risk-averse(2)
The whole price during the preordering period isct=c0+τt for t∈0,T0, where τ>0 is a constant(3)
The demand during the spot selling period is random which obeys probability distribution with probability density functionfx and probability distribution function Fx


## 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 attd with quantity Qd. Then, from the assumption, the retailer’s profit is(1)Πtd,Qd,D=bminD,Qd−ctdQd−γT0−tdctdQd+rQd−D+,where X+=maxX,0, 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.,Qd=D, then the retailer can obtain the maximal profit b−ctdD−γT0−tdctdD. Then, for preorder made at td with quantity Qd, the retailer’s loss is(2)Ltd,Qd,D=b−ctd−γT0−tdctdD−Πtd,Qd,D.Using the fact thatminQd,D=Qd−Qd−D+, the function can be written as(3)Ltd,Qd,D=b−ctd−γT0−tdD−Qd+b−rQd−D+.For the confidence levelα∈0,1, the value-at-risk of the retailer’s loss is(4)VaRαLtd,Qd,D=infy|PrLtd,Qd,D≤y≥α,which represents the minimum loss under the confidence level α. 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:(5)CVaRαLtd,Qd,D=11−αELtd,Qd,D|Ltd,Qd,D⩾VaRαLtd,Qd,D.To facilitate computing of the CVaR, Rockafellar and Uryasev [25] introduced the following auxiliary function:(6)Hαtd,Qd,y=y+11−αELtd,Qd,D−y+,which transforms the optimization problem(7)mintd,QdCVaRαLtd,Qd,D,as the following minimization problem:(8)mintd,Qd,yy+11−αELtd,Qd,D−y+.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(9)mintdminQd,yy+11−αELtd,Qd,D−y+.In the following, we first solve the inner problem w.r.t.Qdandy and then solve the outer problem w.r.t. td. For the inner problem, i.e.,(10)minQd,yy+11−αELtd,Qd,D−y+,we have the following conclusion.Theorem 1.
For fixed preordering timetd and confidence level α∈0,1, the retailer optimal preorder quantity is(11)Qd∗td=c−rF−11−αb−c/b−r+b−cF−11−αb−c/b−r+αb−r,with(12)y∗td=c−rb−cF−11−αb−c/b−r+α−F−11−αb−c/b−rb−r,where c=−γτtd2+τ+γτT0−γc0td+γT0c0+c0.Proof.
For problem (3), we first consider the minimization problem w.r.t. y for fixed Qd and then solve the problem w.r.t. Qd.
ByLtd,Qd,D=b−cD−Qd+b−rQd−D+, the objective function of problem (3) can be written as(13)Hαtd,Qd,y=y+11−α∫0+∞Ltd,Qd,x−y+fxdx=y+11−α∫0Qdcx−Qd−y+fxdx+11−α∫Qd+∞b−cx−Qd−y+fxdx.
We break the discussion into to two cases.Case 1.
y≥c−rQd. In this case,(14)Hαtd,Qd,y=y+11−α∫Qd+y/b−c+∞b−cx−Qd−y+fxdx.
Computing the partial derivative respect toQd yields(15)∂Hαtd,Qd,y∂y=1−11−α1−FQd+yb−c.
From the monotony increasing property ofFx and limx⟶+∞Fx=1, we conclude that ∂Hαtd,Qd,y/∂y≥0 for sufficiently large y. If ∂Hαtd,Qd,y/∂y|y=c−rQd≤0, then the stationary point y=b−cF−1α−Qd is the minimum point of the objective function. Otherwise, Hαtd,Qd,y reaches the minimum in the interval 0,c−rQd, and this boils down to Case 2 discussed as follows.Case 2.
0≤y≤c−rQd. In this case,(16)Hαtd,Qd,y=y+11−α∫0Qd−y/c−rc−rQd−x−yfxdx+11−α∫Qd+y/b−c+∞b−cx−Qd−yfxdx.
Then,(17)∂Hαtd,Qd,y∂y=1−11−α1+FQd−yc−r−FQd+yr−c.
Certainly,∂Hαtd,Qd,y/∂y|y=0=−α/1−α<0. If Q≤b−c/b−rF−1α, then(18)∂Hαtd,Qd,y∂yc−rQd=1−FQd+c−rb−c≥0.
The case boils down to Case1. If Qd≥b−c/b−rF−1α, then the minimum will be attained at the stationary point of the objective function, i.e., the root of the following equation denoted by y2∗:(19)FQd+yb−c−FQd−yc−r=α.
So, for any fixedQd, the optimal solution of the optimization problem minyHαtd,Qd,y is(20)y∗=b−cF−1α−Qd,ifQd≤b−cb−rF−1α,y2∗,ifQd≥b−cb−rF−1α.
Next, we consider minimization problem:(21)minQdHαtd,Qd,y∗.
For this, we break the discussion into two cases.Case 3.
Qd≤b−c/b−rF−1α. In this case, it follows from y∗=b−cF−1α−Qd that(22)Hαtd,Qd,y∗=b−cF−1α−Qd+∫F−1α+∞b−cx−F−1αfxdx.
Using the fact that(23)∂Hαtd,Qd,y∗∂Qd=−b−c<0,we conclude that the function is monotonically decreasing in 0,b−c/b−rF−1α. Hence, function Hαtd,Qd,y∗ reaches minimum in interval b−c/b−rF−1α,+∞, and this boils down to Case 4 discussed as follows.Case 4.
Qd≥b−c/b−rF−1α. In this case, y=y2∗. Then,(24)Hαtd,Qd,y∗=y∗+11−α∫0Qd−y∗/c−rc−rQd−D−y∗fDdD+11−α∫Qd+y∗/b−c+∞b−cD−Qd−y∗fDdD,and(25)∂Hαtd,Qd,y∗∂Qd=11−αc−rFQd−y∗c−r−b−ctd1−FQd+y∗b−c.
Solving equation∂Hαtd,Qd,y∗/∂Qd=0, i.e.,(26)cFQd−y∗c−r+b−cFQd+y∗b−c=b−c,yields the optimal preorder quantity(27)Qd∗td=c−rF−11−αb−c/b−r+b−cF−11−αb−c/b−r+αb−r,and(28)y∗td=c−rb−cF−11−αb−c/b−r+α−F−11−αb−c/b−rb−r.
This completes the proof of the assertion.
Now, consider the out problem of optimization problem (2), i.e., minimization problem(29)mintdy∗+11−αELtd,Qd∗,D−y∗+.
For this, we have the following conclusion.Theorem 2.
For the risk-averse retailer with confidence levelα, the retailer’s optimal preordering time is(30)td∗=t˜d,ift˜d∈0,T0,ϕt˜d≤0,minϕ0,ϕT0,0,ift˜d∈0,T0,ϕ0≤minϕt˜d,ϕT0,T0,ift˜d∈0,Ta0,ϕT0≤minϕt˜d,ϕ0,0,ift˜d∉0,T0,ϕ0≤ϕT0,T0,otherwise,where(31)ϕtd=−γτtd2+τ+γτT0−γc0td+γτT0+c0−rb+γτtd2−τ+γτT0−ic0td−γτT0−c0b−rF−11−αb+γτtd2−τ+γτT0−γc0td−γτT0−c0b−r+α−F−11−αb+γτtd2−τ+γτT0−γc0td−γτT0−c0b−r,and t˜d is the root of the equation ϕ′td=0. Correspondingly, the optimal preorder quantity is(32)Qd∗=−γτtd2+τ+γτT0−γc0td+γτT0+c0−rF−11−αb+γτtd2−τ+γτT0−γc0td−γτT0−c0/b−rb−r+b+γτtd2−τ+γτT0−γc0td−γτT0−c0F−11−αb+γτtd2−τ+γτT0−γc0td−γτT0−c0/b−r+αb−r.Proof.
From Theorem1, we know that the objection function of minimization problem (4) can be written as(33)ϕtd=c−rb−rF−11−αb−c/b−r+α−F−11−αb−c/b−rb−r=−γτtd2+τ+γτT0−γc0td+γτT0+c0−rb+γτtd2−τ+γτT0−γc0td−γτT0−c0b−rF−11−αb+γτtd2−τ+γτT0−γc0td−γτT0−c0b−r+α−F−11−αb+γτtd2−τ+γτT0−γc0td−γτT0−c0b−r.
To compute its minimum, we consider its derivative(34)ϕ′td=b+r+2γτtd2−τ+γτT0−γc0td−γτT0−c0−2γτtd+τ+γτT0−γc0b−rF−11−αb+γτtd2−τ+γτT0−γc0td−γτT0−c0b−r+α−F−11−αb+γτtd2−τ+γτT0−γc0td−γτT0−c0b−r+−γτtd2+τ+γτT0−γc0td+γτT0+c0−rb−r2b+γτtd2−τ+γτT0−γc0td−γτT0−c0f1−αb+γτtd2−τ+γτT0−γc0td−γτT0−c0/b−r+α1−α−γτtd+τ+γT0τ−γc0f1−αb+γτtd2−τ+γτT0−γc0td−γτT0−c0/b−rF−11−αb+γτtd2−τ+γτT0−γc0td−γτT0−c0b−r−F−11−αb+γτtd2−τ+γτT0−γc0td−γτT0−c0b−r+α.
If equationϕ′td=0 is inconsistent, then function ϕt reaches its minimum at 0 or T0; otherwise, we denote t˜d=argminϕtd|ϕ′td=0. Then, if t˜d∉0,T0, function ϕtd reaches its minimum at 0 or T0; if t˜d∈0,T0, then ϕtd reaches its minimum at 0 or t˜d or T0.
From the discussion above, we can obtain the retailer’s optimal preordering time(35)t∗=t˜d,ift˜∈0,T0,ϕt˜d≤0,minϕ0,ϕT0,0,ift˜∈0,T0,ϕ0≤minϕt˜d,ϕT0,T0,ift˜d∈0,T0,ϕT0≤minϕt˜d,ϕ0,0,ift˜d∉0,T0,ϕ0≤ϕT0,T0,otherwise,and the optimal preorder quantity(36)Qd∗=−γτtd2+τ+γτT0−γc0td+γτT0+c0−rF−11−αb+γτtd2−τ+γτT0−γc0td−γτT0−c0/b−rb−r+b+γτtd2−τ+γτT0−γc0td−γτT0−c0F−11−αb+γτtd2−τ+γτT0−γc0td−γτT0−c0/b−r+αb−r.
From Theorems1 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.Algorithm 1: The algorithm to compute the optimal preordering time and the optimal preorder quantity.
Step 1. Input parameters b,c0,τ,r,T0,α,γ,fx,F−1x.Step 2. Define functionϕtd=−γτtd2+τ+γτT0−γc0td+γτT0+c0−rb+γτtd2−τ+γτT0−γc0td−γτT0−c0/b−rF−11−αb+γτtd2−τ+γτT0−γc0td−γτT0−c0/b−r+α−F−11−αb+γτtd2−τ+γτT0−γc0td−γτT0−c0/b−r and solve equation ϕ′td=0 to obtain t˜d. If t˜d<0 or t˜d>T0, set t˜d=0.Step 3. If ϕt˜d≤minϕ0,ϕT0, then set td∗=t˜d, else set td∗=T0, goto next step.Step 4. Compute Qd∗=−γτtd2+τ+γτT0−γc0td+γτT0+c0−rF−11−αb+γτtd2−τ+γτT0−γc0td−γτT0−c0/b−r/b−r+b+γτtd2−τ+γτT0−γc0td−γτT0−c0F−11−αb+γτtd2−τ+γτT0−γc0td−γτT0−c0/b−r+α/b−rStep 5. Output td∗, Qd∗.


## 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 withb=70, c0=25, γ=0.07, τ=0.3, r=15, T0=50, and α=0.95, the demand obeys the exponential distribution with probability density function fx=1/150exp−x/150, x∈0,+∞.
For this problem, by Algorithm1, retailer’s optimal preordering time is td∗=0.37, and optimal preordering quantity is Qd∗=291.
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 Example1, we let retailer’s confidence level α 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 α, the optimal preordering time is decreasing and the retailer’s optimal preordering quantity is also increasing.
For the inventory mechanism, we let the selling priceb 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 b 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 Example1, we let c0 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 c0 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 Example1, we let γ 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 γ goes from 0.05 to 0.09, the retailer’s preordering time is increasing in γ and the retailer’s optimal preordering quantity is also increasing in γ.Figure 2
Influence ofα on the retailer’s preordering time.Figure 3
Influence ofα on the retailer’s preordering quantity.Figure 4
Influence ofb on the retailer’s preordering time.Figure 5
Influence ofb on the retailer’s preordering quantity.Figure 6
Influence ofc0 on the retailer’s preordering time.Figure 7
Influence ofc0 on the retailer’s preordering quantity.Figure 8
Influence ofγ on the retailer’s preordering time.Figure 9
Influence ofγ on the retailer’s preordering quantity.Example 2.
Considering the preorder inventory system withα=0.95, b=70, c0=25, γ=0.07, τ=0.3, and r=15, we let T0 increase from 45 to 55, while the market demand D obeys the exponential distribution with probability density function fx=1/150exp−x/150, x∈0,+∞.
For this inventory mechanism, the numerical results are shown in Figures10 and 11, from which we can see that, when T0 goes from 45 to 55, the retailer’s preordering time is increasing in T0 and the retailer’s optimal preordering quantity is also increasing in T0.Figure 10
Influence ofT0 on the retailer’s preordering time.Figure 11
Influence ofT0 on the retailer’s preordering quantity.Example 3.
Considering the preorder inventory system withα=0.95, b=70, c0=25, γ=0.07, r=15, and T0=50, we let τ increase from 0.2 to 0.4, while the market demand D obeys the exponential distribution with probability density function fx=1/150exp−x/150, x∈0,+∞.
For this inventory mechanism, the numerical results are shown in Figures12 and 13, from which we can see that, when τ goes from 0.2 to 0.4, the retailer’s preordering time is decreasing in τ, and the retailer’s optimal preordering quantity is also increasing in τ, 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 levelα, item selling price b, item initial wholesale price c0, and loan interest rate γ have a larger influence on retailer’s preordering strategy, while the end time of preselling period T0 and wholesale price increasing coefficient τ have a less influence on the retailer’s preordering strategy. The retailer’s optimal preordering time is decreasing in α, c0, and T0, while increasing in b. The retailer’s optimal preordering quantity is increasing in α, γ, and T0, while decreasing in c0.Figure 12
Influence ofτ on the retailer’s preordering time.Figure 13
Influence ofτ on the retailer’s preordering quantity.


## 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.



---

*Source: 7353083-2022-02-02.xml*