Optimal Ordering Policy for Retailers with Bayesian Information Updating in a Presale System

Abstract

In this study, we investigate inventory allocation and pricing strategies for retailers by incorporating demand information into the issue of inventory allocation during the presale period. In a presale system, retailers offer presale goods at a price lower than the retail price. By offering products at a discount, retailers may attract additional demand. In addition, this system enables retailers to reduce the uncertainty of market demand and establish a strategy for inventory allocation based on the results of presales. A Bayesian approach was employed to analyze and update demand information, and inventory allocation was formulated as a newsvendor problem to determine the optimal policy that maximizes retailer profit. A numerical analysis was conducted to validate the effectiveness of the proposed strategy. Results suggest that the proposed strategies can support retailers by more accurately predicting demand and achieving higher profits with less inventory. Furthermore, retailers can experience greater benefits from risk-averse customers than from risk-neutral customers.

1. Introduction

With the rapid development of the online shopping industry, many retailers began to consider e-commerce platforms as important tools for gaining a competitive advantage in the market. E-commerce offers retailers the advantage of strategically utilizing the period of time before regular sales, which typically includes pre-order and presale. Pre-order means that the customer orders the item before it is released to the public and ensures that the customer will obtain the item before it sells out [1]. Presale means that the customer can purchase the item before it is released to the public, while pre-order enables customers to reserve items before they are released to the public. Presale includes a regular price and a presale price discounted from a regular price and offers a different price to the customer based on the sales period [2]. Pre-order programs are the same as advance booking discount programs (ABDP) in that the customer does not know the regular price. Presale programs differ from ABDP because the customer is aware of the regular price and the presale price. In this study, we focus on a methodology that allows retailers to maximize profits in the market through the application of a presale system.

In recent years, large e-commerce companies in China, such as JD, Taobao, TianMao, and Gome, have launched presale services. Such services allow customers to reserve products at a presale price before a regular sales date. A deposit must be paid in advance to prevent no-shows, but if customers are willing to purchase the product, they can purchase it at a price that is lower than the regular price. While customers clearly benefit from the presale system, retailers can also gain an advantage. They can reduce demand uncertainty by obtaining market demand information in advance. Customer valuation and external factors strongly influence the market. Because retailers generally cannot predict external factors accurately, especially for items that are yet to be released, they must increase the inventory level to respond flexibly to the uncertainty of the supply and demand of consumers. Thus, retailers can utilize presale volume information to optimize the inventory of presale items and reduce discrepancies between sales expectations and actual market demand. In addition, retailers can attract new customers from competing companies that do not apply presale services by providing discounted prices during the presale period. If multiple retailers sell a homogeneous product under the same conditions (i.e., lead time, quality, etc.) at different prices, rational consumers choose the retailer that sells the product most inexpensively. Thus, presale services can induce marketing effects and increase retailers’ profits.

The purpose of this study is to propose an optimal inventory allocation and pricing strategy for retailers by incorporating demand information into the issue of inventory allocation during the presale period. We address a two-period newsvendor model in which demand is stochastic and presale demand affects the regular sale demand.

The remainder of the paper is structured as follows: In Section 2, research related to updating demand information and presale strategies is reviewed. In Section 3, retailers develop an optimal sales strategy for the following two scenarios: (1) during a single regular sale period, and (2) when presale is offered to customers with presale prices. The optimal presale price and inventory allocation for presale and regular sale periods are then determined. The optimal sales model is numerically verified in Section 4. Finally, Section 5 concludes the paper.

2. Literature Review

Many studies have investigated presale strategies from both retailers’ and customers’ perspectives [3]. For example, Zhang et al. [4] analyzed the impact of consumers’ strategic behavior on a retailer’s presale profits and found that the profits decreased with customers’ strategic product valuation. In addition, Xie and Shugan [5] investigated the relationship between a product’s marginal cost and the optimal price for a presale under uncertain customer valuation. They indicated that retailers can achieve higher profits with presales when product costs are sufficiently low. You [6] described a scenario in which the retailer offers presale services and market demand is dependent on the retail price only. The retailer’s optimal order quantity and retail price for each period were determined. Shugan and Xie [7] proved that, unlike yield management, advance selling is an effective marketing tool that can diminish competition. Cho and Tang [8] analyzed advance sales strategy by studying advance selling strategies from suppliers to retailers.

This study examines the supplier’s optimal wholesale price and the retailer’s optimal order quantity for each period. Feng et al. [9] investigated consumer preferences when purchasing new products and studied the different price strategies adopted by retailers, such as a skimming price strategy and a penetration price strategy. Hasan et al. [10] proposed a decision-making method for e-commerce retailers who sell deteriorating products under the presale system by studying the optimal price and replenishment cycle when customers purchase products and pay through an online system during the presale period. Cao et al. [11] classified consumers based upon whether they were provided with information about presales and considered their preferences. They proposed four basic models and coordination models and compared the decisions of retailers and third-party platforms when considering the possible strategies adopted by supply chain members. Zhang et al. [12] and Han et al. [13] studied the presale strategies of the suppliers that produce and sell fresh agricultural products to online retailers. Zhang et al. [12] proposed the Stackelberg game model for each presale strategy to maximize revenue and determine optimal service levels and pricing decisions. Han et al. [13] considered the presale strategies for a three-echelon supply chain, and adopted the Stackelberg game method to study the equilibrium strategies of the supplier-led supply chain under centralized and decentralized decision-making. Pei et al. [14] proposed the deposit and final payment sales model for the impact of consumer returns and deposit ratios on retailers’ presale strategies. Ren et al. [15] studied the product service supply chain (PSSC) network optimization model based upon changes in presale service levels and service prices. To solve this model, they proposed a double-layer nested genetic algorithm. Feng et al. [16] extended beyond Feng et al. [9], and considered the influence of retailers’ offline sales efforts on strategic consumer choices. In these studies, the market size was fixed, and retailers did not need to update market demand information. The researchers also considered the presale and regular sale periods separately when making an optimal decision. Other researchers have studied presale services in conjunction with demand information updates.

Tang et al. [17] and McCardle et al. [18] studied a retailer that offered presale services at a discounted price to entice customers to reserve the product in advance of the regular sales season. The retailer then used this market information to update demand forecasts and plan its inventory. Furthermore, Kunnumkal and Topaloglu [19] showed that both retailer and customer can save money when a retailer is willing to offer a discount to the customer during the presale period to reduce demand variability. Zhao and Stecke [1] analyzed a presale pricing strategy based on customer valuation and loss aversion. They found that, for different customer valuations and profit margins, the retailer should offer a different discount. Chu and Zhang [20] demonstrated that the optimal pricing strategy is highly dependent on the amount of information available during the presale period, and optimized the information strategy and presale price. Li and Zhang [21] assumed that the consumer demand during the presale period represents high valuation, and the demand during the regular sales period represents low valuation. Thus, the retailer can forecast the demand for low valuation during a regular sales period using presale demand. Cheng et al. [22] determined that sellers use price discounts as a marketing tool to motivate consumers to make presale purchases. They developed a dynamic version of the Bass model more suitable for fast-changing online markets and considered consumer heterogeneity and uncertainties in market demand. Wang et al. [23] studied the presale strategy for B2C e-commerce sellers in consideration of consumers’ time preference behaviors and developed single-and two-stage models and investigated the effect of presale lead time on sellers’ pricing strategy, sales volume, and revenues. Lan and Zhu [24] studied new product presale strategies of manufacturers under random demand considering that there are both loss averse and rational consumers. Of the scholars [1,17,18,19,20,21,22,23,24] who have studied presale in relation to the information updating issue, most assumed that retailers are already aware of the correlation between presales demand and regular sales demand; a few [21,24] have analyzed cases where retailers are unaware of exactly how presales and regular sales are related.

With the rapid development of information technology, customer demand has diversified and become more personalized, resulting in more volatility in product demand. Therefore, the relationship between presale and sales demand has become more unpredictable than in the past. Specifically, retailers hold little information about market demand for unreleased products. An effective method for retailers to forecast market demand is to estimate it based on the historical sales data of similar products and then revise the demand distribution according to the presale volume.

The use of appropriate forecast models to predict market demand is extremely important for inventory management. To control inventory costs, managers rely heavily on demand prediction and adjust the order quantity to reduce inventory risks and improve inventory performance. Many demand forecasting methods have been widely studied by scholars [25,26,27,28]. Although in a real-life market, retailers can set initial expectations of market demand using historical sales data, they cannot forecast the market demand with certainty. To reflect the variability of market demand, we utilize a Bayesian model to update market demand information. Using this approach enables managers to revise the demand distribution as new demand information becomes available [29]. Choi et al. [30] investigated an optimal two-stage ordering problem for seasonal products where the ordering costs in the second stage are uncertain. Consequently, the retailer ordered twice before the start of the regular sales season. Choi [29] built on Choi et al. [30]’s problem to include inventory and pricing decisions for fashion retailers. Liu et al. [31] investigated ordering decisions for a short-life-cycle product using a bayesian information update framework and demonstrated that a two-stage ordering decision is better than a quick response decision.

In this study, we focus on a retailer who sells products in a regular sales season and offers a presale service to increase profit. In this scheme, the retailer needs to determine the presale price and inventory allocation for presale and regular sale periods. When a retailer offers a presale service, the scenario is similar to a two-period stochastic inventory problem. Petruzzi and Dada [32] considered a single firm selling a product in two countries during a non-overlapping sales season. They optimized the retail price and stocking level for each of the two periods. When studying revenue management with a perishable product, Chew et al. [33] determined the retail price and inventory allocation for two periods, where the price increases with time and demand, depending on the price. Pan et al. [34] considered the situation where a product’s price changes over its lifetime and constructed a two-period model to determine the dynamic pricing and ordering policy where expected demand has a linear relationship with retail price. Zhou and Guo [35] established a two-stage model of presale and spot sale, and proposed a presale scheme with dual-driven factors of price and advertising. They investigated the impact of the reference price on the retailer’s decision-making behavior. In previous studies regarding the two-period issue, either demand during each period is assumed to have a linear relationship with price or the correlation between the demand in each period is given. In this study, we consider a two-period newsvendor model with a dynamic environment and determine the presale price and inventory allocation between presale and regular sales. While the retailer delivers the product to customers during each period in the traditional two-period model, products are not delivered to customers during the presale season. Instead, once the regular sales season starts, presold and regular goods are delivered to customers simultaneously.

This study makes the following contributions: First, many studies have independently addressed information updating and inventory allocation as a two-period problem, but only a few researchers [32,33,34,35] have considered these topics simultaneously. Because these two issues occur together in practice, we incorporate information updating into the two-period stochastic inventory problem. Second, the main reasons retailers employ presales are to predict demand for the regular sales season and to attract new customers. This study considers both of these motivations. However, when a retailer provides a presale service at a discounted price, it may damage its profit level. To avoid this, we determine an optimal inventory allocation and pricing strategy for retailers who intend to offer presales. Third, by considering information updating and inventory allocation simultaneously, we can obtain better managerial insights about market characteristics. The retailer who utilizes information updates can obtain market information from risk-neutral customers. However, the retailer can benefit more from risk-averse customers than risk-neutral customers.

3. Problem Description

The main hypothesis of this study is that updating information for demand through a presale system allows retailers to make higher profits with less inventory. For this reason, we address a two-period newsvendor model with a dynamic environment and determine the presale price and the inventory allocation between presale and regular sale periods. To define the problem, we focus on two main objectives. The first one is the demand update problem, and the second one is the market competition.

For the demand update problem, we suppose that uncertain demand stems from the randomness of consumer demand for the characteristics of the product. It is assumed that demand x follows a normal distribution with mean $\delta$ and standard deviation $\tau$, where mean $\delta$ is determined by $x|\delta :g\left(x|\delta \right)\sim N\left(\delta ,{\tau }^2\right)$. Another source of uncertainty in demand comes from the randomness of demand expectation $\delta$, which also follows a normal distribution with mean $\mu$ and standard deviation $\sigma$, $\theta :h\left(\delta \right)\sim N\left(\mu ,{\sigma }^2\right)$. Combining these, as shown in Figure 1, the prior distribution of total market demand is given by $x:f\left(x\right)={\int }_0^{\infty }g\left(x|\delta \right)·h\left(\delta \right)d\delta =N\left(\mu ,\left({\sigma }^2+{\tau }^2\right)\right)$. According to the retailer’s market experience, the retailer knows the distribution $x_1|\delta :g\left(x|\delta \right)\sim N\left(\delta ,{\tau }^2\right)$ and ${\delta }_1:h\left({\delta }_1\right)\sim N\left({\mu }_1,{{\sigma }_1}^2\right)$. Thus, the distribution of the total market demand that a retailer generally expects is $f_1\left(x_1\right)\sim N\left({\mu }_1,\left({{\sigma }_1}^2+{\tau }^2\right)\right)$. If there are no retailers providing presale services that only sell products during the regular sale period, the demand of the market is $x_1$. To predict the market demand more accurately, the retailer can offer presales at the lower price $P_1$. When Retailer A offers presales, it can observe market information $x_s$ during the presale period, and it updates expected product demand from ${\delta }_1$ to ${\delta }_2$. The retailer also updates the market marginal demand distribution from $f_1\left(x_1\right)$ to $f_2\left(x_2\right)$. Thus, the marginal distribution of ${\mu }_2$ is defined as ${\mu }_2\sim N\left({\mu }_1,\frac{{{\sigma }_1}^4}{{{\sigma }_1}^2+{{\tau }_1}^2}\right)$, where $h\left({\delta }_2\right)\sim N\left({\mu }_2,{{\sigma }_2}^2\right),{\mu }_2=\frac{{{\sigma }_1}^2{x}_s+{\mu }_1{\tau }^2}{{{\sigma }_1}^2+{\tau }^2},{\sigma }_2=\sqrt{\frac{{{\sigma }_1}^2{\tau }^2}{{{\sigma }_1}^2+{\tau }^2}}$, $f_2\left(x_2\right)\sim N\left({\mu }_2,\left({{\sigma }_2}^2+{\tau }^2\right)\right)$.

For the market competition, we assume that retailers sell a homogeneous product to identify the effectiveness of the presale service. We suppose that total market demand for the product is a normal distribution and the demand for each retailer (market share) is defined as the proportion of the total market demand. As shown in Figure 2, the total market demand $D_T$ follows a normal distribution with mean ${\mu }_1$ and variance $\left({{\sigma }_1}^2+{\tau }^2\right)$: $D_T:f_T\left(x_T\right)\sim N\left({\mu }_1,\left({{\sigma }_1}^2+{\tau }^2\right)\right)$. When there are no presales in the market, Retailer A has proportion $\theta$ of the total market demand $D_T$, while demand for all of the other competing retailers B has the proportion $(1-\theta )$. Assuming that the demand for Retailer A and the demand for Retailer B are independent, the total demand for Retailer A is $D_A=\theta D_T$ and follows a normal distribution with $f_A\left(x_A\right)\sim N\left(\theta {\mu }_1,\theta \left({{\sigma }_1}^2+{\tau }^2\right)\right)$. Following the same logic, demand for Retailer B is $D_B=\left(1-\theta \right)D_T$ and follows a normal distribution: $f_B\left(x_B\right)\sim N\left(\left(1-\theta \right){\mu }_1,\left(1-\theta \right)\left({{\sigma }_1}^2+{\tau }^2\right)\right)$. We suppose that Retailer A provides a presale service at lower price $P_1\left(P_1\le P_2\right)$. At the beginning of presale, retailer A will inform the customers of the presale price $P_1$ and the regular sale price $P_2$. Because Retailer A provides presales at price $P_1\left(P_1<P_2\right)$, this will lead proportion $F_B$ of the customers for Retailer B to buy the presale product, so $F_B{D}_B$ customers from the competing companies will move to Retailer A. Following the same logic, the proportion $F_A$ of customers from Retailer A will buy the presale product instead of the regular sale product. Thus, presale demand consists of the proportion of customers from Retailer A and the proportion of customers from Retailer B. Because Retailer A and B sell the homogeneous product, it can be assumed that $F_A=F_B=F$.

At the beginning of the presale period, according to the general sales record, Retailer A believes that the total market demand is $D_T$. Therefore, Retailer A’s total expected demand in the presale period is $D_1=F_A{D}_A+F_B{D}_B=F{D}_T$, where $D_1:f_1\left(x_1\right)\sim N\left(F{\mu }_1,F\left({{\sigma }_1}^2+{\tau }^2\right)\right)$. Therefore, $\left(1-F_A\right)D_A$ customers still purchase the product during the regular sales season at price $P_2$, where $D_A=\theta D_T$. With a presale service, Retailer A can update the market demand distribution $D_T$ to ${\widehat{D}}_T$ according to market information $x_s$ obtained during the presale period. Therefore, with presales, Retailer A’s updated regular sales demand distribution and updated total market demand distribution are $D_2^{\prime }:f_2\left(x_2\right)\sim N\left(\left(1-F\right)\theta {\mu }_2,\left(1-F\right)\theta \left({{\sigma }_2}^2+{\tau }^2\right)\right)$, where ${\mu }_2=\frac{{{\sigma }_1}^2{x}_s+{\mu }_1{\tau }^2}{{{\sigma }_1}^2+{\tau }^2},{\sigma }_2=\sqrt{\frac{{{\sigma }_1}^2{\tau }^2}{{{\sigma }_1}^2+{\tau }^2}}$, and ${\widehat{D}}_T:{\widehat{f}}_T\left({\widehat{x}}_T\right)\sim N\left({\mu }_2,\left({{\sigma }_2}^2+{\tau }^2\right)\right)$.

3.1. Without a Presale Service

As mentioned in Section 3, according to the retailer’s market experience, Retailer A knows the total market demand distribution of $x_1|\delta :g\left(x|\delta \right)\sim N\left(\delta ,{\tau }^2\right)$ and the randomness of demand expectation ${\delta }_1:h\left(\delta \right)\sim N\left({\mu }_1,{{\sigma }_1}^2\right)$. Thus, the marginal distribution of the total market demand follows $D_T:f_T\left(x_T\right)\sim N\left({\mu }_1,\left({{\sigma }_1}^2+{\tau }^2\right)\right)$. Because Retailer A has $\theta$ proportion from the total market demand, if Retailer A does not offer presale and only sells products during the regular sale period, total demand is $D^N=\theta D_T$, where $D^N:f_A\left(x_A\right)\sim N\left(\theta {\mu }_1,\theta \left({{\sigma }_1}^2+{\tau }^2\right)\right)$.

As shown in Figure 3, the total profit during the regular sale period without a presale service is therefore

$$\begin{array}{c}\hfill {\pi }^N=P_{2}min\left(x_A,Q^N\right)+v{\left(Q^N-x_A\right)}^{+}-c{Q}^N,\end{array}$$

(1)

$$\begin{array}{c}\hfill E\left({\pi }^N\right)=\left(P_2-v\right)\left({\int }_0^{Q^N}{x}_1{f}_1\left(x_1\right)d{x}_1+{\int }_{Q^N}^{\infty }{Q}^N{f}_1\left(x_1\right)d{x}_1\right)-\left(c-v\right)Q^N.\end{array}$$

(2)

With the classical newsvendor model, we can derive the following optimal ordering policy.

Theorem 1.

Expected total profit $E\left({\pi }^N\right)$ is strictly concave with $Q^N$. Thus, the total order quantity is $Q^{N*}$, where $Q^{N*}=\theta {\mu }_1+\sqrt{\theta \left({{\sigma }_1}^2+{\tau }^2\right)}{\mathsf{\Phi }}^{-1}\left(\frac{P_2-c}{P_2-v}\right)$. The retailer’s maximum total profit is

$$\begin{array}{c}\hfill E\left({\pi }^{N*}\right)=\left(P_2-c\right)\theta {\mu }_1-\left(P_2-v\right)\sqrt{\theta \left({{\sigma }_1}^2+{\tau }^2\right)}\varphi \left({\mathsf{\Phi }}^{-1}\left(\frac{P_2-c}{P_2-v}\right)\right).\end{array}$$

(3)

3.2. With a Presale Service

When Retailer A offers a presale service at price $P_1\left(P_1\le P_2\right)$, it can attract new customers from competing companies, and customers of Retailer A will switch from the regular sale period to the presale period.

At the beginning of presales, according to the general sales record, Retailer A believes that the total market demand is $D_T$. Thus, the Retailer A’s total demand during the presale period is

$$\begin{array}{c}\hfill \begin{array}{c}\hfill D_1=F_A{D}_A+F_B{D}_B=F{D}_T,\\ \hfill where{D}_1:f_1\left(x_1\right)\sim N\left(F{\mu }_1,F\left({{\sigma }_1}^2+{\tau }^2\right)\right),D_T:f_T\left(x_T\right)\sim N\left({\mu }_1,\left({{\sigma }_1}^2+{\tau }^2\right)\right).\end{array}\end{array}$$

(4)

The total profit from presales is

$$\begin{array}{c}\hfill {\pi }_1=\left\{\begin{array}{c}{p}_1{x}_1-c{q}_1,\left(x_1<q_1\right)\hfill \\ \left(p_1-c\right)q_1,\left(q_1\le x_1\right)\hfill \end{array}\right..\end{array}$$

(5)

In the regular sales season, the retailer prepares $q_2$ for regular sales demand. If there are any leftover products at the end of the presale period, they will be transferred to regular sales. Thus, the total inventory for regular sales is $Q_2=L+q_2,whereL={\left(q_1-x_1\right)}^{+}$. The total expected profit for an entire sales season is

$$\begin{array}{c}\hfill {\pi }^P={\int }_0^{q_1}\left(x_1{P}_1+E\left[{\pi }_{2|L>0}\right]\right)f_1\left(x_1\right)d{x}_1+{\int }_{q_1}^{\infty }\left(q_1{P}_1+E\left[{\pi }_{2|L=0}\right]\right)f_1\left(x_1\right)d{x}_1-c{q}_1.\end{array}$$

(6)

$E\left[{\pi }_{2|L>0}\right]$ is the regular sales profit when there are products left over from presales, and $E\left[{\pi }_{2|L=0}\right]$ is the profit when there are no leftover products. Because the objective is to maximize the total profit, the optimal presale price and the optimal inventory allocation between the presale and the regular sale season must be determined. However, to solve ${\pi }^P$, the profit function for the regular sale period ${\pi }_2$ must be solved. Therefore, dynamic programming is applied to solve the recursive problem and prove the optimal solution for the retailer. To maximize the total profit, the profit function for the regular sale period is solved first to determine the optimal ${q_2^{\prime }}^{*}$ followed by the optimal $q_1^{*}$ and $p_1^{*}$. To study the impact of demand updating on optimal decisions during regular sales, we analyzed regular sales under two scenarios. The first is that there is no demand updating even though the retailer provides presales. It is not uncommon to find this scenario. When the retailer does not analyze the sales data correctly, it cannot forecast the demand for the upcoming season accurately. The second scenario is that there is demand updating during the presale period shown in Figure 4.

If there is no information updating for regular sales demand, the demand distribution of regular sales is

$$\begin{array}{c}\hfill \begin{array}{c}\hfill D_2=\left(1-F\right)D_A=\theta \left(1-F\right)D_T,\\ \hfill where{D}_2:f_2\left(x_2\right)\sim N\left(\theta \left(1-F\right){\mu }_1,\theta \left(1-F\right)\left({{\sigma }_1}^2+{\tau }^2\right)\right).\end{array}\end{array}$$

(7)

If the retailer has the ability to update market demand through presales, we can analyze regular sales demand as follows. With presales, the retailer can acquire demand information $x_s$, and with a Bayesian approach, the retailer updates the expected demand from ${\delta }_1$ to ${\delta }_2$. With $h\left({\delta }_1\right)\sim N\left({\mu }_1,{{\sigma }_1}^2\right)$, $g\left(x|\delta \right)\sim N\left(\delta ,{\tau }^2\right)$, and $x_s$, it is possible to obtain $h\left({\delta }_2\right)\sim N\left({\mu }_2,{{\sigma }_2}^2\right)$, where ${\mu }_2=\frac{{{\sigma }_1}^2{x}_s+{\mu }_1{\tau }^2}{{{\sigma }_1}^2+{\tau }^2}$ and ${\sigma }_2=\sqrt{\frac{{{\sigma }_1}^2{\tau }^2}{{{\sigma }_1}^2+{\tau }^2}}$. Furthermore, with distribution $x|\delta :g\left(x|\delta \right)\sim N\left(\delta ,{\tau }^2\right)$ and $h\left({\delta }_2\right)$, the retailer can update the total market demand distribution from $D_T$ to ${\widehat{D}}_T$ as well, which is ${\widehat{D}}_T^{\prime }:{\widehat{f}}_T\left({\widehat{x}}_T\right)\sim N\left({\mu }_2,\left({{\sigma }_2}^2+{\tau }^2\right)\right)$.

Because Retailer A’s regular sales represent $\theta \left(1-F\right)$ of the total market demand, we can infer that Retailer A’s updated regular sales demand is $D_2^{\prime }=\theta \left(1-F\right)\widehat{D_T}$, and it follows $D_2^{\prime }:\widehat{f_2}\left(\widehat{x_2}\right)\sim N\left(\theta \left(1-F\right){\mu }_2,\theta \left(1-F\right)\left({{\sigma }_2}^2+{\tau }^2\right)\right)$.

Theorem 2.

(a) Information updating with a presale service can increase the accuracy of forecasting, and the retailer can achieve greater profit with a lower inventory with demand updating. (b) The retailer has a unique optimal order quantity for regular sales when the market demand is updated during the presale period, which is

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {q_2^{\prime }}^{*}=\left\{\begin{array}{c}0,\left(0\le x_1\le q_1-{Q_2^{\prime }}^{*}\right)\hfill \\ {Q_2^{\prime }}^{*}-\left(q_1-x_1\right),\left(q_1-{Q_2^{\prime }}^{*}<x_1\le q_1\right)\hfill \\ {Q_2^{\prime }}^{*},\left(q_1<x_1\right)\hfill \end{array}\right.,\\ \hfill where{Q_2^{\prime }}^{*}=\left(1-F\right)\theta {\mu }_2+\sqrt{\theta \left(1-F\right)\left({{\sigma }_2}^2+{\tau }^2\right)}{\mathsf{\Phi }}^{-1}\left(\frac{P_2-c}{P_2-v}\right).\end{array}\end{array}$$

(8)

(c) The retailer’s profit with regular sales with information updating is

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\pi }_2^{\prime }=\left\{\begin{array}{c}{\pi }_2^{\prime }\left(q_1-x_1\right)+c\left(q_1-x_1\right),\left(0\le x_1\le q_1-{Q_2^{\prime }}^{*}\right)\hfill \\ {\pi }_2^{\prime }\left({Q_2^{\prime }}^{*}\right)+c\left(q_1-x_1\right),\left(q_1-{Q_2^{\prime }}^{*}<x_1\le q_1\right)\hfill \\ {\pi }_2^{\prime }\left({Q_2^{\prime }}^{*}\right),\left(q_1<x_1\right)\hfill \end{array}\right.,whereE\left[{{\pi }_2^{\prime }}^{*}\left({Q_2^{\prime }}^{*}\right)\right]=\\ \hfill \left(1-F\right)\theta \left(P_2-c\right){\mu }_2-\left(P_2-v\right)\sqrt{\theta \left(1-F\right)\left({{\sigma }_2}^2+{\tau }^2\right)}\varphi \left({\mathsf{\Phi }}^{-1}\left(\frac{P_2-c}{P_2-v}\right)\right),\\ \hfill andE\left[{\pi }_2^{\prime }\left(q_1-x_1\right)\right]=\left(P_2-c\right)\left(q_1-x_1\right)-\left(P_2-v\right){\int }_0^{q_1-x_1}\left(q_1-x_1-\widehat{x_2}\right)\widehat{f_2}\left(\widehat{x_2}\right)d\widehat{x_2}.\end{array}\end{array}$$

(9)

Proof of Theorem 2.

See Appendix A. □

From Theorem 2, we can assume that the expected order quantity for regular sales when information updating is employed is

$$\begin{array}{c}\hfill E\left[{q_2^{\prime }}^{*}|{q_1}^{*}\right]={\int }_{{q_1}^{*}-{Q_2^{\prime }}^{*}}^{{q_1}^{*}}\left({Q_2^{\prime }}^{*}-\left({q_1}^{*}-x_1\right)\right)f_1\left(x_1\right)d{x}_1+{\int }_{{q_1}^{*}}^{\infty }{Q_2^{\prime }}^{*}{f}_1\left(x_1\right)d{x}_1.\end{array}$$

(10)

After analyzing the optimal inventory allocation for regular sales, the optimal inventory allocation for the presale period can be addressed. In general, if a retailer provides presales at a lower price, many consumers will tend to buy products during the presale period. Thus, if the retailer satisfies all of the demand during the presale period, its profit may be damaged. To avoid this, the retailer always limits the quantity of presale in practice. For example, McDonald’s will inform the customers of the time, price, and quantity of a promotion, and once the stated quantity has sold out, it will stop the campaign even if the campaign has not ended. It can thus be assumed that a certain probability exists that a customer will not obtain the product on sale because of promotion restrictions. Thus, in this study, we assume that the retailer will not satisfy all of the demand during the presale period. We thus propose the optimal inventory levels for presale products, which maximizes total profit.

In the beginning of the presale period, the retailer will inform the customers about the presale price and quantity without market information for the upcoming season. Based on this logic, it can be assumed that when the retailer determines presales quantities $q_1$, it does not know the market information. Thus, regular sales demand follows the distribution before updating: $D_2:f_2\left(x_2\right)\sim N\left(\theta \left(1-F\right){\mu }_1,\theta \left(1-F\right)\left({{\sigma }_1}^2+{\tau }^2\right)\right)$. The expected total profit function when the retailer decides $q_1$ is

$$\begin{array}{c}\hfill \begin{array}{c}\hfill E\left[{\pi }^P\right]={\int }_0^{q_1}\left(p_1{x}_1-c{q}_1\right)f_1\left(x_1\right)d{x}_1+{\int }_{q_1}^{\infty }\left(p_1-c\right)q_1{f}_1\left(x_1\right)d{x}_1\\ \hfill +{\int }_0^{q_1-{Q_2}^{*}}E\left[{\pi }_2\left(q_1-x_1\right)+c\left(q_1-x_1\right)\right]f_1\left(x_1\right)d{x}_1\\ \hfill +{\int }_{q_1-{Q_2}^{*}}^{q_1}E\left[{\pi }_2\left({Q_2}^{*}\right)+c\left(q_1-x_1\right)\right]f_1\left(x_1\right)d{x}_1\\ \hfill +{\int }_{q_1}^{\infty }E\left[{\pi }_2\left({Q_2}^{*}\right)\right]f_1\left(x_1\right)d{x}_1.\end{array}\end{array}$$

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The first and second expressions are the profit from the presale service. The third, fourth, and fifth terms are the expected profit generated from regular sales. The total profit function can be rewritten as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill E\left[{\pi }^P\right]=\left(P_1-c\right)q_1-\left(P_1-c\right){\int }_0^{q_1}\left(q_1-x_1\right)f_1\left(x_1\right)d{x}_1\\ \hfill +{\int }_0^{q_1-{Q_2}^{*}}E\left[{\pi }_2\left(q_1-x_1\right)\right]f_1\left(x_1\right)d{x}_1\\ \hfill +{\int }_{q_1-{Q_2}^{*}}^{\infty }E\left[{\pi }_2\left({Q_2}^{*}\right)\right]f_1\left(x_1\right)d{x}_1.\end{array}\end{array}$$

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Theorem 3.

$E\left[{\pi }^P\right]$ is strictly concave with $q_1$, and the optimal ordering quantity ${q_1}^{*}$ at the beginning of presale is given as ${q_1}^{*}={arg}_{q_1}\left\{J\left(q_1\right)=0\right\}$, where $J\left(q_1\right)=1-F_1\left(q_1\right)+\frac{1}{p_1-c}{\int }_0^{q_1-{Q_2}^{*}}\left[\left(P_2-c\right)-\left(P_2-v\right)F_2\left(q_1-x_1\right)\right]f_1\left(x_1\right)d{x}_1$.

Proof of Theorem 3.

See Appendix B. □

From Theorem 3, it is confirmed that there exists an optimal order quantity at the beginning of the presale period, and ${q_1}^{*}$ can easily be found by solving $J\left(q_1\right)$ function. After ${q_1}^{*}$ is determined, the retailer will provide presales at the presale price $P_1$, obtain market information $x_s$, and update the demand distribution from $D_2$ to $D_2^{\prime }$. With the distribution of $D_2^{\prime }$ and Theorem 2, the retailer orders $E\left[{q_2^{\prime }}^{*}|{q_1}^{*}\right]$ and maximizes its total profit.

According to Theorems 2 and 3, the total optimal order quantity with a presale service is defined as

$$\begin{array}{c}\hfill Q^{P^{*}}={q_1}^{*}+E\left[{q_2^{\prime }}^{*}|{q_1}^{*}\right].\end{array}$$

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In addition, the total expected profit the retailer achieves during the sales season is

$$\begin{array}{c}\hfill \begin{array}{c}\hfill E\left[{\pi }^P\right]=\left(P_1-c\right){q_1}^{*}-\left(P_1-c\right){\int }_0^{{q_1}^{*}}\left({q_1}^{*}-x_1\right)f_1\left(x_1\right)d{x}_1\\ \hfill +{\int }_0^{{q_1}^{*}-E\left[{q_2^{\prime }}^{*}|{q_1}^{*}\right]}E\left[{\pi }_2^{\prime }\left({q_1}^{*}-x_1\right)\right]f_1\left(x_1\right)d{x}_1\\ \hfill +{\int }_{{q_1}^{*}-E\left[{q_2^{\prime }}^{*}|{q_1}^{*}\right]}^{\infty }E\left[{\pi }_2^{\prime }\left(E\left[{q_2^{\prime }}^{*}|{q_1}^{*}\right]\right)\right]f_1\left(x_1\right)d{x}_1.\end{array}\end{array}$$

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3.3. Optimal Pricing Strategy

The two main reasons for providing presales are forecasting the demand for the regular season and attracting additional customers. To obtain demand information during presales, the retailer has to provide a presale price to induce customers to buy products during the presale period. However, when the retailer provides presales at a discounted price, it may hurt its profit. To avoid this, the optimal presale price during the presale period needs to be determined.

It is assumed that the discount rate is $r=\frac{P_1}{P_2}$, where $0\le r\le 1$. If customers feel that the presale price is acceptable given the waiting time, they will pre-order at price $P_1=r{P}_2$. If customers are not sure about their need for the product or have doubts about its quality, the customers may purchase the product during the regular sales season at price $P_1$.

To derive the optimal discount rate, we set $F=1-A{r}^f$, which is the deterministic exponential sales response function [17] widely used in marketing research. In this function, f is the price sensitivity parameter. $0\le f\le 1$ means consumers are not sensitive to price. Thus, if a retailer wants to achieve greater presales, it has to provide a larger discount. In contrast, $1<f$ means customers are likely to buy the presale product even when only a small discount is given. Since $0\le F<1$ forces $0<A\le 1$, when $A=0$, there is no demand transformation regardless of how much discount a retailer provides. Because this is not encountered in real life, we do not consider this case in the paper.

With discount rate r and the response function, we can rewrite the retailer’s inventory allocation quantity ${q_1}^{*}$ and ${Q_2}^{*}$. The retailer decides the presale price before obtaining market information. Thus, when we analyze the optimal discount rate r, there is no information updating in the total profit function. From Theorem 2, the optimal available quantity for regular sales defined with the discount rate is ${Q_2}^{*}=A{r}^f\theta {\mu }_1+\sqrt{A{r}^f\theta \left({{\sigma }_1}^2+{\tau }^2\right)}{\mathsf{\Phi }}^{-1}\left(\frac{P_2-c}{P_2-v}\right)$.

With Theorem 3, we can infer that the optimal inventory allocation ${q_1}^{*}$ defined with the discount rate is ${q_1}^{*}\left(r\right)={arg}_{q_1}\left\{J\left(q_1\left(r\right)\right)=0\right\}$. Therefore, the total profit equation can be defined as r, and the optimal discount rate that maximizes total profit can be proved. The expected total profit for the retailer at the beginning of presale is $E\left[{\pi }^P\right]=\left(P_1-c\right)q_1-\left(P_1-c\right){\int }_0^{q_1}\left(q_1-x_1\right)f_1\left(x_1\right)d{x}_1+{\int }_0^{q_1-{Q_2}^{*}}E\left[{\pi }_2\left(q_1-x_1\right)\right]f_1\left(x_1\right)d{x}_1+{\int }_{q_1-{Q_2}^{*}}^{\infty }E\left[{\pi }_2\left({Q_2}^{*}\right)\right]$ $f_1\left(x_1\right)d{x}_1$.

To analyze the optimal pricing strategy, the total profit has to be defined with the discount rate r. The expected total profit function can be rewritten as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill E\left[{\pi }^P\left(r\right)\right]=\left(r{P}_2-c\right)q_1-\left(r{P}_2-c\right){\int }_0^{q_1\left(r\right)}\left(q_1\left(r\right)-x_1\right)f_1\left(x_1\right)d{x}_1\\ \hfill +{\int }_0^{q_1\left(r\right)-Q_2\left(r\right)}E\left[{\pi }_2\left(q_1\left(r\right)-x_1\right)\right]f_1\left(x_1\right)d{x}_1\\ \hfill +{\int }_{q_1\left(r\right)-Q_2\left(r\right)}^{\infty }E\left[{\pi }_2\left(Q_2\left(r\right)\right)\right]f_1\left(x_1\right)d{x}_1.\end{array}\end{array}$$

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Theorem 4.

Using standard branch, the optimal discount rate is expressed with the Riemann surface. The optimal discount rate $r^{*}$ is given as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill r^{*}=\frac{cB-c+{\int }_0^{-U{\mu }_1\theta -W\sqrt{U\theta V}+q_1}\frac{\left(P_2-c-\left(P_2-v\right)\left(H+T\right)\right)N}{\sqrt{2\pi V\left(1-U\right)}}d{x}_1}{P_2\left(B-1\right)}.\end{array}\end{array}$$

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Proof of Theorem 4.

See Appendix C. □

In summary, when retailers provide a presale service, it need to set the discount rate and allocate inventory at the beginning of presales. Once retailers can gauge the actual market demand during the presale period, they revise the order quantity for regular sales. Using the theorems above, we have proven that presales can reduce demand uncertainty. We also proposed the optimal presale price and inventory allocation for the presale and regular sale periods in a competitive market.

4. Numerical Analysis

In order to compare the total profit with and without presales and to gain greater insights into the presale system, a numerical study is conducted to illustrate the benefits of presales for the retailer.

In this study, the proposed equations are processed using an Intel Core i7-7700HQ at 2.80 GHz with 16 GB of RAM under Windows 10. We ran Sympy to solve the symbolic computation problem. Equations in this study were programmed using Python.

The parameter values are set as $P_2=20$, $c=12$, $v=8$, ${{\sigma }_1}^2=25$, ${\tau }^2=9$, ${\mu }_1=400$, $A=0.8$, $f=0.3$, $\theta =0.8$, $x_1=90$, and $x_s=[380,420]$. From Theorem 4, we can propose the optimal discount rate ($r^{*}=0.89$).

The information about the total market is represented by $x_s$, and the regular sales account for $\theta \left(1-F\right)$ of the total market. As a result, $x_s\theta \left(1-F\right)$ represents the information about regular sales. With a presale service, the retailer obtains demand information during the presale period and revises the inventory allocation for the regular sale period.

Table 1. Impact of presales on the retailer’s profits.

With Presales				Without Presales
${\mathit{x}}_{\mathit{s}}\mathit{\theta }\left(\mathbf{1}-\mathit{F}\right)$	${\mathit{q}}_{\mathbf{1}}$	${{\mathit{q}}_{\mathbf{2}}^{\prime }}^{*}$	$\mathit{E}\left[{\mathit{\pi }}^{\mathit{P}}\right]$	${\mathit{Q}}^{\mathit{N}*}$	$\mathit{E}\left[{\mathit{\pi }}^{\mathit{N}}\right]$
235	232	97	2564.66	322	2537.24
241	232	102	2601.01	322	2537.24
247	232	106	2637.37	322	2537.24
253	232	111	2673.72	322	2537.24
260	232	116	2710.07	322	2537.24

illustrates the impact of presales, with which the retailer can achieve higher profits due to the discounted price and information updating.

$x_1+x_s\theta \left(1-F\right)$ is the actual demand for a retailer who provides presales service. From

Table 2. Impact of information updating on inventory allocation.

${\mathit{x}}_1+{\mathit{x}}_{\mathit{s}}\mathit{\theta }\left(1-\mathit{F}\right)$	${\mathit{q}}_1$	${{\mathit{q}}_2}^{*}$	${{\mathit{q}}_2^{\prime }}^{*}$	${\mathit{Q}}^{{\mathit{P}}^{*}}$	${{\mathit{Q}}^{\mathit{P}}}^{\prime *}$	${{\mathit{Q}}^{\mathit{N}}}^{*}$
325	232	107	97	339	330	322
331	232	107	102	339	334	322
337	232	107	106	339	339	322
344	232	107	111	339	343	322
350	232	107	116	339	348	322

, the retailer’s total ordering quantity for the sales season can be compared. ${Q^P}^{\prime *}$ and ${Q^P}^{*}$ are the total ordering quantities for presales with and without information updating respectively. ${Q^N}^{*}$ is total ordering quantity without presales service. When comparing ${Q^P}^{*}$, ${Q^P}^{\prime *}$, and ${Q^N}^{*}$ to the actual demand, we find that presales with information updating predicts actual demand more accurately. This analysis is consistent with the results corresponding to Theorem 2.

Figure 5 presents the result of sensitivity analysis for market share $\theta$, showing that the retailer’s market share and presales profit have linear relationship. The profit growth rate increases with demand information $x_s$.

The results shown in Figure 6 also confirm that the retailer’s total profit is proportional to demand information $x_s$. In contrast, price sensitivity parameter does not significantly affect the total profit. This can be explained by the choice function $F=1-A{r}^f$. Because the degree of risk averseness of A ranges between 0 and 1, price sensitivity parameter has little effect on the total profit.

To study the impact of the risk averseness of customers on presale profits, we examined the impact of A on the retailer’s total profits and varied the market share to analyze different power structures for the retailer. From the choice function, as shown in Figure 7, we know that F decreases with the customers’ risk aversion (i.e., total profit decreases with F). In real markets, retailers can obtain market information using presale. However, risk-averse customers will tend to buy the product during the regular sales season at full price. Thus, the retailer’s profit will increase.

5. Conclusions

As consumer demand diversifies, minimizing inventory management costs through accurate demand forecasting has become an important issue, especially for retailers who sell items that are to be released. Presale service is an effective strategy for aligning inventory with demand, and this service also attracts customers from competing companies. Therefore, we propose an optimal inventory allocation and pricing strategy for retailers by incorporating demand information into the issue of inventory allocation during the presale period.

In this study, an analysis of two different selling strategies determined that presale services were more profitable for retailers. Furthermore, numerical results provide managerial insight for retailers applying presale services. First, presales with information updating predict market demand more accurately and achieve higher profits with less inventory. When releasing a new product, the retailer faces market demand uncertainty because no historical data exists on the product. Therefore, presale services can play an important role in identifying consumer responses, which is very helpful for efficient inventory management. Second, although the retailer’s total profits increase with market share, the price sensitivity parameter does not significantly affect profit. For products that are sensitive to price increase sales through discounts regardless of how large the discount is, there is a limit to generating large-scale profits. Third, the retailer benefits more from risk-averse customers than risk-neutral customers. Risk-averse customers prefer to purchase quality-guaranteed products, even if prices are high. Therefore, retailers can obtain market signals without providing a large discount rate during the presale period and achieve profits by reducing inventory costs.

In conclusion, the main hypothesis of this study maintains that retailers can reduce the uncertainty of demand and efficiently manage inventory through a presale system. Furthermore, the presale strategy, which considers customer characteristics, increases the retailer’s profit by securing market competitiveness. Therefore, this study provides a useful framework for the analysis of the benefits of a presale system when retailers have access to demand information about a product. However, in this study, we did not consider consumers’ return behaviors and assumed that demand among retailers selling a homogeneous product was independent. In the future, it would be interesting to allow consumers to return goods in the presale system and to consider the relationship between market share and retailers in detail.