3.1. Model Description and Assumptions
An omnichannel brand retailer sells a product to consumers through an online channel (simplify O channel), a store channel (simplify S channel), and a BOPS channel. Customers who purchase through the O channel need to bear the cost of shipping and waiting time. Customers who purchase through the BOPS channel and S channel need to bear the inconvenience costs of the store, but when they purchase through the BOPS channel, there is no time cost for finding the product and waiting for checkout and packaging, as there is when shopping through the S channel. Therefore, customers are less inconvenienced when they shop through the BOPS channel.
When customers shop through the S channel, they are able to experience the product before buying, so the return rate is extremely low. On the other hand, when customers shop through the O channel and the BOPS channel, the return rate is high due to lack of experience before purchasing, and customers have the same probability of return under the O channel and the BOPS channel. If a customer is not satisfied with the purchased product from the O channel, he can apply for a return online and courier the product to the retailer at his own expense. The customer needs to pay extra costs such as return shipping and waiting for the refund time [
51]. If a customer who purchased through the BOPS channel is not satisfied with the product when he picks it up in store, he can return it directly at the store.
The notations used in the model and their definitions are shown in
Table 1.
To simplify calculations and analysis, we make the following assumptions.
- (1)
Since the return rate under the O channel is much higher than the return rate under the S channel [
38,
52], without loss of generality, we assume that the return rate under S channel is 0.
- (2)
Assume that each consumer’s psychological valuation of the product is different, and the valuation is a random variable that follows a uniform distribution in the interval . Its probability density function and cumulative distribution function are and respectively, . The critical psychological valuation of consumers in various channels does not exceed the highest valuation of the distribution interval .
- (3)
Assume that the salvage returned under the O channel and BOPS channel is zero.
- (4)
The shopping cost of consumers under different channels , , and is much lower than the lowest estimate of the distribution interval .
- (5)
The shopping cost under an O channel is higher than the inconvenient cost under an S channel, i.e., .
- (6)
The probability of return under an O channel is lower than the convenience of a BOPS channel relative to an S channel, i.e.,.
According to reference [
53], we can get the consumer surplus when purchasing under different channels as shown in
Table 2.
As can be seen from
Table 2, when considering online returns
, the consumer surplus under O channel is
, and the condition for consumers to purchase under this channel is
. Therefore, the critical psychological value of consumers buying the product under O channel is
, that is , when
, consumers will buy from O channel. the consumer surplus under S channel is
, and the condition for consumers to purchase under this channel is
. Therefore, the critical psychological value of consumers buying the product under S channel is
, that is , when
, consumers will buy from S channel. The consumer surplus under BOPS channel is
, and the condition for consumers to purchase under this channel is
. Therefore, the critical psychological value of consumers buying the product under BOPS channel is
, that is , when
, consumers will buy from a BOPS channel.
Next, we will build the retailer’s pricing and ordering decision model before and after opening BOPS channel, prove the existence of the optimal solution, and obtain the optimal pricing and ordering decisions, based on the behavior of consumers purchasing from different channels.
3.2. Construction and Solution of the Decision Model before Opening a BOPS Channel
Before opening a BOPS channel, the brand retailer only sells their products to consumers through O and S channels. According to the definitions of notations and assumptions in
Table 1, the proportion of the consumers who purchase using the O channel is
, when
, the actual purchase ratio under O channel is
. The proportion of the consumers who purchase under S channel is
, when
, the actual purchase ratio using the S channel is
. Therefore, the proportion of actual purchasers under the two channels of O and S is as follow.
If we assume that , then .
Since the probability density function of the total market demand is , then the probability density function of the actual demand of the two channels O and S is .
Furthermore, we obatin the retailer’s expected profit before opening BOPS channel as follows.
According to the above Equation (2), we write as , as , as , as , then we can get the following Theorem 1.
Theorem 1. Before opening a BOPS channel, when , is a concave function, and the retailer has optimal sales price and optimal order quantity to maximize expected profit.
Theorem 1 shows that under certain conditions, the retailer has optimal joint pricing and ordering decisions before opening BOPS channel.
Due to the complexity of the retailer’s expected profit model, here we use a two-stage optimization technique to find the optimal solution. First we give to find the optimal solution , then substitute into the profit function to find the optimal sales price , and finally substitute into to get the optimal order quantity .
When
is given, we let the first derivative of
with respect to
be 0, that is,
We substitute Equation (4) into expected profit Equation (2), and let the first derivative of
with respect to
be 0, that is,
Thus, we get the optimal sales price as follow.
Furthermore, we substitute the optimal price Equation (6) into Equation (4) to get the optimal order quantity as follows.
Therefore, before opening a BOPS channel, there are optimal joint decisions, as shown in Equations (6) and (7), to maximize the retailer’s expected profit.
According to the retailer’s optimal joint decision before opening a BOPS channel, we can get the following: Lemma 1 and Lemma 2.
Lemma 1. When is given, the optimal order quantity is positively related to the proportion of purchasers under O channel , and the proportion of purchasers under S channel is negatively related to the online return rate , the shopping costs under O channel , the inconvenient costs under S channel , and the maximum expected valuation of the product is negatively correlated with the minimum expected valuation of the product .
Lemma 1 states that, before opening a BOPS channel, the retailer only sells the product to consumers through O and S channels. When the ratio of purchasers under one channel is determined, the higher the ratio of purchasers under the other channel, the greater the total purchase quantity, and the higher the order quantity. Because the online return rate can reduce the customer’s willingness to purchase under the O channel, the shopping costs under the O and S channels also have a certain negative impact on the customer’s willingness to purchase. Therefore, the higher the online return rate and the channel shopping costs, the less the order quantity. The higher the consumer’s minimum expected value of the product, the more worthwhile the product is, so the larger the sales, the higher the order quantity. Conversely, the higher the consumer’s valuation of the product, the higher the price, and the lower the customer’s purchase quantity.
Lemma 2. The optimal price is negatively related to the proportion of purchasers under online channel , the online return rate , the shopping costs under O channel , and the inconvenient costs under S channel , is positively related to the proportion of purchasers under store channel and the maximum expected valuation of the product and has nothing to do with the minimum expected valuation of the product .
Lemma 2 shows that, due to the impact of the return rate, an appropriate price reduction can achieve the effect of small profits and long sales—that is, the proportion of purchasers under the online channel increases, while the S channel is the opposite. Because the customer decides whether to buy after the experience in the store, when the ratio of purchasers is high, the product itself is worth buying, even if the price increases appropriately. When the online return rate and channel shopping costs are high, a proper price reduction can attract consumers to buy. The higher the consumer’s maximum expected valuation of the product, the higher the value of the product itself, and the price should naturally be increased appropriately. Conversely, the product pricing mainly depends on production costs and profit margin and is not affected by the consumer’s minimum expected valuation of the product.
3.3. Construction and Solution of Decision Model After Opening a BOPS Channel
After opening sBOPS channel, the brand retailer sell the product to consumers through three channel—O, S, and BOPS. According to the definitions of notations and assumptions in
Table 1, the proportion of the consumers who purchase under O channel is
; when
, the actual purchase ratio under the O channel is
. The proportion of the consumers who purchase under S channel is
, when
, the actual purchase ratio under the S channel is
. The proportion of consumers who purchase under BOPS channel is
, when
, the actual purchase ratio under the BOPS channel is
.Therefore, the proportion of actual purchasers under the three channels of O, S, and BOPS as follows.
We assume , then .
Similar to
Section 3.2, since the probability density function of the total market demand
is
, then the probability density function of the actual demand of the three channels O, S ,and BOPS is
.
Furthermore, we obatin the retailer’s expected profit after opening a BOPS channel as follows.
According to Equation (9), we write as , as , as , as , we arrive at Theorem 2.
Theorem 2. After opening a BOPS channel, when , is a concave function, and the retailer has optimal sales price and optimal order quantity to maximize expected profit.
Theorem 2 shows that, under certain conditions, the retailer has optimal joint pricing and ordering decisions after opening a BOPS channel.
As in
Section 3.2, we still use the two-stage optimization technique to find the optimal solution. First, we give
to find the optimal solution
, then substitute
into the profit function to find the optimal sales price
, and finally substitute
into
to get the optimal order quantity
.
When
is given, we let the first derivative of
with respect to
be 0, that is,
We substitute equation (11) into expected profit equation (9), and let the first derivative of
with respect to
be 0, that is,
Thus, we get the following optimal sales price:
Furthermore, we substitute the optimal price Equation (13) into Equation (11) to get the optimal order quantity.
Therefore, after opening BOPS channel, there are optimal joint decisions as shown in Equations (13) and (14) to maximize the retailer’s expected profit.
According to the retailer’s optimal joint decision after opening a BOPS channel, we can get Lemma 3 and Lemma 4.
Lemma 3. When is given, the optimal order quantity is negatively related to the proportion of purchasers using the O channel , the online return rate , the shopping costs using the O channel , the inconvenient costs using the S channel , the inconvenience of customers shopping through the BOPS channel compared to the S channel , and the maximum expected valuation of the product , is positively related to the proportion of purchasers using the S channel and the minimum expected valuation of the product .
Unlike Lemma 1, the optimal order quantity before opening a BOPS channel is positively related to the proportion of purchasers under O channel, the optimal order quantity after opening a BOPS channel is negatively related to the proportion of purchasers using the O channel. This is because after opening a BOPS channel, the sum of the proportion of purchasers using the O, S, and BOPS channels is 1. When the proportion of purchasers using the S channel is determined, as the proportion of purchasers using the O channel increases, the proportion of purchasers using the BOPS channel inevitably decreases, that is, some customers who originally intended to buy on BOPS channel transfer to the S channel to purchase. However, because customers using the O channel need to bear the cost of return shipping and waiting for the refund, which affects their purchase demand, so the total purchase demand decreases with the increase of the proportion of purchasers using the O channel. Correspondingly, the retailer should reduce the order quantity to reduce the losses caused by the returns of the O channel and the BOPS channel. When the proportion of purchasers under the O channel is determined, since the S channel is not affected by the return rate, the higher the proportion of purchasers in this channel, the larger the sales and the order quantity. The correlation between the optimal order quantity and other parameters after opening a BOPS channel is the same as in Lemma 1.
Lemma 4. The optimal price is negatively related to the proportion of purchasers under O channel , the online return rate , the shopping costs under O channel , the inconvenient costs under S channel , the inconvenience of customers shopping through BOPS channel compared to S channel , is positively related to the proportion of purchasers under S channel and the maximum expected valuation of the product , and has nothing to do with the minimum expected valuation of the product .
Obviously, Lemma 2 and 4 show that before and after opening a BOPS channel, the correlation between the optimal sales price and other parameters is consistent.
According to Lemma 1 and Lemma 2 in
Section 3.2 and Lemma 3 and Lemma 4 in
Section 3.3, we can get the following conclusions.
Conclusion 1. With the increase in the proportion of O channel purchasers, before opening a BOPS channel, the optimal joint decision of the retailer is to increase the order quantity while lowering the price; after opening a BOPS channel, the retailer’s optimal joint decision is to reduce both the price and the order quantity of the product.
Conclusion 2. Regardless of whether a BOPS channel is opened or not, as the proportion of S channel purchasers increases, the optimal joint decision of the retailer is to simultaneously increase the price and order quantity of the product; with the increase in online return rate, channel shopping costs, and the inconvenience of the BOPS channel compared to the S channel, the retailer’s optimal joint decision is to simultaneously reduce product price and order quantity; with the increase in the highest valuation of the product, the optimal joint decision of the retailer is to reduce the order quantity while increasing the product price.
Obviously, the decision-making behaviors of the retailer’s optimal pricing and ordering in Conclusions 1 and 2 give the direct answer to the second research question. These decisions can enable retailers to meet consumer demand while avoiding a large inventory backlog, thereby maximizing benefits and enhancing the sustainability of healthy business development.