# Digital Coupon Promotion and Inventory Strategies of Omnichannel Brands

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background and Problem Description

#### 1.2. Contributions and Findings

## 2. Literature Review

#### 2.1. Joint Pricing and Inventory Decisions

#### 2.2. Coupon Promotions

#### 2.3. Omnichannel Retailing

## 3. Model

#### 3.1. Homogeneous Online and Store Hassle Costs

**Proposition 1.**

- (1)
- When ${h}_{o}<\mathrm{min}({h}_{b},{h}_{s})$and$p=\delta v-{h}_{o}$, the brand does not offer coupons, then${q}^{B*}=0$.
- (2)
- When${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$,$p=\delta v-{h}_{b}$and$v>\frac{{h}_{o}-{h}_{b}}{1-\delta}$, the brand does not offer coupons, and if$c<(1-\lambda )(\delta v-{h}_{b})+r$, then${q}^{B*}={\overline{F}}^{-1}(\frac{c}{(1-\lambda )(\delta v-{h}_{b})+r})$; otherwise,${q}^{B*}=0$. When${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$,$p=\delta v-{h}_{b}$and$v\le \frac{{h}_{o}-{h}_{b}}{1-\delta}$, the brand does not offer coupons, and if$c<\delta v-{h}_{b}+r$, then${q}^{B*}={\overline{F}}^{-1}(\frac{c}{\delta v-{h}_{b}+r})$; otherwise,${q}^{B*}=0$. When${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$and$p=\delta v-{h}_{o}$, the brand does not offer coupons, and if$c<r$,${q}^{B*}={\overline{F}}^{-1}(\frac{c}{r})$; otherwise,${q}^{B*}=0$.
- (3)
- When${h}_{s}<\mathrm{min}({h}_{o},{h}_{b})$,$p=\delta v-{h}_{s}$and$v>\frac{{h}_{o}-{h}_{s}}{1-\delta}$, the brand does not offer coupons, and if$c<(1-\lambda )(\delta v-{h}_{s})+r$, then${q}^{B*}={\overline{F}}^{-1}(\frac{c}{(1-\lambda )(\delta v-{h}_{s})+r})$; otherwise,${q}^{B*}=0$. When${h}_{s}<\mathrm{min}({h}_{o},{h}_{b})$,$p=\delta v-{h}_{s}$and$v\le \frac{{h}_{o}-{h}_{s}}{1-\delta}$, the brand does not offer coupons, and if$c<\delta v-{h}_{s}+r$, then${q}^{B*}={\overline{F}}^{-1}(\frac{c}{\delta v-{h}_{s}+r})$; otherwise,${q}^{B*}=0$. When${h}_{s}<\mathrm{min}({h}_{o},{h}_{b})$and$p=\delta v-{h}_{o}$, the brand does not offer coupons, and if$c<r$, then${q}^{B*}={\overline{F}}^{-1}(\frac{c}{r})$; otherwise,${q}^{B*}=0$.
- (4)
- When${h}_{o}<\mathrm{min}({h}_{b},{h}_{s})$and$p=v-{h}_{o}$, if${h}_{o}<\delta v$, then${f}^{*}=(1-\delta )v$and${q}^{*}=0$.
- (5)
- When${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$and$p=v-{h}_{b}$, if${h}_{o}<\delta v$,$c<\lambda (v-{h}_{b})+r$and$\lambda >{\lambda}_{1}$, or${h}_{o}<\delta v$and$\lambda (v-{h}_{b})+r\le c<\lambda v+(1-\lambda )\delta v+r-{h}_{b}$, or${h}_{b}<\delta v<{h}_{o}$and$c<\lambda v+(1-\lambda )\delta v+r-{h}_{b}$, then${f}^{*}=(1-\delta )v$and${q}^{*}={\overline{F}}^{-1}(\frac{c}{\lambda v+(1-\lambda )\delta v+r-{h}_{b}})$. If${h}_{o}<\delta v$,$c<\lambda (v-{h}_{b})+r$and$\lambda \le {\lambda}_{1}$, then${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{b}$and${q}^{*}={\overline{F}}^{-1}(\frac{c}{\lambda (v-{h}_{b})+r})$. If${h}_{b}<\delta v$and$\lambda v+(1-\lambda )\delta v+r-{h}_{b}\le c<v-{h}_{b}+r$, or${h}_{b}\ge \delta v$and$c<v-{h}_{b}+r$, the brand does not offer coupons, then${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{v+r-{h}_{b}})$. If$c\ge v-{h}_{b}+r$and${h}_{o}<\delta v$, then${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{b}$and${q}^{*}=0$.
- (6)
- When${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$,$p=v-{h}_{o}$and$v>\frac{{h}_{o}-{h}_{b}}{1-\delta}$, or${h}_{s}<{h}_{b}<{h}_{o}$,$p=v-{h}_{o}$and$v>\frac{{h}_{o}-{h}_{s}}{1-\delta}$, if${h}_{o}<\delta v$,$c<r$and$\lambda >{\lambda}_{2}$, or${h}_{o}<\delta v$and$r\le c<(1-\lambda )(\delta v-{h}_{b})+r$, or${h}_{b}<\delta v<{h}_{o}$and$c<(1-\lambda )(\delta v-{h}_{b})+r$, then${f}^{*}=(1-\delta )v+{h}_{b}-{h}_{o}$and${q}^{*}={\overline{F}}^{-1}(\frac{c}{(1-\lambda )(\delta v-{h}_{b})+r})$. If${h}_{o}<\delta v$,$c<r$and$\lambda \le {\lambda}_{2}$, then${f}^{*}=(1-\delta )v$and${q}^{*}={\overline{F}}^{-1}(\frac{c}{r})$. If${h}_{b}\ge \delta v$and$c<r$, the brand does not offer coupons, then${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{r})$. If${h}_{o}<\delta v$and$c\ge (1-\lambda )(\delta v-{h}_{b})+r$, then${f}^{*}=(1-\delta )v$and${q}^{*}=0$. If${h}_{b}<\delta v<{h}_{o}$and$c\ge (1-\lambda )(\delta v-{h}_{b})+r$, or${h}_{b}\ge \delta v$and$c\ge r$, the brand does not offer coupons, then${q}^{B*}=0$.
- (7)
- When${h}_{s}<{h}_{o}<{h}_{b}$and$p=v-{h}_{s}$, if${h}_{o}<\delta v$and$c<v-{h}_{s}+r$, then${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{s}$and${q}^{*}=\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{s}+r})$. If${h}_{o}<\delta v$and$c\ge v-{h}_{s}+r$, then${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{s}$and${q}^{*}=0$. If${h}_{o}\ge \delta v$and$c<v-{h}_{s}+r$, the brand does not offer coupons, then${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{s}+r})$. When${h}_{s}<{h}_{b}<{h}_{o}$and$p=v-{h}_{s}$, if${h}_{o}<\delta v$,$c<\lambda (v-{h}_{s})+r$and$\lambda >{\lambda}_{3}$, or${h}_{o}<\delta v$and$\lambda (v-{h}_{s})+r\le c<\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}$, or${h}_{b}<\delta v<{h}_{o}$and$c<\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}$, then${f}^{*}=(1-\delta )v+{h}_{b}-{h}_{s}$and${q}^{*}={\overline{F}}^{-1}(\frac{c}{\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}})$. If${h}_{o}<\delta v$,$c<\lambda (v-{h}_{s})+r$and$\lambda \le {\lambda}_{3}$, then${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{s}$and${q}^{*}={\overline{F}}^{-1}(\frac{c}{\lambda (v-{h}_{s})+r})$. If${h}_{b}<\delta v$and$\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}\le c<v-{h}_{s}+r$, or${h}_{b}\ge \delta v$and$c<v-{h}_{s}+r$, the brand does not offer coupons, then${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{v+r-{h}_{s}})$. If$c\ge v-{h}_{s}+r$and${h}_{o}<\delta v$, then${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{s}$and${q}^{*}=0$.
- (8)
- When${h}_{s}<{h}_{o}<{h}_{b}$,$p=v-{h}_{o}$and$v>\frac{{h}_{o}-{h}_{s}}{1-\delta}$, if${h}_{o}<\delta v$and$c<r$, then${f}^{*}=(1-\delta )v$and${q}^{*}=\lambda {\overline{F}}^{-1}(\frac{c}{r})$. If${h}_{o}<\delta v$and$c\ge r$, then${f}^{*}=(1-\delta )v$and${q}^{*}=0$. If${h}_{o}\ge \delta v$and$c<r$, the brand does not offer coupons, then${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{r})$. If${h}_{o}\ge \delta v$and$c\ge r$, the brand does not offer coupons, then${q}^{B*}=0$. ${\lambda}_{1}$,${\lambda}_{2}$and${\lambda}_{3}$are constants defined in theAppendix A.

**Proposition 2.**

- (1)
- When ${h}_{o}<\mathrm{min}({h}_{b},{h}_{s})$and$p=v-{h}_{o}$, or${h}_{s}<{h}_{o}<{h}_{b}$,$p=v-{h}_{o}$and$v>\frac{{h}_{o}-{h}_{s}}{1-\delta}$, the coupon promotion has no effect on store inventory.
- (2)
- When ${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$and$p=v-{h}_{b}$, if${h}_{o}<\delta v$,$c<\lambda (v-{h}_{b})+r$and$\lambda >\mathrm{max}({\lambda}_{1},{\lambda}_{4})$, or${h}_{o}<\delta v$,$\lambda (v-{h}_{b})+r\le c<\lambda v+(1-\lambda )\delta v+r-{h}_{b}$and$\lambda >{\lambda}_{4}$, or${h}_{b}<\delta v<{h}_{o}$,$c<\lambda v+(1-\lambda )\delta v+r-{h}_{b}$and$\lambda >{\lambda}_{4}$, or${h}_{o}<\delta v$,$c<\lambda (v-{h}_{b})+r$and${\lambda}_{5}<\lambda \le {\lambda}_{1}$, the coupon promotion reduces store inventory; otherwise, it increases or has no effect on store inventory.
- (3)
- When${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$,$p=v-{h}_{o}$and$v>\frac{{h}_{o}-{h}_{b}}{1-\delta}$, or${h}_{s}<{h}_{b}<{h}_{o}$,$p=v-{h}_{o}$and$v>\frac{{h}_{o}-{h}_{s}}{1-\delta}$, the coupon promotion increases or has no effect on store inventory.
- (4)
- When${h}_{s}<{h}_{b}<{h}_{o}$and$p=v-{h}_{s}$, if${h}_{o}<\delta v$,$c<\lambda (v-{h}_{s})+r$and$\lambda >\mathrm{max}({\lambda}_{3},{\lambda}_{6})$, or${h}_{o}<\delta v$,$\lambda (v-{h}_{s})+r\le c<\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}$and$\lambda >{\lambda}_{6}$, or${h}_{b}<\delta v<{h}_{o}$,$c<\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}$and$\lambda >{\lambda}_{6}$, or${h}_{o}<\delta v$,$c<\lambda (v-{h}_{s})+r$and${\lambda}_{7}<\lambda \le {\lambda}_{3}$, the coupon promotion reduces store inventory; otherwise, it increases or has no effect on store inventory.

#### 3.2. Heterogeneous Online and Store Hassle Costs

- (1)
- When ${\beta}_{o}+{\beta}_{s}\le 1$, each market segment of the two types of consumers is shown in Figure 1. We denote the proportions of the six segments for the two types of consumers as ${\alpha}_{o}^{{m}_{1}}$, ${\alpha}_{bo}^{{m}_{1}}$, ${\alpha}_{b}^{{m}_{1}}$, ${\alpha}_{so}^{{m}_{1}}$, ${\alpha}_{s}^{{m}_{1}}$, and ${\alpha}_{n}^{{m}_{1}}$, respectively, where $m=BH,BL$. From Figure 1, we can calculate ${\alpha}_{o}^{{m}_{1}}$, ${\alpha}_{bo}^{{m}_{1}}$, ${\alpha}_{b}^{{m}_{1}}$, ${\alpha}_{so}^{{m}_{1}}$, ${\alpha}_{s}^{{m}_{1}}$, ${\alpha}_{n}^{{m}_{1}}$, and show them in Appendix B.
- (2)
- When ${\beta}_{o}+{\beta}_{s}>1$, each market segment of the two types of consumers is depicted in Figure 2. In this situation, we denote the proportions of the six segments for the two types of consumers as ${\alpha}_{o}^{{m}_{2}}$, ${\alpha}_{bo}^{{m}_{2}}$, ${\alpha}_{b}^{{m}_{2}}$, ${\alpha}_{so}^{{m}_{2}}$, ${\alpha}_{s}^{{m}_{2}}$, and ${\alpha}_{n}^{{m}_{2}}$, respectively, where $m=BH,BL$. From Figure 2, we derive ${\alpha}_{o}^{{m}_{2}}$, ${\alpha}_{bo}^{{m}_{2}}$, ${\alpha}_{b}^{{m}_{2}}$, ${\alpha}_{so}^{{m}_{2}}$, ${\alpha}_{s}^{{m}_{2}}$, and ${\alpha}_{n}^{{m}_{2}}$ and show them in Appendix B.

**Proposition 3.**

- (1)
- When ${\beta}_{o}+{\beta}_{s}\le 1$ , the market segmentation of L-type consumers is depicted in Figure 3. In this situation, the proportions of these six segments are denoted as ${\alpha}_{o}^{{L}_{1}}$, ${\alpha}_{bo}^{{L}_{1}}$, ${\alpha}_{b}^{{L}_{1}}$, ${\alpha}_{so}^{{L}_{1}}$, ${\alpha}_{s}^{{L}_{1}}$, and ${\alpha}_{n}^{{L}_{1}}$. From Figure 3, we obtain ${\alpha}_{o}^{{L}_{1}}$, ${\alpha}_{bo}^{{L}_{1}}$, ${\alpha}_{b}^{{L}_{1}}$, ${\alpha}_{so}^{{L}_{1}}$, ${\alpha}_{s}^{{L}_{1}}$, and ${\alpha}_{n}^{{L}_{1}}$ and show them in Appendix B.
- (2)
- When ${\beta}_{o}+{\beta}_{s}>1$, Figure 4 demonstrates the market segmentation of L-type consumers. In this situation, the proportions of these six segments are denoted as ${\alpha}_{o}^{{L}_{2}}$, ${\alpha}_{bo}^{{L}_{2}}$, ${\alpha}_{b}^{{L}_{2}}$, ${\alpha}_{so}^{{L}_{2}}$, ${\alpha}_{s}^{{L}_{2}}$ and ${\alpha}_{n}^{{L}_{2}}$. From Figure 4, we observe that if ${f}_{2}<\frac{({\beta}_{o}+{\beta}_{s}-1)(\delta v-p)}{1-{\beta}_{o}}$, the market segmentation of L-type consumers can be calculated and shown in Appendix B. Similarly, if ${f}_{2}\ge \frac{({\beta}_{o}+{\beta}_{s}-1)(\delta v-p)}{1-{\beta}_{o}}$, the market segmentation of L-type consumers can also be counted and shown in Appendix B.

## 4. Numerical Analysis

#### 4.1. Joint Decision of Coupon Face Value and Store Inventory When the Brand Offers Coupons

#### 4.2. Digital Coupon Promotion Strategy of the Brand: Promote or Not

#### 4.3. Sensitivity Analysis

## 5. Conclusions, Managerial Insights, and Discussions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof of Proposition 1.**

- (1)
- When ${h}_{o}<\mathrm{min}({h}_{b},{h}_{s})$, because the product sales volume should not be zero in the nonpromotional case, the brand sets price $p=\delta v-{h}_{o}$. Then, all consumers shop online; hence, ${q}^{B*}=0$.
- (2)
- When ${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$, if the brand does not conduct a coupon promotion, there are two pricing strategies: $p=\delta v-{h}_{b}$ and $p=\delta v-{h}_{o}$. When $p=\delta v-{h}_{b}$ and $v>\frac{{h}_{o}-{h}_{b}}{1-\delta}$, all consumers prefer using BOPS, while H-type consumers switch to online after finding that the store is out of stock, but L-type consumers directly leave. Then, the brand’s expected profit function is $\pi =(p+r)E\mathrm{min}(D,q)-cq+pE\lambda {(D-q)}^{+}$. The first term represents the profit from consumers who have obtained products by using BOPS. The second item is the inventory cost, and the last item is the profit from the H-type consumers who switch to purchasing online after encountering a stockout in the store. We maximize the profit function and derive that if $c<(1-\lambda )(\delta v-{h}_{b})+r$, then ${q}^{B*}={\overline{F}}^{-1}(\frac{c}{(1-\lambda )(\delta v-{h}_{b})+r})$; otherwise, ${q}^{B*}=0$ since H-type consumers shop online and L-type consumers leave. When $p=\delta v-{h}_{b}$ and $v\le \frac{{h}_{o}-{h}_{b}}{1-\delta}$, all consumers prefer using BOPS and leaving after encountering a stockout in the store. Then, the brand’s expected profit function is $\pi =(p+r)E\mathrm{min}(D,q)-cq$. We can derive that if $c<\delta v-{h}_{b}+r$, then ${q}^{B*}={\overline{F}}^{-1}(\frac{c}{\delta v-{h}_{b}+r})$; otherwise, ${q}^{B*}=0$. When $p=\delta v-{h}_{o}$, all consumers prefer using BOPS and switching to purchasing online after encountering a stockout in the store, and the brand’s expected profit function is $\pi =(p+r)E\mathrm{min}(D,q)-cq+pE{(D-q)}^{+}$. We can calculate that if $c<r$, then ${q}^{B*}={\overline{F}}^{-1}(\frac{c}{r})$; otherwise, ${q}^{B*}=0$.
- (3)
- When ${h}_{s}<\mathrm{min}({h}_{o},{h}_{b})$, if the brand does not conduct a coupon promotion, there are two pricing strategies: $p=\delta v-{h}_{s}$ and $p=\delta v-{h}_{o}$. Similar to (2), we can obtain that when ${h}_{s}<\mathrm{min}({h}_{o},{h}_{b})$, $p=\delta v-{h}_{s}$ and $v>\frac{{h}_{o}-{h}_{s}}{1-\delta}$, if $c<(1-\lambda )(\delta v-{h}_{s})+r$, then ${q}^{B*}={\overline{F}}^{-1}(\frac{c}{(1-\lambda )(\delta v-{h}_{s})+r})$; otherwise, ${q}^{B*}=0$. When ${h}_{s}<\mathrm{min}({h}_{o},{h}_{b})$, $p=\delta v-{h}_{s}$ and $v\le \frac{{h}_{o}-{h}_{s}}{1-\delta}$, if $c<\delta v-{h}_{s}+r$, then ${q}^{B*}={\overline{F}}^{-1}(\frac{c}{\delta v-{h}_{s}+r})$; otherwise, ${q}^{B*}=0$. When ${h}_{s}<\mathrm{min}({h}_{o},{h}_{b})$ and $p=\delta v-{h}_{o}$, if $c<r$, then ${q}^{B*}={\overline{F}}^{-1}(\frac{c}{r})$; otherwise, ${q}^{B*}=0$.

- (1)
- When ${h}_{o}<\mathrm{min}({h}_{b},{h}_{s})$, all consumers prefer the online channel. Then, if $p=v-{h}_{o}$, only H-type consumers shop online. In this instance, coupons can encourage L-type consumers to buy and thus increase the brand’s profit. Therefore, the brand issues coupons.
- (2)
- When ${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$, all consumers prefer using BOPS. If $p=v-{h}_{b}$, only H-type consumers choose to buy in the BOPS channel and leave if they encounter a stockout in the store. Then, the brand offers coupons to encourage L-type consumers to shop in the BOPS or online channel. If $p=v-{h}_{o}$ and $v>\frac{{h}_{o}-{h}_{b}}{1-\delta}$, only H-type consumers tend to use BOPS and switch to online if the store is out of stock. In this situation, the brand’s profit increases if it distributes coupons in the online and BOPS channels to entice new purchases from L-type consumers. In summary, when ${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$, if $p=v-{h}_{b}$, or $p=v-{h}_{o}$ and $v>\frac{{h}_{o}-{h}_{b}}{1-\delta}$, the brand issues coupons.
- (3)
- When ${h}_{s}<\mathrm{min}({h}_{o},{h}_{b})$, all consumers prefer the BOPS channel. Similar to (2), if $p=v-{h}_{s}$, or $p=v-{h}_{o}$ and $v>\frac{{h}_{o}-{h}_{s}}{1-\delta}$, only H-type consumers shop in the store channel, and the brand issues coupons.

- (1)
- When ${h}_{o}<\mathrm{min}({h}_{b},{h}_{s})$ and $p=v-{h}_{o}$, since $\delta v-p+f-{h}_{o}>\mathrm{max}(\delta v-p+f-{h}_{b},\delta v-p-{h}_{s})$, an L-type consumer obtains the maximum utility from buying online with a coupon, and the utility should be nonnegative, that is, $\delta v-p+f-{h}_{o}=f-(1-\delta )v\ge 0$, $f\ge (1-\delta )v$. All consumers shop online. Hence, the brand’s expected profit function is $\pi =(p-(1-\lambda )f)ED$ and decreases with $f$. Then, we can derive that if ${h}_{o}<\delta v$($p>f$),${f}^{*}=(1-\delta )v$ and ${q}^{*}=0$ by maximizing the function.
- (2)
- When ${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$ and $p=v-{h}_{b}$, since $\delta v-p+f-{h}_{b}>\mathrm{max}(\delta v-p+f-{h}_{o},\delta v-p-{h}_{s})$, an L-type consumer obtains the maximum utility from buying in the BOPS channel with a coupon, and the utility should be nonnegative, that is, $\delta v-p+f-{h}_{b}\ge 0$, $f\ge p+{h}_{b}-\delta v$. When an L-type consumer obtains nonnegative utility from purchasing online, i.e., $\delta v-p+f-{h}_{o}\ge 0$, then $f\ge p+{h}_{o}-\delta v$. Therefore, there are two coupon strategies for the brand. One is to issue coupons with a small face value so that L-type consumers leave after encountering a stockout in the BOPS channel. Here, the brand’s expected profit function is $\pi =\lambda (p+r)E\mathrm{min}(D,q)+(1-\lambda )(p+r-f)E\mathrm{min}(D,q)-cq$. The first and second terms are the profits from the H-high and L-type consumers who have obtained products by using BOPS, respectively. In line with Su [61] and He et al. [62], coupon face value and inventory decisions can be derived by maximizing the profit function. Given that the profit function decreases with $f$, we can derive that if ${h}_{b}<\delta v(p-f>0)$ and $c<\lambda v+(1-\lambda )\delta v+r-{h}_{b}$, then ${f}^{*}=p+{h}_{b}-\delta v=(1-\delta )v$ and ${q}^{*}={\overline{F}}^{-1}(\frac{c}{\lambda v+(1-\lambda )\delta v+r-{h}_{b}})$, and both types of consumers buy through the BOPS channel. If ${h}_{b}<\delta v$ and $\lambda v+(1-\lambda )\delta v+r-{h}_{b}\le c<v-{h}_{b}+r$, or ${h}_{b}\ge \delta v$ and $c<v-{h}_{b}+r$, based on the assumption that $r>\frac{(1-\lambda )(\delta v-{h}_{o})ED+c\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{b}+r})-(v-{h}_{b})E\mathrm{min}(\lambda D,\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{b}+r}))}{E\mathrm{min}(\lambda D,\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{b}+r}))}$, the brand does not offer coupons, and only H-type consumers purchase in the BOPS channel, then ${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{v+r-{h}_{b}})$. If $c\ge v-{h}_{b}+r$ and ${h}_{o}<\delta v(p-f>0)$, then ${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{b}$ and ${q}^{*}=0$, and only L-type consumers purchase online. If $c\ge v-{h}_{b}+r$ and ${h}_{o}\ge \delta v$, no consumer will buy the product. This situation is not practical and will not be considered. The other strategy is to issue coupons with a large face value to enable L-type consumers to purchase online after encountering stockouts in the BOPS channel. The brand’s expected profit function is $\pi =\lambda (p+r)E\mathrm{min}(D,q)+(1-\lambda )(p+r-f)E\mathrm{min}(D,q)-cq+(p-f)E(1-\lambda ){(D-q)}^{+}$ and decreases with $f$. Therefore, we can calculate that if ${h}_{o}<\delta v$ and $c<\lambda (v-{h}_{b})+r$, then ${f}^{*}=p+{h}_{o}-\delta v=(1-\delta )v+{h}_{o}-{h}_{b}$ and ${q}^{*}={\overline{F}}^{-1}(\frac{c}{\lambda (v-{h}_{b})+r})$. If ${h}_{o}<\delta v$ and $\lambda (v-{h}_{b})+r\le c<v+r-{h}_{b}$, or ${h}_{o}\ge \delta v$ and $c<v+r-{h}_{b}$, the brand does not offer coupons because $r>\frac{(1-\lambda )(\delta v-{h}_{o})ED+c\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{b}+r})-(v-{h}_{b})E\mathrm{min}(\lambda D,\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{b}+r}))}{E\mathrm{min}(\lambda D,\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{b}+r}))}$, and only H-type consumers purchase in the BOPS channel, then ${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{v+r-{h}_{b}})$. If $c\ge v-{h}_{b}+r$ and ${h}_{o}<\delta v(p-f>0)$, then ${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{b}$, ${q}^{*}=0$, and only L-type consumers purchase online. By comparing the profits under the two coupon strategies, we can summarize the results as follows: when ${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$ and $p=v-{h}_{b}$, if ${h}_{o}<\delta v$, $c<\lambda (v-{h}_{b})+r$ and $\lambda >{\lambda}_{1}$, where ${\lambda}_{1}=\frac{rE\mathrm{min}(D,{\overline{F}}^{-1}(\frac{c}{\lambda (v-{h}_{b})+r}))-(\delta v+r-{h}_{b})E\mathrm{min}(D,{\overline{F}}^{-1}(\frac{c}{\lambda v+(1-\lambda )\delta v+r-{h}_{b}}))+(\delta v-{h}_{o})ED+c({\overline{F}}^{-1}(\frac{c}{\lambda v+(1-\lambda )\delta v+r-{h}_{b}})-{\overline{F}}^{-1}(\frac{c}{\lambda (v-{h}_{b})+r}))}{v(1-\delta )E\mathrm{min}(D,{\overline{F}}^{-1}(\frac{c}{\lambda v+(1-\lambda )\delta v+r-{h}_{b}}))-(v-{h}_{b})E\mathrm{min}(D,{\overline{F}}^{-1}(\frac{c}{\lambda (v-{h}_{b})+r}))+(\delta v-{h}_{o})ED}$, or ${h}_{o}<\delta v$ and $\lambda (v-{h}_{b})+r\le c<\lambda v+(1-\lambda )\delta v+r-{h}_{b}$, or ${h}_{b}<\delta v<{h}_{o}$ and $c<\lambda v+(1-\lambda )\delta v+r-{h}_{b}$, then ${f}^{*}=(1-\delta )v$ and ${q}^{*}={\overline{F}}^{-1}(\frac{c}{\lambda v+(1-\lambda )\delta v+r-{h}_{b}})$. If ${h}_{o}<\delta v$, $c<\lambda (v-{h}_{b})+r$ and $\lambda \le {\lambda}_{1}$, then ${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{b}$ and ${q}^{*}={\overline{F}}^{-1}(\frac{c}{\lambda (v-{h}_{b})+r})$. If ${h}_{b}<\delta v$ and $\lambda v+(1-\lambda )\delta v+r-{h}_{b}\le c<v-{h}_{b}+r$, or ${h}_{b}\ge \delta v$ and $c<v-{h}_{b}+r$, the brand does not offer coupons, then ${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{v+r-{h}_{b}})$. If $c\ge v-{h}_{b}+r$ and ${h}_{o}<\delta v$, then ${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{b}$ and ${q}^{*}=0$.
- (3)
- When ${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$, $p=v-{h}_{o}$ and $v>\frac{{h}_{o}-{h}_{b}}{1-\delta}$, an H-type consumer prefers using BOPS and shops online if the store is out of stock, and an L-type consumer obtains the maximum utility $\delta v-p+f-{h}_{b}$ from buying in the BOPS channel with a coupon. Similar to (2), there are two coupon strategies for the brand. One is to offer coupons with a small face value so that L-type consumers leave after encountering a stockout in the BOPS channel. Here, the brand’s expected profit function is $\pi =\lambda (p+r)E\mathrm{min}(D,q)+(1-\lambda )(p+r-f)E\mathrm{min}(D,q)-cq+pE\lambda {(D-q)}^{+}$ and decreases with $f$. We assume that $r>\frac{c{\overline{F}}^{-1}(\frac{c}{(1-\lambda )(\delta v-{h}_{b})+r})+(1-\lambda )(\delta v-{h}_{b})E{(D-{\overline{F}}^{-1}(\frac{c}{(1-\lambda )(\delta v-{h}_{b})+r}))}^{+}}{E\mathrm{min}(D,{\overline{F}}^{-1}(\frac{c}{(1-\lambda )(\delta v-{h}_{b})+r}))}$. Next, we can derive that if ${h}_{b}<\delta v$ and $c<(1-\lambda )(\delta v-{h}_{b})+r$, then ${f}^{*}=(1-\delta )v+{h}_{b}-{h}_{o}$ and ${q}^{*}={\overline{F}}^{-1}(\frac{c}{(1-\lambda )(\delta v-{h}_{b})+r})$. If ${h}_{o}<\delta v$ and $c\ge (1-\lambda )(\delta v-{h}_{b})+r$, then ${f}^{*}=(1-\delta )v$ and ${q}^{*}=0$, and all consumers purchase online. If ${h}_{b}<\delta v<{h}_{o}$ and $c\ge (1-\lambda )(\delta v-{h}_{b})+r$, the brand does not offer coupons, and only H-type consumers shop online, then ${q}^{B*}=0$. If ${h}_{b}\ge \delta v$ and $c<r$, the brand does not offer coupons, and only H-type consumers purchase in the BOPS channel based on the assumption that $r>\frac{c{\overline{F}}^{-1}(\frac{c}{r})}{E\mathrm{min}(D,{\overline{F}}^{-1}(\frac{c}{r}))}$, then ${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{r})$. If ${h}_{b}\ge \delta v$ and $c\ge r$, the brand does not offer coupons, and only H-type consumers purchase online, then ${q}^{B*}=0$. The other strategy is to issue coupons with a large face value to enable L-type consumers to shop online after facing a stockout in the BOPS channel. The brand’s expected profit function is $\pi =\lambda (p+r)E\mathrm{min}(D,q)+(1-\lambda )(p+r-f)E\mathrm{min}(D,q)-cq+pE\lambda {(D-q)}^{+}+(p-f)E(1-\lambda ){(D-q)}^{+}$ and decreases with $f$. Therefore, we can derive that if ${h}_{o}<\delta v$ and $c<r$, then ${f}^{*}=(1-\delta )v$ and ${q}^{*}={\overline{F}}^{-1}(\frac{c}{r})$, and the two types of consumers buy in the BOPS channel. If ${h}_{o}<\delta v$ and $c\ge r$, then ${f}^{*}=(1-\delta )v$ and ${q}^{*}=0$, and the two types of consumers buy online. If ${h}_{o}\ge \delta v$ and $c<r$, the brand does not offer coupons because $r>\frac{c{\overline{F}}^{-1}(\frac{c}{r})}{E\mathrm{min}(D,{\overline{F}}^{-1}(\frac{c}{r}))}$, and only H-type consumers purchase in the BOPS channel, then ${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{r})$. If ${h}_{o}\ge \delta v$ and $c\ge r$, the brand does not offer coupons, and only H-type consumers shop online, then ${q}^{B*}=0$. By comparing the profits under the two coupon strategies, we can summarize the results as follows: when ${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$, $p=v-{h}_{o}$ and $v>\frac{{h}_{o}-{h}_{b}}{1-\delta}$, if ${h}_{o}<\delta v$, $c<r$ and $\lambda >{\lambda}_{2}$, where ${\lambda}_{2}=\frac{rE\mathrm{min}(D,{\overline{F}}^{-1}(\frac{c}{r}))-(\delta v+r-{h}_{b})E\mathrm{min}(D,{\overline{F}}^{-1}(\frac{c}{(1-\lambda )(\delta v-{h}_{b})+r}))+(\delta v-{h}_{o})ED+c({\overline{F}}^{-1}(\frac{c}{(1-\lambda )(\delta v-{h}_{b})+r})-{\overline{F}}^{-1}(\frac{c}{r}))}{(\delta v-{h}_{o})ED-(\delta v-{h}_{b})E\mathrm{min}(D,{\overline{F}}^{-1}(\frac{c}{(1-\lambda )(\delta v-{h}_{b})+r}))}$, or ${h}_{o}<\delta v$ and $r\le c<(1-\lambda )(\delta v-{h}_{b})+r$, or ${h}_{b}<\delta v<{h}_{o}$ and $c<(1-\lambda )(\delta v-{h}_{b})+r$, then ${f}^{*}=(1-\delta )v+{h}_{b}-{h}_{o}$ and ${q}^{*}={\overline{F}}^{-1}(\frac{c}{(1-\lambda )(\delta v-{h}_{b})+r})$. If ${h}_{o}<\delta v$, $c<r$ and $\lambda \le {\lambda}_{2}$, then ${f}^{*}=(1-\delta )v$ and ${q}^{*}={\overline{F}}^{-1}(\frac{c}{r})$. If ${h}_{b}\ge \delta v$ and $c<r$, the brand does not offer coupons, then ${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{r})$. If ${h}_{o}<\delta v$ and $c\ge (1-\lambda )(\delta v-{h}_{b})+r$, then ${f}^{*}=(1-\delta )v$ and ${q}^{*}=0$. If ${h}_{b}<\delta v<{h}_{o}$ and $c\ge (1-\lambda )(\delta v-{h}_{b})+r$, or ${h}_{b}\ge \delta v$ and $c\ge r$, the brand does not offer coupons, then ${q}^{B*}=0$.
- (4)
- When ${h}_{s}<\mathrm{min}({h}_{o},{h}_{b})$ and $p=v-{h}_{s}$, an H-type consumer prefers to buy in the store channel and leave after encountering a stockout in the store. When ${h}_{s}<{h}_{o}<{h}_{b}$, an L-type consumer should obtain the maximum and nonnegative utility from purchasing online with a coupon, i.e., $\delta v-p+f-{h}_{o}\ge 0$; then, $f\ge (1-\delta )v+{h}_{o}-{h}_{s}$. The brand’s expected profit function is $\pi =(p+r)E\mathrm{min}(\lambda D,q)+(p-f)E(1-\lambda )D-cq$ and decreases with $f$. We can calculate that if ${h}_{o}<\delta v$ and $c<v-{h}_{s}+r$, then ${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{s}$ and ${q}^{*}=\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{s}+r})$. H-type consumers buy offline, while L-type consumers purchase online. If ${h}_{o}\ge \delta v$ and $c<v-{h}_{s}+r$, based on the assumption that $r>\frac{(1-\lambda )(\delta v-{h}_{o})ED+c\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{s}+r})-(v-{h}_{s})E\mathrm{min}(\lambda D,\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{s}+r}))}{E\mathrm{min}(\lambda D,\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{s}+r}))}$, the brand does not offer coupons and only H-type consumers purchase in the store channel, then ${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{s}+r})$. If ${h}_{o}<\delta v$ and $c\ge v-{h}_{s}+r$, then ${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{s}$ and ${q}^{*}=0$, and only L-type consumers shop online. If ${h}_{o}\ge \delta v$ and $c\ge v-{h}_{s}+r$, no product is sold, and this situation is not included here. When ${h}_{s}<{h}_{b}<{h}_{o}$, the brand issues coupons with either a small face value ${f}^{*}=(1-\delta )v+{h}_{b}-{h}_{s}$ or a large face value ${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{s}$. By comparing the profits of the brand under the two coupon strategies and following the previous analysis, we can obtain that when ${h}_{s}<{h}_{o}<{h}_{b}$ and $p=v-{h}_{s}$, if ${h}_{o}<\delta v$ and $c<v-{h}_{s}+r$, then ${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{s}$ and ${q}^{*}=\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{s}+r})$. If ${h}_{o}<\delta v$ and $c\ge v-{h}_{s}+r$, then ${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{s}$ and ${q}^{*}=0$. If ${h}_{o}\ge \delta v$ and $c<v-{h}_{s}+r$, the brand does not offer coupons, then ${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{s}+r})$. When ${h}_{s}<{h}_{b}<{h}_{o}$ and $p=v-{h}_{s}$, if ${h}_{o}<\delta v$, $c<\lambda (v-{h}_{s})+r$ and $\lambda >{\lambda}_{3}$, where ${\lambda}_{3}=\frac{rE\mathrm{min}(D,{\overline{F}}^{-1}(\frac{c}{\lambda (v-{h}_{s})+r}))-(\delta v+r-(1-\lambda ){h}_{b}-\lambda {h}_{s})E\mathrm{min}(D,{\overline{F}}^{-1}(\frac{c}{\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}}))+(\delta v-{h}_{o})ED+c({\overline{F}}^{-1}(\frac{c}{\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}})-{\overline{F}}^{-1}(\frac{c}{\lambda (v-{h}_{s})+r}))}{v(1-\delta )E\mathrm{min}(D,{\overline{F}}^{-1}(\frac{c}{\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}}))-(v-{h}_{s})E\mathrm{min}(D,{\overline{F}}^{-1}(\frac{c}{\lambda (v-{h}_{s})+r}))+(\delta v-{h}_{o})ED}$, or ${h}_{o}<\delta v$ and $\lambda (v-{h}_{s})+r\le c<\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}$, or ${h}_{b}<\delta v<{h}_{o}$ and $c<\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}$, then ${f}^{*}=(1-\delta )v+{h}_{b}-{h}_{s}$ and ${q}^{*}={\overline{F}}^{-1}(\frac{c}{\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}})$. If ${h}_{o}<\delta v$, $c<\lambda (v-{h}_{s})+r$ and $\lambda \le {\lambda}_{3}$, then ${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{s}$ and ${q}^{*}={\overline{F}}^{-1}(\frac{c}{\lambda (v-{h}_{s})+r})$. If ${h}_{b}<\delta v$ and $\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}\le c<v-{h}_{s}+r$, or ${h}_{b}\ge \delta v$ and $c<v-{h}_{s}+r$, the brand does not offer coupons, then ${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{v+r-{h}_{s}})$. If $c\ge v-{h}_{s}+r$ and ${h}_{o}<\delta v$, then ${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{s}$ and ${q}^{*}=0$.
- (5)
- When ${h}_{s}<\mathrm{min}({h}_{o},{h}_{b})$, $p=v-{h}_{o}$ and $v>\frac{{h}_{o}-{h}_{s}}{1-\delta}$, an H-type consumer prefers to buy in the store channel and switches to purchasing online after encountering a stockout in the store. When ${h}_{s}<{h}_{o}<{h}_{b}$, because $v>\frac{{h}_{o}-{h}_{s}}{1-\delta}$, an L-type consumer should obtain the maximum and nonnegative utility from purchasing online with a coupon, i.e., $\delta v-p+f-{h}_{o}\ge 0$; then, $f\ge (1-\delta )v$. The brand’s expected profit function is $\pi =(p+r)E\mathrm{min}(\lambda D,q)+(p-f)E(1-\lambda )D-cq+pE{(\lambda D-q)}^{+}$ and decreases with $f$. We assume that $r>\frac{c{\overline{F}}^{-1}(\frac{c}{r})}{E\mathrm{min}(D,{\overline{F}}^{-1}(\frac{c}{r}))}$. We can derive that if ${h}_{o}<\delta v$ and $c<r$, then ${f}^{*}=(1-\delta )v$ and ${q}^{*}=\lambda {\overline{F}}^{-1}(\frac{c}{r})$. H-type consumers buy offline, while L-type consumers purchase online. If ${h}_{o}<\delta v$ and $c\ge r$, then ${f}^{*}=(1-\delta )v$ and ${q}^{*}=0$, and all consumers buy products online. If ${h}_{o}\ge \delta v$ and $c<r$, the brand does not offer coupons, and only H-type consumers purchase in the store channel because $r>\frac{c{\overline{F}}^{-1}(\frac{c}{r})}{E\mathrm{min}(D,{\overline{F}}^{-1}(\frac{c}{r}))}$, then ${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{r})$. If ${h}_{o}\ge \delta v$ and $c\ge r$, the brand does not offer coupons, and only H-type consumers shop online, then ${q}^{B*}=0$. When ${h}_{s}<{h}_{b}<{h}_{o}$, the brand either issues coupons with a small face value ${f}^{*}=(1-\delta )v+{h}_{b}-{h}_{o}$ or a large face value ${f}^{*}=(1-\delta )v$. Comparing the brand’s profits under the two coupon strategies reveals that the results when ${h}_{s}<{h}_{b}<{h}_{o}$, $p=v-{h}_{o}$, and $v>\frac{{h}_{o}-{h}_{s}}{1-\delta}$ are the same as those when ${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$, $p=v-{h}_{o}$, and $v>\frac{{h}_{o}-{h}_{b}}{1-\delta}$, so they are merged.

**Proof of Proposition 2.**

- (1)
- When ${h}_{o}<\mathrm{min}({h}_{b},{h}_{s})$ and $p=v-{h}_{o}$, then ${f}^{*}=(1-\delta )v$ and ${q}^{*}=0$. If the brand does not offer coupons, then only H-type consumers shop online and ${q}^{B*}=0$. The coupon promotion has no effect on store inventory.
- (2)
- When ${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$ and $p=v-{h}_{b}$, if ${h}_{o}<\delta v$, $c<\lambda (v-{h}_{b})+r$ and $\lambda >{\lambda}_{1}$, or ${h}_{o}<\delta v$ and $\lambda (v-{h}_{b})+r\le c<\lambda v+(1-\lambda )\delta v+r-{h}_{b}$, or ${h}_{b}<\delta v<{h}_{o}$ and $c<\lambda v+(1-\lambda )\delta v+r-{h}_{b}$, then ${f}^{*}=(1-\delta )v$ and ${q}^{*}={\overline{F}}^{-1}(\frac{c}{\lambda v+(1-\lambda )\delta v+r-{h}_{b}})$. If ${h}_{o}<\delta v$, $c<\lambda (v-{h}_{b})+r$ and $\lambda \le {\lambda}_{1}$, then ${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{b}$ and ${q}^{*}={\overline{F}}^{-1}(\frac{c}{\lambda (v-{h}_{b})+r})$. If $c\ge v-{h}_{b}+r$ and ${h}_{o}<\delta v$, then ${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{b}$ and ${q}^{*}=0$. When ${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$ and $p=v-{h}_{b}$, if the brand does not offer coupons and $c<v-{h}_{b}+r$, then ${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{b}+r})$. Only H-type consumers use BOPS and leave if the store is out of stock. If $c\ge v-{h}_{b}+r$, no consumer buys the product in the nonpromotional case, which is not included here. By comparing ${q}^{*}={\overline{F}}^{-1}(\frac{c}{\lambda v+(1-\lambda )\delta v+r-{h}_{b}})$ and ${q}^{*}={\overline{F}}^{-1}(\frac{c}{\lambda (v-{h}_{b})+r})$ with ${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{b}+r})$, we can obtain that when ${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$ and $p=v-{h}_{b}$, if ${h}_{o}<\delta v$, $c<\lambda (v-{h}_{b})+r$ and $\lambda >\mathrm{max}({\lambda}_{1},{\lambda}_{4})$, where ${\lambda}_{4}=\frac{{\overline{F}}^{-1}(\frac{c}{\lambda v+(1-\lambda )\delta v+r-{h}_{b}})}{{\overline{F}}^{-1}(\frac{c}{v-{h}_{b}+r})})$, or ${h}_{o}<\delta v$, $\lambda >{\lambda}_{4}$ and $\lambda (v-{h}_{b})+r\le c<\lambda v+(1-\lambda )\delta v+r-{h}_{b}$, or ${h}_{b}<\delta v<{h}_{o}$, $c<\lambda v+(1-\lambda )\delta v+r-{h}_{b}$ and $\lambda >{\lambda}_{4}$, or ${h}_{o}<\delta v$, $c<\lambda (v-{h}_{b})+r$ and ${\lambda}_{5}<\lambda \le {\lambda}_{1}$, where ${\lambda}_{5}=\frac{{\overline{F}}^{-1}(\frac{c}{\lambda (v-{h}_{b})+r})}{{\overline{F}}^{-1}(\frac{c}{v-{h}_{b}+r})}$, then ${q}^{*}<{q}^{B*}$, and the coupon promotion reduces store inventory. Otherwise, it increases or has no effect on store inventory.
- (3)
- When ${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$, $p=v-{h}_{o}$ and $v>\frac{{h}_{o}-{h}_{b}}{1-\delta}$, or ${h}_{s}<{h}_{b}<{h}_{o}$, $p=v-{h}_{o}$ and $v>\frac{{h}_{o}-{h}_{s}}{1-\delta}$, if ${h}_{o}<\delta v$, $c<r$ and $\lambda >{\lambda}_{2}$, or ${h}_{o}<\delta v$ and $r\le c<(1-\lambda )(\delta v-{h}_{b})+r$, or ${h}_{b}<\delta v<{h}_{o}$ and $c<(1-\lambda )(\delta v-{h}_{b})+r$, then ${f}^{*}=(1-\delta )v+{h}_{b}-{h}_{o}$ and ${q}^{*}={\overline{F}}^{-1}(\frac{c}{(1-\lambda )(\delta v-{h}_{b})+r})$. If ${h}_{o}<\delta v$, $c<r$ and $\lambda \le {\lambda}_{2}$, then ${f}^{*}=(1-\delta )v$ and ${q}^{*}={\overline{F}}^{-1}(\frac{c}{r})$. If ${h}_{o}<\delta v$ and $c\ge (1-\lambda )(\delta v-{h}_{b})+r$, then ${f}^{*}=(1-\delta )v$ and ${q}^{*}=0$. When ${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$, $p=v-{h}_{o}$ and $v>\frac{{h}_{o}-{h}_{b}}{1-\delta}$, or ${h}_{s}<{h}_{b}<{h}_{o}$, $p=v-{h}_{o}$ and $v>\frac{{h}_{o}-{h}_{s}}{1-\delta}$, if the brand does not offer coupons and $c<r$, then ${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{r})$. Only H-type consumers tend to go to the store and switch to online if the store is out of stock. If the brand does not offer coupons and $c\ge r$, then ${q}^{B*}=0$. Only H-type consumers shop online. By comparing ${q}^{*}$ with ${q}^{B*}$, we can obtain that when ${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$, $p=v-{h}_{o}$ and $v>\frac{{h}_{o}-{h}_{b}}{1-\delta}$, or ${h}_{s}<{h}_{b}<{h}_{o}$, $p=v-{h}_{o}$ and $v>\frac{{h}_{o}-{h}_{s}}{1-\delta}$, if ${h}_{o}<\delta v$, $c<r$ and $\lambda >{\lambda}_{2}$, or ${h}_{o}<\delta v$ and $r\le c<(1-\lambda )(\delta v-{h}_{b})+r$, or ${h}_{b}<\delta v<{h}_{o}$ and $c<(1-\lambda )(\delta v-{h}_{b})+r$, then ${q}^{*}={\overline{F}}^{-1}(\frac{c}{(1-\lambda )(\delta v-{h}_{b})+r})>{q}^{B*}$ since ${h}_{b}<\delta v$ and $c<(1-\lambda )(\delta v-{h}_{b})+r$. If ${h}_{o}<\delta v$, $c<r$ and $\lambda \le {\lambda}_{2}$, then ${q}^{*}={\overline{F}}^{-1}(\frac{c}{r})>{q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{r})$. If ${h}_{o}<\delta v$ and $c\ge (1-\lambda )(\delta v-{h}_{b})+r$, then ${q}^{*}={q}^{B*}=0$. In summary, when ${h}_{b}<\mathrm{min}({h}_{o},{h}_{s})$, $p=v-{h}_{o}$ and $v>\frac{{h}_{o}-{h}_{b}}{1-\delta}$, or ${h}_{s}<{h}_{b}<{h}_{o}$, $p=v-{h}_{o}$ and $v>\frac{{h}_{o}-{h}_{s}}{1-\delta}$, the coupon promotion increases or has no effect on store inventory.
- (4)
- When ${h}_{s}<{h}_{o}<{h}_{b}$ and $p=v-{h}_{s}$, if ${h}_{o}<\delta v$ and $c<v-{h}_{s}+r$, then ${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{s}$ and ${q}^{*}=\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{s}+r})$. If ${h}_{o}<\delta v$ and $c\ge v-{h}_{s}+r$, then ${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{s}$ and ${q}^{*}=0$. If the brand does not offer coupons and $c<v-{h}_{s}+r$, then ${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{s}+r})$. Only H-type consumers purchase in the store channel and leave after encountering a stockout in the store. If $c\ge v-{h}_{s}+r$, then no consumer buys the product in the nonpromotional case, which is not included here. Therefore, when ${h}_{s}<{h}_{o}<{h}_{b}$ and $p=v-{h}_{s}$, the coupon promotion has no effect on the store inventory. When ${h}_{s}<{h}_{b}<{h}_{o}$ and $p=v-{h}_{s}$, if ${h}_{o}<\delta v$, $c<\lambda (v-{h}_{s})+r$ and $\lambda >{\lambda}_{3}$, or ${h}_{o}<\delta v$ and $\lambda (v-{h}_{s})+r\le c<\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}$, or ${h}_{b}<\delta v<{h}_{o}$ and $c<\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}$, then ${q}^{*}={\overline{F}}^{-1}(\frac{c}{\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}})$ and ${f}^{*}=(1-\delta )v+{h}_{b}-{h}_{s}$. If ${h}_{o}<\delta v$, $c<\lambda (v-{h}_{s})+r$ and $\lambda \le {\lambda}_{3}$, then ${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{s}$ and ${q}^{*}={\overline{F}}^{-1}(\frac{c}{\lambda (v-{h}_{s})+r})$. If $c\ge v-{h}_{s}+r$ and ${h}_{o}<\delta v$, then ${f}^{*}=(1-\delta )v+{h}_{o}-{h}_{s}$ and ${q}^{*}=0$. If the brand does not offer coupons and $c<v-{h}_{s}+r$, then ${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{v-{h}_{s}+r})$. Only H-type consumers purchase in the store channel and leave after encountering a stockout in the store. The case of $c\ge v-{h}_{s}+r$ is not considered here since it is not practical. By comparing ${q}^{*}$ with ${q}^{B*}$, we can obtain that when ${h}_{s}<{h}_{b}<{h}_{o}$ and $p=v-{h}_{s}$, if ${h}_{o}<\delta v$, $c<\lambda (v-{h}_{s})+r$ and $\lambda >\mathrm{max}({\lambda}_{3},{\lambda}_{6})$, where ${\lambda}_{6}=\frac{{\overline{F}}^{-1}(\frac{c}{\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}})}{{\overline{F}}^{-1}(\frac{c}{v-{h}_{s}+r})}$, or ${h}_{o}<\delta v$, $\lambda >{\lambda}_{6}$ and $\lambda (v-{h}_{s})+r\le c<\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}$, or ${h}_{b}<\delta v<{h}_{o}$, $\lambda >{\lambda}_{6}$ and $c<\lambda v+(1-\lambda )\delta v+r-\lambda {h}_{s}-(1-\lambda ){h}_{b}$, or ${h}_{o}<\delta v$, $c<\lambda (v-{h}_{s})+r$ and ${\lambda}_{7}<\lambda \le {\lambda}_{3}$, where ${\lambda}_{7}=\frac{{\overline{F}}^{-1}(\frac{c}{\lambda (v-{h}_{s})+r})}{{\overline{F}}^{-1}(\frac{c}{v-{h}_{s}+r})}$, then ${q}^{*}<{q}^{B*}$, and the coupon promotion reduces store inventory; otherwise, it increases or has no effect on store inventory.
- (5)
- When ${h}_{s}<{h}_{o}<{h}_{b}$, $p=v-{h}_{o}$ and $v>\frac{{h}_{o}-{h}_{s}}{1-\delta}$, if ${h}_{o}<\delta v$ and $c<r$, then ${f}^{*}=(1-\delta )v$ and ${q}^{*}=\lambda {\overline{F}}^{-1}(\frac{c}{r})$. If ${h}_{o}<\delta v$ and $c\ge r$, then ${f}^{*}=(1-\delta )v$ and ${q}^{*}=0$. If the brand does not offer coupons and $c<r$, then ${q}^{B*}=\lambda {\overline{F}}^{-1}(\frac{c}{r})$. Only H-type consumers purchase in the store channel and switch to online if the store is out of stock. If the brand does not offer coupons and $c\ge r$, then ${q}^{B*}=0$. Only H-type consumers purchase online. In summary, when ${h}_{s}<{h}_{o}<{h}_{b}$, $p=v-{h}_{o}$, and $v>\frac{{h}_{o}-{h}_{s}}{1-\delta}$, the coupon promotion has no effect on store inventory. We merge this situation with situation (1) since the coupon promotion does not affect store inventory in both situations.

**Proof of Proposition 3.**

## Appendix B

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**Figure 1.**Market segmentation of H-type and L-type consumers when ${\beta}_{o}+{\beta}_{s}\le 1$. (

**a**) H-type consumers; (

**b**) L-type consumers.

**Figure 2.**Market segmentation of H-type and L-type consumers when ${\beta}_{o}+{\beta}_{s}>1$. (

**a**) H-type consumers; (

**b**) L-type consumers.

**Figure 3.**Market segmentation of L-type consumers when coupons are offered and ${\beta}_{o}+{\beta}_{s}\le 1$.

**Figure 4.**Market segmentation of L-type consumers when coupons are offered and ${\beta}_{o}+{\beta}_{s}>1$. (

**a**) ${f}_{2}<\frac{({\beta}_{o}+{\beta}_{s}-1)(\delta v-p)}{1-{\beta}_{o}}$; (

**b**) $\text{}{f}_{2}\ge \frac{({\beta}_{o}+{\beta}_{s}-1)(\delta v-p)}{1-{\beta}_{o}}$.

**Figure 7.**Impacts of $p$ and $\delta $ on the brand’s profit difference with different ${\beta}_{o}+{\beta}_{s}$. (

**a**) ${\beta}_{o}={\beta}_{s}=0.4$; (

**b**) ${\beta}_{o}={\beta}_{s}=0.6$.

**Figure 8.**Impacts of $r$ on the brand’s profit with different ${\beta}_{o}+{\beta}_{s}$. (

**a**) ${\beta}_{o}={\beta}_{s}=0.4$; (

**b**) ${\beta}_{o}={\beta}_{s}=0.6$.

**Figure 10.**Impacts of $p$, $\delta $, $\lambda $ and $r$ on store inventory level. (

**a**)$\text{}\delta =0.72,\text{}\lambda =0.8,\text{}r=0.2$; (

**b**)$\delta =0.9,\text{}\lambda =0.8,\text{}r=0.2$; (

**c**)$\text{}\delta =0.72,\text{}\lambda =0.98,\text{}r=0.2$; (

**d**)$\text{}\delta =0.72,\text{}\lambda =0.8,\text{}r=0.5$.

**Figure 11.**Impacts of $p$, $\delta $, $\lambda $ and $r$ on the brand’s profit. (

**a**) $\delta =0.72,\text{}\lambda =0.8,\text{}r=0.2$; (

**b**) $\delta =0.9,\text{}\lambda =0.8,\text{}r=0.2$; (

**c**) $\delta =0.72,\text{}\lambda =0.98,\text{}r=0.2$; (

**d**) $\delta =0.72,\text{}\lambda =0.8,\text{}r=0.5$.

Parameters | |
---|---|

$v$ | An H-type consumer’s valuation of the brand’s product |

$p$ | Selling price of the product |

$\lambda $ | Proportions of H-type consumers |

$\delta $ | Valuation coefficient of an L-type consumer for the product relative to an H-type consumer |

$c$ | Unit cost of store inventory |

$r$ | Unit cross-selling revenue for the brand |

${h}_{o},{h}_{s},{h}_{b}$ | Hassle costs for consumers when they shop in the online, store and BOPS channels |

Decision variables | |

$f$ | Coupon face value |

$q$ | Store inventory level |

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**MDPI and ACS Style**

Zhang, Y.; Hu, X.
Digital Coupon Promotion and Inventory Strategies of Omnichannel Brands. *Axioms* **2023**, *12*, 29.
https://doi.org/10.3390/axioms12010029

**AMA Style**

Zhang Y, Hu X.
Digital Coupon Promotion and Inventory Strategies of Omnichannel Brands. *Axioms*. 2023; 12(1):29.
https://doi.org/10.3390/axioms12010029

**Chicago/Turabian Style**

Zhang, Yue, and Xiaojian Hu.
2023. "Digital Coupon Promotion and Inventory Strategies of Omnichannel Brands" *Axioms* 12, no. 1: 29.
https://doi.org/10.3390/axioms12010029