Optimal advance selling strategy with information provision for omni-channel retailers

Abstract

Demand information and consumer valuation uncertainty of new products have significant impacts on both consumers’ purchasing behavior and retail operations. To address the information transparency for new products launching, this study examines the profitability of omni-channel pre-ordering (i.e., compared to traditional online pre-ordering), a new advance selling strategy for retailers whereby consumers can solve product value uncertainty first and then decide whether to purchase in advance. Our analysis finds that advance selling is not always an appropriate choice for the retailer, but is contingent on related costs (e.g., losses from the costs of returns for retailers and consumers and the hassle cost of solving uncertain value for consumers). Specifically, only when the retailer’s return cost is relatively low and the hassle cost of solving uncertain value is relatively high should the retailer adopt the traditional online pre-ordering strategy. However, when the hassle cost of solving uncertain value is relatively low, the omni-channel pre-ordering strategy is more profitable for the retailer. By contrast, advance selling should not be offered when the retailer’s return cost and the hassle cost of solving uncertain value are both high. Next, we derive the optimal advance selling price and ordering quantity for the regular season for different strategies. Our results reveal that the optimal price varies along with the costs involved in the consumer’s purchasing choice. Finally, we find that the retailer is more likely to order a smaller quantity when the traditional online pre-ordering option is used.

Appendix

1.1 The effect of the fixed cost of building physical showroom

According to Proposition 1, we know that informed consumers’ choices are not influenced by the fixed cost of building physical showroom T. Thus the retailer’s pricing and ordering decisions are also not affected. If considering the fix cost, the retailer’s optimal expected profits under scenarios of no advance selling, traditional online pre-ordering strategy, and omni-channel pre-ordering strategy change to the following:

$$\begin{aligned} \Pi _{A1}^{*}= & {} \left[ \frac{(h+h_r-c)^2-r^2-2r(h+c-r-h_r-2l)}{4(h-l)}\right] \mu _i \nonumber \\&+(p-c){{\bar{F}}}(p)\mu _u-(p-s)\varphi (k)\sigma _u{{\bar{F}}}(p)\sqrt{1-\rho ^2}-T, \end{aligned}$$

(A.1)

$$\begin{aligned} \Pi _{A2}^{*}= & {} \left[ \frac{(h+h_r+r-c)\sqrt{(1-\eta )(h-p)^2+2(h-l)h_r}-\left[ (1-\eta )(h-p)^2+(h-l)(2h_r+r)\right] }{h-l}\right] \mu _i \nonumber \\&+(p-c){{\bar{F}}}(p)\mu _u-(p-s)\varphi (k)\sigma _u{{\bar{F}}}(p)\sqrt{1-\rho ^2}-T, \end{aligned}$$

(A.2)

$$\begin{aligned} \Pi _{15}^{*}= & {} \frac{\left[ h+\frac{h_r}{2}+l-\frac{h_s(h-l)}{h_r}\right] \left[ \frac{h_s(h-l)}{h_r}+\frac{h_r}{2}-l-c\right] -r\left[ \frac{h_s(h-l)}{h_r}-\frac{h_r}{2}-2l\right] }{h-l}\mu _i \nonumber \\&+(p-c){{\bar{F}}}(p)\mu _u-(p-s)\varphi (k)\sigma _u{{\bar{F}}}(p)\sqrt{1-\rho ^2}-T, \end{aligned}$$

(A.3)

$$\begin{aligned} \Pi _{26}^{*}= & {} \left[ \frac{h_s(h-l)}{h_r}{+}\frac{h_r}{2}-l-c\right] \mu _i+(p{-}c){{\bar{F}}}(p)\mu _u{-}(p{-}s) \varphi (k)\sigma _u{{\bar{F}}}(p)\sqrt{1-\rho ^2}-T,\nonumber \\ \end{aligned}$$

(A.4)

$$\begin{aligned} \Pi _{27}^{*}= & {} (h-c-\sqrt{(1-\eta )(h-p)^2+2(h-l)h_s})\mu _i+(p-c){{\bar{F}}}(p)\mu _u \nonumber \\&-(p-s)\varphi (k)\sigma _u{{\bar{F}}}(p)\sqrt{1-\rho ^2}-T. \end{aligned}$$

(A.5)

Next, let $\Delta \Pi _{Ab}^1$ ($\Delta \Pi _{Ab}^2$) denote the difference between the optimal profits from traditional online pre-ordering strategy and no advance selling, i.e., $\Delta \Pi _{Ab}^1=\Pi _{A1}^{*}-\Pi _{basic}$ and $\Delta \Pi _{Ab}^2=\Pi _{A2}^{*}-\Pi _{basic}$. By comparison, we can obtain

$$\begin{aligned} \Delta \Pi _{Ab}^1= & {} \left[ \frac{(h+h_r-c)^2-r^2-2r(h+c-r-h_r-2l)}{4(h-l)}-(p-c){{\bar{F}}}(p)\right] \mu _i \nonumber \\&+(p-s)\varphi (k){{\bar{F}}}(p)\left[ (\sigma _i^2+\sigma _u^2+2\sigma _i\sigma _u\rho )^{\frac{1}{2}} -\sigma _u\sqrt{1-\rho ^2}\right] -T, \end{aligned}$$

(A.6)

$$\begin{aligned} \Delta \Pi _{Ab}^2= & {} \left\{ \frac{(h+h_r+r-c)\sqrt{(1-\eta )(h-p)^2+2(h-l)h_r}-\left[ (1-\eta )(h-p)^2+(h-l)(2h_r+r)\right] }{h-l}\right. \nonumber \\&\left. -(p-c){{\bar{F}}}(p)\right\} \mu _i \nonumber \\&+(p-s)\varphi (k){{\bar{F}}}(p)\left[ (\sigma _i^2+\sigma _u^2+2\sigma _i\sigma _u\rho )^{\frac{1}{2}}-\sigma _u\sqrt{1-\rho ^2}\right] -T. \end{aligned}$$

(A.7)

Similarly, we let $\Delta \Pi _{3b}$ ($\Delta \Pi _{4b}$, $\Delta \Pi _{5b}$, $\Delta \Pi _{6b}$, and $\Delta \Pi _{7b}$) denote the difference between the optimal profits from traditional online pre-ordering strategy and no advance selling, where $\Delta \Pi _{3b}=\Pi _{13}^{*}-\Pi _{basic}$, $\Delta \Pi _{4b}=\Pi _{14}^{*}-\Pi _{basic}$, $\Delta \Pi _{5b}=\Pi _{15}^{*}-\Pi _{basic}$, $\Delta \Pi _{6b}=\Pi _{26}^{*}-\Pi _{basic}$, $\Delta \Pi _{7b}=\Pi _{27}^{*}-\Pi _{basic}$. Because $\Pi _{13}^{*}$, $\Pi _{14}^{*}$ have same expressions with $\Pi _{A1}^{*}$, $\Pi _{A2}^{*}$, respectively, we omit them, and only present

$$\begin{aligned} \Delta \Pi _{5b}^1= & {} \left\{ \frac{\left[ h+\frac{h_r}{2}+l-\frac{h_s(h-l)}{h_r}\right] \left[ \frac{h_s(h-l)}{h_r}+\frac{h_r}{2}-l-c\right] -r\left[ \frac{h_s(h-l)}{h_r}-\frac{h_r}{2}-2l\right] }{h-l}-(p-c){{\bar{F}}}(p)\right\} \mu _i \nonumber \\&+(p-s)\varphi (k){{\bar{F}}}(p)\left[ (\sigma _i^2+\sigma _u^2+2\sigma _i\sigma _u\rho )^{\frac{1}{2}}-\sigma _u\sqrt{1-\rho ^2}\right] -T, \end{aligned}$$

(A.8)

$$\begin{aligned} \Delta \Pi _{6b}^1= & {} \left[ \frac{h_s(h-l)}{h_r}+\frac{h_r}{2}-l-c-(p-c){{\bar{F}}}(p)\right] \mu _i \nonumber \\&+(p-s)\varphi (k){{\bar{F}}}(p)\left[ (\sigma _i^2+\sigma _u^2+2\sigma _i\sigma _u\rho )^{\frac{1}{2}}-\sigma _u\sqrt{1-\rho ^2}\right] -T, \end{aligned}$$

(A.9)

$$\begin{aligned} \Delta \Pi _{7b}^1= & {} \left[ h-c-\sqrt{(1-\eta )(h-p)^2+2(h-l)h_s}-(p-c){{\bar{F}}}(p)\right] \mu _i \nonumber \\&+(p-s)\varphi (k){{\bar{F}}}(p)\left[ (\sigma _i^2+\sigma _u^2+2\sigma _i\sigma _u\rho )^{\frac{1}{2}}-\sigma _u\sqrt{1-\rho ^2}\right] -T. \end{aligned}$$

(A.10)

All potential strategies will be adopted only if they can bring higher profit compared to no advance selling strategy.

Thus, if the fix cost of building physical showroom T is larger than a threshold, the corresponding advance selling strategy will not be adopted by the retailer. Therefore, our results and insights continue to hold qualitatively true when the fixed cost is excluded; thus, without deviating from the purpose of this research, we keep the cost set at zero (i.e., $T=0$).

1.2 Proof of Proposition 1

According to expected utility expressions $U_A$, $U_S$, and $U_W$, we have

$$\begin{aligned} U_A-U_W= & {} E_V\max (V-X, -h_r)-(1-\eta )\int _p^h (v-p)f(v)\, dv \nonumber \\= & {} p+\eta (h-p)+(1-\eta )\int _p^h F(v)\, dv-X-\int _{X-h_r}^h F(v)\, dv, \end{aligned}$$

(A.11)

$$\begin{aligned} U_A-U_S= & {} E_V\max (V-X, -h_r)+h_s-\int _X^h (v-X)f(v)\, dv \nonumber \\= & {} h_s-\int _{X-h_r}^X F(v)\, dv, \end{aligned}$$

(A.12)

and

$$\begin{aligned} U_S-U_W= & {} -h_s+\int _X^h (v-X)f(v)\, dv-(1-\eta )\int _p^h (v-p)f(v)\, dv \nonumber \\= & {} p-h_s+\eta (h-p)+(1-\eta )\int _p^h F(v)\, dv-X-\int _X^h F(v)\, dv. \end{aligned}$$

(A.13)

Informed consumers make best purchase to obtain more expected utility. When the retailer doesn’t provide the physical showroom in the advance selling period, they purchase in advance if and only if $U_A\ge U_W$. Thus we have $X+\int _{X-h_r}^h F(v)\, dv\le p+\eta (h-p)+(1-\eta )\int _p^h F(v)\, dv$. However, when the retailer offer physical showrooms, informed consumers will have three choices: (i) pre-order directly if and only if $U_A\ge U_S$ and $U_A\ge U_W$. Thus we have the constraints $\int _{X-h_r}^{X} F(v)\, dv\le h_s$ and $X+\int _{X-h_r}^h F(v)\, dv\le p+\eta (h-p)+(1-\eta )\int _p^h F(v)\, dv$; (ii) inspect-first-and-then-buy if and only if $U_S>U_A$ and $U_S\ge U_W$. Thus we have the constraints $\int _{X-h_r}^{X} F(v)\, dv>h_s$ and $X+\int _X^h F(v)\, dv\le p-h_s+\eta (h-p)+(1-\eta )\int _p^h F(v)\, dv$. (iii) nobody purchases early if and only if $U_A<U_W$ and $U_S<U_W$. Therefore, we obtain the constraints $X+\int _{X-h_r}^h F(v)\, dv>p+\eta (h-p)+(1-\eta )\int _p^h F(v)\, dv$ and $X+\int _X^h F(v)\, dv>p-h_s+\eta (h-p)+(1-\eta )\int _p^h F(v)\, dv$. $\square$

1.3 Proof of Proposition 2

When the retailer determines his advance selling price X, the objective function and the constraint are given by Eq. (14).

First, differentiating the objective function of Eq. (14) with respect to X, we have

$$\begin{aligned} \frac{\partial {\Pi _A}}{\partial {X}}=\frac{h+h_r+c-r-2X}{h-l}\mu _i, \end{aligned}$$

(A.14)

and

$$\begin{aligned} \frac{\partial ^2{\Pi _A}}{\partial {X^2}}=\frac{-2\mu _i}{h-l}<0. \end{aligned}$$

(A.15)

Second, it’s worth noting that the optimal advance selling price should fulfill $X^{*}\le h+h_r-\sqrt{(1-\eta )(h-p)^2+2h_r(h-l)}$ from the constraint, because we assume consumer’s valuation V is uniformly distributed within the range [l, h]. Therefore, the retailer’s optimal X can belong to the following two cases:

Case 1: When $\frac{\partial {\Pi _A}}{\partial {X}}\vert _{X=h+h_r-\sqrt{(1-\eta )(h-p)^2+2h_r(h-l)}}=\frac{c+\sqrt{(1-\eta )(h-p)^2+2h_r(h-l)}-r-h-h_r}{h-l}\mu _i<0$, (i.e., $r>2\sqrt{(1-\eta )(h-p)^2+2(h-l)h_r}+c-h-h_r$), the optimal advance selling price is $X^*=\frac{h+h_r+c-r}{2}$.

Case 2: When $\frac{\partial {\Pi _A}}{\partial {X}}\vert _{X=h+h_r-\sqrt{(1-\eta )(h-p)^2+2h_r(h-l)}}=\frac{c+\sqrt{(1-\eta )(h-p)^2+2h_r(h-l)}-r-h-h_r}{h-l}\mu _i\ge 0$, (i.e., $r\le 2\sqrt{(1-\eta )(h-p)^2+2(h-l)h_r}+c-h-h_r$), the optimal advance selling price is $X^*=-\sqrt{(1-\eta )(h-p)^2+2h_r(h-l)}+h+h_r$.

Substituting optimal advance selling price into the objective function, we obtain the results in Proposition 2. $\square$

1.4 Proof of Lemma 1

Because $X_1^{*}$ increases with $h_r$, $X_2^{*}$ is convex in $h_r$, let $X_1^{*}=X_2^{*}$, we have

$$\begin{aligned} h_{r1}=c+3h-4l-r-2\sqrt{(1-\eta )(h-p)^2+2c(h-l)+2h^2-6hl-2hr+2lr+4l^2}, \end{aligned}$$

and

$$\begin{aligned} h_{r2}=2\sqrt{(1-\eta )(h-p)^2+2c(h-l)+2h^2-6hl-2hr+2lr+4l^2}+c+3h-4l-r. \end{aligned}$$

And $X^{*}=\min \{X_1^{*}, X_2^{*}\}$, so we obtain Lemma 1 (i).

Because $X_1^{*}$ decreases with r and $X_2^{*}$ is not related to r, the retailer’s optimal pre-order price is non-increasing with the return cost of himself. $\square$

1.5 Proof of Proposition 3

When the retailer determines his advance selling price X, the objective function and the constraint are given by Eq. (17).

First, differentiating the objective function of Eq. (17) with respect to X, we have

$$\begin{aligned} \frac{\partial {\Pi _A}}{\partial {X}}=\frac{h+h_r+c-r-2X}{h-l}\mu _i, \end{aligned}$$

(A.16)

and

$$\begin{aligned} \frac{\partial ^2{\Pi _A}}{\partial {X^2}}=\frac{-2\mu _i}{h-l}<0. \end{aligned}$$

(A.17)

Second, it’s worth noting that optimal advance selling price should fulfill $X^{*}<\frac{h_s(h-l)}{h_r}+\frac{h_r}{2}-l$ and $X^{*}\le h+h_r-\sqrt{(1-\eta )(h-p)^2+2h_r(h-l)}$ from the constraints, because we assume consumer valuation V is uniformly distributed within the range [l, h]. Similar to Proposition 2, the retailer’s optimal advance price $X^{*}=\min \{X_3^{*}, X_4^{*}, X_5^{*}\}$, where $X_3^{*}=\frac{h+h_r+c-r}{2}$, $X_4^{*}=h+h_r-\sqrt{(1-\eta )(h-p)^2+2(h-l)h_r}$, and $X_5^{*}=\frac{h_s(h-l)}{h_r}+\frac{h_r}{2}-l$. Substituting optimal advance selling price into the objective function, we obtain the results in Proposition 3. $\square$

1.6 Proof of Proposition 4

When the retailer determines his advance selling price X, the objective function and the constraint are given by Eq. (21).

Differentiating the objective function of Eq. (21) with respect to X, we have

$$\begin{aligned} \frac{\partial {\Pi _2}}{\partial {X}}=\mu _i>0. \end{aligned}$$

(A.18)

Thus we know that the retailer’s total expected profit in increasing with the advance selling price, but the price is constrained. Solving the constraints, we can obtain two potential optimal advance selling price $X_6^{*}=\frac{h_s(h-l)}{h_r}+\frac{h_r}{2}-l$, $X_7^{*}=h-\sqrt{(1-\eta )(h-p)^2+2(h-l)h_s}$.

Consequently, the retailer’s optimal pre-order price is $X^{*}=\max \{X_6^{*}, X_7^{*}\}$. Substituting optimal advance selling price into the objective function, we obtain the results in Proposition 4. $\square$