Pricing Decisions for an Omnichannel Retailing Under Service Level Considerations

Abstract

An increasing number of retailers are presently moving to omnichannel configurations and embracing modern innovations to integrate the physical store and the online store to provide customers a comprehensive shopping experience. We develop a classical newsvendor model where a retailer buys items from a supplier and distributes them through two market segments, online vs. offline. We seek optimal prices for the product in the two channels under the newsvendor model with a single period, price-based stochastic demand, and cycle service level-based order quantity to maximize the retailer’s profit. Motivated by market share models often used in marketing, we focus on a demand model involving multiplicative uncertainty and interaction between the two sales channels. The pricing problem arising is not to be well behaved because it is difficult to verify the joint concavity in prices of the objective function’s deterministic version. However, we find that the objective function is still reasonably well behaved within the sense that there is a unique solution for our optimal problem. We observe such a situation through the visualization graphs in bounded conditions for prices and find the approximate optimal point.

Appendix

1.1 Appendix I

Proof

The expected profit within the channel i is:

$$\begin{aligned} \varPi _i(Q_i, r_i) = (r_i-s_i)G_i(r)E[\xi ] - (c_i - s_i)Q_i - (r_i-s_i)G_i(r)\int _{\frac{Q_i}{G_i(r)}}^\infty \left( x - \frac{Q_i}{G_i(r)}\right) f_{\xi }(x)dx. \end{aligned}$$

To find the order quantity $Q_i^*$ that maximizes the expected profit associated to the channel i within a given prices $r_i$, we compute the derivative of $\varPi _i(Q_i, r_i)$:

$$\begin{aligned} \frac{\partial \varPi _i(Q_i, r_i)}{\partial Q_i}= & {} - (c_i - s_i) + (r_i-s_i)G_i(r) \frac{1}{G_i(r)} \int _{\frac{Q_i}{G_i(r)}}^\infty f_{\xi }(x)dx \\= & {} - (c_i - s_i) + (r_i-s_i) \left( 1-F_{\xi }\left( \frac{Q_i}{G_i(r)}\right) \right) \\= & {} (r_i-c_i) - (r_i-s_i) F_{\xi }\left( \frac{Q_i}{G_i(r)}\right) . \end{aligned}$$

In addition, the second derivative is negative:

$$\begin{aligned} \frac{\partial ^2 \varPi _i(Q_i, r_i)}{\partial Q_i^2}= & {} - (r_i-s_i) \frac{\partial F_{\xi }\left( \frac{Q_i}{G_i(r)}\right) }{\partial Q_i}\\= & {} - (r_i-s_i)\frac{1}{G_i(r)} f_{\xi }\left( \frac{Q_i}{G_i(r)}\right) < 0. \end{aligned}$$

The function $\varPi _i(Q_i, r_i)$ is therefore concave and is minimal if and only if:

$$\begin{aligned} \frac{\partial \varPi _i(Q_i, r_i)}{\partial Q_i} = 0 \Leftrightarrow (r_i-c_i) - (r_i-s_i) F_{\xi }\left( \frac{Q_i}{G_i(r)}\right) =0 \Leftrightarrow F_{\xi }\left( \frac{Q_i^*(r)}{G_i(r)}\right) = \frac{r_i-c_i}{r_i-s_i}. \end{aligned}$$

This proves the Eq. 7.

1.2 Appendix II

Proof

The induced profit function for the channel i:

$$\begin{aligned} \varPi _i(Q_i^*,r) =&(r_i-s_i)G_i(r)E[\xi ] - (c_i - s_i)Q_i^*(r) - (r_i-s_i)G_i(r)\int _{\frac{Q_i^*(r)}{G_i(r)}}^\infty \left( x - \frac{Q_i^*(r)}{G_i(r)}\right) f_{\xi }(x)dx \\ =&(r_i-s_i)G_i(r)E[\xi ] - (c_i - s_i)Q_i^*(r) \\&- (r_i-s_i)G_i(r) \left( E[\xi ] - \int _{-\infty }^{\frac{Q_i^*(r)}{G_i(r)}} xf_{\xi }(x)dx - \frac{Q_i^*(r)}{G_i(r)} \left( 1-F_{\xi }\left( \frac{Q_i^*(r)}{G_i(r)}\right) \right) \right) \\ =&- (c_i - s_i)Q_i^*(r) + (r_i-s_i)G_i(r) \int _{-\infty }^{\frac{Q_i^*(r)}{G_i(r)}} xf_{\xi }(x)dx + (r_i-s_i)Q_i^*(r) \\&- (r_i-s_i)Q_i^*(r) F_{\xi }\left( \frac{Q_i^*(r)}{G_i(r)}\right) \\ =&(r_i - c_i)Q_i^*(r) + (r_i-s_i)G_i(r) \int _{-\infty }^{\frac{Q_i^*(r)}{G_i(r)}} xf_{\xi }(x)dx - (r_i-s_i)Q_i^*(r) \frac{r_i - c_i}{r_i-s_i}\\ =&(r_i-s_i)G_i(r)\int _{-\infty }^{\frac{Q_i^*(r)}{G_i(r)}} xf_{\xi }(x)dx. \end{aligned}$$

Since $\varPi (Q^*,r) = \sum _i\varPi _i(Q^*_i,r)$, the Eq. 8 holds.

1.3 Appendix III

Proof

For each index i, we have

$$\begin{aligned} \int _{\frac{\hat{Q}_i}{G_i(r)}}^\infty \left( x - \frac{\hat{Q}_i}{G_i(r)}\right) f_{\xi }(x)dx= & {} \int _{\frac{\hat{Q}_i}{G_i(r)}}^\infty xf_{\xi }(x)dx - \frac{\hat{Q}_i}{G_i(r)} \int _{\frac{\hat{Q}_i}{G_i(r)}}^\infty f_{\xi }(x)dx \\= & {} E[\xi ] - \int _{-\infty }^{\frac{\hat{Q}_i}{G_i(r)}} xf_{\xi }(x)dx - \frac{\hat{Q}_i}{G_i(r)} \left( 1-\int _{-\infty }^{\frac{\hat{Q}_i}{G_i(r)}} f_{\xi }(x)dx\right) \\= & {} E[\xi ] - \int _{-\infty }^{\frac{\hat{Q}_i}{G_i(r)}} xf_{\xi }(x)dx - \frac{\hat{Q}_i}{G_i(r)} \left( 1-F_{\xi }\left( \frac{\hat{Q}_i}{G_i(r)}\right) \right) . \end{aligned}$$

Thus

$$\begin{aligned} \varPi _i(\hat{Q}_i,r) =&(r_i-s_i)G_i(r)E[\xi ] - (c_i - s_i)\hat{Q}_i - (r_i-s_i)G_i(r)\int _{\frac{\hat{Q}_i}{G_i(r)}}^\infty \left( x - \frac{\hat{Q}_i}{G_i(r)}\right) f_{\xi }(x)dx \\ =&(r_i-s_i)G_i(r)E[\xi ] - (c_i - s_i)\hat{Q}_i \\&- (r_i-s_i)G_i(r) \left( E[\xi ] - \int _{-\infty }^{\frac{\hat{Q}_i}{G_i(r)}} xf_{\xi }(x)dx - \frac{\hat{Q}_i}{G_i(r)} \left( 1-F_{\xi }\left( \frac{\hat{Q}_i}{G_i(r)}\right) \right) \right) \\ =&- (c_i - s_i)\hat{Q}_i + (r_i-s_i)G_i(r) \int _{-\infty }^{\frac{\hat{Q}_i}{G_i(r)}} xf_{\xi }(x)dx + (r_i-s_i)\hat{Q}_i - (r_i-s_i)\hat{Q}_i F_{\xi }\left( \frac{\hat{Q}_i}{G_i(r)}\right) \\ =&(r_i - c_i)\hat{Q}_i - (r_i-s_i)CSL_i\hat{Q}_i + (r_i-s_i)G_i(r) \int _{-\infty }^{\frac{\hat{Q}_i}{G_i(r)}} xf_{\xi }(x)dx. \end{aligned}$$

Since $\varPi (\hat{Q},r) = \sum _i\varPi _i(\hat{Q}_i,r)$, the Eq. 12 is proved.