Service supply chain coordination with platform effort-induced demand

Abstract

In this study we investigate the effort of a service platform in cooperation with a hotel and its influence on the hotel’s decision about the quantity of reserved rooms. We begin with a newsvendor hotel facing two kinds of customers: D-customers and C-customers. The D-customers order rooms from the hotel’s front desk while the C-customers book rooms through a service platform, i.e., Ctrip.com. The hotel makes decisions about how many rooms to allot to Ctrip.com to achieve optimal profit. Two newsvendor settings are proposed that depict the demands of the two parties independently. We discuss a fixed payment contract and a cost sharing contract and find that the cost sharing contract achieves channel coordination and that the division of profit depends upon the hotel’s payment to Ctrip.com. We then extend our study by taking into consideration the effort Ctrip.com exerts. We verify that a cost sharing contract can achieve channel coordination imperfectly if the idle cost of a vacant reserved room is shared between the hotel and Ctrip.com. Further, given that the cost of effort is shared, the channel is coordinated and a win–win is guaranteed as well if the terms of the new cost sharing contract are properly chosen.

Appendix: Explaining and proofs

Proof of Lemma 1

For

$$\begin{aligned} \pi _c (Q)= & {} \int _0^{K-Q} {\left[ \left( p_h -c_v\right) x-c_u (K-Q-x)\right] f_h (x)dx}\\&+\,\int _{K-Q}^{+\infty } {\left( p_h -c_v\right) (K-Q)f_h (x)dx} {\,+\int _0^Q {\left[ \left( p_t -c_v\right) x-c_u (Q-x)\right] f_t (x)dx} } \\&+\,\int _Q^{+\infty } {\left( p_t -c_v\right) Q f_t (x)dx} \end{aligned}$$

Taking derivative of $\pi _c (Q)$ gives

$$\begin{aligned} {\pi }^{\prime }_c (Q)=\left( p_h -c_v +c_u\right) F_h (K-Q)-\left( p_t -c_v +c_u\right) F_t (Q)-\left( p_h -p_t\right) \end{aligned}$$

(EC.1)

Obviously, ${\pi }^{\prime }_c (Q)$ is continuous and decreasing in Q.

• (a) If ${\pi }^{\prime }_c (0)\le 0$, $\pi _c (Q)$ has a unique maximizer and 0 is the unique solution.

• (b) If ${\pi }^{\prime }_c (0)\,{\ge }\,0$, since ${\pi }^{\prime }_c (K)=-(p_t -c_v +c_u)F_t (K)-(p_h -p_t)<0, \pi _c (Q)$ is quasi-concave and has a unique maximizer.

Together we know $\pi _c (Q)$ has a unique maximizer.

Proof of Lemma 2

(i) For

$$\begin{aligned} \pi _w^h (Q)= & {} \int _0^{K-Q} {\left[ \left( p_h -c_v\right) x-c_u (K-Q-x)\right] f_h (x)dx}\\&\quad +\,\int _{K-Q}^{+\infty } {\left( p_h -c_v\right) (K-Q)f_h (x)dx} \\&\quad {+\,\int _0^Q {\left[ \left( p_t -w-c_v\right) x-c_u (Q-x)\right] f_t (x)dx} } \\&\quad +\,\int _Q^{+\infty } {\left( p_t -w-c_v\right) Qf_t (x)dx} \end{aligned}$$

The first-order-condition yields

$$\begin{aligned} {\pi }_w^{{\prime }h} (Q)=\left( p_h -c_v +c_u\right) F_h (K-Q)-\left( p_t -w-c_v +c_u\right) F_t (Q)-\left( p_h +w-p_t\right) \end{aligned}$$

(EC.2)

Similar to proof of Lemma 1, $\pi _w^h (Q)$ is quasi-concave and has a unique maximizer.

(ii) The derivative of $\pi _w^h (Q)$ at $Q=Q_c$ is

$$\begin{aligned} {\pi }_w^{{\prime }h} (Q_c)= & {} \left( p_h -c_v +c_u \right) F_h (K-Q_c)\\&\quad -\,\left( p_h -c_v\right) -\left( p_t -w-c_v +c_u \right) F_t (Q_c)+\left( p_t -w-c_v\right) \\= & {} {\pi }^{\prime }_c (Q_c)+wF_t (Q_c)-w \\= & {} -w\left( 1-F_t (Q_c)\right) <0 \end{aligned}$$

Since $\pi _w^h (Q)$ is increasing first and then decreasing in Q. Hence there must be $Q_w \le Q_c$.

Proof of Lemma 3

(i) For

$$\begin{aligned} \pi _\beta ^h (Q)= & {} \int _0^{K-Q} {\left[ \left( p_h -c_v\right) x-c_u (K-Q-x)\right] f_h (x)dx}\\&\quad +\,\int _{K-Q}^{+\infty } {\left( p_h -c_v\right) (K-Q)f_h (x)dx} \\&\quad {+\,\int _0^Q {\left[ \left( p_t -w-c_v\right) x-c_u (Q-x)\right] f_t (x)dx} } \\&\quad +\,\int _Q^{+\infty } {\left( p_t -w-c_v\right) Qf_t (x)dx} \\&\quad +\,\int _0^Q {\beta c_u (Q-x)f_t (x)dx} \\ \end{aligned}$$

Consider the first-order condition for $\pi _\beta ^h (Q)$:

$$\begin{aligned} {\pi }_\beta ^{{\prime }h} (Q)&=\left( p_h -c_v +c_u\right) F_h (K-Q)-\left( p_t -w-c_v +c_u -\beta c_u\right) F_t (Q)\\&\quad -\left( p_h +w-p_t\right) \end{aligned}$$

(EC.3)

The proof is similar to that of Lemma 1 and omitted.

(ii) The derivative of $\pi _\beta ^h (Q)$ at $Q=Q_w$ is

$$\begin{aligned} {\pi }_\beta ^{{\prime }h} (Q_c)= & {} (p_h -c_v +c_u)F_h (K-Q_w)-(p_t -w-c_v +c_u)F_t (Q_w)\\&\quad -\,(p_h +w-p_t)+\beta c_u F_t (Q_w) \\= & {} {\pi }_w^{{\prime }h} (Q_w)+\beta c_u F_t (Q_w) \\= & {} \beta c_u F_t (Q_w)>0 \end{aligned}$$

Thus $Q_\beta \ge Q_w$.

(iii) Let us change $\beta$ to $\beta +\delta$, where $\delta >0$. From (EC.3), we have

$$\begin{aligned} {\pi }_\beta ^{{\prime }h{*}} (Q)= & {} (p_h -c_v +c_u)F_h (K-Q)-(p_t -w-c_v +c_u -\beta c_u)F_t (Q)\\&\quad -\,(p_h +w-p_t)+\delta c_u F_t (Q) \end{aligned}$$

Let $Q=Q_\beta$, we have $\left. {{\pi }_w^{{\prime }h{*}} (Q)} \right| _{Q=Q_\beta } =\delta F_t (Q_c)>0$. For ${\pi }_\beta ^{{\prime }h{*}} (Q)$ is also continuous and decreasing in Q, we know that $Q_\beta ^{*} >Q_\beta$, where $Q_\beta ^{*}$ is the solution to ${\pi }_\beta ^{{\prime }h{*}} (Q)=0$. Then we know $Q_\beta$ is increasing in $\beta$.

Proof of Theorem 1

Let $w^{{*}}\in (0,p_t -c_v),\beta ^{{*}}=\frac{w^{{*}}(1-F_t (Q_c))}{c_u F_t (Q_c)}$, we have

$\left. {{\pi }_\beta ^{{\prime }h} (Q)} \right| _{Q=Q_c } =\pi _c (Q_c )+[w^{{*}}F_t (Q_c)-w^{{*}}+\beta ^{{*}}c_u F_t (Q_c )]=0,$ which means that $Q_\beta =Q_c$. In other words, the cost sharing contract achieves channel coordination. Furthermore, the resulting profit to Ctrip.com is $\pi _\beta ^t =\frac{w^{{*}}}{F_t (Q_c)}\int _0^{Q_c } {xf_t (x)dx} =w^{{*}}r$.

Proof of Lemma 4

For

If $\pi _\beta ^h \ge \pi _w^h$, we have $\beta \le \beta _1 =\frac{w\left[ E\left( Q_\beta \wedge D_t\right) -E\left( Q_w \wedge D_t\right) \right] }{c_u \left[ Q_\beta -E\left( Q_\beta \wedge D_t\right) \right] }$; If $\pi _\beta ^t \ge \pi _w^t$,we have

$$\begin{aligned} \beta \ge \beta _2 =\frac{\left( p_h -c_v +c_u\right) \left[ E\left( Q_w \wedge D_t \right) -E\left( Q_\beta \wedge D_t\right) \right] +\left( p_t -w-c_v +c_u\right) \left[ E\left( (K-Q_w)\wedge D_h \right) -E\left( (K-Q_\beta )\wedge D_h\right) \right] }{c_u \left[ Q_\beta -E\left( Q_\beta \wedge D_t \right) \right] }. \end{aligned}$$

Clearly,

$$\begin{aligned}&\beta _2 =\beta _1 +\frac{\left( p_h -c_v +c_u\right) \left[ E\left( Q_w \wedge D_t \right) -E\left( Q_\beta \wedge D_t\right) \right] +\left( p_t -c_v +c_u\right) \left[ E\left( (K-Q_w)\wedge D_h \right) -E\left( (K-Q_\beta )\wedge D_h\right) \right] }{c_u \left[ Q_\beta -E\left( Q_\beta \wedge D_t \right) \right] } \\&\quad =\beta _1 +\frac{\pi _c (Q_\beta )-\pi _c (Q_w)}{c_u \left[ Q_\beta -E\left( Q_\beta \wedge D_t\right) \right] } \end{aligned}$$

Since $Q_w \le Q_\beta$, so $\pi _c (Q_\beta )-\pi _c (Q_w)\ge 0$, thus $\beta _2 \ge \beta _1$. Let $\beta _l =\max \{0,\beta _1 \}, \beta _u =\min \{1,\beta _2 \}$. Obviously, when $\beta \in [\beta _l,\beta _u]$, we have $\pi _\beta ^h \ge \pi _w^h$ and $\pi _\beta ^t \ge \pi _w^t$.

Proof of Lemma 5

Solving (7) and (8), we get optimal Q and e. Assuming $(Q_1,e_1)$ and $(Q_2,e_2)$ are the optimal solutions of $\Pi _c (Q,e)$, if $(Q_1,e_1)\ne (Q_2,e_2)$, then we have three cases: (i) $\frac{Q_1 }{e_1 }=\frac{Q_2 }{e_2 }$; (ii) $\frac{Q_1 }{e_1 }<\frac{Q_2 }{e_2 }$; (iii) $\frac{Q_1 }{e_1 }>\frac{Q_2 }{e_2 }$.

• Case (i): if $\frac{Q_1 }{e_1 }=\frac{Q_2 }{e_2 }$, from (2), we know $e_1 =e_2$, then $Q_1 =Q_2$, that is $(Q_1,e_1)=(Q_2,e_2 )$;

• Case (ii): if $\frac{Q_1 }{e_1 }<\frac{Q_2 }{e_2 }$, from (2), we know $e_1 <e_2$, from (1), we know $Q_1 >Q_2$, then there must be $\frac{Q_1 }{e_1 }>\frac{Q_2 }{e_2 }$;

• Case (iii): if $\frac{Q_1 }{e_1 }>\frac{Q_2 }{e_2 }$, from (2), we know $e_1 >e_2$, from (1), we know $Q_1 <Q_2$, then there must be $\frac{Q_1 }{e_1 }<\frac{Q_2 }{e_2 }$;

Obviously, all three cases are contradictory and false, so there must be $(Q_1,e_1)=(Q_2,e_2)$.

Proof of Theorem 2

Since the objective functions of both players are continuous and concave, the existence of an equilibrium is established from the Theorem (Section 1.2) in Fudenberg and Tirole (1991). The proof of uniqueness is similar to that of Lemma 4 and omitted.

Proof of Theorem 3

Let $w^{{*}}=(p_t -c_v +c_u)F_t ({\overline{{Q}}_c }/{\overline{{e}}_c })$ and $\beta ^{{*}}=[{(p_t -c_v +c_u)}{(1-F_t ({\overline{{Q}}_c }/{\overline{{e}}_c }))]}/{c_u }$, we can see that functions (5) and (6) are the same as functions (1) and (2) respectively. Obviously, the contract can achieve channel coordination.

Proof of Theorem 4

• (a) For $\lambda ={[\overline{{e}}_c \int _0^{{\overline{{Q}}_c }/{\overline{{e}}_c }} {xf_t (x)} dx-{V(\overline{{e}}_c)}/{(p_t -c_v +c_u )}]}/{F_t ({\overline{{Q}}_c }/{\overline{{e}}_c })}$,

if $\overline{{e}}_c \int _0^{{\overline{{Q}}_c }/{\overline{{e}}_c }} {xf_t (x)} dx-{V(\overline{{e}}_c)}/{(p_t -c_v +c_u) }>0$ is established, then $\lambda >0$.

Let $G(e)=(p_t -c_v +c_u)e\int _0^{{Q_c }/{e_c }} {xf_t (x)dx-V(e)}$, the first-order—condition of G(e) is

${G}^{\prime }(e)=(p_t -c_v +c_u)\int _0^{{Q_c }/{e_c }} {xf_t (x)dx-{V}^{\prime }(e)} ,$ clearly, ${G}^{\prime }(e)$ is decreasing in e. From function (2), we know ${G}^{\prime }(e_c)=0$, then $G(e_c)=\mathop {\max }\limits _e G(e)$. Since $G(0)=0$, so $G(e_c)>0$, that is $\overline{{e}}_c \int _0^{{\overline{{Q}}_c }/{\overline{{e}}_c }} {xf_t (x)} dx-{V(\overline{{e}}_c)}/{(p_t -c_v +c_u) }>0$, consequently, $\lambda >0$.

• (b) If conditions (1), (2) and (3) are established, we can see functions (7) and (8) are the same as functions (1) and (2) respectively, which reveals that the contract $(w^{{*}},\beta _1^{*},\beta _2^{*})$ can achieves channel coordination.

• (c) The resulting profit to Ctrip.com is

  $$\begin{aligned} \Pi _{\beta _1,\beta _2 }^{t{*}} =\frac{w^{{*}}}{F_t ({\overline{{Q}}_c }/{\overline{{e}}_c })}\left[ \overline{{e}}_c \int _0^{{\overline{{Q}}_c }/{\overline{{e}}_c }} {xf_t (x)} dx-{V(\overline{{e}}_c)}/{p_t -c_v +c_u }\right] =w^{{*}}\lambda \end{aligned}$$