Coordination of a socially responsible two-stage supply chain under price-dependent random demand

Abstract

This paper investigates the coordination problem of a supply chain (SC) composed of a manufacturer exhibiting corporate social responsibility (CSR) and a retailer faced with random demand. The random demand is made up of the multiplication of price-dependent demand and random demand factor (RDF), plus the CSR-dependent demand. The centralized decision problem of the SC is an extension of the existing price setting newsvendor problem (PSNP). It is found that the sufficient condition for the quasi-concavity of expected profit (EP) on PSNP can not ensure the quasi-concavity of EP of the SC. Then, the concavity condition of EP related to the CSR effect factor is presented in the case of uniformly distributed RDF and linear demand in price, and the concavity of EP is proven under centralized decision. For decentralized decision under manufacturer’s Stackelberg game, the manufacturer determines wholesale price and its CSR investment, and then the retailer decides the order quantity and the retail price. The standard revenue-sharing (RS) contract is found not able to coordinate the SC, so a modified RS (MRS) contract is proposed to coordinate the SC. Finally, numerical examples illustrate the validity of the theoretical analysis and the coordination effectiveness of the MRS contract via Matlab.

Appendixes

1.1 A. Proof of Lemma 2

From the results of PSNP in Yao et al. (2006), we know that

$$\begin{aligned} \pi ^{o}(p)=d(p)\left[ {(p-c)F^{-1}(\rho )-p\int \nolimits _A^{F^{-1}(\rho )} {F(x)dx}}\right] \end{aligned}$$

when there is no CSR in the SC. Comparing $\pi ^{o}(p)$ with $\pi ^{c}(p)$ in Eq. (10), we have $\pi ^{c}(p)=\pi ^{o}(p)+K^{\mathrm {2}}(p-c)^{2}/4$, thus $\partial ^{2}\pi ^{c}/\partial p^{2}=\partial ^{2}\pi ^{o}/\partial p^{2}+K^{2}/2$. Because of

$$\begin{aligned} \partial \pi ^{c}/\partial p=pd^{\prime }(p)\int \nolimits _A^{F^{-1}(\rho )} {xf(x)dx} +d(p)\left[ {\int \nolimits _A^{F^{-1}(\rho )} {xf(x)dx+\frac{c}{p}F^{-1}(\rho )}} \right] +\frac{K^{2}}{2}(p-c), \end{aligned}$$

when $p = c$ so that $\rho = 0$, we have $\partial \pi ^{c}/\partial p|_{p=c} =d(c)A> 0$. When $p={\overline{p}}$ so that $d(p)= 0$, we have

$$\begin{aligned} \partial \pi ^{c}/\partial p|_{p={\overline{p}}} ={\overline{p}}d^{\prime }({\overline{p}})\int \nolimits _A^{F^{-1}(\rho )} {xf(x)dx}+\frac{K^{2}}{2}({\overline{p}}-c). \end{aligned}$$

Note that $d^{\prime }({\overline{p}})<0$ under the condition that d(p) has IPE, and therefore ${\overline{p}} d^{\prime }({\overline{p}} )\int \nolimits _A^{F^{-1}(\rho )} {xf(x)dx} <0$. Because $\partial \pi ^{c}(p)$ is continuous function w.r.t. p, only if

$$\begin{aligned} K^{2}({\overline{p}}-c)/2<\left| {{\overline{p}} d^{\prime }({\overline{p}})\int \nolimits _A^{F^{-1}(({\overline{p}} -c)/{\overline{p}})}{xf(x)dx}}\right| , \end{aligned}$$

there exist $\partial \pi ^{c}/\partial p=0$ in the range $[c,{\overline{p}} ]$. To ensure $\partial \pi ^{c}(p)$ being concave w.r.t p, we need $\partial ^{2}\pi ^{c}/\partial p^{2}<0$. In light of Lemma 1, we know that $\partial ^{2}\pi ^{o}/\partial p^{2}<0$ and $\partial ^{2}\pi ^{o}/\partial p^{2}|_{p=p^{c}} <0$. Note that $\partial ^{2}\pi ^{c}/\partial p^{2}=\partial ^{2}\pi ^{o}/\partial p^{2}+K^{2}/2$ when $p = p^{c}$. If $K^{2}/2<\left| {\partial ^{2}\pi ^{o}/\partial p^{2}} \right| _{p=p^{c}}$, then $\partial ^{2}\pi ^{c}/\partial p^{2}<0$, where $p^{c}$ is the optimal price satisfying $\partial \pi ^{c}/\partial p=0$. Hence, we have Lemma 2.

1.2 B. Proof of Theorem 1

First, we show that there exists a p in the range $[c_m + c_r ,{\overline{p}}]$ satisfying $\partial \pi ^{c}(p)/\partial p=0$ if $K^{2}<2bA$. The expression of $\partial \pi ^{c}(p)/\partial p$ can be derived from Eq. (11) and is shown as the left-hand side in Eq. (12). From Eq. (12), we know that $\rho = 0$ when $p = c$ and then $\partial \pi ^{c}(p)/\partial p|_{p=c} =(a-bc)A>0$. When $p={\overline{p}}, d({\overline{p}})=0$, i.e., $a-b{\overline{p}} =0$ and $\rho$ reaches it maximum. Let’s denote maximal $\rho$ by $\bar{{\rho }}(0<\bar{{\rho }}< 1)$. Then,

$$\begin{aligned} \partial \pi ^{c}(p)/\partial p|_{p=\bar{{p}}} =K^{2}(\bar{{p}}-c)/2-b\bar{{p}}[N\bar{{\rho }}^{2}/2+A\bar{{\rho }}]=\bar{{p}}\bar{{\rho }}[(K^{2}/2-bA)-b\bar{{\rho }}N/2]. \end{aligned}$$

It is clear that $\partial \pi ^{c}(p)/\partial p|_{p=\bar{{p}}}<0$ if $(K^{2}/2-bA)-b\bar{{\rho }}N/2<0$. Due to $\bar{{\rho }}>0 , \partial \pi ^{c}(p)/\partial p|_{p=\bar{{p}}} <0$ if $(K^{2}/2-bA)<0$, i.e., $K^{2}<2bA$. Since $\partial \pi ^{c}(p)/\partial p$ is continuous w.r.t. p, there must exist a solution of $\partial \pi ^{c}(p)/\partial p=0$ in the range $[c_m \hbox {+}c_r ,{\overline{p}} ]$.

Below, we show that there exists a unique p maximizing $\pi ^{c}(p)$; in another words, $\pi ^{c}(p)$ is strictly concave w.r.t p, if $b>a(B-A)/(cB)$. From the expression shown on left-hand side in Eq. (12), we have $\partial ^{2}\pi ^{c}(p)/\partial p^{2}=K^{2}/2+(a-bp)Nc^{2}/p^{3}-b[N\rho ^{2}+2A\rho +2c(N\rho +A)/p]$. Note that $N\rho ^{2}+2A\rho +2c(N\rho +A)/p=\rho (N\rho +A)+\frac{c}{p}(N\rho +A)+A\rho +\frac{c}{p}(N\rho +A)N\rho +A+A\rho +(1-\rho )(N\rho +A)=-N(\rho -1)^{2}+A+B$. Then, we have $\partial ^{2}\pi ^{c}(p)/\partial p^{2}=K^{2}/2+(a-bp)Nc^{2}/p^{3}+bN(\rho -1)^{2}-bA-bB$. Since the optimal price $p > c$ and $1> \rho > 0,\partial ^{2}\pi ^{c}(p)/\partial p^{2}<=K^{2}/2+(a-bc)N/c+bN-bA-bB=(K^{2}/2-bA)+(aN/c-bB)$. Because of $K^{2}/2-bA< 0$, we have $\partial ^{2}\pi ^{c}(p)/\partial p^{2}<0$ if $aN/c-bB\le 0$, i.e., $b\ge a(B-A)/(cB)$. In summary, $\pi ^{c}(p)$ is strictly concave w.r.t p if $K^{2}<2bA$ and $b\ge a(B-A)/(cB)$. According to the first-order conditions, we know that the optimal price $p^{c}$ is the solution of $\partial \pi ^{c}(p^{c})/\partial p^{c}=0$, i.e., $p^{c}$ is determined by Eq. (12).

1.3 C. Proof of Theorem 4

Since the manufacturer fist moves in manufacturer’s Stackelberg game under standard RS contract, let’s consider the optimal CSR investment y determined by the manufacturer. Under given retial price p and order quantity $q^{{ rs}}$ that will be determined by the retailer in the later, we have $\pi _m^{{ rs}} =(1-\phi ){ pE}[\min (q^{{ rs}},D)]-(c_m -\omega )q^{{ rs}}-y$, and then the optimal CSR investment is $y^{{ rs}} =[K_m ((1-\phi )p+\omega ^{{ rs}}-c_m )/2]^{2}$ using the first-order condition $\partial \pi _m^{{ rs}} /\partial y=0$. From Proposition 3, we know that the optimal CSR investment under centralized decision is $y^c =[K(p-c_m -c_r )/2]^{2}$. Under channel coordination, $y^{{ rs}} =y^c$ and then $(1-\phi )p+\omega ^{{ rs}}-c_m =p-c_m -c_r$, i.e., $\omega ^{{ rs}}=\phi p-c_r$.

For the retailer, we have $\pi _r^d =\phi { pE}[\min (q,D)]-(\omega +c_r )q$, and then the optimal order quantity is $q^{{ rs}}=F^{-1}[(\phi p-\omega ^{{ rs}}-c_r )/(\phi p)](a-bp)+K\sqrt{y^{{ rs}}}$ given $y^{{ rs}}$ and $\omega ^{{ rs}}$ announced by the manufacturer. From Proposition 2, we know that the optimal order quantity under centralized decision is $q^{c}=F^{-1}[(p-\omega ^{d}-c_r )/p](a-bp)+K\sqrt{y^c}$. Under channel coordination, $q^{{ rs}} =q^c$ and then $(\phi p-\omega ^{d}-c_r )/(\phi p)=(p-c_m -c_r )/p$, i.e., $\omega ^{{ rs}}=\phi (c_m +c_r )-c_r$.

To satisfy $y^{{ rs}} =y^c$ and $q^{{ rs}} =q^c$, we need $\phi p-c_r =\phi (c_m +c_r )-c_r$, i.e., $p=c_m +c_r$. Note that $p^{c}>c=c_m +c_r$, which contradicts with $p=c_m +c_r$. Hence, standard RS is not able to coordinate the SC.

1.4 D. Proof of Theorem 5

Substituting $\omega =\phi (c_m +c_r )-c_r$ into Eq. (17), we have $[(1-\phi )p+\omega ^{{ rs}}-c_m ]/(1-\phi ) = p-c_m -c_r$ and then $y^{{ rs}} =[\frac{K}{2}(p-c_m -c_r ]^{2}$. Compared Eq. (9) in Proposition 3 with $y^{{ rs}} =[\frac{K}{2}(p-c_m -c_r ]^{2}$, it is cleat that $y^{{ rs}}=y^{c}$. Substituting $\omega =\phi (c_m +c_r)-c_r$ into Eq. (16), we have $(\phi p-\omega ^{{ rs}}-c_r ) /(\phi p)=(p-c_m -c_r) /p$ and then $q^{{ rs}}=F^{-1}[(p-c_m -c_r )/p]d(p)+K\sqrt{y^{{ ys}}}$. Comparing Eq. (8) in Proposition 2 with $q^{{ rs}}=F^{-1}[(p-c_m -c_r )/p]d(p)+K\sqrt{y^{{ ys}}}$, we have $q^{{ rs}}=q^{c}$ owing to $y^{{ rs}}=y^{c}$. Substituting $\omega =\phi (c_m +c_r )-c_r$ into Eq. (18), Eq. (18) can be rewritten as $\frac{K^{2}}{2}(p-c_m -c_r )+(a-bp)[N\frac{(p-c_m -c_r )}{p^{2}}+\frac{A}{p}](c_m +c_r )+(a-2bp)[N\frac{(p-c_m -c_r )^{2}}{2p^{2}}+A\frac{(p-c_m -c_r )}{p}]=0$, which is identical with Eq. (12), hence $p^{{ rs}}=p^{c}$. So, the parameter $\omega =\phi (c_m +c_r )-c_r$ can motivate each member of the SC to maximize its expected profit meanwhile maximizing the expected profit of the SC. To satisfy $\omega > 0$, we need $\phi (c_m +c_r )-c_r >0$ and therefore $c_r /(c_m +c_r )<\phi$.