Exploring the strategies of online and offline recycling channels in closed-loop supply chain under government subsidy

Abstract

The influence of online and offline dual recycling channels in a closed-loop supply chain (CLSC) is investigated in our work. The CLSC models of three recycling modes (single online recycling, single offline recycling, and dual recycling channels of online and offline) are established, respectively, and the impact of government subsidies on the pricing decision-making and recycling mode selection of channel members is therefore researched. The study found that the remanufacturer sets appropriate recycling price and transfer price to coordinate online and offline recycling channels and maximize its profits in the dual recycling modes; collector’s offline recycling faces the competitive threat of remanufacturers’ online recycling under the dual recycling modes, so the collector prefers a single offline recycling mode; the relationship between the collection quantities of the three modes depends on consumers’ preference of the online recycling channel. In the meanwhile, it is illustrated that government subsidy plays a positive role in promoting the recycling and remanufacturing willingness of remanufacturers and collectors. This work provides practical insights for the CLSC system to make recycle decisions.

Appendices

Appendix 1

Derivation process of model MC Using the inverse regression method, for the collector’s profit function in Equation (10), we have $\partial {\varPi}_C^{MC}/\partial {p}_c=k\left(b-{c}_c-2{p}_c+\theta {p}_d\right)/\theta \left(1-\theta \right)=0$, ${\partial}^2{\varPi}_C^{MC}/\partial {p_c}^2=-2\left(\frac{1}{\theta }+\frac{1}{1-\theta}\right)<0$. Hence, we determine that the collector’s profit function is a strictly concave function of p_c, and has a unique optimal profit.

Then, we compute the Hessian Matrix of the objective function of the remanufacturer’s profit regarding p_n, p_d, and b _,which is expressed as

$${H}_M^{MC}==\left(\begin{array}{ccc}-2a& 0& 0\\ {}0& -\frac{k}{\theta \left(1-\theta \right)}& \frac{k}{1-\theta}\\ {}0& \frac{k}{1-\theta }& -\frac{k\left(2-\theta \right)}{1-\theta}\end{array}\right)<{0}_{\circ }$$

Hence, ${\varPi}_M^{MC}$ is strictly concave in p_n, p_d, and b, by deriving p_n, p_d, and b, we have

$$\left\{\begin{array}{c}\partial {\varPi}_M^{MC}/\partial {p}_n=1+a{c}_n-2a{p}_n=0\\ {}\begin{array}{c}\partial {\varPi}_M^{MC}/\partial b=k\left[\left(1-\theta \right)\left({c}_n-{c}_r\right)-2b+\theta {c}_d+{c}_c+2\theta {p}_d\right]/2\theta \left(1-\theta \right)=0\\ {}\partial {\varPi}_M^{MC}/\partial {p}_d=k\left[\left(1-\theta \right)\left({c}_n-{c}_r\right)+2b-\left(2-\theta \right){c}_d-{c}_c-2\left(2-\theta \right){p}_d\right]/2\left(1-\theta \right)=0\end{array}\end{array}\right.$$

Finally, we substitute the optimal price into Equation (10) and Equation (11), and we can obtain the optimal profits of each channel member.

Appendix 2

Proof of proposition 1

Since ${p}_n^{M\ast }=\frac{1+a{c}_n}{2a}$, ${p}_n^{C\ast }=\frac{1+a{c}_n}{2a}$, and ${p}_n^{MC\ast }=\frac{1+a{c}_n}{2a}$, hence we have ${p}_n^{M\ast }={p}_n^{C\ast }={p}_n^{MC\ast }$, similarly, it can be proved that D^M∗ = D^C∗ = D^MC∗.

Proof of proposition 2

Because ${p}_C^{MC\ast }-{p}_C^{C\ast }=\frac{\left(1+\theta \right)\varDelta -{c}_c+\left(1-\tau \right)s-\theta \left({c}_d-\tau s\right)}{4}-\frac{\varDelta -{c}_c+s}{4}=\frac{\theta \varDelta -\tau s-\theta \left({c}_d-\tau s\right)}{4}$, therefore, it can be concluded that when $\theta >\frac{\tau s}{\varDelta +\tau s-{c}_d}$ , ${p}_C^{MC\ast }>{p}_C^{C\ast }$.Similarly, other size relationship of prices could be certified.

Proof of proposition 3

Because ${Q}_r^{MC\ast }-{Q}_r^{C\ast }=k\frac{\left(1+\theta \right)\varDelta -{c}_c+\left(1-\tau \right)s-\theta \left({c}_d-\tau s\right)}{4\theta }-\frac{k\left(\varDelta -{c}_d+s\right)}{4\theta }=\frac{k}{4\theta}\left[\theta \left(\varDelta -{c}_d+\tau s\right)+{c}_d-{c}_c-\tau s\right]$, from the equation, it could be seen that when $\theta >\frac{c_c+\tau s-{c}_d}{\varDelta +\tau s-{c}_d}$, ${Q}_r^{MC\ast }>{Q}_r^{C\ast }$. Therefore, other size of relationship of collection quantity under different recycling mode could be decided.

Proof of proposition 4

Since ${\varPi}_C^{MC\ast }=\frac{k{\left[\left(1-\theta \right)\varDelta -{c}_c+\left(1-\tau \right)s+\theta \left({c}_d-\tau s\right)\right]}^2}{16\theta \left(1-\theta \right)}<M=\frac{k{\left[\left(1-\theta \right)\varDelta -{c}_c+\left(1-\tau \right)s+\theta \left({c}_d-\tau s\right)\right]}^2}{16\theta {\left(1-\theta \right)}^2},$ therefore, to prove ${\varPi}_C^{C\ast }>{\varPi}_C^{MC\ast }$, we need to certify ${\varPi}_C^{C\ast }>M$. Because ${\varPi}_C^{C\ast }>M\Rightarrow \frac{k{\left(\varDelta -{c}_c+s\right)}^2}{16\theta }>\frac{k{\left[\left(1-\theta \right)\varDelta -{c}_c+\left(1-\tau \right)s+\theta \left({c}_d-\tau s\right)\right]}^2}{16\theta {\left(1-\theta \right)}^2}$, ⇒(Δ − c_c + s)(1 − θ) > (1 − θ)Δ − c_c + (1 − τ)s + θ(c_d − τs),

⇒θc_c − θs >  − τs + θ(c_d − τs) ⇒ (c_d + s − τs − c_c)θ < τs, it could reach that when $\theta <\frac{\tau s}{c_d+\left(1-\tau \right)s-{c}_c}$, ${\varPi}_C^{C\ast }>{\varPi}_C^{MC\ast }.$