Research on services decision-making in closed-loop supply chain dominated by a logistics provider

Abstract

Remanufacturing is an advanced form of recycling in circular economy. In order to promote the development of remanufacturing, the government gradually uses subsidy policies to regulate and intervene related enterprises. In this paper, we assume that a closed-loop supply chain consists of an original equipment manufacturer (OEM) producing new products from raw materials, a remanufacturer producing re-manufactured products from used items directly collected from customers, and a logistics provider which sells and distributes two products as a monopolist in the given market. By constructing game model in which logistics provider is a leader and OEM and remanufacturer are the equal status followers, we solve chain members’ optimal services decision-making under government subsidy. Finally, considering the government subsidy, we analyze the impact of remanufacturer service level and the logistics provider service scope on members’ equilibrium decision-making. Analysis shows that government subsidy policies are always profitable for enterprises. The OEM and the logistics provider have the same choice. They would choose the subsidy policy for the service scope of the logistics supplier in order to improve the service level of new products and the service scope of the logistics supplier. The remanufacturer would believe that subsidizing the service level of remanufactured products is more effective in improving his own profits.

Appendix

• 1. Symbols and Their Meaning

  	Symbol	Meaning
  Symbol	$j$	$j\in \{r,n\}$, $r$, and $n$ represent remanufactured products and new products, respectively
  	$b$	Remanufacturer’s recovery cost
  	$c$	Each manufacturer’s product cost
  	$d_{j}$	Market demand for product $j$, $j \in \{ r,n\}$
  	$m$	Manufacturer’s service investment cost
  	$m_{l}$	Logistics costs of logistics providers
  	$s_{j}$	The service level of product $j$
  	$p$	For the selling price of the product, it is assumed that the new product will sell for the same price as the remanufactured product
  	$l$	The service scope of logistic providers
  	$\delta$	Cost savings in remanufacturing
  	$\tau$	Degree of remanufacturing
  	$\pi$	Profit function
  	$\alpha_{j}$	The market basis of product j refers to the market size when the product price is fixed and there is no service
  	$\beta_{s}$	Represents the service flexibility of market demand
  	$\gamma_{s}$	The degree of competition for services
  Subscript	$r,n$	$r$ and $n$ represent remanufactured products and new products, respectively
  	$s$	Service
  	$l$	Logistic providers
  Superscript	$M$	Manufactures
  	$^{b}$	The optimal reaction function
  	*	Nash Equilibrium

• 2. Proof of Corollary 1

  $$\frac{\partial {l}^{*}}{\partial {m}_{l}}=\frac{(-{\alpha }_{n}-{\alpha }_{r}+2{\beta }_{p}p)h{m}^{2}}{{[2{\beta }_{s}h({\beta }_{s}-{\gamma }_{s})(2p-2c-2h+\delta \tau )-{m}_{l}m]}^{2}}<0,$$
  $$\frac{{\partial {s_n}^*}}{{\partial {m_l}}} = \frac{{( - p + c + h){\beta _s}({\alpha _n} + {\alpha _r} - 2{\beta _p}p)h{m^{}}}}{{{{[2{\beta _s}h({\beta _s} - {\gamma _s})(2p - 2c - 2h + \delta \tau ) - {m_l}m]}^2}}} < 0$$
  $$\frac{{\partial {s_r}^*}}{{\partial {m_l}}} = \frac{{( - p + c + h - \delta \tau ){\beta _s}({\alpha _n} + {\alpha _r} - 2{\beta _p}p)h{m^{}}}}{{{{[2{\beta _s}h({\beta _s} - {\gamma _s})(2p - 2c - 2h + \delta \tau ) - {m_l}m]}^2}}} < 0$$

  Due to $\frac{\partial {l}^{*}}{\partial {m}_{l}}<0,\frac{\partial {s}_{n}^{*}}{\partial {m}_{l}}<0,\frac{\partial {s}_{r}^{*}}{\partial {m}_{l}}<0,$ ${l}^{*},{s}_{n}^{*},{s}_{r}^{*}$ decreases with the increase of $m_{l}$.

• 3. Proof of Corollary 2 

  $$\frac{\partial {l}^{*}}{\partial m}=\frac{-2({\alpha }_{n}+{\alpha }_{r}-2{\beta }_{p}p){h}^{2}{\beta }_{s}({\beta }_{s}-{\gamma }_{s})(2p-2c-2h+\delta \tau )}{{[2{\beta }_{s}h({\beta }_{s}-{\gamma }_{s})(2p-2c-2h+\delta \tau )-{m}_{l}m]}^{2}},$$
  $$\frac{\partial {s}_{n}{}^{*}}{\partial m}=\frac{(-p+c+h){\beta }_{s}({\alpha }_{n}+{\alpha }_{r}-2{\beta }_{p}p)h{m}_{l}}{{[2{\beta }_{s}h({\beta }_{s}-{\gamma }_{s})(2p-2c-2h+\delta \tau )-{m}_{l}m]}^{2}}<0$$
  $$\frac{\partial {s}_{r}{}^{*}}{\partial m}=\frac{(-p+c+h-\delta \tau ){\beta }_{s}({\alpha }_{n}+{\alpha }_{r}-2{\beta }_{p}p)h{m}_{l}}{{[2{\beta }_{s}h({\beta }_{s}-{\gamma }_{s})(2p-2c-2h+\delta \tau )-{m}_{l}m]}^{2}}<0,\left(p+\delta \tau >c+h\right)$$

  Because $\frac{\partial {s}_{n}^{*}}{\partial m}<0,\frac{\partial {s}_{r}^{*}}{\partial m}<0$, ${s}_{n}^{*},{s}_{r}^{*}$ decrease with the increase of $m$. When ${\beta }_{s}>{\gamma }_{s},\frac{\partial {l}^{*}}{\partial m}<0;{\beta }_{s}<{\gamma }_{s},\frac{\partial {l}^{*}}{\partial m}>0$, that is, when ${\beta }_{s}>{\gamma }_{s}$, $l^{*}$ decrease with the increase of $m$; When $\beta_{s} < \gamma_{s}$, $l^{*}$ increases as $m$ increases; When $\beta_{s} = \gamma_{s}$, $l^{*} = \frac{{(\alpha_{n} + \alpha_{r} - 2\beta_{p} p)h}}{{m_{l} }}$ is not affected by the change of $m$.

• 4. Proof of Corollary 3

  $$\frac{\partial {l}^{*}}{\partial \tau }=\frac{2({\alpha }_{n}+{\alpha }_{r}-2{\beta }_{p}p){h}^{2}m{\beta }_{s}({\beta }_{s}-{\gamma }_{s})\delta }{{[2{\beta }_{s}h({\beta }_{s}-{\gamma }_{s})(2p-2c-2h+\delta \tau )-{m}_{l}m]}^{2}}, \frac{\partial {s}_{n}{}^{*}}{\partial \tau }=\frac{2(p-c-h){\beta }_{s}{}^{2}({\alpha }_{n}+{\alpha }_{r}-2{\beta }_{p}p){h}^{2}({\beta }_{s}-{\gamma }_{s})\delta }{{[2{\beta }_{s}h({\beta }_{s}-{\gamma }_{s})(2p-2c-2h+\delta \tau )-{m}_{l}m]}^{2}},$$
  $$\frac{\partial {s}_{r}{}^{*}}{\partial \tau }=\frac{-{\beta }_{s}({\alpha }_{n}+{\alpha }_{r}-2{\beta }_{p}p)h[2{\beta }_{s}h({\beta }_{s}-{\gamma }_{s})(2p-2c-2h)-{m}_{l}m]}{{[2{\beta }_{s}h({\beta }_{s}-{\gamma }_{s})(2p-2c-2h+\delta \tau )-{m}_{l}m]}^{2}}>0,$$

  When ${\beta }_{s}>{\gamma }_{s}$, $\frac{\partial {l}^{*}}{\partial \tau }>0,\frac{\partial {s}_{n}^{*}}{\partial \tau }>0,$ ${l}^{*}$ increases as $\tau$ increases, and $s_{n}^{*}$ increases with the increase of $\tau$; when $\beta_{s} < \gamma_{s}$, $\frac{{\partial l^{*} }}{\partial \tau } < 0,\frac{{\partial s_{n}^{*} }}{\partial \tau } < 0,$ $l^{*}$ decreases with the increase of $\tau$, and $s_{n}^{*}$ decreases with the increase of $\tau$; When $\beta_{s} = \gamma_{s}$, $l^{*} = {\raise0.7ex\hbox{${(\alpha_{n} + \alpha_{r} - 2\beta_{p} p)h}$} \!\mathord{\left/ {\vphantom {{(\alpha_{n} + \alpha_{r} - 2\beta_{p} p)h} {m_{l} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${m_{l} }$}}$, $s_n^* = (p-c-h)\beta_s(\alpha_n+\alpha_r - 2\beta_p p)h/-m_l m$, $l^{*} ,s_{n}^{*}$, and $\tau$ are not related. Since $\frac{{\partial s_{r}^{*} }}{\partial \tau } > 0,$ $s_{r}^{*}$ increases with the increase of $\tau$. The effect of $\delta$ on $l^{*} ,s_{n}^{*} ,s_{r}^{*}$ is the same as that of $\tau$.

• 5. Proof of corollary 4

  $$\frac{\partial {l}^{*}}{\partial c}=\frac{4(-{\alpha }_{n}-{\alpha }_{r}+2{\beta }_{p}p){h}^{2}m{\beta }_{s}({\beta }_{s}-{\gamma }_{s})}{{[2{\beta }_{s}h({\beta }_{s}-{\gamma }_{s})(2p-2c-2h+\delta \tau )-{m}_{l}m]}^{2}},\frac{\partial {s}_{n}{}^{*}}{\partial c}=\frac{{\beta }_{s}({\alpha }_{n}+{\alpha }_{r}-2{\beta }_{p}p)h[2{\beta }_{s}h\delta \tau ({\beta }_{s}-{\gamma }_{s})-{m}_{l}m]}{{[2{\beta }_{s}h({\beta }_{s}-{\gamma }_{s})(2p-2c-2h+\delta \tau )-{m}_{l}m]}^{2}}<0.$$
  $$\frac{\partial {s}_{r}{}^{*}}{\partial c}=\frac{-{\beta }_{s}({\alpha }_{n}+{\alpha }_{r}-2{\beta }_{p}p)h[2{\beta }_{s}h\delta \tau ({\beta }_{s}-{\gamma }_{s})+{m}_{l}m]}{{[2{\beta }_{s}h({\beta }_{s}-{\gamma }_{s})(2p-2c-2h+\delta \tau )-{m}_{l}m]}^{2}},$$

  When ${\beta }_{s}>{\gamma }_{s}$,$\frac{\partial {l}^{*}}{\partial c}<0,\frac{\partial {s}_{r}^{*}}{\partial c}<0,$ ${l}^{*}$ decreases with the increase of $c$, and $s_{r}^{*}$ decreases with the increase of $c$; When $\beta_{s} < \gamma_{s}$, $\frac{{\partial l^{*} }}{\partial c} > 0,$ $l^{*}$ expands with the increase of $c$, and $\frac{{\partial s_{r}^{*} }}{\partial c}$ depends on the size of $[2\beta_{s} h\delta \tau (\beta_{s} - \gamma_{s} ) + m_{l} m]$. Since $\frac{{\partial s_{n}^{*} }}{\partial c} < 0,$ $s_{n}^{*}$ decreases with the increase of $c$. When $\beta_{s} = \gamma_{s}$, $l^{*}$ and $c$ are not related, $\frac{{\partial s_{r}^{*} }}{\partial c} = \frac{{\partial s_{n}^{*} }}{\partial c} < 0,$ $s_{n}^{*} ,s_{r}^{*}$ decreases with the increase of $c$, and the decreasing trend is the same.