A green vendor-managed inventory analysis in supply chains under carbon emissions trading mechanism

Abstract

One of the most important strategies for reducing carbon emissions is to optimize firms’ operation decisions in business practices. This paper proposes a green vendor-managed inventory (a green VMI) model with a supplier and a manufacturer under a carbon emissions trading mechanism. The proposed model integrates both environmental and economic goals under a carbon emissions constraint, and then the members’ optimal decisions are obtained. Comparing this model with the traditional VMI model, this paper finds that, in the green VMI model, whether the supplier should sell or buy carbon credit depends on the carbon cap. Further, the impacts of the carbon cap and the carbon emissions factors on the optimal decisions, the carbon emissions, and the total costs in the supply chain are examined analytically. Finally, numerical experiments are performed to verify the theoretical results. It is shown that, after introducing the carbon trading mechanism, the VMI model could increase the total cost of the supply chain under some specified set of parameters.

Appendices

Appendix 1

Proof of Proposition 1.

Note that model T is a special case of model G, and the total cost of the supply chain in model T is as follows:

$${\text{TC}}_{0}^{\text{T}} = (S_{0} + S_{1} )\frac{D}{{Q_{1}^{\text{T}} }} + \frac{1}{2}Q_{1}^{\text{T}} h_{1} + \frac{D}{{Q_{0}^{\text{A}} }}S_{P} + \frac{{Q_{1}^{\text{T}} }}{2}h_{0} \left[ {(m^{\text{T}} - 1) - (m^{\text{T}} - 2)\frac{D}{P}} \right].$$

Taking the first-order derivative of TC ^T₀ , there are the following equations:

$$\frac{{\partial {\text{TC}}_{0}^{\text{T}} }}{\partial m} = - \frac{1}{{(m)^{2} }} \cdot \frac{{S_{P} D}}{{Q_{1}^{\text{T}} }} + \frac{1}{2}Q_{1}^{\text{T}} h_{0} \frac{P - D}{P},$$

(12)

$$\frac{{\partial TC_{0}^{\text{T}} }}{{\partial Q_{1}^{\text{T}} }} = - \frac{D}{{(Q_{1}^{\text{T}} )^{2} }}\left( {S_{0} + S_{1} + \frac{{S_{P} }}{m}} \right) + \frac{1}{2}h_{1} + \frac{1}{2}h_{0} \left[ {(m - 1) - (m - 2)\frac{D}{P}} \right].$$

(13)

Let (12) and (13), respectively, equal to zero and joining together, and the optimal decisions under model T can be obtained as follows:

$$Q_{1}^{{{\text{T}}^{*} }} = \frac{1}{{m^{T} }}\sqrt {\frac{{2PDS_{P} }}{{h_{0} (P - D)}}} = \sqrt {\frac{{2PD(S_{0} + S_{1} )}}{{2Dh_{0} - h_{0} P + h_{1} P}}} ,$$

(14)

$$m^{{{\text{T}}^{*} }} = \sqrt {\frac{{S_{P} (2Dh_{0} - h_{0} P + h_{1} P)}}{{h_{0} (P - D)(S_{0} + S_{1} )}}} ,$$

$$Q_{0}^{{{\text{T}}^{*} }} = Q_{1}^{{{\text{T}}^{*} }} \cdot m^{{{\text{T}}^{*} }} = \sqrt {\frac{{2PDS_{P} }}{{h_{0} (P - D)}}} .$$

Remember that $Q_{0}^{G*} = \sqrt {\frac{{2PDS_{P} }}{{(h_{0} + C \cdot g)(P - D)}}}$, comparing $Q_{0}^{{{\text{T}}^{*} }}$ and $Q_{0}^{{{\text{G}}^{*} }}$ directly, Proposition 1 is proved.

Appendix 2

Proof of Proposition 2.

From Eq. (10) and Eq. (11), $Q_{1}^{{{\text{G}}^{*} }} = \sqrt {\frac{{(2PD + S_{0} + S_{1} + C \cdot e)}}{{(2D - P)(h_{0} + C \cdot g) + h_{1} P}}}$ is derived.

Besides $Q_{1}^{{{\text{T}}^{ * } }} = \sqrt {\frac{{2PD(S_{0} + S_{1} )}}{{2Dh_{0} - h_{0} P + h_{1} P}}}$. When $Q_{1}^{{{\text{G}}^{*} }} = Q_{1}^{{{\text{T}}^{*} }}$, there is $\frac{{(2PD + S_{0} + S_{1} + C \cdot e)}}{{(2D - P)(h_{0} + C \cdot g) + h_{1} P}} = \frac{{2PD(S_{0} + S_{1} )}}{{2Dh_{0} - h_{0} P + h_{1} P}}$. So the condition $\frac{e}{g} = \frac{{2D(S_{0} + S_{1} )}}{{2Dh_{0} - h_{0} P + h_{1} P}}$ in case (1) of Proposition 2 is obtained.

Similarly, when $Q_{1}^{{{\text{T}}^{*} }} < Q_{1}^{{{\text{G}}^{*} }}$, there is $\frac{e}{g} > \frac{{2D(S_{0} + S_{1} )}}{{2Dh_{0} - h_{0} P + h_{1} P}}$ and when $Q_{1}^{{{\text{T}}^{*} }} > Q_{1}^{{{\text{G}}^{*} }},$ there is $\frac{e}{g} < \frac{{2D(S_{0} + S_{1} )}}{{2Dh_{0} - h_{0} P + h_{1} P}}$.

Appendix 3

Proof of Proposition 3.

According to Eq. (6), $X = \alpha - \left\{ {\frac{eD}{{Q_{1}^{{{\text{G}}^{*} }} }} + \frac{g}{2}Q_{1}^{{{\text{G}}^{*} }} \left[ {\left( {1 - \frac{D}{p}} \right)m^{ * } + \frac{2D}{P}} \right]} \right\}$. Let $\alpha_{0} = \frac{eD}{{Q_{1}^{{{\text{G}}^{*} }} }} + \frac{g}{2}Q_{1}^{{{\text{G}}^{*} }} \left[ {\left( {1 - \frac{D}{p}} \right)m^{ * } + \frac{2D}{P}} \right]$. Thus X = α − α ₀. When X < 0, X > 0, X = 0, there are, respectively, α < α ₀, α > α ₀, α = α ₀.

Appendix 4

Proof of Proposition 4.

According to Eqs. (9), (10), and (11), given a fixed carbon price C, the carbon cap α has no effect on $Q_{1}^{{{\text{G}}^{*} }}$ and $Q_{1}^{{{\text{G}}^{*} }} ,Q_{0}^{{{\text{G}}^{*} }}$. Also, the total amount CF(Q) of carbon emissions remains constant from Eq. (4).

While in Eq. (5) there is CF(Q) + X = α, CF(Q) remains constant when α decreases. Therefore, the transfer quantity X decreases when α decreases. According to Eq. (7), because X decreases when α decreases, the total cost TC₀ increases with the decrease in α when other terms remain unchanged.