Abstract
Previous research indicates that it is feasible to incorporate carbon considerations into production and inventory management by making informed decisions based on total costs and other relevant factors. In this article, the inventory model of single-vendor single-buyer with fixed demand is considered and carbon emission costs are added to the model according to the two policies of carbon tax and emission trading, and the model is also discussed through environmental concerns. The objective of this study is to establish an equilibrium between the costs associated with single-vendor single-buyer (SV-SB) and carbon emissions, utilizing carbon tax and carbon trading policies to regulate the quantity of carbon generated. The results show using the carbon trading policy is more economical although it is much more complicated in implementation than carbon tax policy.
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Appendix 1
Appendix 1
In order to proof the convexity of the objective functions, it is required demonstrating that their second derivatives are positive. In this segment, we compute the second derivative for each of the primary functions.
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1.
The model of a single vendor and a single buyer
This function exhibits a positive second derivative, indicating the presence of a minimum point.
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Carbon emission function model for a single vendor and a single buyer
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A SV-SB function model under carbon tax policy
$$ \begin{gathered} Tc_{t} = \frac{D}{q}\left( {\frac{{PC + n.TC + t\left( {k_{0} + ne_{0} } \right)}}{{\left( {1 + \left( {n - 1} \right)r} \right)}}} \right) \hfill \\ + \frac{q}{2}\left( {h_{1} \frac{{2D + \left( {P - D} \right)\left( {1 + \left( {n - 1} \right)r} \right)}}{P} + \left( {h_{2} - h_{1} } \right)\frac{{\left( {1 + \left( {n - 1} \right)r^{2} } \right)}}{{\left( {1 + \left( {n - 1} \right)r} \right)}} + \frac{2gtD}{P} + \frac{{gt\left( {P - D} \right)\left( {1 + \left( {n - 1} \right)r} \right)}}{P}} \right) \hfill \\ + t\left( {kD + eD + 2g_{0} } \right) \hfill \\ \end{gathered} $$$$ \begin{gathered} \frac{{d Tc_{t} }}{dq} = \frac{ - D}{{q^{2} }}\left( {\frac{{PC + n.TC + t\left( {k_{0} + ne_{0} } \right)}}{{1 + \left( {n - 1} \right)r}}} \right) \hfill \\ + \frac{1}{2}\left( {h_{1} \frac{{2D + \left( {P - D} \right)\left( {1 + \left( {n - 1} \right)r} \right)}}{P} + \left( {h_{2} - h_{1} } \right)\frac{{\left( {1 + \left( {n - 1} \right)r^{2} } \right)}}{{\left( {1 + \left( {n - 1} \right)r} \right)}} + \frac{2gtD}{P} + \frac{{gt\left( {P - D} \right)\left( {1 + \left( {n - 1} \right)r} \right)}}{P}} \right) \hfill \\ \end{gathered} $$$$ \frac{{ d^{2} Tc_{t} }}{{d q^{2} }} = - \frac{2D}{{q^{3} }}\left( {\frac{{PC + n.TC + t\left( {k_{0} + ne_{0} } \right)}}{{\left( {1 + \left( {n - 1} \right)r} \right)}}} \right) > 0 $$
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A SV-SB function model under carbon trade policy
$$ \begin{gathered} TC = \frac{D}{q}\left( {\frac{{PC + n.TC + t\left( {k_{0} + ne_{0} } \right)}}{{\left( {1 + \left( {n - 1} \right)r} \right)}}} \right) \hfill \\ + \frac{q}{2}\left( {h_{1} \frac{{2D + \left( {P - D} \right)\left( {1 + \left( {n - 1} \right)r} \right)}}{P} + \left( {h_{2} - h_{1} } \right)\frac{{\left( {1 + \left( {n - 1} \right)r^{2} } \right)}}{{\left( {1 + \left( {n - 1} \right)r} \right)}} + \frac{2gtD}{P} + \frac{{gC\left( {P - D} \right)\left( {1 + \left( {n - 1} \right)r} \right)}}{P}} \right) - C\alpha \hfill \\ \end{gathered} $$$$ \frac{{ d^{2} Tc_{t} }}{{d q^{2} }} = \frac{D}{{q^{3} }}\left( {\frac{{PC + n.TC + C\left( {k_{0} + ne_{0} } \right)}}{{\left( {1 + \left( {n - 1} \right)r} \right)}}} \right) > 0 $$
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Sepehriar, A., Eslamipoor, R. An economical single-vendor single-buyer framework for carbon emission policies. J Bus Econ (2024). https://doi.org/10.1007/s11573-024-01192-w
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DOI: https://doi.org/10.1007/s11573-024-01192-w