## 3 Mathematical Model
### Formulation of the Mathematical Model
#### Carbon emissions cost (CEC)  
Cost of carbon emissions is estimated when penalty is added to the production firms on the basis of CO\({}_{2}\) produced due to the burning of fossils fuel. The costs penalized on the carbon emission by the production, outsourcing and reworking operations are expressed as in Equation (20), where \(CE\) is the total carbon emission, which is represented in more detail in Equations (22)-(26) [51]:  
\[CEC = \gamma_{5}CE. \tag{20}\]  
From the costs given in Equations (6) to (20), total cost of production (\(TC\)) can be given as in the form of Equation (21):  
\[ \begin{array}{llll}TC\left(Goal\ A\right)&=&\sum_{i=1}^{J}\sum_{t=1}^{T}(FC _{ij}+MC_{ij}\times MQ_{i}+CC_{ij}\times CQ_{ji})+\sum_{i=1}^{J}\sum_{t=1}^{T} HC_{i}\times(MQ_{jt}+CQ_{ji})\\ &+&\sum_{i=1}^{J}\sum_{t=1}^{T}mC_{ji}+\sum_{j=1}^{J}\sum_{t=1}^{T}WC_{i} \times(MQ_{jt}+CQ_{ji})+\sum_{i=1}^{J}\sum_{t=1}^{T}\gamma_{i}\times EP_{i} \times(MQ_{jt}+CQ_{ji})\\ &+&\sum_{i=1}^{J}\sum_{t=1}^{T}\gamma_{i}\times EP_{i}\times OQ_{ijt}\times(1 -\alpha_{ijt})\times\beta_{ijt}+\sum_{i=1}^{J}\sum_{t=1}^{T}\gamma_{i}\times CP _{ij}\times(MQ_{jt}+CQ_{ji})\\ &+&\sum_{i=1}^{J}\sum_{t=1}^{T}\gamma_{i}\times EP_{i}\times OQ_{ijt}\times(1 -\alpha_{ijt})\times\beta_{ijt}+\sum_{i=1}^{J}\sum_{t=1}^{T}\gamma_{i}\times CP _{ij}\times(MQ_{jt}+CQ_{ji})\\ &+&\sum_{i=1}^{J}\sum_{t=1}^{J}\sum_{t=1}^{T}\gamma_{i}\times CP_{ij}\times OQ _{ijt}\times(1-\alpha_{ijt})\times\beta_{ijt}+\sum_{i=1}^{J}\sum_{t=1}^{T}\gamma _{i}\times CP_{ij}\times(MQ_{jt}+CQ_{ji})\\ &+&\sum_{i=1}^{J}\sum_{t=1}^{J}\sum_{t=1}^{T}\gamma_{i}\times S_{j}\times OQ_{ ijt}\times(1-\alpha_{ijt})\times\beta_{ijt}+\sum_{i=1}^{J}\sum_{t=1}^{T}\gamma _{i}Cq_{ijt}\times(1-\alpha_{ijt})\times\beta_{ijt}\\ &+&\sum_{i=1}^{J}\sum_{t=1}^{J}\sum_{t=1}^{J}OC_{ijt}\times QQ_{ijt}\times(1 -\alpha_{ijt})\times\beta_{ijt}+\sum_{i=1}^{J}\sum_{t=1}^{J}\gamma_{i}Cq_{ijt} \times OQ_{ijt}\times(1-\alpha_{ijt})\times\beta_{ijt}\\ &+&\gamma_{5}CE+d_{2}^{+}-d_{2}^{+}.\end{array}  \tag{21}\]  
The second objective of the model is to minimize the carbon emissions generated by the manufacturing processes of the automobile industry. Carbon emissions are calculated as total carbon emissions from in-house, outsourcing and reworking operations depending on energy source. Carbon emissions for the complete manufacturing process can be optimized by the expression as given in Equation (22):  
\[ \begin{array}{llll}\text{Carbon emission goal (B)}&=&\text{ Carbon emission due to electricity in manufacturing, outsourcing,}\\ &&\text{and reworking operations}+\text{emission due to usage of solar energy}\\ &+&\text{emission due to burning of diesel}+\text{emisssion by burning gasoline}\\ &+&d_{2}^{-}-d_{2}^{+}.\end{array}  \tag{22}\]  
Similarly, Equation (22) represents the theoretical form to minimize the carbon emissions. To express this mathematically, it can be broken down into three equations: Equations (23) to (25). Carbon produced from in-house manufacturing (\(CEI\)) is the sum of emissions due to electricity, diesel, and gasoline utilized in-house production operations, given as in Equation (21):  
\[ \begin{array}{llll}CEI&=&\sum_{j=1}^{J}\sum_{t=1}^{T}(\alpha_{1}\times EP_{j} +\alpha_{2}\times DP_{j}+\alpha_{3}\times GP_{j}+\alpha_{4}\times SP_{j})\\ &+&(MQ_{jt}+CQ_{jt}).\end{array}  \tag{23}\]Carbon emissions due to consumption of electricity, diesel and gasoline in outsourcing operations (\(CEO\)) by the suppliers can be calculated as in Equation (24):  
\[ \begin{array}{rcl}CEO&=&\sum_{i=1}^{l}\sum_{j=1}^{l}\sum_{t=1}^{T}((\alpha_{1} \times EO_{ij}+\alpha_{2}\times DO_{ij}+\alpha_{4}\times SO_{ij}\\ &+&\alpha_{3}\times GO_{ij})\times OQ_{jit}).\end{array}  \tag{24}\]  
The proportion of carbon emissions is due to reworking operations (\(CER\)) generated by the supplier. In the model, a proportion of defective parts is transported to scrap because they can not be reworked and are required to be sold at a discounted rate to be disposed of in a sustainable way.