## 3 Mathematical Model
### Formulation of the Mathematical Model
#### Carbon emissions cost (CEC)
Such is not subjected to reworking operations and so will not generate carbon emissions, reducing the amount of emissions by \(OQ_{jit}\approx\beta_{jit}\ast\alpha_{jit}\) quantity. Carbon emissions as a result of reworking operations can be given as in Equation (25):  
\[ \begin{array}{rcl}CER&=&\sum_{i=1}^{l}\sum_{j=1}^{l}\sum_{t=1}^{T}(\alpha_{1} \times ER_{j}+\alpha_{2}\times DR_{j}+\alpha_{3}\times GR_{j}+\alpha_{4}\times SR _{j})\\ &&OQ_{jit}\times\beta_{jit}\times(1-\alpha_{ijt}).\end{array}  \tag{25}\]  
Using Equations (23) to (25), the total carbon emission \((CE)\) produced by the system can be given as in Equation (26):  
\[ \begin{array}{rcl}CE\ (Goal\ B)&=&\sum_{i=1}^{l}\sum_{t=1}^{T}(\alpha_{1} \times EP_{j}+\alpha_{2}\times DP_{j}+\alpha_{3}\times GP_{j}+\alpha_{4}\times SP _{j})\times(MQ_{it}+CQ_{jit})\\ &+&\sum_{i=1}^{l}\sum_{j=1}^{l}\sum_{t=1}^{T}(\alpha_{1}\times EO_{ij}+\alpha_{ 2}\times DO_{ij}+\alpha_{3}\times GO_{ij}+\alpha_{4}\times SO_{ij})\times OQ_{jit }\\ &+&\sum_{i=1}^{l}\sum_{j=1}^{l}\sum_{t=1}^{T}(\alpha_{1}\times ER_{j}+\alpha_{ 2}\times DR_{j}+\alpha_{3}\times GR_{j}+\alpha_{4}\times SR_{j})\\ &\times&OQ_{ijt}\times\beta_{jit}\times(1-\alpha_{ijt})+d_{2}^{-}-d_{2}^{+} \end{array}  \tag{26}\]  
The third objective is to optimize the model by minimizing the cost of energy consumed in manufacturing. Primary sources of energy are electricity and solar utilized to perform in-house, reworking, and outsourcing operation. Therefore, the objective function in the form of costs of electricity and solar energy is given as in Equation (27):  
\[ \begin{array}{rcl}\text{Energy cost goal (C)}&=&\text{ Electricity utilized in in manufacturing, outsourcing}\\ &\text{and reworking operations}+\text{Utilization of solar energy}\\ &+&d_{3}^{-}-d_{3}^{+}.\end{array}  \tag{27}\]  
Electricity cost can be broken down into the portions of energy utilized in in-house manufacturing (\(EI\)), in reworking operations (\(ER\)) and to perform outsourcing operations (EO), as given in Equations (28)-(30):  
\[EI = \sum_{j=1}^{l}\sum_{t=1}^{T}(\gamma_{1}\times EP_{j}\times(MQ_{ jt}+CQ_{jit})), \tag{28}\] \[EO = \sum_{i=1}^{l}\sum_{j=1}^{l}(\gamma_{1}\times EO_{ij}\times OQ_{jit }),\] (29) \[ER = \sum_{i=1}^{l}\sum_{j=1}^{l}\sum_{t=1}^{T}(\gamma_{1}\times ER_{ j}\times(1-\alpha_{ijt})\times\beta_{jit}\times OQ_{jit}). \tag{30}\]  
Similarly, the energy requirement of the production system is also fulfilled by a solar energy system to perform in-house (\(SI\)), reworking (\(SR\)), and outsourcing operations (\(SO\)), as given in Equations (31)-(33):  
\[SI = \sum_{j=1}^{l}\sum_{t=1}^{T}(\gamma_{4}\times SP_{j}\times(MQ_{ jt}+CQ_{jit})), \tag{31}\] \[SO = \sum_{i=1}^{l}\sum_{j=1}^{l}\sum_{t=1}(\gamma_{4}\times SO_{ij} \times OQ_{jit}),\] (32) \[SR = \sum_{i=1}^{l}\sum_{j=1}^{l}\sum_{t=1}^{T}(\gamma_{4}\times SR_{ j}\times(1-\alpha_{ijt})\times\beta_{jit}\times OQ_{jit}). \tag{33}\]From Equations (28)-(33), the overall equation to minimize total cost energy (\(TEC\)) is expressed as in Equation (34):  
\[ \begin{array}{rcl}TEC\ (Goal\ C)&=&\sum_{i=1}^{J}\sum_{t=1}^{T}(\gamma_{1}\times E \dot{P}_{j}\times(MQ_{ji}+CQ_{ji}))+\sum_{i=1}^{J}\sum_{i=1}^{J}(\gamma_{1} \times EO_{ij}\times QQ_{ijt})\\ &+&\sum_{i=1}^{J}\sum_{j=1}^{T}\sum_{t=1}^{T}(\gamma_{1}\times E\dot{R}_{j} \times(1-\alpha_{ijt})\times\beta_{ji}\times QQ_{jit})+\sum_{j=1}^{J}\sum_{t=1} ^{T}(\gamma_{4}\times SP_{j}\times(MQ_{ji}+CQ_{ji}))\\ &+&\sum_{i=1}^{J}\sum_{i=1}^{J}\sum_{t=1}(\gamma_{4}\times SO_{ij}\times QQ_{ ijt})+\sum_{i=1}^{J}\sum_{j=1}^{J}\sum_{t=1}^{T}(\gamma_{4}\times SR_{j} \times(1-\alpha_{ijt})\\ &\times&\beta_{jit}\times QQ_{ijt})+d_{3}^{-}-d_{3}^{+}.\end{array}  \tag{34}\]