## 3 Mathematical Model
### Formulation of the Mathematical Model  
The multiobjective model is formulated to minimize the total production cost as the first objective, carbon emission as the second, and cost of energy as the third objective. The objective of the proposed WGP model is to minimize the deviations from the goals:  
\[Minimize\ w_{1}d_{1}^{+}+w_{2}d_{2}^{+}+w_{3}d_{3}^{+}. \tag{3}\]  
Equation (3) depicts the minimization of the deviations around the targeted goals. \(d_{1}^{+}\), \(d_{2}^{+}\) and \(d_{3}^{+}\) indicate the deviational variables of total cost of production, carbon emission and energy cost goals, respectively. \(w_{1}\), \(w_{2}\) and \(w_{3}\) are the corresponding weights. Since deviations have different units, i.e., $ and kg, it is not possible to sum them directly due to incommensurability. In order to add these factors, normalization methods are needed. The proposed model uses the percentage normalization method to standardize the different units to a single function and is represented by dividing the deviations by their corresponding target goals. Thus, the objective function of the proposed model using WGP is given in Equation (4):  
\[Minimize\ w_{1}\frac{d_{1}^{+}}{\textit{Coal\ A}}+w_{2}\frac{d_{2}^{+}}{\textit {Coal\ B}}+w_{3}\frac{d_{3}^{+}}{\textit{Coal\ C}}. \tag{4}\]  
The first goal of the proposed model is to minimize the total cost of production, which covers fixed cost and variable cost. Fixed cost includes capital and setup cost, while variable cost includesfor manufacturing cost, crashing cost, maintenance cost, holding cost, warehousing cost, energy cost, rework with scrap, and cost of MQL along with deviational variables, as in Equation (5):  
\[ \begin{array}{rcl}\text{Total cost of production (A)}&=&\text{Fixed cost}+\text{ Manufacturing cost}+\text{Cashing cost}+\text{Holding cost}\\ &+&\text{Maintenance cost}+\text{ Warehouse cost}+\text{Electricity cost}\\ &+&\text{Cost of solar energy}+\text{Cost of diesel}+\text{Cost of gasoline}\\ &+&\text{Reworking cost}+\text{Outsourcing cost}+\text{Cost of MQL system}\\ &+&\text{Scrap cost}+\text{Carbon emission cost}+d_{1}^{-}-d_{1}^{+}.\end{array}  \tag{5}\]  
The ultimate objective of the industry is to minimize the production cost by any source. The managers need to verify each cost separately and carefully to determine which cost contributes the most to the total cost. This is why each must be defined and calculated individually, as given in Equations (6) to (19). Fixed cost (FC)The cost is related to the initial cost required to run the production system. It may include setup cost, changeover, and tool setting cost. This cost is fixed, independent of the production quantity. It is dependent on time and consists of initial investment and setup cost of the system in this production model, as given in Equation (6):  
\[ \begin{array}{rcl}FC&=&\sum_{j=1}^{J}\sum_{t=1}^{T}(FC_{it}).\end{array}  \tag{6}\]  
Manufacturing cost (MC)Specifically, this is the cost incurred in the manufacturing operations of automobile parts, which is a variable cost and highly dependent on the production quantity. The cost comprises the sum of all costs utilized on resources required to manufacture product. It includes machining and processing cost, utilities required for machines, and labor cost. It is associated with the manufacturing operations of in-house manufacturing and can be formulated in the form of Equation (7):  
\[ \begin{array}{rcl}MC&=&\sum_{j=1}^{J}\sum_{t=1}^{T}(MC_{it}\times MQ_{it}). \end{array}  \tag{7}\]  
Crashing cost (CC)Crashing cost is used to improve the service level of the product by adding resources in the form of labor, machines, and energy. This cost is utilized to reduce the crashing lead time required to manufacture the given product. The crashing cost is incurred on the calculated amount of extra quantity required to avoid shortages. It is produced to compensate the extra demand in terms of backorders. The equation of crashing cost is given as in Equation (8):  
\[