# Designing Sustainable Flexible Manufacturing Cells with Multi-Objective Optimization Models +
## 3 Material and Methods
### Developed Goal Programming Model
In the objective function (1), \(d_{1}^{+},d_{2}^{+},d_{3}^{+},d_{4}^{+},d_{5}^{+},d_{6}^{+},d_{7}^{+},d_{8}^{+}\), respectively, are the decision variables that represent positive deviations from the targets. In Equations (2)-(9), \(d_{1}^{-},d_{2}^{-},d_{3}^{-},d_{4}^{-},d_{5}^{-},d_{6}^{-},d_{7}^{-},d_{8}^{-}\), respectively, are negative deviations from the targets. Constraints  
\[\sum_{e=1}^{E}\sum_{w=1}^{W}\sum_{r=1}^{R}\sum_{q=1}^{Q}\sum_{s=1}^{S}t_{wraps}x_{ ewr}GH_{wraps}D_{ew}pw_{wraps}F+\left(d_{1}^{-}-d_{1}^{+}\right)=HD_{1} \tag{2}\]  
\[\sum_{e=1}^{E}\sum_{w=1}^{W}\sum_{r=1}^{R}\sum_{q=1}^{Q}\sum_{s=1}^{S}t_{wraps}x_ {ewr}GH_{wraps}D_{ew}pw_{wraps}CE_{es}F+\left(d_{2}^{-}-d_{2}^{+}\right)=HD_{2} \tag{3}\]  
The first goal constraint, Equation (2), aims to not exceed the carbon emission target value, as shown by \(HD_{1}\). Conversion factors can vary according to factors such as fuel types and materials used. In this study, the carbon emissions released from the machines are calculated based on the energy consumption values of the machines. In Equation (3), the second goal constraint of the model, aims to not exceed the target value of the total carbon emission cost. This cost item varies depending on factors such as the power values of the machines, processing times, and part demands. \[\sum_{e=1}^{E}\sum_{w=1}^{W}\sum_{r=1}^{R}\sum_{q=1}^{Q}\sum_{s=1}^{S}t_{wraps }x_{ewr}N_{eup}GH_{wraps}D_{ew}O_{es}+\left(d_{3}^{-}-d_{3}^{+}\right)=HD_{3} \tag{4}\]  
Equation (4) aims to not exceed the target value of the total operation cost. The total operation cost is calculated according to factors such as part demands, part processing times, and the number of machines in the cell. \[\sum_{e=1}^{E}\sum_{w=1}^{W}\sum_{r=1}^{R}\Bigg{[}\left(\sum_{q=1}^{Q}z_{ewrq} \right)-1\Bigg{]}D_{ew}HA_{ew}x_{ewr}+\left(d_{4}^{-}-d_{4}^{+}\right)=HD_{4} \tag{5}\]  
\[\sum_{e=1}^{E}\sum_{q=1}^{Q}\sum_{i=1}^{I}V_{qi}EM_{qqi}+\left(d_{5}^{-}-d_{5} ^{+}\right)=HD_{5} \tag{6}\]  
Equation (5), the fourth goal constraint, aims to not pass over the target value of the total cost of movement between cells. The goal constraint indicated by Equation (6) aims to not pass over the target value of the total training cost of workers assigned to cells in each period. \[\sum_{e=1}^{E}\sum_{q=1}^{Q}\sum_{s=1}^{S}NA_{eqs}NAC_{es}+\left(d_{6}^{-}-d_{ 6}^{+}\right)=HD_{6} \tag{7}\]  
\[\sum_{e=1}^{E}\sum_{q=1}^{Q}\sum_{s=1}^{S}NR_{eqs}NRC_{es}+\left(d_{7}^{-}-d_{ 7}^{+}\right)=HD_{7} \tag{8}\]  
Equations (7) and (8) are the goal constraints related to the total cost items generated during cell design. Equation (7) aims to not exceed the target value of total cost of the number of machines added to cells. Equation (8) aims to not exceed the target value of the total cost of the item associated with removing machines from cells. \[\sum_{e=1}^{E}\sum_{q=1}^{Q}\sum_{i=1}^{I}\sum_{l=1}^{L}V_{qi}Y_{il}B_{l}+ \left(d_{8}^{-}-d_{8}^{+}\right)=HD_{8} \tag{9}\]  
Equation (9) aims to not exceed the target value of the total bonus wage that workers assigned to cells receive according to their abilities. In this study, the economic dimension of sustainability is also considered when designing the cells. \[\sum_{r=1}^{R}x_{ewr}=1\forall e,w \tag{10}\]In this study, it is assumed that the model has routing flexibility and each part has alternative routes. Equation (10) shows that the parts can choose only one of their alternative routes in each period. \[\sum_{w=1}^{W}\sum_{r=1}^{R}z_{ewmq}x_{ewr}\geq 1\forall e,q \tag{11}\]  
\[\sum_{e=1}^{E}\sum_{w=1}^{W}\sum_{r=1}^{R}\sum_{q=1}^{Q}\sum_{s=1}^{S}t_{wraps}x_ {ewr}GH_{wraps}D_{ew}pw_{wraps}CE_{es}F+\left(d_{2}^{-}-d_{2}^{+}\right)=HD_{2} \tag{3}\]0  
Equation (11) shows that at least one part is processed in the selected route in each cell in each period. Equation (12) ensures that the machines cannot exceed their time capacity for each period and each cell.