# Designing Sustainable Flexible Manufacturing Cells with Multi-Objective Optimization Models +
## 3 Material and Methods
### Developed Goal Programming Model  
The developed goal programming model consists of several components, including sets, parameters, decision variables, objective function, and constraint equations. These elements are defined as follows: \begin{tabular}{l l} Indices: & \\ \(W\) & Set of part types (\(w\in W\)). \\ \(R\) & Set of alternative routes (\(r\in R\)). \\ \(Q\) & Set of cells (\(i\in Q\)). \\ \(S\) & Set of machine types (\(s\in S\)). \\ \(I\) & Set of worker types (\(l\in I\)). \\ \(L\) & Set of skills (\(l\in L\)). \\ \(E\) & Set of periods (\(i\in E\)). \\ Parameters: & \\ \(D_{w}\) & The demand of part \(w\) in period \(e\). \\ \(K_{w}\) & The time capacity of machine \(s\) in period \(e\). \\ \(l_{wap}\) & The machine time of part \(w\) on machine \(s\) in cell \(q\) using alternative route \(r\). \\ \(pw_{wap}\) & The amount of power consumed while machining part \(w\) on machine \(s\) in cell \(q\) using alternative route \(r\). \\ \(HA_{wap}\) & The cost of moving part \(w\) between cells in a period \(e\). \\ \(CE_{w}\) & The machine \(s\) carbon emission cost per period \(e\). \\ \(O_{m}\) & The machine \(s\) processing/operation cost per period \(e\). \\ \(NAC_{en}\) & The cost of adding machine \(s\) to cells per period \(e\). \\ \(NRC_{en}\) & The cost of removing machines from cells per period \(e\). \\ \(EH_{api}\) & The training costs of worker \(i\) in cell \(q\) in period \(e\). \\ \(ES_{gl}\) & The time that worker \(i\) spends on an operation of skill \(l\). \\ \(HY_{gl}\) & The limit value of skill \(l\) in cell \(q\) in period \(e\). \\ \(TES_{gal}\) & The limit time that workers with skill \(l\) in cell \(q\) in period \(e\) can spend. \\ \(IK_{eq}\) & The maximum number of workers of cell \(q\) in period \(e\). \\ \(HAL_{eq}\) & The lower bound for the number of machines in cell \(q\) in period \(e\). \\ \(HUL_{eq}\) & The upper limit for the number of machines in cell \(q\) in period \(e\). \\ \(F\) & The carbon emission conversion factor. \\ \(LB_{q}\) & The carbon emission limit value of cell \(q\) in period \(e\). \\ \(B_{q}\) & The bonus wage to be received by \(j\) skilled worker. \\ \(EST_{gal}\) & The training time received by worker \(i\) in the skill \(l\) in period \(e\). \\ \(z_{newq}\) & \begin{tabular}{} \end{tabular} & \begin{tabular}{} \end{tabular} \\ \(TY_{2}\) & \begin{tabular}{} \end{tabular} & \begin{tabular}{} \end{tabular} \\ \(GH_{new}\) & \begin{tabular}{} \end{tabular} \\ \(HD_{1}\), \(HD_{2}\), \(HD_{3}\), \(HD_{4}\), \(HD_{5}\), \(HD_{6}\), \(HD_{7}\), and \(HD_{8}\), respectively, represent the target values for the objective items. \\ \begin{tabular}{l l} Decision Variables \\ \(x_{new}\) & \begin{tabular}{} \end{tabular} & \begin{tabular}{} \end{tabular} \\ \(V_{eq}\) & \
\begin{tabular}{} \end{tabular} \\ \(N_{eq}\) & The number of machine \(s\) assigned to cell \(q\) in period \(e\). \\ \(N_{eq}\) & The number of machine \(s\) added to cell \(q\) in period \(e\). \\ \(NR_{eq}\) & The number of machine \(s\) removed from cell \(q\) in period \(e\). \\ \(IS_{eq}\) & The number of workers in cell \(q\) in period \(e\). \\ \(EI_{S}\) & The total number of workers in the system in period \(e\). \\ \(TA_{i}\) & The total training time that worker \(i\) will receive during all periods according to the worker's abilities. \\ \end{tabular} \\ \end{tabular}  \\ \end{tabular} \begin{tabular}{l l} The parameters \(K_{es}\), \(t_{waps}\), \(ES_{gl}\), \(TES_{gal}\), and \(EST_{all}\) have the same time unit in the model. Additionally, the parameters \(HA_{ew}\), \(CE_{es}\), \(O_{es}\), \(NAC_{es}\), \(NRC_{es}\), \(EM_{qui}\), and \(B_{l}\) have the same currency unit. \\ Objective Function \\ \(\text{Min}\ =\ d_{1}^{+}+d_{2}^{+}+d_{3}^{+}+d_{4}^{+}+d_{5}^{+}+d_{6}^{+}+d_{7}^{+}+d_ {8}^{+}\) \\ \end{tabular}  (1)  
The objective function of the goal programming model is given by Equation (1). Using Equation (1), the minimization of the sum of deviations from the handled targets is ensured.