# Designing Sustainable Flexible Manufacturing Cells with Multi-Objective Optimization Models +
## 4 Results and Discussion
In this study, the LINGO 20.0 optimization program was employed to solve the multi-objective integer mathematical programming models, which addresses the design of sustainable and flexible manufacturing cells. The developed goal programming, \(\epsilon\)-constraint, and AUGMECON mathematical programming models were solved separately using the LINGO 20.0 optimization program using a MacBook Air (M1, 2020) with 8 GB of RAM. The global optimal results were obtained in 12 min and 38 s, 18 min and 31 s, and 19 min and 51 s for the goal programming method, the \(\epsilon\)-constraint method, and the AUGMECON method, respectively. In the results obtained in the goal programming model, the positive and negative deviation values from the related targets are obtained as \(d_{1}^{+}=\) 5,330,653, \(d_{2}^{+}=\) 52,162,290, \(d_{3}^{+}=\) 14,190,360, \(d_{4}^{+}=\) 3,317,127, \(d_{5}^{+}=\) 0, \(d_{6}^{+}=\) 0, \(d_{7}^{+}=\) 0, \(d_{8}^{+}=\) 25,000, \(d_{1}^{-}=\) 0, \(d_{2}^{-}=\) 0, \(d_{3}^{-}=\) 0, \(d_{4}^{-}=\) 0, \(d_{5}^{-}=\) 6413, \(d_{6}^{-}=\) 4506, \(d_{7}^{-}=\) 4613, \(d_{8}^{-}=\) 0. Additionally, the objective function value of the goal programming model is obtained as 75,025,430. In Table A3 seen in Appendix B, the results related to optimal routes obtained  
\begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**The Limit Value of Workers with**}} & \multicolumn{4}{c}{**The Limit Time of Workers with**} & \multicolumn{4}{c}{**Training Cost of Workers**} \\ \multicolumn{1}{c}{} & \multicolumn{3}{c}{**Skills**} & \multicolumn{4}{c}{**Skills**} & \multicolumn{4}{c}{**Workers**} \\ \hline
**1. Period** & **1** & **2** & **3** & **1** & **2** & **3** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline Cell 1 & 1 & 2 & 0 & 25 & 20 & 0 & 50 & 65 & 75 & 45 & 35 & 70 \\ Cell 2 & 2 & 1 & 15 & 30 & 25 & 30 & 80 & 55 & 65 & 35 & 25 \\ Cell 3 from these three methods are presented separately. The optimal machine assignments results for the goal programming, \(\varepsilon\)-constraint, and AUGMECON methods are shown in Tables 11-13, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Training Time According to Skill**} \\ \hline  & Worker & Skill 1 & Skill 2 & Skill 3 \\ \cline{2-5} \multirow{3}{*}{1. Period} & 1 & 15 & 10 & 0 \\  & 2 & 0 & 15 & 0 \\  & 3 & 17 & 0 & 15 \\  & 4 & 0 & 0 & 11 \\  & 5 & 0 & 18 & 10 \\  & 6 & 18 & 0 & 0 \\ \hline  & Worker & Skill 1 & Skill 2 & Skill 3 \\ \cline{2-5} \multirow{3}{*}{2. Period} & 1 & 10 & 8 & 0 \\  & 2 & 0 & 18 & 0 \\  & 3 & 17 & 0 & 13 \\  & 4 & 0 & 0 & 11 \\  & 5 & 0 & 13 & 0 \\  & 6 & 17 & 0 & 0 \\ \hline  & Worker & Skill 1 & Skill 2 & Skill 3 \\ \cline{2-5} \multirow{3}{*}{3.Period} & 1 & 10 & 8 & 0 \\  & 2 & 0 & 13 & 0 \\  & 3 & 15 & 0 & 18 \\  & 4 & 0 & 0 & 12 \\  & 5 & 0 & 12 & 0 \\  & 6 & 23 & 0 & 0 \\ \hline  & Worker & Skill 1 & Skill 2 & Skill 3 \\ \cline{2-5} \multirow{3}{*}{4.Period} & 1 & 15 & 17 & 0 \\  & 2 & 0 & 18 & 0 \\  & 3 & 22 & 0 & 13 \\  & 4 & 0 & 0 & 16 \\  & 5 & 0 & 16 & 0 \\  & 6 & 20 & 0 & 0 \\ \hline  & Worker & Skill 1 & Skill 2 & Skill 3 \\ \cline{2-5} \multirow{3}{*}{5.Period} & 1 & 11 & 7 & 0 \\  & 2 & 0 & 16 & 0 \\  & 3 & 15 & 0 & 14 \\  & 4 & 0 & 0 & 10 \\  & 5 & 0 & 23 & 0 \\  & 6 & 15 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Training times received by workers according to skills. The optimal worker assignments for multi-objective approaches are shown in Table 14. In Table 15, the number of workers in cell \(q\) in the period \(e\) is shown for each multi-objective approach. Moreover, for each multi-objective approach it was determined that \(EIS_{1}=6\), \(EIS_{2}=6\), \(EIS_{3}=6\), \(EIS_{4}=6\), \(EIS_{5}=6\), \(TA_{1}=111\), \(TA_{2}=80\), \(TA_{3}=159\), \(TA_{4}=60\), \(TA_{5}=82\), and \(TA_{6}=93\).