## 6 Conclusions  
This study introduces a mixed-integer mathematical programming model with routing flexibility to design flexible manufacturing cells. This model minimizes the weighted sum of the objective function items containing the total numbers of intracellular and intercellular movements, tool utilization, machine-system workload unbalance, and worker-system workload unbalance. The sample problem is developed and then solved using the LINGO 17.0 optimization program under the branch and bound algorithm. The optimal global solution of the developed model containing the objective function value and the values of the decision variables, such as optimal routes among the alternative routes of parts, is obtained in less than 1 s. Generally, the other results ensured from this study are as follows:  
Figure 5: Effects of tool capacities. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & **Number** & **Variable** & **Number** \\ \hline \(x_{pr}\) & P \(\times\) R & \(hl_{c}\) & C \\ \(u_{cm}\) & C \(\times\) M & \(A_{c}\) & C \\ \(ul_{ci}\) & C \(\times\) I & \(B_{c}\) & C \\ \hline \multicolumn{4}{c}{Total number = [(P \(\times\) R) + (C \(\times\) M) + (C \(\times\) I) + (3 \(\times\) C)]} \\ \end{tabular}
\end{table} 
Table 12: Total number of decision variables in terms of indices of the developed model. According to the selected alternative routes of parts, it is determined that the parts can be processed on machines located in the same cell or different cells. This way, the total number of intracellular and intercellular movements is minimized. * The total number of tools used in the system is minimized according to the selected alternative routes for parts. * Balancing the workloads of the machines and workers in cells contributes to improving the system's performance. * By calculating the idle waiting rates of workers in each cell, the average idle waiting rates of the cells are minimized. In this way, the system-worker unbalance is minimized, and the workers are used efficiently in line with their capacities. * As can be seen from the sensitivity analysis of the sample problem, changes in the demands of the parts and the capacities of the machines, tools, and workers can change the objective function value. * The developed mathematical programming model can be an effective mechanism for solving the design and planning problems of flexible manufacturing cells. Under the constraints considered, a mathematical programming model can serve as an effective mechanism for solving the design and planning problems of flexible manufacturing cells. This model simultaneously provides a balancing of machine workloads and a balanced use of the workforce, reduces intracellular and intercellular movements of parts in the system, and minimizes the number of tools used. The developed model enabled the identification of the optimal routes for parts with alternative routings within the context of flexible manufacturing cells. Additionally, the model effectively optimized the allocation of parts and tools to the respective machines, thereby enhancing overall efficiency and performance. The integration of these factors contributes to the design of flexible manufacturing cells, serving as an effective mechanism for planning problems. In future studies, deterministic parameters such as demand values and processing times can be considered stochastic or fuzzy parameters in the model. The approach presented in this study can be expanded by including different objective elements in the model by using the epsilon-constraint, which is one of the optimization methods. In addition, heuristic or metaheuristic algorithms can be utilized when the size of the model instance expands and when achieving the optimal solution within a reasonable time frame becomes unfeasible. Workers' social and physical factors are ignored while ensuring cell labor-workload balance. Future studies may include factors such as workers' skills, learning abilities, and teamwork.