## 2 Problem Definition and Formulation
\begin{array}{ll}1,\text{ if route $r$ for part $p$ is chosen}\\ 0,\text{ otherwise}\\ \text{ the utilization rate of machine $m$ in cell $c$}\\ u_{ci}\\ hl_{c}\\ A_{c}\\ B_{c}\end{array} \right.\]  
In the model, the parameters \(t_{prm}\), \(W_{c}\), \(tl_{prc}\), and \(ts_{i}\) have the same time unit. Objective function:  
\[\min\left\{ \begin{array}{ll}AG_{1}\sum\limits_{p=1}^{P}\sum\limits_{r=1}^{R}z_ {pr}x_{pr}+AG_{2}\sum\limits_{p=1}^{P}\sum\limits_{r=1}^{R}(q_{pr}-1)x_{pr}\\ +AG_{3}\frac{1}{c}\sum\limits_{c=1}^{C}(B_{c}-A_{c})\\ +AG_{4}\sum\limits_{p=1}^{P}\sum\limits_{r=1}^{R}\sum\limits_{c=1}^{C}\sum \limits_{m=1}^{M}\sum\limits_{j=1}^{J}s_{prcm}x_{pr}+AG_{5}\frac{1}{c}\sum \limits_{c=1}^{C}(L_{c}-hl_{c})\\ \end{array} \right\} \tag{1}\]  
The objective function of the mathematical model is shown by Equation (1). The objective function minimizes the weighted sum of movements within and between cells in flexible manufacturing cells, machine-to-system workload unbalances, tool usage, and worker-system workload unbalance items under certain constraints. A weighted approach is used for different terms in the objective function. The first part of Equation (1) calculates the intracellular movements using alternative routings of parts. The second part of Equation (1) calculates the number of intercellular movements using alternative routings of parts. As stated in Logendran [12], if a part is required to visit n cells (n \(\geq\) 1), then it contributes to (n \(-\) 1) intercellular moves. The third part of Equation (1) calculates the total machine-system workload unbalances. The fourth part of Equation (1) calculates the total number of tools on all machines in cells according to the optimum routes of parts. The final part of Equation (1) calculates the total worker-system workload unbalances. The system-based average waiting time of workers in cells is minimized. Each term of the objective function has a user-specified weight. Constraints:  
\[\sum\limits_{r=1}^{R}x_{pr}=1\ \forall p \tag{2}\]  
\[\sum\limits_{p=1}^{P}\sum\limits_{r=1}^{R}(t_{prm}x_{pr}D_{p})/K_{m}=u_{cm} y_{cm}\forall c,m \tag{3}\]  
\[A_{c}\leq u_{cm}\ \ \ \ \forall c,m \tag{4}\]  
\[u_{cm}\leq B_{c}\ \ \ \ \forall c,m \tag{5}\]  
\[\sum\limits_{m=1}^{M}y_{cm}\geq 1\ \ \ \ \forall c \tag{6}\]  
\[\sum\limits_{m=1}^{M}y_{cm}\leq W_{c}\ \ \ \ \forall c \tag{7}\]  
\[\sum\limits_{p=1}^{P}\sum\limits_{r=1}^{R}\left(s_{prcm}x_{pr}\right)\leq tk_{ cmj}\ \ \ \ \forall c,m,j \tag{8}\]\[\sum_{p=1}^{p}\sum_{r=1}^{R}\left(tl_{prci}x_{pr}D_{p}\right)/ts_{i}=ul_{ci}ts_{ ci}\forall c,i \tag{9}\]  
\[\sum_{i=1}^{I}(ul_{ci}ts_{ci})=hl_{c}\forall c \tag{10}\]  
\[\sum_{i=1}^{I}(ul_{ci}ts_{ci})\leq L_{c}\forall c \tag{11}\]  
\[\sum\limits_{p=1}^{P}\sum\limits_{r=1}^{R}(t_{prm}x_{pr}D_{p})/K_{m}=u_{cm} y_{cm}\forall c,m \tag{3}\]0  
\[\sum\limits_{p=1}^{P}\sum\limits_{r=1}^{R}(t_{prm}x_{pr}D_{p})/K_{m}=u_{cm} y_{cm}\forall c,m \tag{3}\]1  
\[\sum\limits_{p=1}^{P}\sum\limits_{r=1}^{R}(t_{prm}x_{pr}D_{p})/K_{m}=u_{cm} y_{cm}\forall c,m \tag{3}\]2  
\[\sum\limits_{p=1}^{P}\sum\limits_{r=1}^{R}(t_{prm}x_{pr}D_{p})/K_{m}=u_{cm} y_{cm}\forall c,m \tag{3}\]3  
Equation (2) states that only one of its alternative routes for parts can be selected. The utilization of each machine in each cell \((u_{cm})\) is calculated by Equation (3). While calculating the utilization of each machine in each cell, the demand and processing time for each part and the capacity of each machine are considered, along with alternative routings of parts. While Equation (4) provides the minimum machine utilization rate in each cell, Equation (5) allows the maximum machine utilization rate to be calculated. The machine-system workload unbalance specified in the objective function is minimized with the minimum and maximum utilization rates. Thus, using Equations (3)-(5), the machine-system unbalance stated in the objective function is calculated. Equation (6) indicates that each cell must contain one machine.