## 2 Problem Definition and Formulation  
In the study, a mixed-integer linear mathematical programming model is developed. This model, including the weighted approach, aims to minimize intercellular part movements, intracellular part movements, machine-system workload imbalance, labor-system workload imbalance, and several tools on all machines in cells providing optimum alternative routes for parts. The model assumes that there is routing flexibility for all parts of the system. Therefore, parts can be processed on different machines in different cells using different tool types according to selected alternative routings. [PERSON] [67] and [PERSON] and [PERSON] [68] emphasize the importance of alternative routings in their studies on the reconfiguration of flexible manufacturing cells. They indicate that excluding alternative routes for parts can lead to increases in the optimum total reconfiguration cost. The assumptions of this study are as follows:  
* The processing time of each part on machines is known, and all parts have a constant and known demand. * Parts can be produced on different machines. Parts have routing flexibility. * Intercellular and intracellular part movements are known. * The capacity of each machine, the total number of tool types in each machine in the cells, and the capacity of each tool on each machine are known and constant. * The numbers of machines, parts, tools, cells, and workers are given in the system. * The number of workers in each cell is known. * Workers' capacities are constant and known. Mathematical Programming Model  
The mathematical model describing the characteristics of the problem can be formulated using the following notation:  
Indices: \begin{tabular}{l l l l} \(p\) & part types & \(p=1,\ldots,p\) & \(p\) indicates the number of part types \\ \(r\) & alternative routes & \(r=1,\ldots,R\) & \(R\) indicates the number of alternative routes \\ \(c\) & cells & \(c=1,\ldots,C\) & \(C\) indicates the number of cells \\ \(m\) & machines & \(m=1,\ldots,M\) & \(M\) indicates the number of machines \\ \(j\) & tool types & \(j=1,\ldots,J\) & \(J\) indicates the number of tool types \\ \(i\) & worker types & \(i=1,\ldots,J\) & \(I\) indicates the number of worker types \\ \end{tabular}   
Parameters: \begin{tabular}{l l l} \(D_{p}\) & the demand for part \(p\) \\ \(q_{pr}\) & number of visited cells according to chosen alternative route \(r\) for part \(p\) \\ \(z_{pr}\) & number of movements within cells according to chosen alternative route \(r\) for part \(p\) \\ \(t_{prcm}\) & process time of part \(p\) according to chosen alternative route \(r\) on machine \(m\) in cell \(c\) \\ \(y_{cm}\) & number of machines \(m\) in cell \(c\) \\ \(K_{m}\) & capacity of machine \(m\) \\ \(W_{c}\) & machine capacity of cell \(c\) \\ \(L_{c}\) & labor capacity of cell \(c\) \\ \(s_{prcmj}\) & tool type \(j\) number on machine \(m\) in cell \(c\) by route \(r\) of part \(p\) \\ \(tk_{cmj}\) & capacity of tool type \(j\) on machine \(m\) in cell \(c\) \\ \(t_{prci}\) & working time of worker type \(i\) in cell \(c\) using route \(r\) of part \(p\) \\ \(ts_{i}\) & capacity of worker type \(i\) \\ \(ts_{cli}\) & number of worker type \(i\) in cell \(c\) \\ \(AG_{1}\) & The weighted factor for the total number of intracellular movements \\ \(AG_{2}\) & The weighted factor for the total number of intercellular movements \\ \(AG_{3}\) & The weighted factor for the total machine-system workload unbalance \\ \(AG_{4}\) & The weighted factor for the total number of used tools \\ \(AG_{5}\) & The weighted factor for the total worker-system workload unbalance \\ \end{tabular}   
Decision variables:  
\[x_{pr}\]  
\[\left\{ \begin{array}{ll}1,\text{ if route $r$ for part $p$ is chosen}\\ 0,\text{ otherwise}\\ u_{cm}\end{array} \right.\]  
\[ \begin{array}{ll}u_{cm}\\ ul_{ci}\\ hl_{c}\\ A_{c}\\ B_{c}\end{array} \]  
\[\left\{