## Materials and Methods
### Mathematical Models
#### 2.3.3 Part Assignment to Cells  
The solution of PMP-type model 3 identifies only the machine cells. Once the machine cells are formed, parts need to be assigned to their best associated cells in such a way that the objective of CF is optimized. The classic part assignment rule that has been widely used is the maximum density rule, assigning a part to the cell in which it has most operations. Since different part assignment rules affect the solution quality of CF, several modified part assignment rules have been proposed [50; 59; 60; 61; 62; 63; 64]. A critical drawback of these rules is that the solution quality due to the assignment of a part depends on the assignment of other parts since they are heuristic rules. Some authors [50; 53] have established sufficient conditions for optimal part assignment that do not depend on the assignment of other parts for the SCF problem. Several heuristic part assignment procedures for the GCF problem considering the number of EEs and voids have been proposed in the literature [59; 61; 62; 63]; however, due to the existence of alternative process plans and replicate machines, no optimal part assignment rules for the GCF problem have been proposed. [PERSON] [63] used the nonbinary generalized PMIM to develop a heuristic part assignment procedure for the GCF problem. In this paper, we use a modified heuristic part assignment procedure based on the binary generalized PMIM to classify parts into eight categories as follows:  
* A type I strongly nonexceptional part (SNEP) for which a unique process plan has the most 1s in a unique machine cell without EEs;
* A type II SNEP for which multiple process plans have the most 1s in a unique machine cell without EEs;
* A type I neutrally nonexceptional part (NNEP) for which a unique process plan has the most 1s in more than one machine cell without EEs;
* A type II NNEP for which multiple process plans have the most 1s in more than one machine cell without EEs;
* A type I weakly exceptional part (WEP) for which a unique process plan has the most 1s in a unique machine cell with EEs;
* A type II WEP for which multiple process plans have the most 1s in a unique machine cell with EEs;
* A type I neutrally exceptional part (NEP) for which a unique process plan has the most 1s in more than one machine cell with EEs;
* A type II NEP for which multiple process plans have the most 1s in more than one machine cell with EEs. To determine the specific type of a part and assign the best process plan to its best associated cell in such a way that the GE is maximized, the following measures need to be calculated for all process plans of parts:1. The numbers of EEs due to the assignment of each process plan to each cell;
2. The number of voids due to the assignment of each process plan to each cell;
3. The number of cells which have the most 1s for completing the required operations due to the assignment of each process plan to each cell;
4. The total number of operations (1s) contained in each cell with all the parts assigned until the current stage;
5. The number of parts assigned to each cell until the current stage; and
6. The number of operations processed by the machines in each cell. Criteria 1 and 3 contribute to the independent cell configuration with the least inter-cell moves. Criteria 2 and 6 contribute to the compact cell configuration with high machine utilization. Criteria 4 and 5 contribute to the balanced cell configuration with an even workload among cells. A specific type of a part can be determined from criteria 1 and 3. Figure 2 presents a flow chart showing the procedure which identifies the type of a part. The assignment of a type I SNEP or type I WEP is straightforward: its unique candidate process plan is assigned to its associated unique cell. However, the assignment of remaining part types requires tie-breaking rules to select the best candidate process plan and its best associated cell.