## 1 Introduction
Three different strategies have been used to execute the CF algorithm [52]: the part family identification (PGI) strategy forms part families first and then groups machines, the machine group identification (MGI) strategy creates machine cells first and then allocate parts to cells, and the part family/machine grouping (PF/MG) strategy forms machine cells and part families simultaneously. Most soft and hard computing approaches for CF use the MGI strategy to execute CF algorithms since it usually takes enormous computation time to implement the PF/MG strategy even for intermediate-size incidences. The PMP-based approach for CF also uses the MGI strategy to create cells. Therefore, once machine cells are obtained from the PMP solution, part families need to be formed by allocating parts to the best cells. [PERSON] and [PERSON] [50] and [PERSON] et al. [53] have established sufficient conditions for the optimal assignment of parts to machine cells given a partition of machines of a SCF problem. However, it should be noted that their sufficient conditions may not guarantee the optimal assignment of parts maximizing the GE in the GCF problem due to the existence of alternative process plans and/or replicate machines. Motivated by the drawbacks of extant studies attacking the GCF problem, this paper proposes an effective hard computing approach using the PMP-type model to solve large-sized GCF problems. Our hard computing approach has the following distinctive features compared to previous hard computing approaches dealing with the GCF problem:  
* Two new linear 0-1 mathematical models of GCF are formulated: an exact model that directly maximizes the GE and a PMP-type model that indirectly maximizes the GE. Because the exact model contains too many binary variables and constraints, the PMP-type model is used to solve large-sized GCF instances optimally. According to the computational experiments applied to large GCF instances with over 10,000 binary variables, our PMP-type model solves those large GCF instances optimally within one second using the LINGO MILP solver. * Since the PMP-type approach uses MGI strategy to form machine cells first, a subsequent part allocating step is needed to form the corresponding part families. In this paper, a systematic heuristic part assignment procedure based on a new classification scheme of part types with alternative process plans is used to assign the best process plan of each part to its best cell. A subsequent refinement procedure is the used to further improve the block diagonal solution by reassigning improperly assigned exceptional machines (EMs) in such a way that the GE is maximized. The computational burden of implementing these extra procedures is negligibly small since they accomplish a high-quality CF within 0.2 s, even for the largest GCF instances tested in our computational experiments. * Unlike many comparative studies of SCF using the standard data set provided in [PERSON] and Resende [19], studies of GCF lack the standard data set. Our computational experiment has been conducted over the widest range of GCF incidences that have ever appeared in the CF-related literature. Our collection of the GCF incidences can be used as a standard data set for subsequent benchmark tests in the future.