## 1 Introduction
The first is an exact method that attempts to find the best cell configuration by directly maximizing the objective function of CF. The grouping efficacy (GE) measure [10] has been widely used as an objective function of CF. Since the GE takes a fractional function, the CF problem with the GE objective function results in a 0-1 nonlinear fractional programming problem. Thus, maximizing the GE directly has attracted many researchers since the early 2010s [11; 12; 13; 14; 15; 16; 17; 18]. In order to evaluate the performance of their exact methods, 35 small to intermediate-size benchmark incidences [19] have been widely used for benchmark testing, and their solutions have been compared. However, some instances were not solved optimally even under the time limit of 100,000 s using the CPLEX MILP solver. The other approach aims to indirectly maximize the GE or other alternative performance measures by using the classic or modified \(p\)-median problem (PMP). Since [PERSON] [20; 21] first introduced the PMP on a network of nodes and arcs, the PMP has been widely studied and extended to many practical situations including the location of plants, warehouses, distribution centers, hubs, and public service facilities [22]. [PERSON] and [PERSON] [23] used Balinski-type constraints [24] to present an integer linear programming (ILP) formulation of the PMP. Unfortunately, since the original [PERSON] and [PERSON] model (ORSM) defined on an \(n\)-node network contains \(n^{2}\) binary variables and \(n^{2}+1\) constraints, it is computationally infeasible to exactly solve the ORSM even for moderately sized networks. Therefore, many attempts have been made to formulate equivalent PMP models including fewer binary variables and constraints than the original ORSM [25; 26; 27; 28; 29; 30; 31]. Those reduced PMP models have been solved using hard computing techniques on MILP solvers, and their performances have been compared to those of past PMP models. [PERSON] [32; 33] first proposed using the PMP-type model as an alternative mathematical programming model for the CF, replacing exact methods. However, the PMP itself does not explicitly optimize the objective of CF in the same way as the GE. Nevertheless, the PMP grasps the clustering nature of CF and presents a flexible framework by allowing additional constraints reflecting realistic aspects to be introduced [34]. In this context, the PMP matches the CF problem well and shows good solution performance for small to intermediate-size SCF/GCF instances [35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49]. Recently, [PERSON] et al. [34] proposed a flexible PMP-based approach for solving large-sized 0-1 SCF problems and used the Xpress MILP solver to optimally solve most of the SCF instances available in the literature within one second. However, few studies reporting successful applications of the hard computing approach of the PMP-type model to large-sized GCF instances have appeared in the literature. There are two main reasons for this:  
* First, with regard to SCF, the 35 standard incidences available in the literature have been widely used for benchmark testing for the last 20 years, and a recent study adopting a hybrid algorithm [50] has reported the best optimal solutions with huge time-savings. However, there are few open large GCF data sets available in the literature regarding the standard instances for benchmark testing and the performance comparison of the solution algorithms used. As far as the present author knows, the largest example of an open GCF available in the literature has at most 55 machines, 60 part types, and a total of 124 process plans [51]. * Second, execution strategies for CF and complicated aspects inherent in the GCF problem itself make the solution quality of CF methods very sensitive to subsequent part assignment or improvement procedures that are necessary to follow after machine cells are obtained.