## 1 Introduction  
The cell formation (CF) problem has attracted researchers in academia as well as practitioners in the field since it was introduced as a part of group technology (GT) [1]. The initial step of CF is to create machine cells and their associated part families. A machine cell is a collection of functionally dissimilar machines which are grouped together and dedicated to process its associated part family, which is a collection of parts which are similar with regard to their geometric shape and size or processing requirements. By creating efficient cells, the maximum operations of the machines within cells (intra-cell operations) and minimum transfers of parts from one cell to another (inter-cell operations) are achieved. This leads numerous operational benefits, such as a reduction in setup time, work-in-process inventories, improvement in quality and a high degree of flexibilities to product demand changes [2]. Since the CF is an NP-hard problem [3, 4], a number of approaches and methods have been proposed to solve the CF problem effectively. [PERSON] and [PERSON] [5] provided a recent review of the CF solution methodologies. The CF problems are classified into two categories: the standard CF (SCF) problem, considering only one process plan for each part, and the generalized CF (GCF) problem, considering alternative process plans for each part. Both problems can include replicate machines; i.e., extra copies for a machine type. GCF is more complicated than SCF since SCF is a special case of GCF. When a part has alternative process plans, operations can be performed on different types of machines or extra copies of a machine type. By considering alternative process plans and replicate machines, more independent cells and higher machine utilization due to reduced inter-cell flows can be achieved [6]. The first step in solving CF problems is to construct the mathematical model which is best suited to achieving the objectives of a specific CF. However, this usually leads to a huge model that has many integer and continuous variables, constraints, and/or nonlinear functions. As the number of machines and parts directly influencing the size of CF problem increases, the optimal solution methodology fails to solve large CF instances [7]. More specifically, if a mathematical model of CF contains nonlinear and/or multi-objective functions over multiple periods, it is very difficult to optimally solve that model, although those nonlinear functions can be linearized. Therefore, a number of soft computing approaches relying on local search methodologies such as artificial intelligence, heuristic/meta-heuristic, or hybrid algorithms have been proposed. Soft computing approaches attempt to find good or acceptable solutions to the proposed mathematical model of CF in a short computation time at the expense of the global optimum. Most soft computing approaches use specific mathematical models to set up the CF problem rather than solving them optimally. In this regard, almost all soft computing approaches for CF are heuristic in nature. However, the cell design experience with industrial experts shows that designers would rather spend more time to achieve an optimal or near optimal solution than use a heuristic approach to get an inferior solution [8]. On the contrary, hard computing approaches can use exact search algorithms such as branch-and-bound to solve the mathematical models of CF optimally if reasonable computation time is allowed. The application of hard computing approaches for CF significantly has relied on recent advances in computer hardware and commercially available mixed integer linear programming (MILP) solvers, such as CPLEX, LINGO, or Gurobi. [PERSON] et al. [9] stated that hard computing approaches can yield optimal solutions to large MILP problems with a reasonable running time if appropriate mathematical models are constructed. Recently, two hard computing approaches for solving the mathematical models of CF have been mentioned in the CF-related literature.