  The Cell Formation Problem has been studied as an optimization problem in
manufacturing for more than 90 years. It consists of grouping machines and
parts into manufacturing cells in order to maximize loading of cells and
minimize movement of parts from one cell to another. Many heuristic algorithms
have been proposed which are doing well even for large-sized instances.
However, only a few authors have aimed to develop exact methods and most of
these methods have some major restrictions such as a fixed number of production
cells for example. In this paper we suggest a new mixed-integer linear
programming model for solving the cell formation problem with a variable number
of manufacturing cells. The popular grouping efficacy measure is used as an
objective function. To deal with its fractional nature we apply the Dinkelbach
approach. Our computational experiments are performed on two testsets: the
first consists of 35 well-known instances from the literature and the second
contains 32 instances less popular. We solve these instances using CPLEX
software. Optimal solutions have been found for 63 of the 67 considered problem
instances and several new solutions unknown before have been obtained. The
computational times are greatly decreased comparing to the state-of-art
approaches.
