Production capacity modelling of alternative, nonidentical, flexible machines

Analyzing the production capacity of a flexible manufacturing system consisting of a number of alternative, nonidentical, flexible machines, where each machine is capable of producing several different part types simultaneously (by flexibly allocating its production capacity among these part types), is not a trivial task. The production capacity set of such a system is naturally expressed in terms of the machine-specific production rates of all part types. In this paper we also express it in terms of the total production rates of all part types over all machines. More specifically, we express the capacity set as the convex hull of a set of points corresponding to all possible assignments of machines to part types, where in each assignment each machine allocates all its capacity to only one part type. First, we show that within each subset of assignments having a given number of machines assigned to each part type, there is a unique assignment that corresponds to an extreme point of the capacity set. Then, we propose a procedure for generating all the extreme points and facets of the capacity set. Numerical experience shows that when the number of part types is less than four, the size of the capacity set (measured in terms of the number of variables times the number of constraints) is smaller, if the capacity set is expressed in terms of the total production rates of all part types over all machines than if it is expressed in terms of the machine-specific production rates of all part types. When the number of part types is four or more, however, the opposite is true.