Some underlying mathematical definitions and principles for cellular manufacturing

This paper uses set theory to analyze cellular manufacturing systems. We develop a structural architecture to investigate the previous literature. We divide all research problems into three types-what, why, and how. We advocate studying these three types of problems from two managerial perspectives- philosophy and science. To fully and deeply understand cellular manufacturing, both three-problems and two-perspectives are important. We use the developed architecture to review cellular manufacturing literature and point out the weakness in the previous studies. This review motivates the research of this paper. We use simple but rigorous set theory to analyze two what-problems. Underlying concepts, such as part family, machine cell, cellular manufacturing system, bottleneck machine are defined. Following these definitions, we deduce sufficient and necessary conditions for perfectly partitioned cellular manufacturing systems. Two sufficient and necessary conditions of perfect partition have been discovered-prior perfect partition theorems check the potential independent manufacturing cells within an arbitrary manufacturing system; in contrast, posterior perfect partition theorem verifies a perfectly partitioned cellular manufacturing system. To our knowledge, we are among the first to apply rigorous mathematical models to these two types of what-problems for cellular manufacturing. We try to provide mathematical vocabularies to discuss common problems inherent in the design of cellular manufacturing systems.