We consider a robotic cell, consisting of a flow-shop in which the machines
are served by a single central robot. We concentrate on the case where onl
y one part type is produced and want to analyze the conjecture of Sethi, Sr
iskandarajah, Sorger, Blazewicz and Kubiak. This well-known conjecture clai
ms that the repetition of the best one-unit production cycle will yield the
maximum throughput rate in the set of all possible robot moves. The conjec
ture holds for two and three machines, but the existing proof by van de Klu
ndert and Crama for the three-machine case is extremely tedious.
We adopt the theoretical background developed by Crama and van de Klundert.
Using a particular state graph, the k-unit production cycles are represent
ed as special paths and cycles for which general properties and bounds for
the m-machine robotic cell can be obtained. By means of these concepts, we
shall give a concise proof for the validity of the conjecture for the three
-machine case.