Cycles and permutations in robotic cells

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 only one part type is produced and analyze the validity of the so-called one-cycle conjecture by Sethi, Sriskandarajah, Sorger, Blazewicz and Kubiak. This conjecture claims that the repetition of the best one-unit production cycle will yield the maximum throughput rate in the set of all possible cyclic robot moves. sent a new algebraic approach, unifying the former rather tedious proofs for the known results on pyramidal one-cycles and two- and three-machine cells. In this framework, counterexamples will be constructed, showing that the conjecture is not valid for four and more machines. st present examples for a general four-machine cell, for which the two-unit production cycles dominate the one-unit cycles. Then we consider in particular the so-called regular cells, where all machines are equidistant, since the one-cycle conjecture has originally been formulated for this case. Here, we prove that for four-machine cells, two-unit production cycles are still dominated by one-unit production cycles. Then we describe a counterexample with a three-unit production cycle, thus, settling completely the one-cycle conjecture.