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In this paper, we consider the scheduling problem on m-machine chain-reentrant flow shop problem with two competing agents. In which, all the jobs must pass through all the machines in the same predetermined order, then return back to the first machine for their last operation. The objective is to minimize the overall completion time of one agent subject to an upper bound on the makespan of the second agent. NP-hardenness proof is provided for the two-machine flow shop problem with the two competing agents with respect to the makespan criteria, even if each job has the same processing time on the machines, while a polynomial time algorithm is presented for the m-machine with unit processing time. We also develop a mathematical programming model and a heuristic for the resolution.
