Solving a Permutation Flow Shop Problem with Blocking and Transportation Delays

Given a job set J = f1; 2; :::; ng where each job has to be processed non preemptively on m machines M1; M2;..., Mm in that order. The processing time of job j on machine Mi is pij : At time t = 0; all jobs are available at an input device denoted by M0: After completion, each job must be taken from Mm to an output device denoted Mm+1 (for convenience, we set pm+1;j = 0;8j 2 J): The transfer between machine Mi and Mk (i; k = 0; :::;m + 1) is performed by means of a single robot and takes ik units of time. The machines have no input or output bu¤ering facilities. Consequently, after processing a job j on machine Mi (i = 1; :::;m), this latter remains blocked until the robot picks j and transfers it to Mi+1: Such a move could only be performed if machine Mi+1 is free (that is, no job is being processed by or is waiting at Mi+1). At any time, each machine can process at most one job and each job can be processed on at most one machine. Moreover, the robot can transfer at most one job at any time. The problem is to …nd a processing order of the n jobs, the same for each machine (because of the blocking constraint, passing is not possible), such that the time Cmax at which all the jobs are completed (makespan) is minimized. In the sequel, we partially relax the constraint requiring that the robot can transfer at most one job at any time and we propose to investigate the ‡ow shop problem with blocking and transportation delays. This problem might be viewed as a generalization of the much studied ‡ow shop with blocking (Ronconi, 2005). An overview of the literature on ‡ow shop scheduling blocking can be found in Hall and Sriskandarajah (1996).