In this paper, we propose exact and heuristic solution approaches based on dynamic programming for an open lot streaming problem. We also present the first application of Lagrangian relaxation to compute strong lower bounds to such a problem. The application concerns the minimization of the total flow time for the discrete version of a single-job lot streaming problem from the literature in a two-machine flow shop with attached setup times. Computational results on benchmark instances illustrate the effectiveness of the proposed approaches and give evidence of the strength of the Lagrangian relaxation lower bounds.

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We address the tactical fixed job scheduling problem with spread-time constraints. In such a problem, there are a fixed number of classes of machines and a fixed number of groups of jobs. Jobs of the same group can only be processed by machines of a given set of classes. All jobs have their fixed start and end times. Each machine is associated with a cost according to its machine class. Machines have spread-time constraints, with which each machine is only available for L consecutive time units from the start time of the earliest job assigned to it. The objective is to minimize the total cost of the machines used to process all the jobs. For this strongly NP-hard problem, we develop a branch-and-price algorithm, which solves instances with up to 300 jobs, as compared with CPLEX, which cannot solve instances of 100 jobs. We further investigate the influence of machine flexibility by computational experiments. Our results show that limited machine flexibility is sufficient in most situations.