We study an optimization problem that originates from the packaging industry, and in particular in the process of blown film extrusion, where a plastic film is used to produce rolls of different dimensions and colors. The film can be cut along its width, thus producing multiple rolls in parallel, and setup times must be considered when changing from one color to another. The optimization problem that we face is to produce a given set of rolls on a number of identical parallel machines by minimizing the makespan. The problem combines together cutting and scheduling decisions and is of high complexity. For its solution, we propose mathematical models and heuristic algorithms that involve a nontrivial decomposition method. By means of extensive computational experiments, we show that proven optimality can be achieved only on small instances, whereas for larger instances good quality solutions can be obtained especially by the use of an iterated local search algorithm
This paper addresses a single machine total weighted tardiness (TWT) batch-scheduling problem in which jobs have release dates, nonidentical sizes, and are compatible between each other. We propose two integer linear programming models: the first one is a time-indexed formulation (TIF), and the second is an innovative time-size-indexed formulation (TSIF). Although TIF clearly outperforms the existing formulation for the problem, TSIF is capable of producing much stronger bounds in practice. The latter also enables us to develop an efficient column-generation (CG) algorithm. The pricing subproblem corresponds to a resource-constrained shortest path problem that is solved using a bucket graph–based labeling algorithm. The solutions of such a subproblem may contain cycles (reprocessing of jobs), and thus, a memory mechanism called dynamic arc-based ng-sets is employed in the labeling with a view toward avoiding some of them. Moreover, we also implement a preprocessing scheme based on Lagrangian relaxation to perform variable fixing. Extensive computational experiments were carried out in 810 benchmark instances. The proposed CG algorithm is capable of solving instances with up to 100 jobs to optimality. In addition, we believe that this is the first exact approach for a TWT batch-scheduling variant capable of systematically solving instances with up to 50 jobs. High-quality results are also reported for three special cases of the problem—more precisely, when (i) the penalty weights are unitary, (ii) there are no release dates, and (iii) all due dates are set to zero and, hence, the objective becomes equivalent to minimizing the weighted completion time.
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