Summary: This paper considers a generalization of the permutation flow shop problem that combines the scheduling function with the planning stage. In this problem, each work center consists of parallel identical machines. Each job has a different release date and consists of ordered operations that have to be processed on machines from different machine centers in the same order. In addition, the processing times of the operations on some machines may vary between a minimum and a maximum value depending on the use of a continuously divisible resource. We consider a nonregular optimization criterion based on due dates which are not a priori given but can be fixed by a decision-maker. A due date assignment cost is included into the objective function. For this type of problems, we generalize well-known approaches for the heuristic solution of classical problems and propose constructive algorithms based on job insertion techniques and iterative algorithms based on local search. For the latter, we deal with the design of appropriate neighborhoods to find better quality solution. Computational results for problems with up to 20 jobs and 10 machine centers are given.