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This paper considers the un-restricted case of the Common Due-Date (CDD) problem with controllable  processing times. The problem consists of scheduling jobs with controllable processing times on a single  machine against a common due-date to minimize the overall earliness/tardiness and the compression penalties  of the jobs. The objective of the problem is to find the processing sequence of jobs, the optimal reduction  in the processing times of the jobs and their completion times. In this work, we first present and prove an  essential property for the controllable processing time CDD problem for the un-restricted case along with an  exact linear algorithm for optimizing a given job sequence for a single machine with a run-time  complexity of O(n), where n is the number of jobs. Henceforth, we implement our polynomial algorithm in  conjunction with a modified Simulated Annealing (SA) algorithm and Threshold Accepting (TA) to obtain the  optimal/best processing sequence while comparing the two heuristic approaches, as well. The implementation  is carried out on appended CDD benchmark instances provided in the OR-library.

Un-restricted Common Due-Date Problem with Controllable Processing Times - Linear Algorithm for a Given Job Sequence