Minimising mean squared deviation of job completion times about a common due date in multimachine systems

In this paper we consider the problem of scheduling n jobs on two identical parallel machines in order to minimise the mean squared deviation (MSD) of job completion times about a given common due date. When due dates are small and large deviations of job completion times from the due dates are undesirable, it becomes necessary to consider parallel machines. MSD comes under the category of non-regular performance measures, which penalises jobs that are early as well as late. In this paper we develop a lower bound on MSD for a given partial schedule and present a branch and bound algorithm to solve the problem. Optimal solutions for problem instances up to 35 jobs have been obtained for different values of due dates and the results of computational testing are presented. Based on our experiments we observe that when the due date exceeds a certain value the second machine becomes undesirable. We also propose a heuristic to provide quick solutions for problems of larger size. [Received: 25 July 2009; Revised: 30 March 2010, 1 June 2010; Accepted: 5 June 2010]