Parallel Jobs Scheduling with a Specific Due Date: a Semi-Definite Relaxation-Based Algorithm

One of the assumptions made in classical scheduling theory is that a job is always executed by one machine at a time. Since this assumption is not always true, in this paper, a relatively new concept of job scheduling, namely parallel jobs, is considered, in which a job can be executed by one or more machine at the same time. While the analytical conditions can be easily stated for some simple models, a graph model approach is required when conflicts of processor usage are present. The main decisions and solving steps are: (i) converting the scheduling problem to graph model; (ii) dividing jobs into independent sets: in this phase, we propose a semi-definite relaxation-based algorithm in which we use graph coloring concept; (iii) sequencing the independent sets as a single-machine scheduling in which jobs in such a system are job sets formed by using a semi-definite relaxation solution and determining the problem as a schedule that minimizes the sum of the tardiness of jobs. In this regard, after grouping the jobs by a semi-definite programming relaxation-based algorithm, we used the rounding algorithm for graph coloring. We also proposed a variable neighborhood search algorithm for sequencing the obtained job sets in order to minimize the sum of the tardiness. Experimental results show that this methodology is interesting by obtaining good results.