Using Quadratic Programming to Solve High Multiplicity Scheduling Problems on Parallel Machines

We introduce and analyze several models of scheduling n different types (groups) of jobs on m parallel machines, where in each group all jobs are identical. Our main goal is to exhibit the usefulness of quadratic programming approaches to solve these classes of high multiplicity scheduling problems, with the total weighted completion time as the minimization criterion. We develop polynomial algorithms for some models, and strongly polynomial algorithms for certain special cases. In particular, the model in which the weights are job independent, as well as the generally weighted model in which processing requirements are job independent, can be formulated as an integer convex separable quadratic cost flow problem, and therefore solved in polynomial time. When we specialize further, strongly polynomial bounds are achievable. Specifically, for the weighted model with job-independent processing requirements if we restrict the weights to be machine independent (while still assuming different machine speeds), an O (mn + n log n) algorithm is developed. If it is also assumed that all the machines have the same speed, the complexity of the algorithm can be improved to O (m log m + n log n). These results can be extended to related unweighted models with variable processing requirements in which all the machines are available at time zero.