A New LP Algorithm for Precedence Constrained Production Scheduling

The precedence constrained production scheduling problem is the problem of scheduling the performance of jobs over a number of scheduling periods subject to precedence constraints among the jobs. The jobs can each be performed in a number of ways, and it also needs to be determined which processing option (or options) is to be chosen for each job. There can also be arbitrary side constraints among these variables. The side constraints typically represent either period capacity constraints, or profile constraints on the aggregate product produced in each period. These problems, as they occur in the mining industry, typically have a small number of side constraints often well under 100, but may contain millions of jobs and tens of millions of precedences. Thus despite the fact that the integrality gap is often small in practice, the LP itself is beyond the practical reach of commercial software. We present a new iterative lagrangian-based algorithm for solving the LP relaxation of this problem. This algorithm can be proven to converge to optimality and in practice we have found that even for problems with millions of variables and tens of millions of constraints, convergence to proved optimality is usually obtained in under 20 iterations, with each iteration requiring only a few seconds to solve with current computer hardware.