Parallel Machine Scheduling Through Column Generation: Minimax Objective Functions

In this paper we describe a solution framework for a number of scheduling problems in which the goal is to find a feasible schedule that minimizes some objective function of the minimax type on a set of parallel, identical machines, subject to release dates, deadlines, and/or generalized precedence constraints. We determine a lower bound on the objective function in the following way. We first turn the minimization problem into a decision problem by putting an upper bound on the outcome value; the question is then ‘Does there exist a feasible schedule for the given instance?’. The question of feasibility is identical to ‘Are m machines enough to feasibly accommodate all jobs?’. We turn this decision problem into a minimization problem again by asking for the minimum number of machines that we need. This can be formulated as an integer linear programming problem that resembles the ILP-formulation of the cutting stock problem. For this problem we determine a high quality lower bound by applying column generation to the LP-relaxation; if this lower bound is more than m, then we can conclude infeasibility. To speed up the process, we compute an intermediate lower bound based on the outcome of the pricing problem. As the pricing problem is intractable for many variants of the original scheduling problem, we mostly solve it approximately by applying local search, but once in every 50 iterations, we solve it to optimality by formulating it as an integer linear programming problem using a time-indexed formulation that we solve using CPLEX such that we can compute our intermediate lower bound. After having derived the lower bound on the objective function of the original scheduling problem, we try to find a matching upper bound by identifying