Local Search for Project Scheduling with Convex Objective Functions

We consider the scheduling of projects subject to temporal and resource constraints such that a continuous convex objective function in the start times of activities is minimized. Optimal solutions to this problem represent local minimizers on polytopes belonging to inclusion–minimal feasible orders which reflect precedence relationships between activities. Thus, a natural approach to the problem is to enumerate appropriate orders for each of which a local minimizer of the objective function on the corresponding polytope is determined by a descent algorithm. We propose different neighborhoods on the set of orders which arise from removing, replacing, or adding pairs in the respective covering relation. For the resource–constrained weighted earliness–tardiness problem, a corresponding tabu search procedure has been implemented which uses a primal and a dual first–order descent algorithm for the computation of stationary points. 1 Preliminaries Let N = 〈V,E; δ〉 be an activity–on–node project network with set of nodes V , set of arcs E, and corresponding arc weights δ. By R we denote the set of resources required for the execution of the project. The start times Si of activities i ∈ V have to be determined such that a continuous convex objective function f of schedule S = (Si)i∈V is minimized, the temporal constraints Sj ≥ Si+δij belonging to arcs 〈i, j〉 ∈ E are met, the project is started at time zero and completed by a deadline d, and the requirements rk(S, t) of resources k ∈ R do not exceed the corresponding resource capacities Rk at any point in time t. This problem, designated as PS|temp, d|f and m, 1|gpr, δn|nonreg in the triple classifications devised by Brucker et al. (1999) and Herroelen et al. (1998), respectively, can be stated as follows: