Job sequencing with one common and multiple secondary resources: An A⁎/Beam Search based anytime algorithm

Abstract

We consider a sequencing problem that arises, for example, in the context of scheduling patients in particle therapy facilities for cancer treatment. A set of non-preemptive jobs needs to be scheduled, where each job requires two resources: (1) a common resource that is shared by all jobs and (2) a secondary resource, which is shared with only a subset of the other jobs. While the common resource is only required for a part of the job's processing time, the secondary resource is required for the whole duration. The objective is to minimize the makespan. First we show that the tackled problem is NP-hard and provide three different lower bounds for the makespan. These lower bounds are then exploited in a greedy construction heuristic and a novel exact anytime A * algorithm, which uses an advanced diving mechanism based on Beam Search and Local Search to find good heuristic solutions early. For comparison we also provide a basic Constraint Programming model solved with the ILOG CP optimizer. An extensive experimental evaluation on two types of problem instances shows that the approach works even for large instances with up to 2000 jobs extremely well. It typically yields either optimal solutions or solutions with an optimality gap of less than 1%.

A Variable Neighborhood Search for the Job Sequencing with One Common and Multiple Secondary Resources Problem

“…In this work, we build up on the recent findings of Horn et al [HRB19] on solution methods for the JSOCMSR problem and investigate alternative, heuristic solution methods. To this end, a VNS is developed with the goal of being able to obtain nearly optimal solutions for a wide range of large instances.…”

“…Finally, the devised algorithm and variations thereof are then analyzed in a computational study and compared to the novel anytime A * algorithm proposed by Horn et al [HRB19]. The algorithms are compared with respect to different instance classes, sizes, number of secondary resources and their temporal behavior.…”

“…In their work, Horn et al [HRB17,HRB19] devised a novel anytime A * algorithm to solve different instances with up to 2000 jobs almost to proven optimality. They primarily focused on two different classes of artificially generated instances.…”

“…The results of a computational study comparing the devised algorithms with respect to baseline algorithms of [HRB19] are presented in Chapter 8. In this chapter, the outcome of different experiments are presented, where among other things quality and temporal characteristics of the algorithms are investigated.…”

“…Based on this lower bound, they proposed a greedy construction heuristic as well as a novel A * algorithm incorporating a recurring diving mechanism, which both showed their effectiveness compared to a MILP formulation solved by ILOG CPLEX. In subsequent work [HRB19], the lower bound was further improved and their A * algorithm was extended by a beam search subprocedure to obtain nearly optimal solutions for previously intractable instances. Since the MILP formulation showed to be not very efficient, a constraint programming model, solved by ILOG CP Optimizer, was used as a baseline instead.…”