Approximability results for the resource-constrained project scheduling problem with a single type of resources

• Published in: Annals of operations research Vol. 213; no. 1; pp. 115 - 130
• Main Authors: Gafarov, Evgeny R., Lazarev, Alexander A., Werner, Frank
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.02.2014; Springer; Springer Nature B.V
• Subjects: Algorithms; Approximation; Approximation theory; Business and Management; Combinatorics; Industrial project management; Methods; Operations research; Operations Research/Decision Theory; Optimization techniques; Production planning; Production scheduling; Project management; Schedules; Scheduling; Scheduling (Management); Studies; Theory of Computation; Lower bounds; Project scheduling; Makespan; Upper bounds
• DOI: 10.1007/s10479-012-1106-5
• ISSN: 0254-5330

In this paper, we consider the well-known resource-constrained project scheduling problem. We give some arguments that already a special case of this problem with a single type of resources is not approximable in polynomial time with an approximation ratio bounded by a constant. We prove that there exist instances for which the optimal makespan values for the non-preemptive and the preemptive problems have a ratio of O (log n ), where n is the number of jobs. This means that there exist instances for which the lower bound of Mingozzi et al. has a bad relative error of O (log n ), and the calculation of this bound is an NP-hard problem. In addition, we give a proof that there exists a type of instances for which known approximation algorithms with polynomial time complexity have an approximation ratio of at least equal to , and known lower bounds have a relative error of at least equal to O (log n ). This type of instances corresponds to the single machine parallel-batch scheduling problem 1| p − batch , b =∞| C max .