Related Machine Scheduling Research Articles

Branch-and-price approach for robust parallel machine scheduling with sequence-dependent setup times

This paper studies a machine scheduling problem that minimizes the worst-case total tardiness for unrelated parallel machines with sequence-dependent setup and uncertain processing times. We propose a robust optimization reformulation of the related machine scheduling problem and discuss several important properties of the mathematical model and the reformulation approach. The proposed model generalizes robust parallel machine scheduling problems by including sequence-dependent setup times and ellipsoidal uncertainty sets. Another key contribution of the paper is to show that scheduling problems usually have alternative optimal solutions for the worst-case tardiness objective, whose performance under nominal processing times may vary or vice a versa. This issue has been addressed by studying the Pareto efficient extensions of the proposed robust optimization models to provide solutions that are immune to changes in processing times. A branch-and-price algorithm has been developed to solve realistically sized instances in less than one hour, which a commercial solver cannot achieve. Numerical results show the effectiveness of the proposed approach since realistically sized instances such as (4 machines, 32 jobs) and (150 machines, 300 jobs) can be solved to optimality within the time limit, and the (average) objective function value improvement made by the robust approach can get as high as 56% compared with the (nominal) optimal solutions that ignore uncertainty in problem data.

Related machine scheduling with machine speeds satisfying linear constraints

Related machine scheduling with machine speeds satisfying linear constraints

Competitive Algorithms from Competitive Equilibria

We introduce and study a general scheduling problem that we term the Polytope Scheduling problem (PSP). In this problem, jobs can have different arrival times and sizes, and the rates assigned by the scheduler to the jobs are subject to arbitrary packing constraints. The PSP framework captures a variety of scheduling problems, including the classical problems of unrelated machines scheduling, broadcast scheduling, and scheduling jobs of different parallelizability. It also captures scheduling constraints arising in diverse modern environments ranging from individual computer architectures to data centers. More concretely, PSP models multidimensional resource requirements and parallelizability, as well as network bandwidth requirements found in data center scheduling. We show a surprising result—there is a single algorithm that is O (1) competitive for all PSP instances when the objective is total completion time, and O (1) competitive for a large sub-class of PSP instances when the objective is total flow time. This algorithm simply uses the well-known Proportional Fairness (PF) algorithm to perform allocations each time instant. Though P F has been extensively studied in the context of maximizing fairness in resource allocation, we present the first analysis in adversarial and general settings for optimizing job latency. Further, P F is non-clairvoyant, meaning that the algorithm doesn’t need to know jobs sizes until their completion. We establish our positive results by making novel connections with Economics, in particular, the notions of market clearing, Gross Substitutes, and Eisenberg-Gale markets. We complement these positive results with a negative result: We show that for the total flow time objective, any non-clairvoyant algorithm for general PSP has a strong lower bound on the competitive ratio unless given a poly-logarithmic speed augmentation. This motivates the need to consider sub-classes of PSP when studying flow time. The sub-class for which we obtain positive results not only captures several well-studied models, such as scheduling with speedup curves and related machine scheduling, but also captures as special cases hitherto unstudied scheduling problems, such as single source flow routing, routing multicast (video-on-demand) trees, and resource allocation with substitute resources.

New Approximation Bounds for Lpt Scheduling

We provide new bounds for the worst case approximation ratio of the classic Longest Processing Time (Lpt) heuristic for related machine scheduling (Q||C max ). For different machine speeds, Lpt was...

New Approximation Bounds for Lpt Scheduling

We provide new bounds for the worst case approximation ratio of the classic Longest Processing Time (Lpt) heuristic for related machine scheduling (Q||C max ). For different machine speeds, Lpt was first considered by Gonzalez et al. (SIAM J. Comput. 6(1):155–166, 1977). The best previously known bounds originate from more than 20 years back: Dobson (SIAM J. Comput. 13(4):705–716, 1984), and independently Friesen (SIAM J. Comput. 16(3):554–560, 1987) showed that the worst case ratio of Lpt is in the interval (1.512,1.583), and in (1.52,1.67), respectively. We tighten the upper bound to $1+\sqrt{3}/3\approx1.5773$, and the lower bound to 1.54. Although this improvement might seem minor, we consider the structure of potential lower bound instances more systematically than former works. We present a scheme for a job-exchanging process, which, repeated any number of times, gradually increases the lower bound. For the new upper bound, this systematic method together with a new idea of introducing fractional jobs, facilitated a proof that is surprisingly simple, relative to the result. We present the upper-bound proof in parameterized terms, which leaves room for further improvements.

Scheduling and constraint propagation

The resource-constrained project-scheduling problem (RCPSP) and some of its generalizations are defined. Furthermore, constraint propagation techniques for these problems and related machine scheduling problems are introduced and possible applications of these techniques in connection with lower bound calculations, branch-and-bound methods, and heuristics are discussed.