Constraint programming model and biased random-key genetic algorithm for the single-machine coupled task scheduling problem with exact delays to minimize the makespan

TL;DR

The paper tackles the single-machine coupled-task scheduling problem with exact delays, aiming to minimize the makespan $C_{max}$. It introduces a constraint programming model and a biased random-key genetic algorithm BRKGA-R-S-LS that integrates an initial-solution generator, periodic restarts and shakes, and a tailored local search. Empirical results on 240 instances (up to 100 jobs) show CP-8T often matches or surpasses state-of-the-art MIPs, while BRKGA-R-S-LS delivers high-quality approximate solutions under tight time limits and outperforms the CP model in those settings; the BRKGA components (shaking and LS) significantly improve performance. The work demonstrates the value of hybridizing CP and BRKGA strategies for constrained scheduling, with practical implications for fast, high-quality solutions in complex scheduling environments.

Abstract

We consider the strongly NP-hard single-machine coupled task scheduling problem with exact delays to minimize the makespan. In this problem, a set of jobs has to be scheduled, each composed of two tasks interspersed by an exact delay. Given that no preemption is allowed, the goal consists of minimizing the completion time of the last scheduled task. We model the problem using constraint programming (CP) and propose a biased random-key genetic algorithm (BRKGA). Our CP model applies well-established global constraints. Our BRKGA combines some successful components in the literature: an initial solution generator, periodical restarts and shakes, and a local search algorithm. Furthermore, the BRKGA's decoder is focused on efficiency rather than optimality, which accelerates the solution space exploration. Computational experiments on a benchmark set containing instances with up to 100 jobs (200 tasks) indicate that the proposed BRKGA can efficiently explore the problem solution space, providing high-quality approximate solutions within low computational times. It can also provide better solutions than the CP model under the same computational settings, i.e., three minutes of time limit and a single thread. The CP model, when offered a longer running time of 3600 seconds and multiple threads, significantly improved the results, reaching the current best-known solution for 90.56% of these instances. Finally, our experiments highlight the importance of the shake and local search components in the BRKGA, whose combination significantly improves the results of a standard BRKGA.

Figures (15)

• Figure 1: Illustration of the single machine coupled task scheduling problem
• Figure 2: Decoder example.
• Figure 3: Schedule comparison for instance 5_4_L_gen of the benchmark set of csbenchmarkKhaSalChe20.
• Figure 4: Optimal insertion for instance 5_4_L_gen of the benchmark set of csbenchmarkKhaSalChe20. Optimality requires right-shifting jobs 1 and 5.
• Figure 5: Schedule comparison for instance 5_10_L_gen csbenchmarkKhaSalChe20.