As an essential scheduling problem with several practical applications, the parallel machine scheduling problem (PMSP) with family setups constraints is difficult to solve and proven to be NP-hard. To this end, we present a deep reinforcement learning (DRL) approach to solve a PMSP considering family setups, aiming at minimizing the total tardiness. The PMSP is first modeled as a Markov decision process, where we design a novel variable-length representation of states and actions, so that the DRL agent can calculate a comprehensive priority for each job at each decision time point and then select the next job directly according to these priorities. Meanwhile, the variable-length state matrix and action vector enable the trained agent to solve instances of any scales. To handle the variable-length sequence and simultaneously ensure the calculated priority is a global priority among all jobs, we employ a recurrent neural network, particular gated recurrent unit, to approximate the policy of the agent. The agent is trained based on Proximal Policy Optimization algorithm. Moreover, we develop a two-stage training strategy to enhance the training efficiency. In the numerical experiments, we first train the agent on a given instance and then employ it to solve instances with much larger scales. The experimental results demonstrate the strong generalization capability of the trained agent and the comparison with three dispatching rules and two metaheuristics further validates the superiority of this agent.

Erosion is one of the most common forms of material failure and equipment damage in gas transmission pipelines. Shale gas fields use hydraulic fracturing whereby solid particles are often carried in the gas flow, and the pipeline is in a high-pressure state, which is more likely to cause erosion. The prediction of particle erosion regulation in gas-solid two-phase flow is an effective means to ensure the safe operation of shale gas fields. In this paper, an integrated CFD-DPM model is established to investigate the erosion of 90° elbow in a shale gas field under gas-solid two-phase flow, employing the realizable k-ε turbulence model, discrete phase model, and erosion rate prediction model. The reliability of the proposed numerical models is verified by comparing the predicted data with the experimental data. Moreover, the effects of six important factors on maximum erosion rate are analyzed, including gas velocity, mass flow rate of sand particles, particle diameter, shape coefficient of sand particles, pipeline diameter, elbow radius of curvature. Specifically, the results indicate that the gas velocity, mass flow rate and shape coefficient of sand particles are positively correlated with the maximum erosion rate, while the pipe diameter and the elbow radius of curvature are negatively correlated with the maximum erosion rate. A new correlation was developed, which included four dimensionless groups, namely Reynolds number, diameter ratio, density ratio and particle number. The correlation can be used to predict maximum corrosion rate of elbows. This work can provide data reference and theoretical basis for mitigating the erosion rate of pipelines and managing the integrity of gas pipelines.