This paper studies a two-agent scheduling problem on mixed batch machines in parallel. A mixed batch machine can process several jobs simultaneously as a batch, as long as the number of jobs in the batch does not exceed the machine capacity. The processing time of a mixed batch is the weighted sum of the maximum processing time and the total processing time of jobs in the batch. The objective is to minimise the weighted sum of two agents' makespans. We present four approximation algorithms based on two strategies: the machine-centric strategy and the agent-centric strategy. For each strategy, a full batch longest processing time (FBLPT) rule and a longest processing time greedy (LPTG) rule are used. We conduct theoretical analyses based on the worst-case performance ratio to provide the provable guarantees on the performances of the algorithms, and simulation analyses based on randomly generated instances to evaluate the average performances of the algorithms. Furthermore, we verify the consistency between the theoretical and simulation results. The algorithms using agent-centric strategy perform better than ones using machine-centric strategy. Finally, we provide managerial insights for the problem by analysing the technological parameters of batches, importance of agents, and demand seasonality.

本文研究了混合批处理机器上并行的双智能体调度问题。混合批次机器可以同时处理多个作业作为一个批次，只要批次中的作业数量不超过机器容量即可。混合批次的处理时间是最大处理时间与批次中作业的总处理时间的加权和。目标是最小化两个代理的 makespan 的加权和。我们基于两种策略提出了四种近似算法：以机器为中心的策略和以代理为中心的策略。对于每个策略，使用完整批处理最长处理时间 (FBLPT) 规则和最长处理时间贪婪 (LPTG) 规则。我们基于最坏情况下的性能比进行理论分析，为算法的性能提供可证明的保证，并基于随机生成的实例进行仿真分析，以评估算法的平均性能。此外，我们验证了理论和模拟结果之间的一致性。使用以代理为中心策略的算法比使用以机器为中心策略的算法表现更好。最后，我们通过分析批次的技术参数、代理的重要性和需求季节性来提供对该问题的管理见解。