A new class of multi agent single machine scheduling problems is introduced, where each job is associated with a self interested agent with a utility function decreasing in completion time. We aim to achieve a fair solution by maximizing the minimum utility across all agents. We study the problem's complexity and propose solution methods for several variants. For the general case, we present a binary search procedure to find the largest possible minimum utility, as well as an exact greedy based alternative. Variants with release and due dates are analyzed, showing strong NP hardness for arbitrary release dates, but weak NP hardness for a single release date job, and polynomial solvability when all jobs share processing times. For all these cases we also study the corresponding problem of finding efficient solutions where the sum of utilities is maximized. We also examine settings where linear utility functions can be adjusted within budget constraints, exploring the impact on optimal schedules when intercepts or slopes are modified. From a single agent perspective, we investigate the effect of improving one agent's utility in the overall solution. Adding a new job to be inserted with the best possible utility gives rise to rescheduling problems, where different lower bounds depending on the utilities of the original fair schedule are imposed. Finally, we consider a bi level setting where a leader wants to enforce a certain target schedule by modifying utility functions while the follower computes a fair solution for the modified instance. Our work contributes to scheduling theory, multi agent systems, and algorithmic fairness, highlighting fairness oriented objectives in competitive scheduling.

翻译：摘要：本文引入了一类新的多智能体单机调度问题，其中每个作业关联一个自私智能体，其效用函数随完工时间递减。我们旨在通过最大化所有智能体的最小效用来实现公平解。研究了问题的计算复杂性，并为若干变体提出了求解方法。针对一般情形，我们提出了一种二分搜索程序来寻找最大可能的最小效用，以及一种基于贪心的精确替代方法。分析了具有释放时间和截止日期的变体，结果表明：任意释放时间下问题为强NP困难，单一释放时间作业时为弱NP困难，而所有作业加工时间相同时可在多项式时间内求解。针对所有情形，我们还研究了相应的效率导向问题，即最大化效用总和。此外，我们考察了在线性效用函数可在预算约束内调整的场景，探索截距或斜率修改对最优调度的影响。从单一智能体视角，我们研究了改进某智能体效应对整体解的影响。以最优可能效用插入新作业引发了重调度问题，其中需施加基于原始公平调度效用的不同下界。最后，我们考虑了双层设定：领导者通过修改效用函数强制执行特定目标调度，而跟随者针对修改后的实例计算公平解。本工作对调度理论、多智能体系统及算法公平性做出贡献，突出竞争性调度中的公平性目标。