We study the problem of allocating a set of indivisible operations to a set of agents in a fair and efficient manner while also minimizing the makespan. We first present the Operation Trading Algorithm that generates allocations satisfying the DEQx (Duplicated Equitability up to any operation) fairness criterion while also guaranteeing an upper bound of 2 on the makespan for identical agents. The pairwise approach used in this algorithm has the added advantages of being decentralizable and robust. We then define a relaxed version of the DEQ1 (Duplicated Equitability upto some operation) fairness criterion called partial-DEQ1. A market based algorithm is presented to achieve partial-DEQ1 along with Pareto Optimality. Following this, it is shown that the algorithm also guarantees an upper bound of 1.618 on the makespan for 2 non-identical agents. Parametric Pruning further improves the upper bound to 1.5 which is theoretically the best possible upper bound. To the best of our knowledge, these are the first algorithms designed to achieve the mentioned fairness criteria. The algorithms additionally guarantee upper bounds on the makespan. Finally, we show the efficacy of the algorithms in generating allocations with near optimal makespans by numerically evaluating the algorithms on random instances. Note to Practitioners—This paper provides fast algorithms that can be used for task allocation among any number of agents. The algorithms generate allocations that satisfy fairness criteria along with minimizing the makespan, thus achieving two practically important criteria simultaneously. Additionally, some of the algorithms introduced can be implemented in a reactive manner thus making it easy to deal with sudden changes. The algorithms can also be implemented in a decentralized manner in environments where maintaining a centralized situational awareness is difficult due to communication and machine failures.

中文翻译：

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自主代理的运营公平分配和完工时间最小化


我们研究以公平有效的方式将一组不可分割的操作分配给一组代理的问题，同时最小化完工时间。我们首先提出操作交易算法，该算法生成满足 DEQx（任何操作的重复公平性）公平标准的分配，同时还保证相同代理的完工时间上限为 2。该算法中使用的成对方法具有分散性和鲁棒性的额外优点。然后，我们定义 DEQ1（某些操作的重复公平性）公平性标准的宽松版本，称为部分 DEQ1。提出了一种基于市场的算法来实现部分 DEQ1 和帕累托最优。接下来，结果表明该算法还保证 2 个不同代理的完工时间上限为 1.618。参数剪枝进一步将上限提高到 1.5，这在理论上是可能的最佳上限。据我们所知，这些是第一个旨在实现上述公平标准的算法。该算法还保证了完工时间的上限。最后，我们通过在随机实例上对算法进行数值评估，展示了算法在生成具有接近最佳完工时间的分配方面的功效。从业者须知——本文提供了可用于任意数量代理之间的任务分配的快速算法。该算法生成的分配既满足公平标准，又最小化完工时间，从而同时实现两个实际重要的标准。此外，引入的一些算法可以以反应方式实现，从而可以轻松应对突然的变化。 这些算法还可以在由于通信和机器故障而难以维持集中式态势感知的环境中以分散的方式实现。