Equitable Allocation of Operations and Makespan Minimization for Autonomous Agents

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.