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In this paper, we address the problem of online dynamic multi-robot task allocation (MRTA) problem. In the existing literature, several works investigated this problem as a multi-objective optimization (MOO) problem and proposed different approaches to solve it including heuristic methods. Existing works attempted to find Pareto-optimal solutions to the MOO problem. However, to the best of authors’ knowledge, none of the existing works used the task quality as an objective to optimize. In this paper, we address this gap, and we propose a new method, distributed multi-objective task allocation approach (DYMO-Auction), that considers tasks’ quality requirement, along with travel distance and load balancing. A robot is capable of performing the same task with different levels of perfection, and a task needs to be performed with a level of perfection. We call this level of perfection quality level. We designed a new utility function to consider four competing metrics, namely the cost, energy, distance, type of tasks. It assigns the tasks dynamically as they emerge without global information and selects the auctioneer randomly for each new task to avoid the single point of failure. Extensive simulation experiments using a 3D Webots simulator are conducted to evaluate the performance of the proposed DYMO-Auction. DYMO-Auction is compared with the sequential single-item approach (SSI), which requires global information and offline calculations, and with Fuzzy Logic Multiple Traveling Salesman Problem (FL-MTSP) approach. The results demonstrate a proper matching with SSI in terms of quality satisfaction and load balancing. However, DYMO-Auction …