Federated learning (FL) is a promising distributed learning framework designed for privacy-aware applications of resource-constrained devices. Without sharing data, FL trains a model on each device locally and builds the global model on the server by aggregating the trained models. To reduce the communication overhead, only a portion of client devices participate in each round of training. Random selection is the most common way of selecting client devices for training data in a round of FL. However, random client selection uses distributed data and computational resources inefficiently, as it does not take into account the hardware specifications and data distribution among clients. This paper proposes FedGRA, an adaptive fair client selection algorithm designed for FL applications with unbalanced, non-Identically and Independently Distributed (IID) data running on client devices with heterogeneous computing resources. FedGRA dynamically adjusts the set of selected clients at each round of training based on clients' trained models and their available computational resources. To find an optimal solution, we model the client selection problem of FL as a multi-objective optimization by using Grey Relational Analysis (GRA) theory. To examine the performance of our proposed method, we implement our contribution on Amazon Web Services (AWS) by using 50 Elastic Compute Cloud (EC2) instances with 4 different hardware configurations. The evaluation results reveal that our contribution improves convergence significantly and reduces the average client's waiting time compared to state-of-the-art methods.

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