Fairness plays a vital role in crowd computing by attracting its workers. The power of crowd computing stems from a large number of workers potentially available to provide high quality of service and reduce costs. An important challenge in the crowdsourcing market today is the task allocation of crowdsourcing workflows. Requester-centric task allocation algorithms aim to maximize the completion quality of the entire workflow and minimize its total cost, which are discriminatory for workers. The crowdsourcing workflow needs to balance two objectives, namely, fairness and cost. In this study, we propose an alternative greedy approach with four heuristic strategies to address such an issue. In particular, the proposed approach aims to monitor the current status of workflow execution and use heuristic strategies to adjust the parameters of task allocation. We design a two-phase allocation model to accurately match the tasks with workers. The F-Aware allocates each task to the worker that maximizes the fairness and minimizes the cost. We conduct extensive experiments to quantitatively evaluate the proposed algorithms in terms of running time, fairness, and cost by using a customer objective function on the WorkflowSim, a well-known cloud simulation tool. Experimental results based on real-world workflows show that the F-Aware, which is 1% better than the best competitor algorithm, outperforms other optimal solutions in finding the tradeoff between fairness and cost.