Abstract Task allocation plays a vital role in crowd computing by determining its performance. 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. Task allocation aims to maximize the completion quality of the entire workflow and minimize its total cost. Trust can affect the quality of the produced results and costs. Selecting workers with high levels of trust could provide better solution to the workflow and increase the budget. Crowdsourcing workflow needs to balance the two conflicting objectives. In this paper, we propose an alternative greedy approach with four heuristic strategies to address the 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. T-Aware allocates each task to the worker that maximizes the trust level, while minimizing the cost. We conduct extensive experiments to quantitatively evaluate the proposed algorithms in terms of running time, task failure ratio, trust and cost using a customer objective function on WorkflowSim, a well-known cloud simulation tool. Experimental results based on real-world workflows show that T-Aware outperforms other optimal solutions on finding the tradeoff between trust and cost, which is 3 to 6% better than the best competitor algorithm.