Grid computing is the aggregation of the power of heterogeneous, geographically distributed computing resources to provide high-performance computing. To benefit from the grid computing capabilities, effectual scheduling algorithms are primarily essential. This paper presents a GA-based approach, called Grid Workflow Tasks Scheduling Algorithm (GWTSA), for scheduling workflow tasks on grid services based on users’ QoS (quality of service) constraints in terms of cost and time. For a given set of inter-dependent workflow tasks, it generates an optimal schedule, which minimizes the execution time and cost, such that the optimized time be within the time constraints (deadline) imposed by the user. In GWTSA, the workflow tasks are modeled as a DAG, which is divided, then the optimal sub-schedules of all task divisions are computed and used to obtain the execution schedule of the entire workflow. A GA-based technique is employed in GWTSA to compute the optimal execution sub-schedule for each branch division that consists of a set of sequential tasks. In this technique, the chromosome represents a branch division, where each gene holds the id of the service provider chosen to execute the corresponding task in the branch; and the fitness function is formulated as a multi-objective function of time and cost, this gives users the ability to determine their requirements if speed against cost or vice versa, by changing the weighting coefficients in the fitness function. The paper also exhibits the experimental results of assessing the performance of GWTSA with workflow samples of different sizes.