We develop a multi-objective model for the resource allocation problem in a dynamic PERT network, where the activity durations are exponentially distributed random variables and the new projects are generated according to a Poisson process. This dynamic PERT network is represented as a network of queues, where the service times represent the durations of the corresponding activities and the arrival stream to each node follows a Poisson process with the generation rate of new projects. It is assumed that the mean time spent in each service station is a non-increasing function and the direct cost of each activity is a non-decreasing function of the amount of resource allocated to it. The decision variables of the model are the allocated resource quantities. To evaluate the distribution function of total duration for any particular project, we apply a longest path technique in networks of queues. Then, the problem is formulated as a multi-objective optimal control problem that involves three conflicting objective functions. The objective functions are the project direct cost (to be minimized), the mean of the project completion time (min) and the variance of the project completion time (min). Finally, the goal attainment method is applied to solve a discrete-time approximation of the original optimal control problem. We also computationally investigate the trade-off between accuracy and the computational time of the discrete-time approximation technique.