We develop a discrete-time approximation technique dealing with the time-cost trade-off problem in PERT networks. It is assumed that the activity durations are independent random variables with generalized Erlang distributions, in which the mean duration of each activity is a non-increasing function of the amount of resource allocated to it. It is also assumed that the amount of resource allocated to each activity is controllable. Then, we construct an optimal control problem with three conflicting objective functions. Solving this optimal control problem, optimally, is impossible. Therefore, a discrete-time approximation technique is applied to solve the original multi-objective optimal control problem, using goal attainment method. To show the advantages of the proposed technique, we also develop a Simulated Annealing (SA) algorithm to solve the problem, and compare the discrete-time approximation results against the SA and also the genetic algorithm results.