The design of inspection schedules is a complex optimization problem that requires the reliability to be assessed. The solution to this problem can be found balancing the costs associated to inspection/repair activities against the benefits related to the faultless operation of the infrastructure. The optimization aims at minimizing the total cost, obtained as the combination of maintenance and failure costs, by tuning some design parameters, such as the number, time and quality of inspections. The reliability is assessed making use of probability boxes, i.e. by accounting for both variability and imprecision. The use of probability boxes relaxes the assumption of exact input probability distributions, which is always too strong given that these distributions are very often estimated within a degree of confidence, or elicited from a finite set of experimental data. The optimization problem is formulated as a time-dependent reliability-based optimization problem, where both objective and constraint functions require the evaluation of upper and lower reliability bounds. The solution to this problem represents a real technological challenge, as the reliability assessment by means of p-boxes is a computationally intensive task, which may take up to few days to be completed on last generation processors. In this paper, an efficient and generally applicable numerical technique, which is capable of producing a solution in a very short amount of time (≤1 hour), is proposed. The technique combines a forced Monte Carlo simulation method with an optimization strategy, which makes the interval reliability assessment particularly efficient. The efficiency and accuracy of the proposed technique is shown by means of a literature example involving a fatigue-prone weld in a bridge girder.