Fouling mitigation is paramount to maintaining the reliable and efficient operation of a heat exchanger network (HEN). From the operational perspective, fouling can be mitigated by changing the flow distribution in the network (control actions), or performing periodic cleanings of the units (scheduling actions). Flow control and scheduling have usually been considered independently, ignoring their interaction. This paper presents an online methodology and implementation that integrates control and scheduling decisions for fouling mitigation in HEN, using first principle models of the heat exchangers subject to fouling. A multiloop NMPC/MHE scheme is proposed to estimate the current state of the HEN, and then define the optimal flow distribution and cleaning schedule over a moving horizon. It is shown that this online scheme reacts rapidly to disturbances and copes with model-plant mismatch by updating the model parameters at an appropriate frequency. The methodology is demonstrated on a real industrial case study involving crude oil fouling in the preheat train of a refinery. Application of the methodology shows that (i) significant economic benefits result relative to the actual historical operation, (ii) the online integration achieves a lower operating cost than that of the optimization of control or scheduling individually, (iii) the effect of disturbances is important and the scheme rejects them efficiently, (iv) updating the prediction models deals effectively with plant-model mismatch and process variability, and gives a sufficiently accurate representation of the underlying process, and (v) the computational effort required to solve all optimization problems is low and allows for the practical online implementation of the scheme.