Abstract Heat exchanger network synthesis with a stagewise superstructure gives rise to a mixed integer nonlinear problem. This problem is further complicated if discontinuous cost functions are considered. Three different formulations namely Big-M (BM), Convex-Hull (CH) and Multi-M (MM) are used to formulate the discontinuous cost functions. The BM formulation is believed to be the simplest but least efficient among the three methods. The CH formulation, on the other hand, performs better than the BM method by disaggregating the exchanger area into sub-areas. Due to the disaggregation of the exchanger area, the CH formulation requires more continuous variables but produces a tighter relaxation than the BM method. Instead of disaggregating the exchanger area, the MM formulation disaggregates the BM parameter for each pair of cost segments produces the same relaxation as the CH formulation for problems. Since the MM formulation does not need to disaggregate exchanger areas, it requires less continuous variables than the CH formulation. In this paper, performances in terms of problem relaxation, objective values and computing times of the three formulations are compared.