Usually, heat exchanger network (HEN) problems focus on the design of networks that achieve full energy integration. In this work, we analyze the HEN design problem by considering fully integrated HENs (which achieve the target of minimum utility consumption) and partially integrated alternatives. Within this context, economic, environmental, and safety indicators are analyzed to detect which design alternative is the most suitable one. The framework is applied for the design of an HEN of an industrial process to produce ethylene and propylene using natural gas as feedstock. It is shown how a partially integrated network outperforms the one with minimum energy consumption for some of the metrics considered in this work. It is shown how a fully integrated network does not provide a significant improvement from economic and environmental aspects. Furthermore, the safety indicator shows that there is a point after which if more heat exchangers are added, then the HEN becomes more hazardous.