Application of a multi-objective optimization model for the design of Piano Key Weirs with a fixed dam height

Piano Key Weirs (PKWs) have recently been used as new or rehabilitation options in the world because of their advantages in hydraulic performance and construction costs. However, designing an efficient PKW is challenging due to a large and com-plex set of geometric and hydraulic parameters. Therefore, reaching an optimal PKW design depends on the examination of various geometric combinations and hydraulic parameters. In this study, we applied a multi-objective optimization model known as Non-dominated Sorting Genetic Algorithm-II to determine an optimal design by maximizing hydraulic discharge while minimizing the volume of the concrete. Here, we evaluated the capability of our approach in two separate case studies, one which represented as a rehabilitated weir and other as a new design. Both studies show that the developed approach could significantly reduce the concrete volume per unit of discharge. Moreover, the results show similar patterns in terms of the hydro-economic behavior of the model, which were discussed in three distinguished regions. The unique characteristics of these regions were elaborated, and their most cost-effective values of design parameters were identified. Finally, we dis-cussed how the proposed model and findings of this study could be used for improving the preliminary design of PKWs in practice.