Labyrinth Weir (LW) is a popular control structure that passes a significantly higher flow rate compared to the linear weirs. In order to approach the optimal design of a trapezoidal LW, a multi-objective problem is defined to concurrently minimize the LW consumed concrete volume and maximize its discharge capacity. Simultaneously, a Radial Basis function Neural Networks (RBFNN) is designed and used for estimating LW discharge coefficient (Cd) according to the existing experimental results. An improved multi-objective particle swarm optimization (MOPSO) algorithm named TOPSIS Fuzzy MOPSO (TFMOPSO) is proposed to solve the LW optimization problem. This algorithm utilizes the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) to rank the solutions, while a fuzzy inference system is developed to select the algorithm strategy for finding two leaders among the non-dominated solutions. The performance of the proposed TFMOPSO has been tested on the optimization problem of the LW of the Ute dam. The results of TFMOPSO, along with three other state-of-the-art multi-objective algorithms, are explored in terms of hypervolume, coverage, and spacing metrics. It is demonstrated that the TFMOPSO outperforms other algorithms and studies for solving the LW multi-objective optimization problem for the case of Ute dam. Also, RBFNN is found to be one of the most appropriate approaches among studied algorithms in estimating the discharge coefficient of LW, while Pareto optimal solutions from TFMOPSO exhibit a significant improvement compared to the original design of Ute dam LW.

Soft Computing Techniques (SCT) are extensively used to estimate Labyrinth Weir's (LW) flow-rate. Due to the multiplicity of these techniques, identifying the most competent SCT is indispensable. This study aims to estimate the flow-rate of a sharp-crest triangular LW as a function of its side leg angle α and total head ratio (H/P) through several SCTs such as Adaptive Neuro-Fuzzy Inference System (ANFIS), Multi-Layer Perceptron (MLP), Support Vector Regression, and Radial Basis Function Neural Network. Additionally, these SCTs' potential combinations with Firefly Optimization Algorithm (FA) and Particle Swarm Optimization (PSO) are also investigated and used for estimation. The conducted experimental studies on LW examined a wide range of H/P in some limited α values. Correspondingly, all the proposed models and techniques are incapable of estimating the flow rate for intermediate α values without interpolation. Therefore, SCT's Interpolation accuracy is of the utmost importance. Besides the standard evaluation in the testing stage, a novel approach is utilized to evaluate the SCT's accuracy in the interpolation task. The SCTs are evaluated based on several statistical criteria, the Taylor diagram, Kruskal-Wallis, and Mann-Whitney tests. It is concluded that the competence of an SCT in the testing stage cannot guarantee its accuracy in the interpolation task. Subsequently, ANFIS-PSO and MLP-FA show the highest accuracy in the testing stage and interpolation task, respectively. Eventually, according to a systematic investigation in the implemented diagnostic test results, two rankings are presented for the applied SCTs based on their performance in the testing stage and their interpolation accuracy.