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Water cycle algorithm (WCA) is a population-based metaheuristic algorithm, inspired by the water cycle process and movement of rivers and streams towards sea. The WCA shows good performance in both exploration and exploitation phases. Further, the relationship between improvised exploitation and each parameter under asymmetric interval is derived and an iterative convergence of WCA is proved theoretically. In this paper, CEC'15 computationally expensive benchmark problems (i.e., 15 problems) have been considered for efficiency measurement of WCA accompanied with other optimisers. Also, a new discretisation strategy for the WCA has been proposed and applied along with other optimisers for solving combinatorial Internet shopping optimisation problem. By applying complexity analysis, it shows that using the WCA intricacy from dimension 10–30 is increased for almost three times. Proposing a unique discretisation approach along with providing iterative convergence proof can be considered as novelty of this research. By observing the attained numerical results, the WCA could find the minimum average error of CEC'15 in 12 and 8 out of 15 cases for dimensions 10 and 30, respectively. Experimental optimisation results for a wide range computationally expensive problems reveal the effectiveness and advantage of WCA for solving both continuous and discrete optimisation problems. [ABSTRACT FROM AUTHOR]

This paper presents a novel approximation technique based on metaheuristics and weighted residual function (WRF) for tackling singular boundary value problems (BVPs) arising in engineering and science. With the aid of certain fundamental concepts of mathematics, Fourier series expansion, and metaheuristic optimisation algorithms, singular BVPs can be approximated as an optimisation problem with boundary conditions as constraints. The target is to minimise the WRF (i.e. error function) constructed in approximation of BVPs. The scheme involves generational distance metric for quality evaluation of the approximate solutions against exact solutions (i.e. error evaluator metric). Four test problems including two linear and two non-linear singular BVPs are considered in this paper to check the efficiency and accuracy of the proposed algorithm. The optimisation task is performed using three different optimisers including the particle swarm optimisation, the water cycle algorithm, and the harmony search algorithm. Optimisation results obtained show that the suggested technique can be successfully applied for approximate solving of singular BVPs. [ABSTRACT FROM AUTHOR]