Simultaneous optimization of several competing objectives requires increasing the capability of optimization algorithms. This paper proposes the multi-objective moth swarm algorithm, for the first time, to solve various multi-objective problems. In the proposed algorithm, a new definition for pathfinder moths and moonlight was proposed to enhance the synchronization capability as well as to maintain a good spread of non-dominated solutions. In addition, the crowding-distance mechanism was employed to select the most efficient solutions within the population. This mechanism indicates the distribution of non-dominated solutions around a particular non-dominated solution. Accordingly, a set of non-dominated solutions obtained by the proposed multi-objective algorithm is kept in an archive to be used later for improving its exploratory capability. The capability of the proposed MOMSA was investigated by a set of multi-objective benchmark problems having 7 to 30 dimensions. The results were compared with three well-known meta-heuristics of multi-objective evolutionary algorithm based on decomposition (MOEA/D), Pareto envelope-based selection algorithm II (PESA-II), and multi-objective ant lion optimizer (MOALO). Four metrics of generational distance (GD), spacing (S), spread (Δ), and maximum spread (MS) were employed for comparison purposes. The qualitative and quantitative results indicated the superior performance and the higher capability of the proposed MOMSA algorithm over the other algorithms. The MOMSA algorithm with the average values of CPU time = 2771 s, GD = 0.138, S = 0.063, Δ = 1.053, and MS = 0.878 proved to be a robust and reliable model for multi-objective optimization.
Deriving optimal operation policies for multi-reservoir systems is a complex engineering problem. It is necessary to employ a reliable technique to efficiently solving such complex problems. In this study, five recently-introduced robust evolutionary algorithms (EAs) of Harris hawks optimization algorithm (HHO), seagull optimization algorithm (SOA), sooty tern optimization algorithm (STOA), tunicate swarm algorithm (TSA) and moth swarm algorithm (MSA) were employed, for the first time, to optimal operation of Halilrood multi-reservoir system. This system includes three dams with parallel and series arrangements simultaneously. The results of mentioned algorithms were compared with two well-known methods of genetic algorithm (GA) and particle swarm optimization (PSO) algorithm. The objective function of the optimization model was defined as the minimization of total deficit over 223 months of reservoirs operation. Four performance criteria of reliability, resilience, vulnerability and sustainability were used to compare the algorithms’ efficiency in optimization of this multi-reservoir operation. It was observed that the MSA algorithm with the best value of objective function (6.96), the shortest CPU run-time (6738 s) and the fastest convergence rate (< 2000 iterations) was the superior algorithm, and the HHO algorithm placed in the next rank. The GA, and the PSO were placed in the middle ranks and the SOA, and the STOA placed in the lowest ranks. Furthermore, the comparison of utilized algorithms in terms of sustainability index indicated the higher performance of the MSA in generating the best operation scenarios for the Halilrood multi-reservoir system. The application of robust EAs, notably the MSA algorithm, to improve the operation policies of multi-reservoir systems is strongly recommended to water resources managers and decision-makers.
Optimal operation of reservoirs is one of the most important issues in water resources management. In this study, a novel metaheuristic optimisation algorithm, called the water cycle algorithm (WCA), was used to derive operating policy for a multi-reservoir system. In the first step, the performance of the model was successfully assessed through several benchmark functions. The WCA was then used to derive the optimal operation of four- and ten-reservoir systems. It was then applied to the monthly operation of Golestan and Voshmgir consecutive dams located in Gorganrood basin in the north of Iran. In this way, the objective function was defined as minimising the total deficit for the study period. The WCA results were compared with the results of other developed evolutionary algorithms, including a genetic algorithm, harmony search algorithm, particle swarm optimisation and imperialist competitive algorithm. The results showed that, for cases of four- and ten-reservoir systems, the best solutions achieved by the WCA were 306·3918 and 1172·4197, which had differences of 0·5% and 1% compared with the global optimum solutions, respectively. In addition, it was found that the WCA was superior to other algorithms in calculating the annual deficit of the Golestan–Voshmgir multi-reservoir system.
In this study, the capability of the recently introduced moth swarm algorithm (MSA) was compared with two robust metaheuristic algorithms: the harmony search (HS) algorithm and the imperialist competitive algorithm (ICA). First, the performance of these algorithms was assessed by seven benchmark functions having 2–30 dimensions. Next, they were compared for optimization of the complex problem of four-reservoir and 10-reservoir systems operation. Furthermore, the results of these algorithms were compared with nine other metaheuristic algorithms. Sensitivity analysis was performed to determine the appropriate values of the algorithms’ parameters. The statistical indices coefficient of determination (R2), root mean square error (RMSE), mean absolute error (MAE), mean square error (MSE), normalized MSE (NMSE), mean absolute percentage error (MAPE), and Willmott’s index of agreement (d) were used to compare the algorithms’ performance. The results showed that MSA was the superior algorithm for solving all benchmark functions in terms of obtaining the optimal value and saving CPU usage. ICA and HS were ranked next. When the dimensions of the problem were increased, the performance of ICA and HS dropped but MSA has still performed extremely well. In addition, the minimum CPU usage and the best solutions for the optimal operation of the four-reservoir system were obtained by MSA, with values of 269.7 seconds and 308.83, which are very close to the global optimum solution. Corresponding values for ICA were 486.73 seconds and 306.47 and for HS were 638.61 seconds and 264.61, which ranked them next. Similar results were observed for the 10-reservoir system; the CPU time and optimal value obtained by MSA were 722.5 seconds and 1,195.58 while for ICA they were 1,421.62 seconds and 1,136.22 and for HS they were 1,963.41 seconds and 1,060.76. The R2 and RMSE values achieved by MSA were 0.951 and 0.528 for the four-reservoir system and 0.985 and 0.521 for the 10-reservoir system, which demonstrated the outstanding performance of this algorithm in the optimal operation of multi-reservoir systems. In a general comparison, it was concluded that among the 12 algorithms investigated, MSA was the best, and it is recommended as a robust and promising tool in the optimal operation of multi-reservoir systems.
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