Abstract
There is an increasing trend in the use of multi-objective evolutionary algorithms (MOEAs) to solve multi-objective optimization problems of the allocation of water resources. However, typically the outcome is a set of Pareto optimal solutions which make up a trade-off surface between the objective functions. For decision makers to choose a satisfactory alternative from a set of Pareto-optimal solutions, this paper suggests a new method based on least squares support vector machine (LSSVM) and k-means clustering for ranking the optimal solutions for the multi-objective allocation of water resources. First, the k-means clustering method was adopted to reduce the large set of solutions to a few representative solutions. Then, to capture and represent the decision maker's preferences as well as to select the most desirable alternative, the LSSVM method was applied to obtain the utility value for each representative solution. According to the magnitude of the utility values, the final priority orders of the representative solutions were determined. Finally, this methodology was applied to rank the Pareto optimal solution set obtained from the multi-objective optimization problems of water resources allocation for the water-receiving areas of the South-to-North Water Transfer Project in Hebei Province, China. Moreover, the comparisons of the proposed method with the information entropy method and the artificial neural network (ANN) model were given. The results of the comparison indicate that the proposed method has the ability to rank the non-dominated solutions of the multi-objective operation optimization model and that it can be employed for decision-making on water allocation and management in a river basin.
Highlights
As a limited natural resource, water is increasingly demanded for various purposes, and how to allocate water from a river basin is among the most widely discussed issues in water resources management [1]
The radius basis function (RBF) network model was developed using the same training dataset, testing dataset and normalization technique employed for the least squares support vector machine (LSSVM)
Compared with the performance between the RBF network model and LSSVM model shown in Table 2, it is clear that the RBF network and LSSVM have close performance during the training and testing periods, the LSSVM model, overall, has performed better than the RBF network model
Summary
As a limited natural resource, water is increasingly demanded for various purposes, and how to allocate water from a river basin is among the most widely discussed issues in water resources management [1]. There have been different optimization methods introduced into the literature for optimal allocation of water resources. These methods have been applied with various degrees of success, based on mathematical programming such as linear and dynamic programming [4,5,6], and. Water 2017, 9, 257 more recently on evolutionary algorithms, such as strength Pareto EA (SPEA), non-dominated sorting genetic algorithm (NSGA-II) and multi-objective particle swarm optimization (MOPSO) [7,8]. Considering the characteristics of water resources allocation, such as nonlinearity, discreteness, non-convexity and high dimension decisions, multi-objective evolutionary algorithms (MOEAs) are proven to be more suitable for discovering and exploiting the critical tradeoffs of multi-objective water resource allocation problems, due to their efficiency and ease in handling non-linear and non-convex relationships of real-world problems, compared to the capacity of traditional optimization techniques [1,2,3,9]
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