Improving multidimensional normal cloud model to evaluate groundwater quality with grey wolf optimization algorithm and projection pursuit method

Groundwater quality is related to several uncertain factors. Using multidimensional normal cloud model to reduce the randomness and ambiguity of the integrated groundwater quality evaluation is important in environmental research. Previous optimizations of multidimensional normal cloud models have focused on improving the affiliation criteria of the evaluation results, neglecting the weighting scheme of multiple indicators. In this study, a new multidimensional normal cloud model was constructed for the existing one-dimensional normal cloud model (ONCM) by combining the projection-pursuit (PP) method and the Grey Wolf Optimization (GWO) algorithm. The effectiveness and robustness of the model were analyzed. The results showed that compared with ONCM, the new multidimensional normal cloud model (GWOPPC model) integrated multiple evaluation parameters, simplified the modeling process, and reduced the number of calculations for the affiliation degree. Compared with other metaheuristic optimization algorithms, the GWO algorithms converged within 20 iterations during 20 simulations showing faster convergence speed, and the convergence results of all objective functions satisfy the iteration accuracy of 0.001, which indicates that the algorithm is more stable. Compared to the traditional entropy weights (0.27, 0.23, 0.47, 0.44, 0.29, 0.59, 0.12) or principal component weights (0.38, 0.33, 0.42, 0.34, 0.47, 0.29, 0.38), the weight allocation scheme provided by the GWOPP method (0.50, 0.48, 0.05, 0.38, 0.02. 0.51 and 0.32) considers the density of the distribution of all samples in the data set space. Among all 55 groundwater samples, the GWOPPC model has 21 samples with lower evaluation ratings than the fuzzy evaluation method, and 28 samples lower than the Random Forest method or the WQI method, indicating that the GWOPPC model is more conservative under the conditions of considering fuzziness and randomness. This method can be used to evaluate groundwater quality in other areas to provide a basis for the planning and management of groundwater resources.