## 1 Introduction
In recent years, meta-heuristic algorithms have been widely used to solve multi-objective optimization problems, and effectively applied to engineering practice, which are classified into two dominant classes: swarm intelligence and evolutionary. Swarm intelligence algorithms imitate the intelligence of swarms, flocks and herds of creatures while evolutionary algorithms imitate the mechanisms of evolution in nature. [PERSON] et al. [12] proposed a novel optimization algorithm named multi-objective salp swarm algorithm (MSSA) for solving optimization problems multiple objectives by imitate swarm behaviour of salps in oceans. MSSA had been tested on mathematical optimization functions and the results show that it is able to refine the initial random solutions effectively and convergence and is successfully applied to the multi-objective optimization problem of maximizing the efficiency and minimizing the cavitation of the propeller to achieve the optimal blade shape. [PERSON] et al. [13] proposed a bi-swarm particle swarm optimizer with novel neighbourhood topology strategy (BPSO-NT) to solve the optimization of intermodal transportation planning in the field of logistics. Ten well-known benchmark functions are utilized to test the performance of BPSO-NT and the results show that the proposed BPSO-NT is superior to others PSO variants algorithm. In addition, BPSO-NT has achieved good results in solving practical engineering problems. [PERSON] et al. [14] presented an intelligent technique using modified genetic algorithms, combined with SuperMC code, to find the optimum loading pattern for the nuclear fuel, which is favourable to maximize effective multiplication factor and neutron flux densities inside the reactor core under design limits. In 2002, [PERSON] et al. [15] proposed a modified version of NSGA, the well-known NSGA-II algorithm, which has a superior comprehensive performance compared with other evolutionary algorithms and is widely used in optimizing the centrifugal impellers. [PERSON] et al. [16] used NSGA-II to optimise the conflict between pressure head and hydraulic efficiency. The tests proved that the optimised mini-pump increased the pressure head by about 24% and the hydraulic efficiency by 4.75% compared with the original design prototype, and the flow field analysis showed that the optimised model could alleviate the impeller wake and improve the pressure distribution and flow direction. [PERSON] et al. [17] used NAGS-II to optimise the design of the automotive torque converter impeller. The results show that the zero-order polynomial of the Gaussian model is the best prediction for the stall torque ratio. They proposed an effective solution for the design of the hydraulic torque converter impeller geometry. However, the NSGA-II algorithm is excessively focused on convergence, leading to controversy over its performance in the population diversity. [PERSON] et al. [18] pointed out that the defect of NSGA-II is the selecting principle of crowding distance which is based on the cubic distance to the adjacent points. For instance, in the ZDT problems, with continuous evolution, more and more points would cluster to a small range. The diversity of individuals is lost which leads to a local optimum. [PERSON] et al. [19] indicated that the crowding distance operator in NSGA-II can guarantee the diversity along the non-dominated front, but it may lose the horizontal diversity when wiping off excessive individuals. Due to the wide range of FC operating conditions and the higher requirements for aerodynamic performance under different operating conditions, the multi-objective problems using the classical NSGA-II algorithm easily fall into the local optimal solution. In order to improve the population diversity of solution sets for multiple operating conditions performance of FC centrifugal impeller, a modified NSGA-II optimisation algorithm (i.e., DNSGA) is proposed, which has a novel classification method for non-dominated ranking and crowding distance operator (i.e., SD operator with weights).