## 1 Introduction  
The efficient global optimization (EGO) algorithm, which was first proposed by [PERSON] et al. [1] in 1998, has proved to be very efficient for dealing with expensive-to-evaluate optimizations [2; 3; 4; 5]. Benefited from the fundamental concepts of the Kriging surrogate model and the expected improvement (EI) criterion, EGO can always capture a high-quality optimal solution with relatively less objective function evaluations (FEs) compared to other popular optimization algorithms, such as genetic algorithms (GAs) [6]. Because of this advantage, the corresponding theory has been applied to a variety of modern engineering applications [7; 8; 9; 10; 11]. One must notice that the original EI criterion is initially designed for single-objective optimization (SOO) problems and cannot be applied to more important multi-objective optimization (MOO) problems directly. It is well known that real engineering optimization problems usually come up with not only one single objective but also a set of objectives, which usually conflict with each other. Motivated by the outstanding behaviors of the traditional EGO algorithm in dealing with SOO problems, different methods have been developed by modifying the EGO algorithm to be able to cope with MOO problems [12; 13; 14], particularly for expensive-to-evaluate problems. For instance, the method, namely ParEGO, was developed by [PERSON] [12] for coping with MOO problems. Different aggregate single-objectives are updated iteration by iteration using the augmented Tchebycheff functions with different random vectors to balance the multiple sub-objectives. A similar method called MOEA/D-EGO was later proposed by [PERSON] et al. [14]. These methods, which are realized in a way of transforming the MOO problem into an SOO problem associated with any available single-objective EGO optimizers, do show promise in the test cases appearing in the literature mentioned. However, because of the usage of several probabilistic parameters of corresponding optimization problems, they are usually not easy to directly apply to real time-consuming engineering optimizations [15; 16]. In addition, only one optimal solution captured as a result of the transformed SOO problem is mostly weight dependence. An alternative to avoiding the shortages mentioned is to capture a set of solutions distributed at the Pareto front of the MOO problem. The main challenge in capturing the distribution of Pareto solutions is modifying the EI infill criterion of the EGO algorithm to be suitable for updating Pareto solutions efficiently. For example, [PERSON] et al. [13] developed the Euclidean distance-based EI (EI-ELU) infill criterion by maximizing the minimum distance between the expected central point of probability distribution and the Pareto points. A more recent hypervolume indicator-based criterion, which was proposed by [PERSON] and [PERSON] [17], is more of interest for MOO problems. In recent decades, many practical contributions have been made to hypervolume-based infill metrics, such as works in hypervolume indicator [17; 18], hypervolume improvement [19], S-metric [2; 20], lower confidence bound (LCB) [2; 21], expected hypervolume improvement (EHVI) [18; 22], truncated expected hypervolume improvement (TEHVI) [23], and modified expected hypervolume improvement (MEHVI) [24]. In general, the hypervolume-based metric has been reported to be an outstanding metric for improving accuracy and distribution on the Pareto front of the MOO problem [24; 25]. However, the calculation of such metrics is still struggling with enhancing the accuracy of computed hypervolume and reducing the computational complexity implementation. In view of accuracy enhancement, several notable methods have been successfully developed for computing hypervolume, such as the LebMeasure method [21; 26], Overmars, and Yap method [27], hypervolume by slicing objectives (HSO) method [28], IHSO method [29], HypE method [30], and box decomposition method [31].