## 1 Introduction
[9] considered cutting energy consumption and non-cutting energy consumption; established a multi-objective mathematical model with energy consumption, machining surface quality, and machining time as the optimization objectives; and solved the optimal Pareto solution set. [PERSON] et al. [10] introduced a dynamic multiple-swarm particle swarm algorithm that preserves infeasible particles generated by the particle swarm algorithm and used the separation index method to continuously adjust the range of constraints in the multi-objective problem. They then selected the optimal machining energy and machining cost, the model for which was verified using the AD-35 CNC lathe. [PERSON] et al. [11] considered the long cycle time, high energy consumption, and high wear of CNC grinding machines; established a mathematical model with quality cost and energy loss as the optimization objectives; and used the NSGA-II algorithm for optimization. The outcomes demonstrated a 30.2% reduction in carbon emissions and an 18.4% decrease in total cost while ensuring accuracy. Although previous researchers have made some progress in the multi-objective optimization of cutting parameters for CNC lathe, the above optimization algorithms have limitations in terms of optimization ability and efficiency. For example, [PERSON] et al. [8] used a combination of grey correlation analysis and response surface analysis to compare and select the optimal solution within a pre-set limited combination of machining parameters, ignoring other combinations within the constraint interval. The quality of the optimal solution needs to be further improved. The dynamic multi-swarm particle swarm algorithm proposed by [PERSON] et al. [10] mainly adopts the calculation steps of particle swarm optimization, while traditional particle swarm algorithms have low accuracy, are prone to divergence, and the quality of their corresponding search results is not high. Therefore, further research is needed to improve the multi-objective optimization method used to optimize the cutting parameters of CNC lathe. The continuous ant colony optimization algorithm (ACOR) was originally proposed by [PERSON] [12]. It is more suitable for solving continuous function optimization problems than the original ant colony algorithm. The ACOR has been widely used to address engineering problems such as ship longitudinal motion parameter optimization [13] and the design of lightweight crane girders [14] and provides a new solution with which to address processing energy consumption in CNC workshops. However, the ACOR easily falls into a local optimum and cannot easily exit the local optimum region due to its high convergence speed. The black hole optimization algorithm (BH) [15] uses high-quality solutions to guide the optimization direction, can fully search in a limited space, and helps to improve the diversity of solutions. Starting from the disadvantages and the limits of the above algorithms, this paper deals with a substantial improvement, consisting of the combination of the black hole and ant colony algorithms. The new algorithm, suggested to be named BK-ACO\({}_{R}\), is able to overcome the limits (mainly relating to falling into local optima) of the algorithms used before and, at the same time, is suited for the multi-objective optimization of cutting energy consumption and material surface roughness while machining on a CNC lathe.