Multi-objective Optimization of High Speed Milling Parameters Based on Genetic Algorithm

In order to optimize the high-speed milling parameters and reduce the experimental work in the actual processing, a multi-objective optimization model is established. The improved genetic algorithm NSGA-II combined with penalty function method is used to solve the mathematical model, and the optimal cutting parameters are obtained when the high-speed milling conditions are known, which provides guidance for the process decision-making of high-speed milling.


Introduction
Processing efficiency, processing cost and product quality are the basic standards to evaluate the advantages and disadvantages of CNC processing technology [1][2][3]. In the actual production and processing, the selection of cutting parameters is related to the quality, production cost and processing efficiency of the processed parts, directly. Under the existing processing conditions, according to the requirements of the parts, the most suitable cutting parameters are selected to achieve the best economic index, that is to achieve the multi-objective optimization of cutting parameters [4][5], which can process the parts that meet the requirements faster and better under the same conditions, and can greatly reduce the test workload used to determine the more optimized milling parameters in actual production.
However, the selection of cutting parameters is affected by many and complex variables, so the optimization process is more complex. In the actual mold high-speed milling production, the constraints and optimization objectives are various, belonging to a typical multi-objective multi constraint optimization problem. Therefore, with the help of mathematical modeling knowledge, this paper establishes the multi-objective optimization mathematical model of mold high-speed milling parameters, and selects the appropriate method to solve the mathematical model, so as to obtain the optimal high-speed milling parameters to meet the processing efficiency, quality and cost at the same time.

Multi-objective Optimization Mathematical Model of Milling Quantity
Optimization problem is to search for the optimal solution for some targets in some possible choices. In the actual production process, the optimization objectives are incompatible and cannot be realized simultaneously. However, multi-objective optimization is not to hope that these goals can be achieved at the same time, but to obtain a set of relatively satisfactory solutions for each target by determining the relative importance of these objectives according to the actual situation. Therefore, the optimization of the high-speed milling amount of die can be described by transforming the multiobjective optimization problem.
The three elements of establishing mathematical model are [6]: decision variables, objective functions and constraints.

Optimize Decision Variables
The independent parameters that need to be optimized in optimization problems are called optimization decision variables. When the conditions of high-speed milling are determined, the milling parameters such as milling speed , feed per tooth , axial milling depth and radial milling c v z f p a spacing are the most important factors affecting the realization of high-speed milling. Therefore, in e a the optimization model of high-speed milling parameters, the optimal decision vector is as follows:

Optimization Objective Function
In order to obtain the optimal parameters of high-speed milling, the optimization objective function including optimization decision variables should be established firstly, and the optimal decision variables can be found by solving the maximum or minimum value of the objective function. In this paper, processing time, surface roughness and processing cost are selected as the optimization objectives.

.1. Productivity Objective Function of High Speed
Milling.Productivity is generally reflected by processing time, and cutting time accounts for the main part of processing time. This paper uses cutting time to describe productivity. The cutting time model of flat end mill is [7]:

Objective Function of Surface Roughness in High Speed
Milling.The surface roughness of highspeed milling is the main parameter to measure and evaluate the machined surface quality. In this paper, the surface roughness is used as the index to evaluate the machining quality. The surface roughness model of flat end mill with fillet is as follows: : Objective Function of High Speed Milling Cost. Low cost and high profit is the ultimate goal of machining enterprises, this paper takes the processing cost as one of the optimization goals. Empirical formula for tool life of flat end milling cutter [7]: -correction factor related to cutting condition; -coefficient related to cutting condition;

Optimization Constraints and Functions
In the process of mold high-speed milling, due to the limitations of processing equipment and processing conditions, the range of milling parameters can be selected is limited. Therefore, it is necessary to consider the limitations of these factors on the selection of milling parameters. In this paper, the constraint conditions are as follows: (1)Torque constraint of machine tool spindle : -maximum allowable torque of high speed milling machine.
(2)Power constraint of machine tool spindle : -the total efficiency of the high-speed machine tool ; -the rated power of the machine tool.

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(4) Axial cutting depth constraint :  nZ -Minimum and maximum feed per tooth allowed in high speed milling centers.

Algorithm for Solving Multi-objective Optimization Model of Milling Parameters
Genetic algorithm uses random search for multiple points, which can effectively use historical information to find better offspring, and can be used to solve large and complex optimization problems [8]. Because genetic algorithm itself does not have the ability to deal with constrained functions, the constrained optimization problem must be transformed into an unconstrained optimization problem when genetic algorithm is used to optimize functions with constraints. For this reason, this paper selects the improved genetic algorithm NSGA-II combined with penalty function method [9][10] for constraint processing, and the multi-objective optimization mathematical model of mold high-speed  (1), which can be used as the fitness function of multi-objective genetic algorithm NSGA-II. (1) Among them, constraint 7 is used as the initialization and coding range of optimization variables.

Optimization Example
In order to facilitate comparative analysis, this paper uses the literature data, NSGA-II combined with penalty function method to solve the optimized milling parameters. The specific processing conditions and parameters are as follows [7]

Algorithm Solving Process
Floating point coding method is used to code the four parameters to be optimized in high-speed milling. According to the constraints of constraint 7 and the milling parameters recommended in the tool manual, the initial values are , , respectively. If the penalty parameter is , ,the fitness function expression can be obtained. NSGA-II algorithm is used to solve the mathematical function constrained by penalty function, and the algorithm program combined with an example is implemented in Matlab platform.

Optimization Results and Analysis
In this example, set the population size of the algorithm as 500, the number of iterations as 500, the crossover rate = 0.9, and the mutation rate = 0.02, and optimize the high-speed milling c p m p parameters of the mold. The operation effect is shown in Figure 1.  Figure 1. Optimization effect of milling parameter optimization example It can be seen from the operation in Figure 1 that the optimal fitness value fluctuates greatly at the beginning of iteration. With the evolution of the population, after about 20 iterations, the fitness value of each objective function value changes very little, and the corresponding three objective function values of individuals have converged to a very small region. By weighing the relative proportion of the three objective functions in the finishing stage, a group of optimal parameters can be selected in the area where the fitness of the objective value changes little. The comparison table of milling parameters and optimization objectives is shown in Table 1. 2.73 Compared with the above two sets of optimization target results, it can be concluded that a set of milling parameters obtained by the optimization solution can reduce the processing efficiency (expressed by processing time) by 3.2%, improve the surface roughness by 55.6%, and reduce the processing cost by 37.5%, only because of the reduction of milling feed, the processing efficiency is reduced.
Because the main goal of finishing stage is to improve the processing quality as much as possible, the requirement of processing efficiency is not very high. Therefore, the optimized milling parameters basically achieve the expected optimization goal, which solves the multi-objective optimization selection problem of mold high-speed milling parameters.
Therefore, the multi-objective optimization mathematical model of high-speed milling parameters and its solution algorithm established in this paper can better optimize the high-speed milling parameters and achieve the expected optimization objectives, which also has certain guiding significance for production practice.

Conclusion
In this paper, a multi-objective optimization mathematical model of mold high-speed milling parameters is established, and NSGA-II algorithm is used to solve the mathematical model constrained by penalty function method. Combined with an example, Matlab is used to compile the algorithm program, and the optimization objective function values corresponding to two groups of parameters before and after optimization are compared, which shows that the multi-objective optimization mathematical model and algorithm of mold high-speed milling parameters established in this paper are feasible It provides a new idea for the optimization of high-speed milling parameters under specific processing conditions, and also reduces the experimental workload in actual production.