# Application of probabilistic multi-objective optimization in material milling

In the PMOO, a new concept of “preferable probability” is introduced, and the objectives are preliminarily divided into two basic types, i.e., beneficial and unbeneficial type. The total preferable probability is the product of all partial preferable probabilities, which is the decisive indicator.
Published in Mathematics

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At present, there are some methods for multi-objective optimization, such as simple additive weighting (SAW),  weighted aggregation and product evaluation (WASPAS), preference method similar to ideal solution (TOPSIS), VIKOR method (VISEKriterijumska optimizacija i kom promisno Resenje), analytic hierarchy process (AHP), Multi-objective optimization method based on ratio analysis (MORA), compound proportional evaluation (COPRAS), proximity index value (PIV), preference selection index (PSI), preference selection index (PSIe) for determining weight by entropy method, as well as Pareto solution, etc. These methods have been applied in some fields.

However, there are inherent defects in the above methods. For example, in the linear weighting method, if the objective functions f1(x), f2(x), …, fp(x) are "added” with the weight coefficient wj, the objectivity of the selection of the weight coefficient, the “virtual ideal point” and other human factors, the need for “normalization” when the dimensions of the objectives are different, and the rationality of the normalized denominator are all unknown. In fact, the “additive” is a union in set theory, and it is the “sum” of events in probability theory instead of the intrinsic intention of “simultaneous optimization”; Pareto solution can only give a set of solutions instead of an exact solution.

Therefore, it can be seen that the previous methods of multi-objective (attribute) optimization is not promised. In fact, the intrinsic intention of multi-objective optimization is the “simultaneously optimization” of multiple objectives. The multiple objectives in the optimization are within a system actually, so the optimization of these multiple objectives is actually the optimization of a system, which is the integral optimization of the system.

From the perspective of probability theory, it is the “product” of the probability of each objective (attribute) and the “intersection” of each objective in set theory. Therefore, the probability theory method can be adopted to deal with the issue of optimization problem with multiple objectives.

In this paper, material milling is taken as example, orthogonal (Taguchi) design with twenty-seven experiments is employed, in which the number of insert, cutting piece materials and tool tip radius, cutting speed, feed speed and cutting depth are taken as input variables; Minimizing surface roughness (SR) and maximizing material removal rate (MRR) of the workpiece are taken as the target responses simultaneously. Finally, the optimum status corresponding to the highest total preferable probability is obtained in viewpoint of system theory.

Continuous Optimization
Mathematics and Computing > Mathematics > Optimization > Continuous Optimization