A probabilistic multi-objective optimization mechanical design

Optimization design of machinery is usually a multi-objective one inevitably. At present, the popular mechanical optimization design is limited by the intrinsic shortcomings of the previous multi-objective optimization methods, which leads to the difficulty of non-comprehensive and nonsystematic optimal solutions in the viewpoint of probability theory. In the linear weighting “additive” method, there is inherent problems of normalization and introduction of subjective factors, and the final results depend on the normalization method to a great extent; the Pareto solution set is a “set” instead of an exact solution. In this paper, the probability—based multi-objective optimization, discretization with uniform design and sequential optimization are combined to establish a new approach of multi-objective optimization mechanical design based on probability theory; the probabilistic multi-objective optimization is used to transform the multi-objective optimization problem into single-objective optimization one from the perspective of probability theory; the discretization by means of uniform design provides an effective sampling to simplify the mathematical processing, which is especially important for dealing with multi-objective optimization problems with continuous objective functions; the sequential optimization algorithm is used to conduct the successive deep optimization. Furthermore, the implementation steps are illustrated with two examples. The results show that the approach can not only give excellent optimization results, but also provide a relatively simple processing.


Introduction
In mechanical optimization design, there are many objectives and several input variables usually. The sub-objectives of the optimization are often contradictory, so it makes the mechanical optimization design a multi-objective optimization problem in general. Optimization design of machinery in general is to take the performance function, strength and economy of mechanical system as optimization objectives (attributes) and aims to let the objective functions reach to their optimal value (state) at a set of appropriate design variables, which is under condition of the shape and geometric dimension of mechanical products as constraints. In some cases the purpose of mechanical optimization design is to reduce the volume (weight) of mechanical equipment, the material consumption, the manufacturing costs, and to be green and environmentally friendly on one hand, but also to improve the quality and working performance of products.
At present, the popular optimization design of machinery is limited by the inherent shortcomings of the common multi-objective optimization methods, therefore it is difficult to achieve comprehensive and systematic optimal results in the viewpoint of probability theory. You et al. optimized the worm gear transmission mechanism by taking the minimum volume of the worm gear crown and the minimum relative sliding speed of the tooth surface simultaneously as the goals [1], which is used in the lifting system of large-scale vacuum injection molding machine, they obtained the "satisfactory solution" by linear weighting "additive" method. Zhong et al. aimed at the optimizing wall-climbing robot design by taking the minimizing mass and displacement and maximum stress as the optimization objectives [2], they adopted response surface method and Pareto solution set to conduct their optimization, and at last they selected the "final solution". However, in the linear weighting "additive" method, there is intrinsic problem of normalization and introduction of subjective factors, and the final result depends on the normalization method to a great extent [3][4][5]. Different normalization methods may lead to completely different results. In addition, in some algorithms, the beneficial performance index and the unbeneficial performance index are treated unequally.
From the point of view of set theory, the "additive" algorithm in previous methods for multi-objective optimization corresponds to the form of "union". Therefore, the above algorithm can only be regarded as a semiquantitative method in a sense. The Pareto solution set is a "set" instead of an exact solution.
Based on above analysis, it can be seen that an appropriate scheme for optimization design of machinery is still in need. In this paper it combines the probabilistic multiobjective optimization, the uniform design, the sequential optimization together to propose a new approach of multiobjective optimization design of machinery on basis of probability theory first; furthermore, the design of worm gear transmission mechanism and the design of wall-climbing robot are taken as examples to resolve them again, and comparative analysis of the relevant results is performed.

Probability-based multi-objective mechanical optimization design scheme
In this section, the probability theory, the discretization with uniform design and the sequential optimization are organically combined, which establishes a reasonable scheme for solving multi-objective optimization mechanical design problem. The probabilistic multiobjective optimization (PMOO) is used to transform the multi-objective optimization problem into singleobjective optimization problem from the perspective of probability theory; the discretization by means of uniform design provides an effective sampling method to simplify the mathematical processing, which is especially important for dealing with multi-objective optimization problems with continuous objective functions; the sequential optimization algorithm is used for successive deep optimization. The systematic implementation is the integral combination of the procedures in Sects. 2.1 and 2.2.

Approach based on the perspective of probability theory
Preliminarily, the probabilistic multi-objective optimization is used to transform the multi-objective optimization problem into single-objective optimization one from the viewpoint of probability theory. From the perspective of probability theory, the whole event of "multi-objective simultaneous optimization" corresponds to the product of each single objective (event). As to the problem of multi-objective optimization design of machinery, each objective (attribute) is actually a goal of PMOO [3][4][5]. Furthermore, according to the role and preference of performance of a candidate in the optimization [3][4][5], all the performance and utility indexes of the candidate are preliminarily divided into two categories, i.e., beneficial type and unbeneficial type. Specifically, the evaluation of the partial preferable probability P ij of beneficial indicators and unbeneficial indicators can be carried out according to the procedure of Fig. 1 [3][4][5].
In Fig. 1, P ij expresses the partial preferable probability of the j-th performance utility indicator of the i-th alternative scenario, X ij ; n indicates the total number of the alternative scheme; m indicates the total number of the performance (objective); x i reflects the arithmetic value of the j-th Fig. 1 Procedure of the evaluation of probabilistic multi-objective optimization performance utility indicator; X jmax and X jmin represent the maximum and minimum values of the j-th performance utility indicator, respectively; α j and β j show the normalized factors of the j-th performance utility indicator X ij in beneficial status and unbeneficial status, individually; the determination of the type of beneficial status or unbeneficial status of the j-th performance utility indicator X ij depends on its specific rule or preference of the index in the instant problem; P i reflects the total (overall) preferable probability of the i-th alternative scheme.

Discretization by means of uniform design and sequential optimization
Successively, the discretization could provide an effective sampling to simplify the mathematical processing by means of uniform design.
In multi-objective optimization mechanical design problems, the objective function is continuous sometimes, it needs to conduct a discretization so as to simplify the mathematical processing. The discretization can be performed by using uniform design [6][7][8]. The methods of uniform experimental design (UED) and good lattice point (GLP) could provide a set of discrete point in the super space of input variables, which could be used to conduct approximate assessment of definite integral and maximum problems of a function through a limited number of sampling points [6][7][8]. Such limited number of sampling points are uniformly distributed in the integration domain with low discrepancy [6][7][8]. The characteristic of uniform point set makes its convergence speed much faster than that of Monte Carlo sampling method [6][7][8], so it is a reasonable approximate calculation algorithm -"quasi-Monte Carlo method". In order to use this uniformly distributed point set properly, Fang specially developed a series of uniform design tables to arrange the position of "point" in the super space [8,9]. For multi-objective optimization problems, sequential optimization algorithm (SNTO) can be further used for successive deep optimization [10,11]. Finally, through integral combination of the probability-based multiobjective optimization, the discretization by means of uniform experimental design, and the sequential optimization, a scheme for solving multi-objective optimization mechanical design problem can be set up rationally.

Solving the problem of mechanical optimization design
Multi-objective optimization mechanical design has practical important significance in modern technologies [12][13][14][15]. In this section, two examples are given to illustrate the procedures of the implementation steps of scheme for solving multi-objective optimization mechanical design problem in the following sections.

Design of worm gear transmission mechanism
You et al. conducted a research of worm gear transmission mechanism of lifting system of large vacuum injection molding machine [1], the target indexes include the minimum volume (V) of worm gear crown and the minimum relative sliding speed (v s ) of tooth surface; while the worm end face modulus m, the worm diameter coefficient q and the worm head number z 1 were taken as independent design variables, and the weight linear "additive" algorithm was used for optimization. After modeling and analysis, the objective functions f 1 and f 2 are obtained, The constraints are, Because it involves three input variables, m, q and z 1 , at least 19 discrete points are needed for the simplifying treatment by using the discretization method as suggested [ Table 2 gives the values of functions f 1 and f 2 at discrete points, as well as the evaluation results of partial preferable probability and total preferable probability at discrete points and ranking. As can be seen from Table 2, the 11th sampling point has the largest total preferable probability P t , so the optimal result is not far from the 11th sampling point, and it can be used as the first order approximate result of this optimization. According to Table 2, the corresponding f 1 *(z 1 , q, m) = 32,508.11, and f 2 *(z 1 , q, m) = 1.6358, which are obviously better than the results obtained by You et al. with f 1 (z 1 , q, m)  Besides, if the You's weight factors are adopted [1], i.e., w 1 = 0.8333 and w 2 = 0.1667, the evaluation results in Table 3 can be obtained. As can be seen from Table 3, the 11th sampling point still has the largest total preferable probability P t , and the approximate evaluation result of the optimized point remains unchanged.

Design of wall-climbing robot
Zhong et al. designed the wall-climbing robot by taking the minimum mass (M), the minimum deformation (T 1 ) and the minimum maximum stress (T 2 ) as the optimization objectives [2], and adopted response surface method and Pareto solution set to optimize and select the "final solution". The range of input variables x 1 , x 2 and x 3 are, x 1 : 432-528 mm; x 2 : 1 3 54-66 mm; x 3 : 9-11 mm, respectively. Specifically, MOGA multi-objective genetic optimization algorithm (NSGA-H) and the second generation non-gene dominating genetic algorithm are adopted. Its initial population generation used 3000 samples, while each iteration needs 600 samples.
The maximum allowable genetic algebra was 20 generations, and the variation coefficient was set to 0.01 and the cross coefficient was set to 0.98. The maximum allowed Pareto ratio is set to 60%, and the convergence stability is set to 2%. Finally, the first group is selected from the three candidate points, and the optimized mass M is 9.51 kg, the maximum deformation T 1 is 0.53 mm, and the maximum stress T 2 is 6.55 MPa as its "final solution". In the following, the probability-based multi-objective optimization method, the uniform design and sequential optimization are used to re-solve the problem, and furthermore the optimized results are compared with the consequences of Zhong.
In Zhong's experimental design, the combination of parameter sensitivity screening and Latin hypercube test is employed [2], which produced 15 sample points, and the results are summarized in Table 4. Table 5 shows the evaluation results of the preferable probability and total preferable probability of M, T 1 and T 2 values and the ranking of these 15 sample points. As can be seen from Table 5, the 6th sampling point has the largest total preferable probability P t , so the optimization point should be near the sampling point No. 6.
The regressed formulae of total preferable probability P t , M, T 1 and T 2 vs. input variables x 1 , x 2 and x 3 are expressed as, T 1 = 4.837274 − 0.02588x 1 + 0.024731x 2 + 0.330507x 3 T 2 = −16.7851 − 0.39669x 1 + 1.943064x 2 Near the sampling point No. 6, sequential optimization can be conducted to implement deep optimization. Here, the uniform design table U* 19 (19 7 ) is used for sequential optimization again, and the results are shown in Table 6, the evaluation of P t for each sampling point is conducted with the partial preferable probability of predicted values of M, T 1 and T 2 by using Eq. (5) through Eq. (7) at each discrete sampling point. The P t × 10 4 of zero-th step is 4.4535. Table 6 shows that the maximum total preferable probability P t decreases slowly. If the pre-assigned quantity δ that we set for c (k) =(P t (k−1)−P t (k))/P t (k−1)) is δ = 4%, then this sequential optimization can be terminated at step 4. At this point, the optimized result is at x 1 = 523.0789 mm, x 2 = 54.0211 mm and x 3 = 9.0158 mm, and the corresponding optimization objective values are mass, M* = 9.46 kg, maximum deformation T 1 * = 0.50 mm, and maximum stress T 2 * = 6.14 MPa. These values of M * , T 1 * and T 2 * are calculated by using Eq. (5) through Eq. (7). These results are obviously superior to the "final solution" given by Zhong's, and the optimization process is relatively simple as well.

Conclusion
From above optimization process, it can be concluded that, as to multi-objective mechanical optimization design, the probability-based multi-objective optimization method, discretization of uniform design and sequential optimization could be combined integrally to formulate a new approach of probability-based multi-objective mechanical optimization design. The new approach of probability-based multi-objective mechanical optimization design could not only obtain excellent optimization results, but also make the optimization process simpler. More prosperous achievements of utilizing and testing the newly proposed mechanical optimization design will be expected in the future in more examples. Step k Rang(mm) Optimum "coordinate" Max P t × 10 4 c (k)