To improve the accuracy and efficiency of multiobjective design optimization for a multicomponent system with complex nonuniform loads, an efficient surrogate model (the decomposed collaborative optimized Kriging model, DCOKM) and an accurate optimal algorithm (the dynamic multiobjective genetic algorithm, DMOGA) are presented in this study. Furthermore, by combining DCOKM and DMOGA, the corresponding multiobjective design optimization framework for the multicomponent system is developed. The multiobjective optimization design of the carrier roller system is considered as a study case to verify the developed approach with respect to multidirectional nonuniform loads. We find that the total standard deviation of three carrier rollers is reduced by 92%, where the loading distribution is more uniform after optimization. This study then compares surrogate models (response surface model, Kriging model, OKM, and DCOKM) and optimal algorithms (neighbourhood cultivation genetic algorithm, nondominated sorting genetic algorithm, archive microgenetic algorithm, and DMOGA). The comparison results demonstrate that the proposed multiobjective design optimization framework is demonstrated to hold advantages in efficiency and accuracy for multiobjective optimization.

Multicomponent system is defined as the complex mechanism system comprising a plurality of rigid and flexible components, which is an indispensable part in mechanical equipment, such as excavator and loader [_{i}(_{j} is the _{j}] is the allowable load under _{u} is the _{u} and _{u} are the upper and bottom boundaries of the parameter

To solve the MODO model shown in (

The first technique is to establish an efficient surrogate model to approximate the multiobjective and multiconstraint. Only needing a small amount of black-box function calls, the surrogate model can be established and is promising to reduce simulation cost [

The second technique is to develop an accurate multiobjective optimal algorithm to resolve the complex MODO model. Due to the complex interaction effects between various objective functions in the multicomponent system, the global optimal solution of the MODO model is almost impossible to acquire at the same time. Therefore, an accurate optimal algorithm, which can precisely search for Pareto optimal frontier and acquire nondominated solutions, is urgently required. At present, a variety of multiobjective optimization algorithms are developed, such as multi-island genetic algorithm (MIGA) [

The objective of this paper is to develop an efficient and accurate multiobjective design optimization framework for the multicomponent system with nonuniform loads. By fusing the proposed surrogate model (i.e., DCOKM) and optimal algorithm (i.e., DMOGA), the urgently needed multiobjective design optimization framework is established. The innovation of this paper lies in that the corresponding efficient methods are firstly proposed for the multiobjective optimization design of the multicomponent system with nonuniform loading. Regarding the multiobjective design optimization of the carrier roller system as study case, the effectiveness of the presented framework is validated. In what follows, Section

By integrating the Kriging model (KM) with the best unbiasedness ability and the improved particle swarm optimization (IPSO) with global search capability [_{1}, _{2}, …, _{d}] indicates the input variable; ^{T}(^{T}(_{1}(_{2}(_{m}(_{1}, _{2}, …, _{m}] is the regression coefficient; ^{2}). Herein, the covariance measure of _{i} and _{j} denote the _{i} and _{j}, respectively; _{k} is the

Moreover, the optimal correlation parameter

Then, the corresponding regression coefficient ^{2} of OKM can be estimated by

Clearly, the approximation accuracy of OKM mostly depends on the correlation parameter

Searching algorithm is a key factor to find the optimal solution

We assume that the particle position is assigned as the correlation parameter _{i} is the current particle velocity; _{i} is the current particle position; _{i} is the current individual extremum; _{1} and _{2} are the individual and population leaning factors, respectively; _{1} and _{2} are the random numbers during time domain [0,1]; _{max} is the maximum inertia weight; _{min} is the minimum inertia weight; _{avg} is the average fitness value; _{min} is the minimum fitness value; _{1s} and _{2s} are the initial individual learning factor and initial population learning factor, respectively; and _{1n} and _{2n} are the individual learning factor and population learning factor in the largest iterations, respectively.

With finite iteration times of the improved PSO algorithm, the specified searching precision is met and the optimal correlation parameters _{1}, _{2}, …, _{n}).

With the predicted response

Noticeably, by combining the nonlinear approximation ability of KM and the powerful global searching ability of the improved PSO algorithm, the presented OKM holds the potential to improve approximate efficiency and accuracy. The flowchart of OKM is drawn in Figure

Flowchart of OKM. The establishment process of the OKM model is described.

Multiobjective design optimization of the multicomponent system refers to multiobjective (i.e., the force standard deviation in the first component _{F1}, in the third component _{F3}, and in all rollers _{MF}, etc.) and multiconstraint (i.e., the sum of mean force in all rollers _{MF} and the sum of maximum force in all rollers _{Fmax}, etc.), which is difficult to perform the nonlinear approximation with high efficiency and acceptable accuracy. To address the multiobjective and multiconstraint (MOMC) problem, we develop a decomposed collaborative optimized Kriging model (DCOKM) with respect to the decomposed collaborative strategy and optimized Kriging model, to further address the strong nonlinearity issues and enhance approximation efficacy. For the MOMC approximation with the presented DCOKM, an entire complex MOMC system with all input variables and output responses is decomposed into many single-objective single-constraint subsystems, in which each submodel has few input variables and output responses. Considering Latin hypercube sampling technique and OKM, the decomposed OKM for many single-objective single-constraints is established. Subsequently, considering collaborative sampling technique and decomposed OKM, the DCOKM is mathematically constructed. The mathematical modeling process of DCOKM is summarized.

Assuming that the multiobjective and multiconstraint problem involves ^{(t)} indicates the ^{(t)} is

With a given predicted point

Equation (^{(t)}, and

The resolving algorithm is a critical factor for solving the MODO model and obtaining the optimal solutions. NSGA-II is a vital nondominated fast sorting algorithm in multiobjective optimal design, which can effectively control the population distribution and reduce the resolving complexity. However, due to randomness and uncertain fluctuation effects in simulated binary crossover and polynomial mutation operator, its convergence precision is still unacceptable for the complex MODO model. To tackle with this problem, by designing the arithmetic crossover operator and Poisson mutation operator, we propose a dynamic multiobjective genetic algorithm (DMOGA) based on the traditional NSGA-II. The objective of the DMOGA is to reduce the uncertain fluctuation effects of operators and gain ideal Pareto optimal frontier. The design thought of the arithmetic crossover operator and Poisson mutation operator is presented as follows.

To improve the population diversity and avoid over propagation of excellent solutions, an arithmetic crossover operator is presented by combining the information of nondominated sorting levels. The arithmetic crossover operator can keep the high-ranking individuals in parent generation and increase the low-ranking individuals in offspring population. Therefore, the quality of offspring population and the diversity of whole population shall be improved. For two given parent vectors _{1}, _{2}, …, _{m}) and _{i} and _{i} is the random number in [0,1], which is expanded as_{r} and _{r} indicate the nondominated sorting level of individual _{d} and _{d} are the crowding distance of individual

To change the disadvantage of low convergence speed caused by polynomial mutation parameters, a new Poisson distribution mutation operator is proposed, which can overstep the local optimum and thereby is conducive to search for global solutions. Assuming that the individual before mutation is expressed as _{1}, _{2}, …, _{n}) and the individual after compilation is expressed as _{i} is the _{k} is the Poisson random number, which is introduced as

To obtain the Pareto optimal frontier and acquire nondominated solutions, the DMOGA algorithm is proposed by designing the arithmetic crossover operator and Poisson mutation operator. The essential process of DMOGA algorithm is summarized as follows:

Generate initial _{n}, and set iteration times

Evaluate fitness values of all individuals, and rank them with Pareto dominance and crowding distances

Select individuals from _{n} by the binary tournament method

Obtain the child _{n} by the arithmetic crossover operator and Poisson mutation operator

Generate population _{n} by merging parent and child individuals, evaluate fitness value of all individuals, and perform fast nondominated sorting of _{n}

Calculate crowding degree and crowding distance, and select individuals to form the new species group _{n+1}

Terminate the algorithm when the accuracy requirement is met; otherwise, the algorithm will back to (2)

Acquire Pareto optimal surface and global optimal solution

By absorbing the global search capability of the arithmetic crossover operator and fast convergence ability of the Poisson mutation operator, the proposed DMOGA can accurately solve the multiobjective design optimization model and quickly obtain the Pareto solution set. To achieve the tradeoff effect between multiple objectives and optimal solution, a Utopia-Pareto directing adaptive (UPDA) search scheme is adopted by capturing plenty of characteristics and utilizing the ordering information of Pareto solutions [

For the multicomponent system optimal design problem, we develop a multiobjective design optimization framework based on a surrogate model (i.e., DCOKM) to approximate multiple responses and an optimal algorithm (i.e., DMOGA) to acquire optimal solution set. Herein, the improved PSO algorithm and Kriging model are first absorbed into decomposed collaborative strategy, to enhance the approximation efficacy of the surrogate model; then, the arithmetic crossover operator and Poisson mutation operator are designed, to acquire the efficient and accurate convergence of the optimal algorithm. Therefore, by combining the DCOKM and DMOGA, the calculation accuracy and efficiency of the multiobjective design optimization model is promising to be greatly improved. The multiobjective design optimization framework is shown in Figure

Multiobjective design optimization framework.

In this section, a carrier roller system from the track driving device of the excavator is selected as an engineering case to verify the effectiveness of the presented multiobjective design optimization framework. It should be noted that all computations are performed on an Inter(R) Core (TM) Desktop Computer (i7-9700K CPU 3.6 GHz and 16 GB RAM).

A typical carrier roller system mainly includes three components, and each component endures multidirectional nonuniform loading. The change of structural size for one component will cause the change of the contact force between the whole track and the roller, and then influence the load of other components, which will lead to the load change among the multiple component systems. The load transfer path of the carrier roller system is shown in Figure _{F1}, in the third component _{F3}, and in all rollers _{MF} (i.e., the sum of maximum force) appears significant fluctuations, while the sum of mean force in all rollers _{MF} and the sum of maximum force in all rollers _{Fmax} show little variations.

Load transfer path of the multicomponent system.

Load analysis of the first carrier roller: (a) forward force

Load distribution of forward force

Load distribution of radial force

Load distribution of axial force

Load distribution of resultant force FM of the carrier roller system: (a) simulation history and (b) boxplot curve.

Load distribution of standard deviation

Load distribution of standard deviation

Load distribution of standard deviation

Load fluctuation with different stiffness value. (a) Force standard deviations. (b) Sum of mean force and maximum force.

Therefore, by minimizing the responses of _{F1}, _{F3}, and _{MF}, the nonuniform appearance would be reduced and the load distribution will become more uniform. Therefore, to minimize the nonuniform loading of the multicomponent system, by regarding the force standard deviations (i.e., _{F1}, _{F3}, and _{MF}) as optimal objectives, force mean values (i.e., _{MF} and _{Fmax}) and other constraints as constraint functions, and the structural sizes (i.e., height of first component _{1}, height of third component _{3}, and displacement condition _{2}) as design variables, the multiobjective design optimization model is established as illustrated in the following equation:_{MF}] and [_{Fmax}] indicate the allowable sum of mean force and allowable sum of maximum force in all rollers, respectively. In this study, [_{MF}] and [_{Fmax}] are set as 59013 N and 131842 N, respectively.

To obtain the objective responses and constraint function values of the multiobjective design optimization model, the kinematic and dynamic equations are solved as follows.

Based on Lagrange multiplier modeling technique [_{1}, _{2}, and _{3}; _{1}, _{2}, and _{3};

Subsequently, by solving the constraint equation with the gear prediction-correction algorithm, the velocity and acceleration at time _{n} can be obtained as_{k} and _{l} are the generalized coordinates of the

Consequently, by solving the Lagrange multiplier equation, the constraint and response reaction force in (

Obviously, it is time-consuming to solving the nonlinear dynamic equations for thousands of times. Therefore, to build the surrogate model of the above dynamic equations, a handful of the input variables (_{1}, _{2}, and _{3}) are extracted by Latin hypercube sampling, and the corresponding output responses (_{F1}, _{F3}, _{MF}, _{MF}, and _{Fmax}) are obtained by running the above dynamic equations. The extracted samples would be considered as training samples for surrogate modeling, which are shown in Table

Extracted training samples.

Sample | _{1} | _{2} | _{3} | _{F1} | _{F3} | _{MF} | _{MF} | _{Fmax} |
---|---|---|---|---|---|---|---|---|

1 | 390 | 39 | 446 | 5478.36 | 6088.866 | 2096.75 | 34708.48 | 70669.73 |

2 | 433 | 106 | 462 | 6752.66 | 6670.62 | 4389.78 | 49454.32 | 93768.56 |

3 | 409 | 48 | 406 | 4213.39 | 4146.539 | 4882.84 | 34017.66 | 63289.52 |

4 | 418 | 82 | 453 | 5718.21 | 5777.563 | 1403.27 | 43320.44 | 85526.51 |

5 | 393 | 86 | 459 | 6297.80 | 6242.555 | 2301.85 | 41561.77 | 89275.85 |

6 | 449 | 91 | 448 | 5431.01 | 5322.861 | 5860.59 | 47852.8 | 89216.56 |

7 | 424 | 140 | 451 | 8671.58 | 7605.778 | 2878.48 | 51373.97 | 113861.80 |

8 | 473 | 108 | 488 | 7110.48 | 6871.813 | 12331.36 | 57903.74 | 112841.70 |

9 | 485 | 76 | 470 | 6382.03 | 6362.146 | 11143.59 | 52054.45 | 93507.65 |

10 | 457 | 128 | 409 | 4517.62 | 5344.153 | 10078.74 | 49212.41 | 112440.51 |

11 | 451 | 23 | 432 | 4282.40 | 5153.78 | 4734.30 | 37107.03 | 76721.86 |

12 | 423 | 55 | 443 | 6500.60 | 5537.399 | 1825.19 | 39707.02 | 95494.29 |

13 | 481 | 123 | 400 | 3658.48 | 4148.80 | 13708.95 | 51039.61 | 99463.75 |

14 | 458 | 98 | 403 | 3675.75 | 3830.37 | 9571.45 | 44760.41 | 85159.23 |

15 | 441 | 60 | 485 | 6235.71 | 5748.82 | 6150.34 | 46056.87 | 111631 |

16 | 479 | 136 | 450 | 8958.53 | 8488.76 | 12654.48 | 58537.90 | 119643 |

17 | 472 | 31 | 395 | 2550.09 | 3079.38 | 8237.67 | 36571.72 | 65595.75 |

18 | 462 | 69 | 464 | 6185.57 | 6892.61 | 7400.77 | 47623.44 | 105196.80 |

19 | 429 | 90 | 434 | 5715.60 | 6410.48 | 3553.80 | 43720.39 | 85185.63 |

20 | 426 | 47 | 468 | 5295.95 | 5673.30 | 2924.82 | 40827.15 | 77965.02 |

21 | 414 | 107 | 392 | 2459.88 | 3156.21 | 8467.26 | 39682.11 | 78960.68 |

22 | 413 | 30 | 452 | 4183.20 | 4876.16 | 520.430 | 36114.36 | 63734.86 |

23 | 456 | 56 | 397 | 4436.30 | 4444.57 | 7942.65 | 38563.35 | 83004.92 |

24 | 428 | 103 | 487 | 6741.81 | 7158.78 | 6310.21 | 51060.07 | 95940.30 |

25 | 477 | 24 | 465 | 4936.89 | 6018.77 | 7260.10 | 41893.87 | 72833.50 |

26 | 399 | 129 | 472 | 9084.01 | 6987.26 | 3386.82 | 49349.60 | 117109.30 |

27 | 475 | 26 | 440 | 4903.20 | 6094.30 | 6697.05 | 39920.45 | 67536.79 |

28 | 430 | 97 | 410 | 4272.04 | 5065.08 | 6294.49 | 42083.32 | 87140.54 |

29 | 431 | 57 | 398 | 3964.19 | 4139.24 | 6453.212 | 36342.50 | 70549.19 |

30 | 480 | 63 | 404 | 5216.05 | 6164.22 | 10156.87 | 42814.12 | 120803.00 |

By importing the generated samples into the presented DCOKM method, we perform the surrogate modeling for multiobjective and multiconstraint. Considering the improved PSO algorithm, the optimal correlation parameter _{max} = 0.9, minimum inertia weight _{min} = 0.4, initial individual learning factor _{1s} = 2.5, individual learning factor in the largest iterations _{1n} = 0.5, initial population learning factor _{2s} = 0.5, and population learning factor in the largest iterations _{2n} = 2.5, the training process of DCOKM is shown in Figure

Optimal correlation parameter of DCOKM.

Response | Related parameters | ||
---|---|---|---|

_{1} | _{2} | _{3} | |

_{F1} | 38.83488 | 40.38819 | 42.71815 |

_{F3} | 3.086433 | 3.201069 | 3.394138 |

_{MֿF} | 48.52073 | 50.46128 | 53.37277 |

_{MF} | 49.97663 | 51.97569 | 54.97429 |

_{Fmax} | 49.98934 | 51.98891 | 54.98827 |

Training process of DCOKM.

Absolute errors of three objectives: (a) _{F1.}, (b) _{F3.}, and (c) _{MֿF.}

Absolute errors of two constraints: (a) _{MF} and (b) _{Fmax}.

By employing the built DCOKM to approximate the multiobjective and multiconstraint, the DMOGA is executed to solve the established MODO model shown in (_{MF} and _{F1}, and _{MF} and _{F3} both show negative correlation and _{F1} and _{F3} show positive correlation. To achieve the reasonable balance between three objectives, we sort _{F1} and _{F3} as ascending order and _{MF} as descending order. Then, the middle point in the Pareto front curve is selected as the optimal solution of multiobjective optimization [_{1}, _{2}, and _{3} corresponding to the optimal solution are chosen as 412, 10, and 459, respectively.

Pareto optimal frontier of MODO.

To validate the optimal results, we compare the optimal parameters before and after optimization in Table _{MF} decreased by 92%. To check the loading uniformity after optimization, we further compare the loading distribution before and after optimization in Figures

Comparison of results before and after optimization.

Optimization index | Before optimization | After optimization | Reduce the proportion (%) | Remark |
---|---|---|---|---|

_{1} | 467 | 412 | — | — |

_{2} | 40 | 10 | — | — |

_{3} | 467 | 459 | — | — |

_{F1} | 8735 | 2904 | 67 | Reduce track run out and impact load |

_{F3} | 8156 | 4382 | 46 | Reduce track run out and impact load |

_{MF} | 10987 | 924 | 92 | Better load uniformity |

_{MF} | 59013 | 33924 | 55 | Reduce wear failure rate |

_{Fmax} | 131842 | 57477 | 43 | Reduce fracture failure rate |

Simulation history of three carrier rollers before and after optimization: (a) before optimization and (b) after optimization.

Resultant force of three carrier rollers before and after optimization: (a) before optimization and (b) after optimization.

Because of the particularity of the study case (practical engineering problem), there is no standard solution for reference. To compare the effectiveness of the proposed method, some advanced surrogate models and optimization algorithms are selected to perform the multiobjective design of the carrier roller system and then compared with the presented model (DCOKM) and algorithm (DMOGA). Herein, to support the feasibility of the proposed surrogate model (i.e., DCOKM), the multiobjective and multiconstraint approximation of the carrier roller system MODO is also studied based on the modeling methods of the direct Monte Carlo simulation (MCS), response surface (RS) model, KM, OKM, and the presented DCOKM; Furtherly, to validate the effectiveness of the developed optimal algorithm (i.e., DMOGA), the multiobjective design optimization of the carrier roller system is also performed based on NCGA algorithm, AMGA algorithm, NSGA-II algorithm, and DMOGA algorithm. Note that to ensure the rationality of algorithm comparison, all of optimal algorithms are using DCOKM to approximate the objectives and constraints in optimal algorithm comparisons. The approximation efficacy of different models is listed in Table

Approximation efficacy of different models.

Methods | Fitting surrogate models | Testing surrogate models | Precision | ||
---|---|---|---|---|---|

Fitting number | Fitting time (s) | Testing number | Testing time (s) | ||

MCS | — | — | — | — | 1 |

RS | 245 | 2.16 × 10^{7} | 200 | 15.623 | 0.9451 |

KM | 180 | 1.55 × 10^{7} | 200 | 13.532 | 0.9792 |

OKM | 130 | 1.12 × 10^{7} | 200 | 11.367 | 0.9875 |

DCOKM | 90 | 7.78 × 10^{6} | 200 | 9.2476 | 0.9952 |

Computing efficiency and accuracy of different algorithms.

Methods | Before optimization (×10^{4} N) | After optimization (×10^{4} N) | Reduction | Time (s) | ||||||
---|---|---|---|---|---|---|---|---|---|---|

_{1} | _{2} | _{3} | _{1} | _{2} | _{3} | _{1} | _{2} | _{3} | ||

NCGA | 0.8735 | 0.8156 | 1.0987 | 0.3332 | 0.4620 | 0.1023 | 0.61 | 0.43 | 0.90 | 18.23 |

AMGA | 0.8735 | 0.8156 | 1.0987 | 0.3146 | 0.4657 | 0.1275 | 0.64 | 0.43 | 0.88 | 16.52 |

NSGA-II | 0.8735 | 0.8156 | 1.0987 | 0.3065 | 0.4640 | 0.1112 | 0.65 | 0.43 | 0.89 | 15.72 |

DMOGA | 0.8735 | 0.8156 | 1.0987 | 0.2904 | 0.4382 | 0.0924 | 0.67 | 0.46 | 0.92 | 13.31 |

_{1} indicates the _{F1}, _{2} indicates the _{F3}, and _{3} indicates the _{MF}.

As illustrated in Table

As revealed in Table

Therefore, with the integration of the surrogate model (i.e., DCOKM) and optimal algorithm (i.e., DMOGA), the proposed multiobjective design optimization framework is validated to improve optimal efficiency while maintaining acceptable optimal accuracy and thereby is an effective way for the multiobjective optimization design of the multicomponent system.

The purpose of this study is to develop an efficiently and accurately multiobjective optimization design framework for the multicomponent system. Firstly, to efficiently approximate the multiobjective and multiconstraint, a surrogate model named as the decomposed collaborative optimized Kriging model (DCOKM) is proposed, by absorbing the strengths of the improved PSO algorithm and Kriging model into decomposed collaborative strategy. Then, to accurately solve the multiobjective design optimization model, an optimal algorithm named as the dynamic multiobjective genetic algorithm (DMOGA) is developed, by designing the arithmetic crossover operator and Poisson mutation operator. Regarding the carrier roller system MODO as a study case, the validity and feasibility of multiobjective optimization design framework is proved. Some conclusions are summarized as follows:

From the multiobjective design optimization of the carrier roller system, we discover that the total standard deviation of three carrier rollers is reduced by 92 %, and loading distribution is more uniform after optimization

DCOKM has higher accuracy and efficiency than the ordinary Kriging model and optimized Kriging model and is an effective surrogate model for the multiobjective design optimization

The optimal solutions of DMOGA are better than the NSGA-II algorithm due to the superior properties of the arithmetic crossover operator and Poisson mutation operator

The comparison of methods reveals that the proposed multiobjective optimization design framework (i.e., DCOKM and DMOGA) possesses high-efficiency and high-accuracy in multiobjective optimal design of the multicomponent system

All data generated or analyzed during this study are included in this published article.

The authors declare that there are no conflicts of interest regarding the publication of this article.

This paper was cosupported by the National Natural Science Foundation of China (grant nos. 51975028 and 51575024).