Robust Optimization of Spring Design under Variable Manufacturing Tolerances

Grzegorz Sroka; Slawomir T. Wierzchon

cs.NEOct 3, 2026

Robust Optimization of Spring Design under Variable Manufacturing Tolerances

Authors: Grzegorz Sroka, Slawomir T. Wierzchon

Organizations: Department of Analysis Nonlinear, Rzeszów University of Technology, Powstańców Warszawy 12, 35-959 Rzeszów, Poland · Institute of Computer Science, Polish Academy of Sciences, ul. Jana Kazimierza 5, 01-248 Warsaw, Poland

Abstract

A reliable assessment of an optimizer robustness requires tracking controlled changes in parameters to identify sources of algorithmic weakness. Such an approach was applied to the problem of designing tension/compression springs with three continuous variables describing the spring geometry. Three additional independent binary parameters reflect material variability, manufacturing-related geometric deviations, and an additional safety margin. Without expanding the decision space, these parameters can be combined in eight ways to generate variants that allow for direct comparison. In this study we distinguish between robustness effects and the dimension of the problem, which allows tracking changes in feasibility while keeping the nominal mass objective constant. Seven optimization methods and eight hybridized their variants are used for the analysis. The hybridization mechanism applies the operator selected by the prediction procedure presented in Section 3. All methods are evaluated under the same experimental conditions. Each method is run thirty times for each variant, with a maximum of 30000 function evaluations permitted per run. The final quality assessment takes into account the geometry of the feasibility region, search trajectories, performance variations across landscapes and the computational cost of fixed objectives. Together, these factors allow us to determine the quality of the solution, reproducibility, feasibility control, and the effectiveness of optimization. An interesting observation was that hybridization can mitigate cost or feasibility losses associated with a specific configuration. While it compensates for a specific weakness in the underlying mechanism, it does not provide universal resilience.

Figures & tables

Configuration	$M$	$P$	$R$	Comparison purpose
$C(000)$	0	0	0	Nominal TCSD; all $MPR$ mechanisms disabled
$C(100)$	1	0	0	Material variability only
$C(010)$	0	1	0	Geometric perturbation only
$C(001)$	0	0	1	Requirements margin only
$C(110)$	1	1	0	Material and geometric effects
$C(101)$	1	0	1	Material and requirements effects

Table 1: Configurations of the proposed

MPR

-parameterized TCSD benchmark.

No.	Algorithm	Hyperparameter setting
	CMA-ES [ 27 ]	$\lambda=20$ , $\mu=10$ , $\sigma^{(0)}=0.3(U-L)$ ( $L/U$ : lower/upper bounds).
	DE [ 65 ] /hDE	$DE/rand/1/bin:$ $NP=20$ , $F=0.5$ , $CR=0.9$ .
	L-SHADE [ 68 ] /hL-SHADE	$N_{\mathrm{init}}=18\cdot dim$ , $N_{\min}=4$ , $r_{\mathrm{arc}}=2.6$ , $p=0.11$ , $H=6$ .
	MPA [ 20 ] /hMPA	$P=0.5$ , $\mathrm{FADs}=0.2$ , $\alpha=1.5$ ; 3-equal search phases.
	SPSO2011 [ 13 ] /hSPSO2011	$w=1/(2\ln 2)$ , $c_{1}=c_{2}=0.5+\ln 2$ , $K=3$ .
	WOA [ 44 ] /hWOA	$a:2\rightarrow 0$ ; random search updates.

Table 2: Hyperparameter settings of the investigated base algorithms.

Figure 1: Objective-violation geometry across the MPR configurations.

Panel of Figure 3 — Figure 3: Fixed-target empirical runtime distributions across eight $MPR$ configurations for $\Delta f$ .

Figure 4: Aggregated empirical runtime distributions over the complete set of $408$ $MPR$ configuration-target evaluation units.

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