⚡ electrical engineering

A robust multi-objective approach in additive manufacturing

This paper proposes a robust multi-objective optimization approach using the NSGA-II algorithm to determine the optimal 3D part orientation in fused deposition modeling by balancing support area, build time, surface robustness, and surface quality under uncertainty.

Original authors: Marina Araújo de Matos, Ana Maria A. C. Rocha, Lino A. Costa, Ana I. Pereira

Published 2026-08-19

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Marina Araújo de Matos, Ana Maria A. C. Rocha, Lino A. Costa, Ana I. Pereira

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a factory floor where machines do not cut away material to shape a part, but instead build it up, layer by tiny layer, like a stack of paper that hardens into a solid object. This is additive manufacturing, a technology that has moved from rapid prototyping to producing everything from medical implants to car components. The process works by melting plastic and extruding it through a heated nozzle, tracing out a shape in three dimensions. However, the way a designer places that object on the printer's bed before starting the job is far from trivial. The angle at which the object sits dictates how the machine moves, how much time the job takes, and how much extra material is needed to hold up overhanging sections. If the object is tilted the wrong way, the printer might need to build a temporary scaffold of plastic to support the part, which must be broken off later, often leaving a rough surface. If the angle is poor, the final part might look blocky or take hours longer to print than necessary. Finding the perfect angle is a balancing act, because improving one aspect, like speed, often worsens another, like surface smoothness.

In a recent study, researchers set out to solve this balancing act not just for a single perfect scenario, but for the messy reality of the real world. They focused on a specific challenge: what happens if the printer does not follow the instructions exactly? In a perfect simulation, a machine might rotate a part to exactly 90 degrees, but in a physical factory, a slight vibration or a tiny error in the motor could shift that angle by a few degrees. Most previous methods for finding the best printing angle assumed the machine would be perfect. The team, led by Marina Araújo de Matos and colleagues at the University of Minho and the Polytechnic Institute of Bragança, asked a different question: can we find printing angles that remain good even if the machine is slightly off? They applied a robust multi-objective optimization approach to a 3D model of a fin, a shape often used in engineering studies. Their goal was to find a set of angles that would keep the part's quality high, the printing time low, and the need for support structures minimal, even if the actual angle ended up being slightly different from the planned one.

To do this, the researchers used a computer algorithm inspired by natural evolution to test thousands of different angles. They did not just look for the single best angle; they looked for a group of solutions that offered the best possible trade-offs between four competing goals. The first goal was to minimize the area where the part touched the temporary support structures, because less contact means less damage when the supports are removed. The second was to reduce the total build time, which depends heavily on how tall the object stands relative to the printer. The third was to lower the surface roughness, which is the microscopic unevenness caused by the layering process. The fourth was to improve the overall surface quality, which suffers when the layers create a jagged, stair-step appearance on curved surfaces. The algorithm ran 30 separate times, generating a vast collection of potential solutions. From these, the team identified 214 distinct options that were not dominated by any other option, meaning you could not improve one of the four goals without making another worse.

The researchers then filtered these 214 options down to 23 representative solutions that showed the full range of possibilities. They found that some angles were excellent for minimizing the support area but resulted in longer print times, while others were incredibly fast but left the surface rough. Crucially, they discovered that many of the angles that looked perfect in previous studies were actually fragile; a tiny shift in the angle would cause the quality to drop sharply. The new approach, however, identified angles that were stable. For instance, one solution required the part to be oriented at 90 degrees on one axis and 90 on the other, which minimized the support area and surface quality issues but took the longest to print. Another solution, oriented at 68 degrees and 180 degrees, offered the fastest print time but required the most support material. The team also found several new angles that had not been identified in earlier work, proving that the robust method could find hidden gems that standard methods missed.

To prove that these computer-generated solutions actually worked in the real world, the team selected one specific angle, where the part was tilted at 10 degrees on one axis and 165 degrees on the other, and printed it. They then printed a second version of the same part, but this time they intentionally shifted the angle by 3 degrees, simulating a small error that might happen during a real print job. They measured the time it took to print both parts and the amount of plastic filament used. The results were striking: the two parts took almost the same amount of time to print, differing by less than two percent, and they used nearly the same amount of material. When the researchers removed the support structures and looked at the final parts, there was no visible difference between the one printed with the perfect angle and the one printed with the slightly off angle. The surface finish, the shape, and the structural integrity remained consistent.

This experiment confirmed that the robust approach works. The angle chosen by the algorithm was not just a theoretical ideal; it was a practical solution that could withstand the small imperfections inherent in any manufacturing process. The study showed that by designing for uncertainty, engineers can find printing orientations that are reliable and consistent, rather than just optimal on paper. While the research focused on a single fin-shaped object, the method provides a clear path for improving the reliability of 3D printing for more complex parts. The work suggests that the future of additive manufacturing lies not just in finding the fastest or cheapest way to print, but in finding the most stable way to do so, ensuring that the final product looks and performs exactly as intended, regardless of the tiny variations that occur in the real world.

Technical Summary: A Robust Multi-Objective Approach in Additive Manufacturing

Problem Statement
Additive Manufacturing (AM), particularly the Fused Deposition Modeling (FDM) technique, offers significant advantages in producing complex geometries with reduced material waste and post-processing requirements. However, the quality of the final part is heavily dependent on the build orientation. The selection of the optimal orientation is a critical multi-objective problem, as it simultaneously influences conflicting factors such as the amount of support material required, total build time, surface roughness, and overall surface quality. While previous studies have utilized multi-objective optimization to find Pareto-optimal solutions, these deterministic approaches often fail to account for uncertainties and small perturbations in decision variables (rotation angles) that occur during the actual printing process. Consequently, solutions that appear optimal in simulation may prove sensitive to minor deviations, leading to suboptimal manufacturing outcomes.

Methodology
This study proposes a robust multi-objective optimization framework to address the build orientation problem for a 3D "Fin" model. The core objective is to identify solutions that remain effective and feasible despite small perturbations in the decision variables.

Optimization Formulation: The problem is defined as minimizing four effective objective functions:
1. Effective Support Area ( $SA_{eff}$ ): The total contact area between external supports and the printed object.
2. Effective Build Time ( $BT_{eff}$ ): Derived from the model height and the number of layers required for a specific orientation.
3. Effective Surface Roughness ( $RA_{eff}$ ): An indicator of overall surface quality influenced by the angle between surface normals and the vertical direction.
4. Effective Surface Quality ( $SQ_{eff}$ ): Measured by integrating the cusp height (staircase effect) over the entire surface.
Robustness Definition: Instead of optimizing the raw objective functions, the approach minimizes the mean effective objective function ( $f^{eff}_j$ ), calculated by integrating the objective function over a $\delta$ -neighborhood ( $B_\delta(\theta)$ ) of the decision variables ( $\theta_x, \theta_y$ ). This ensures the solution is insensitive to small variations within the defined neighborhood.
Algorithm: The study employs the Non-dominated Sorting Genetic Algorithm II (NSGA-II), an elitist multi-objective genetic algorithm.
Experimental Setup:
- Model: A 3D CAD "Fin" model (121.5 × 53.9 × 16 mm) with a layer thickness of 0.2 mm.
- Parameters: A population size of 50, 600 generations, and 30 independent runs. The perturbation value ( $\delta$ ) was set to 3 degrees.
- Validation: The Simplify3D software was used for visualization. An experimental validation was conducted using an Ender-3 3D printer, manufacturing a representative solution (Solution L: $\theta_x=10^\circ, \theta_y=165^\circ$ ) and its perturbed counterpart ( $\theta_y=168^\circ$ ).

Key Results
The optimization process yielded 214 non-dominated solutions, from which 23 representative solutions were selected to characterize the Pareto front.

Pareto Front Analysis: The results demonstrate clear trade-offs between the objectives.
- Solution A ( $90^\circ, 90^\circ$ ) exhibited the lowest Support Area and Surface Quality values but the highest Build Time.
- Solution W ( $68^\circ, 180^\circ$ ) achieved the highest values for Support Area and Surface Quality.
- Solution T ( $90^\circ, 180^\circ$ ) provided the lowest Build Time but the highest Surface Roughness.
- Solution B ( $180^\circ, 109^\circ$ ) achieved the lowest Surface Roughness.
Correlations: The analysis confirmed a positive correlation between Support Area ( $SA_{eff}$ ) and Surface Quality ( $SQ_{eff}$ ); as one increases, the other tends to increase.
Comparison with Previous Work: When compared to the deterministic approach in Matos et al. [17], the robust approach identified new solutions (E, I, L, O) that were not previously found. Conversely, solutions identified in the previous work with very small support areas (near $0^\circ, 0^\circ$ ) were not obtained here, as they were deemed non-robust. Solutions T and W were confirmed as robust alternatives.
Experimental Validation: The physical printing of Solution L and its perturbed version ( $\bar{L}$ ) showed negligible differences. The build time and filament consumption differed by less than 2%. Visual inspection after support removal revealed no perceptible differences between the two parts, confirming that the selected orientation is robust to small angular deviations.

Significance and Claims
The paper claims to present the first robust multi-objective optimization approach specifically applied to the build orientation problem in additive manufacturing. Its primary significance lies in shifting the focus from finding theoretically optimal solutions to finding robust solutions that maintain performance under the inevitable uncertainties of the printing process.

The authors assert that their approach successfully generates a set of stable, well-balanced trade-off solutions. By validating the results experimentally, the study demonstrates that robust optimization can identify build orientations that are less sensitive to noise and perturbations compared to deterministic methods. This provides decision-makers with a more reliable set of alternatives for 3D printing, ensuring that the chosen orientation remains effective even if slight deviations occur during the manufacturing phase. The study concludes that this method offers a practical advantage for industrial applications where consistency and reliability are paramount.

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