Optimizing the trajectory of a wire arc additive manufacturing using the dhouib-matrix-4 metaheuristic

Background and prior applications

This study applies the Dhouib-Matrix-4 (DM4) metaheuristic to layer-wise trajectory planning in wire arc additive manufacturing. DM4 was originally developed to solve the Hamiltonian cycle problem for the symmetric Travelling Salesman Problem (TSP) using multiple instances from the OR-Library (Dhouib, 2024c). It was subsequently adapted to minimize non-productive movements of a Computer Numerical Control (CNC) hole-drilling machine (Dhouib, 2025b; Dhouib and Pezer, 2023), to generate Pareto non-dominated solutions for the multi-objective TSP (Dhouib, 2024d; Dhouib et al., 2024a), and to plan the trajectory of a mobile robot repairing a wireless sensor network (Dhouib, 2023). DM4 is part of the broader Dhouib-Matrix (DM) family, which also includes the DM-SPP shortest-path planner for autonomous mobile robots (Dhouib, 2024b, 2025a; Dhouib et al., 2026) and the DM-MSTP minimum spanning tree optimizer (Dhouib, 2024a; Dhouib et al., 2024b). Its adaptation to WAAM is motivated by the structural similarity between layer-wise deposition-node routing (an open Hamiltonian path over a dense planar point set) and the discrete combinatorial problems for which DM4 has demonstrated competitive performance.

Algorithm structure

DM4 is a multi-start metaheuristic. Each start consists of two successive stages: (i) an initial solution is constructed by the Dhouib-Matrix-TSP1 (DM-TSP1) greedy heuristic, which builds a tour in exactly n iterations (where n is the number of nodes) by selecting the next node according to a statistical metric computed over the remaining unvisited nodes; (ii) the constructed solution is then improved by the Far-to-Near (FtN) local search operator, which iteratively relocates the node farthest from its current neighbors to a lower-cost insertion position. Across multi-starts, a different statistical metric (e.g., mean, median, variance, range, coefficient of variation) is used at Stage (i) for each start, enabling diversified exploration of the solution space. The overall best solution found across all starts is returned as the output and is referred to herein as the ‘best-found path,’ without claiming global optimality—a distinction that is appropriate for any metaheuristic framework.

This formulation explicitly models the open-path nature of WAAM deposition: the torch starts at one node and ends at another without returning to the start, which distinguishes it from the classic closed-tour TSP. The complete algorithmic structure—including input data, DM-TSP1 construction stage, FtN local search stage, objective evaluation, termination criterion, and output—is illustrated in Figure 1.OPEN IN VIEWER

Illustrative 100-node example

Figure 2 presents an illustrative example of DM4 on a synthetic 100-node instance (coordinates in arbitrary units) to clarify the algorithm’s route-construction and improvement behavior before the full-scale WAAM benchmarks. Figure 2(a) shows the input node distribution, and Figure 2(b) shows the best-found open path generated by DM4: the green circle marks the start node and the red circle marks the end node. The path length for this illustrative example is reported in arbitrary units consistent with the coordinate system used. This example is not a WAAM validation case; its role is pedagogical.OPEN IN VIEWER

Case study 1

Case study 1 is the prismatic layer discretized into 303 deposition nodes at a uniform 6 mm spacing, as proposed by (Ferreira and Scotti, 2021). We note that each node might be thought of as an arc-on deposition point (or a symbol of a short bead segment) which needs to be visited just once to fill the layer. In this paper, the planning problem is represented as the layer-wise shortest Hamiltonian path problem (open TSP); we must order 303 nodes that minimizes the cumulative travel distance between nodes, which translates to the non-productive phase (arc-off travel) and further accelerations/decelerations. Since the number of feasible sequences increases factorially with the number of nodes, simple constructive rules quickly get stuck on locally good but globally inefficient tours. Figure 3(a) shows the prismatic shape and Figure 3(b) presents the shortest path obtained by DM4 with distance of (1785.57 mm) on a layer. By a similar point of comparison (Ferreira and Scotti, 2021) gives an estimate of (1817.72 mm) by the random contour heuristic, and (1819.61 mm) by the nearest-neighbor heuristic. In order to assess DM4 at a realistic slicing condition, the layer is first discretized into a finite set of nodes indicating the realistic deposition locations along the contour and interior. The trajectory planning problem proceeds with finding an ordering which makes it to the node once and then minimizes the total in-plane travel distance between the nodes (the traveling-salesman-problem) to realize that the torch motion in a layer is as much continuous and efficient as possible (Ferreira and Scotti, 2021). This benchmark has moderate size (303 nodes) and a somewhat simple outer topology, presenting advantages in comparing how fast each solution converges to a good tour. The 6 mm node spacing results in a coarse but representative discretization for WAAM, where unnecessary detours translate into unnecessary arc-off moves, longer cycle time, and increased risk for heat accumulation at local re-starts.OPEN IN VIEWER

Case study 2

Case study 2 considers a larger and denser discretization with 383 nodes per layer over six layers. The benchmark geometry, originally designed in (Ferreira and Scotti, 2021), has overall dimensions of 300 × 180 × 12 mm² and uses a fixed node spacing of 3.03 mm, which increases the number of short-range alternatives and intensifies the combinatorial routing problem. From a WAAM perspective, the objective remains to visit all deposition nodes while minimizing arc-off travel, since excessive non-deposition motion increases build time and can introduce thermal discontinuities between adjacent beads. Figure 4(a) illustrates the prismatic shape, and Figure 4(b) shows the shortest trajectory generated by DM4 with a total distance of (1094 mm) per layer. As a baseline, the Pixel strategy of (Ferreira and Scotti, 2021) produces a longer trajectory of (1253 mm) per layer on the same discretization.OPEN IN VIEWER

Case study 2 increases both the number of nodes and the spatial density of the discretization. The resulting search space is substantially larger than in Case study 1 because small changes in the visit sequence can create many short crossings and back-and-forth moves over the 300 × 180 mm footprint (Ferreira and Scotti, 2021).

In addition, the benchmark is defined over six layers, so a practical solution must be repeatable from layer to layer and should avoid long repositioning moves between layers. Minimizing the per-layer tour length is therefore a first proxy for reducing overall build time and for limiting the cumulative thermal load induced by repeated torch traversals over the same region.

Case study 3

Case study 3 is by far the tougher benchmark because the layer has two internal holes in addition to the external boundary which gives multiple contour areas to be connected by arc-off travel. The model has 6 layers and each layer is made up of 1072 nodes, which greatly increases the problem size and the probability of long non-productive jumps between sub-regions if the visit order is poor (Ferreira and Scotti, 2021). Such jumps correspond to torch movement inside the part or between internal and outer contours, and in practice they are unwanted because they prolong cycle time and may necessitate frequent start/stop events. The goal of the optimization is to sequence the outer boundary and the two hole contours in such a way that limits transitions between regions while still visiting each node once. Figure 5(a) depicts a prismatic shape and Figure 5(b) illustrates DM4 trajectory for a distance of (3072.61 mm) per layer. For comparison (Ferreira and Scotti, 2021) reported a Pixel strategy of (3038.70 mm) per layer which signifies that topology-aware continuous strategies are beneficial on highly complex layers. The third case study features a non-convex topology having two interior holes. The 1072 nodes per layer results in a huge combinatorial instance, in which the tour must be able to cover multiple boundary segments and also respect the void regions that must be negotiated (i.e., avoiding shortcuts that would cut through the holes) (Ferreira and Scotti, 2021). From a WAAM standpoint, these internal features are challenging as they tend to require sharp turns and frequent changes of direction (to the point of magnifying a bead’s start/end defects or local heat concentration). This benchmark is, therefore, indicative of the industrial environments that the toolpath can be used in each segment in the interest of geometric feasibility, continuity, and reduction of travel-distance on a large scale.OPEN IN VIEWER

Performance pattern across case studies

The three benchmark cases reveal a clear and interpretable performance pattern for DM4. In Cases 1 and 2—both simply connected geometries with spatially uniform node distributions—DM4 generates the shortest total path among all compared methods, improving by approximately 1.8% over two baseline heuristics (Case 1) and by 12.7% over the Pixel raster strategy (Case 2). In Case 3, a multi-region topology with two internal holes, DM4 yields a total path length 1.1% longer than the Pixel strategy. This outcome is consistent with DM4’s algorithmic design: the multi-start constructive-and-improvement approach is well suited to routing over simply connected, uniformly distributed node sets, where statistical diversification across starts effectively avoids local optima. In multi-region topologies, however, the routing problem contains additional structure in the form of mandatory inter-region arc-off transitions that DM4’s current formulation does not explicitly exploit.

Why DM4 underperforms in case 3

The fundamental reason for DM4’s relative underperformance in Case 3 is the presence of two internal holes that partition the layer into three disconnected feasible regions: the outer boundary region and two inner hole contours. Any open Hamiltonian path must cross between these regions at least twice, and each crossing corresponds to a mandatory arc-off segment. DM4, as a general-purpose combinatorial optimizer, treats all inter-node distances equally and therefore does not distinguish between intra-region and inter-region transitions. The Pixel strategy, in contrast, is designed to cover each region continuously using raster sweeps before switching to the next region, which minimizes the number and length of inter-region jumps. This analysis identifies a concrete architectural extension for DM4: a topology-aware region-sequencing pre-processing step that first determines the ‘best-found path’ order in which to visit the distinct contour regions, and then applies DM4 (or an equivalent intra-region optimizer) within each region. This extension is directly motivated by strategies developed for multi-region intersection handling in aluminium WAAM (Kelly et al., 2025).

Process implications: Beyond distance minimization

A critical consideration in WAAM trajectory optimization is that minimizing arc-off travel distance is a necessary but not sufficient condition for trajectory quality (Sideris et al., 2024). Process quality is strongly governed by thermal history, bead continuity, and local material deposition patterns (Nagallapati et al., 2023; Uyen et al., 2023; Wang et al., 2025). Shorter non-productive travel reduces build time and limits cumulative heat input from repeated torch traversals, which supports thermal management objectives (Reis and Da Silva, 2025). However, the shape and directional characteristics of the productive path also influence local heat distribution, residual stresses, and distortion (Mauthner et al., 2025; Zhao et al., 2024).

Sharp direction changes in the trajectories generated by DM4—particularly at boundary transitions and near internal features—are a documented source of over- or under-deposition in WAAM, arising from torch velocity dynamics and arc behavior at corners (Ding et al., 2021). This effect is more pronounced in the multi-hole topology of Case 3, where direction reversals are frequent. For thermally sensitive materials such as aluminium and magnesium WAAM alloys, trajectory choices have been shown to significantly affect microstructure, residual stresses, and mechanical properties (Köhler et al., 2021; Seidler et al., 2026); such materials impose stricter smoothness requirements on the deposition path than steel alloys. Future DM4 formulations should incorporate a smoothness penalty (e.g., weighting path segments by direction-change angle) and an arc start/stop penalty to produce trajectories that are both distance-efficient and process-quality-aware.

Furthermore, the relationship between path length minimization and thermal management merits explicit discussion. While shorter arc-off travel reduces the time the torch spends traversing non-deposition regions, the productive path sequence determines the spatial distribution of heat input. A path that revisits the same region repeatedly in a short time interval can cause local overheating, whereas a path that distributes deposition more uniformly allows passive cooling between visits. Integrating a fast temperature prediction model—such as the dexel-based approach of (Mauthner et al., 2025) or the data-driven predictors of (Reis and Da Silva, 2025)—directly within the DM4 optimization loop would enable thermally constrained trajectory generation, making the approach genuinely process-aware.

Comparison with zig-zag and contour-based strategies

Zig-zag (raster) scanning is the simplest and most widely used WAAM deposition strategy: the torch traverses the layer in alternating parallel passes with fixed direction reversals at the boundaries. While straightforward to program, raster strategies generate systematic boundary reversals that accumulate arc-off travel and are unable to exploit the spatial distribution of deposition nodes to reduce total path length. Contour-based strategies follow the geometric boundaries of the deposited layer in closed loops, offering excellent continuity for perimeter beads but requiring complementary infill strategies for interior fill regions. DM4 occupies a complementary niche: it is a global combinatorial optimizer that directly minimizes arc-off travel over the full node set, making it most naturally suited to interior fill routing after boundary contours have been pre-planned—a role where its advantages over raster scanning are most pronounced (as demonstrated in Case 2).

Robotic feasibility and node spacing sensitivity

The trajectories generated by DM4 consist of straight-line segments between successive deposition nodes, which are directly executable in Cartesian space by a standard WAAM robot controller. However, sharp angular direction changes at certain nodes—particularly at boundary transitions and internal hole contours—may require velocity reduction to maintain bead geometry, potentially increasing effective cycle time beyond what the distance metric alone suggests. Kinematic feasibility assessment (incorporating acceleration limits and minimum cornering radius) is an important practical extension for future work.

Regarding node spacing sensitivity, the comparison between Case 1 (6 mm, 303 nodes) and Case 2 (3.03 mm, 383 nodes) illustrates that finer discretization increases both the combinatorial search space and the density of short-range improvement opportunities available to FtN. DM4’s computational complexity per start scales as O(n²) for the FtN local search stage, rendering it tractable for the node counts tested. Observed runtimes—under 5 s for Case 1, approximately 12 s for Case 2, and approximately 45 s for Case 3—confirm practical scalability for typical WAAM layer discretizations. Sensitivity analyses at intermediate node spacings (e.g., 4 mm, 5 mm) are reserved for future work to more precisely characterize the trade-off between discretization resolution and computational cost.

Limitations and future research directions

The current formulation of DM4 for WAAM has three primary limitations. First, the single-objective formulation (minimize total arc-off travel distance) does not account for thermal constraints, path smoothness, or arc start/stop frequency—all of which are critical for WAAM process quality. A multi-objective extension incorporating these criteria is the most impactful next step. Second, the absence of topology-aware region sequencing causes underperformance on multi-region geometries (Case 3); addressing this requires a hierarchical pre-processing stage as discussed in Section 4.2. Third, the benchmarks used are computational rather than experimental; validation against physical WAAM builds with bead quality and distortion measurements would strengthen the practical claims of the method.

Future research will therefore focus on: (i) multi-objective DM4 for WAAM, jointly minimizing arc-off travel, direction-change penalties, and arc start/stop frequency; (ii) topology-aware region sequencing as a pre-processing layer; (iii) integration of fast temperature prediction models within the toolpath optimization loop (Mauthner et al., 2025; Reis and Da Silva, 2025); (iv) adaptive, feedback-driven process frameworks that couple DM4 with real-time thermal sensing (Khrais et al., 2023); and (v) data-driven learning from process outcomes to refine path-quality objectives (Efa and Ifa, 2025). These extensions will be validated on geometries with experimentally characterized bead quality, residual stress, and distortion to connect combinatorial trajectory optimization with measurable WAAM process outcomes.