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Portfolio of Solving Strategies in CEGAR-based Object Packing and Scheduling for Sequential 3D Printing

Pavel Surynek

TL;DR

This work parallelizes the existing CEGAR-SEQ algorithm that solves the sequential object arrangement and scheduling for sequential 3D printing by expressing it as a linear arithmetic formula which is then solved by a technique inspired by counterexample guided abstraction refinement (CEGAR).

Abstract

Computing power that used to be available only in supercomputers decades ago especially their parallelism is currently available in standard personal computer CPUs even in CPUs for mobile telephones. We show how to effectively utilize the computing power of modern multi-core personal computer CPU to solve the complex combinatorial problem of object arrangement and scheduling for sequential 3D printing. We achieved this by parallelizing the existing CEGAR-SEQ algorithm that solves the sequential object arrangement and scheduling by expressing it as a linear arithmetic formula which is then solved by a technique inspired by counterexample guided abstraction refinement (CEGAR). The original CEGAR-SEQ algorithm uses an object arrangement strategy that places objects towards the center of the printing plate. We propose alternative object arrangement strategies such as placing objects towards a corner of the printing plate and scheduling objects according to their height. Our parallelization is done at the high-level where we execute the CEGAR-SEQ algorithm in parallel with a portfolio of object arrangement strategies, an algorithm is called Porfolio-CEGAR-SEQ. Our experimental evaluation indicates that Porfolio-CEGAR-SEQ outperforms the original CEGAR-SEQ. When a batch of objects for multiple printing plates is scheduled, Portfolio-CEGAR-SEQ often uses fewer printing plates than CEGAR-SEQ.

Portfolio of Solving Strategies in CEGAR-based Object Packing and Scheduling for Sequential 3D Printing

TL;DR

This work parallelizes the existing CEGAR-SEQ algorithm that solves the sequential object arrangement and scheduling for sequential 3D printing by expressing it as a linear arithmetic formula which is then solved by a technique inspired by counterexample guided abstraction refinement (CEGAR).

Abstract

Computing power that used to be available only in supercomputers decades ago especially their parallelism is currently available in standard personal computer CPUs even in CPUs for mobile telephones. We show how to effectively utilize the computing power of modern multi-core personal computer CPU to solve the complex combinatorial problem of object arrangement and scheduling for sequential 3D printing. We achieved this by parallelizing the existing CEGAR-SEQ algorithm that solves the sequential object arrangement and scheduling by expressing it as a linear arithmetic formula which is then solved by a technique inspired by counterexample guided abstraction refinement (CEGAR). The original CEGAR-SEQ algorithm uses an object arrangement strategy that places objects towards the center of the printing plate. We propose alternative object arrangement strategies such as placing objects towards a corner of the printing plate and scheduling objects according to their height. Our parallelization is done at the high-level where we execute the CEGAR-SEQ algorithm in parallel with a portfolio of object arrangement strategies, an algorithm is called Porfolio-CEGAR-SEQ. Our experimental evaluation indicates that Porfolio-CEGAR-SEQ outperforms the original CEGAR-SEQ. When a batch of objects for multiple printing plates is scheduled, Portfolio-CEGAR-SEQ often uses fewer printing plates than CEGAR-SEQ.
Paper Structure (10 sections, 4 equations, 7 figures, 2 algorithms)

This paper contains 10 sections, 4 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contribution
  4. Background
  5. A Linear Arithmetic Model
  6. A Portfolio of Scheduling Strategies
  7. Experimental Evaluation
  8. Comparison of Underlying Solvers
  9. Benefit of Portfolio
  10. Conclusion

Figures (7)

  • Figure 1: Standard 3D printing slice by slice and sequential 3D printing where objects are completed one by one shown in Prusa Slicer prusa-slicer-2025. The ordering of objects for sequential printing is shown by numbers. Printer extruder and gantry must avoid previously printed objects in the sequential case (printing of the last object is shown). The sequential arrangement and schedule has been obtained by the CEGAR-SEQ algorithm DBLP:conf/iros/SurynekBMK25.
  • Figure 2: Object arrangement strategies (tactics): (i) objects are placed towards the center of the printing plate $P_P$ ( Center tactic), (ii) objects are placed towards the maximum $x$ and minimum $y$ coordinates w.r.t. $P_P$ ( Max-X-Min-Y tactic).
  • Figure 3: Comparison of solving SEQ-PACK+S by the Gecode solver and the z3 solver on random cuboids on a 200mm $\times$ 200mm plate. The right part shows cactus plots of runtimes (lower plot is better).
  • Figure 4: Sorted runtimes of Portfolio-CEGAR-SEQ with various portfolios on printer parts.
  • Figure 5: Average number of printing plates used by a sub-optimal variant of Portfolio-CEGAR-SEQ under four different portfolios of composite strategies.
  • ...and 2 more figures