Compact SAT Encoding for Power Peak Minimization

The Simple Assembly Line Balancing Problem with Power Peak Minimization (SALBP-3PM) minimizes maximum instantaneous power usage while assigning $n$ tasks to $m$ workstations and determining execution schedules within given cycle time constraints. This NP-hard problem couples workstation assignment, temporal sequencing, and power aggregation, presenting significant computational challenges for exact optimization methods. Existing Boolean Satisfiability (SAT) and Maximum Satisfiability (MaxSAT) approaches suffer from baseline encodings generating $O(m^2)$ clauses per precedence edge. We introduce a Compact SAT Encoding (CSE) achieving $O(m)$ clauses per transitive precedence edge using sequential counter techniques. We instantiate four optimization variants: Clause-Based iterative SAT, Pseudo-Boolean (PB) Constraint iterative SAT, MaxSAT, and Incremental SAT. Comprehensive experimental evaluation on benchmark instances demonstrates consistent performance improvements over state-of-the-art approaches, enabling exact optimization on previously intractable industrial-scale instances. The encoding principles generalize to other assembly line balancing variants and broader scheduling problems with precedence constraints.

💡 Research Summary

This paper presents a significant advancement in solving the Simple Assembly Line Balancing Problem with Power Peak Minimization (SALBP-3PM), an NP-hard problem that integrates task assignment, temporal scheduling, and power consumption aggregation to minimize the maximum instantaneous power usage in an assembly line.

The core contribution is the introduction of a Compact SAT Encoding (CSE) that dramatically improves upon existing Boolean Satisfiability (SAT) and Maximum Satisfiability (MaxSAT) approaches. Prior encodings suffered from a quadratic blow-up, requiring O(m²) clauses per precedence constraint (where ’m’ is the number of workstations), which hindered scalability. The novel CSE overcomes this by employing sequential counter techniques. It introduces auxiliary Boolean variables R_j,k (task j is assigned to a workstation ≤ k) and T_j,τ (task j starts at time ≤ τ). Using these variables, the encoding reformulates precedence and temporal non-overlap constraints, reducing the clause complexity to a linear O(m) per transitive precedence edge. This compaction leads to substantially smaller SAT formulas, enabling modern Conflict-Driven Clause Learning (CDCL) solvers to reason much more efficiently.

Building upon this foundational encoding, the authors instantiate and compare four distinct optimization frameworks:

1. Clause-Based iterative SAT: An extension of prior work (Py et al., 2024) that iteratively solves feasibility SAT problems for a given peak power bound, adding blocking clauses to exclude solutions with the current peak, and tightening the bound until unsatisfiability proves optimality.
2. Pseudo-Boolean (PB) Constraint iterative SAT: This variant replaces blocking clauses with global Pseudo-Boolean constraints (e.g., the sum of powers of tasks active at a time must be less than a target). PB constraints can often express complex conditions more concisely than equivalent sets of clauses.
3. MaxSAT: This approach integrates the compact encoding with the logarithmic peak representation from Zheng et al. (2024). It encodes the power peak using binary bits and formulates power limit constraints as soft clauses with weights. A single call to a MaxSAT solver then directly searches for the optimal solution minimizing the peak.
4. Incremental SAT: This is a performance-focused variant that preserves the solver’s internal state (learned clauses, variable activities) between iterations where only the peak bound changes. Reusing this computational context eliminates redundant search, offering substantial speed-ups.

The methodology includes a preprocessing step to compute the transitive closure of the precedence graph and an initialization phase that uses a quick SAT call to find a feasible solution, providing a tight upper bound for the peak power, which is more effective than using coarse analytical bounds.

A comprehensive experimental evaluation was conducted on 89 established benchmark instances. The proposed methods, all utilizing the CSE, were compared against state-of-the-art solvers including commercial ILP solvers (CPLEX, Gurobi) and the previous best SAT/MaxSAT approaches. The results demonstrate consistent and often dramatic performance improvements. The Incremental SAT variant emerged as the most robust, successfully solving all instances, including many industrial-scale “hard” instances that caused timeouts for all competing methods. The compact encoding’s reduction in formula size directly translated to faster solver runtimes and better scalability.

In conclusion, this work provides a breakthrough in exact optimization for SALBP-3PM. The Compact SAT Encoding effectively tackles the core inefficiency of prior logic-based models, and the suite of optimization variants offers flexible and powerful solving strategies. The principles of the encoding are general and can be adapted to other assembly line balancing variants and a broader class of scheduling problems with precedence and resource aggregation constraints, marking an important methodological contribution to the field.