In Constraint Programming, solving discrete minimization problems with hard and soft constraints can be done either using (i) soft global constraints, (ii) a reformulation into a linear program, or (iii) a reformulation into local cost functions. Approach (i) benefits from a vast catalog of constraints. Each soft constraint propagator communicates with other soft constraints only through the variable domains, resulting in weak lower bounds. Conversely, the approach (ii) provides a global view with strong bounds, but the size of the reformulation can be problematic. We focus on approach (iii) in which soft arc consistency (SAC) algorithms produce bounds of intermediate quality. Recently, the introduction of linear constraints as local cost functions increases their modeling expressiveness. We adapt an existing SAC algorithm to handle linear constraints. We show that our algorithm significantly improves the lower bounds compared to the original algorithm on several benchmarks, reducing solving time in some cases.
This paper presents an energy-aware Ant Colony Optimization (ACO) framework specifically designed for P-Time labeled Petri Net systems operating under partial observation. As energy consumption becomes increasingly critical in industrial automation, traditional optimization approaches often fail to adequately handle the temporal dynamics and observational limitations inherent in real-world manufacturing systems. We propose an advanced ACO framework that integrates MAX-MIN Ant System techniques with multi-objective optimization strategies specifically tailored for P-Time Petri Nets. Our approach incorporates sophisticated constraint handling mechanisms for temporal intervals, partial observation uncertainty, and energy cost minimization. The experimental validation demonstrates significant energy savings (achieving 1610.67 units compared to 2524.47 units with baseline approaches) and computational efficiency suitable for real-time industrial deployment.
