Distributed and Asynchronous Coordination of a Mixed-Integer Linear System via Surrogate Lagrangian Relaxation

With the emergence of Internet of Things that allows communications and local computations, and with the vision of Industry 4.0, a foreseeable transition is from centralized system planning and operation toward decentralization with interacting components and subsystems, e.g., self-optimizing factories. In this paper, a new “price-based” decomposition and coordination methodology is developed to efficiently coordinate subsystems such as machines and parts, which are described by Mixed-Integer Linear Programming (MILP) formulations, in a distributed and asynchronous way. To ensure low communication requirements, exchanges between the “coordinator” and subsystems are limited to “prices” (Lagrangian multipliers) broadcast by the coordinator, and to subsystem solutions sent to the coordinator. Asynchronous coordination, however, may lead to convergence difficulties since the order in which subsystem solutions arrive at the coordinator is not predefined as a result of uncertainties in communication and solving times. Under realistic assumptions of finite communication and solve times, convergence of our method is proved by innovatively extending Lyapunov Stability Theory. Numerical testing of generalized assignment problems through simulation demonstrates that the method converges fast and provides near-optimal results, paving the way for self-optimizing factories in the future.