Search for: All records

Job shops are an important production environment for low-volume high-variety manufacturing. Its scheduling has recently been formulated as an Integer Linear Programming (ILP) problem to take advantages of popular Mixed-Integer Linear Programming (MILP) methods, e.g., branch-and-cut. When considering a large number of parts, MILP methods may combinatorial difficulties. To address this, a critical but much overlooked issue is formulation tightening. The idea is that if problem constraints can be transformed to directly delineate the problem convex hull in the data preprocessing stage, then a solution can be obtained by using linear programming methods without combinatorial difficulties. The tightening process, however, is fundamentally challenging because of the existence of integer variables. In this paper, an innovative and systematic approach is established for the first time to tighten the formulations of individual parts, each with multiple operations, in the data preprocessing stage. It is a major advancement of our previous work on problems with binary and continuous variables to integer variables. The idea is to first link integer variables to binary variables by innovatively combining constraints so that the integer variables are uniquely determined by the binary variables. With binary and continuous variables only, it is proved that the vertices of the convex hull can be obtained based on vertices of the linear problem after relaxing binary requirements. These vertices are then converted to tightened constraints for general use. This approach significantly improves our previous results on tightening individual operations. Numerical results demonstrate significant benefits on solution quality and computational efficiency. This approach also applies to other ILP problems with similar characteristics and fundamentally changes the way how such problems are formulated and solved. 

Sub-hourly Unit Commitment (UC) problems have been suggested as a way to improve power system efficiency. Such problems, however, are much more difficult than hourly UC problems. This is not just because of the increased number of period to consider, but also because of much reduced unit ramping capabilities leading to more complicated convex hulls. As a result, state-of-the-art and practice methods such as branch-and-cut suffer from poor performance. In this paper, our recent Surrogate Absolute-Value Lagrangian Relaxation (SAVLR) method, which overcame major difficulties of standard Lagrangian Relaxation, is enhanced by synergistically incorporating the concept of Ordinal Optimization (OO). By using OO, solving subproblems becomes much faster. Testing of Midcontinent ISO (MISO)’s problem with 15 minutes as the time interval over 36 hours involving about 1,100 units and 15000 virtuals demonstrates that the new method obtains near-optimal solutions efficiently and significantly outperforms branch-and-cut. 

With the emergence of the 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 article, a new ``price-based'' decomposition and coordination methodology is developed to efficiently coordinate a system consisting of distributed subsystems such as machines and parts, which are described by mixed-integer linear programming (MILP) formulations, in an asynchronous way. The novel method is a dual approach, whereby the coordination is performed by updating Lagrangian multipliers based on economic principles of ``supply and demand.'' To ensure low communication requirements within the method, exchanges between the ``coordinator'' and subsystems are limited to ``prices'' (Lagrangian multipliers) broadcast by the coordinator and to subsystem solutions sent at 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, the convergence of our method is proven by innovatively extending the 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. Accompanying CPLEX codes and data are included. 

Unit Commitment is an important problem faced by independent system operators. It is usually formulated as a Mixed Binary Linear Programming (MBLP) problem, and is believed to be NP hard. To solve UC problems efficiently, an idea is through formulation tightening. If constraints can be transformed to directly delineate an MBLP problem’s convex hull during data preprocessing, then the problem can be solved by using linear programming methods. The resulting formulation can be reused for other data sets, tremendously reducing computational requirements. To achieve the above goal, both unit- and system-level constraints are tightened with synergistic combination in this paper. Unit-level constraints are tightened based on existing cuts and novel “constraint-and-vertex conversion” and vertex projection processes. To tighten system-level constraints, selected cuts are applied and some potentially powerful cuts are identified. Numerical results demonstrate the effectiveness of tightening unit- and system-level constraints.