Job-shop scheduling is an important but difficult problem arising in low-volume high-variety manufacturing. It is usually solved at the beginning of each shift with strict computational time requirements. To obtain near-optimal solutions with quantifiable quality within strict time limits, a direction is to formulate them in an Integer Linear Programming (ILP) form so as to take advantages of widely available ILP methods such as Branch-and-Cut (B&C). Nevertheless, computational requirements for ILP methods on existing ILP formulations are high. In this paper, a novel ILP formulation for minimizing total weighted tardiness is presented. The new formulation has much fewer decision variables and constraints, and is proven to be tighter as compared to our previous formulation. For fast resolution of large problems, our recent decomposition-and-coordination method “Surrogate Absolute-Value Lagrangian Relaxation” (SAVLR) is enhanced by using a 3-segment piecewise linear penalty function, which more accurately approximates a quadratic penalty function as compared to an absolute-value function. Testing results demonstrate that our new formulation drastically reduces the computational requirements of B&C as compared to our previous formulation. For large problems where B&C has difficulties, near-optimal solutions are efficiently obtained by using the enhanced SAVLR under the new formulation.

ob-shop scheduling is an important but difficult problem arising in low-volume high-variety manufacturing. It is usually solved at the beginning of each shift with strict computational time requirements. For fast resolution of the problem, a promising direction is to formulate it in an Integer Linear Programming (ILP) form so as to take advantages of widely available ILP methods such as Branch-and-Cut (B&C). Nevertheless, computational requirements on ILP methods for existing ILP formulations are high. In this paper, a novel ILP formulation is presented. In the formulation, a set of binary indicator variables indicating whether an operation begins at a time slot on a machine group or not is selected as decision variables, and all constraints are innovatively formulated based on this set of variables. For fast resolution of large problems, our recent decomposition-and-coordination method “Surrogate Absolute-Value Lagrangian Relaxation” (SAVLR) is enhanced by using a 3-segment piecewise linear penalty function, which more accurately approximates a quadratic penalty function as compared to an absolute-value function. Testing results demonstrate that our new formulation drastically reduces the computational requirements of B&C as compared to our previous formulation. For large problems where B&C has difficulties, near-optimal solutions are efficiently obtained by using the enhanced SAVLR under the new formulation.

Job-shop scheduling is an important but difficult combinatorial optimization problem for low-volume and high-variety manufacturing, with solutions required to be obtained quickly at the beginning of each shift. In view of the increasing demand for customized products, problem sizes are growing. A promising direction is to take advantage of Machine Learning (ML). Direct learning to predict solutions for job-shop scheduling, however, suffers from major difficulties when problem scales are large. In this paper, a Deep Neural Network (DNN) is synergistically integrated within the decomposition and coordination framework of Surrogate Lagrangian Relaxation (SLR) to predict good-enough solutions for subproblems. Since a subproblem is associated with a single part, learning difficulties caused by large scales are overcome. Nevertheless, the learning still presents challenges. Because of the high-variety nature of parts, the DNN is desired to be able to generalize to solve all possible parts. To this end, our idea is to establish “surrogate” part subproblems that are easier to learn, develop a DNN based on Pointer Network to learn to predict their solutions and calculate the solutions of the original part subproblems based on these predictions. Moreover, a masking mechanism is developed such that all the predictions are feasible. Numerical results demonstrate that good-enough subproblem solutions are predicted in many iterations, and high-quality solutions of the overall problem are obtained in a computationally efficient manner. The performance of the method is further improved through continuous learning.

Unit Commitment (UC) is important for power system operations. With increasing challenges, e.g., growing intermittent renewables and intra-hour net load variability, traditional mathematical optimization could be time-consuming. Machine learning (ML) is a promising alternative. However, directly learning good solutions is difficult in view of the combinatorial nature of UC. This paper synergistically integrates ML within our recent decomposition and coordination method of Surrogate Lagrangian Relaxation to learn subproblem solutions of deterministic UC. Compared to the original problem, a subproblem is much easier to learn. Nevertheless, because of many types of constraints, finding “good enough” subproblem solutions for a given set of multipliers is still challenging. To overcome this, dimensionality is reduced via aggregating multipliers and removing unnecessary variables. Multiplier distributions are novelly specified for effective training. Furthermore, to exploit solutions obtained in daily operations, offline supervised learning and online self-learning are unified at the switching of the innovatively designed loss functions. Although testing of this first attempt to integrate two methods is limited to the IEEE 118-bus system, results demonstrate that ML obtains good-enough subproblem solutions efficiently, leading to near-optimal overall solutions. Our method opens a new direction for integrating ML and mathematical optimization to solve complicated UC and beyond.

This paper summarizes the technical activities of the IEEE Task Force on Solving Large Scale Optimization Problems in Electricity Market and Power System Applications. This Task Force was established by the IEEE Technology and Innovation Subcommittee to first review the state-of-the-art of the security-constrained unit commitment (SCUC) business model, its mathematical formulation, and solution techniques in solving electricity market clearing problems. The Task Force then investigated the emerging challenges of future market clearing problems and presented efforts in building benchmark mathematical and business models.

Electricity prices determined from economic dispatch without considering fixed costs may cause high uplift payments. With fixed costs, however, a price is not a monotonic function of demand, affecting market transparency. To overcome these, convex hull (CH) pricing has recently been introduced for unit commitment with fixed costs. Several CH pricing methods were presented, and a feasible cost was used to quantify the CH price quality. The associated difficulties are 1. high computational effort required to obtain a feasible cost and 2. the associated duality gap may not be tight enough to provide accurate measure. In this paper, a novel measure to quantify the quality of CH prices is presented by establishing an upper bound to the optimal dual value approaching it from above. Near-optimal CH prices are efficiently obtained by using Surrogate Lagrangian Relaxation (SLR), meanwhile, the upper bound decreases fast due to convergence of SLR. Testing results on the IEEE 118-bus system without transmission capacities indicate that the novel quality measure reaches a value of less than 0.1% in seconds – much more accurate and faster than the measure provided by a feasible cost – demonstrating the high quality of the upper bound and the efficiency of SLR.

Unit Commitment (UC) is an important problem in power system operations. It is traditionally scoped for 24 hours with one-hour time intervals. To improve system flexibility by accommodating the increasing net-load variability, sub-hourly UC has been suggested. Such a problem is larger and more complicated than hourly UC because of the increased number of periods and reduced unit ramping capabilities per period. The computational burden is further exacerbated for systems with large numbers of virtual transactions leading to dense transmission constraints matrices. Consequently, the state-of-the-art and practice method, branch-and-cut (B&C), suffers from poor performance. In this paper, our recent Surrogate Absolute-Value Lagrangian Relaxation (SAVLR) is enhanced by embedding ordinal-optimization concepts for a drastic reduction in subproblem solving time. Rather than formally solving subproblems by using B&C, subproblem solutions that satisfy SAVLR’s convergence condition are obtained by modifying solutions from previous iterations or solving crude subproblems. All virtual transactions are included in each subproblem to reduce major changes in solutions across iterations. A parallel version is also developed to further reduce the computation time. Testing on MISO’s large cases demonstrates that our ordinal-optimization embedded approach obtains near-optimal solutions efficiently, is robust, and provides a new way on solving other MILP problems.

Unit commitment (UC) is an important problem solved on a daily basis within a strict time limit. While hourly UC problems are currently considered, they may not be flexible enough with the fast-changing demand and the increased penetration of intermittent renewables. Sub-hourly UC is therefore recommended. This, however, will significantly increase problem complexity even under the deterministic setting, and current methods may not be able to obtain good solutions within the time limit. In this paper, deterministic sub-hourly UC is considered, with the innovative exploitation of soft constraints – constraints that do not need to be strictly satisfied, but with predetermined penalty coefficients for their violations. The key idea is the “surrogate optimization” concept that ensures multiplier convergence within “surrogate” Lagrangian relaxation as long as the “surrogate optimality condition” is satisfied without the need to optimally solve the “relaxed problem.” Consequently, subproblems can still be formed and optimized when soft constraints are not relaxed, leading to a drastically reduced number of multipliers and improved performance. To further enhance the method, a parallel version is developed. Testing results on the Polish system demonstrate the effectiveness and robustness of both the sequential and parallel versions at finding high-quality solutions within the time limit.

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 experience 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 much difficulty. 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 tight 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.

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.