Lagrangian Relaxation for Mixed-Integer Linear Programming: Importance, Challenges, Recent Advancements, and Opportunities

Operations in areas of importance to society are frequently modeled as Mixed-Integer Linear Programming (MILP) problems. While MILP problems suffer from combinatorial complexity, Lagrangian Relaxation has been a beacon of hope to resolve the associated difficulties through decomposition. Due to the non-smooth nature of Lagrangian dual functions, the coordination aspect of the method has posed serious challenges. This paper presents several significant historical milestones (beginning with Polyak's pioneering work in 1967) toward improving Lagrangian Relaxation coordination through improved optimization of non-smooth functionals. Finally, this paper presents the most recent developments in Lagrangian Relaxation for fast resolution of MILP problems. The paper also briefly discusses the opportunities that Lagrangian Relaxation can provide at this point in time

Toward Robust Manufacturing Scheduling: Stochastic Job-Shop Scheduling

Manufacturing plays a significant role in promoting economic development, production, exports, and job creation, which ultimately contribute to improving the quality of life. The presence of manufacturing defects is, however, inevitable leading to products being discarded, i.e. scrapped. In some cases, defective products can be repaired through rework. Scrap and rework cause a longer completion time, which can contribute to the order being shipped late. In addition, complex manufacturing scheduling becomes much more challenging when the above uncertainties are present. Motivated by the presence of uncertainties as well as combinatorial complexity, this paper addresses the challenge illustrated through a case study of stochastic job-shop scheduling problems arising within low-volume high-variety manufacturing. To ensure on-time delivery, high-quality solutions are required, and near-optimal solutions must be obtained within strict time constraints to ensure smooth operations on the job-shop floor. To efficiently solve the stochastic job-shop scheduling (JSS) problem, a recently-developed Surrogate "Level-Based" Lagrangian Relaxation is used to reduce computational effort while efficiently exploiting the geometric convergence potential inherent to Polyak's step-sizing formula thereby leading to fast convergence. Numerical testing demonstrates that the new method is more than two orders of magnitude faster as compared to commercial solvers

Applying Machine Based Decomposition in 2-Machine Flow Shops

The Shifting Bottleneck (SB) heuristic is among the most successful approximation methods for solving the Job Shop problem. It is essentially a machine based decomposition procedure where a series of One Machine Sequencing Problems (OMSPs) are solved. However, such a procedure has been reported to be highly ineffective for the Flow Shop problems (Jain and Meeran 2002). In particular, we show that for the 2-machine Flow Shop problem, the SB heurisitc will deliver the optimal solution in only a small number of instances. We examine the reason behind the failure of the machine based decomposition method for the Flow Shop. An optimal machine based decomposition procedure is formulated for the 2-machine Flow Shop, the time complexity of which is worse than that of the celebrated Johnsons Rule. The contribution of the present study lies in showing that the same machine based decomposition procedures which are so successful in solving complex Job Shops can also be suitably modified to optimally solve the simpler Flow Shops.

Mathematical Modelling and Methods for Load Balancing and Coordination of Multi-Robot Stations

The automotive industry is moving from mass production towards an individualized production, individualizing parts aims to improve product quality and to reduce costs and material waste. This thesis concerns aspects of load balancing and coordination of multi-robot stations in the automotive manufacturing industry, considering efficient algorithms required by an individualized production. The goal of the load balancing problem is to improve the equipment utilization. Several approaches for solving the load balancing problem are suggested along with details on mathematical tools and subroutines employed.Our contributions to the solution of the load balancing problem are fourfold. First, to circumvent robot coordination we construct disjoint robot programs, which require no coordination schemes, are flexible, admit competitive cycle times for several industrial instances, and may be preferred in an individualized production. Second, since solving the task assignment problem for generating the disjoint robot programs was found to be unreasonably time-consuming, we model it as a generalized unrelated parallel machine problem with set packing constraints and suggest a tailored Lagrangian-based branch-and-bound algorithm. Third, a continuous collision detection method needs to determine whether the sweeps of multiple moving robots are disjoint. We suggest using the maximum velocity of each robot along with distance computations at certain robot configurations to derive a function that provides lower bounds on the minimum distance between the sweeps. The lower bounding function is iteratively minimized and updated with new distance information; our method is substantially faster than previously developed methods. Fourth, to allow for load balancing of complex multi-robot stations we generalize the disjoint robot programs into sequences of such; for some instances this procedure provides a significant equipment utilization improvement in comparison with previous automated methods