A key challenge in inventory management is to identify policies that optimally replenish inventory from multiple suppliers. To solve such optimization problems, inventory managers need to decide what quantities to order from each supplier given the net inventory and outstanding orders so that the expected backlogging, holding, and sourcing costs are jointly minimized. Inventory management problems have been studied extensively for more than 60 years, and yet even basic dual-sourcing problems, in which orders from an expensive supplier arrive faster than orders from a regular supplier, remain intractable in their general form. In addition, there is an emerging need to develop proactive, scalable optimization algorithms that can adjust their recommendations to dynamic demand shifts in a timely fashion. In this work, we approach dual sourcing from a neural network–based optimization lens and incorporate information on inventory dynamics and its replenishment (i.e., control) policies into the design of recurrent neural networks. We show that the proposed neural network controllers (NNCs) are able to learn near-optimal policies of commonly used instances within a few minutes of CPU time on a regular personal computer. To demonstrate the versatility of NNCs, we also show that they can control inventory dynamics with empirical, nonstationary demand distributions that are challenging to tackle effectively using alternative, state-of-the-art approaches. Our work shows that high-quality solutions of complex inventory management problems with nonstationary demand can be obtained with deep neural network optimization approaches that directly account for inventory dynamics in their optimization process. As such, our research opens up new ways of efficiently managing complex, high-dimensional inventory dynamics. History: Accepted by Ram Ramesh, Area Editor for Data Science & Machine Learning. Funding: This work was supported by Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (NCCR Automation) [Grant P2EZP2 191888] and the Army Research Office [Grant W911NF-23-1-0129]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0136 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0136 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Despite the significant attention they have drawn, big-bucket lot-sizing problems remain notoriously difficult to solve. Previous literature contained results (computational and theoretical) indicating that what makes these problems difficult are the embedded single-machine, single-level, multiperiod submodels. We therefore consider the simplest such submodel, a multi-item, two-period capacitated relaxation. We propose a methodology that can approximate the convex hulls of all such possible relaxations by generating violated valid inequalities. To generate such inequalities, we separate two-period projections of fractional linear programming solutions from the convex hulls of the two-period closure we study. The convex hull representation of the twoperiod closure is generated dynamically using column generation. Contrary to regular column generation, our method is an outer approximation and can therefore be used efficiently in a regular branch-and-bound procedure. We present computational results that illustrate how these two-period models could be effective in solving complicated problems.

We introduce horizon decomposition in the context of Dantzig-Wolfe decomposition, and apply it to the capacitated lot-sizing problem with setup times. We partition the problem horizon in contiguous overlapping intervals and create subproblems identical to the original problem, but of smaller size. The user has the flexibility to regulate the size of the master problem and the subproblem via two scalar parameters. We investigate empirically which parameter configurations are efficient, and assess their robustness at different problem classes. Our branch-and-price algorithm outperforms state-of-the-art branch-and-cut solvers when tested to a new data set of challenging instances that we generated. Our methodology can be generalized to mathematical programs with a generic constraint structure.

We study the multi-item capacitated lot sizing problem with setup times. Based on two strong reformulations of the problem, we present a transformed reformulation and valid inequalities that speed up column generation and Lagrange relaxation. We demonstrate computationally how both ideas enhance the performance of our algorithm and show theoretically how they are related to dual space reduction techniques. We compare several solution methods and propose a new efficient hybrid scheme that combines column generation and Lagrange relaxation in a novel way. Computational experiments show that the proposed solution method for finding lower bounds is competitive with textbook approaches and state-of-the-art approaches found in the literature. Finally, we design a branch-and-price-based heuristic and report computational results. The heuristic scheme compares favorably or outperforms other approaches.

Conservation of angular momentum that introduces nonholonomic behavior, underactuation and time dependence, makes the trajectory planning of gymnastic robots difficult. By defining appropriate values for the initial angular momentum, a method is developed that can lead a mechanism to a desired final configuration from an initial given one, in prescribed time. This method is optimization-based and fully exploits the initial mechanism angular momentum. Obstacle avoidance during flight is achieved by setting additional constraints. The method results in smooth, small in magnitude, and therefore easily applicable joint torques.