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Multi-echelon inventory optimization (MEIO) plays a key role in a supply chain seeking to achieve specified customer service levels with a minimum capital in inventory. In this work, we propose a generalized MEIO model based on the Guaranteed Service approach to allocate safety stock levels across the network at the lowest holding cost. This model integrates several existing and some novel features that are usually present in pharmaceutical multi-echelon supply chains into a single model: review periods, manufacturing facilities, hybrid nodes (nodes with both internal and external demand), minimum order quantities (MOQ), and different service level performance indicators (fill rate and cycle service levels). We include a polynomial regression to approximate fill rates as a possible target measure to set safety stocks. To improve efficiency, we propose a nonlinear programming model to support decision making, which can be reformulated as a Quadratically Constrained Program (QCP), which leads to order of magnitude reductions in computational time. The performance of the model is evaluated by solving illustrative and real-world cases, and is validated with simulation.

We present a Julia package, DisjunctiveProgramming.jl, that extends the functionality in JuMP.jl to allow modeling problems via logical propositions and disjunctive constraints. Such models can then be reformulated into Mixed-Integer Programs (MIPs) that can be solved with the various MIP solvers supported by JuMP. To do so, logical propositions are converted to Conjunctive Normal Form (CNF) and reformulated into equivalent algebraic constraints. Disjunctions are reformulated into mixed-integer constraints via the reformulation technique specified by the user (Big-M or Hull reformulations). The package supports reformulations for disjunctions containing linear, quadratic, and nonlinear constraints., Comment: 6 pages

Optimization problems with discrete-continuous decisions are traditionally modeled in algebraic form via (non)linear mixed-integer programming. A more systematic approach to modeling such systems is to use Generalized Disjunctive Programming (GDP), which extends the Disjunctive Programming paradigm proposed by Egon Balas to allow modeling systems from a logic-based level of abstraction that captures the fundamental rules governing such systems via algebraic constraints and logic. Although GDP provides a more general way of modeling systems, it warrants further generalization to encompass systems presenting a hierarchical structure. This work extends the GDP literature to address three major alternatives for modeling and solving systems with nested (hierarchical) disjunctions: explicit nested disjunctions, equivalent single-level disjunctions, and flattening via basic steps. We also provide theoretical proofs on the relaxation tightness of such alternatives, showing that explicitly modeling nested disjunctions is superior to the traditional approach discussed in literature for dealing with nested disjunctions., Comment: 35 pages

Reinforcement learning (RL) has been widely applied to game-playing and surpassed the best human-level performance in many domains, yet there are few use-cases in industrial or commercial settings. We introduce OR-Gym, an open-source library for developing reinforcement learning algorithms to address operations research problems. In this paper, we apply reinforcement learning to the knapsack, multi-dimensional bin packing, multi-echelon supply chain, and multi-period asset allocation model problems, as well as benchmark the RL solutions against MILP and heuristic models. These problems are used in logistics, finance, engineering, and are common in many business operation settings. We develop environments based on prototypical models in the literature and implement various optimization and heuristic models in order to benchmark the RL results. By re-framing a series of classic optimization problems as RL tasks, we seek to provide a new tool for the operations research community, while also opening those in the RL community to many of the problems and challenges in the OR field., Comment: 29 pages, 10 figures

One of the most crucial challenges faced by the Li-ion battery community concerns the search for the minimum time charging without irreversibly damaging the cells. This can fall into solving large-scale nonlinear optimal control problems according to a battery model. Within this context, several model-based techniques have been proposed in the literature.However, the effectiveness of such strategies is significantly limited by model complexity and uncertainty. Additionally, it is difficult to track parameters related to aging and re-tune the model-based control policy. With the aim of overcoming these limitations, in this paper we propose a fast-charging strategy subject to safety constraints which relies on a model-free reinforcement learning framework. In particular, we focus on the policy gradient-based actor-critic algorithm, i.e., deep deterministic policy gradient (DDPG), in order to deal with continuous sets of actions and sets. The validity of the proposal is assessed in simulation when a reduced electrochemical model is considered as the real plant. Finally, the online adaptability of the proposed strategy in response to variations of the environment parameters is highlighted with consideration of state reduction., Comment: 8 pages

Here, we consider a regularized mean-field game model that features a low-order regularization. We prove the existence of solutions with positive density. To do so, we combine a priori estimates with the continuation method. In contrast with high-order regularizations, the low-order regularizations are easier to implement numerically. Moreover, our methods give a theoretical foundation for this approach., Comment: 15 pages