The curse of dimensionality gives rise to prohibitive computational requirements that render infeasible the exact solution of large- scale stochastic control problems. We study an efficient method based on linear programming for approximating solutions to such prob(cid:173) lems. The approach "fits" a linear combination of pre- selected basis functions to the dynamic programming cost- to- go function. We develop bounds on the approximation error and present experi(cid:173) mental results in the domain of queueing network control, providing empirical support for the methodology.

Same-Day Delivery (SDD) services aim to maximize the fulfillment of online orders while minimizing delivery delays but are beset by operational uncertainties such as those in order volumes and courier planning. Our work aims to enhance the operational efficiency of SDD by focusing on the ultra-fast Order Dispatching Problem (ODP), which involves matching and dispatching orders to couriers within a centralized warehouse setting, and completing the delivery within a strict timeline (e.g., within minutes). We introduce important extensions to ultra-fast ODP such as order batching and explicit courier assignments to provide a more realistic representation of dispatching operations and improve delivery efficiency. As a solution method, we primarily focus on NeurADP, a methodology that combines Approximate Dynamic Programming (ADP) and Deep Reinforcement Learning (DRL), and our work constitutes the first application of NeurADP outside of the ride-pool matching problem. NeurADP is particularly suitable for ultra-fast ODP as it addresses complex one-to-many matching and routing intricacies through a neural network-based VFA that captures high-dimensional problem dynamics without requiring manual feature engineering as in generic ADP methods. We test our proposed approach using four distinct realistic datasets tailored for ODP and compare the performance of NeurADP against myopic and DRL baselines by also making use of non-trivial bounds to assess the quality of the policies. Our numerical results indicate that the inclusion of order batching and courier queues enhances the efficiency of delivery operations and that NeurADP significantly outperforms other methods. Detailed sensitivity analysis with important parameters confirms the robustness of NeurADP under different scenarios, including variations in courier numbers, spatial setup, vehicle capacity, and permitted delay time.

On-demand ride-pooling (e.g., UberPool) has recently become popular because of its ability to lower costs for passengers while simultaneously increasing revenue for drivers and aggregation companies. Unlike in Taxi on Demand (ToD) services -- where a vehicle is only assigned one passenger at a time -- in on-demand ride-pooling, each (possibly partially filled) vehicle can be assigned a group of passenger requests with multiple different origin and destination pairs. To ensure near real-time response, existing solutions to the real-time ride-pooling problem are myopic in that they optimise the objective (e.g., maximise the number of passengers served) for the current time step without considering its effect on future assignments. This is because even a myopic assignment in ride-pooling involves considering what combinations of passenger requests that can be assigned to vehicles, which adds a layer of combinatorial complexity to the ToD problem. A popular approach that addresses the limitations of myopic assignments in ToD problems is Approximate Dynamic Programming (ADP). Existing ADP methods for ToD can only handle Linear Program (LP) based assignments, however, while the assignment problem in ride-pooling requires an Integer Linear Program (ILP) with bad LP relaxations. To this end, our key technical contribution is in providing a general ADP method that can learn from ILP-based assignments. Additionally, we handle the extra combinatorial complexity from combinations of passenger requests by using a Neural Network based approximate value function and show a connection to Deep Reinforcement Learning that allows us to learn this value-function with increased stability and sample-efficiency. We show that our approach outperforms past approaches on a real-world dataset by up to 16%, a significant improvement in city-scale transportation problems.

Unmanned aircraft systems can perform some more dangerous and difficult missions than manned aircraft systems. In some highly complicated and changeable tasks, such as air combat, the maneuvering decision mechanism is required to sense the combat situation accurately and make the optimal strategy in real-time. This paper presents a formulation of a 3-D one-on-one air combat maneuvering problem and an approximate dynamic programming approach for computing an optimal policy on autonomous maneuvering decision making. The aircraft learns combat strategies in a Reinforcement Leaning method, while sensing the environment, taking available maneuvering actions and getting feedback reward signals. To solve the problem of dimensional explosion in the air combat, the proposed method is implemented through feature selection, trajectory sampling, function approximation and Bellman backup operation in the air combat simulation environment. This approximate dynamic programming approach provides a fast response to a rapidly changing tactical situation, learns in long-term planning, without any explicitly coded air combat rule base.

Real-world problems of operations research are typically high-dimensional and combinatorial. Linear programs are generally used to formulate and efficiently solve these large decision problems. However, in multi-period decision problems, we must often compute expected downstream values corresponding to current decisions. When applying stochastic methods to approximate these values, linear programs become restrictive for designing value function approximations (VFAs). In particular, the manual design of a polynomial VFA is challenging. This paper presents an integrated approach for complex optimization problems, focusing on applications in the domain of operations research. It develops a hybrid solution method that combines linear programming and neural networks as part of approximate dynamic programming. Our proposed solution method embeds neural network VFAs into linear decision problems, combining the nonlinear expressive power of neural networks with the efficiency of solving linear programs. As a proof of concept, we perform numerical experiments on a transportation problem. The neural network VFAs consistently outperform polynomial VFAs, with limited design and tuning effort.

A self-learning optimal control algorithm for sequential manufacturing processes with time-discrete control actions is proposed and evaluated with simulated deep drawing processes. The necessary control model is built during consecutive process executions under optimal control via Reinforcement Learning, using the measured product quality as reward after each process execution. Prior model formation, which is required by state-of-the-art algorithms like Model Predictive Control and Approximate Dynamic Programming, is therefore obsolete. This avoids the difficulties in system identification and accurate modelling, which arise with processes subject to non-linear dynamics and stochastic influences. Also runtime complexity problems of these approaches are avoided, which arise when more complex models and larger control prediction horizons are employed. Instead of using pre-created process- and observation-models, Reinforcement Learning algorithms build functions of expected future reward during processing, which are then used for optimal process control decisions. The learning of such expectation functions is realized online by interacting with the process. The proposed algorithm also takes stochastic variations of the process conditions into consideration and is able to cope with partial observability. A method for the adaptive optimal control of partially observable fixed-horizon manufacturing processes, based on Q-learning is developed and studied. The resulting algorithm is instantiated and then evaluated by application to a time-stochastic optimal control problem in metal sheet deep drawing, where the experiments use FEM-simulated processes. The Reinforcement Learning based control shows superior results over the model-based Model Predictive Control and Approximate Dynamic Programming approaches.

We propose a new class of computationally fast algorithms to find close to optimal policy for Markov Decision Processes (MDP) with large finite horizon T.The main idea is that instead of planning until the time horizon T, we plan only up to a truncated horizon H << T and use an estimate of the true optimal value function as the terminal value. Our approach of finding the terminal value function is to learn a mapping from an MDP to its value function by solving many similar MDPs during a training phase and fit a regression estimator. We analyze the method by providing an error propagation theorem that shows the effect of various sources of errors to the quality of the solution. We also empirically validate this approach in a real-world application of designing an energy management system for Hybrid Electric Vehicles with promising results.

Reinforcement learning has gained wide popularity as a technique for simulation-driven approximate dynamic programming. A less known aspect is that the very reasons that make it effective in dynamic programming can also be leveraged for using it for distributed schemes for certain matrix computations involving non-negative matrices. In this spirit, we propose a reinforcement learning algorithm for PageRank computation that is fashioned after analogous schemes for approximate dynamic programming. The algorithm has the advantage of ease of distributed implementation and more importantly, of being model-free, i.e., not dependent on any specific assumptions about the transition probabilities in the random web-surfer model. We analyze its convergence and finite time behavior and present some supporting numerical experiments.

This paper presents a novel non-parametric approximate dynamic programming (ADP) algorithm that enjoys graceful, dimension-independent approximation and sample complexity guarantees. In particular, we establish both theoretically and computationally that our proposal can serve as a viable alternative to state-of-the-art parametric ADP algorithms, freeing the designer from carefully specifying an approximation architecture. We accomplish this by developing a kernel-based mathematical program for ADP. Via a computational study on a controlled queueing network, we show that our non-parametric procedure is competitive with parametric ADP approaches.

Large Markov decision processes (MDPs) are common in reinforcement learning and operations research and are often solved by approximate dynamic programming (ADP). Many ADP algorithms have been developed and studied, often with impressive empirical performance. However, because many ADP methods must be carefully tuned to work well and offer insufficient theoretical guarantees, it is important to develop new methods that have both good theoretical guarantees and empirical performance. Approximate linear programming (ALP)--an ADP method--has been developed with the goal of achieving convergence and good theoretical guarantees (de Farias & van Roy, 2003). Approximate bilinear programming (ABP) improves on the theoretical properties of ALP at the cost of additional computational complexity (Petrik & Zilberstein, 2009, 2011).