Wim Nuijten

Riccardo Lo Bianco, Remco Dijkman, Wim Nuijten et al.

Dynamic task assignment involves assigning arriving tasks to a limited number of resources in order to minimize the overall cost of the assignments. To achieve optimal task assignment, it is necessary to model the assignment problem first. While there exist separate formalisms, specifically Markov Decision Processes and (Colored) Petri Nets, to model, execute, and solve different aspects of the problem, there is no integrated modeling technique. To address this gap, this paper proposes Action-Evolution Petri Nets (A-E PN) as a framework for modeling and solving dynamic task assignment problems. A-E PN provides a unified modeling technique that can represent all elements of dynamic task assignment problems. Moreover, A-E PN models are executable, which means they can be used to learn close-to-optimal assignment policies through Reinforcement Learning (RL) without additional modeling effort. To evaluate the framework, we define a taxonomy of archetypical assignment problems. We show for three cases that A-E PN can be used to learn close-to-optimal assignment policies. Our results suggest that A-E PN can be used to model and solve a broad range of dynamic task assignment problems.

Minshuo Li, Yaoxin Wu, Pavel Troubil et al.

Complex real-world optimization problems often involve both discrete decisions and nonlinear relationships between variables. Many such problems can be modeled as polynomial-objective integer programs, encompassing cases with quadratic and higher-degree variable interactions. Nonlinearity makes them more challenging than their linear counterparts. In this paper, we propose a hypergraph neural network (HNN) based method to solve polynomial-objective integer programming (POIP). Besides presenting a high-degree-term-aware hypergraph representation to capture both high-degree information and variable-constraint interdependencies, we also propose a hypergraph neural network, which integrates convolution between variables and high-degree terms alongside convolution between variables and constraints, to predict solution values. Finally, a search process initialized from the predicted solutions is performed to further refine the results. Comprehensive experiments across a range of benchmarks demonstrate that our method consistently outperforms both existing learning-based approaches and state-of-the-art solvers, delivering superior solution quality with favorable efficiency. Note that our experiments involve both polynomial objectives and constraints, demonstrating our HNN's versatility for general POIP problems and highlighting its advancement over the existing literature.

Riccardo Lo Bianco, Remco Dijkman, Wim Nuijten et al.

Dynamic task assignment concerns the optimal assignment of resources to tasks in a business process. Recently, Deep Reinforcement Learning (DRL) has been proposed as the state of the art for solving assignment problems. DRL methods usually employ a neural network (NN) as an approximator for the policy function, which ingests the state of the process and outputs a valuation of the possible assignments. However, representing the state and the possible assignments so that they can serve as inputs and outputs for a policy NN remains an open challenge, especially when tasks or resources have features with an infinite number of possible values. To solve this problem, this paper proposes a method for representing and solving assignment problems with infinite state and action spaces. In doing so, it provides three contributions: (I) A graph-based feature representation of assignment problems, which we call assignment graph; (II) A mapping from marked Colored Petri Nets to assignment graphs; (III) An adaptation of the Proximal Policy Optimization algorithm that can learn to solve assignment problems represented through assignment graphs. To evaluate the proposed representation method, we model three archetypal assignment problems ranging from finite to infinite state and action space dimensionalities. The experiments show that the method is suitable for representing and learning close-to-optimal task assignment policies regardless of the state and action space dimensionalities.

Thijs Overmeer, Tim Janssen, Wim P. M. Nuijten

This paper introduces the first Expected Possession Value (EPV) benchmark and a new and improved EPV model for football. Through the introduction of the OJN-Pass-EPV benchmark, we present a novel method to quantitatively assess the quality of EPV models by using pairs of game states with given relative EPVs. Next, we attempt to replicate the results of Fernández et al. (2021) using a dataset containing Dutch Eredivisie and World Cup matches. Following our failure to do so, we propose a new architecture based on U-net-type convolutional neural networks, achieving good results in model loss and Expected Calibration Error. Finally, we present an improved pass model that incorporates ball height and contains a new dual-component pass value model that analyzes reward and risk. The resulting EPV model correctly identifies the higher value state in 78% of the game state pairs in the OJN-Pass-EPV benchmark, demonstrating its ability to accurately assess goal-scoring potential. Our findings can help assess the quality of EPV models, improve EPV predictions, help assess potential reward and risk of passing decisions, and improve player and team performance.

Igor G. Smit, Yaoxin Wu, Pavel Troubil et al.

Neural combinatorial optimization (NCO) has gained significant attention due to the potential of deep learning to efficiently solve combinatorial optimization problems. NCO has been widely applied to job shop scheduling problems (JSPs) with the current focus predominantly on deterministic problems. In this paper, we propose a novel attention-based scenario processing module (SPM) to extend NCO methods for solving stochastic JSPs. Our approach explicitly incorporates stochastic information by an attention mechanism that captures the embedding of sampled scenarios (i.e., an approximation of stochasticity). Fed with the embedding, the base neural network is intervened by the attended scenarios, which accordingly learns an effective policy under stochasticity. We also propose a training paradigm that works harmoniously with either the expected makespan or Value-at-Risk objective. Results demonstrate that our approach outperforms existing learning and non-learning methods for the flexible JSP problem with stochastic processing times on a variety of instances. In addition, our approach holds significant generalizability to varied numbers of scenarios and disparate distributions.

Igor G. Smit, Jianan Zhou, Robbert Reijnen et al.

Job shop scheduling problems (JSSPs) represent a critical and challenging class of combinatorial optimization problems. Recent years have witnessed a rapid increase in the application of graph neural networks (GNNs) to solve JSSPs, albeit lacking a systematic survey of the relevant literature. This paper aims to thoroughly review prevailing GNN methods for different types of JSSPs and the closely related flow-shop scheduling problems (FSPs), especially those leveraging deep reinforcement learning (DRL). We begin by presenting the graph representations of various JSSPs, followed by an introduction to the most commonly used GNN architectures. We then review current GNN-based methods for each problem type, highlighting key technical elements such as graph representations, GNN architectures, GNN tasks, and training algorithms. Finally, we summarize and analyze the advantages and limitations of GNNs in solving JSSPs and provide potential future research opportunities. We hope this survey can motivate and inspire innovative approaches for more powerful GNN-based approaches in tackling JSSPs and other scheduling problems.