This paper introduces a reinforcement learning approach to optimize the Stochastic Vehicle Routing Problem with Time Windows (SVRP), focusing on reducing travel costs in goods delivery. We develop a novel SVRP formulation that accounts for uncertain travel costs and demands, alongside specific customer time windows. An attention-based neural network trained through reinforcement learning is employed to minimize routing costs. Our approach addresses a gap in SVRP research, which traditionally relies on heuristic methods, by leveraging machine learning. The model outperforms the Ant-Colony Optimization algorithm, achieving a 1.73% reduction in travel costs. It uniquely integrates external information, demonstrating robustness in diverse environments, making it a valuable benchmark for future SVRP studies and industry application.

This article provides an overview of stochastic process and fundamental mathematical concepts that are important to understand. Stochastic variable is a variable that moves in random order. Exchange rates, interest rates or stock prices are stochastic in nature. Stochastic variables can follow wiener or Itos process. I will start by explaining what stochastic process is.

This paper studies the theoretical predictive properties of classes of forecast combination methods. The study is motivated by the recently developed Bayesian framework for synthesizing predictive densities: Bayesian predictive synthesis. A novel strategy based on continuous time stochastic processes is proposed and developed, where the combined predictive error processes are expressed as stochastic differential equations, evaluated using Ito's lemma. We show that a subclass of synthesis functions under Bayesian predictive synthesis, which we categorize as non-linear synthesis, entails an extra term that "corrects" the bias from misspecification and dependence in the predictive error process, effectively improving forecasts. Theoretical properties are examined and shown that this subclass improves the expected squared forecast error over any and all linear combination, averaging, and ensemble of forecasts, under mild conditions. We discuss the conditions for which this subclass outperforms others, and its implications for developing forecast combination methods. A finite sample simulation study is presented to illustrate our results.

Decision-making problems can be modeled as combinatorial optimization problems with Constraint Programming formalisms such as Constrained Optimization Problems. However, few Constraint Programming formalisms can deal with both optimization and uncertainty at the same time, and none of them are convenient to model problems we tackle in this paper. Here, we propose a way to deal with combinatorial optimization problems under uncertainty within the classical Constrained Optimization Problems formalism by injecting the Rank Dependent Utility from decision theory. We also propose a proof of concept of our method to show it is implementable and can solve concrete decision-making problems using a regular constraint solver, and propose a bot that won the partially observable track of the 2018 {\mu}RTS AI competition. Our result shows it is possible to handle uncertainty with regular Constraint Programming solvers, without having to define a new formalism neither to develop dedicated solvers. This brings new perspective to tackle uncertainty in Constraint Programming.

Among the languages used for representing goals, actions and their consequences on the world for decision making and planning, GDL (Game Description Language) has the ability to represent complex actions in potentially uncertain and competitive environments. The aim of this paper is to exploit stochastic constraint networks in order to provide compact representations of strategic games, and to identify optimal policies in those games with generic forward checking method. From this perspective, we develop a compiler allowing to translate games, described in GDL, into instances of the Stochastic Constraint Optimization Problem (SCSP). Our compiler is proved correct for the class GDL of games with complete information and oblivious environment. The interest of our approach is illustrated by solving several GDL games with a SCSP solver.

Many decision problems contain uncertainty. Data about events in the past may not be known exactly due to errors in measuring or difficulties in sampling, whilst data about events in the future may simply not be known with certainty. For example, when scheduling power stations, we need to cope with uncertainty in future energy demands. As a second example, nurse rostering in an accident and emergency department requires us to anticipate variability in workload. As a final example, when constructing a balanced bond portfolio, we must deal with uncertainty in the future price of bonds. To deal with such situations, [27] proposed an extension of constraint programming, called stochastic constraint programming, in which we distinguish between decision variables, which we are free to set, and stochastic (or observed) variables, which follow some probability distribution. A semantics for stochastic constraint programs based on policies was proposed and backtracking and forward checking algorithms to solve such stochastic constraint programs were presented.

To model combinatorial decision problems involving uncertainty and probability, we introduce stochastic constraint programming. Stochastic constraint programs contain both decision variables (which we can set) and stochastic variables (which follow a probability distribution). They combine together the best features of traditional constraint satisfaction, stochastic integer programming, and stochastic satisfiability. We give a semantics for stochastic constraint programs, and propose a number of complete algorithms and approximation procedures. Finally, we discuss a number of extensions of stochastic constraint programming to relax various assumptions like the independence between stochastic variables, and compare with other approaches for decision making under uncertainty.