Scalarisation functions are widely employed in MORL algorithms to enable intelligent decision-making. However, these functions often struggle to approximate the Pareto front accurately, rendering them unideal in complex, uncertain environments. This study examines selected Multi-Objective Reinforcement Learning (MORL) algorithms across MORL environments with discrete action and observation spaces. We aim to investigate further the limitations associated with scalarisation approaches for decision-making in multi-objective settings. Specifically, we use an outer-loop multi-policy methodology to assess the performance of a seminal single-policy MORL algorithm, MO Q-Learning implemented with linear scalarisation and Chebyshev scalarisation functions. In addition, we explore a pioneering inner-loop multi-policy algorithm, Pareto Q-Learning, which offers a more robust alternative. Our findings reveal that the performance of the scalarisation functions is highly dependent on the environment and the shape of the Pareto front. These functions often fail to retain the solutions uncovered during learning and favour finding solutions in certain regions of the solution space. Moreover, finding the appropriate weight configurations to sample the entire Pareto front is complex, limiting their applicability in uncertain settings. In contrast, inner-loop multi-policy algorithms may provide a more sustainable and generalizable approach and potentially facilitate intelligent decision-making in dynamic and uncertain environments.

Matching in two-sided markets such as ride-hailing has recently received significant attention. However, existing studies on ride-hailing mainly focus on optimising efficiency, and fairness issues in ride-hailing have been neglected. Fairness issues in ride-hailing, including significant earning differences between drivers and variance of passenger waiting times among different locations, have potential impacts on economic and ethical aspects. The recent studies that focus on fairness in ride-hailing exploit traditional optimisation methods and the Markov Decision Process to balance efficiency and fairness. However, there are several issues in these existing studies, such as myopic short-term decision-making from traditional optimisation and instability of fairness in a comparably longer horizon from both traditional optimisation and Markov Decision Process-based methods. To address these issues, we propose a dynamic Markov Decision Process model to alleviate fairness issues currently faced by ride-hailing, and seek a balance between efficiency and fairness, with two distinct characteristics: (i) a prediction module to predict the number of requests that will be raised in the future from different locations to allow the proposed method to consider long-term fairness based on the whole timeline instead of consider fairness only based on historical and current data patterns; (ii) a customised scalarisation function for multi-objective multi-agent Q Learning that aims to balance efficiency and fairness. Extensive experiments on a publicly available real-world dataset demonstrate that our proposed method outperforms existing state-of-the-art methods.

Robust optimisation is a well-established framework for optimising functions in the presence of uncertainty. The inherent goal of this problem is to identify a collection of inputs whose outputs are both desirable for the decision maker, whilst also being robust to the underlying uncertainties in the problem. In this work, we study the multi-objective case of this problem. We identify that the majority of all robust multi-objective algorithms rely on two key operations: robustification and scalarisation. Robustification refers to the strategy that is used to account for the uncertainty in the problem. Scalarisation refers to the procedure that is used to encode the relative importance of each objective to a scalar-valued reward. As these operations are not necessarily commutative, the order that they are performed in has an impact on the resulting solutions that are identified and the final decisions that are made. The purpose of this work is to give a thorough exposition on the effects of these different orderings and in particular highlight when one should opt for one ordering over the other. As part of our analysis, we showcase how many existing risk concepts can be integrated into the specification and solution of a robust multi-objective optimisation problem. Besides this, we also demonstrate how one can principally define the notion of a robust Pareto front and a robust performance metric based on our ``robustify and scalarise'' methodology. To illustrate the efficacy of these new ideas, we present two insightful case studies which are based on real-world data sets.

The goal of multi-objective optimisation is to identify the Pareto front surface which is the set obtained by connecting the best trade-off points. Typically this surface is computed by evaluating the objectives at different points and then interpolating between the subset of the best evaluated trade-off points. In this work, we propose to parameterise the Pareto front surface using polar coordinates. More precisely, we show that any Pareto front surface can be equivalently represented using a scalar-valued length function which returns the projected length along any positive radial direction. We then use this representation in order to rigorously develop the theory and applications of stochastic Pareto front surfaces. In particular, we derive many Pareto front surface statistics of interest such as the expectation, covariance and quantiles. We then discuss how these can be used in practice within a design of experiments setting, where the goal is to both infer and use the Pareto front surface distribution in order to make effective decisions. Our framework allows for clear uncertainty quantification and we also develop advanced visualisation techniques for this purpose. Finally we discuss the applicability of our ideas within multivariate extreme value theory and illustrate our methodology in a variety of numerical examples, including a case study with a real-world air pollution data set.

Multi-objective reinforcement learning (MORL) is a relatively new field which builds on conventional Reinforcement Learning (RL) to solve multi-objective problems. One of common algorithm is to extend scalar value Q-learning by using vector Q values in combination with a utility function, which captures the user's preference for action selection. This study follows on prior works, and focuses on what factors influence the frequency with which value-based MORL Q-learning algorithms learn the optimal policy for an environment with stochastic state transitions in scenarios where the goal is to maximise the Scalarised Expected Return (SER) - that is, to maximise the average outcome over multiple runs rather than the outcome within each individual episode. The analysis of the interaction between stochastic environment and MORL Q-learning algorithms run on a simple Multi-objective Markov decision process (MOMDP) Space Traders problem with different variant versions. The empirical evaluations show that well designed reward signal can improve the performance of the original baseline algorithm, however it is still not enough to address more general environment. A variant of MORL Q-Learning incorporating global statistics is shown to outperform the baseline method in original Space Traders problem, but remains below 100% effectiveness in finding the find desired SER-optimal policy at the end of training. On the other hand, Option learning is guarantied to converge to desired SER-optimal policy but it is not able to scale up to solve more complex problem in real-life. The main contribution of this thesis is to identify the extent to which the issue of noisy Q-value estimates impacts on the ability to learn optimal policies under the combination of stochastic environments, non-linear utility and a constant learning rate. In conclusion, this study presents several alternative methods that may be more suitable to overcome noisy Q value estimate issue and also find SER optimal policy in MOMDPs with stochastic transitions.