How people spend their finite time budget of 24 hours on daily activities is linked to their wellbeing. Yet, how to best allocate time to optimise multi-dimensional wellbeing (physical, mental and cognitive) remains unknown. Here, we utilise a number of (objective) functions derived using compositional data analysis and a large child cohort ( \(n>1{,}000\) ), to predict how time allocation is associated with wellbeing outcomes such as body mass index, life satisfaction and cognition. We develop and advocate joint cumulative distribution function constraints to ensure the feasible solutions do not extrapolate the sampled data for which the objective function is derived from. Moreover, we incorporate quality diversity (QD) approaches to study these objective functions. We define two types of behavioural spaces (BSs), one based on the activities, called the variable-based behavioural space (VBS), and the other based on the objectives, called the objective-based behavioural space (OBS). The VBS allows us to generate a set of high-quality solutions with different activity durations, while the OBS allows us to tradeoff different wellbeing dimensions against each other. We also demonstrate a web application, Time allocation optimiser, for creating personalised, optimised time-use plans.

How people spend their finite time budget of 24 hours on daily activities is linked to their wellbeing. Yet, how to best allocate time to optimise multi-dimensional wellbeing (physical, mental and cognitive) remains unknown. Here, we utilise a number of (objective) functions derived using compositional data analysis and a large child cohort (n > 1000), to predict how time allocation is associated with wellbeing outcomes such as body mass index, life satisfaction, and cognition. We develop and advocate joint cumulative distribution function constraints to ensure the feasible solutions do not extrapolate the sampled data for which the objective function is derived from. Moreover, we incorporate quality diversity (QD) approaches to study these objective functions. We define two types of behavioural spaces (BSs), one based on the activities, called the variable-based BS, and the other based on objectives. The variable-based BS aids in studying solution space and generating a set of high-quality solutions with different variable values, while the objective-based BS is beneficial in diversifying the objective values for a number of objective functions while optimising another. We also demonstrate a web application, Time allocation optimiser, providing personalised, optimised time-use plans.

In real-world optimisation, it is common to face several sub-problems interacting and forming the main problem. There is an inter-dependency between the sub-problems, making it impossible to solve such a problem by focusing on only one component. The travelling thief problem (TTP) belongs to this category and is formed by the integration of the travelling salesperson problem (TSP) and the knapsack problem (KP). In this paper, we investigate the inter-dependency of the TSP and the KP by means of quality diversity (QD) approaches. QD algorithms provide a powerful tool not only to obtain high-quality solutions but also to illustrate the distribution of high-performing solutions in the behavioural space. We introduce a multi-dimensional archive of phenotypic elites (MAP-Elites) based evolutionary algorithm using well-known TSP and KP search operators, taking the TSP and KP score as the behavioural descriptor. MAP-Elites algorithms are QD-based techniques to explore high-performing solutions in a behavioural space. Afterwards, we conduct comprehensive experimental studies that show the usefulness of using the QD approach applied to the TTP. First, we provide insights regarding high-quality TTP solutions in the TSP/KP behavioural space. Afterwards, we show that better solutions for the TTP can be obtained by using our QD approach, and it can improve the best-known solution for a number of TTP instances used for benchmarking in the literature.

Computing diverse solutions for a given problem, in particular evolutionary diversity optimisation (EDO), is a hot research topic in the evolutionary computation community. This paper studies the Boolean satisfiability problem (SAT) in the context of EDO. SAT is of great importance in computer science and differs from the other problems studied in EDO literature, such as KP and TSP. SAT is heavily constrained, and the conventional evolutionary operators are inefficient in generating SAT solutions. Our approach avails of the following characteristics of SAT: 1) the possibility of adding more constraints (clauses) to the problem to forbid solutions or to fix variables, and 2) powerful solvers in the literature, such as minisat. We utilise such a solver to construct a diverse set of solutions.

In real-world optimisation, it is common to face several sub-problems interacting and forming the main problem. There is an inter-dependency between the sub-problems, making it impossible to solve such a problem by focusing on only one component. The traveling thief problem (TTP) belongs to this category and is formed by the integration of the traveling salesperson problem (TSP) and the knapsack problem (KP). In this paper, we investigate the inter-dependency of the TSP and the KP by means of quality diversity (QD) approaches. QD algorithms provide a powerful tool not only to obtain high-quality solutions but also to illustrate the distribution of high-performing solutions in the behavioural space. We introduce a MAP-Elite based evolutionary algorithm using well-known TSP and KP search operators, taking the TSP and KP score as behavioural descriptor. Afterwards, we conduct comprehensive experimental studies that show the usefulness of using the QD approach applied to the TTP. First, we provide insights regarding high-quality TTP solutions in the TSP/KP behavioural space. Afterwards, we show that better solutions for the TTP can be obtained by using our QD approach and it can improve the best-known solution for a wide range of TTP instances used for benchmarking in the literature.

There has been a growing interest in the evolutionary computation community to compute a diverse set of high-quality solutions for a given optimisation problem. This can provide the practitioners with invaluable information about the solution space and robustness against imperfect modelling and minor problems' changes. It also enables the decision-makers to involve their interests and choose between various solutions. In this study, we investigate for the first time a prominent multi-component optimisation problem, namely the Traveling Thief Problem (TTP), in the context of evolutionary diversity optimisation. We introduce a bi-level evolutionary algorithm to maximise the structural diversity of the set of solutions. Moreover, we examine the inter-dependency among the components of the problem in terms of structural diversity and empirically determine the best method to obtain diversity. We also conduct a comprehensive experimental investigation to examine the introduced algorithm and compare the results to another recently introduced framework based on the use of Quality Diversity (QD). Our experimental results show a significant improvement of the QD approach in terms of structural diversity for most TTP benchmark instances.

Evolutionary algorithms based on edge assembly crossover (EAX) constitute some of the best performing incomplete solvers for the well-known traveling salesperson problem (TSP). Often, it is desirable to compute not just a single solution for a given problem, but a diverse set of high quality solutions from which a decision maker can choose one for implementation. Currently, there are only a few approaches for computing a diverse solution set for the TSP. Furthermore, almost all of them assume that the optimal solution is known. In this paper, we introduce evolutionary diversity optimisation (EDO) approaches for the TSP that find a diverse set of tours when the optimal tour is known or unknown. We show how to adopt EAX to not only find a high-quality solution but also to maximise the diversity of the population. The resulting EAX-based EDO approach, termed EAX-EDO is capable of obtaining diverse high-quality tours when the optimal solution for the TSP is known or unknown. A comparison to existing approaches shows that they are clearly outperformed by EAX-EDO.

Computing diverse sets of high-quality solutions has gained increasing attention among the evolutionary computation community in recent years. It allows practitioners to choose from a set of high-quality alternatives. In this paper, we employ a population diversity measure, called the high-order entropy measure, in an evolutionary algorithm to compute a diverse set of high-quality solutions for the Traveling Salesperson Problem. In contrast to previous studies, our approach allows diversifying segments of tours containing several edges based on the entropy measure. We examine the resulting evolutionary diversity optimisation approach precisely in terms of the final set of solutions and theoretical properties. Experimental results show significant improvements compared to a recently proposed edge-based diversity optimisation approach when working with a large population of solutions or long segments.