Optimisation problems involving multiple objectives are commonly found in real-world applications. The existence of conflicting objectives produces trade-offs where a solution can be better with... Show moreOptimisation problems involving multiple objectives are commonly found in real-world applications. The existence of conflicting objectives produces trade-offs where a solution can be better with respect to one objective but requires a compromise in the other objectives. In many real-world problems the relationship between objectives is unknown or uncertain, and it is common to find problems with non-conflicting objectives. Understanding these relationships has been proven useful in different ways. The search efficiency of a multi-objective optimisation algorithm can benefit if objectives that are not essential to describe the Pareto-optimal front are omitted during the search procedure. Analysts and decision makers might get a better understanding about exiting synergies between the objectives, in turn facilitating the decision-making process of identifying the best solution. One particular useful technique to capture the relationships between objective functions is to rely on correlation measures. This chapter explores the literature of finding correlations among objective functions in solving multi-objective optimisation problems. Particularly, we focus on innovization and objective reduction approaches. We explain different statistical correlation measures and also provide details of benchmark and real-world optimisation problems solved by exploiting the correlations. This chapter provides an insight in solving multi-objective optimisation problems by considering the correlation among objective functions. Show less
Next to the Pareto dominance relation, alternative order relations can be useful in many-objective optimization. In particular, it is interesting to extend the Pareto dominance relation in order to... Show moreNext to the Pareto dominance relation, alternative order relations can be useful in many-objective optimization. In particular, it is interesting to extend the Pareto dominance relation in order to make more pairs comparable and decrease the size of the optimal set (for discrete approximations in continuous optimization or discrete problems), which tends to grow exponentially with an increasing number of objectives. Here, we review some basic concepts in order theory with a focus on the concept of an order extension. Moreover, we will define and discuss properties of some orders that have been proposed as alternatives to the Pareto dominance relation in the context of dealing with larger number of objective functions. The chapter compares the different concepts and reveals commonalities between them. It will be shown that many of the order extensions proposed are special cases of cone orders. Therefore, the chapter puts particular emphasize on the concept of dominance cones and demonstrates how different ways of defining dominance cones, such as by means of angles or by trade-offs, can be mapped onto each other. Show less
Bossema, F.G.; Zwetsloot, C.P.A..; Smeets, I. 2023
This chapter describes the development of a math trail for high school students. In 2016, we developed this trail through Leiden (The Netherlands) during a student project for the Science... Show moreThis chapter describes the development of a math trail for high school students. In 2016, we developed this trail through Leiden (The Netherlands) during a student project for the Science Communication and Society specialization, a track for master students at the Faculty of Science at Leiden University. Our aim was to provide a guided trail through the city that links everyday sights to mathematical concepts within the curriculum of high school students between 13 and 15 years old. The entire project was carried out in 3 weeks. We did background research, consisting of literature reviews, target audience surveys with school children, and focus groups with teachers. Based on the conclusions from this background research, we developed questions that suited both the goal to make the math trail a fun experience that makes math less abstract and the goal to include questions from across the curriculum. In this chapter, we would like to share our insights from the background research and our experiences in developing a math trail. We moreover aim to provide those who are interested in designing a math trail in their city with a practical step-by-step plan and checklist. Show less
Mariot, L.; Jacobovic, D.; Bäck, T.H.W.; Hernandez-Castro J. 2022
