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Approximation of Multiobjective Optimization Problems
We study optimization problems with multiple objectives. Such problems are pervasive across many diverse disciplines -- in economics, engineering, healthcare, biology, to name but a few -- and heuristic approaches to solve them have already been deployed in several areas, in both academia and industry. Hence, there is a real need for a rigorous investigation of the relevant questions. In such problems we are interested not in a single optimal solution, but in the tradeoff between the different objectives. This is captured by the tradeoff or Pareto curve, the set of all feasible solutions whose vector of the various objectives is not dominated by any other solution. Typically, we have a small number of objectives and we wish to plot the tradeoff curve to get a sense of the design space. Unfortunately, typically the tradeoff curve has exponential size for discrete optimization problems even for two objectives (and is typically infinite for continuous problems). Hence, a natural goal in this setting is, given an instance of a multiobjective problem, to efficiently obtain a ``good'' approximation to the entire solution space with ``few'' solutions. This has been the underlying goal in much of the research in the multiobjective area, with many heuristics proposed for this purpose, typically however without any performance guarantees or complexity analysis. We develop efficient algorithms for the succinct approximation of the Pareto set for a large class of multiobjective problems. First, we investigate the problem of computing a minimum set of solutions that approximates within a specified accuracy the Pareto curve of a multiobjective optimization problem. We provide approximation algorithms with tight performance guarantees for bi-objective problems and make progress for the more challenging case of three and more objectives. Subsequently, we propose and study the notion of the approximate convex Pareto set; a novel notion of approximation to the Pareto set, as the appropriate one for the convex setting. We characterize when such an approximation can be efficiently constructed and investigate the problem of computing minimum size approximate convex Pareto sets, both for discrete and convex problems. Next, we turn to the problem of approximating the Pareto set as efficiently as possible. To this end, we analyze the Chord algorithm, a popular, simple method for the succinct approximation of curves, which is widely used, under different names, in a variety of areas, such as, multiobjective and parametric optimization, computational geometry, and graphics
Knowledge creation and visualisation by using trade-off curves to enable set-based concurrent engineering applications
Inefficiencies that could be avoided during the product development process account for a large percentage of the manufacturing cost. To introduce innovative, high-quality products in a time- and cost-efficient manner, companies need to improve the performance of their product development processes. Set-based concurrent engineering (SBCE) has the capability of addressing this issue if the right knowledge-environment is provided. Trade-off curves (ToCs) are effective tools to provide this environment through knowledge creation and visualisation. However, there are several challenges that designers face during their product development activities such as rework, inaccurate decisions, and failure in design performance, which eventually cause waste. Therefore, the aim of this thesis is to eliminate waste by developing a systematic approach for generating and using ToCs. These then serve as a guide for designers to support their decision-making and achieve an efficient product development performance in an SBCE environment. To achieve this aim, qualitative research methods were employed. Following an extensive literature review, industrial field study and industrial applications, three processes were developed to generate ToCs and validated with five industrial case studies.
The process for generating knowledge-based ToCs describes how to create and visualise knowledge that is obtained from historical data and/or experience. This process facilitates the reuse of knowledge about existing products, in order to reduce the requirement for resources (e.g. product development time). The process for generating physics-based ToCs describes an approach to creating knowledge that is obtained from understanding the physics and functionality of the product under development. Thus, the practitioners gain sufficient confidence for identifying a compromise between conflicting design parameters. Finally, the process for using ToCs within the SBCE process model presents a technique to use generated knowledge-based and physics-based ToCs in order to enable key SBCE activities. These activities are (1) Identifying the feasible design area, (2) Developing a design-set, (3) Comparing possible design solutions, (4) Narrowing down the design-set and (5) Achieving the final optimal design solution.
For validation, the developed processes were applied in five industrial case studies, and two expert judgements were obtained. Findings showed that ToCs are essential tools in several aspects of new product development, specifically by reducing the lead time through enabling more confident and accurate decisions. Additionally, it was found that through ToCs, the conflicting relationships between the characteristics of the product can
be understood and communicated effectively among the designers. This facilitated the decision-making on an optimal design solution in a remarkably short period of time. The design performance of this optimal design increased by nearly 60% in a case study of a surface jet pump. Furthermore, it was found that ToCs have the capability of storing useful data for knowledge creation and reusing the created knowledge for the future projects