Discrete representation of non-dominated sets in multi-objective linear programming

In this paper we address the problem of representing the continuous but non-convex set of nondominated points of a multi-objective linear programme by a finite subset of such points. We prove that a related decision problem is NP-complete. Moreover, we illustrate the drawbacks of the known global shooting, normal boundary intersection and normal constraint methods concerning the coverage error and uniformity level of the representation by examples. We propose a method which combines the global shooting and normal boundary intersection methods. By doing so, we overcome their limitations, but preserve their advantages. We prove that our method computes a set of evenly distributed non-dominated points for which the coverage error and the uniformity level can be guaranteed. We apply this method to an optimisation problem in radiation therapy and present illustrative results for some clinical cases. Finally, we present numerical results on randomly generated examples

Multiobjective navigation of external radiotherapy plans based on clinical criteria

This study considers a navigation method for finding the most preferable radiotherapy plan from a discrete set using planner-defined clinical criteria. The method is based on repeatedly solving an optimisation model to identify a plan that best satisfies the aspiration values set by the planner. During navigation, the planner iteratively adjusts the aspiration values to match the preference information learned from previous plans until the most preferable plan is identified. The use of soft constraints to model aspiration values enables navigation among a discrete set and allows the planner to freely specify the aspiration values without producing an infeasible model. We demonstrate the use of the model by applying it to a prostate cancer case. This illustrates that improvements in optimisation criteria do not necessarily lead to improvements in clinical criteria. Hence the method obviates the need to simultaneously monitor both optimisation and clinical criteria in current navigation systems. Instead, the direct use of clinical criteria for navigation aids the planner to quickly identify the most preferable plan

Integrating column generation in a method to compute a discrete representation of the non-dominated set of multi-objective linear programmes

In this paper we propose the integration of column generation in the revised normal boundary intersection (RNBI) approach to compute a representative set of non-dominated points for multi-objective linear programmes (MOLPs). The RNBI approach solves single objective linear programmes, the RNBI subproblems, to project a set of evenly distributed reference points to the non-dominated set of an MOLP. We solve each RNBI subproblem using column generation, which moves the current point in objective space of the MOLP towards the non-dominated set. Since RNBI subproblems may be infeasible, we attempt to detect this infeasibility early. First, a reference point bounding method is proposed to eliminate reference points that lead to infeasible RNBI subproblems. Furthermore, different initialisation approaches for column generation are implemented, including Farkas pricing. We investigate the quality of the representation obtained. To demonstrate the efficacy of the proposed approach, we apply it to an MOLP arising in radiotherapy treatment design. In contrast to conventional optimisation approaches, treatment design using column generation provides deliverable treatment plans, avoiding a segmentation step which deteriorates treatment quality. As a result total monitor units is considerably reduced. We also note that reference point bounding dramatically reduces the number of RNBI subproblems that need to be solved

Multi-objective decision-making for dietary assessment and advice

Unhealthy diets contribute substantially to the worldwide burden of non-communicable diseases, such as cardiovascular diseases, cancers, and diabetes. Globally, non-communicable diseases are the leading cause of death, and numbers are still rising, which makes healthy diets a global priority. In Nutrition Research, two fields are particularly relevant for formulating healthier diets: dietary assessment, which assesses food and nutrient intake in order to investigate the relation between diet and disease, and dietary advice, which translates food and nutrient recommendations into realistic food choices. Both fields face complex decision problems: which foods to include in dietary assessment or advice in order to pursue the multiple objectives of the researcher or fulfil the requirements of the consumer. This thesis connects the disciplines of Nutrition Research and Operations Research in order to contribute to formulating healthier diets. In the context of dietary assessment, the thesis proposes a MILP model for the selection of food items for food frequency questionnaires (a crucial tool in dietary assessment) that speeds up the selection process and increases standardisation, transparency, and reproducibility. An extension of this model gives rise to a 0-1 fractional programming problem with more than 200 fractional terms, of which in every feasible solution only a subset is actually defined. The thesis shows how this problem can be reformulated in order to eliminate the undefined fractional terms. The resulting MILP model can solved with standard software. In the context of dietary advice, the thesis proposes a diet model in which food and nutrient requirements are formulated via fuzzy sets. With this model, the impact of various achievement functions is demonstrated. The preference structures modelled via these achievement functions represent various ways in which multiple nutritional characteristics of a diet can be aggregated into an overall indicator for diet quality. Furthermore, for Operations Research the thesis provides new insights into a novel preference structure from literature, that combines equity and utilitarianism in a single model. Finally, the thesis presents conclusions of the research and a general discussion, which discusses, amongst others, the main modelling choices encountered when using MODM methods for optimising diet quality. Summarising, this thesis explores the use of MODM approaches to improve decision-making for dietary assessment and advice. It provides opportunities for better decision-making in research on dietary assessment and advice, and it contributes to model building and solving in Operations Research. Considering the added value for Nutrition Research and the new models and solutions generated, we conclude that the combination of both fields has resulted in synergy between Nutrition Research and Operations Research.</p