{"title":"Combining discrete and continuous information for multi-criteria optimization problems","authors":"","doi":"10.1007/s00186-024-00849-0","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>In multi-criteria optimization problems that originate from real-world decision making tasks, we often find the following structure: There is an underlying continuous, possibly even convex model for the multiple outcome measures depending on the design variables, but these outcomes are additionally assigned to discrete categories according to their desirability for the decision maker. Multi-criteria deliberations may then take place at the level of these discrete labels, while the calculation of a specific design remains a continuous problem. In this work, we analyze this type of problem and provide theoretical results about its solution set. We prove that the discrete decision problem can be tackled by solving scalarizations of the underlying continuous model. Based on our analysis we propose multiple algorithmic approaches that are specifically suited to handle these problems. We compare the algorithms based on a set of test problems. Furthermore, we apply our methods to a real-world radiotherapy planning example.</p>","PeriodicalId":49862,"journal":{"name":"Mathematical Methods of Operations Research","volume":"32 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Methods of Operations Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00186-024-00849-0","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In multi-criteria optimization problems that originate from real-world decision making tasks, we often find the following structure: There is an underlying continuous, possibly even convex model for the multiple outcome measures depending on the design variables, but these outcomes are additionally assigned to discrete categories according to their desirability for the decision maker. Multi-criteria deliberations may then take place at the level of these discrete labels, while the calculation of a specific design remains a continuous problem. In this work, we analyze this type of problem and provide theoretical results about its solution set. We prove that the discrete decision problem can be tackled by solving scalarizations of the underlying continuous model. Based on our analysis we propose multiple algorithmic approaches that are specifically suited to handle these problems. We compare the algorithms based on a set of test problems. Furthermore, we apply our methods to a real-world radiotherapy planning example.
期刊介绍：
This peer reviewed journal publishes original and high-quality articles on important mathematical and computational aspects of operations research, in particular in the areas of continuous and discrete mathematical optimization, stochastics, and game theory. Theoretically oriented papers are supposed to include explicit motivations of assumptions and results, while application oriented papers need to contain substantial mathematical contributions. Suggestions for algorithms should be accompanied with numerical evidence for their superiority over state-of-the-art methods. Articles must be of interest for a large audience in operations research, written in clear and correct English, and typeset in LaTeX. A special section contains invited tutorial papers on advanced mathematical or computational aspects of operations research, aiming at making such methodologies accessible for a wider audience.
All papers are refereed. The emphasis is on originality, quality, and importance.
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