Understanding knee points in bicriteria problems and their implications as preferred solution principles

Abstract A knee point is almost always a preferred trade-off solution, if it exists in a bicriteria optimization problem. In this article, an attempt is made to improve understanding of a knee point and investigate the properties of a bicriteria problem that may exhibit a knee on its Pareto-optimal front. Past studies are reviewed and a couple of new definitions are suggested. Additionally, a knee region is defined for problems in which, instead of one, a set of knee-like solutions exists. Edge-knee solutions, which behave like knee solutions but lie near one of the extremes on the Pareto-optimal front, are also introduced. It is interesting that in many problem-solving tasks, despite the existence of a number of solution methodologies, only one or a few of them are commonly used. Here, it is argued that often such common solution principles are knee solutions to a bicriteria problem formed with two conflicting goals of the underlying problem-solving task. The argument is illustrated on a number of tasks, such as regression, sorting, clustering and a number of engineering designs. Keywords: knee pointpreferred solutionsbicriteria problemsevolutionary algorithmsNSGA-II Acknowledgements The study is funded by the Department of Science and Technology, Government of India, under the SERC-Engineering Sciences scheme (No. SR/S3/MERC/091/2009). Notes The word ‘knee’ is appropriate for a scenario in which both objectives are to be maximized, as then the front would look like a bended leg and the point of interest would lie on the knee part of the bended leg. The word ‘elbow’ is more appropriate for minimization problems but, owing to its wide-spread use, these solutions will be referred to as ‘knee’ points even for minimization problems.