{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T12:44:45Z","timestamp":1753879485126,"version":"3.41.2"},"reference-count":34,"publisher":"ASME International","issue":"6","license":[{"start":{"date-parts":[[2021,5,13]],"date-time":"2021-05-13T00:00:00Z","timestamp":1620864000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.asme.org\/publications-submissions\/publishing-information\/legal-policies"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51605286"],"award-info":[{"award-number":["51605286"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["asmedigitalcollection.asme.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2021,12,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Multi-objective optimization (MOO) problems are encountered in many applications, of which bi-objective problems are frequently met. Despite the computational efforts, the quality of the Pareto front is also a considerable issue. An evenly distributed Pareto front is desirable in certain cases when a continuous representation of the Pareto front is needed. In this paper, a new approach called circle intersection (CI) is proposed. First, the anchor points are computed. Then in the normalized objective space, a circle with a proper radius of r centering at one of the anchor points or the latest obtained Pareto point is drawn. Interestingly, the intersection of the circle and the feasible boundary can be determined whether it is a Pareto point or not. For a convex or concave feasible boundary, the intersection is exactly the Pareto point, while for other cases, the intersection can provide useful information for searching the true Pareto point even if it is not a Pareto point. A novel MOO formulation is proposed for CI correspondingly. Sixteen examples are used to demonstrate the applicability of the proposed method and results are compared to those of normalized normal constraint (NNC), multi-objective grasshopper optimization algorithm (MOGOA), and non-dominated sorting genetic algorithm (NSGA-II). Computational results show that the proposed CI method is able to obtain a well-distributed Pareto front with a better quality or with less computational cost.<\/jats:p>","DOI":"10.1115\/1.4050471","type":"journal-article","created":{"date-parts":[[2021,3,11]],"date-time":"2021-03-11T16:36:11Z","timestamp":1615480571000},"update-policy":"https:\/\/doi.org\/10.1115\/crossmarkpolicy-asme","source":"Crossref","is-referenced-by-count":0,"title":["A Circle Intersection Method for Bi-Objective Optimization"],"prefix":"10.1115","volume":"21","author":[{"given":"Jianhua","family":"Zhou","sequence":"first","affiliation":[{"name":"School of Mechanical Engineering, Shanghai Dian Ji University, 300 Shui Hua Road, Shanghai 201306, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mian","family":"Li","sequence":"additional","affiliation":[{"name":"University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaojin","family":"Fu","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Shanghai Dian Ji University, 300 Shui Hua Road, Shanghai 201306, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"33","published-online":{"date-parts":[[2021,5,13]]},"reference":[{"key":"2021051316372115500_CIT0001","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/978-1-4899-3734-6","volume-title":"Multicriteria Optimization in Engineering and in the Sciences","author":"Stadler","year":"1988"},{"issue":"7","key":"2021051316372115500_CIT0002","doi-asserted-by":"publisher","first-page":"1329","DOI":"10.2514\/3.11771","article-title":"Multiobjective Optimization of Large-Scale Structures","volume":"31","author":"Grandhi","year":"1993","journal-title":"AIAA J."},{"issue":"1","key":"2021051316372115500_CIT0003","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1007\/BF01197559","article-title":"A Closer Look at Drawbacks of Minimizing Weighted Sums of Objectives for Pareto Set Generation in Multicriteria Optimization Problems","volume":"14","author":"Das","year":"1997","journal-title":"Struct. 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