{"id":"https://openalex.org/W4401387394","doi":"https://doi.org/10.1109/tevc.2024.3440571","title":"Exact Calculation and Properties of the <i>R</i>2 Multiobjective Quality Indicator","display_name":"Exact Calculation and Properties of the <i>R</i>2 Multiobjective Quality Indicator","publication_year":2024,"publication_date":"2024-08-07","ids":{"openalex":"https://openalex.org/W4401387394","doi":"https://doi.org/10.1109/tevc.2024.3440571"},"language":"en","primary_location":{"id":"doi:10.1109/tevc.2024.3440571","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tevc.2024.3440571","pdf_url":null,"source":{"id":"https://openalex.org/S93787993","display_name":"IEEE Transactions on Evolutionary Computation","issn_l":"1089-778X","issn":["1089-778X","1941-0026"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Evolutionary Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5011598685","display_name":"Andrzej Jaszkiewicz","orcid":"https://orcid.org/0000-0002-8337-888X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Andrzej Jaszkiewicz","raw_affiliation_strings":["Institute of Computing Science, Pozna&#x0144; University of Technology, Pozna&#x0144;, Poland","Institute of Computing Science, Pozna&#x0144; University of Technology, Poland"],"raw_orcid":"https://orcid.org/0000-0002-8337-888X","affiliations":[{"raw_affiliation_string":"Institute of Computing Science, Pozna&#x0144; University of Technology, Pozna&#x0144;, Poland","institution_ids":[]},{"raw_affiliation_string":"Institute of Computing Science, Pozna&#x0144; University of Technology, Poland","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5090952989","display_name":"Piotr Zielniewicz","orcid":"https://orcid.org/0000-0001-6996-3671"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Piotr Zielniewicz","raw_affiliation_strings":["Institute of Computing Science, Pozna&#x0144; University of Technology, Pozna&#x0144;, Poland"],"raw_orcid":"https://orcid.org/0000-0001-6996-3671","affiliations":[{"raw_affiliation_string":"Institute of Computing Science, Pozna&#x0144; University of Technology, Pozna&#x0144;, Poland","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.0209,"has_fulltext":false,"cited_by_count":7,"citation_normalized_percentile":{"value":0.91454971,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"29","issue":"4","first_page":"1227","last_page":"1238"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10050","display_name":"Multi-Criteria Decision Making","score":0.9350000023841858,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10050","display_name":"Multi-Criteria Decision Making","score":0.9350000023841858,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/quality","display_name":"Quality (philosophy)","score":0.5400934815406799},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.47446733713150024},{"id":"https://openalex.org/keywords/multi-objective-optimization","display_name":"Multi-objective optimization","score":0.4285238981246948},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.428295761346817},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4162842035293579},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3854734003543854},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08731284737586975}],"concepts":[{"id":"https://openalex.org/C2779530757","wikidata":"https://www.wikidata.org/wiki/Q1207505","display_name":"Quality (philosophy)","level":2,"score":0.5400934815406799},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.47446733713150024},{"id":"https://openalex.org/C68781425","wikidata":"https://www.wikidata.org/wiki/Q2052203","display_name":"Multi-objective optimization","level":2,"score":0.4285238981246948},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.428295761346817},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4162842035293579},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3854734003543854},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08731284737586975},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tevc.2024.3440571","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tevc.2024.3440571","pdf_url":null,"source":{"id":"https://openalex.org/S93787993","display_name":"IEEE Transactions on Evolutionary Computation","issn_l":"1089-778X","issn":["1089-778X","1941-0026"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Evolutionary Computation","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320321147","display_name":"Ministry of Higher Education","ror":"https://ror.org/0512bh102"},{"id":"https://openalex.org/F4320322733","display_name":"Ministerstwo Edukacji i Nauki","ror":"https://ror.org/05dwvd537"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":43,"referenced_works":["https://openalex.org/W1491287429","https://openalex.org/W1588375755","https://openalex.org/W2001646999","https://openalex.org/W2025822274","https://openalex.org/W2038420231","https://openalex.org/W2040622444","https://openalex.org/W2050541690","https://openalex.org/W2050914643","https://openalex.org/W2069192243","https://openalex.org/W2075016777","https://openalex.org/W2090797242","https://openalex.org/W2098907614","https://openalex.org/W2101991430","https://openalex.org/W2103640719","https://openalex.org/W2106334424","https://openalex.org/W2108968575","https://openalex.org/W2122297020","https://openalex.org/W2133275519","https://openalex.org/W2767528495","https://openalex.org/W2793738399","https://openalex.org/W2803331670","https://openalex.org/W2888529829","https://openalex.org/W2893653004","https://openalex.org/W2897286683","https://openalex.org/W2912436365","https://openalex.org/W2921981915","https://openalex.org/W2962840996","https://openalex.org/W2962987192","https://openalex.org/W2963054067","https://openalex.org/W3037787799","https://openalex.org/W3046910774","https://openalex.org/W3082605505","https://openalex.org/W3119402155","https://openalex.org/W3125481533","https://openalex.org/W3129740799","https://openalex.org/W3176248073","https://openalex.org/W3183976931","https://openalex.org/W4206797488","https://openalex.org/W4291418344","https://openalex.org/W4294811723","https://openalex.org/W4381249952","https://openalex.org/W4385189066","https://openalex.org/W6636952118"],"related_works":["https://openalex.org/W4318426920","https://openalex.org/W2055791821","https://openalex.org/W2153873031","https://openalex.org/W570431388","https://openalex.org/W2637862190","https://openalex.org/W2611723089","https://openalex.org/W2337862504","https://openalex.org/W3082547637","https://openalex.org/W1994436307","https://openalex.org/W2164920121"],"abstract_inverted_index":{"Quality":[0],"indicators":[1],"play":[2],"an":[3,205],"essential":[4],"role":[5],"in":[6,19,124,239,261],"evolutionary":[7],"multiobjective":[8],"optimization":[9],"(EMO).":[10],"Most":[11],"likely":[12],"the":[13,31,56,59,110,117,125,131,137,153,218,232],"most":[14],"often":[15],"used":[16],"quality":[17,85,101],"indicator":[18,102],"EMO":[20,207],"is":[21,36,64,79,103,152],"hypervolume,":[22],"due":[23],"to":[24,30,55,73,81,109,136,221,228,259],"its":[25,52],"strict":[26],"monotonicity":[27],"with":[28,107,165,192,208],"respect":[29,108],"dominance":[32,111],"relation.":[33],"However,":[34],"hypervolume":[35,63,134,209,260],"not":[37,235],"free":[38],"of":[39,47,58,140,199,243],"some":[40,74],"weak":[41],"points.":[42],"For":[43],"example,":[44],"a":[45,225,253,257],"number":[46],"recent":[48],"papers":[49],"pointed":[50],"out":[51],"high":[53,226],"sensitivity":[54],"specification":[57],"reference":[60,118],"point.":[61],"Furthermore,":[62,128],"based":[65],"on":[66],"fully":[67],"geometric":[68,240],"reasoning":[69],"which":[70],"may":[71],"lead":[72],"undesired":[75],"results.":[76],"Thus,":[77],"it":[78],"worth":[80],"consider":[82],"also":[83,104,197],"other":[84],"indicators.":[86],"In":[87,169],"this":[88,151],"article,":[89],"we":[90,129,178],"prove":[91],"that":[92,180],"another":[93],"well-known":[94],"<inline-formula":[95,141,158,181,211,247],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[96,142,159,182,212,248],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">":[97,143,160,183,213,249],"<tex-math":[98,144,161,184,214,250],"notation=\"LaTeX\">$R2$":[99,145,162,185,215,251],"</tex-math></inline-formula>":[100,146,163,186,252],"strictly":[105],"monotonic":[106],"relation":[112],"when":[113],"calculated":[114],"exactly":[115],"and":[116,175,210],"point":[119],"strongly":[120],"dominates":[121],"any":[122],"solution":[123],"evaluated":[126],"set.":[127],"adapt":[130],"improved":[132],"quick":[133],"algorithm":[135,156],"exact":[138,155],"calculation":[139,164],"indicator.":[147],"To":[148],"our":[149],"knowledge,":[150],"first":[154],"for":[157,189],"publicly":[166],"available":[167],"implementation.":[168],"addition,":[170],"through":[171],"both":[172],"theoretical":[173],"analysis":[174],"computational":[176],"experiments,":[177],"show":[179],"performs":[187],"consistently":[188],"Pareto":[190,200],"fronts":[191,201],"different":[193],"shapes.":[194],"We":[195],"discuss":[196],"differences":[198],"representations":[202],"generated":[203],"by":[204,231],"indicator-based":[206],"</tex-math></inline-formula>,":[216],"where":[217],"latter":[219],"tends":[220],"generate":[222],"solutions":[223],"having":[224],"chance":[227],"be":[229],"preferred":[230],"decision":[233],"maker,":[234],"necessarily":[236],"uniformly":[237],"distributed":[238],"sense.":[241],"All":[242],"these":[244],"results":[245],"make":[246],"sound":[254],"alternative":[255],"or":[256],"complement":[258],"EMO.":[262]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":1}],"updated_date":"2026-06-13T07:54:00.901334","created_date":"2024-08-08T00:00:00"}
