{"id":"https://openalex.org/W2783758083","doi":"https://doi.org/10.1111/itor.12502","title":"Cybersecurity investments with nonlinear budget constraints and conservation laws: variational equilibrium, marginal expected utilities, and Lagrange multipliers","display_name":"Cybersecurity investments with nonlinear budget constraints and conservation laws: variational equilibrium, marginal expected utilities, and Lagrange multipliers","publication_year":2018,"publication_date":"2018-01-08","ids":{"openalex":"https://openalex.org/W2783758083","doi":"https://doi.org/10.1111/itor.12502","mag":"2783758083"},"language":"en","primary_location":{"id":"doi:10.1111/itor.12502","is_oa":false,"landing_page_url":"https://doi.org/10.1111/itor.12502","pdf_url":null,"source":{"id":"https://openalex.org/S267729","display_name":"International Transactions in Operational Research","issn_l":"0969-6016","issn":["0969-6016","1475-3995"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Transactions in Operational Research","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/A5039387072","display_name":"Gabriella Colajanni","orcid":"https://orcid.org/0000-0002-8183-241X"},"institutions":[{"id":"https://openalex.org/I39063666","display_name":"University of Catania","ror":"https://ror.org/03a64bh57","country_code":"IT","type":"education","lineage":["https://openalex.org/I39063666"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Gabriella Colajanni","raw_affiliation_strings":["Department of Mathematics and Computer Science University of Catania Italy","Department of Mathematics and Computer Science, University of Catania, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science University of Catania Italy","institution_ids":["https://openalex.org/I39063666"]},{"raw_affiliation_string":"Department of Mathematics and Computer Science, University of Catania, Italy","institution_ids":["https://openalex.org/I39063666"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5019303003","display_name":"Patrizia Daniele","orcid":"https://orcid.org/0000-0002-8170-9382"},"institutions":[{"id":"https://openalex.org/I39063666","display_name":"University of Catania","ror":"https://ror.org/03a64bh57","country_code":"IT","type":"education","lineage":["https://openalex.org/I39063666"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Patrizia Daniele","raw_affiliation_strings":["Department of Mathematics and Computer Science University of Catania Italy","Department of Mathematics and Computer Science, University of Catania, Italy"],"raw_orcid":"https://orcid.org/0000-0002-8170-9382","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science University of Catania Italy","institution_ids":["https://openalex.org/I39063666"]},{"raw_affiliation_string":"Department of Mathematics and Computer Science, University of Catania, Italy","institution_ids":["https://openalex.org/I39063666"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5021747750","display_name":"Sofia Giuffr\u00e9","orcid":"https://orcid.org/0000-0001-8503-9630"},"institutions":[{"id":"https://openalex.org/I59725666","display_name":"University of Reggio Calabria","ror":"https://ror.org/041sz8d87","country_code":"IT","type":"education","lineage":["https://openalex.org/I59725666"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Sofia Giuffr\u00e8","raw_affiliation_strings":["DIIES Mediterranea University of Reggio Calabria Italy","DIIES, Mediterranea University of Reggio Calabria, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"DIIES Mediterranea University of Reggio Calabria Italy","institution_ids":["https://openalex.org/I59725666"]},{"raw_affiliation_string":"DIIES, Mediterranea University of Reggio Calabria, Italy","institution_ids":["https://openalex.org/I59725666"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5072294897","display_name":"Anna Nagurney","orcid":"https://orcid.org/0000-0002-1645-5363"},"institutions":[{"id":"https://openalex.org/I24603500","display_name":"University of Massachusetts Amherst","ror":"https://ror.org/0072zz521","country_code":"US","type":"education","lineage":["https://openalex.org/I24603500"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Anna Nagurney","raw_affiliation_strings":["Department of Operations and Information Management, Isenberg School of Management University of Massachusetts Amherst MA 01003 USA","Department of Operations and Information Management, Isenberg School of Management, University of Massachusetts, Amherst, MA, 01003 USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Operations and Information Management, Isenberg School of Management University of Massachusetts Amherst MA 01003 USA","institution_ids":["https://openalex.org/I24603500"]},{"raw_affiliation_string":"Department of Operations and Information Management, Isenberg School of Management, University of Massachusetts, Amherst, MA, 01003 USA","institution_ids":["https://openalex.org/I24603500"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5019303003","https://openalex.org/A5021747750","https://openalex.org/A5039387072","https://openalex.org/A5072294897"],"corresponding_institution_ids":["https://openalex.org/I24603500","https://openalex.org/I39063666","https://openalex.org/I59725666"],"apc_list":{"value":3660,"currency":"USD","value_usd":3660},"apc_paid":null,"fwci":7.3999,"has_fulltext":false,"cited_by_count":44,"citation_normalized_percentile":{"value":0.97228055,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"25","issue":"5","first_page":"1443","last_page":"1464"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10734","display_name":"Information and Cyber Security","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/1710","display_name":"Information Systems"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10734","display_name":"Information and Cyber Security","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/1710","display_name":"Information Systems"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10400","display_name":"Network Security and Intrusion Detection","score":0.9944999814033508,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11864","display_name":"Supply Chain Resilience and Risk Management","score":0.9883999824523926,"subfield":{"id":"https://openalex.org/subfields/1408","display_name":"Strategy and Management"},"field":{"id":"https://openalex.org/fields/14","display_name":"Business, Management and Accounting"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/lagrange-multiplier","display_name":"Lagrange multiplier","score":0.898530125617981},{"id":"https://openalex.org/keywords/variational-inequality","display_name":"Variational inequality","score":0.753230631351471},{"id":"https://openalex.org/keywords/conservation-law","display_name":"Conservation law","score":0.5568585395812988},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5454505085945129},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5234231948852539},{"id":"https://openalex.org/keywords/economics","display_name":"Economics","score":0.5094776749610901},{"id":"https://openalex.org/keywords/nash-equilibrium","display_name":"Nash equilibrium","score":0.45536842942237854},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.4405458867549896},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.361766517162323},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.314761221408844}],"concepts":[{"id":"https://openalex.org/C73684929","wikidata":"https://www.wikidata.org/wiki/Q598870","display_name":"Lagrange multiplier","level":2,"score":0.898530125617981},{"id":"https://openalex.org/C161999928","wikidata":"https://www.wikidata.org/wiki/Q4556320","display_name":"Variational inequality","level":2,"score":0.753230631351471},{"id":"https://openalex.org/C3445786","wikidata":"https://www.wikidata.org/wiki/Q205805","display_name":"Conservation law","level":2,"score":0.5568585395812988},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5454505085945129},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5234231948852539},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.5094776749610901},{"id":"https://openalex.org/C46814582","wikidata":"https://www.wikidata.org/wiki/Q23389","display_name":"Nash equilibrium","level":2,"score":0.45536842942237854},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.4405458867549896},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.361766517162323},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.314761221408844},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"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:10.1111/itor.12502","is_oa":false,"landing_page_url":"https://doi.org/10.1111/itor.12502","pdf_url":null,"source":{"id":"https://openalex.org/S267729","display_name":"International Transactions in Operational Research","issn_l":"0969-6016","issn":["0969-6016","1475-3995"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Transactions in Operational Research","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":33,"referenced_works":["https://openalex.org/W791078186","https://openalex.org/W1516366611","https://openalex.org/W1543439990","https://openalex.org/W1972221945","https://openalex.org/W1977489026","https://openalex.org/W1977794212","https://openalex.org/W1983653915","https://openalex.org/W1992885019","https://openalex.org/W1994176432","https://openalex.org/W2015320990","https://openalex.org/W2026981261","https://openalex.org/W2037683468","https://openalex.org/W2039230544","https://openalex.org/W2044631974","https://openalex.org/W2056075452","https://openalex.org/W2067050450","https://openalex.org/W2078206210","https://openalex.org/W2088000393","https://openalex.org/W2098564215","https://openalex.org/W2131849901","https://openalex.org/W2148932990","https://openalex.org/W2180201744","https://openalex.org/W2286305083","https://openalex.org/W2330024298","https://openalex.org/W2343984271","https://openalex.org/W2517715553","https://openalex.org/W2548478361","https://openalex.org/W2562157208","https://openalex.org/W2583805138","https://openalex.org/W2596181392","https://openalex.org/W3160050409","https://openalex.org/W4234761190","https://openalex.org/W4321430205"],"related_works":["https://openalex.org/W2371300278","https://openalex.org/W2367026916","https://openalex.org/W2803931294","https://openalex.org/W2282604764","https://openalex.org/W2500870930","https://openalex.org/W2218535193","https://openalex.org/W4389370903","https://openalex.org/W2468739564","https://openalex.org/W2171981925","https://openalex.org/W2501065092"],"abstract_inverted_index":{"Abstract":[0],"In":[1],"this":[2,100],"paper,":[3],"we":[4,51],"propose":[5],"a":[6,40,60],"new":[7],"cybersecurity":[8,114],"investment":[9,115],"supply":[10],"chain":[11],"game":[12],"theory":[13],"model,":[14],"assuming":[15],"that":[16,89],"the":[17,20,28,53,70,73,76,80,83,90,129],"demands":[18],"for":[19,49],"product":[21,112],"are":[22],"known":[23],"and":[24,79,103,117,125],"fixed":[25],"and,":[26],"hence,":[27],"conservation":[29,77],"law":[30],"of":[31,72,75,82,95],"each":[32,96],"demand":[33],"market":[34],"is":[35,39],"fulfilled.":[36],"The":[37],"model":[38,44],"generalized":[41],"Nash":[42],"equilibrium":[43,111],"with":[45,59,109,121],"nonlinear":[46],"budget":[47],"constraints":[48],"which":[50,56],"define":[52],"variational":[54,61],"equilibrium,":[55],"provides":[57],"us":[58],"inequality":[62],"formulation.":[63],"We":[64,87],"construct":[65],"an":[66],"equivalent":[67],"formulation,":[68],"enabling":[69],"analysis":[71],"influence":[74],"laws":[78],"importance":[81],"associated":[84],"Lagrange":[85,101,118],"multipliers.":[86],"find":[88],"marginal":[91],"expected":[92],"transaction":[93],"utility":[94],"retailer":[97],"depends":[98],"on":[99],"multiplier":[102],"its":[104],"sign.":[105],"Finally,":[106],"numerical":[107],"examples":[108],"reported":[110],"flows,":[113],"levels,":[116],"multipliers,":[119],"along":[120],"individual":[122],"firm":[123],"vulnerability":[124],"network":[126],"vulnerability,":[127],"illustrate":[128],"obtained":[130],"results.":[131]},"counts_by_year":[{"year":2025,"cited_by_count":9},{"year":2024,"cited_by_count":6},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":9},{"year":2021,"cited_by_count":6},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}