Two-stage robust mixed integer programming problem with objective uncertainty
Many real problems with uncertain parameters can be modeled as two-stage robust mixed integer programming problems (RMIPs). Due to the complex nature of this kind of problems, this paper focuses on the two-stage RMIPs with objective uncertainty. Based on the results that the augmented Lagrangian is...
Uloženo v:
| Vydáno v: | Optimization letters Ročník 12; číslo 5; s. 959 - 969 |
|---|---|
| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.07.2018
|
| Témata: | |
| ISSN: | 1862-4472, 1862-4480 |
| On-line přístup: | Získat plný text |
| Tagy: |
Přidat tag
Žádné tagy, Buďte první, kdo vytvoří štítek k tomuto záznamu!
|
| Abstract | Many real problems with uncertain parameters can be modeled as two-stage robust mixed integer programming problems (RMIPs). Due to the complex nature of this kind of problems, this paper focuses on the two-stage RMIPs with objective uncertainty. Based on the results that the augmented Lagrangian is a strong dual for integer programming Boland and Eberhard (Math Program 150(2):491–509,
2015
), we present the upper and lower bounds. In a special case, we show that the two-stage RMIPs can be equivalently reformulated as a solvable minimax problem. |
|---|---|
| AbstractList | Many real problems with uncertain parameters can be modeled as two-stage robust mixed integer programming problems (RMIPs). Due to the complex nature of this kind of problems, this paper focuses on the two-stage RMIPs with objective uncertainty. Based on the results that the augmented Lagrangian is a strong dual for integer programming Boland and Eberhard (Math Program 150(2):491–509,
2015
), we present the upper and lower bounds. In a special case, we show that the two-stage RMIPs can be equivalently reformulated as a solvable minimax problem. |
| Author | Zhang, Ning |
| Author_xml | – sequence: 1 givenname: Ning orcidid: 0000-0001-5487-6524 surname: Zhang fullname: Zhang, Ning email: ningzhang_2008@yeah.net organization: Department of Mathematics, National University of Singapore, Future Resilient Systems, Singapore ETH Centre |
| BookMark | eNp9kM1KAzEURoNUsK0-gLt5gWhuJjNJllL8g4qbug7JNDOmdJKSpNb26Z1SceGiq_tx-c6FeyZo5IO3CN0CuQNC-H0CqCTBBDgG4DU-XKAxiJpixgQZ_WVOr9AkpRUhNYCUY_S22AWcsu5sEYPZplz07tsuC-ez7WwsNjF0Ufe9890xm7Xti53Ln0UwK9tk92WLrW9szHog9tfostXrZG9-5xR9PD0uZi94_v78OnuY44YKkTEH2UpruRS1AF1WghtZyUrThiwla4lhjLS8rOmwF1qWhmpaWijFUrPKsLacIn6628SQUrStalzW2QWfo3ZrBUQdraiTFTVYUUcr6jCQ8I_cRNfruD_L0BOThq4frKhV2EY_PHgG-gGBiXh_ |
| CitedBy_id | crossref_primary_10_1287_ijoc_2019_0928 |
| Cites_doi | 10.1016/0012-365X(79)90147-X 10.1007/BF01585518 10.1007/BF01386316 10.1007/s10107-014-0818-5 10.1007/s10107-016-1012-8 10.1287/opre.2013.1172 10.1287/opre.2015.1365 10.1007/s10107-003-0454-y 10.1287/moor.1100.0444 10.1287/opre.2015.1392 10.1515/9781400873173 10.1515/9781400831050 10.1137/1016053 10.1016/0012-365X(77)90028-0 10.1287/moor.1090.0440 10.1007/BF01583794 10.1287/opre.1080.0605 10.1007/s10107-014-0763-3 10.1287/ijoc.2016.0696 10.1109/CDC.2011.6161382 |
| ContentType | Journal Article |
| Copyright | Springer-Verlag GmbH Germany 2017 |
| Copyright_xml | – notice: Springer-Verlag GmbH Germany 2017 |
| DBID | AAYXX CITATION |
| DOI | 10.1007/s11590-017-1176-z |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Engineering Mathematics |
| EISSN | 1862-4480 |
| EndPage | 969 |
| ExternalDocumentID | 10_1007_s11590_017_1176_z |
| GrantInformation_xml | – fundername: National Research Foundation Singapore grantid: FI 370074011 funderid: http://dx.doi.org/10.13039/501100001381 |
| GroupedDBID | -5D -5G -BR -EM -Y2 -~C .VR 06D 0R~ 0VY 123 1N0 203 2J2 2JN 2JY 2KG 2KM 2LR 2VQ 2~H 30V 4.4 406 408 409 40D 40E 5VS 67Z 6NX 8TC 8UJ 95- 95. 95~ 96X AAAVM AABHQ AACDK AAHNG AAIAL AAJBT AAJKR AANZL AARHV AARTL AASML AATNV AATVU AAUYE AAWCG AAYIU AAYQN AAYTO AAYZH ABAKF ABBXA ABDZT ABECU ABFTV ABHQN ABJNI ABJOX ABKCH ABMNI ABMQK ABNWP ABQBU ABSXP ABTEG ABTHY ABTKH ABTMW ABULA ABWNU ABXPI ACAOD ACBXY ACDTI ACGFS ACHSB ACHXU ACKNC ACMDZ ACMLO ACOKC ACOMO ACPIV ACSNA ACZOJ ADHHG ADHIR ADINQ ADKNI ADKPE ADRFC ADTPH ADURQ ADYFF ADZKW AEBTG AEFQL AEGAL AEGNC AEJHL AEJRE AEMSY AENEX AEOHA AEPYU AESKC AETLH AEVLU AEXYK AFBBN AFGCZ AFLOW AFQWF AFWTZ AFZKB AGAYW AGDGC AGJBK AGMZJ AGQEE AGQMX AGRTI AGWIL AGWZB AGYKE AHAVH AHBYD AHKAY AHSBF AHYZX AIAKS AIGIU AIIXL AILAN AITGF AJBLW AJRNO AJZVZ ALMA_UNASSIGNED_HOLDINGS ALWAN AMKLP AMXSW AMYLF AMYQR AOCGG ARMRJ AXYYD AYJHY B-. BA0 BAPOH BDATZ BGNMA BSONS CAG COF CS3 CSCUP DDRTE DNIVK DPUIP DU5 EBLON EBS EIOEI EJD ESBYG FERAY FFXSO FIGPU FINBP FNLPD FRRFC FSGXE FWDCC GGCAI GGRSB GJIRD GNWQR GQ6 GQ7 GQ8 GXS H13 HF~ HG5 HG6 HLICF HMJXF HQYDN HRMNR HZ~ IJ- IKXTQ IWAJR IXC IXD IXE IZIGR IZQ I~X I~Z J-C J0Z J9A JBSCW JCJTX JZLTJ KDC KOV LLZTM M4Y MA- N9A NPVJJ NQJWS NU0 O9- O93 O9J OAM P2P P9M PF0 PT4 QOS R89 R9I ROL RPX RSV S16 S1Z S27 S3B SAP SDH SHX SISQX SJYHP SMT SNE SNPRN SNX SOHCF SOJ SPISZ SRMVM SSLCW STPWE SZN T13 TSG TSK TSV TUC U2A UG4 UOJIU UTJUX UZXMN VC2 VFIZW W23 W48 WK8 YLTOR Z45 Z7R Z7Y Z83 Z88 ZMTXR ~A9 AAPKM AAYXX ABBRH ABDBE ABFSG ABRTQ ACSTC AEZWR AFDZB AFHIU AFOHR AHPBZ AHWEU AIXLP ATHPR AYFIA CITATION |
| ID | FETCH-LOGICAL-c288t-719f9ee798681a3587b9595a2c0d94f0b440f73627b98a93b2a23e138da45b4f3 |
| IEDL.DBID | RSV |
| ISICitedReferencesCount | 1 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000436825400003&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 1862-4472 |
| IngestDate | Sat Nov 29 02:44:09 EST 2025 Tue Nov 18 22:24:42 EST 2025 Fri Feb 21 02:36:59 EST 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 5 |
| Keywords | Mixed integer programming Objective uncertainty Two-stage robust optimization Augmented Lagrangian |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c288t-719f9ee798681a3587b9595a2c0d94f0b440f73627b98a93b2a23e138da45b4f3 |
| ORCID | 0000-0001-5487-6524 |
| PageCount | 11 |
| ParticipantIDs | crossref_citationtrail_10_1007_s11590_017_1176_z crossref_primary_10_1007_s11590_017_1176_z springer_journals_10_1007_s11590_017_1176_z |
| PublicationCentury | 2000 |
| PublicationDate | 20180700 2018-7-00 |
| PublicationDateYYYYMMDD | 2018-07-01 |
| PublicationDate_xml | – month: 7 year: 2018 text: 20180700 |
| PublicationDecade | 2010 |
| PublicationPlace | Berlin/Heidelberg |
| PublicationPlace_xml | – name: Berlin/Heidelberg |
| PublicationTitle | Optimization letters |
| PublicationTitleAbbrev | Optim Lett |
| PublicationYear | 2018 |
| Publisher | Springer Berlin Heidelberg |
| Publisher_xml | – name: Springer Berlin Heidelberg |
| References | Benders (CR1) 1962; 4 Boland, Eberhard (CR11) 2015; 150 Chen, Zhang (CR12) 2009; 57 Bertsimas, Goyal (CR5) 2010; 35 Meyer (CR16) 1974; 7 Blair, Jeroslow (CR9) 1979; 25 Ben-Tal, El Ghaoui, Nemirovski (CR3) 2009 Bertsimas, Iancu, Parrilo (CR4) 2010; 35 Feizollahi, Ahmed, Sun (CR13) 2017; 161 Simons, Du, Pardalos (CR19) 1995 Wets (CR21) 1974; 16 Blair, Jeroslow (CR10) 1982; 23 Bertsimas, Bidkhori (CR6) 2015; 153 CR20 Ben-Tal, Goryashko, Guslitzer, Nemirovski (CR2) 2004; 99 Iancu, Sharma, Sviridenko (CR15) 2013; 61 Rockafellar (CR18) 1970 Blair, Jeroslow (CR8) 1977; 19 Bertsimas, Georghiou (CR7) 2015; 63 Hanasusanto, Kuhn, Wiesemann (CR14) 2015; 63 Postek, Hertog (CR17) 2016; 28 NL Boland (1176_CR11) 2015; 150 1176_CR20 D Bertsimas (1176_CR4) 2010; 35 CE Blair (1176_CR10) 1982; 23 RT Rockafellar (1176_CR18) 1970 GA Hanasusanto (1176_CR14) 2015; 63 D Bertsimas (1176_CR5) 2010; 35 MJ Feizollahi (1176_CR13) 2017; 161 CE Blair (1176_CR9) 1979; 25 RJB Wets (1176_CR21) 1974; 16 CE Blair (1176_CR8) 1977; 19 K Postek (1176_CR17) 2016; 28 RR Meyer (1176_CR16) 1974; 7 X Chen (1176_CR12) 2009; 57 D Bertsimas (1176_CR6) 2015; 153 DA Iancu (1176_CR15) 2013; 61 JF Benders (1176_CR1) 1962; 4 D Bertsimas (1176_CR7) 2015; 63 A Ben-Tal (1176_CR3) 2009 S Simons (1176_CR19) 1995 A Ben-Tal (1176_CR2) 2004; 99 |
| References_xml | – volume: 25 start-page: 7 issue: 1 year: 1979 end-page: 19 ident: CR9 article-title: The value function of a mixed integer program: II publication-title: Discrete Math. doi: 10.1016/0012-365X(79)90147-X – volume: 7 start-page: 223 issue: 1 year: 1974 end-page: 235 ident: CR16 article-title: On the existence of optimal solutions to integer and mixed-integer programming problems publication-title: Math. Program. doi: 10.1007/BF01585518 – volume: 4 start-page: 238 issue: 1 year: 1962 end-page: 252 ident: CR1 article-title: Partitioning procedures for solving mixed-variables programming problems publication-title: Nume. Math. doi: 10.1007/BF01386316 – volume: 153 start-page: 577 issue: 2 year: 2015 end-page: 594 ident: CR6 article-title: On the performance of affine policies for two-stage adaptive optimization: a geometric perspective publication-title: Math. Program. doi: 10.1007/s10107-014-0818-5 – volume: 161 start-page: 365 issue: 1 year: 2017 end-page: 387 ident: CR13 article-title: Exact augmented Lagrangian duality for mixed integer linear programming publication-title: Math. Program. doi: 10.1007/s10107-016-1012-8 – volume: 61 start-page: 941 issue: 4 year: 2013 end-page: 956 ident: CR15 article-title: Supermodularity and affine policies in dynamic robust optimization publication-title: Oper. Res. doi: 10.1287/opre.2013.1172 – volume: 63 start-page: 610 issue: 3 year: 2015 end-page: 627 ident: CR7 article-title: Design of near optimal decision rules in multistage adaptive mixed-integer optimization publication-title: Oper. Res. doi: 10.1287/opre.2015.1365 – volume: 99 start-page: 351 issue: 2 year: 2004 end-page: 376 ident: CR2 article-title: Adjustable robust solutions of uncertain linear programs publication-title: Math. Program. doi: 10.1007/s10107-003-0454-y – volume: 35 start-page: 363 issue: 2 year: 2010 end-page: 394 ident: CR4 article-title: Optimality of affine policies in multistage robust optimization publication-title: Math. Oper. Res. doi: 10.1287/moor.1100.0444 – volume: 63 start-page: 877 issue: 4 year: 2015 end-page: 891 ident: CR14 article-title: K-adaptability in two-stage robust binary programming publication-title: Oper. Res. doi: 10.1287/opre.2015.1392 – year: 1970 ident: CR18 publication-title: Convex Analysis doi: 10.1515/9781400873173 – year: 2009 ident: CR3 publication-title: Robust Optimization doi: 10.1515/9781400831050 – volume: 16 start-page: 309 issue: 3 year: 1974 end-page: 339 ident: CR21 article-title: Stochastic programs with fixed recourse: the equivalent deterministic program publication-title: SIAM Rev. doi: 10.1137/1016053 – volume: 19 start-page: 121 issue: 2 year: 1977 end-page: 138 ident: CR8 article-title: The value function of a mixed integer program: I publication-title: Discrete Math. doi: 10.1016/0012-365X(77)90028-0 – volume: 35 start-page: 284 issue: 2 year: 2010 end-page: 305 ident: CR5 article-title: On the power of robust solutions in two-stage stochastic and adaptive optimization problems publication-title: Math. Oper. Res. doi: 10.1287/moor.1090.0440 – volume: 23 start-page: 237 year: 1982 end-page: 273 ident: CR10 article-title: The value function of an integer program publication-title: Math. Program. doi: 10.1007/BF01583794 – volume: 57 start-page: 1469 issue: 6 year: 2009 end-page: 1482 ident: CR12 article-title: Uncertain linear programs: extended affinely adjustable robust counterparts publication-title: Oper. Res. doi: 10.1287/opre.1080.0605 – start-page: 1 year: 1995 end-page: 23 ident: CR19 article-title: Minimax theorems and their proofs publication-title: Minimax and Applications – ident: CR20 – volume: 150 start-page: 491 issue: 2 year: 2015 end-page: 509 ident: CR11 article-title: On the augmented Lagrangian dual for integer programming publication-title: Math. Program. doi: 10.1007/s10107-014-0763-3 – volume: 28 start-page: 553 issue: 3 year: 2016 end-page: 574 ident: CR17 article-title: Multistage adjustable robust mixed integer optimization via iterative splitting of the uncertainty set publication-title: INFORMS J. Comput. doi: 10.1287/ijoc.2016.0696 – volume: 35 start-page: 284 issue: 2 year: 2010 ident: 1176_CR5 publication-title: Math. Oper. Res. doi: 10.1287/moor.1090.0440 – volume: 16 start-page: 309 issue: 3 year: 1974 ident: 1176_CR21 publication-title: SIAM Rev. doi: 10.1137/1016053 – volume: 28 start-page: 553 issue: 3 year: 2016 ident: 1176_CR17 publication-title: INFORMS J. Comput. doi: 10.1287/ijoc.2016.0696 – volume: 23 start-page: 237 year: 1982 ident: 1176_CR10 publication-title: Math. Program. doi: 10.1007/BF01583794 – volume: 7 start-page: 223 issue: 1 year: 1974 ident: 1176_CR16 publication-title: Math. Program. doi: 10.1007/BF01585518 – volume-title: Convex Analysis year: 1970 ident: 1176_CR18 doi: 10.1515/9781400873173 – volume-title: Robust Optimization year: 2009 ident: 1176_CR3 doi: 10.1515/9781400831050 – volume: 25 start-page: 7 issue: 1 year: 1979 ident: 1176_CR9 publication-title: Discrete Math. doi: 10.1016/0012-365X(79)90147-X – volume: 161 start-page: 365 issue: 1 year: 2017 ident: 1176_CR13 publication-title: Math. Program. doi: 10.1007/s10107-016-1012-8 – volume: 99 start-page: 351 issue: 2 year: 2004 ident: 1176_CR2 publication-title: Math. Program. doi: 10.1007/s10107-003-0454-y – volume: 35 start-page: 363 issue: 2 year: 2010 ident: 1176_CR4 publication-title: Math. Oper. Res. doi: 10.1287/moor.1100.0444 – volume: 61 start-page: 941 issue: 4 year: 2013 ident: 1176_CR15 publication-title: Oper. Res. doi: 10.1287/opre.2013.1172 – volume: 4 start-page: 238 issue: 1 year: 1962 ident: 1176_CR1 publication-title: Nume. Math. doi: 10.1007/BF01386316 – volume: 153 start-page: 577 issue: 2 year: 2015 ident: 1176_CR6 publication-title: Math. Program. doi: 10.1007/s10107-014-0818-5 – volume: 63 start-page: 610 issue: 3 year: 2015 ident: 1176_CR7 publication-title: Oper. Res. doi: 10.1287/opre.2015.1365 – volume: 19 start-page: 121 issue: 2 year: 1977 ident: 1176_CR8 publication-title: Discrete Math. doi: 10.1016/0012-365X(77)90028-0 – volume: 57 start-page: 1469 issue: 6 year: 2009 ident: 1176_CR12 publication-title: Oper. Res. doi: 10.1287/opre.1080.0605 – volume: 150 start-page: 491 issue: 2 year: 2015 ident: 1176_CR11 publication-title: Math. Program. doi: 10.1007/s10107-014-0763-3 – volume: 63 start-page: 877 issue: 4 year: 2015 ident: 1176_CR14 publication-title: Oper. Res. doi: 10.1287/opre.2015.1392 – ident: 1176_CR20 doi: 10.1109/CDC.2011.6161382 – start-page: 1 volume-title: Minimax and Applications year: 1995 ident: 1176_CR19 |
| SSID | ssj0061199 |
| Score | 2.1073508 |
| Snippet | Many real problems with uncertain parameters can be modeled as two-stage robust mixed integer programming problems (RMIPs). Due to the complex nature of this... |
| SourceID | crossref springer |
| SourceType | Enrichment Source Index Database Publisher |
| StartPage | 959 |
| SubjectTerms | Computational Intelligence Mathematics Mathematics and Statistics Numerical and Computational Physics Operations Research/Decision Theory Optimization Original Paper Simulation |
| Title | Two-stage robust mixed integer programming problem with objective uncertainty |
| URI | https://link.springer.com/article/10.1007/s11590-017-1176-z |
| Volume | 12 |
| WOSCitedRecordID | wos000436825400003&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVAVX databaseName: Springer LINK customDbUrl: eissn: 1862-4480 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0061199 issn: 1862-4472 databaseCode: RSV dateStart: 20070101 isFulltext: true titleUrlDefault: https://link.springer.com/search?facet-content-type=%22Journal%22 providerName: Springer Nature |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1LS8QwEA66etCDb3F9kYMnJdA0aZMcRVw86CK4yt5K0iSy4m6l7frYX2_62nVBBT22TEqZpvNNO998A8CJtVpjPw4RMYwiaplEnPAYEaW5ZVbp0JSNwtes2-X9vrit-7izhu3elCTLSD1rdnPIW5CoGMKYhWiyCJaCQmym-ES_e2jCb4iroZGYF-1AlPlNKfO7S8yD0XwltASYzvq_bm0DrNX5JDyvNsAmWDCjLbD6RWXQHd1MpVmzbXDTe0uQSwkfDUwTNc5yOBy8Gw1L3QiTwpqwNXRLYT1tBhY_a2GinqrgCB0UVkSC_GMH3HcuexdXqJ6pgGKf8xwxLKwwhgkecixJwJkSgQikH3taUOspSj3LHKq581wKonzpE4MJ15IGilqyC1qjZGT2ALTOu4TRIJCaUx9LpSwVmksScypMSNvAa5wbxbXgeDH34jmaSSUXfouc3wrN8TCatMHpdMlLpbbxm_FZ8zSi-sXLfrbe_5P1AVhxmRGveLmHoJWnY3MEluPXfJClx-WG-wRhqNH- |
| linkProvider | Springer Nature |
| linkToHtml | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV3dS8MwEA86BfXBb3F-5sEnJbA2aZM8ijgmbkNwim8laRKZuFXazo_99aZfmwMV9LHhEsolubv2fvc7AE6MUcpxQx9hTQkihgrEMAsRlooZaqTydV4o3KbdLnt44DdlHXdSod2rlGRuqafFbtbzZiAqihyH-mg8DxZI1mUn-0S_va_Mr-8UTSMdlpUDEepWqczvlph1RrOZ0NzBNNf-9WrrYLWMJ-F5cQA2wJweboKVLyyD9qkzoWZNtkCn9xYhGxI-ahhHcpSkcNB_1wrmvBE6hiVga2CnwrLbDMx-1sJIPhXGEVpXWAAJ0o9tcNe87F20UNlTAYUuYymiDjdca8qZzxyBPUYl97gn3LChODENSUjDUOvV7DgTHEtXuFg7mClBPEkM3gG1YTTUuwAaq11MiecJxexGCCkN4YoJHDLCtU_qoFEpNwhLwvGs78VzMKVKzvQWWL1lnON-MK6D08mUl4Jt4zfhs2o3gvLiJT9L7_1J-hgstXqddtC-6l7vg2UbJbECo3sAamk80odgMXxN-0l8lB--T0MR1OI |
| linkToPdf | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1bS8MwFA7eEH3wLt7Ng09KcGnSJnkUdShuY-iUvZWkSUTRTdp62683vW0KKoiPLSelnFzOac93vg-APWu1xl4UIGIYRdQyiTjhESJKc8us0oHJG4UbrNXi3a5olzqnSYV2r0qSRU9DxtLUSw-ftD0cNb65KJwBqhjCmAVoMA4mqfuQyTBdl1c31VEc4EJAEvOsNYgyryprfveIr4Hpa1U0Dzb1-X-_5gKYK_NMeFQsjEUwZnpLYPYT-6C7ag4pW5Nl0Oy89pFLFW8NjPvqOUnh492b0TDnkzAxLIFcj24oLFVoYPYTF_bVfXFoQhciC4BB-r4CruunneMzVGotoMjjPEUMCyuMYYIHHEvic6aEL3zpRTUtqK0pSmuWuWjn7nMpiPKkRwwmXEvqK2rJKpjo9XtmDUDrPE0Y9X2pOfWwVMpSobkkEafCBHQd1CpHh1FJRJ7pYTyEIwrlzG-h81vGRR6Eg3WwPxzyVLBw_GZ8UM1MWG7I5GfrjT9Z74Lp9kk9bJy3LjbBjEueeAHd3QITafxstsFU9JLeJfFOvg4_AFZL3cY |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Two-stage+robust+mixed+integer+programming+problem+with+objective+uncertainty&rft.jtitle=Optimization+letters&rft.au=Zhang%2C+Ning&rft.date=2018-07-01&rft.issn=1862-4472&rft.eissn=1862-4480&rft.volume=12&rft.issue=5&rft.spage=959&rft.epage=969&rft_id=info:doi/10.1007%2Fs11590-017-1176-z&rft.externalDBID=n%2Fa&rft.externalDocID=10_1007_s11590_017_1176_z |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1862-4472&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1862-4472&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1862-4472&client=summon |