{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:53:55Z","timestamp":1760147635804,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,16]],"date-time":"2023-02-16T00:00:00Z","timestamp":1676505600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Indonesian Ministry of Education, Culture, Research, and Technology","award":["2064\/UN6.3.1\/PT.00\/2022"],"award-info":[{"award-number":["2064\/UN6.3.1\/PT.00\/2022"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>In this paper, the implementation of the Benders decomposition method to solve the Adjustable Robust Counterpart for Internet Shopping Online Problem (ARC-ISOP) is discussed. Since the ARC-ISOP is a mixed-integer linear programming (MILP) model, the discussion begins by identifying the linear variables in the form of continuous variables and nonlinear variables in the form of integer variables. In terms of Benders decomposition, the ARC-ISOP model can be solved by partitioning them into smaller subproblems (the master problem and inner problem), which makes it easier for computational calculations. Pseudo-codes in Python programming language are presented in this paper. An example case is presented for the ARC-ISOP to determine the optimal total cost (including product price and shipping cost) and delivery time. Numerical simulations were carried out using Python programming language with case studies in the form of five products purchased from six shops.<\/jats:p>","DOI":"10.3390\/computation11020037","type":"journal-article","created":{"date-parts":[[2023,2,16]],"date-time":"2023-02-16T04:51:46Z","timestamp":1676523106000},"page":"37","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Benders Decomposition Method on Adjustable Robust Counterpart Optimization Model for Internet Shopping Online Problem"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2073-1754","authenticated-orcid":false,"given":"Diah","family":"Chaerani","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Padjadjaran, Sumedang 45363, Indonesia"}]},{"given":"Shenya","family":"Saksmilena","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Padjadjaran, Sumedang 45363, Indonesia"}]},{"given":"Athaya Zahrani","family":"Irmansyah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Padjadjaran, Sumedang 45363, Indonesia"}]},{"given":"Elis","family":"Hertini","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Padjadjaran, Sumedang 45363, Indonesia"}]},{"given":"Endang","family":"Rusyaman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Padjadjaran, Sumedang 45363, Indonesia"}]},{"given":"Erick","family":"Paulus","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Faculty of Mathematics and Natural Science, Universitas Padjadjaran, Sumedang 45363, Indonesia"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"148","DOI":"10.3390\/computation10090148","article-title":"Face Detection & Recognition from Images & Videos Based on CNN & Raspberry Pi","volume":"10","author":"Zamir","year":"2022","journal-title":"Computation"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"9977252","DOI":"10.1155\/2021\/9977252","article-title":"An optimized and efficient routing protocol application for IoV","volume":"2021","author":"Afzal","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"14572","DOI":"10.1109\/JIOT.2021.3068056","article-title":"Energy efficient fog computing for 6G enabled massive IoT: Recent trends and future opportunities","volume":"9","author":"Malik","year":"2021","journal-title":"IEEE Internet Things J."},{"key":"ref_4","unstructured":"Kumar, S. (2015). Online Shopping-A Literature Review. Natl. Conf. Innov. Trends Comput., 129\u2013131."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"385","DOI":"10.2478\/v10006-010-0028-0","article-title":"Internet shopping optimization problem","volume":"20","author":"Kovalyov","year":"2010","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Yakovlev, S., Kartashov, O., and Podzeha, D. (2022). Mathematical Models and Nonlinear Optimization in Continuous Maximum Coverage Location Problem. Computation, 10.","DOI":"10.3390\/computation10070119"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"239","DOI":"10.5267\/j.uscm.2021.9.004","article-title":"Contribution of robust optimization on handling agricultural processed products supply chain problem during COVID-19 pandemic","volume":"10","author":"Chaerani","year":"2022","journal-title":"Uncertain Supply Chain. Manag."},{"key":"ref_8","unstructured":"Rao, S.S. (2009). Engineering Optimization: Theory and Practice, John Wiley & Sons Inc. [4th ed.]."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"15","DOI":"10.15722\/jds.15.2.201702.15","article-title":"Internet shopping optimization problem with delivery constraints","volume":"15","author":"Chung","year":"2017","journal-title":"J. Distrib. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"012074","DOI":"10.1088\/1742-6596\/1722\/1\/012074","article-title":"Adjustable robust counterpart optimization model for internet shopping online problem","volume":"1722","author":"Chaerani","year":"2021","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_11","first-page":"678","article-title":"A Systematic Review on Integer Multi-objective Adjustable Robust Counterpart Optimization Model Using Benders Decomposition","volume":"6","author":"Irmansyah","year":"2022","journal-title":"JTAM J. Teor. Dan Apl. Mat."},{"key":"ref_12","unstructured":"Bisschop, J. (1999). AIMMS Optimization Modeling, Paragon Decision Technology, B.V."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"581","DOI":"10.5267\/j.ijdns.2021.11.006","article-title":"Systematic literature review on adjustable robust counterpart for internet shopping optimization problem","volume":"6","author":"Mahrudinda","year":"2022","journal-title":"Int. J. Data Netw. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"264","DOI":"10.7326\/0003-4819-151-4-200908180-00135","article-title":"Preferred Reporting Items for Systematic Reviews and Meta-Analyses: The PRISMA Statement","volume":"6","author":"Moher","year":"2009","journal-title":"Ann. Intern. Med."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1515\/amcs-2016-0028","article-title":"Exact and heuristic approaches to solve the Internet shopping optimization problem with delivery costs","volume":"26","author":"Musial","year":"2016","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"key":"ref_16","unstructured":"Kingsford, C. (2017). CMSC 451: Max-Flow Extensions Circulations with Demands, University of Maryland."},{"key":"ref_17","first-page":"91","article-title":"Adjustable robust counterpart optimization model for maximum flow problems with box uncertainty","volume":"141","author":"Agustini","year":"2020","journal-title":"World Sci. News"},{"key":"ref_18","first-page":"19","article-title":"Penyelesaian Metode Dekomposisi Benders pada Model Optimisasi Robust Masalah Mixed Integer Linear Programming Dua Tahap yang melibatkan Variabel Recourse","volume":"16","author":"Chaerani","year":"2020","journal-title":"J. Mat. Integr."},{"key":"ref_19","unstructured":"Tokopedia (2022, June 15). Pencarian Produk Skincare Merek \u2018Emina\u2019 di Kota Bandung. Available online: https:\/\/www.tokopedia.com\/search?fcity=165&srp_component_id=02.01.00.00&st=product&q=emina."},{"key":"ref_20","unstructured":"(2022, October 19). What is Secondary Data? + [Examples, Sources, & Analysis]. Available online: https:\/\/www.formpl.us\/blog\/secondary-data#:~:text=Secondary%20data%20is%20the%20data,been%20collected%20in%20the%20past."},{"key":"ref_21","unstructured":"Tokopedia (2022, June 22). Pencarian Produk Skincare Merek \u2018Emina\u2019 di Berbagai Kota. Available online: https:\/\/www.tokopedia.com\/search?st=product&q=emina&srp_component_id=01.07.00.00."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/11\/2\/37\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:37:57Z","timestamp":1760121477000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/11\/2\/37"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,16]]},"references-count":21,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["computation11020037"],"URL":"https:\/\/doi.org\/10.3390\/computation11020037","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2023,2,16]]}}}