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Within these takeout systems, riders have a role throughout the order fulfillment process. Their behaviors involve multiple key time points, and accurately predicting these critical moments in advance is essential for enhancing both user retention and operational efficiency on such platforms. This paper first proposes a time chain simulation method, which simulates the order fulfillment in segments with an incremental process by combining dynamic and static information in the data. Subsequently, a GRU-Transformer architecture is presented, which is based on the Transformer incorporating the advantages of the Gated Recurrent Unit, thus working in concert with the time chain simulation and enabling efficient parallel prediction before order creation. Extensive experiments conducted on a real-world takeout food order dataset demonstrate that the Mean Squared Error of the prediction results of GRU-Transformer with time chain simulation is reduced by about 9.78% compared to the Transformer. Finally, according to the temporal inconsistency analysis, it can be seen that GRU-Transformer with time chain simulation still has a stable performance during peak periods, which is valuable for the intelligent takeout system.<\/jats:p>","DOI":"10.3390\/systems13060457","type":"journal-article","created":{"date-parts":[[2025,6,10]],"date-time":"2025-06-10T08:18:43Z","timestamp":1749543523000},"page":"457","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Study on Predicting Key Times in the Takeout System\u2019s Order Fulfillment Process"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-1494-1819","authenticated-orcid":false,"given":"Dongyi","family":"Hu","sequence":"first","affiliation":[{"name":"School of Big Data Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9781-3101","authenticated-orcid":false,"given":"Wei","family":"Deng","sequence":"additional","affiliation":[{"name":"School of Big Data Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zilong","family":"Jiang","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Technology, Guizhou University of Finance and Economics, Guiyang 550025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yong","family":"Shi","sequence":"additional","affiliation":[{"name":"School of Economics and Management, University of Chinese Academy of Sciences, Beijing 100190, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,10]]},"reference":[{"key":"ref_1","unstructured":"(2025, April 10). 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