{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T02:46:13Z","timestamp":1769049973112,"version":"3.49.0"},"reference-count":28,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,19]],"date-time":"2025-08-19T00:00:00Z","timestamp":1755561600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012456","name":"National Social Science Foundation of China","doi-asserted-by":"publisher","award":["2024-SKJJ-C-027"],"award-info":[{"award-number":["2024-SKJJ-C-027"]}],"id":[{"id":"10.13039\/501100012456","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012456","name":"National Social Science Foundation of China","doi-asserted-by":"publisher","award":["202350A010"],"award-info":[{"award-number":["202350A010"]}],"id":[{"id":"10.13039\/501100012456","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012456","name":"National Social Science Foundation of China","doi-asserted-by":"publisher","award":["2025500330"],"award-info":[{"award-number":["2025500330"]}],"id":[{"id":"10.13039\/501100012456","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012456","name":"National Social Science Foundation of China","doi-asserted-by":"publisher","award":["202350A020"],"award-info":[{"award-number":["202350A020"]}],"id":[{"id":"10.13039\/501100012456","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012456","name":"National Social Science Foundation of China","doi-asserted-by":"publisher","award":["202550A030"],"award-info":[{"award-number":["202550A030"]}],"id":[{"id":"10.13039\/501100012456","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Naval University of Engineering independent research projects","award":["2024-SKJJ-C-027"],"award-info":[{"award-number":["2024-SKJJ-C-027"]}]},{"name":"Naval University of Engineering independent research projects","award":["202350A010"],"award-info":[{"award-number":["202350A010"]}]},{"name":"Naval University of Engineering independent research projects","award":["2025500330"],"award-info":[{"award-number":["2025500330"]}]},{"name":"Naval University of Engineering independent research projects","award":["202350A020"],"award-info":[{"award-number":["202350A020"]}]},{"name":"Naval University of Engineering independent research projects","award":["202550A030"],"award-info":[{"award-number":["202550A030"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Products with complex structures are structurally intricate and involve multiple professional fields and engineering construction elements, making it difficult for a single contractor to independently develop and manufacture such complex structural products. Therefore, during the research, development, and production of complex products, collaboration between manufacturers and suppliers is essential to ensure the smooth completion of projects. In this process, a complex supply chain network is often formed to achieve collaborative cooperation among all project participants. Within such a complex supply chain network, issues such as delayed delivery, poor product quality, or low resource utilization by any participant may trigger the bullwhip effect. This, in turn, can negatively impact the delivery cycle, product cost, and quality of the entire complex product, causing it to lose favorable competitive positions such as quality advantages and delivery advantages in fierce market competition. Therefore, this paper firstly explores the mechanism of complex product manufacturing and the supply network of complex product manufacturing, in order to grasp the inherent structure of complex product manufacturing with a focus on identifying symmetrical properties among supply chain nodes. Secondly, a complex product supply chain network model is constructed with the Graphical Evaluation and Review Technique (GERT), incorporating symmetry constraints to reflect balanced resource allocation and mutual dependencies among symmetrical nodes. Then, from the perspective of supply chain, we focus on identifying the shortcomings of supply chain suppliers and optimizing the management cost of the whole supply chain in order to improve the quality of complex products, delivery level, and cost saving level. This study constructs a Restricted Grey GERT (RG-GERT) network model with constrained outputs, integrates moment-generating functions and Mason\u2019s Formula to derive transfer functions, and employs a hybrid algorithm (genetic algorithm combined with non-linear programming) to solve the multi-objective optimization problem (MOOP) for joint optimization of delivery time, quality, and cost. Empirical analysis is conducted using simulated data from Y Company\u2019s aerospace equipment supply chain, covering interval parameters such as delivery time [5\u201330 days], cost [40,000\u2013640,000 CNY], and quality [0.85\u20131.0], validated with industry-specific constraints. Empirical analysis using Y Company\u2019s aerospace supply chain data shows that the model achieves a maximum customer satisfaction of 0.96, with resource utilization efficiency of inefficient suppliers improved by 15\u201320% (p &lt; 0.05) after secondary optimization. Key contributions include (1) integrating symmetry analysis to simplify network modeling; (2) extending GERT with grey parameters for non-probabilistic uncertainty; (3) developing a two-stage optimization framework linking customer satisfaction and resource efficiency.<\/jats:p>","DOI":"10.3390\/sym17081354","type":"journal-article","created":{"date-parts":[[2025,8,19]],"date-time":"2025-08-19T07:56:06Z","timestamp":1755590166000},"page":"1354","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Joint Optimization of Delivery Time, Quality, and Cost for Complex Product Supply Chain Networks Based on Symmetry Analysis"],"prefix":"10.3390","volume":"17","author":[{"given":"Peng","family":"Dong","sequence":"first","affiliation":[{"name":"Department of Management Engineering and Equipment Economics, Naval University of Engineering, Wuhan 430033, China"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-4861-2614","authenticated-orcid":false,"given":"Weibing","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Management Engineering and Equipment Economics, Naval University of Engineering, Wuhan 430033, China"}]},{"given":"Kewen","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Management Engineering and Equipment Economics, Naval University of Engineering, Wuhan 430033, China"}]},{"given":"Enze","family":"Gong","sequence":"additional","affiliation":[{"name":"Department of Management Engineering and Equipment Economics, Naval University of Engineering, Wuhan 430033, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,19]]},"reference":[{"key":"ref_1","first-page":"56","article-title":"Shared Quality Improvement Cost Contract for Complex Product Supply Chain","volume":"24","author":"Wu","year":"2019","journal-title":"Ind. Eng. Manag."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"8889903","DOI":"10.1155\/2020\/8889903","article-title":"Quality Risk Propagation of Complex Product Collaborative Manufacturing Supply Chain Network Based on CN and SoV","volume":"2020","author":"Li","year":"2020","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_3","first-page":"17","article-title":"Study of an Evolutionary Game between Shipyards and Suppliers about Product Quality","volume":"30","author":"Xue","year":"2021","journal-title":"Oper. Res. Manag. Sci."},{"key":"ref_4","first-page":"62","article-title":"Research on Risk Transmission of Emergencies in Supply Chain Based on an Improved GERT Network","volume":"31","author":"Jing","year":"2022","journal-title":"Oper. Res. Manag. Sci."},{"key":"ref_5","first-page":"112","article-title":"Optimization Model for the Joint of Time and Probability under the Controlled Result Output of the GERT Network","volume":"33","author":"Geng","year":"2015","journal-title":"Syst. Eng."},{"key":"ref_6","first-page":"103","article-title":"Hotspot Analysis of collaboration Mechanism in Complex Equipment Supply Chain","volume":"39","author":"Chen","year":"2019","journal-title":"Sci. Technol. Manag. Res."},{"key":"ref_7","first-page":"19","article-title":"Research on Factors Affecting Knowledge Input of Complex Equipment \u201cMain-manufacturer-Core-supplier\u201d Collaborative Development","volume":"50","author":"Pei","year":"2020","journal-title":"Math. Pract. Theory"},{"key":"ref_8","first-page":"1","article-title":"Supply chain vulnerability assessment for manufacturing industry","volume":"326","author":"Sharma","year":"2021","journal-title":"Ann. Oper. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"106212","DOI":"10.1016\/j.cie.2019.106212","article-title":"An integrated fuzzy QFD and grey decision-making approach for supply chain collaborative quality design of large complex products","volume":"140","author":"Wang","year":"2020","journal-title":"Comput. Ind. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"102873","DOI":"10.1016\/j.tre.2022.102873","article-title":"Unraveling the cheap talk\u2019s informativeness of product quality in supply chains: A lying aversion perspective","volume":"166","author":"Jiang","year":"2022","journal-title":"Transp. Res. Part E"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"122744","DOI":"10.1016\/j.techfore.2023.122744","article-title":"Designing a reverse supply chain network with quality control for returned products: Strategies to mitigate free-riding effect and ensure compliance with technology licensing requirements","volume":"195","author":"Sahar","year":"2023","journal-title":"Technol. Forecast. Soc. Change"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"108738","DOI":"10.1016\/j.compeleceng.2023.108738","article-title":"TruCert: Blockchain-based trustworthy product certification within autonomous automotive supply chains","volume":"109","author":"Alsadi","year":"2023","journal-title":"Comput. Electr. Eng."},{"key":"ref_13","first-page":"330","article-title":"Multi-objective Optimization Model of Supply Chain Management for Complex Product Based on Multi-parameter CF-GERT Network","volume":"31","author":"Su","year":"2025","journal-title":"Comput. Integr. Manuf. Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"123199","DOI":"10.1016\/j.jclepro.2020.123199","article-title":"A multi-objective optimization approach for green and resilient supply chain network design: A real-life case study","volume":"278","author":"Aliakbar","year":"2021","journal-title":"J. Clean. Prod."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"108095","DOI":"10.1016\/j.cie.2022.108095","article-title":"A hybrid bi-objective optimization approach for joint determination of safety stock and safety time buffers in multi-item single-stage industrial supply chains","volume":"168","author":"Silva","year":"2022","journal-title":"Comput. Ind. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"108131","DOI":"10.1016\/j.cie.2022.108131","article-title":"A three-objective optimization model for mid-term sustainable supply chain network design","volume":"168","author":"Bortolini","year":"2022","journal-title":"Comput. Ind. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"110774","DOI":"10.1016\/j.asoc.2023.110774","article-title":"A meta-heuristic-based algorithm for designing multi-objective multi-echelon supply chain network","volume":"147","author":"Mohammed","year":"2023","journal-title":"Appl. Soft Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4018\/IJISSCM.316168","article-title":"Collaborative Bullwhip Effect-Oriented Bi-Objective Optimization for Inference-Based Weighted Moving Average Forecasting in Decentralized Supply Chain","volume":"16","author":"Tliche","year":"2023","journal-title":"Int. J. Inf. Syst. Supply Chain. Manag."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1080\/24725854.2023.2217248","article-title":"A multi-agent reinforcement learning model for inventory transshipments under supply chain disruption","volume":"56","author":"Kim","year":"2024","journal-title":"IISE Trans."},{"key":"ref_20","first-page":"113","article-title":"Supply Chain Quality Transaction Mechanism for Complex Product Concerning Profit-reputation Utility Function","volume":"36","author":"Liu","year":"2018","journal-title":"Syst. Eng."},{"key":"ref_21","first-page":"145","article-title":"Supply Chain Quality Incentive Contract for Complex Product from Dual Perspective","volume":"24","author":"Liu","year":"2019","journal-title":"Ind. Eng. Manag."},{"key":"ref_22","first-page":"1973","article-title":"Quality Cost Optimization Method for Complex Product Supply Chain Considering Quality Value","volume":"34","author":"Wang","year":"2019","journal-title":"Control Decis."},{"key":"ref_23","first-page":"5","article-title":"Research on Test Quality Control of Weapon Equipment Based on GERT Network Model","volume":"41","author":"Bai","year":"2022","journal-title":"Control Theory Appl."},{"key":"ref_24","first-page":"136318","article-title":"Fuzzy-grey RG-GERT: A hybrid model for uncertain supply chain networks","volume":"392","author":"Wang","year":"2024","journal-title":"J. Clean. Prod."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1016\/j.jbusres.2020.09.009","article-title":"Artificial intelligence in supply chain management: A systematic literature review","volume":"122","author":"Toorajipour","year":"2021","journal-title":"J. Bus. Res."},{"key":"ref_26","first-page":"309","article-title":"GERT network optimization with consideration of \u201ctime-resource\u201d on large aircraft collaborative development","volume":"34","author":"Wang","year":"2019","journal-title":"Control Decis."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Li, Y., Dong, P., and Ye, W. (2023). Joint duration-cost-quality optimization model for complex product supply chains under contingency conditions. PLoS ONE, 18.","DOI":"10.1371\/journal.pone.0292010"},{"key":"ref_28","first-page":"61","article-title":"Time-cost-quality Joint Optimization Model of Construction Project Based on the Interval GERT Network","volume":"36","author":"Han","year":"2018","journal-title":"Syst. Eng."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1354\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:30:30Z","timestamp":1760034630000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1354"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,19]]},"references-count":28,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["sym17081354"],"URL":"https:\/\/doi.org\/10.3390\/sym17081354","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,19]]}}}