{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T10:41:38Z","timestamp":1773916898023,"version":"3.50.1"},"reference-count":53,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T00:00:00Z","timestamp":1773705600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["72021002"],"award-info":[{"award-number":["72021002"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["71771048"],"award-info":[{"award-number":["71771048"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["71832001"],"award-info":[{"award-number":["71832001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["72071144"],"award-info":[{"award-number":["72071144"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["72471174"],"award-info":[{"award-number":["72471174"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Systems"],"abstract":"<jats:p>Supply chain (SC) disruption risk assessment and mitigation have attracted significant attention in both academia and practice. However, existing research predominantly focuses on unidirectional disruption propagation, either forward or backward, despite the reality that risks can propagate bi-directionally in complex supply chain networks. Furthermore, conventional assessment tools often concentrate on conceptualizing and quantifying risks, while risk mitigation requires mathematical optimization approaches. To bridge these gaps, this paper proposes a novel pressure wave-based approach inspired by fluid mechanics to assess bi-directional disruption propagation in cluster supply chain networks (CSCNs). The method conceptualizes disruptions as pressure signals that transmit between SC partners and explicitly quantifies disruption severity through wave intensity. By employing mathematical optimization, we develop a framework that assists managers in optimizing risk mitigation strategies, including inventory buffering and cross-chain cooperation. Numerical experiments demonstrate the effectiveness of the proposed method in explaining risk influencing factors, mitigating disruption risks, and achieving dynamic restructuring of SC structures. The results show that our approach reduces the Cluster Propagation Vulnerability Index (CPVI) by up to 40% compared to baseline models without optimization decisions.<\/jats:p>","DOI":"10.3390\/systems14030316","type":"journal-article","created":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T12:42:16Z","timestamp":1773751336000},"page":"316","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Pressure Wave Propagation Optimization Models for Supply Chain Risk Mitigation"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3190-5008","authenticated-orcid":false,"given":"Ming","family":"Liu","sequence":"first","affiliation":[{"name":"School of Economics & Management, Tongji University, Shanghai 200092, China"},{"name":"Laboratory of High Quality Urban Development and Strategic Decision, Tongji University, Shanghai 200092, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiawei","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Economics & Management, Tongji University, Shanghai 200092, China"},{"name":"Laboratory of High Quality Urban Development and Strategic Decision, Tongji University, Shanghai 200092, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yueyu","family":"Ding","sequence":"additional","affiliation":[{"name":"Urban Mobility Institute, Tongji University, Shanghai 201804, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"22135","DOI":"10.1021\/acs.est.4c10523","article-title":"Systematic risks of the global lithium supply chain network: From static topological structures to cascading failure dynamics","volume":"58","author":"Ouyang","year":"2024","journal-title":"Environ. Sci. Technol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2154","DOI":"10.1080\/00207543.2013.858836","article-title":"The Ripple effect in supply chains: Trade-off \u2018efficiency-flexibility-resilience\u2019 in disruption management","volume":"52","author":"Ivanov","year":"2014","journal-title":"Int. J. Prod. Res."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2539","DOI":"10.1080\/00207543.2017.1374575","article-title":"Entropy-based model for the ripple effect: Managing environmental risks in supply chains","volume":"56","author":"Levner","year":"2018","journal-title":"Int. J. Prod. Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"6938","DOI":"10.1080\/00207543.2014.917769","article-title":"Modelling of cluster supply network with cascading failure spread and its vulnerability analysis","volume":"52","author":"Zeng","year":"2014","journal-title":"Int. J. Prod. Res."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"666","DOI":"10.1080\/24725854.2023.2253881","article-title":"A network-of-networks adaptation for cross-industry manufacturing repurposing","volume":"56","author":"Dolgui","year":"2024","journal-title":"IISE Trans."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"8651","DOI":"10.1080\/00207543.2024.2347561","article-title":"Resilience of interdependent supply chain networks design and protection under the ripple effect","volume":"62","author":"Zhang","year":"2024","journal-title":"Int. J. Prod. Res."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"107921","DOI":"10.1016\/j.ijpe.2020.107921","article-title":"OR-methods for coping with the ripple effect in supply chains during COVID-19 pandemic: Managerial insights and research implications","volume":"232","author":"Ivanov","year":"2021","journal-title":"Int. J. Prod. Econ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1016\/j.physa.2016.06.058","article-title":"An ant colony based resilience approach to cascading failures in cluster supply network","volume":"462","author":"Wang","year":"2016","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1117","DOI":"10.1016\/j.ejor.2020.09.053","article-title":"Ripple effect in the supply chain network: Forward and backward disruption propagation, network health and firm vulnerability","volume":"291","author":"Li","year":"2021","journal-title":"Eur. J. Oper. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1080\/09535314.2013.872081","article-title":"Estimation of production capacity loss rate after the great east Japan earthquake and tsunami in 2011","volume":"26","author":"Kajitani","year":"2014","journal-title":"Econ. Syst. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"566","DOI":"10.2307\/258918","article-title":"The missing link: A transformational view of metaphors in organizational science","volume":"16","author":"Tsoukas","year":"1991","journal-title":"Acad. Manag. Rev."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1111\/j.1745-493X.2009.03166.x","article-title":"On social network analysis in a supply chain context","volume":"45","author":"Borgatti","year":"2009","journal-title":"J. Supply Chain. Manag."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Paul, S.K., Sarker, R.A., and Essam, D. (2014, January 9\u201312). Managing supply disruption in a three-tier supply chain with multiple suppliers and retailers. Proceedings of the 2014 IEEE International Conference on Industrial Engineering and Engineering Management, Selangor, Malaysia.","DOI":"10.1109\/IEEM.2014.7058627"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1411","DOI":"10.1007\/s10479-020-03640-6","article-title":"Viable supply chain model: Integrating agility, resilience and sustainability perspectives-lessons from and thinking beyond the COVID-19 pandemic","volume":"319","author":"Ivanov","year":"2020","journal-title":"Ann. Oper. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1080\/00207543.2020.1740348","article-title":"Robust optimisation for ripple effect on reverse supply chain: An industrial case study","volume":"59","year":"2021","journal-title":"Int. J. Prod. Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"110873","DOI":"10.1016\/j.cie.2025.110873","article-title":"Quantifying performance indicators in perishable food supply chain networks: Assessing dynamic performance under ripple effects","volume":"201","author":"Karanam","year":"2025","journal-title":"Comput. Ind. Eng."},{"key":"ref_17","unstructured":"Angerhofer, B.J., and Angelides, M.C. (2000, January 10\u201313). System dynamics modelling in supply chain management: Research review. Proceedings of the 2000 Winter Simulation Conference Proceedings (Cat. No. 00CH37165), Orlando, FL, USA."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1016\/S0360-8352(02)00080-3","article-title":"Supply chain simulation with discrete\u2013continuous combined modeling","volume":"43","author":"Lee","year":"2002","journal-title":"Comput. Ind. Eng."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"6963","DOI":"10.1080\/00207543.2014.986303","article-title":"Integration of aggregate distribution and dynamic transportation planning in a supply chain with capacity disruptions and the ripple effect consideration","volume":"53","author":"Ivanov","year":"2015","journal-title":"Int. J. Prod. Res."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Ovezmyradov, B. (2022). Product availability and stockpiling in times of pandemic: Causes of supply chain disruptions and preventive measures in retailing. Ann. Oper. Res., 1\u201333.","DOI":"10.1007\/s10479-022-05091-7"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1285","DOI":"10.1080\/00207543.2019.1627438","article-title":"Does the ripple effect influence the bullwhip effect? An integrated analysis of structural and operational dynamics in the supply chain","volume":"58","author":"Dolgui","year":"2020","journal-title":"Int. J. Prod. Res."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1109\/TCSS.2022.3215260","article-title":"Risk propagation decision-making for product and supply chain change systems under COVID-19: An assessment-to-control support scheme","volume":"11","author":"Cao","year":"2022","journal-title":"IEEE Trans. Comput. Soc. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3284","DOI":"10.1080\/00207543.2019.1661538","article-title":"Ripple effect modelling of supplier disruption: Integrated Markov chain and dynamic Bayesian network approach","volume":"58","author":"Hosseini","year":"2020","journal-title":"Int. J. Prod. Res."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"102683","DOI":"10.1016\/j.omega.2022.102683","article-title":"An optimization approach for multi-echelon supply chain viability with disruption risk minimization","volume":"112","author":"Liu","year":"2022","journal-title":"Omega"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"104264","DOI":"10.1016\/j.tre.2025.104264","article-title":"Analysis of the ripple effects of disruptions on multimodal container terminals operations: A System Dynamics approach","volume":"202","author":"Zhang","year":"2025","journal-title":"Transp. Res. Part E Logist. Transp. Rev."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Liu, M., Ding, Y., Chu, F., Zheng, F., and Chu, C. (2025). Additive manufacturing for improving supply chain resilience under the ripple effect. Int. J. Prod. Res., 1\u201336.","DOI":"10.1080\/00207543.2025.2537344"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"102972","DOI":"10.1016\/j.omega.2023.102972","article-title":"Robust actions for improving supply chain resilience and viability","volume":"123","author":"Liu","year":"2024","journal-title":"Omega"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1016\/j.ijpe.2019.03.018","article-title":"Resilient supplier selection and optimal order allocation under disruption risks","volume":"213","author":"Hosseini","year":"2019","journal-title":"Int. J. Prod. Econ."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"4693","DOI":"10.1080\/00207543.2021.1934745","article-title":"An analysis of the ripple effect for disruptions occurring in circular flows of a supply chain network","volume":"60","author":"Park","year":"2022","journal-title":"Int. J. Prod. Res."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"5795","DOI":"10.1080\/00207543.2018.1467059","article-title":"Bayesian network modelling for supply chain risk propagation","volume":"56","author":"Ojha","year":"2018","journal-title":"Int. J. Prod. Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3111","DOI":"10.1109\/TEM.2020.3026465","article-title":"Conceptualization and measurement of supply chain resilience in an open-system context","volume":"69","author":"Hosseini","year":"2020","journal-title":"IEEE Trans. Eng. Manag."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1111\/jscm.12257","article-title":"Building novel supply chain theory using \u201cmetaphorical imagination\u201d","volume":"58","author":"Stephens","year":"2022","journal-title":"J. Supply Chain. Manag."},{"key":"ref_33","unstructured":"Cerabona, T., Lauras, M., Faug\u00e8re, L., Gitto, J.P., Montreuil, B., and Benaben, F. (2020). A Physics-Based Approach for Managing Supply Chain Risks and Opportunities Within Its Performance Framework. Proceedings of the Boosting Collaborative Networks 4.0: 21st IFIP WG 5.5 Working Conference on Virtual Enterprises, PRO-VE 2020, Valencia, Spain, 23\u201325 November 2020, Springer. Proceedings 21."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1080\/13675567.2017.1324834","article-title":"Supply chain immune system: Concept, framework, and applications","volume":"20","author":"Srinivasan","year":"2017","journal-title":"Int. J. Logist. Res. Appl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2904","DOI":"10.1080\/00207543.2020.1750727","article-title":"Viability of intertwined supply networks: Extending the supply chain resilience angles towards survivability. A position paper motivated by COVID-19 outbreak","volume":"58","author":"Ivanov","year":"2020","journal-title":"Int. J. Prod. Res."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"103081","DOI":"10.1016\/j.omega.2024.103081","article-title":"Supply chain resilience: Conceptual and formal models drawing from immune system analogy","volume":"127","author":"Ivanov","year":"2024","journal-title":"Omega"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"10804","DOI":"10.1109\/JSEN.2021.3058507","article-title":"Modeling dynamic pressure of gas pipeline with single and double leakage","volume":"21","author":"Ndalila","year":"2021","journal-title":"IEEE Sens. J."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"13651","DOI":"10.1109\/ACCESS.2026.3657041","article-title":"A pressure wave-based approach for assessing evolving disruption risks in cluster supply chain under bidirectional propagation","volume":"14","author":"Liu","year":"2026","journal-title":"IEEE Access"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"102806","DOI":"10.1016\/j.omega.2022.102806","article-title":"Post-pandemic adaptation and development of supply chain viability theory","volume":"116","author":"Ivanov","year":"2023","journal-title":"Omega"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.pursup.2017.10.004","article-title":"Supply chain vulnerability assessment: A network based visualization and clustering analysis approach","volume":"24","author":"Blackhurst","year":"2018","journal-title":"J. Purch. Supply Manag."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1287\/mnsc.1060.0515","article-title":"On the value of mitigation and contingency strategies for managing supply chain disruption risks","volume":"52","author":"Tomlin","year":"2006","journal-title":"Manag. Sci."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1476","DOI":"10.1111\/poms.12887","article-title":"Increasing supply chain robustness through process flexibility and inventory","volume":"27","author":"Wang","year":"2018","journal-title":"Prod. Oper. Manag."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1080\/00207543.2020.1841318","article-title":"A new robust dynamic Bayesian network approach for disruption risk assessment under the supply chain ripple effect","volume":"59","author":"Liu","year":"2021","journal-title":"Int. J. Prod. Res."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1627","DOI":"10.1007\/s10479-021-04047-7","article-title":"Exiting the COVID-19 pandemic: After-shock risks and avoidance of disruption tails in supply chains","volume":"335","author":"Ivanov","year":"2024","journal-title":"Ann. Oper. Res."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"102516","DOI":"10.1016\/j.tre.2021.102516","article-title":"Optimal recovery model in a used batteries closed-loop supply chain considering uncertain residual capacity","volume":"156","author":"Liu","year":"2021","journal-title":"Transp. Res. Part E Logist. Transp. Rev."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"4898","DOI":"10.1080\/00207543.2018.1467062","article-title":"A review on supply chain contracting with information considerations: Information updating and information asymmetry","volume":"57","author":"Shen","year":"2019","journal-title":"Int. J. Prod. Res."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"693","DOI":"10.4208\/aamm.2014.5.s6","article-title":"A Remark on the Courant-Friedrichs-Lewy Condition in Finite Difference Approach to PDE\u2019s","volume":"6","author":"Abe","year":"2014","journal-title":"Adv. Appl. Math. Mech."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.physrep.2005.10.009","article-title":"Complex networks: Structure and dynamics","volume":"424","author":"Boccaletti","year":"2006","journal-title":"Phys. Rep."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1247","DOI":"10.1038\/s41567-023-02132-1","article-title":"More is different in real-world multilayer networks","volume":"19","year":"2023","journal-title":"Nat. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1038\/s42254-023-00676-y","article-title":"Robustness and resilience of complex networks","volume":"6","author":"Artime","year":"2024","journal-title":"Nat. Rev. Phys."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Salas, D., and Raman, R. (2025, January 11\u201312). Fuzzy Petri-Net-Based Risk Assessment Model for AI Adoption in SME Auto-Component Clusters. Proceedings of the 2025 2nd International Conference on New Frontiers in Communication, Automation, Management and Security (ICCAMS), Bangalore, India.","DOI":"10.1109\/ICCAMS65118.2025.11233866"},{"key":"ref_52","unstructured":"CNBC (2024). Apple Made $14 Billion Worth of iPhones in India in Shift from China, CNBC."},{"key":"ref_53","unstructured":"CNBC (2025). Apple Airlifted iPhones Worth a Record $2 Billion from India in March as Trump Tariffs Loomed, CNBC."}],"container-title":["Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-8954\/14\/3\/316\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T05:26:04Z","timestamp":1773897964000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-8954\/14\/3\/316"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,17]]},"references-count":53,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2026,3]]}},"alternative-id":["systems14030316"],"URL":"https:\/\/doi.org\/10.3390\/systems14030316","relation":{},"ISSN":["2079-8954"],"issn-type":[{"value":"2079-8954","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,17]]}}}