{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T17:19:24Z","timestamp":1774459164621,"version":"3.50.1"},"reference-count":49,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,7,28]],"date-time":"2022-07-28T00:00:00Z","timestamp":1658966400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In today\u2019s competitive world, it is essential to provide a new method through which maximum efficiency can be created in the production and supply cycle. In many production environments, sending goods directly from the producer to the consumer brings many problems. Therefore, an efficient transport system should be established between producers and consumers. Such a system is designed in the field of supply chain management knowledge. Supply chain management is the evolutionary result of warehousing management and is one of the important infrastructural foundations of business implementation, in many of which the main effort is to shorten the time between the customer\u2019s order and the actual delivery of the goods. In this research, the supply chain consists of three levels. Suppliers are placed on the first level, cross-docks on the second level, and factories on the third level. In this system, a number of suppliers send different raw materials to several different cross-docks. Each channel is assigned to a cross-dock for a specific product. The main goal of this article is to focus on optimizing the planning of incoming and outgoing trucks with the aim of minimizing the total operation time within the supply chain. The arrival rate of goods from suppliers to the cross-dock is stochastic with a general probability distribution. On the other hand, the time required to prepare and send the goods is random with a general probability distribution. The service time in each cross-dock depends on the number of its doors. Therefore, each cross-dock can be modeled as a G\/G\/m queueing system where m represents the number of doors. The mathematical model of the research has been developed based on these assumptions. Since the problem is NP-hard, the time to solve it increases drastically with the increase in the dimensions of the problem. Therefore, three metaheuristics, including multi-objective water flow, non-dominated sorting genetic, and a multi-objective simulated annealing algorithm have been used to find near-optimal solutions to the problem. After adjusting the parameters of the algorithms using the Taguchi method, the results obtained from the algorithms were analyzed with a statistical test and the performance of the algorithms was evaluated. The results vividly demonstrate that non-dominated sorting genetics is the best of all.<\/jats:p>","DOI":"10.3390\/a15080265","type":"journal-article","created":{"date-parts":[[2022,7,28]],"date-time":"2022-07-28T20:49:28Z","timestamp":1659041368000},"page":"265","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Modelling Cross-Docking in a Three-Level Supply Chain with Stochastic Service and Queuing System: MOWFA Algorithm"],"prefix":"10.3390","volume":"15","author":[{"given":"Parinaz","family":"Rostami","sequence":"first","affiliation":[{"name":"Department of Industrial Engineering, Faculty of Industrial and Mechanical Engineering, Islamic Azad University, Qazvin 34199-15195, Iran"}]},{"given":"Soroush","family":"Avakh Darestani","sequence":"additional","affiliation":[{"name":"Guidhall School of Business and Law, London Metropolitan University, London N7 8DB, UK"}]},{"given":"Mitra","family":"Movassaghi","sequence":"additional","affiliation":[{"name":"Department of Industrial Engineering, Faculty of Industrial and Mechanical Engineering, Islamic Azad University, Qazvin 34199-15195, Iran"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"47","DOI":"10.5772\/9717","article-title":"A Review: Mathematical Modles for Cross Docking Planning","volume":"2","author":"Agustina","year":"2010","journal-title":"Int. J. Eng. Bus. Manag."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"827","DOI":"10.1016\/j.omega.2012.01.005","article-title":"Cross-docking: State of the art","volume":"40","author":"Valckenaers","year":"2012","journal-title":"J. Omega"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1080\/713682769","article-title":"Effective Cross Docking for Improving Distribution Efficiencies","volume":"3","author":"Apte","year":"2000","journal-title":"Int. J. Logist. Res. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"747","DOI":"10.1007\/s10845-008-0125-1","article-title":"Designing an integrated multi-echelon agile supply chain network: A hybrid taguchi-particle swarm optimization approach","volume":"19","author":"Bachlaus","year":"2008","journal-title":"J. Intell. Manuf."},{"key":"ref_5","unstructured":"Yu, W. (2002). Operational Strategies for Cross Docking Systems. [Ph.D. Thesis, Iowa State University]."},{"key":"ref_6","unstructured":"Magableh, G., Rossetti, M., and Mason, S. (2005, January 4). Modeling and Analysis of a Generic Cross-Docking Facility. Proceedings of the 37th Winter Simulation Conference, Orlando, FL, USA."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Ley, S., and Elfayoumy, S. (2007, January 20\u201323). Cross Dock Scheduling Using Genetic Algorithms. Proceedings of the 2007 International Symposium on Computational Intelligence in Robotics and Automation, Jacksonville, FL, USA.","DOI":"10.1109\/CIRA.2007.382928"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1016\/j.ejor.2006.10.047","article-title":"Scheduling of inbound and outbound trucks in cross docking systems with temporary storage","volume":"184","author":"Yu","year":"2008","journal-title":"Eur. J. Oper. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.ejor.2007.10.051","article-title":"Minimizing the makespan in a two-machine cross-docking flow shop problem","volume":"193","author":"Chen","year":"2009","journal-title":"Eur. J. Oper. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.cie.2009.06.006","article-title":"Scheduling trucks in cross-docking systems: Robust meta-heuristics","volume":"58","author":"Vahdani","year":"2010","journal-title":"Comput. Ind. Eng."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.compchemeng.2015.02.003","article-title":"The heterogeneous vehicle routing and truck scheduling problem in a multi-door cross-dock system","volume":"76","author":"Dondo","year":"2015","journal-title":"Comput. Chem. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.compchemeng.2014.06.015","article-title":"Vehicle routing scheduling problem with cross docking and split deliveries","volume":"69","author":"Moghadam","year":"2014","journal-title":"Comput. Chem. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"7065","DOI":"10.1016\/j.apm.2015.03.005","article-title":"Two-level vehicle routing with cross-docking in a three-echelon supply chain: A genetic algorithm approach","volume":"39","author":"Ahmadizar","year":"2015","journal-title":"Appl. Math. Model."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1016\/j.cie.2016.01.018","article-title":"Open vehicle routing problem with cross-docking","volume":"94","author":"Yu","year":"2016","journal-title":"Comput. Ind. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1016\/j.cie.2016.03.023","article-title":"A bi-objective truck scheduling problem in a cross-docking center with probability of breakdown for trucks","volume":"96","author":"Amini","year":"2016","journal-title":"Comput. Ind. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.compchemeng.2015.04.039","article-title":"A branch-and-price approach to evaluate the role of cross-docking operations in consolidated supply chains","volume":"80","author":"Mendez","year":"2015","journal-title":"Comput. Chem. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Utama, D.N., Zaki, F.A., Munjeri, I.J., and Putri, N.U. (2016, January 15\u201316). A water flow algorithm based optimization model for road traffic engineering. Proceedings of the International Conference on Advanced Computer Science and Information Systems (ICACSIS), Malang, Indonesia.","DOI":"10.1109\/ICACSIS.2016.7872734"},{"key":"ref_18","first-page":"1","article-title":"A Multi-Objective Imperialist Competitive Algorithm for Vehicle Routing Problem in Cross-docking Networks with Time Windows","volume":"11","author":"Nahavandi","year":"2018","journal-title":"J. Ind. Syst. Eng."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1016\/j.cie.2018.03.007","article-title":"Capacitated and multiple cross-docked vehicle routing problem with pickup, delivery, and time windows","volume":"119","author":"Ahkamiraad","year":"2018","journal-title":"Comput. Ind. Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/s40092-017-0203-0","article-title":"Two phase genetic algorithm for vehicle routing and scheduling problem with cross-docking and time windows considering customer satisfaction","volume":"14","author":"Baniamerian","year":"2018","journal-title":"J. Ind. Eng. Int."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/j.cie.2018.01.009","article-title":"Truck scheduling in a cross docking systems with fixed due dates and shipment sorting","volume":"117","author":"Molavi","year":"2018","journal-title":"Comput. Ind. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/j.ejor.2018.11.033","article-title":"A hybrid Lagrangian metaheuristic for the cross-docking flow shop scheduling problem","volume":"275","author":"Fonseca","year":"2018","journal-title":"Eur. J. Oper. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"102015","DOI":"10.1016\/j.omega.2018.12.004","article-title":"A comparative study of formulations for a cross-dock door assignment problem","volume":"91","author":"Gelareh","year":"2018","journal-title":"Omega"},{"key":"ref_24","first-page":"2","article-title":"A bi-objective model for pickup and delivery pollution routing problem with integration and consolidation shipments in cross-docking system","volume":"8","author":"Musavi","year":"2018","journal-title":"Iran. J. Oper. Res."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1016\/j.cor.2018.01.022","article-title":"A comparison of formulations and relaxations for cross-dock door assignment problems","volume":"94","author":"Nassief","year":"2018","journal-title":"Comput. Oper. Res."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.cie.2018.02.012","article-title":"Integration of parts transportation without cross docking in a supply chain","volume":"118","author":"Shaelaie","year":"2018","journal-title":"Comput. Ind. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1016\/j.asoc.2018.11.029","article-title":"Modified variable neighborhood search and genetic algorithm for profitable heterogeneous vehicle routing problem with cross-docking","volume":"75","author":"Baniamerian","year":"2019","journal-title":"Appl. Soft Comput."},{"key":"ref_28","first-page":"296","article-title":"Truck Scheduling in a Cross-Docking Terminal by Using Novel Robust Heuristics","volume":"32","author":"Seyedi","year":"2019","journal-title":"Int. J. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"752","DOI":"10.1016\/j.ejor.2019.04.028","article-title":"Integrated scheduling and assignment of trucks at unit-load cross-dock terminals with mixed service mode dock doors","volume":"278","author":"Rijal","year":"2019","journal-title":"Eur. J. Oper. Res."},{"key":"ref_30","first-page":"207","article-title":"Integrating truck and workforce scheduling in a cross-dock: Analysis of different workforce coordination policies","volume":"90","author":"Corsten","year":"2019","journal-title":"J. Bus. Econ."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1016\/j.ijpe.2019.02.017","article-title":"A Delayed Start Parallel Evolutionary Algorithm for just-in-time truck scheduling at a cross-docking facility","volume":"212","author":"Dulebenets","year":"2019","journal-title":"Int. J. Prod. Econ."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1016\/j.apm.2019.01.047","article-title":"The vehicle routing and scheduling problem with cross-docking for perishable products under uncertainty: Two robust bi-objective models","volume":"70","author":"Rahbari","year":"2019","journal-title":"Appl. Math. Model."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"106048","DOI":"10.1016\/j.ress.2018.01.007","article-title":"Redundancy allocation in series-parallel systems under warm standby and active components in repairable subsystems","volume":"192","author":"Hadipour","year":"2019","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1007\/s10845-016-1229-7","article-title":"An intelligent truck scheduling and transportation planning optimization model for product portfolio in a cross-dock","volume":"30","author":"Khorshidian","year":"2019","journal-title":"J. Intell. Manuf."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.jclepro.2018.12.106","article-title":"Assignment and scheduling trucks in cross-docking system with energy consumption consideration and trucks queuing","volume":"213","author":"Fard","year":"2019","journal-title":"J. Clean. Prod."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"106832","DOI":"10.1016\/j.cie.2020.106832","article-title":"A new bi-objective vehicle routing-scheduling problem with cross-docking: Mathematical model and algorithms","volume":"149","author":"Goodarzi","year":"2020","journal-title":"Comput. Ind. Eng."},{"key":"ref_37","first-page":"2","article-title":"Cross-docks scheduling with multiple doors using fuzzy approach","volume":"79","author":"Movassaghi","year":"2020","journal-title":"Eur. Transp."},{"key":"ref_38","first-page":"2397","article-title":"Vehicle routing problem in a kanban controlled supply chain system considering cross-docking strategy","volume":"20","author":"Goodarzi","year":"2020","journal-title":"Oper. Res."},{"key":"ref_39","first-page":"201","article-title":"Truck-to-door sequencing in multi-door cross-docking system with dock repeat truck holding patter","volume":"11","author":"Ardakani","year":"2020","journal-title":"Int. J. Ind. Eng. Comput."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"106391","DOI":"10.1016\/j.cie.2020.106391","article-title":"Parallel-machine scheduling methodology for a multi-dock truck sequencing problem in a cross-docking center","volume":"143","author":"Nogueira","year":"2020","journal-title":"Comput. Ind. Eng."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"102060","DOI":"10.1016\/j.tre.2020.102060","article-title":"A healthcare logistic network considering stochastic emission of contamination: Bi-objective model and solution algorithm","volume":"142","author":"Nikzamir","year":"2020","journal-title":"Transp. Res. Part E Logist. Transp. Rev."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"107299","DOI":"10.1016\/j.cie.2021.107299","article-title":"A routing and scheduling problem for cross-docking networks with perishable products, heterogeneous vehicles and split delivery","volume":"157","author":"Asian","year":"2021","journal-title":"Comput. Ind. Eng."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"105271","DOI":"10.1016\/j.cor.2021.105271","article-title":"Network scheduling problem with cross-docking and loading constraints","volume":"132","author":"Castellucci","year":"2021","journal-title":"Comput. Oper. Res."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"105513","DOI":"10.1016\/j.cor.2021.105513","article-title":"An M\/M\/c queue model for vehicle routing problem in multi-door cross-docking environments","volume":"138","author":"Goodarzi","year":"2021","journal-title":"Comput. Oper. Res."},{"key":"ref_45","first-page":"1569","article-title":"Reliable cross-docking location problem under the risk of disruptions","volume":"21","author":"Goodarzi","year":"2021","journal-title":"Oper. Res."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"105205","DOI":"10.1016\/j.cor.2020.105205","article-title":"Adaptive neighborhood simulated annealing for the heterogeneous fleet vehicle routing problem with multiple cross-docks","volume":"129","author":"Yu","year":"2021","journal-title":"Comput. Oper. Res."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"107240","DOI":"10.1016\/j.cie.2021.107240","article-title":"Truck scheduling optimization at a cold-chain cross-docking terminal with product perishability considerations","volume":"156","author":"Theophilus","year":"2021","journal-title":"Comput. Ind. Eng."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"102210","DOI":"10.1016\/j.tre.2020.102210","article-title":"The two-echelon production routing problem with cross-docking satellites","volume":"147","author":"Qiu","year":"2021","journal-title":"Transp. Res. Part E Logist. Transp. Rev."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"105526","DOI":"10.1016\/j.cor.2021.105526","article-title":"A multi-tiered vehicle routing problem with global cross-docking","volume":"137","author":"Smith","year":"2022","journal-title":"Comput. Oper. Res."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/8\/265\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:58:28Z","timestamp":1760140708000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/8\/265"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,28]]},"references-count":49,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["a15080265"],"URL":"https:\/\/doi.org\/10.3390\/a15080265","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,28]]}}}