“…From then until now, it has been gaining prestige as one of the most powerful exact algorithms to solve a broad range of NP-hard problems, such as vehicle routing ( Corréa, Langevin, & Rousseau, 2007 ), facility location ( Boland, Fischetti, Monaci, & Savelsbergh, 2016 ), logistics network design ( Cordeau, Pasin, & Solomon, 2006 ), transportation ( Coelho & Laporte, 2014 ), and inventory vehicle routing ( Alkaabneh, Diabat, & Gao, 2020 ). What is more, many studies in the literature have used the BD algorithm to solve the vehicle routing problem and its derivatives, such as location-routing ( Çalık, Oulamara, Prodhon, & Salhi, 2021 ), location-inventory-routing ( Zheng, Yin, & Zhang, 2019 ), and production-inventory-routing ( Cordeau, Laganà, Musmanno, & Vocaturo, 2015 ) problems. As a result, several efficient valid inequalities have been presented for these problems that could limit the solution space and improve the convergence speed of the BD.…”