Providing a Lagrangian relaxation algorithm to solve a reliable location- inven-tory model by considering disruption

Nowadays, supply chains have been facing significant economic forfeitures because of unpredicted disruptions. Furthermore, managers try to design sustainable and reliable supply chains. In this paper, we present an inventory-location model to propound a reliable three echelon supply chain which includes a production plant, distribution centers, and retailers. The production plant distributes a single product to retailers through distribution centers that are at risk of disruption. We considered reactive (consider backup distribution center for each retailer) and proactive (distribution center fortification) activates to enhance the supply chain's reliability. The proposed model indicates the location of distribution centers (DCs), the DCs that must be fortified, the allocation of retailers to DCs, and the inventory policy of DCs. The problem is formulated as a nonlinear integer programming model. Since our model is an NP-hard problem, we provide a Lagrangian relaxation algorithm to solve it. Numerical examples demonstrate the computational efficiency of the proposed solution algorithm. Results show that, with increasing the budget of fortification, the total expected cost will decrease. A higher inventory cost leads to an increase in the number of opened DCs, while higher ordering cost and the transportation cost from production plant to DCs decrease the number of opened DCs. Among other results, the number of opened DCs is positively affected by the cost of transporting from DCs to retailers.