Lagrangian relaxation method for solving a new time-dependent production–distribution planning model

In today’s competitive business environment, organizations must decide how to handle the processing of their logistics equipment economically. One of the vital logistical concerns is distribution planning that is especially crucial depending on the facilities and goods being used. When it comes to perishable goods, this problem assumes double the significance. The position of the warehouse and the route of the vehicles make up the distribution planning problem. These two problems are considered concurrently and solved in the location-routing mathematical model. This paper aims to provide a production and distribution strategy to serve clients and consumers better. This research attempts to produce as efficiently as possible while providing prompt customer service, which is crucial in today’s corporate environment. This study uses three-level supply chains for perishable goods to create a supply chain network that minimizes costs. In this case, time-dependent demands refer to requests that may be made when the vehicle will arrive. Places and routes in this area are designed to meet all needs. In general, it is desirable to have factories and distribution centers in known locations, know the service’s opening and closing hours, and know how to manage the flow of materials and goods as they are stored in distribution centers and for retailers (clients). Additionally, it is desirable to route vehicle that connects the various levels of the supply chain and ensures that vehicles travel on schedule overall. First, the supply chain model represented as non-linear programming is transformed into linear programming to solve it using the CPLEX solver of GAMS commercial software and the Lagrangian relaxation (LR) method. Then, this model is verified using numerical examples and related parameters to see how it impacts the variables and the objective function’s result. The results show the capability of the LR method.