  The Dynamic Modified Stochastic p-Median Problem (DMS-p-MP) is an important
problem in supply chain network design, as it deals with the optimal location
of facilities and the allocation of demand in a dynamic and uncertain
environment. In this research paper, we propose a mixed-integer linear
formulation for the DMS-p-MP, which captures the key features of the problem
and allows for efficient solution methods. The DMS-p-MP adds two key features
to the classical problem: (1) it considers the dynamic nature of the problem,
where the demand is uncertain and changes over time, and (2) it allows for the
modification of the facility locations over time, subject to a fixed number of
modifications. The proposed model is using robust optimization in order to
address the uncertainty of demand by allowing for the optimization of solutions
that are not overly sensitive to small changes in the data or parameters. To
manage the computational challenges presented by large-scale DMS-p-MP networks,
a Lagrangian relaxation (LR) algorithm is employed. Our computational study in
a real-life case study demonstrates the effectiveness of the proposed
formulation in solving the DMS p-Median Problem. According to the results, the
number of opened and closed buildings remains unchanged as the time horizon
increases. This is due to the periodic nature of our demand. This formulation
can be applied to real-world problems, providing decision-makers with an
effective tool to optimize their supply chain network design in a dynamic and
uncertain environment.
